zotero-db/storage/TKQAUTRP/.zotero-ft-cache

THE PHYSICAL PRINCIPLES
OF THE QUANTUM THEORY
By WERNER HEISENBERG
Professor of Physics, University of Leipzig Translated into English by
CARL ECKART AND FRANK C. HOYT Department 'of Physics, University of Chicago
uwtdi'ii'tL uit-iJiViL IA> 'NMIOITI. L I 3 r"7 A n Y
AOO.^O. *Z?*Z;„
DOVER PUBLICATIONS, INC.

COPYRIGHT I 9 3 0 BY THE UNIVERSITY OF CHICAGO ALL RIGHTS RESERVED. PUBLISHED JULY 193O
Printed and bound in the United Slates of America

LIQUIDS AND' SOFT SOLIDS

145

One has first to determine the true fluidity of the material at rest, as explained above. If we know this we redraw the consistency curve in accordance with the equation

/<»Pw- L+m %_ -„q>' , —*

• '

2<mP«,i +. <Po~ <' Po — • * . (31)

The first term of the series on the right side, i.e. for n = 0, is

identical with the first term for a liquid showing no wall effect. In

the case of a simple Newtonian liquid showing a wall effect, this

curve is therefore the correct consistency curve of the bulk of the

material and may accordingly be named the " bulk " consistency

curve. The curve is identical with the consistency curve of the bulk

of a generalized Newtonian liquid showing a wall effect, if either

small P*s are considered or \\R is small. For larger P's and lfR's

there is no way of plotting consistency curves independently of the

dimensions of the apparatus, if the law of flow of the material is

not known.

Regions of very small shearing velocities require special consideration. Very thorough investigations were undertaken by Scott Blair and collaborators (compare Scott Blair Introduction [35], pp. 32-34). The consistency curve of clay and soil pastes was found to divide itself into four regions. In region I there is no flow, region I I is a straight line, region I I I is markedly curved and region IV becomes asymptotically straight. In regions I and IV one recognizes the picture of Fig. 4. Region II is due to slippage. It was Scott Blair's contribution to find Region I, i.e. the fact that a definite stress is necessary to start slippage.

9. Lubricating and Roughening Wall Layers
When considering wall effects in the foregoing, we have spoken of " slippage " as occurring in a two-phase system, with the dispersion medium forming the wall layer. But this is taking too narrow a view of wall effects. As Bingham early (1922) pointed out, it is " quite possible that the fluidity of a liquid near the surface is not identical with that within the body of the liquid ". Bartsch [36] found, e.g. that when oleic acid is in contact with a metal surface a layer about 10,000 molecules thick adheres to the wall. Such molecules are at the same time oriented. Generally there may be two sorts of effects. The molecules may be either oriented parallel to the wall, which should reduce friction, or they may become oriented normal, which must increase friction. The first made a " lubricating ",

146

WALL EFFECTS

the second a " roughening " layer. If we introduce in (29)

d = Rjn

(32)

Fig. 4 shows tp0 as a function of ljn for different values of a.

F o r a < I the wall layer acts as lubricant, for OL>1 as a

roughener. It must, however, be said that it is doubtful

whether these effects are pronounced enough to affect viscosity

measurements in the case of simple liquids. An attempt by

n" R
Fia. V H I , 4. Apparent zero fluidities of liquids in the presence of wall effects as determined in capillary instruments. <pa fluidity of the wall layer. Jt radius of capillary. d thickness of wall layer.
R. Bulkley [37] in the case of different oils, using capillary viscometers and raising the precision of the measurements to unusual levels, had negative results.
When a liquid does not wet the solid, as in the case of mercury, one cannot properly speak of slippage in the sense in which this word is ordinarily used. In this case there exists a layer of some other medium, either air or the vapour of the liquid, between the material and the solid wall. In the case of air at room temperature <pw> 5,000. The thickness d must be >10~8 cm., which is the order of magnitude of the diameter

SUMMARY

147

of simple molecules. Bingham and Thompson (1928) determined the viscosity of mercury in a capillary of B — 0-012 cm. Therefore in this case <p' — <p (compare (29) ) > 0-026. The observed fluidity of mercury at room temperature <66.
m' — tp
Therefore ;—>0-024 per cent. This is less than the
accuracy of the Bingham instrument, which is about 0*1 per cent. The presence of the air layer can therefore not make itself felt and the viscosity of mercury can be determined from the usual formulas assuming that the mercury adheres to the solid wall [1]. Erk [2] has pointed out that the viscosity of air or vapour, which is in a state of adsorption on the solid surface, must be assumed as much higher and may even have a " roughening " influence.

10. Summary

In the flow of a dispersed system, the dispersion medium may

form a lubricating layer between the body of the liquid and a

solid wall. This is especially so with suspensions which are

soft plastic solids. In liquids in contact with solids there may

be wall layers acting either as " lubricants " or " rougheners ".

In the presence of a wall layer the apparent fluidity at rest

will be in the case of a plastic solid in the

capillary

[

rotating-oylinder

instrument

(

l~2dj{Re-R%)
I+W(2-W

J

-l, V

' }

and in the case of a liquid, the fluidity at rest of which is q>0>

9>/=9>o+?d.7-(i-<W]

9*' = Vo + 2<P» [ ' -

When plotting calculated <p0' values against varying

IjR

| IfRi, keeping " a " constant

extrapolation of the resultant curve to zero gives <p0' = 0 in the case of a plastic solid and <p0 in the case of a liquid.
L 2

CHAPTER IX
VOLUME ELASTICITY AND VISCOSITY
1. Epistomological Digression
W E have said in Chapter I that under isotropic pressure all materials are elastic and only elastic, and we have called this an axiom of rheology. We did not give any proof for this statement. We relied upon the fact that, being an axiom, it is, as the Oxford Dictionary puts it, " a self-evident proposition, not requiring demonstration, but assented to as soon as stated ". This use of the term axiom, going back (again on the authority of the Oxford Dictionary) to the year 1600, is, however, not in accordance with modern ideas. We now, even in geometry, and still more so in rheology, do not accept the truth of a proposition without recourse to experience and'if, as in the case of an axiom, experience is not drawn upon, the axiom cannot be more than the definition (an " implicit" definition) of a word appearing in the proposition. If we now look back to our statement that " under isotropic pressure all materials are elastic and only elastic " we find that there is no word in it which we have not already defined before, and, therefore, as it stands, the statement cannot be true. And so it is. We arrived at the statement by considering such materials as steel, plasticene and water. But if we had put a piece oiwood under a high hydrostatic pressure we would have observed, after release of the pressure, a Permanent Set, i.e. an instantaneous deformation which is not recoverable. The reason is that there are pores in the wood and, under high pressure, stresses are produced which partly crush the material around the hollow spaces of the pores, forcing it into the pores. This local deformation which exceeds the strength of the material, is not reco rered on the removal of the pressure, and the aggregate effect of many such microscopic deformations becomes manifest in the macroscopic set. To make our statement true we have to re-word it as follows : " Under isotropic pressure all nonporous materials are elastic and only elastic ", and in this form the axiom is an implicit (rheological) definition for " non-
148

THE BULK MODULUS

149

porous ". If we now read this statement together with the second rheological axiom that " every real material possesses all rheological properties ", which, as we now realize, is an implicit definition for " real material ", we must conclude that no real material is non-porous and we shall not be surprised to learn that Eyring [38] has propounded a theory of the viscosity of liquids which assumes that even a liquid possesses " holes ".
Some readers may think this digression to be out of place, but the situation as described means that the assertions which I have made and shall make about properties of materials are never quite correct. They are always only provisional, to be amplified or qualified later. If the reader will keep this in mind, I will be able to go ahead, speaking often in a loose way, without having to add to every quantitative statement the qualification " approximately ". Therefore, let it be granted that there are materials, which on the whole, are under isotropic pressure and tension elastic and only elastic, and let us examine what the laws of such elasticity may be.

2. The Bulk Modulus

If a hydrostatic pressure is applied to a body, its volume V is reduced and its density p, which is the mass per unit volume, or

P = ™JV

(1)

increased. Conversely, an isotropic tension (which it is much

more difficult to apply) will increase the volume and reduce the

density. Let V0 be the volume for vanishing isotropic tension p = 0 and let A VQ be the increase of volume produced by p, then an Elastic Bulk Modulus K is usually defined by

p=KAVJV0

(2)

from which

AV0=pJK.V0

(3)

and therefore

V = V0 + AV0 = V0{l+p!*<) . . . (4) Equation (2) is regarded as a special case of Hooke's law, but this definition of the bulk modulus is objectionable for the reason that the latter cannot be constant. We denned the coefficient of viscosity in a certain way and said that in certain liquids it will be a constant, while in others it may vary with

150 VOLUME ELASTICITY AND VISCOSITY

the stress. Rheological coefficients should generally be denned in such a way t h a t for certain simple materials they are constants. The coefficient K as denned above, can, however, under no circumstances be a constant because then the volume V0 would vanish for p = — K, with the density increasing to infinity. Actually, if K is defined as in (2), it is generally found to increase with increasing compression ; i.e. in order to produce the same increase of compression the pressure has to be comparatively more increased the greater the compression from which we start. For instance, Bridgman (1923) found that /c-1 varies for metals in accordance with the following empirical formulas :

for iron K"1 = 20"7 {5-87 - 2-10 x 10^p) . . . A

for copper = 10~7 (7-32 - 2-7 x lO^p) . . . A (5)

for lead

= 10~7 {23-73 = 17-25 x 10^p) . . . J

where p is the hydrostatic pressure measured in kg. cm.-2 u p to 20,000 kg. cm.-2.

Other approximate bulk moduli are listed in the Table.

TABLE I X , 1. Bulk Moduli

Material Ether Alcohol Water Tuff
K X I O " 8 in 0-08 0-1 0-2 0-5 megabars

Clay Glass Mercury Iron

1

4

5-4

15

Steel 18

3. Volumetric Strain

We can derive from (2) a more suitable definition by keeping in mind that this equation was put up to suit a comparatively small range of stresses.
Let that range be infinitely small, then (2) becomes

dp=KdV0iV0

(6)

Now let this equation be assumed to be valid not only for the volume Y0> but for any volume V, then

dp = KdVjV

(7)

and this equation defines a bulk modulus which, in principle, oould be a constant. Because let K be a constant, then (6) can be integrated and gives

" p=Kln(V}V0)

(8)

or

V = VQe*>*

(9)

VOLUMETRIC STRAIN

151

In this case, therefore, V vanishes forp = — co or, practically

never, the negative sign denoting a pressure.

We call the quantity

ev=\n(VjV0)

(10)

the Volumetric Strain or Cubical Dilatation and we can

accordingly write (7) in the form

P =K%

(11)

which is the complete analogy to (I, d). Introducing again

7 = V0 + AV„ ev = In (1 + AVJV0) and if AVJV0 is small, this expression can be developed into a series, the first two terms

of which are

ev = AVJV0 - h{AV0jV0Y . . . . (12)

In a first approximation we find again (2), but in a second

approximation we have

p = KA V0(V0 - K\2 . (A VJV0)* . . . (13)
Indicating by K' an " apparent " bulk modulus in accordance with (2), we find by introducing

from which and

AVJV0=PIK' p =PKIK' - KI2.P2}K'2
K' = K — PKJ2K
K' =KI2[1 + V1-2PIK']

.... •

(14) (15) (16)
(17)

K' l he

J^

'/!<•

s

2 it 1i i1 10 '-?

Fig. I X , 1. Apparent bulk modulus (*') and apparent coefficient of compressibility (1/*') as a function of the hydrostatic pressure

152 VOLUME ELASTICITY AND VISCOSITY
If we plot K against p we get a curve as shown in Fig. 1. The Apparent Coefficient of Compressibility («'_1) has also been plotted. It can be seen from these curves that the assumption that K in (11) is a constant gives a better approximation to the compressibility of materials because the compressibility in accordance with this formula decreases with increasing pressure. However, comparison with the empirical equations (5) shows that quantitatively for large pressures the position is still far from satisfactory. Following Hencky [39] we therefore proceed to apply two corrections.
4. The Reduced Stress For the first correction we must remember that the resistance
of the material to the external forces comes from reactive forces between the molecules or atoms, as the case may be, caused by a change in their stable distances. The forces, accordingly, are bound up with the quantity of matter. The stresses, on the other hand, are denned in respect of the areas through which the forces act and a unit isotropic pressure will be a unit force applied upon the sides of a cube of unit length of edge and unit volume, say 1 cm.3 That cubic centimetre of volume, however, contains in the strained state, assuming tension and V>V0, fewer molecules or atoms than in the unstrained state, in short, less matter. Conversely, the unity of matter in the unstrained state will in the strained state take up a volume increased in the proportion pjp where p0 is the density in the unstrained, p the density in the strained state. To calculate from the stresses the reactive force caused by the same piece of matter, we must increase the stress in the same proportion, i.e. we must replace the stress as defined in Chapter I by the Reduced Stress
Pred=~P = VIV0.p=Vrelp . . . (18)*
where Vnl is the Relative Volume = VjV0. This makes (11)
p = K]nVrJVrel = Keve~<* . . . . (19)
where e is the basis of " natural " logarithms.
* Note that V^j here is an entirely different quantity from what IB denoted in the same way in Chapter V.

THE LIMITING RELATIVE VOLUME

153

5. The Limiting Relative Volume

The second correction refers to the feature of (11) or (19) that with infinite pressure the volume of the body would be compressed to zero. There must, however, be a limiting volume containing all the molecules even if in a collapsed and crumpledup state (the latter was realized by Bridgmann reaching pressures up to 20,000 kg. cm.-2). Hencky, on applying (19) to Bridgman's observations, found that for a small number of solids this was a satisfactory formula. Most of the experiments, however, suggested that there was a Limiting Relative Volume W which could not be reached by finite pressure. He therefore suggested the formula

p - w f 4 ^ - . . . . <*»

v rel

-TV rel

which he compared with Bridgman's observations and especially

of those substances which Bridgman had marked out as very

irregular and therefore particularly fit for a crucial test. Near

atmospheric pressure (20) does not give as good results as at

higher pressures and the simpler formula (11) may be used

there. Hencky concluded that the individual structure of the

molecule apparently has much influence at low pressures, but it seems t h a t a t pressure from 10,000 kg. cm."2 upwards even

gases acquire a constant bulk modulus in accordance with (20).

The following Table gives the results of his calculations. Equa-

tion (20) can be considered as expressing the modulus K of (11)

b y (K) (1 — *?)/( Vrel — W). While now K is a variable coefficient

and not a constant, (K) and W are supposed to be constants.

TABLE I X , 2. Hencky's Bulk Modulus (K) and Limiting Volume {¥) of certain Substances calculated with Hencky^ Theory (20) from Bridgman's Observations.

Substance

Temperature Centigrade

K X 10* In kg/cm -1

*

Iron . . . .

30

Rubidium

50

Mercury

20

Water

40

Alcohol

20

Hydrogen

65

Helium

65

170 1-66 24-9
2-34 4-17 0-82 0-47

0-8 0-44 0-75 0-50 0-52 nearly zero
ij

154 VOLUME ELASTICITY AND VISCOSITY

6. Pulsations
We have said in Section 4 that the internal forces with which the material reacts to the external forces are due to the displacements of the molecules or atoms from their stable relative positions. To effect such displacement, external work has to be expended which is converted into the potential elastic energy of the strained body. Let us look into this process in more detail.
If we apply a finite isotropic pressure p upon a body,* its particles will be brought into movement in accordance with (I, a), i.e. after some time through the operation of the acceleration a, they will attain certain velocities v and will by virtue of these possess kinetic energy of the magnitude

EK = mv2j2

(21)

At the same time they will undergo by displacements u and this will produce in their surroundings elastic stresses in accordance with (11) or (20) or some other rheological equation. Let us assume the body to be a sphere of radius Roi the centre of which is fixed, then, for reasons of symmetry, we may assume that the direction of the different w's and v's passes through the centre. To every p there corresponds a certain definite ev and therefore a certain definite radius R at which the internal stresses p raised in the body through the strain will balance the external pressure p. If p is applied from the start to the full extent, this means that before R0 is reduced to R there is no equilibrium; this causes the appearance of kinetic energy. When the sphere is compressed to V, p balances p, but the particles will move on towards the centre because they have kinetic energy. This increases p beyond p and there are unbalanced stresses acting away from the centre. This, in turn, causes a negative acceleration a or retardation — a, a diminution of v and ultimately a state of rest when v vanishes. That state of rest, however, does not correspond to a state of equilibrium. The stress p exceeds the external pressure p; rest lasts only through an element of time ; the body starts to expand. We need not continue this story ; the result will be voluminal oscillations or pulsations around the position of

* The bar above jj indioates the external foroe.

VOLUMETRIC STRAINWORK

155

equilibrium ev = pJK ; a position which, however, is not one of rest.

7. Volumetric Strainwork
If we want to determine the compressibility of a material in a suitable test, we shall therefore have to apply the external pressure very slowly, practically infinitely slowly, i.e. we shall have to increase the pressure from zero to p so slowly that it is always equal to p and the body passes continuously through states of equilibrium. Considering now the sphere of radius R which is at rest under the action of the isotropic tension p

Fia. IX, 2. Volumetric strainwork.

p isotropic tension.

AR increase of radius.

applied upon its surface A ; if pis slightly increased the radius will slightly be increased by AR and the forces acting upon the surface perform work. The force acting upon an element of the surface (AA) is p.AA and the work p.AA, AR (compare Fig. 1). Adding up all elements of work over the whole surface of the sphere, we get p.A.AR. If AR is very small, the increase of volume is dV = A dR and, therefore,

dW = pdV

(22)

or the elementary work per unit volume

dw =pdV}V =pev

(23)

and the power per unit volume

w= swjdt = pev

(24)

156 VOLUME ELASTICITY AND VISCOSITY

If p is gradually increased from zero to p, the work performed per unit volume is

w = Cpdev

(25)

Jo

If the load p is decreased from p to zero,

V)' = pdev = — pdev = — w . . . (26)

J'o

Jo

and the strainwork is completely regained. The only condition

for this reversibility is that there should be a one-to-one relation

between # and ev) but the form of the relation, whether straight line or curved, is immaterial. Generally, in a perfectly elastic

body, all strainwork performed by the external forces through

states of equilibrium is stored up in the body as potential energy

and completely regained upon removal of the external forces, if

this is likewise carried out through states of equilibrium.

If p is expressed as a function of ev by means of a Theological equation, the integration can be carried out. For instance, if

we use (11),

w = Ke^/2

(27)

or

w=p2j2K

(28)

FIG. IX, 3. Fundamental rheological curve for infinitely slow compression. Area o a b measures the strainwork. e„ volumetric strain. p isotropic stress. Ic modulus of compressibility.

DAMPING OF FREE PULSATIONS

157

The reader will note that (22) is the analogue of (III, 41); (25) of (III, 42); (27) of (III, 43); and (28) of (III, 44).
If we plot correlated observed values of p against evi the resultant curve is a graphical representation of the rheological equation (compare Fig. IX, 3). If the curve is a straight line, K in (11) is a constant; otherwise K is variable. Comparing (23), we see that the elementary work per unit volume is represented in the Figure by an elementary strip parallel to the p-axis and of width det. The total work performed when increasing the pressure from zero to p is therefore represented by the area enclosed by the curve, the c„ axis and the ordinate p. If all stored-up energy is to be regained as the material expands with decreasing pressure, the points representing descending observations must lie on the same curve or, as we said before, p must be a one-valued function of ev.
8. Damping of Free Pulsations
Let us now consider what will happen if, having compressed a sphere, by gradually applying an isotropic pressure p, from the radius R0 to the radius M, the external pressure is suddenly released. Now, in expanding, the internal stresses meet no resistance against which to perform work. The particles will therefore acquire kinetic energy and oscillations will start. Because of the absence of external forces, the oscillations are called free oscillations. The elastic potential energy (Ev) is converted into kinetic energy {Ek), which gradually increases until it reaches a maximum for R — R0 when the elastic energy vanishes. Then the sphere continues to expand with kinetic energy being converted into potential energy, etc., all the time the law of conservation of energy requiring that the sum of both is constant. This, however, cannot go on indefinitely. The process just described of recurring conversion of kinetic into potential energy requires that the velocities and accelerations or retardations of all particles are at all times directed towards the centre of the sphere. But if the oscillations started in this way, imperfections in the structure of the material making up the sphere, such as pores, local variations of density, or, generally, heterogeneities and local aeolotropies, etc., will soon make themselves felt. While the sum of the kinetic energies of all particles will still conform to the law of conservation, the

158 VOLUME ELASTICITY AND VISCOSITY

velocities will gradually become less oriented towards the centre and more and more acquire a random orientation. The oscillations of the sphere as a whole will gradually be replaced by individual oscillations of the particles which macroscopically manifest themselves in an increase of temperature. This is called Damping of Oscillations, with which we shall deal in more detail in Chapter XV. But as we have said in Section 9 of Chapter III, a rheological experiment must be isothermal. This can be realized by connecting the body with a large heat reservoir kept at constant temperature. The heat into which the kinetic energy of the oscillations is gradually converted is drawn off into the reservoir and dissipated {Ed). The law of conservation of energy now requires

E„ + Ek + Ed const

(29)

No other form of energy appears in rheology. What we have

derived here on the example of isotropic stress and voluminal

oscillations is, as will be shown later, valid for every other kind

of stress and deformation. Equation (29), together with the

proviso that in every process Ed can only increase, form the energetic basis of rheology.

9. The Coefficient of Volume Viscosity

If there is dissipation of energy whenever the cubical dilatation is not produced infinitely slowly, or whenever there is a finite rate of dilatation, ev, this implies a sort of viscosity -qv which we may call Volume Viscosity. The complete isotropic rheological equation is accordingly

p = Kev + yev

(30)

It should be noted that in deriving (30) we did not say

whether we were speaking of a liquid or a solid. This is in

accordance with the first rheological axiom which (in other

words) says that for simple changes of volume or density it is

irrelevant whether the material is a solid or liquid. A Kquid,

therefore, mwt have two kinds of viscosity, viz. the ordinary

Newtonian viscosity in shear rj and the volume viscosity 7}v. One could, of course, in principle assume that for a certain class

of liquids r)v vanishes and we may call this class of liquids the Stokes Liquid, because this was what Stokes (1851) assumed

when deriving the famous Stokes-Navier differential equation

ELASTIC AFTER-EFFECT & TOTAL STRAINWORK 159
of viscous flow named after him and Navier (1823). Until recently this was generally assumed to conform to actual conditions, but Tisza [40] has pointed out t h a t v\v must in real liquids be rather high and I have pointed to other consequences connected with a vanishing 77. which are not likely to conform to experience and about which we shall say more in Chapter X.
10. Elastic After-effect and Total Strainwork Equation (30) yields an interesting consequence. Let a
certain volumetric strain [ej be produced by an isotropic external tension p which is then removed. If r}0 is large so

FJG. I X , 4. Elastic after effect in volumetric strain upon removal of stress, /time, T time of lagging, © basis of natural logarithms.

that no oscillations, but only an aperiodic movement results (or, conversely, if inertia-forces can be neglected) we find from (30) for p =0

Ke. + i f c e . = 0

(31)

and introducing

ec = dejdt

(32)

7}v dejev = -tcdt

(33)

which by integration yields

Vo)nev = -Kt + C

(34)

160 VOLUME ELASTICITY AND VISCOSITY

The integration constant G is determined from % = [ej for t = o and we ultimately have

e. + Kle-'V*

(35)

The quotient TJJK is of the dimension of time (T) which is a measure of the lagging of the elastic strain, r may be called Time of Lagging. I n accordance with (35), the original volume is recovered (i.e. ev = 0) when tjr = co, which is t = co for any finite T. The curve corresponding to (35) is shown in Fig. IX, 4.

When the load is removed, the original volume for which ev = 0 is therefore generally not instantaneously regained; this takes time and, theoretically, even infinite time. Conversely, when the load is applied, it also takes time (and again theoretically infinite time) until the cubical dilatation corresponding to the load is attained. This phenomenon is called Elastic After-Effect. It should be noted that the elastic aftereffect does not constitute an imperfection of elasticity, according to both definitions of perfect elasticity, viz. (i) total disappearance of strain on removal of load, (ii) complete conservation of strainwork in infinitely slow deformation. If, however, the deformation is produced with a finite velocity, p a r t of the strainwork is dissipated through volume viscosity.

Volume viscosity in liquids makes itself felt in the absorption and dispersion of ultrasonic waves, the rate of which is higher than if shear viscosity (rj) only was present. I n liquids — Kev is called the hydrostatic pressure. I t is, as follows from (30), not identical with the isotropic component of the stress.

In the general case (30), the strainpower per unit volume is

& = M> = K%% + Vv%2

•

• (36)

The first term on the right-hand side reverses its sign when ev decreases and cv is negative. On de-straining, this work is therefore regained and the first term accordingly constitutes the conserved part of the strainwork. The second term is always positive, whether ev increases (positive e j or decreases (negative e„). On de-straining as well as on straining, work is expended and the second term accordingly constitutes the dissipated part of the strainwork. The total strainwork is

w = j w dt = | \ev dev + | 7}vev*dr . . (37)

SUMMARY

161

We shall come back to the Theological equation of form (30) in Chapter XIV showing that it embodies other properties besides those dealt with in the present chapter. Here let us only add the remark that we have introduced another qualification into our first axiom. Under isotropic stress non-porous materials are elastic only if the stress is applied infinitely dowly. Otherwise they also show a viscous resistance.

11. Summary
The rheological behaviour of every homogeneous material solid or liquid, conforms under isotropic stress p to the Theological equation
p = KCV + 7}vev . . . . (IX, a)
where ev, the volumetric strain,
e , = m ( 7 / F , ) « l n ( F M l ) . . . (IX, b) I n liquids KBV is called " hydrostatic pressure." In cases where the density p of the material is much changed by pressure, p must be replaced by the reduced stress

P*i--P P
The volumetric strain-power is wv = K%% + 7]vev2 . . . .

(IX, o) (IX, d)

of which the first part is conserved, the second dissipated. If K is constant, the conserved (potential) energy is

Ev=Ke*j2=p*i2K

. . . . (IX, e)

I t follows from (IX, a) that there is an elastic after-effect with lagging time

r = *?>

(IX, f)

so that a strain [e„] disappears upon removal of the load in accordance with the equation

% =

feje-*

(IX, g)

where it is assumed that T is constant. Where K is a variable coefficient, Hencky has proposed

1- W

W being the relative limiting volume and (K) a constant.

or.

ii

CHAPTER X
SIMPLE TENSION AND SIMPLE EXTENSION
1. Simple Stresses and Deformations
So far we have become acquainted with two simple cases of stress and deformation. They are simple shearing stress and simple isotropic stress on the one hand—simple shear and simple cubical dilatation on the other. These are co-ordinated pairs, simple shearing stress producing simple shear, and simple isotropic stress producing simple cubical dilatation. There exist, however, two other important cases of stress and deformation with which we have not yet dealt and which are of an entirely different character. They are simple tension on the one hand and simple extension on the other. Simple tension is a one-dimensional stress and simple extension is a one-dimensional deformation, but the latter is not produced by the former.
Simple tension and its opposite, simple pressure, can both be included under the term simple Normal Stress, designated by pn, where the subscript n indicates " normal". A simple normal stress is produced by either simple Pull ( + Pn) or simple Push (— Pn) upon a prismatic bar, acting in the direction normal to, and passing through the centre of the cross-section.* Under the action of such simple pull, the bar is elongated ; at the same time, however, it contracts laterally. A one-dimensional traction is here accompanied by a three-dimensional deformation.
Simple extension, or its opposite, simple compression, both included under the term simple Normal Deformation, (e„), because the displacement is in the direction normal to the crosssection, is not so easily realised. A plastic material filling a cylinder, where the latter is so rigid that it does not extend in diameter, would, under pressure from a piston, be deformed in the manner of simple compression. Here the walls of the cylinder will also produce pressure upon the material and the
* If the force does not pasa through the centre of the cross-section, bending is caused, with which we shall deal m Chapter XII.

YOUNG'S MODULUS

163

one-dimensional deformation is therefore accompanied by a threedimensional stress. However, in order to produce in a plastic material a measurable compression, the pressure would have to be so high that the walls of the cylinder, too, would give way slightly, albeit elastically. Nevertheless, in porous materials, simple pressure may produce simple compression. One such material is cork, another porous rubber; still another, as we shall learn in Chapter XIII, is concrete. These, however, are exceptions.
2. Young's Modulus
The solution of this puzzle—viz. that simple normal stress does not correspond to simple normal deformation or strain (and vice versa)—is that " simple " normal stress and deformation are not as simple as one may think.
Let us first consider simple normal stress, for instance as produced by the simple pull of a cylinder or prismatic steel bar in the so-called " tensile test ". This is the predominant test for metals, but has also been in use for such materials as cement, pitch, bitumen, flour dough, etc. In this test a short bar, say of mild steel, of length l0 is fixed between two pairs of jaws (or some similar device), one stationary and the other movable, and is elongated by a gradually increasing load P n . If A is the cross-sectional area of the bar, a traction is produced equal to

acting in the longitudinal direction, or normal to the crosssection, and which may be assumed as uniformly distributed over the cross-section. The assumption is not correct near the ends, where the bar is fixed between the jaws, but it will be valid at some distance from the ends, especially if the bar is slender.* Up to a certain point, the elongation Al is proportional to the load and therefore follows Hooke's law. For this part we write in analogy to (I, 5)

Aljl0 = (PJA)le . '

(2)

where the coefficient c is called Young's Modulus.

The ratio Aljl0 is usually taken as the measure of the normal strain, here called Extension, positive or negative. For small

* This ia the Principle of St, Venant about which compare Ten Lectures, pp. 58-59, 65.
H 2

164 SIMPLE TENSION AND SIMPLE EXTENSION
elongations Al there is no objection to this. For large elongations, however, this definition breaks down ; firstly, for the same reason as advanced in Section 3 of Chapter IX in respect of the volumetric strain; and, secondly, because there is no reason why Aljl0 should be a more correct measure than Alfl, where I = l0 + Al. For Al — l0, the first gives 100 per cent, the second a 50 per cent, increase.* A consistent definition would result from relatmg a differential of increase to the instantaneous length so that the element of strain is dljl and the total strain

en=\dlll = {lnlil0)

(3)

A

in which formula both I and l0 are of equal standing. This logarithmic measure of extension was first introduced
by Roentgen, of X-ray fame. It was first systematically employed by Hencky (1929, 1) (compare also Ten Lectures, pp. 23-25).
For small elongations

en =

I 4~Al ln(2/Z0)=ln^—

=]n(l+Al}l0)=Alll0-lWIh)2+

• W

With this definition in mind we write for the Hooke solid

e«=Pnfc

(5)

Values of Young's modulus for different materials are listed in the following Table :—

Material

Lead Concrete Tin

Glass

ZlDC

Copper

Wrought lion

Steel

e X 10"5 in in megabars 1 2 4 7 8 12 18 25

* Recently a local paper quoted the President of an American College as having said that " the price of automatic pilot devices had dropped by several hundred per cent, during the war " I t must be assumed that he did not intend to say t h a t every buyer of a pilot device is being paid a premium.

POISSON'S RATIO

165

3. Poisson's Ratio

As we have already pointed out, the pull P n produces in the bar not only an axial extension but, generally, at the same time

also lateral contractions ee of magnitude.

ec = - oen

(6)

where a is a material constant called " Poisson's ratio ". If en is negative, ee will be positive or be a lateral expansion. Such lateral expansion cannot be observed in a cork-cylinder under

pressure, in which case a = 0.

Both ec and en are to be measured logarithmically, and (6) therefore defines Poisson's ratio for finite strains as well.

|d+e.)

-!ci-se,y Pri
FIG. X, 1. Simple pull of a prismatic bar.
The strain has been assumed as very small, so that " a " can be put for

166 SIMPLE TENSION" AND SIMPLE EXTENSION
We have thereby introduced for the Hooke solid two new constants, e and a, so that altogether we have four constants y, /c, e and <J. However, the values for y and K completely determine the rheological behaviour of a Hooke solid and it must therefore be possible to express e and a by the former. We regard y and / c a s " fundamental ", e and a as " derived " constants.
Poisson's ratio is a pure number. Approximate values of Poisson's ratio for different materials are listed in the following Table :—

Material Cork Aluminium Zinc Glass Steel Copper Lead Rubber

<7

0

013

0-20 0-24 0-30 0-32 0-40 0-49

Let us consider an axial section of the bar shown in Fig. X, 1. Generally, when being extended, the volume of the bar will be increased. At the same time, as can be seen from that figure, certain right angles will be changed and certain parallel planes accordingly sheared against each other. Therefore, the simple normal stress pn will be accompanied by a volumetric strain ev and a tangential strain et. Similarly, it is obvious that a simple extension en is accompanied by an isotropic stress p and a tangential stress pt.
Our task will consist in finding the stresses p and pt equivalent to pn in the first case and the strains ev and et equivalent to e„ in the second case.
4. Principal Strains and Deformations
If the bar is not itself a prism, it can be considered as made up of a large number of " elementary " prisms, the deformation of which we shall presently investigate.
Let the prism of edges a, b, c be extended or contracted in the directions of its edges. These directions are at right angles to each other before deformation and remain so after deformation. If we look at Fig. I, 1, we see that in the case of simple shear there also exist three such directions. They are called the Principal Axes of Deformation. In the case of simple shear, as again to be seen from Fig. I, 1, there is not only deformation but also rotation. One axis (the one normal to the plane of the

PRINCIPAL STRAINS AND DEFORMATIONS 167

figure) is fixed, but the other two are rotated by the same angle around the third. In our case, where the straining of the prism is in the direction of the edges, there is no such rotation. A deformation without rotation is called Pure Deformation.
We denote the directions of the principal axes by i, j and h. If these directions are the same throughout the body, the deformation is said to be homogeneous. This is the case in simple extension.
Now, let the edges of the prism before strain be aQi boi c0, so t h a t V0 = a0, b0, c0, and after strain a, b, c, so t h a t 7 = abc.
The principal strains or deformations are

e, = ln(a/a0), ei = ln(6/60), ek =In(c/c0) . . (7) Therefore

e , = l n ( F / F J = \n{abcja0b0c0) = \n{aja0) + ln(6/6J + In(c/c0) =

et + es + ek

(8)

If we define Mean Normal Strain or Deformation em by

«* = («< + c, + en)/3

(9)

we have

ev=3em

(10)

If we deduct from each of the three principal strains the mean strain in the following manner

eoi = ei ~ em> %} = ei ~ em» eok = efc — em • • ( H )
three other principal strains result, the cubical dilatation of which vanishes in accordance with

e<* = %i + %i + eok = ei + ei + ek-3em = 0 . (12)
Therefore eoi. eoj. eojQ are the principal strains of the distortion resulting from the strains or, more generally, deformations et e}i ek. Every combination of principal deformations ef, et, ek, can be resolved into a dilatation ev = e{ + e} -f- ek and a distortion et - ej3, es - eJ3, ek-eJ3.
We shall indicate the distortional deformation by the subscript " 0 ".
I n the case of simple normal stress (or more correctly, traction) pn, the principal strains are en in the longitudinal and — aen in the two transverse directions and, therefore,

168 SIMPLE TENSION AND SIMPLE E X T E N S I O N

ev=en{l-2a); em=eJ3=en(l-2o)l3

^

eo<=eoi = - 0 € n - e m = - ( J + a ) / 3 . e n ; e , * = e „ - e m = ^

(13)

2(l+o)eJ3i

For an incompressible material, c„ must vanish or a = J, in

which case eoi = eoi = - eJ2 ; eoft = en. In the case of simple normal strain in the ^-direction we have

e, = ei = 0 ; ek = en ; ev = en

5. Principal Stresses
The foregoing considerations in respect of strains can be repeated with suitable modifications in respect of stresses. Here Principal Stress Axes exist and in isotropic materials these coincide with the principal axes of strain, so that we can use for them the same subscripts, i, j , h. The Mean Normal Stress

Prn = (Pt+Pi + Pk)l3

(15)

is that part of the stress which, when applied isotropically in the

manner of a hydrostatic stress, causes the cubical dilatation

(with positive or negative sign as the case may be) of the

material, while the normal stresses

Pox =Pi ~Pm\ Pot =Pi ~Pm\ Pok = Pk ~ Pm • (16)

cause the distortion. In the case of simple normal tension

Pk =Pn\ Pi =Pi = 0; p =pm = pJ3^, Po* = Pa = - PJ3 ; Pok = 3p»/3 /
Now if the cubical dilatation et, as calculated in (13), is caused by the mean stresses pm as expressed in (16), Equation (IX, a) requires in the case of rest {ev —0),p — xev, oipnj3 = /cen (1 —2a) and because of (5)

K = €j3{l ~2a)

(18)

6. The Definition of Finite Shearing Strain
Let us now consider the shear produced by simple normal stress. In Fig. X, 1 a square has been inscribed on the longitudinal section of the prism, which by the extension en is converted into a rhomb. Let us first assume the extension to be very small; in fact so small that we may use the first term only in the development (4) of c„ ; then the originally right angle of the square will be slightly changed to, say, 8. In Section 5 of Chapter I we have defined the change of

T H E DEFINITION OF FINITE SHEARING STRAIN 169

a right angle produced by shear, if small, as the shearing strain. We find from the Figure

tg$l2 = (1 - ae„)j{l + e„)

(19)

The change of the right angle is (90 — 8) and its tangent is equal to the cotangent of 0, or tg(90 — 6) = ctg6 = Ijtgd.
From known formula

tgS = {2tgBj2)i(l - tg^)

(20)

and introducing igBj2 from (19) ,,fl _ o (-* ~ ge") <J + en)
*'-*<;+«.)'-<*-*.)•

/ ; • • (21)

Neglecting powers of en higher than the first, this gives

Ijtgd = en(l + a)

(22)

Therefore, for a small change of the right angle into 6,

et^ejl+a)

(23)*

A

FIG. X, 2. The maximum shearing strain in pure extension.

V2~ . V2

en

a / 2 = — _ e n + 2y/2

We now make a new definition for the shearing strain. We postulate that if c„ is large so that the complete (3) applies, e^ is defined by (23). This definition is consistent and for small strains conforms with the usual definition of the classical theory of elasticity.

7. The Law of Elastic Distortion The shear of the square inscribed into the rectangle of Fig.
X, 1 is produced by a shearing stress resulting from the action of
* Por those readers who prefer geometrical methods. Fig. X, 2 shows another way of deriving (23).

170 SIMPLE TENSION AND SIMPLE E X T E N S I O N
forces PJ2 acting on all sides of the square. These forces which act in the longitudinal direction, have tangential com-
p __ ponents Pt = -^IV2. The tangential forces act upon areas
— —— . c and the tangential stresses are therefore

* = PJ ~ o = PJ2bc.

Introducing PJbc = pn, this gives Pt=Pj2

^ (24)

For a Hookean solid we can go back to (I, d) and get

pJ2=yen(l + o)

(25)

which gives, because of (5),

2y=el(l + o)

(26)

From the two equations (18) and (26) we can now express the derived coefficients e and a through the fundamental coefficient Kand y :—

e = 9KYJ{3K + y) ; a = (3K-2y)l2(3K+y) . . (27)

For an incompressible material K = oo and therefore

e = 9yl(3 + yJK) =3yyo=\ From (17) we find

. . . (27')

Pa =Pa = ~ i/3.ett ; pQh = 2j3.een . . . (28) and considering (13)

poi=el(l+o).eoi;

^ = € / ( J + f f ) . e e i ; pok=ej(l-\-a).eok . (29)

Because of (26), these now become
Pa = 2y%i; Pd = 2y%3; p0* = h%*. • • • (3°) and from the symmetry of the expressions it is clear t h a t the relations (30) express Hooke's law for elastic distortion, independent of the conditions of simple normal stress from which they were derived. We can accordingly write more concisely

Pon=2yeon

(30')

where n now stands for any direction of the normal to any section however oriented.

T H E LAW OF ELASTIC DISTORTION 171

If the distortion is accompanied by a cubical dilatation, very much changing the density of the body, the stresses p0 must be
replaced by the reduced stresses — p0 in accordance with (IX,c).
P

8. Strain-work in Simple Tension and Simple Extension

Let us go back to the tensile test of a steel bar extended elastically in simple tension with which we started in Section 2. In extending the bar, the pull P n will perform work. In the testing machine, the speed of extension is moderate and the change will be through states of equilibrium so that kinetic energy is not generated. Therefore all work of the external forces will be converted into strain-work. Consider again an elementary prism with edges a, b, c (where a is in the direction of the axis of the test-piece) which is in equilibrium under the action of the simple normal stress pn. Ifpn is slightly increased, the edge a will be increased by Aa, while b and c will be decreased by Ab and Ac respectively. However, while in the direction of a, the force pnbc is acting, no forces act in the directions b and c. The increment of work will therefore be

AW = pnbcAa = pnabc.Aaja . . . (31)

If Aa is very small, Aaja = en and the element of work per unit volume

dw = pnden = €enden

(32)

This equation is perfectly analogous to (IX, 23) and

analogous conclusions can be drawn which need not all be enunciated here.

For the simple Hooke body, where e is constant, (32) gives

by integration

w=cen*l2=pn*l2c

(33)

In simple extension, while there are stresses in all three directions a, b and c, displacement is only in the direction a.

We therefore arrive at the simple expression (32), but here (5) is not applicable. To calculate the stress-strain relation, we start from (14) from which

p = Kev = K.en ; pok = 2yeon = 4}3.yen . . (34) and therefore
** = *„* + * = « » ( « + 4/3.y) • • • (35)

172 SIMPLE TENSION AND SIMPLE EXTENSION

This gives for constant coefficients

10 = (K + 4j3 . y). e*j2

(36)

F o r an incompressible material (K = co) both pk and w become infinite, which means that simple extension is not possible with an incompressible material.

9. The Coefficient of Viscous Traction
We said in Section 2 that the tensile test is used not only for metals but also for such materials as pitch. Of course, in the latter case the rheologics of the test is entirely different. I t is generally assumed t h a t if a rod of mild steel is loaded below the yield point it extends elastically and then stays put to the end of times. This, now, is a statement where the epistomological considerations of Section 1 of Chapter I X come in. I t has been maintained that even at room temperature mild steel will flow slowly or " creep ". About creep we shall speak in Chapter XIII. B u t even if mild steel should flow at the same temperature as pitch, there still remains a very definite quantitative difference between both. I n the case of a rod of pitch it is not necessary to employ exceedingly sensitive instruments to discover the continuous elongation ; it is visible to the naked eye—so much so t h a t Trouton (1906) regarded pitch as a very viscous liquid. This, too, is a gross simplification and we shall therefore consider here the continuous elongation under constant load of a rod made of a Newtonian liquid of very high viscosity instead of pitch, which, as we shall see in Chapter XIV, manifestly possesses other properties in addition to viscosity.
For such exceedingly viscous liquids, which to all appearances are solids, the two methods for the determination of viscosity, dealt with in Chapter I, are inapplicable. Whilst the viscosity of bitumen has been determined in rotation and tube viscometers, it may reach such values that it would either take tremendous force to produce a flow measurable in short time, or an impracticable long time under ordinary forces. In such cases (and for similar materials) Trouton applied such testing methods as are used for the determination of the elasticity of solids, e.g. the tensile test. He loaded a rod made of the material under test and measured the velocity of its extension. I t is

THE COEFFICIENT OF VISCOUS TRACTION 173

clear that the load per unit area of the section {pn) divided by the velocity of extension (e„) must be a measure of the viscosity. Trouton called it the coefficient of viscous traction.

A=i>„/en

.(37)

10. The Mohr-Circle for Simple Tension
The question now arises : how is this quantity related to the coefficient of viscosity 17 as defined by (I, f) ? In the latter equation the coefficient of viscosity 77 is defined by considering a case of simple shear, i.e. a case where neither the strain nor

FIG. X, 3. Stresses in simple tension. P„„ normal component of pull P„. P„, tangential component of pull P„.
the stress in respect of a given surface element has a normal component, i.e. e„ = 0 and pn = 0. Moreover, simple shear is a laminar displacement and therefore not accompanied by any change of volume, which means that we also have ev = 0.
Now, is there any section in the loaded rod inclined against the cross-section at the angle <f>, where the only stress is a shearing stress ? If we examine Fig. X, 3, it is clear that in every

174 SIMPLE TENSION AND SIMPLE EXTENSION

section through the rod, the pull P is equivalent to a tangential force

Pn,=P„sin^

(38)

together with a normal force

Pnn=Pncos^

(39)

Let the cross-section of the rod be unity, then the area of the

inclined section is

A+ = 7/cos <f>

(40)

and the tractions in the inclined section

Pnt = PntlA4. = P n sin ^ cos <j> = ?^ sin 2<j> (41)
p n n = PnJA4 = p n c o s ^ = ^(1 + cos 2<j>)

Therefore, when <j> varies from zero to 90°, the normal component of the traction acting upon the inclined section increases from zero to pni while the tangential component goes from zero to zero, passing through a maximum for ^ = 45°

nn
FIG. 3C 4. Mohr's Circle for simple tension. <j> slope of inclined section.
when max p^ = pJ2) while in that section pnn also = pJ2. These relations can be pictured graphically in a co-ordinate system where the pnn are abscissas and pnt ordinates. As can be seen from Fig. X, 4, comparing (41), all points representing connected values of pnn and pnt lie on a circle.

THE MOHR-CIRCLE FOR SIMPLE TENSION 175
This is a special case of the so-called Mohr circle (compare Ten Lectures, pp. 123-128).* The reply to the question, whether in any section the only stress is a shearing stress, is therefore in the negative.
11. Trouton's Derivation of the Relation between A and -q The reply being in the negative, we cannot directly apply
(I, f). However, have we forgotten our first Theological axiom that, in order to examine rheological properties, we have to consider distortions ? Let us correct this oversight.
This is usually done as follows, the method apparently being due to Lord Kelvin. Let us resolve the longitudinal tensile stress pn into three superposed tensile stresses, each equal to pJ3. The strains produced will not be affected by this procedure. Also let each side face be subjected to two opposite normal stresses, each equal to pJ3; these stresses will not produce any strain, and thus original strains remain unchanged. We can now group the stresses applied to the prism in any way we please, and the resultant strains must be identical with those produced by tensile stresses applied to the end faces. Each side face is subjected to a tensile stress pJ3, indicated in Fig. X, 5 by a black arrow ; grouping these stresses with the component tensile stresses pJ3 applied to the end faces, we obtain a uniform dilatational stress pJ3. We then group the stresses indicated by white arrows and those indicated by alternatively black and white arrows together, f
As can be seen from the Inset 1 of Fig. X, 5, these stresses—• black and alternate arrows separately—are equivalent to shearing stresses equal to any of them (and therefore = pJ3) and making angles of 45° with the length of the prism (compare Inset 2). Therefore, if a pyramid as shown in Inset 3 is cut out of the prism, it is in equilibrium under the action of the forces shown, with the traction pn requiring the superposition of a hydrostatic tension pn. In his calculation of the coefficient
* This gives us the opportunity to explain, with an example, the difference between " t r a c t i o n " and "stress". We have used both indiscriminately, but the stress is represented by the complete circle, which contains all tractions (Pnn» Pnt)> while the tractions are represented by points on the circle. This shows that stress is an entity of a higher order than traction.
t The reader will recognise that this is a geometrical derivation of what we found analytically in Section 5.

176 SIMPLE TENSION AND SIMPLE EXTENSION of viscous traction, Trouton followed this method as is evident from the quotation : " The tractional force applied to a rod may be resolved, as is usual in questions of elasticity, into two
equal shears ( = shearing stresses or tractions, M.R.) which are situated at right angles to each other and at 45° to the direction of traction, along with a uniform force of dilatation.

TROUTON'S DERIVATION

177

The value of either shearing stress and also of the dilatation stress {hydrostatic tension) is in each case one-third of that of the tractive stress (pn)"
It should be noted that Trouton is wrong in stating that the

Fio. X, 6 Graphical determination of angle between planes of shear ABa and ABb.
Arrows and figures 1 to 1G indicate procedure of construction.
two shears are " at right angles to each other ". Their horizontal projections are at right angles to each other, but not they themselves ; because the planes in which the shears act make
D.F.

178 SIMPLE TENSION AND SIMPLE'EXTENSION
an angle which is necessarily greater than 90°.* Trouton continues : " I n the first instance on the application of the tractive force, the resolved effects produced corresponding to these resolved stresses, will consist of a dilatation and of shearing strain. It can only be to the flow resulting from the latter that the continued elongation of the rod is due. Nothing similar can take place in the case of the stress of dilatation, which can only have an initial effect." I.e. should the material be compressible, and this it will generally be, then hydrostatic tension will only change its density immediately after the application of the traction and this will be all hydrostatic tension can produce : it will have no influence upon the flow. " The continued application of each shear will produce a corresponding flow given in each case by pt — r}6t, where pt is the shearing stress, t\ the coefficient of viscosity, and et the rate of change of direction of any line in the material in the plane of the shear, as it passes through the direction normal to the shearing stress " (see Inset 1 in Fig. X, 5). This, however, involves two assumptions, which have not been expressly stated : Firstly the assumption that added hydrostatic pressure or tension does not effect the value of the coefficient of viscosity. This is only approximately correct. Secondly, it should be noted that (I, f) defines rj for the case of one simple shear, while here we have two shears superposed. But what is worse, these two shears are not at right angles to each other, as was in Trouton's mind when advancing his argument. Trouton finishes as follows : " The resulting flow in the direction of the axis is obtained by adding the resolved components of the two flows in that direction; so that resolving the two effects, adding the components, and reducing the axial alongation to that per unit length, we find that e„ = et. Since pt = -qet and pt = pj3, where pn is the tractive force per square centimeter, we get 7] = Xj3, so that the coefficient of viscosity is equal to one-third of the coefficient of viscous traction."
This can be seen as follows :— We consider a square with sides of unit length, subjected to simple shear. The square shown in Fig. X, 7, in dotted lines
* I t is approximately = U8£°. For those of my readers who studied Descriptive Geometry and liked it, I have shown in Fig. X, 6, how the angle can be determined on that method.

TROUTON'S DERIVATION

179

becomes a rhomboid, as shown in full lines. If the displacement BB' = CC is very small, the originally right angle between the diagonals AG and BD, which after strain are AC' and B'D, remains a right angle. AG, however, is extended to AC and BD contracted to B'D. Simple shear is therefore equivalent to an extension and contraction without shear in two directions which are at right angles to each other and inclined by 45° against the direction of shear. As can be seen BB" = BB'jV2. On the other hand DB" is equal to DB' with a good enough approximation.
Furthermore, since BD = y/2, we have (DB — DB')jDB =

(DB-DB")JDB = BB"jDB = j$lV2 = BB'j2. Similarly

it is found that {AC - AC)jAC is equal to CC\2. As AB is unity, BB' = CC = et and a simple shear et is equivalent to an extension etj2 combined with a contraction eJ2 in directions which are mutually perpendicular, while either is inclined at an angle of 45° to the direction of the. shearing displacement.
Therefore, comparing Fig. X, 7, we find that from the action of

Fia. X , 7. Simple shear of square ABCD into rhomboid A, B', C , D. 4 indicates right angle (exactly or approximately.)
the white arrows the prism will extend by etf2 and from the action of the alternately black and white arrows likewise by et}2. The total extension is therefore twice eJ2 or = et. Now e\ can be calculated from (I, f) and is equal to ptjn. However, as pt = pJ3, we have et = pj3r) and therefore also
From the definition of the coefficient of viscous traction (1)
K 2

180 SIMPLE TENSION AND SIMPLE EXTENSION

on the other hand, e „ = pJX. Comparing the two last equa-

tions, we obtain

A = 3r)

(42)

12. The General Relation between A and -q

We have reproduced Trouton's proof of the relation between the coefficient of viscous traction and the coefficient of viscosity because most rheologists have followed the same lines. I t must, however, be said that at the same time many have felt some uneasiness. The subject has its difficulties, and I have therefore dealt with it by applying exact methods of tensor analysis, reducing assumptions to a minimum [39']. The result is that Trouton's relation (42) is not generally correct, but perhaps correct in most practical cases. To indicate this, we shall write Trouton's special A with an asterisk, so A*.

I shall here give the mathematical derivation in an elementary form. This will give me the opportunity to correct a mistake of mine in Ten Lectures, p . 49.
In Trouton's experiment we have to distinguish two stages. The stress due to the pull pn has, in accordance with (17), an isotropic component pm = pJ3. When the pull is applied there will be an initial stage which starts with an accelerated and ends with a retarded movement of the particles with—in general—pulsations in between. During this initial stage the material expands, the measure of the cubical dilatation a t every movement being ev. This cubical dilatation produces a n elastic isotropic reaction — KCV. I t is accompanied by a viscous resistance = t\ev due to volume viscosity. When ev has so much increased that *e„ = pJ3, the elastic isotropic reaction balances the isotropic component of the pull pn. The cubical dilatation then ceases to increase and e\ therefore vanishes. Then the second stage sets in, in which the movement is steady.
Now let us assume that the rate of deformation of the prism or the velocities of flow of its particles are so small that their kinetic energy may be neglected ; in which case there will be no pulsations but only an aperiodic movement leading to the second stage, when the movement becomes steady.
For the isotropic component of the traction we have accordingly (compare (IX, a))

PJ2 = «e* + Vv%

(43)

GENERAL RELATION B E T W E E N A AND v 181

For viscous distortion, because of the perfect analogy of (I, d) and (I, f), there are equations in force analogous to those of elastic distortion (30'), viz.

Pan = &?«„» In our case (compare (17))

(44)

Po* = 2pJ3 while (compare (10) and (11))

Therefore

Kh = efc - eJ3 = en - eJ3

pJ3=r)(en-eJ3) and

(45) . . . . (46)
(47)

i.=3in-Pnh Introducing this expression for ev into (43) we find

(48)

pJ3 = Kev + 7}v(3en - pjrj) . . . . (49) or

tn=(Pnl3-K%)}3Vv + pJ2v • • • (50)

The coefficient of viscous traction A is defined b y (37) in accordance with which we find
X-l=tnIPn = tt-3KeJpn)l9r)v + ll3r} . . (51)

or

*-(*7^/(' + 7^--<«>
Equation (52) is the viscous analogy to the first of (27), viz.
e ~—£=—. The analogy is, however, not exact. While « = 3y
for k = oo and only in this case, the situation is very different in the case of A.
I n interpreting (52) one should keep in mind that 1 — 3KeJp„ is either >0 in the first stage of Trouton's experiment, or = 0 in the second stage. F o r t h e volume viscosity we have 0 ^ Vv ^ ° ° - ft *s c l e a r th&t the volume viscosity cannot be negative as " otherwise the more alternate expansion and compression, alike in all directions, of a fluid, instead of demanding the exertion of work upon it, would cause it to give work out " (Stokes). As can be seen, if r}v' vanishes. A

182 SIMPLE TENSION AND SIMPLE EXTENSION

also vanishes in the first stage. On the other hand, A/17 = 3 in both stages if rjv = 00 and also in the second stage, when 1 — 3icejpn = 0, whatever the magnitude of T)V. Because -qv cannot be negative, A/77 cannot exceed the value 3. Therefore

0 ^ A £ 3T)

(55)

with A* = 3v in the second stage only.

13. The Stokes Liquid
These results are of interest in connection with the rheologics of the classical viscous liquid or what we called the Stokes Liquid. This was derived by Stokes by assuming t\v = 0, but in the first stage of the Trouton experiment a vanishing volume viscosity would mean A = 0 or vanishing viscous resistance against extension of a liquid cylinder no matter how high the ordinary viscosity -q of the liquid, a result at variance with our ideas of viscous flow. Tisza [40] has recently drawn attention to the following quotation from Stokes : " O f course we may at once put 17^ = 0 if we assume that in the case of a uniform motion of dilatation the pressure at any instant depends only on the actual density and temperature at that instant and not at the rate at which the former changes with the time. In most cases in which it would be interesting to apply the theory of friction of fluids, the density of the fluid is either constant or may without sensible error be regarded as constant, or else changes slowly with the time. In the first two cases the results would be the same and in the third nearly the same whether r}v were equal to zero or not. Consequently, if theory and experiments should in such cases agree, the experiments must not be regarded as confirming that part of the theory which relates to supposing 7)v to be equal to zero." Stokes does not seem to have realised the consequences which his assumption of vanishing volume viscosity carries in respect of the viscous resistance in simple tension, and he was definitely mistaken in equating the influence of either ev = 0 or r}v = 0 on experimental results. Examination of (52) shows that the same result follows from either e\ = 0 or TJV = 00 and not t\v — 0. I was misled by Stokes to make the same mis-statement in my Ten Lectures, p. 49, lines 17-20. If we assume with Stokes vanishing volume viscosity, the material would in the first

THE STOKES LIQUID

183

stage expand purely elastically and instantaneously in no time as is natural in the absence of the damping influence of a volume viscosity. If, on the other hand, 7jv were so great that it may be put = oo, the expansion would be infinitely delayed and the second stage never reached. Tisza deduces from observations on supersonic adsorption in certain liquids a value of yj-q = 2,000. This, when introduced in (54), is practically infinite. It is true, as Trouton found, that in the second stage A/17 = 3; but it cannot be assumed a priori tliat in an actual experiment with some real material that second stage is reached during the experiment. This depends entirely upon the magnitude of r]v.
Assuming that the second stage has in fact been reached in his experiments, Trouton calculated the values for A* shown in the following table :—

Material

Pitch

A*xl0-10 in poises

3-6-4-3

Mixture of pitch and

Tar

More tor

1-29

0-67

Shocmnker'a Wax
0-0059

14. Summary
In every deformed body there exists at every point a triple of normal axes i, j , k which were at right angles, too, before deformation. They are called principal axes of deformation (or strain, as the case may be). As the right angles between them do not change, they are not sheared and the deformation consists of positive or negative extensions in the direction of the principal axes or normal to the planes containing them. They are accordingly normal deformations (en) called principal deformations and denoted by et, cy, ek. In an isotropic material these deformations are connected with normal stresses (pn) called principal stresses and denoted by pif pit pk.
If the directions of i, j , k in space are the same before and after deformation, the latter is called pure deformation.
If the directions i, j , k and the magnitudes of et, eJ} ek are the same throughout the body, the deformation is said to be homogeneous.

184 SIMPLE TENSION AND SIMPLE EXTENSION

In a prism with edges a, 6, c oriented in the directions i, j , h respectively with lengths of edges a0, b0> c0 before deformation, the principal deformations are extensions

e, = In(a/aD), e, = ln(6/60), ek = ln(c/c0) . . (X, a) and the cubical dilatation as defined by (IX, 6)

% = ei + et + Ct

( x , b)

The principal deformations

««« = e( - « , A %i = e, - eJ3, eok = ek - eJ3 . (X, c)

for which e0{ -+- eoi + e0jt vanishes, constitute a distortion. I n accordance with t h e first axiom of rheology (compare
Section 3 of Chapter I) the characteristic rheological equation of a material connects eoi, eoj, eok, with stresses poi,poi,pok which are to be derived from plf p^ pk through

Po* = P*-P>Poj =Pi-P>Poie =P*-P

• (X, d)

where the isotropic stress p is the mean of the principal stresses

P=(pi + Pi+pk)f3 • . • . (X,e) In the case of simple tension, pn,

P = Pnfi ; Pa =Po, = ~ PJ3 ; Pok = 2pJ3 - (X, f)
The isotropic stress p is connected with the cubical dilatation e„ by means of (IX, a), which is t h e isotropic rheological equation valid for every kind of material.
For the distortional part in the case of the Hooke solid

Pen = 2ye„n
I n the case of the Newtonian liquid

(X, g)

Pon=2v%n

(X,h)

When a prismatic bar of length l0 is acted upon b y a pull ( + Pn) or a push (— Pn) in the direction of its axis (direction k) they produce a simple one-dimensional stress pn. This causes a three-dimensional pure deformation. In a Hooke-solid this

consists of a longitudinal extension without shear

e„ = In (Z/ZJ

(X, i)

or for small extensions
«««4yi.

(x,i'>

which is connected with pn through

e* = Pj<

(X,j)

SUMMARY

185

and lateral contractions of the magnitude

% = ~ ™n

(X, k)

In sections inclined by 45° against the axis there is a maxi-

mum shearing strain

et=en{l + a)

(X,J)

The " derived " elastic coefficients e and <J can be calculated

from the " fundamental " coefficients K and y in accordance

with

€ = 9KYI{3K + y)

(X,m)

o = (3K-2y)j2(3K+y)

. . . (X,n)

When a viscous liquid is subjected to simple normal stress

the analogy to e is Trouton's coefficient of viscous traction

A-i>n/en and the analogy to (X, 1)

(X,o)

_ A =

1 - j -3KeJpn . . . . (X, p)

V + 1 - 3KeJPn
When e„ vanishes and the movement becomes steady, ev = PJ3K, and A* = 3r\. This is also so whenever ev = 0.
A Newtonian hquid in the special condition when e„ vanishes is called a Stokes Liquid.
Simple extension is a one-dimensional extension which generally is connected with a three-dimensional stress. In the case of simple shear c() the principal deformations are e< = et — etj2 while ek normal to the plane of shear vanishes. The axes i and j are inclined at angles of 45° to the direction of the shearing displacement.
The strain-work per unit volume in simple tension is

w» = \Pn den= yenden . . . . (X, q)

which for the simple Hooke body when e is constant

wH = cen*l2=pn*l2t

. . . . <X,r)

CHAPTER XI
WORK -HARDENING
1. Technical and Rheological Test Curves
W E have dealt in the preceding Chapter with the tensile test and have mentioned that this is the predominant test for metals. Among these the most important from the economical point of view is, without doubt, mild steel. Let us therefore look more closely into the tensile test of mild steel.
A record of the observations consists in plotting the load (P) against the elongation (Al) or vice versd. In Section 5 of Chapter V we have called such a curve, where directly measured quantities are plotted, a Technical Curve. It forms the basis of the theoretical analysis; the latter aims at expressing the rheological properties of the material under test by (i) a rheological equation and (ii) the numerical values of its rheological coefficients. For accomplishing this task there are certain difficulties to be overcome. First of all, there is a difference depending upon which one of the two variables P and Al is taken as the independent and which one as the dependent variable. If we gradually increase the load, the rod of mild steel behaves first elastically, more or less as a Hooke solid ; but with a certain load the yield point is reached, when the rod starts to flowplastically at a more or less constant load. If, disregarding this, we continue to increase the load, there will be no equilibrium, the material will flow at an accelerated rate and soon break. This difficulty does not arise if the elongation is taken as independent variable. Here the load first increases ; then remains constant; then again increases ; then, after what is called the " necking " of the piece, decreases, until the piece breaks. This is the arrangement in most tensile testing machines for metals, where one end of the rod is fixed, the other travels at constant rate ; and where Al is therefore the independent variable, and the co-related P is observed.
Another method of taking account of the difficulty consists in considering the traction p and not the load P as the independent

TECHNICAL AND RHEOLOGICAL TEST CURVES 187
variable. With increasing elongation the cross-sectional area gradually decreases. If we let the traction gradually increase, this will often mean that the load must be decreased. If the elongation continues under constant traction, as it is supposed in a St. Venant body or in viscous traction, the load must be gradually decreased in the same ratio as the cross-sectional area decreases. There exist a number of devices to effect this automatically. One, by Andrade for pull, is described in Ten Lectures, p. 83. For another one, also for pull, compare Andrade and Chalmers (1932). Two methods for push, keeping compression stresses constant, are described by Scott Blair in Survey, pp. 120-122. Compare also Caffyn [41].
A curve in which stress is plotted against deformation in such a manner that the curve is independent of the dimensions of the apparatus may be called, in contradistinction to the technical curve, a Rheological Test-Curve. A consistency curve, in which the stress and deformation are consistency variables, is such a rheological test curve (provided there is no walleffect). A rheological test curve will not necessarily be independent of the kind of apparatus. If the variables are such that this is the case, the curve may be called the Fundamental Rheological Curve of the material. This is a graphical representation of the rheological equation of the material. The upper part of Fig. I l l , 3, a, shows the fundamental rheological curve of the Bingham body, while Fig. I l l , 4, shows two of its rheological test curves.
2. Strain-hardening
I have mentioned the necking of a mild steel bar after prolonged plastic extension. When necking starts, the rod ceases to be cylindrical or prismatic and obtains a shape which makes it less suitable as a test-piece ; stresses and deformations being no longer more or less evenly distributed over the length of the test-piece, but now concentrated at the section of striction where the cross-sectional area is a minimum.
If one comes to think of it, it is rather strange that the cylindrical shape can be maintained in any tensile test, either throughout the test or up to a certain elongation. The material tested (whatever it may be) cannot be of uniform quality over the length of the bar ; its quality must at some section be lowest.

188

WORK-HARDENING

At the same time it will not be absolutely cylindrical, but possess somewhere a minimum cross-sectional area. Therefore when loaded, it will first give way at one or the other of those sections. If so, the cross-sectional area will there be reduced and become markedly less than along the rest of the test piece. This will increase the stress at that section and worsen the position.* If this does not actually happen there must be some counteracting influence. The counteracting influence is strain hardening.^ Strain Hardening is the rise of the yield point of a material with increased deformation.I When the material first gives way at the weakest section it is there strained. When the strain brings with it strain-hardening, the material at that section may become harder than throughout the rest of the length and, even with increased local stress, deformation will not continue at the section but start somewhere else, i.e. at the next worst place. By attacking one section after the other, deformation is equally distributed over the length of the rod, the shape of which is finally again cylindrical. Therefore, if a bar of any material remains cylindrical (or prismatic) in a tensile test, this is an indication that the material possesses the property of strain-hardening.

3. The " Stress-Strain " Curve of Mild Steel
Let us go back to the technical curve for the tensile test of
mild steel. If we plot the load P against the elongation
JZ, this gives from the start a different curve for every
different diameter of bar of the same material. Therefore,
denoting by A0 the original cross-sectional area and by l0 the original length of the test-piece, the " stress " PjA is plotted
against the " s t r a i n " Aljl0 in what is called § the " stress-
* The tensile test, accordingly, proceeds through states of unstable equilibrium ; in contradistinction to the compression test where the equilibrium is stable, any local deformation resulting in a reduction of stress.
t This is the usual designation. It should, however, be noted that after the first yield point has been exceeded we should (in accordance with our terminology) not speak of " strain " but of " deformation ". We shall later propose to replace " strain hardening " by work hardening, a term which is more appropriate.
J This is the customary case. Schofield and Scott Blair [42] have observed strain hardening in flour dough where the viscosity increases with strain.
§ The terms " nominal stress ", " nominal strain " and " nominal stressstrain curve " are also used, to distinguish them from the " true " stress, strain and stress-strain curve, about which more will presently be said.

" S T R E S S - S T R A I N " CURVE OF MILD STEEL 189 strain curve " of mild steel. While this curve is more or less independent of the dimensions of the test-piece, it can still not be regarded as a rheological test curve, the designation of " stress-strain curve " being, as we shall presently see, not fully justified.
As we have said, the elongation Al is at first proportional to
P/W/A A
tag/cm"2 i cm2 7000 -
ihi
0.05 Fio. X I , 1. Technical " stress-strain curve " of mild steel. " a " first yield point. " h " yield point after strain-hardening. A cross-sectional area. P load. £1 elongation. (The elastic strain has been exaggerated.)
the load, until the yield point {a in Fig. XI, 1) is reached. This is the elastic or Hooke-range. After that the elongation grows quickly at approximately constant load. This is the plastic range. When the point c is reached the load must again be increased, but at a much lower rate than in the Hooke range. This is the strain-hardening range. When the point b is reached the load must be decreased ; the rod starts " necking " until it finally breaks at point e. Plotting PjA0 against Aljl0.

190

WORK-HARDENING

results in the curve o-a-c-d-b-e of Fig. 1. However, it is clear that this does not tell us much about the Theological properties of the material and even gives a misleading picture, because it is not obvious why, if the test-piece actually breaks at the load Pet it can sustain the same load at an earlier point d without breaking ; and later on at point b an even higher load. The explanation is, of course, that with increasing elongation the cross-sectional area is gradually reduced and, therefore, the stress increases even for the same load. If we plot the stress PjA against Mjloy the curve o-a-c'-b'-e' results and this curve looks more reasonable in that there is no fall of stress with increasing strain. However, the strain is here understood as the ratio of elongation to the total length of the test-piece, while after necking elongation takes place at the section of striction and should be related to the length of the latter. For this reason the denotion " stress-strain curve " is not applicable after the point " 6 ".

4. The St. Venant Range
Our description of what we called in the preceding section the plastic range, but what in accordance with our nomenclature we shall call the Simple St. Venant Range, was rather superficial. When the bar first yields, an immediate drop of the load to a lower level can be observed and yielding continues at that lower level. To describe this phenomenon, Bach [43] coined the terms " upper " and " lower " yield point. It has often been maintained that the phenomenon is not due to intrinsic rheological properties of the material, but to a " machine-effect" resulting from the combined mechanics of test-piece-cum-testing-machine. It has been stated that the rise of the yield point is caused by the inertia of the machine and is due to the sudden stretching of the specimen and the inability of the machine to react satisfactorily to this quick change. This question of the influence of the mechanical behaviour of the testing machine upon the observational results is certainly of importance. When the reaction of the apparatus is of the same order of magnitude as the phenomenon to be observed, these can hardly be separated. This has lead in the realm of the submicroscopic, where the observed is an electron

THE ST. VENANT RANGE

191

and the means of observation a photon, to Heisenberg's uncertainty principle. If, in the realm of the macroscopic, the testing machine is a steel structure and the test-piece a steel rod introduced into that structure, conditions become similar. We cannot go into the problem here. I t has been dealt with in great detail in a book by W. Spath [44]. In order to find out about the reality of the upper yield point, Edwards, Phillips and Liu [45] used a simple arrangement of direct loading by adding weights on a pan attached to the lower end

St V

£ FRICTION
'/////////// W//////////////////A

n
\

STATIC FRICTION

KINETIC FRICTION

I

Al

FIG. XI, 2. Model for a St. Venant body.

of the test-piece, avoiding mechanical devices for applying loads so that the difficulties from inertia arising from machine bearings, levers and other features of design would be overcome. It was found that, once plastic deformation commenced, it continued at much lower loads than those initially required. The upper yield point could be raised by careful and very slow loading.
This suggests for the St. Venant body a model as shown in Fig. XI, 2. A weight rests upon a table top and there is sob'd friction between both, a friction which is higher at rest and lower when movement takes place. A string is connected to

192

WORK-HARDENING

the weight and a gradually increasing pull P applied to its free end.* First the string is slack, then it is tightened.
As long as the pull is smaller than static friction, the weight remains at rest. When P reaches and slightly exceeds static friction, the weight starts to move, and in order to maintain equilibrium with constant velocity of movement, P must be reduced to kinetic friction. The resulting load-movement diagram is shown in the lower half of Fig. XI, 2. If we introduce

$ FRICTION
7,

St V
>w 777
////////•
i

H WftMMM— P

STATIC FRICTION

KINETIC FRICTION

Al

IHOOKE.

—1,RAN6EJ

ST. VENANT RANGE

FIG. XI, 3. Model for a combined Hooke-St. Venant body.

the stress pn for the load P and the strain en for the movement Alt we see that the model qualitatively reproduces the behaviour of the test-piece during the St. Venant range. To include the Hooke range, we introduce an elastic spring as a model for the Hooke solid and replace the string by the sprin°or couple the StV element with the H element in series, an operation which we indicate symbolically by StV — H. In

* If the reader has seen Fig. VIII, 1, (a) in Ten Lectures or the figure in m y paper [46], he will notice that I have now changed the model by introducing the " string " as a structural element. This was suggested to me by Mr. W. Fuchs and has many advantages.

THE STRAIN-HARDENING RANGE

193

Fig. XT, 3, the model and the load-movement diagram for this arrangement are shown.

5. The Strain-hardening Range
I t is not difficult to adapt this model so as to include the strain-hardening range. Fig. XI, 4 shows a model and the

*mfr*-

LIMIT OF STRAIN HARDENING
-A I
•INCREASE OF DISTANCE BETWEEN WEIGHTS CONNECTING STRING THK3HTENED
Fio. X I , 4. Model for a generalised St. Venant body.
respective load diagram. The model consists of a number of St. Venant elements connected in series and preceded by an H element. As deformation proceeds, or more strings are tightened, more and more weights are called into action, with the yield point rising. I t has been found that in mild steel, as in other materials showing strain-hardening, this does not go on indefinitely as the deformation increases. In the tensile test the process is interrupted by the breakage of the test-piece at about 20 per cent, overall extension or 30 per cent, local extension, but in wire drawing that same small steel rod can be converted into a thin wire by drawing it cold through a number of decreasing holes. By this process its diameter is reduced to a fraction and its length increased by many thousands

194

WORK-HARDENING

per cent. The process has a limit only when the wire thins to crystaUite dimensions. While, therefore, the deformation is practically without limit, no such parallel unhmited increase in yield point can be observed. The yield point gradually increases with diminishing rate of increase until a Maximum Yield stress, O^ is reached. This implies the existence of a Limit of strain-hardening. To this fact there corresponds in the model a limited number of decreasing weights.

6. The Tensile Strength
The models of Figs. X I , 2,3 and 4 do not include considerations of " strength ". We shall deal with strength in Chapter X V I I I . However, as the main object of the tensile test for mild steel is the determination of its " strength " we cannot avoid including a preliminary treatment of strength already in the present Chapter.
The tensile test is carried out in accordance with specifications for which we may take as typical the British Standard Specification (B.S.S.) No. 785—1938 for rolled steel bars and hard-drawn steel wire for concrete reinforcement. That specification asks (Clause 5) for " the tensile breaking strength, yield point (where specified) and elongation ". The terms are not defined, but there can be no question that the " yield p o i n t " is the stress PJA0 (Fig. X I , 1) and the "elongation" (extension) Aljl0. There is, however, a curious point about the "tensile breaking strength ". The specification says that it " shall conform to the requirements specified in Appendix A, page 18 ", but if we turn to that Appendix no tensile breaking strength is specified, but something which is called Ultimate Tensile Stress. The question naturally arises : are these two one and the same and how is it that they are named differently ?
British terminology is not as definite about the concepts involved as is American. In the United States the stress PbjA0 is called Ultimate Tensile Strength and the stress PJA0 Tensile Breaking Strength. In the well-known English textbook, The Strength of Materials, by E. S. Andrews, Ph\A0 is called Commercial Maximum Stress. The Germans call PJA0 Zvgfestigkeit. There can be no doubt that what the British Standard Specification requires is PjA0. In view, however, of the fact that at the Load Pb no breaking occurs, the term

THE TENSILE STRENGTH

195

" tensile breaking strength " is not very fortunate, especially if it is not defined.
The purpose of the present discussion is neither philological nor legalistic. There can be no question that in the use of a steel rod or bar as a structural member in tension, what governs its usefulness is the magnitude of the load Pb. The load which the member is to take is given. It does not help that the steel member can resist higher stresses when the point b has been passed (compare Fig. X I , 1), if the load has at the same time to be decreased—because this cannot be done. There are, however, other uses of mild steel and the question then arises : Is the stress PbjA0 or, more correctly, PJA of physical significance ? Does it tell us something of the intrinsic properties of the steel under test ? What is the necking due to ? Until the point b is reached the cylindrical rod remains cylindrical, albeit with increased length and reduced cross-section. Why does this not continue down to actual breaking ? And why does breakage occur at a certain extension, while in the wire-drawing process that extension can be infinitely enlarged ?
We shall presently discuss these questions using the matter provided by experiments which I first undertook in collaboration with W. Fuchs and H. Hberg at the Laboratory for Testing Materials of the Standards Institution of Palestine at Tel-Aviv [47] and continued in collaboration with A. Freudenthal [48] and which will be described in the following sections.

7. A Close-up of the Tensile Test Piece
A test-piece was shaped as shown in Fig. XI, 5. I t was assumed that the yield stress of the steel was about 3,000 kg/cm.2 and

n———Ui—

u-

j—i

\* 1.- IM em M
I
FIG. XI, 5. Original form and dimensions of test piece.

the test-piece was first loaded with 4,500 kg., giving a stress of 1,460 kg./cm.2 After that the rod was unloaded and the length found to be the same within 1/100 part of a mm. This, there-
O 2

196

WORK-HARDENING

fore, was a purely elastic strain. After that, experiment 2 was undertaken.
In experiment 2, point a ofFig. XI, 1 was reached and exceeded at a load of between 8,900 and 9,000 kg., giving a stress of about 2,900 kg./cm.2 The rod was again unloaded and measured.

F I G . X I , 6. Lateral contractions of testpiece in linear scale.
The testpiece was turned down several times. I shows the contractions of the original testpiece ; I I , I I I , IV of testpieces turned down. The outline of the last shape before turning down has been hatched vertically. Numbers indicate the experiment (as listed in the Tablo) in which the deformation was effected. For the meaning of " a " compare section 11 below.
There was now a permanent elongation of the central piece of 1-65 mm. which gives an extension = elongation/original length of 1-26 per cent, within a hundredth of 1 per cent. Elongations were measured in two longitudinal sections at right angles and differed slightly. They were averaged.
This procedure was continued. The rod was loaded until the yield point was reached, then unloaded, then measured.

A CLOSE-UP OF TENSILE TEST P I E C E 197
There were large intervals of time between experiments ; not on purpose, but due to the conditions under which the investigation was undertaken. However, as will be explained later, this brought to better light the phenomenon of " ageing " of steel.
Table I gives the data of the experiments made. With the sixth experiment necking set in, the point b in Fig. XI, 1 being reached. The tensile strength, as usually defined, was found to be 4,120 kg./cm.2 In the same Table the reduced crosssectional areas as determined by measurement have also been entered, and the " true " stress calculated accordingly. The " true " tensile strength was 5,200 kg./cm.2 but it should be kept in mind that actually no breaking occurred at this stress. There was no point in continuing the experiments in the same way. One knew what was to be expected : the rod would cease to be extended over the whole length, it would thin down at the section of striction where it would locally be extended ; finally the rod would break at that place at a certain smaller load but higher stress.
TABLE I. Data for Eheological Tensile Test Curve of Mild Steel

Experi- Date ment 0 - 2 6 43 No. In dajs

Load P
in kg

Stress P/A0
in kg/cm'

Elongain cm

Unit Exten-
sion Al/la fn%

Initial dimensions for scries

Smnllcst cross-
scctlona]
area A
in cm'

" True "
stress P/A in Kg/cm'

1 1 2 3 4

3

5 6

n 7 8 9 10

3rd Series

11 12 13 14 IB

n 10 17

1 18 19 20 21
§ 22

0

4500 1400

0

0 D , = l 982 cm 3-085 1400

0

8050 2000 0-165 1 2 6 A , - 3 085cm' 3 040 2038

46

0010 2020 0 263 2 01 /„ - 1 3 - 1 cm 2-080 3030

113

9730 3150 0 379 2-90

2 979 3260

103 12380 4020 1202 9-18

£814 4400

193 12720 4730 1801 14-20

2 430 6200

266 10000 4780 0180 1 2 6 D , - 1 636 cm 278 10900 5180 0-392 2-65 ^ , - 2 - 1 0 2 c m 1 S23 11020 5240 0-558 3-77 /„ -14-8cm 341 11220 5340 0-701 5-14

350

7700 5400 0-000 0-43 D , = l 333cm

371

7900 5490 0-118 0-76 d , = l 438 cm"

443

8140 5G60 0 220 1-42 /, —15 5 cm

NOTE:

440

7970 5550 0 287 1-85

Elongation AI

471

8030 5590 0 3(18 109

and

smallest

crosa - sectional

5(12

5850 5710 0 077 II 18 D,>-=1 ] 11 tin nrta A arc thoso

512

5850 6710 0-121 0 70 A , = l 0'J4nn' n(tallied at the

/, —15 9 cm end of the experi-

ment.

543

5150 5870

D, - 1 tlSScta

553

5100 5810 0152 0-95 A, = 0 878 cm'

560

5240 5960 0-218 1-36 /, - 1 0 0 cm

572

5100 5810

5S0

5220 5940

198

WORK-HARDENING

When measuring the length after eaoh experiment, the diameters along the length were also measured to 1/100 mm. In the uppermost part of Fig. XI, 6 the shapes of the rod up to the necking are shown in exaggerated scale. As can be seen, the statement in most textbooks that " until the point b is reached the cylindrical rod remains cylindrical " is correct only as far as observation with the naked eye goes. Actually the rod loses its perfect cylindrical shape with the first permanent deformation in experiment 2.
In experiment 3 the section where the diameter had been reduced before toll minimum, did not move ; deformation took place where the diameter had not been changed and the stress was therefore the smallest. In experiment 4 also the shallow microscopic neck formed in experiment 2 was not changed ; deformation took place at the ends which had not moved in either experiment 2 or 3. As we said in Section 2, the point where the diameter is most reduced in each experiment wanders along the length of the rod, to find the material which has so far been least strained. Just before necking, in experiment 5, the shape was again fairly cylindrical.
8. The Cause of Necking We can now understand what causes necking. As can be
seen from Fig. XI, 1, the rate of strain-hardening, i.e. the rise in yield point per unit increase of strain, falls off between points c and b. At the same time, with the reduction of the crosssectional area, the stress increases and has its maximum at the smallest section. At that section the yield point has also been raised most. There is accordingly a competition between the increase in yield-point and increase in stress at the smallest section. As long as the former prevails the deformation wanders away to some other place where the yield point is lower. As soon, however, as the rate of strain-hardening has become so low that the latter prevails, deformation continues at the same place and necking sets in.
This answered the question as to the cause of necking. At the same time it showed in which way the experiments were to be continued. After necking has set in the stresses are not more or less uniformly distributed over the length of the rod, but concentrated at the point of striction. Therefore, before continuing the experiments, the cylindrical form was restored.

AGEING

199

This was done by turning down the test-piece from a diameter of 1*982 cm. to 1*636 cm. The experiments were then continued as before.

9. Ageing
Now necking occurred at once, but at an entirely different place from that in experiment 6 (compare Fig. XT, 6). When the

FIG. XI, 7. Test-piece after the 9th experiment, showing three strictures. (Compare Fig. XI, 6.)

200

WORK-HARDENING

test-piece was given a " rest", it further hardened at the place where it had been strained most or, as is said, it " aged *\ Therefore in the nest experiment the deformation did not continue at the same place but occurred elsewhere, etc. In the end we had three strictions in the one test-piece, a rather unusual phenomenon (compare Fig. XI, 7). After that, the testpiece was again turned down and this was repeated several

F I G . X I , 8. Superposed lateral contractions of all testpieces. Numbers correspond with number of experiment in Table. Roman numerals refer to the series, of which each represents a new turningdown. Note how the testpiece in the first experiment to which it was subjected in series I I (No. 7) remembers, so to speak, where its least contraction had taken place before it was turned down after series I {and similarly in further experiments). The fifth testpiece has again been turned down and is now perfectly cylindrical. Necking was expected at point A and so happened in experiment 21.
times. In Fig. XI, 8, all deformations have been plotted in one figure. As can be seen, the material well " remembers " its previous history. Here also deformation wanders from point to point: from a more to a less strained section, even if the test-piece, after turning down, is perfectly cylindrical.
10. Local Deformations
It is clear that the value for M has no significance after necking has set in. Before that it gives a good average of the deformation in the different cross-sections. After necking, however, all deformation occurs at one place, while the rest of the rod is not deformed. Locally, as we pointed out in Section 3, the deformation will be quite considerable ; while its contribution to the increase in length will be very small. The apparent high rate of strain hardening between points b' and e' in Fig. XI, 1 is therefore misleading. Actually the length

LOCAL DEFORMATIONS

201

on the abscissa related to these two points will be much longer

in terms of deformation than shown here in terms of Aljl0.

To get a true picture one has to relate the stress at a section

to the local deformation which manifests itself in the reduction

of the diameter. Let us imagine the rod divided into disks of

originally equal thickness d0. The elongation Al is the sum of the increases of thicknesses of these disks. Every such

increase in thickness is accompanied by a decrease in diameter

and we may assume (as a first approximation) that in this

process the volume of the disk is not changed. Let d be the

thickness of the disk after extension, then its contribution to

the total axial extension is in, the correct logarithmic measure

en = \x\(djd0). Furthermore, let D0 be the original diameter and D the reduced diameter of the disk, then constancy of

volume requires that

D*d=D0*d0

(1)

or

\n{DHjD0U0) = 2 \n{DjD0) + ln(dfd9) = 0 . . (2)

from which with

In(Z>/Z>0) = cfi

(3)

e„ = - 2ec

(4)*

It is, therefore, possible to calculate the local axial extension from the local relative contraction. The former is twice as large as the latter.

11. The Rheological Tensile Test Curve
For the rheological test curve we have to plot local stresses
D27T
Pj—r- against local strains en = \n{djd0). This requires the
tracing of the different disks in the consecutive experiments as they change their distance from the axis.
Tn Fig. X I , 6 I have shown how we can follow the course of one such disk, marked " a," and it obviously is a cumbersome process. In Fig. X I , 9 the successive extended lengths of the test-piece have therefore been reduced to the original length taking account of local extensions and in this way the disks
* The same result is obtained from (X, 8) considering that the length and two diameters at right angles are principal axes of strain and therefore er = e„ + 2ef. If er vanishes, e„ = — 2ee.

202

WORK-HARDENING

keep their places. At the same time the diameters have been plotted in logarithmic scale and the strains, or rather deformations, can therefore be read directly from the figure. K i n the first series of experiments (I) any one disk suffered a reduction of its diameter from DQ to Dx and was then turned down to a diameter D2 which by deformation was reduced to Da, etc., its

20 T Sfains i7i?egari/Amic fiteasure can be rtacC Jirzcfy from /4/s-

19

FIG. X I . 9. Lateral deformations of testpiece. The contractions shown m Fig. X I , 5 are here plotted in logaritlimic scale and the lateral deformation can be read off directly.
final deformation is ec = Iv^DJD^ + ln(D3/Z)2) - ( - . . . which is therefore simply additive. This is not the case with the usual definition of strains and deformations. In Fig. X I , 9 all deformations have been superposed, in accordance with this principle, upon those of the first series as if no turning down had taken place.
It is now easy to plot local yield stresses against local strains, which has been done in Fig. XI, 10. No special precautions were taken either to exclude ageing or to ensure a definite and equal

THE RHEOLOGICAL TENSILE TEST CURVE 203
degree of ageing in the course of the experiments and there is therefore considerable scattering of points. Nevertheless the
Ic.9/cm

7000-

X

6300 6000 3300 3000 4500 4000 •

..'Ft
J** A
• AA

S A

V

y

-**A"A
A A

•^•^

i rmopatrr v mm Ajowmio POINT » JTAEWBEIOwnELflPOM

3300 -

%? V
3000 -

2800 •

•

0 0.02 004 0.06 Q08 010 012 OU 016 019 020 022 0 2 * Q26 028 Q30 032 034 036

F I G . X I , 10. Kheological stress-deformation omve. The true yield stress has been plotted against double the local contraction.
curve can serve as the basis for the discussion of the problem of the determination of the rheological equation governing the strain-hardening of a material, i.e. a relation between 0- and e.

12. The Mechanism of Strain-hardening of a Polycrystal
Before attacking this problem we must enquire to what cause the strain-hardening of mild steel may be due. First of all it should be mentioned that single crystals of metals are in all cases very soft. Metals, as we ordinarily know them, are, however, assemblages of very small crystals ; they are polycrystals. It is to this fact that they owe their hardness. At the same time it has been found that the hardness of a metal is greater the finer its crystalline structure (compare Andrade [49]). Therefore, whatever makes for smaller crystals will make for greater hardness and strain-hardening, on this view, is due to the diminution of the crystals through individual rupture on plastic deformation.
But then the question arises : why should a polycrystalline structure make for hardness ? To answer this question two

204

WORK-HARDENING

theories were proposed, one by Beilby and Rosenhain and the other by Ludwik. In accordance with the first, between the crystals of a metal there exists a layer of metal in the amorphous state—" the intercrystalline cement"—to which the hardness is attributed. This theory was discarded because no difference in hardness in the interior of the crystals and in the intercrystalline cement could be found. The second theory attributes the greater hardness of the polycrystalline metal to the random arrangement of adjacent crystals, with their glide planes in different directions, which leads to a, jamming, which in turn renders glide more difficult. This theory is now generally accepted. It has been proposed also by Jeffries and Archer under the name of the Slip-Interference Theory. We cannot here go more fully into these questions, but the reader may look up the very instructive and easily understandable article by Sir Lawrence Bragg [50] from which Fig. XI, 11 is reproduced.*
This shows a model for a single crystal and for a polycrystal made of small soap-bubbles, each one representing an " atom ". It can be seen at once why the first can be easily deformed by gliding between rows of bubbles, while in the second gliding is more difficult through the different orientations of the rows.

13. The Bauschinger Effect
While, therefore, rupture of individual crystals increases the resistance against gliding and raises the yield point, nevertheless a weakness is introduced in the surface along which rupture has taken place. This weakness possibly makes itself felt in the so-called Bauschinger effect. After plastic deformation and a reversal of direction of loading (from tension to compression or vice versa or from twisting in one sense to twisting in the opposite sense) comparatively large plastic deformations are produced by very small loads ; in other words, the yield point for the opposite kind of stress is lowered. But when a mild steel bar is twisted in one sense beyond the yield point there will be small internal rupture surfaces in the crystals along which there will be minute gliding. If the twisting is reversed, some of the ruptures will open up in the same way as the strands in a rope open up when the rope is twisted against its thread. This must cause large deformations or, * in other
* Endeavour, 1943, 2, 43-51: by permission.

THE BAUSCHINGER EFFECT

205

words, a Bauschinger effect in torsion. The explanation for the reversal from tension to compression is similar. Now, when we deformed the mild steel test-piece beyond the yield point and then allowed it to rest before reloading, we found that the yield point is raised and it is raised the more the longer the rest. This phenomenon is the ageing of which we spoke before. Ageing in this sense of the word is accelerated by heating to low temperatures up to, but not exceeding about 300° C. and the Bauschinger effect disappears. This is the tempering of the strain-hardened steel. It can be explained as brought about by a healing of internal rupture surfaces and the term " ageing " is therefore rather misleading. Rupture has taken place because the distance between the atoms on both sides of an interface is increased so much that they leave the range of atomic cohesion. Now, due to the heat energy of the body, every atom is in constant vibrations, the amplitude of which is determined by the temperature. If the amplitude of a vibration is sufficiently great, an atom on one side of the rupture surface may come within the range of attraction of an atom on the other side of the surface and a connection is made across the rupture. In this way the rupture heals. The process will also take place at ordinary temperature (albeit at a lower rate), because the vibrations are not all of the same amplitude but statistically distributed around some mean magnitude, and from time to time an extraordinarily great amplitude will effect a connection and local healing of the rupture.
When, however, the temperature is raised above 300° C. the vibrations become so strong that they not only heal the ruptures, but the atoms re-arrange themselves in their most stable grids. This is reerystallisation ; the crystals increase in size and the yield point is lowered until all strain-hardening may vanish. This is the annealing of the strain-hardened steel.

14. In Search of a Law of Strain-hardening
We are now in a better position to take up the problem which we put to ourselves in the closing sentence of Section 11. The tensile test of mild steel shows that the yield stress increases with increasing deformation. The question is, how is the former related to the latter ?

206

WORK-HARDENING

Let us go back to our test as represented graphically in Fig. XI, 1. If we, after reaching p o i n t " / " on the curve, unload the test-piece, a certain elastic strain is recovered, corresponding to the difference of abscissas of points " / " and "g", while the deformation og is plastic and permanent. Now, again increasing the load to the amount corresponding to point " / " » we reach approximately the same point (denoted in the figure by " h ") in an elastic straining with the same elastic modulus as in the first loading. This shows itself in the figure by the slope of gh being the same as that of oa. The curve a-c—b-e is therefore the geometrical locus of all the yield points corresponding to the successive deformations.* Nevertheless, as becomes clear for reasons which we similarly encountered before in two other cases,t the yield stress cannot depend directly upon the deformation. We mentioned in Section 13 the raising of the yield stress through a twisting of the bar. It is immediately obvious that this effect cannot depend upon whether we twist the bar clockwise or anti-clockwise. The yield stress &, must therefore be an even function of the tangential deformation et or a function of the et2. Let us remember (compare Section 7, Chapter III) that 9> itself is calculated by taking the root of another quantity, the plastic resilience Evl which itself is an even function of the stress. It will also help us in our problem if we recollect that strain-hardening is also called work-hardening. What governs the rise of the yield point is clearly the work expended in the plastic deformation and not the deformation itself. Let us imagine a giant of such strength that he would be able to knead mild steel as we knead flour dough. Let us hand him a steel ball which he will knead into all sorts of shapes, at the end restoring the sphere. When he has handed back the ball to us, its deformation is nil: all distortions, positive ana" negative, having cancelled each other. The strain-work, however, has all the time increased to a distinct amount. If, in order to make our considerations more definite, we assume that the deformations are simple shears, alternatively in positive and negative directions, the deformational work in terms of strain is, in accordance with (III, 43), = yet2j2.

* The existence of a yield point is therefore best revealed on unloading a testpiece.
f Vide Section 5 of Chapter I I I and Section 2 of Chapter VII.

IN SEARCH OF A LAW OF STRAIN-HARDENING 207

The resilience on the other hand is, in accordance with (III, d) = ptzj2y, and the plastic resilience in accordance with (III, b) = $2j2y. The modulus of elasticity being in that
process constant (as we said before), we may lump it together
with the numerical factor 2. In the case of tangential deformations the law of work-hardening would therefore relate &t2 to ef and should be expressed as a functional relationship &(2 =f(ef). We may denote by 8 the limiting strain corresponding to S-, so
that 6t is the ultimate shearing strain when the yield point is reached, or

*t=y9t

(5)

In other words, therefore, the law of work-hardening must

relate the square of the tangential strain to the square of the

tangential deformation.*

15. The Mises-Hencky Flow Condition for Simple Tension

We cannot apply such a functional relationship directly to our case, which is one of simple tension and not of simple shear. We must first express the Mises-Hencky flow condition for the case of simple tension. As we remarked in Section 1 of Chapter X, simple tension is not as simple as one may think. We have shown in Section 5 of that Chapter and in Fig. V, 5, that a simple tension pn can be considered as the result of the superposition of an isotropic tension p = pJ3 and a distortional stress the components pf which are a tension 2pJ3 in the axial direction and two compressions pj3 normal to the axis and at right angles to each other. I n accordance with the first axiom of rheology, the distortional stress only produces the plastic deformation and the plastic resilience Evl must be calculated not from the total strain-work of simple tension but only from its distortional component. The first is, in accordance with (X, r), = pn2j2e. The latter can be calculated as follows:—
In accordance with (X, g) we have

eon=PoJ2y

(5')

Therefore the distortional work

U>o = )Ponte0n= y - | P 0 n ^ o n = Pon*!*/ - - • (6)
* Keeping in mind the de6nition of strain as the recoverable part of the deformation.

208

WORK-HARDENING

Now, in the oase of simple extension, in accordance with

(X,f)

P0l =Poi = ~ PJ3 : Pou = 2pJ3 . . . . (V) and, therefore,

w0=l!4y.PT?i9.(l + l + 4)=pn*l6y. . . (8)

If we increase the tension pn until we reach the yield point,

when pn = &„, the strain-work w0 reaches the plastic resilience

Evl and therefore

K = + VW*

(«)

On the other hand from (III, b)

E*i=*t*IZy

(10)

which makes

&n = &tV3 = 1-73 a„, or &t = 0-578 &n . . (11)

and if we determine &t from a torsion experiment we can predict the yield stress $n in a tensile experiment with the same material. We may compare (11) with St. Venant's flow condition,* i.e. the condition that the shearing stress reaches a certain maximum (compare Section 6 of Chapter III). We know from Mohr's circle, Fig. X, 4, that in the case of a simple tension pn, the shearing stress reaches its maximum in a section inclined at 45° to the axis of the test-piece and is there equal to pJ2. The yield point will therefore be reached in simple tension when pn — &n = 2&t. The normal yield stress in accordance with St. Venant would accordingly be 2/1-73 = 1-15 times the yield stress in accordance with Mises-Hencky. The difference is not great. Geiringer and Prager [51] are of the opinion that St. Venant's flow condition furnishes in certain cases, e.g. the first starting of flow of mild steel, the best description and must be abandoned only when full plastic flow has developed. I hope, however, to have made it clear that, theoretically, the Mises-Hencky flow condition is highly superior to the St. Venanfc flow condition. Speaking of flow conditions, I may mention that Beltrami in 1885 postulated that the yield point is reached when the strain-work reaches a certain limit. This hypothesis was soon refuted by experimental evidence and we know why : it contradicts our first rheological axiom ; the

* This condition was postulated independently by Couloumb (1801) and Treaca (1863)

A " P R I M I T I V E " LAW OF WORK-HARDENING 209

dilatational strain-work should not cause any appreciable plastic flow.

16. A " P r i m i t i v e " Law of Work-hardening

We said at the end of Section 14 that the law of strain-

hardening would be one relating 8t2 to et2, or, as we can now say, 6n2 to e„2. I n physical language we express the plastic

resilience, which is the resilience at the yield point, as a function

of what we may call the Hardening Work ; the first is propor-

tional to 8t2 or 0n2, the second to e2 or e2. If we plot 8n2 against en2 the resulting curve has a striking resemblance to

the one (Fig. V, 8) for the variable fluidity of a Hquid as a

function of pt2, with 82 corresponding to q>0, 8^2 to (p^ and en2 to pt2. We may therefore tentatively write down in analogy

to (VII, i)

02 = dj _ (0w« _ 0o2)e-«V*

(12)

or in another form

Evl = EDl<0 - (Epta> - Ept0)e-"k& . . . (13)
where Epl is the plastic resilience which gradually, but more and more slowly, increases from Epl0, the plastic resilience in the annealed state, to Evlv>, the maximum plastic resilience which can be stored up in the material after it undergoes the maximum work-hardening. The hardening work wh is the work of the external forces spent in overcoming internal plastic friction and in changing the structure of the metal by breaking down its constituent crystal grains. The coefficient ^ is therefore quite analogous to the coefficient x '> i* also is a Coefficient of Structural Stability. As the expressions e2/^ and wh[ij/ must be dimensionless, the quantity ^ has the dimension of work, and *l> (compare (III, d ) ) the dimension of a work per unit rigidity.
Equation (13) was proposed by me as a " primitive " workhardening law at the Paris Congress for Applied Mechanics, 1946.
We found in Chapter VII that x was not a constant. Neither can we expect ^ or ^ to be constants. From what has been said in Section 12 it is clear that the larger the crystal the less its resistance against slip. With the continuous breaking down of the crystals $ and f must therefore increase.

OF,

*

210

WORK-HARDENING

In the second of the researches mentioned at the end of Section 6 above [48'], a strain-hardening curve was obtained first by discontinuous pulling in a tensile testing machine, and then by successive wire-drawing operations. A hot-rolled low-carbon steel rod of 8 mm. diameter was drawn down to a wire of 0-8 mm. diameter resulting in an extensional deformation of 4-6 in the logarithmic measure. It was found that the work-hardening curve consisted of three e-eurves of the form of Equations (12) and (13) following each other with short transitional curves between them. The transition from the first to the second e-curve occurred at the maximum tensile load (point b in Fig. XI, 1); that from the second to the third at an extension at which the tensile test fracture (point e in Fig. XI, 1) takes place. In the paper, an " atomistic " interpretation is given, using ideas developed by Sir Lawrence Bragg [50], which, however, belong to what may be termed Metarheology.

17. Summary
In a static test, i.e. one proceeding through states of equilibrium, mild steel shows a Hooke range, a simple St. Venant range reached through an " upper " and " lower " yield point, and a " generalised " St. Venant range in which the yield point increases at a decreasing rate, reaching an upper limit. An elastic spring can serve as a model for the Hooke-body (indicated symbolically by H) and a weight resting on a table, with solid friction between them, as a model for the St. Venant body (StV). Both coupled in series {H-StV) represent the Hooke and simple St. Venant range, while H-{StV)x-{StV) 2-.. .'-(StV)n includes the work-hardening or generalised St. Venant range, provided the series (StV)lt . . . {StV)n is arranged in monotonously decreasing order.
The distortional work in simple tension is

«>o»=Pn2I6Y and, therefore, the yield stress

(XI, a)

which gives

&n = ±V&^>
fr.=2-73»,

(XI, b)
(XI, o)

SUMMARY

211

in accordance with Mises-Hencky, while

*.=«**

(XI, d)

in accordance with St. Venant.

The rise of the yield point in the work-hardening range is due

to the breaking up of large into small crystals. For a hypo-

thetical polycrystal consisting of large crystals of equal size

which in each slip are converted into small crystals of minimum size, the rheological equation

S2 - V - (*«2 - V ) © " " * • • (XI, e) is proposed, where ^ is a coefficient of structural stability. For a real polycrystal $ gradually increases and the equation must be amended, taking into account a distribution of crystals of different sizes.

P 2

CHAPTER XII
BENDING AND TORSION*
1. Homogeneous Deformation and Stress
IN the foregoing we have become acquainted with three cases of simple deformation, viz., simple shear, simple cubical dilatation and simple extension. These three are Homogeneous Deformations, i.e. they are the same throughout the body. If we assume that the prism of Fig. I, 2 has no weight and subdivide it into a number of smaller prisms, each one of these is deformed in exactly the same manner as the large prism. In other words, et or ev or en in each one of the three cases does not depend upon the ordinates of the particles of the body. As the stress is related to the deformation by means of a rheological _ equation in which the co-ordinates do not appear, homogeneity of deformation carries with it homogeneity of stress.
On the other hand, if we subdivide a prism which has weight into horizontal layers, the weight of the material will cause a pressure increasing from zero at the top to a maximum at the bottom of the prism. It is therefore clear that homogeneous stress, and consequently homogeneous deformation, are possible only in the absence of weight, or, more generally, of Body Forces, such as inertia. Only if external forces acting upon the surface of the body, or Surface Forces, prevail over body forces so that the latter can be neglected, is homogeneous stress and consequently homogeneous deformation possible.
Generally, we must relate the rheological equation to an " infinitely " small volume element to which we apply the laws of mechanics (I, a) or (I, b) and (I, c), deriving, by integration, the rheological behaviour of the whole body. Sometimes, as
* The &ubject-matter of this Chapter can be found in essence m any of the numerous textbooks on the Strength of Materials ; I have, however, included it for the benefit of those readers who have not had an engineering training in order to make the book self-contained, aa in the followmg chapters reference is made to certain equations derived here.

HOMOGENEOUS DEFORMATION AND STRESS 213
in laminar distortion, we can treat a portion of the body, the lamina, finite in two dimensions but infinitely small in the third, as the volume element. This was the case when we dealt with the flow through the tube and in the rotation instrument where et was assumed to be constant throughout the length of the cylinder and the same in all radial sections, depending upon r only. This may be called ' quasi-homogeneous " deformation. In homogeneous deformation we do not need to relate the rheological equation to the volume element. If the deformation is homogeneous, the whole body can be considered as an "element"; there is no need for integration and all the rheologics of the body is contained in its rheological equation. This is the case in simple shear, simple dilatation and simple extension.
2. Simple Bending
There is an important case of " simple " deformation which is neither homogeneous nor quasi-homogeneous. It is Simple Bending.
Let a prismatic bar be subjected at both ends to couples which are in equilibrium, the plane in which the couples act passing through the axis (x) of the beam. Such couples are called " bending couples " (see Fig. XII, 1).
In order to find the deformation and the stresses by an elementary method, we reason as follows :—
For the sake of simplicity of argument let us first assume that the cross-section of the bar is symmetrical about an axis (y) and that the plane in which the bending couples act passes through this axis of symmetry. This plane is therefore the xy plane of the co-ordinate system. Now let the beam be divided into a number of shorter lengths.
As each length is in equilibrium, this requires that the internal stresses give rise to couples equal to the couples of the external forces. With exactly the same forces acting on each free body there is no reason why one should be strained differently from another. Yet, if all are strained in the same manner, the axes of all pieces are bent to the same curvature. As the length of the pieces can be taken as infinitely small, this implies that the whole beam is bent to a curve of constant curvature, i.e. to a circle, the plane of which is the x-y plane. Furthermore, it is

214

BENDING AND TORSION

Fia. X I I , 1. Simple bending of a prismatic beam. The first length has been given a rotation without altering the deformation.
obvious that the plane end sections of the pieces must remain plane and normal to the bent axis otherwise they could not fit together to complete the beam. This is known as Bernoulli's assumption (1705).
We now consider such a piece of length dx. As has been said above, the internal stresses acting upon an imagined crosssection are equivalent to a couple and to a couple only. The resultant of the stresses therefore vanishes, and the short prismatic piece is, accordingly, as a wJiole, neither extended nor compressed. The length dx of the axis will therefore not be changed and as it is bent, as we have seen, to a circular curve, the (< fibres " of the prism of length dx which are parallel to

SIMPLE BENDING

215

the axis are either extended or shortened in accordance with the equation

dx — dx = (M -f y) dQ — dx

(1)

If the arc to which the beam is bent is flat, the strains will be small and the bending strain eb will be
e6 = (dx — dx)jdx — (M + y) dQjdx — 1 . . (2)
However, dQ — dxjR, and, therefore,

% = yfR

(3)

These strains are directed Twrmal to the cross-section. They are therefore normal strains as dealt with in Chapter X, and if the material is a Hooke body, Hooke's law in the form (X, j) can be applied, or

The stresses are accordingly distributed linearily over the cross-section in triangles. Let the distance from the axis of the outermost fibre in tension be h1 and of the innermost fibre in compression A 2 > t n e n t n e maximum tensile stress is ekJR, and the maximum compressive stress ckJM.
Now let us assume the cross-section to be a rectangle of width 6, then the stresses give a resultant force in the indirection

I n our case, as there is no external force in the a;-direction, Px must vanish or h1 = k2 and the axis of the beam the length of which is not changed, or the so-called Neutral Axis, will pass through the centre of the section. This will be so in the case of every cross-section symmetrical about the 2-axis, but the reader will easily find that in a triangular cross-section the neutral axis will be nearer to the base and that, generally, the neutral axis will pass through the centre of gravity of the section.
The stresses must, however, give a finite bending couple Mbt the couple of the external forces.
Consider a strip of the cross-section having a length 6 parallel

216

BENDING AND TORSION

to the z-direction and the width dy. The increment of force

acting on this strip will be r=bdy and its moment in respect of

the axis \-^bdy)y = -^y2dy. The bending moment is

therefore Mb = ^Ufdy =JLJ0*

The reader will have no difficulty in finding that in the more general case

Mh = eIlR

(6)

where I is the moment of inertia of the cross-section about the neutral axis.
From (6) we find the radius of curvature

-B = e!\Mh

(7)

FIG. X I I , 2. Deformation of the cross-section of a bent beam.

SIMPLE BENDING

217

and, therefore, considering (3) and (4)

eb = Mbyjd ; pb = Mbyjl . . . . (8)

The maximum stress is found for the greatest value of y = h-^

in tension and y — h2 in compression. If the neutral axis passes through the geometrical centre of the section,

ht = h2 = hj2. The quantity SM = 21jk is called the Section Modulus and

max.#ft = MbjSM

(9)

For a rectangular section i" = bh3jl2 and SM = bh2j6 and,

therefore,

max.^b = 6MJbh2

(9')

While the pb stresses are the only stresses acting on the cross-section, the eb strains are not, of course, the only strains. With the extension of the fibres on the positive y-side, there goes hand-in-hand a contraction in the 2-direction which is its Ija part. With the shortening of the fibres on the negative y-sides, there goes hand-in-hand an extension in ^-direction. An originally rectangular cross-section therefore becomes of a shape as shown in Fig. XII, 2.
The " deflection " of the beam (d) can be calculated from well-known geometrical properties of a circle, according to which

{Lj2f = d{2R - d)

(10)

or, if d is small,

L2j4 =2Bd

(11)

and

d = L*j8R = LWj8eI

(12)

If the bending couple M is known and L and J are determined and d is measured, Young's modulus e can be calculated from

€ = L*MI8Id

(13)

Simple bending is produced if a beam rests on two supports and loads are suspended outside the supports as shown in Fig. XII, 3. Only the piece of the beam between the supports is bent to a circle, because only in that part are the external forces equivalent to a constant couple M = P.I provided the weight of the beam can be neglected.

218

BENDING AND TORSION

FIG. X I I , 3. Scheme for simple bending of a beam.

Bernoulli's assumption is correct only in this single case of simple bending. It is not correct in any other case. E.g. in a beam resting upon two supports and bent by its own weight, plane cross-sections do not remain plane and the neutral-axis is not bent to a circle.
The general problem of bending has been treated by Reiner [52]. It transcends the possibilities of elementary methods. If, however, it is desired to find such an " overall " macroquantity as the deflection d of the beam, when the exact microdistribution of local strains and stresses is irrelevant, (8) gives a sufficient approximation.
If the curve is not a circle, its curvature at any point is, from known principles

^"[JHTFF

(14)

SIMPLE BENDING

219

where y' = dyjdx and y" = dhjjdx2. If the curvature is
moderate and the arc to which the beam is bent is accordingly flat, dyjdx can be neglected and IjR approximately = dh/jdx2.
This makes (8)

s—•«

(15)*

from which the deflection can be calculated if Mb is known as a function of x.

3. Bending under its Own Weight

Let the weight of the beam itself be w per unit length, then the reactions from the supports are wLj2 and, therefore, the bending moment at the point x

wL

x

x = ~2X ~ ^2

(16)

(compare Fig. XII, 4).

REACTION FROM SUPPORT*^ foAL
l

m± A

VElWyjAx I

Fio. X I I , 4. Bending of a beam by a distributed load.

Therefore

w dh/jdx* = - g - j * ( £ - * ) • • •

(17)

which gives by integrating twice
v--jgrwfxI**-*M + 0* + 0* • • (18)
where Gx and Cz are integration constants.
» The negative sign is due to d being positive for a negative curvature.

220

BENDING AND TORSION

We determine these from the kinematic boundary conditions wL3
y = 0 for x = 0 and x = L, which give C2 — 0, Cx = 24eF , so
that y = wx}24el. (£3 + a;3 - 2Lx*). . . . (19)
and d=yl* = us = 5wL4l384tI . . . . (20)

4. Bending by a Concentrated Load in the Centre

Here the reactions from the supports are P\2 and the moment

Mx = Pxj2

(21)

Compare Fig. XII, 5.

REACTION PA FROM SUPPORT

REACTION P/z FROM SUPPORT

A^pilfmf.Wml/Am-}\

FIG. X I I , 5. Bending of a beam by a concentrated load in the centre.

Therefore

dhjidx* = - Px\2*I

(22)

which gives by integrating twice

y = - PX*\12<LI + C1x + C2. . . . (23)

The deflection line consists of two parts which meet at the centre without a break. The tangent to the deflection line at the centre is therefore horizontal. The integration constants can accordingly be determined from y = 0 for x— 0 and dyjdx = 0 for x = L\2. We find C% = 0, C t = Plt/lSd, and therefore
y = PL* 116*1. (L* - 4x2}3) . . . . (24) This makes

d = Vz-1,12 = PL*f48eI

(25)

BENDING BY TWO CONCENTRATED LOADS 221
5. Bending by Two Concentrated Loads arranged Symmetrically about the Centre
Here the reactions from the supports are M = P, and the moment is (compare Fig. XII, 6)

REACT/ON 'P' FROM SUPPORT p
<x
A Y////////////A x£
7

REACTION P' FROM SUPPORT
-&

FIG. X I I , 6. Bending of a beam by two concentrated loads arranged symmetrically about the centre.

Mn=Px
between x =0 and x = (L — a)j2, and
Mx=P(L-a)j2
between both loads. Therefore d*yjdx* = - Pxj€l

(26) (27) (28)

between x = 0 and x = (L — a)\2, while

Ahjl&x* = - P(L ~a)l2

(29)

between the loads. Integration of both differential equations gives

y = _ Px*l6tl -f Cx x + C2 . . . . (30)

y = - P(£ - a)x*l4el + C3 x + C4 . . (31)
Prom the condition y = 0 for x = 0 we find C3 — 0. From the condition dyjdx = 0 for x = LJ2, C3 = PL{L — a)j4. The other two integration constants can be calculated from the condition that at x = {L — a)j2 the deflection and the slope must be the same whether calculated from (30) or (31), i.e.

222

BENDING AND TORSION

P{L - af , n L-a

48<d

+Cl"2~

+ P{L - af 16*1

PL{L 8el

af +

Ct

(32)

and

P(L - af c = P(L - af PL(L - a) (33)

8el

4cl

4el

We find

P(L* - q«)

P(L - af

°1-

8S*-?1 » ° 4 - _ 48el

(34)

and therefore

2/ = - iV/oW + P(£2 - a2)xj8el y = - P(L - a)x2j4el + P i ( L - o)a/4e/ - P{L -

(35) afj48€I

The maximum deflection follows from the second equation

for x = Lj2 with

d = P ( i - a)(2i2 + 2£a - a2)/4Se/

(36)

6. Simple Elastic Torsion
Another case of simple deformation is simple torsion. It is laminar, as shown in Fig. I, I, and accordingly quasi-homo-

FIG. X I I , 7. Torsion of a circular cylinder.

SIMPLE ELASTIC TORSION

223

geneous. Let the circular cylinder of radius R and length I be fixed at one end and, at the other end, loaded by a torsional couple Mt. The cylinder will be twisted, i.e. the section z = I will be rotated against the section z = 0 by an angle, say Q. The problem is to express Q in terms of Mz and the dimensions of the cylinder, i.e. the geometrical quantities R and I.
In the elementary method which we are following in the present book we cannot solve this problem without the introduction of an assumption similar to Bernoulli's assumption. This assumption can be expressed in different ways. The first to attempt a solution of the problem of torsion was Coulomb and he started from the assumption that shearing stresses only act in the cross-sections and that they are proportional to their distances from the axis and directed normal to the radius, i.e. he started from the equation

Pt=cr

(37)

where c is a constant.

From this he found by applying Hooke's law

et=crly

(38)

Comparing Fig. XII, 7 with Fig. I, 2, we see that we can express et, introducing dz for H, as follows :—•

et = dujdz

(39)

or

CT

du=-dz

(40)

y

and by integration

u = crjy.z

(41)

considering that u = 0 for z = 0. I n order to find c we compose the stresses taken over the
section to the torsional couple

Mz = f (j>fir-ndr)r = 2TTC\ r3dr = -ncrWft . . (42)

and get c = 2MJTTR* so that (compare with (37))

pt = ZMzr\TTW

(43)

Equation (41), on the other hand, gives

u =2MJirBKrly.z

(44)

224

BENDING AND TORSION

The free end-section is rotated through the angle

Q = ujr\z=l = 2MJIirR*y

(45)

and this equation connects the observable macromechanical quantities Mz and Q. Q may be called the rotational displacement.

Equation (45) can also be written

Mz = d ^ \ l = QDjl T . . . (45')

where Djl is the "restoring m o m e n t " of (2, 45) and D the " torsional resistance " of Table I I I , Chapter V.
From (41) it can be "seen that" the section % = z will be rotated against the section z = 0 by an angle of the magnitude ujr =cjy, z ~ Qjt.z.
The quantity Qjt is called the twist. I n our case, the twist is constant. Generally where it is not constant, the twist is measured by the relative rotation of one slice against the other or by dQjdz. From (41) it also follows t h a t if the cylinder is imagined to be composed of " slices " of thickness dz, the slices are displaced as wJwles so that the displacements have neither radial nor longitudinal components, the only component being a rotational displacement. This confirms the picture of Fig. I I , 1.
We could also have started with this, as a natural kinematical assumption, and would have found from

ug = Crz

(46)

et = du$jdz =Cr

(47)

and, in accordance with Hooke's law, again

Pt=Yet=yOr

(48)

where

yC = c

(49)

While it might seem natural to assume that in torsion the shearing stress is proportional to the distance from the axis ; or that plane cross-sections are rotated against each other, but remain plane (both assumptions, as has just been shown, being identical), it can easily be shown that Coulomb's assumption cannot be valid for any shape of section other than a circular

SIMPLE ELASTIC TORSION

225

section. If the section is other than circular its edge makes an angle with the radius vector r; and the shearing stress^, which has the direction of 8, or is normal to r, must then have a component normal to the linear element of the edge (see Fig. XII, 8), i.e. in the direction of n. However, in accordance with the law of corresponding shearing stresses {compare Section 6, Chapter I), for every shearing stress in the cross-

FIG. X I I , 8. Torsion of a rod of non-circular cross-section.

section there exists an equal shearing stress in a longitudinal section normal to the cross-section. In our case, if there existed a shearing stress in the cross-section having the direction n, there would have to exist a shearing stress acting upon the side in a direction normal to n. We have, however, assumed that no surface forces act upon the side of the cylinder and, therefore, such a shearing stress as mentioned last does not exist. Therefore, pt cannot have a component in the direction n. Hence pt cannot he directed normal to r and it has for non-circular sections generally the two components p0 and pr.
The general solution for non-circular sections was found by St. Venant (1855). St. Venant assumed, as we have assumed

D.F.

<»

226

BENDING AND TORSION

above, that no forces act upon the sides of the prism. Actually, however, in order to twist a rod it must be fixed at the ends in such a way that surface forces are exerted upon short lengths of the sides. I t is true that at some distance from the fixing-points, the sides will be free from forces and there the stresses will be those of St. Venant's theory. These stresses may, however, not be the maximum stresses and if the material fails it will do so at the fixed ends. Therefore, for calculations of strengths, a better approximation to actual conditions than provided by St. Venant's theory is required. In three papers I have made an attempt at such a generalisation of St. Venant's theory by considering the action of torsional surface forces applied upon the sides of a cylinder or prism [52', 53, 54].

7. Plastic Torsion

If an elastic rod is subjected to torsion, we find (compare (45) ) that in order to rotate its free end through an angle Q a torque

Mz = ir&yQftl

(50)

has to be applied. From this equation it follows that there

is a linear relationship between Mz and Q. If, however, Q is gradually increased through an increase of the external forces,

we find that this linear relationship has a limit and that the

rod either breaks, when it is brittle, or that it can from a

certain stage on be further considerably increased by a very

small increase of Ms> when the material is ductile or plastic. We may therefore try to apply St. Venant's law (I, e) to this

case.

The application of St. Venant's law is easy enough if we have

a case of homogeneous stress. Then the yield point is reached

in all parts of the body at the same time and while, before

reaching the yield point, the deformation of the whole body was

elastic, after passing the yield point the deformation of the

whole body is plastic. I t is not so in cases of heterogeneous

stress. In such cases there must be a surface or surfaces

dividing the body in two parts, the rheological equation of

which will be either (I, d) or (I, e). The form of the dividing

surface is generally not known and this causes the greatest

mathematical difficulties. In certain cases, however, the form

of the surface suggests itself. This is especially so if we are

PLASTIC TORSION

227

dealing with a laminar displacement. The form of the surface can then be assumed, the assumption can be verified and the position of the surface found from the condition that for the dividing surface the tangential stresses of the elastic part are all equal in magnitude and equal to &t. We followed this procedure in deriving the Buckingham-Reiner and the Reiner and Riwlin equations. We shall take the same course in the present case.
As we have shown in the preceding Section, the tangential stress in the twisted elastic rod is

pt = 2Mzrl7rR* = yQrjI

(51)

The tangential stress has therefore the same value in every point of a cylindrical surface with the radius r and its maximum value for r = R

max. pt = 2MJirR*

(52)

The dividing surface will accordingly be a coaxial cylindrical surface. If Q is gradually increased, maximum pt also increases, until it reaches the value &t and the other stresses are increased correspondingly. If Q is still further increased, maximum pt cannot, in accordance with St. Venant's law, increase beyond $t and a plastic body develops as the outer shell of the cylinder, enclosing an elastic co-axial cylinder. As the tangential stress pt of this elastic core is the same over one and the same concentric cylindrical surface, the dividing surface between the plastic and the elastic parts must, as has been said before, be a cylinder of, say, radius r0. Within this cylinder the stresses will still follow Hooke's law as given in (51) and r0 can be calculated from

*t=yrfill

(53)

to be

r0=HfrQ

(54)

In the outer shell the tangential stresses will all be equal to §t. Analogous to (42) we then have

M = 27r[Rptr2dr = J&r|~J "yr^jl.rHr + J &tr2drj =
& r * ^ » / 3 - w V * W f l 8 * • (55) Fig. X I I , 9 shows Mx as a function of Q.
a 2

228

BENDING AND TORSION

It is interesting to note that while in the case represented in Fig. I, 5, b, when the shear has exceeded the value &/y, the material starts to flow under constant stress ; in our case, when plastic deformation starts there is no flow, but the external

2*tfF^
COMBINED HOOKE-StVEIWNT RANGE FIG. XII, 9. Plastic torque-twist diagram.
forces may even be increased and there will still be an arresting deformation up to a maximum moment (max. M. = 2&tirR*j3). This is due to the fact that in the present case there always remains a solid elastic core of ch^inishing, but never vanishing, radius r0. It should, however, be noted that we have treated our problem as a case of equilibrium. The torsional moment of the external forces can, of course, be greater than maximum M3. In this case equilibrium is not possible and also not steady flow. If Mx>2&tTTRzj3, there will be accelerated flow under the action of a torsional moment which is Mz — 2Q-tirR3j3. 8. Viscosity-Elasticity Analogies
In Section 1 of Chapter III we have pointed out the analogy of (I, d) and (I, f) in accordance with which, if we know the

VISCOSITY-ELASTICITY ANALOGIES

229

solution of a problem of elasticity, we can write down at once the

solution of the analogous problem of viscosity. These equa-

tions refer to cases of shear, but in Seotion 12 of Chapter X we

have shown that a similar analogy exists also in simple tension.

In shear, 77 corresponds to y ; in simple tension in the second

stage after the cubical dilatation has reached its maximum,

Trouton's coefficient of viscous traction A corresponds to

Young's modulus e in the case of an incompressible material.

If, for instance, we place a beam made of, say, fairly hard

bitumen upon two supports and load the beam in such a way

that simple bending is produced, the beam will gradually and

continuously sag and, as long as the deflection is not too great,

the rate of sagging, d can be found from (12) to be

d = L2Mj8\I = L*MI24*iI

(56)

If the bending is by a weight w per unit length of the beam, the rate of deflection is in accordance with (20)
d = 5wL*j384)J = 5wLilll52vI . . . (57)
A concentrated load P in the centre will produce a rate of deflection in accordance with (25) of

d =PL*148\I =PL*jl44r)I . . . . (58)

Bending by two concentrated loads P arranged symmetrically about the centre with a as the distance between them will produce in accordance with (36) a rate of deflection
d = P(L - a){2L2 + 2La - a2)j48\I = P(L - a) {21? + 2La- a2)ll44r)I . (59)

These are all examples where the main stress is " normal ". Torsion of a viscous liquid gives, in accordance with (45)

& = 2ML\ITB.S

(6°)

9. The Advantages of Bending and Torsion Tests and Hooke's Lucky Chance
The bending and torsion tests are often more suitable for the determination of rheological constants than simple tension. In a rheological test the kinematical quantity observed is seldom directly a strain (or a deformation) or rate of strain (or rate of deformation). It is rather a displacement or rate of displacement. In simple tension, where the deformation is

230

BENDING AND TORSION

'pure, the total displacement is the sum of the elementary displacements. In the bending of a rod, where there is rotation of the elements, the displacements are increased along the length of the rod as in a rotating pointer arrangement. Take, for instance, a short rod of any elastic material in your hand and apply with the other hand some force. If the force is a pull in the direction of the axis of the rod, the displacement of the free end will hardly be noticeable. If the force is applied at the free end in the direction normal to the axis, there will be a noticeable displacement, provided the rod is not too stiff. To make this example more definite, let us assume the rod to be of mild steel of a cross-section 1 mm. square and 10 cm. long. Applying a pull of 100 g., the extension will be in accordance with (X, j), en — 3 x 10~6 and, therefore, in accordance with (X, i') the displacement of the tree end Al = 3 x 10~s cm. Applying the same force in the direction normal to the axis, the deflection will be found to be the same as if twice the load had been applied in the centre of a beam of double length supported at both ends. Tins, in accordance with (25) is d = 2P(8L5)/48€l and considering I = bhz\12 = 1(12 mm.\ we shall find d to be slightly over 1 cm. or a displacement one million times that of the first case.
Because of the viscosity-elasticity analogies mentioned in the preceding section, similar considerations apply in respect of the determination of coefficients of viscosity. Here also the " sagging-beam method " is more striking than simple traction.
In torsion also, due to the cumulative effect of rotations, the displacement is much greater than in simple shear.
The disadvantage of both the bending and torsion test is the heterogeneity of deformation and stress which means that the rheological test curve is not a fundamental curve in the meaning of Section 1, Chapter XI.
These considerations are relevant when considering the elastic deformation of a helical spring. Even when every element of the spring undergoes infinitesimal strain only, the combined and additive effect of rotations due to bending and torsion of the elements will produce a very pronounced displacement of the ends under an axial pull. • Even should the material of the spring not follow Hooke's law of proportionality, the displacement will follow such a law because deviations from the

SUMMARY

231

law become noticeable at finite strain only. I t was Hooke's lucky chance to experiment with springs, to measure displacements, and to relate force and displacement. Had he attempted a law between stress and strain, he would have failed because (i) he would not have been able to calculate either stress or strain in a spring, a problem which was solved by Kirchhoff centuries later only, (ii) had he therefore chosen the case of simple tension of a straight rod, the mechanics of which were accessible to him, his means of measurement would not have been accurate enough. Most rheological laws were discovered in such loose ways.

10. Summary
If a beam of elastic material is acted upon at both ends by bending couples Mb which are in equilibrium, it is bent to a circular arc of radius R which is

R = djMb

(XII, a)

The deflection of the beam, if small, is given by

d = L*MJ8€l

(XH, b)

The equation of the neutral axis can generally be calculated for small deflections from the differential equation
d*yjdx2 = - MJel . . . . (XII, c)

By integration we find for the deflection of a beam through

bending from its own weight w

d = 5wLil384€l

(XII, d)

from a load P concentrated in the centre d = PLs}48eI

(XII, e)

and from two concentrated equal loads P with a the distance

between them

d=P(L-

a) (2L* + 2La- a2)j48d . (XH, f)

The twist of a circular rod of elastic material by the torsional

moment Mt is

Qfl = 2Mtl7rR*y . . . . (XII, g)

and the stress

pt = 2Mplir&

(XII, h)

232

BENDING AND TORSION

If the material is plastic there is, in the case of equilibrium Mt = 2*^13 - irn5l6ys& . . . (XII, i)
The viscosity-elasticity analogy permits the calculation of the rate of deflection from (XII, b to f) by replacing d by d and e by A and of the rate of twist from (XII, g) by replacing Q by & and y by 77.

CHAPTER XIJI
CREEP
1. Cement Stone a Liquid, not a Solid—Glass a Solid, not a Liquid ?
COMMERCIAL cement is a powder which on mixing with water hardens to an artificial stone. Many will be astonished to hear t h a t this stone, which looks solid enough, flows, if given enough time, and is therefore actually a liquid. One may well ask : if cement stone is a liquid, what is a liquid ? The answer is : a liquid flows, and flow is a continuous deformation under constant stress. A solid either does not flow at all, or it flows plastically. Plastic flow requires a stress larger than a definite stress, the yield stress, and a plastic solid does not flow under the action of a stress below that yield stress. The limiting stress below which the material does not flow may be quite low : this makes a Soft Solid. Such a soft solid may be mistaken for a liquid for quite a long time, as was the case with oil paint. In contradistinction to the plastic solid, a liquid flows under any stress, however small.*
If this definition is accepted, cement stone is, as will presently be described, at least up to an age of five or six years, a liquid. (After that, it possibly has hardened to a solid.)
But if cement stone is a liquid, one may become doubtful, whether there are any solids at all. There is an inclination among physicists to admit a single crystal only to the status of " true solid " and to consider every amorphous material as a liquid. Glass, for instance, is spoken of as an " undercooled liquid " ; but it is interesting to hear that " the 200-inch mirror for Mount Palomar is made of glass not for any optical property . . . but for" its mechanical properties. In this case we are concerned with permanence of shape. . . . For in that mirror the silver or aluminium is on the front face, not the back one. . . . The glass is a purely mechanical support for the
* Never forget that first axiom of rheolopy. If I spenk of " iwiy " stress, I exclude, of course, an isotropic stream.
233

234

CREEP

almost infinitesimally thin mirror. We use glass in this case, not because it is transparent, but because its general rigidity and permanence of shape are better than, steel or concrete " [55].
If this is so, glass would be—at least Theologically—a solid, not a liquid. However, let us see what Lord Rayleigh had to say on this subject: " I have tried the following experiment: A piece of optically flat crown glass 3*5 cm. long and 1*5 cm. broad and 0-3 cm. thick, was supported on wood at the extreme ends, and the middle was loaded with 6 kg. applied by means of wooden chisel edge. I t remained in position from April 6th, 1938, to December 13th, 1939. At the end of that time the glass was taken out and tested on an optical flat b y means of interference fringes. I t was found to have been bent, the sagita of the arc amounting to 2-5 bands or 1*25 waves, t h a t is about 6 x 10~5 cm." [56]. With the help of (III, e) and the viscosity elasticity analogy it is easy to calculate the viscosity of that particular glass at room temperature :

We have

rie=PL^\lUdI

(1)

where, in our case, 1 = bhz\12 = 1-5 X 0-3*jl2 = 0-0034 cm.4 ; P = 6,000 x 981 = 5,886 X 10* dyne ; L = 3-5 cm.; d = 6 X 10-5/616 x24 x60 X60 = 125 X 10~™ cm/sec. We find rje — 6'3 x 1016 poises, using the subscript c in ??„ to indicate Creep Viscosity. This is a viscosity not essentially different from the ordinary liquid shear viscosity, creep not being essentially different from slow viscous flow.
In the above-mentioned paper [55], Preston finds. ij0 by extrapolating to room temperature a graph showing viscosities of glass at high temperatures to be around 10*° or 1070 poises and then demonstrates that in our universe such a viscosity is without meaning. If Lord Rayleigh's experiment can be trusted, this shows the well-known dangers of extrapolation. We shall presently calculate the creep viscosity of concrete and find it to be about 3 X 1017 poises and this confirms t h a t the " permanence of shape (of glass) is somewhat better than . . . concrete." I am not sure that it is better than steel.

Comparatively speaking, therefore, cement stone and concrete may be considered as liquids and glass as a solid—but

CEMENT STONE A LIQUID, NOT A SOLID 235
actually this brings us back to what we said in the first Chapter, namely that the strictly defined rheological divisions belong to ideal abstract bodies and not to real materials. If we say that concrete is a liquid, every builder will laugh at us and the structural engineer dismiss the idea as fantastic ; if we say that glass is a solid, the theoretical physicists will consider us to be simple and crude. But neither the one nor the other can answer if we say : " a Hookean material is a solid, a Newtonian is a liquid ". And we shall presently have to postulate another liquid to take account of the fact that while both concrete and glass flow slowly or creep, both are also elastic, a property absent in a Newtonian liquid.
2. The Permanent Deformation of Concrete The discovery of the creep of cement stone was made through
the discovery of the creep of a still more (t solid " material, of which cement is a constituent, viz. the creep of concrete and even reinforced concrete.
When a concrete prism of height h is loaded, it is compressed,
izu-
X
hi
a. • Ahp

1ATION

TIME

Fio. XIEI, 1. Creep and permanent set.
JA„ permanent deformation. Aht permanent set. Ahe creep.

i.e. shortened by, say Ah. When the load is removed, part of

the reduction of height is recovered at once (Ake). After a day or two we find that still more is recovered, and this process goes

on for several days up to a maximum {Aha). A residuum {Ahv) is not recovered even if we wait for a long time.

We accordingly have,

Ah=Ahc + Aha + Ahv

(1)

Ake is the ordinary elastic strain, and the phenomenon to

236

CREEP

which Aha is due is called delayed elasticity, about which we shall speak in Chapter XV. Ahv is called the Permanent Deformation, but needs a further analysis.
If we keep our load in position for, say, a year, we shall see t h a t Ake and Aha are not affected, but AhP increases with time. If we plot the Ahv against time and extrapolate to time t = o, we get a permanent Ahs and a variable Ahc so that

Ahv=Ahs + Ahe

(2)

Aks is called the Permanent Set and creep is the phenomenon to which Ahc is due (compare Fig. X I I I , 1). The same considerations can be applied to the deflections of a loaded beam, where h then stands for " deflection ".
The permanent deformation {Ahv) was described by Bach as

FIG. X H I , 2. Creep of a reinforced concrete beam in a building. Reinforced concrete in elevation shown dotted, in section shown full.
early as 1888, but as its increase in time was not noticed, it was thought to be entirely due to the permanent set (Aha).*
During the first world war, American investigators first observed and described creep. McMillan in 1921 reported on a concrete column in compression which at the end of a period of 600 days still showed a deformation proceeding at the average rate. In 1928 Eaber first described creep in England. He called it " plastic yield " but it is important to keep in mind
* We shall say more about the permanent set in Section 11 below.

THE PERMANENT DEFORMATION OF CONCRETE 237

that oreep is not what we called plastic deformation. The

plastic deformation of metals is easily produced by impact.

Tn contradistinction, creep is slow viscousflowof a very viscous

liquid and not the plastic deformation of a plastic solid. For

instance, bitumen willflowslowly, i.e. will creep, but cannot be

quickly deformed plastically. If an attempt is made to

produce the permanent deformation quickly by impact, the

bitumen breaks in a brittle manner. Also, a plastically

deformable body can sustain loads up to the yield point

without further deformation, while a creeping material has no

yield point: it flows under the smallest load.

Faber discovered creep in a case very similar to what I had

occasion to observe in a building in Jerusalem some years later.

The building is a monolithic reinforced concrete frame with

concrete panels later filled in (compare Fig. XIII, 2).

The floor A was heavily loaded. It was carried by beams of

large spans (B and Bx). After a few years, a crack (C) was noticed. The calculation and design of beam B were checked

and found in order. It was also found that the width of the

crack was of such magnitude that the sagging of the beam

could not have been elastic. An elastic deflection of such

magnitude would have been accompanied by stresses exceeding

the strength of the concrete. Actually the following had

happened : the heavily loaded floor had caused high stresses

in the beams and these in turn had caused creep, i.e. a deflection

of the beams increasing with time. The deflection was too

small to be noticed at one of the intermediate beams (Bj), but

at the front wall the rigid panel {P) which was fixed between

the pillars {P1 and P2) had not partaken in the movement of beam B and the joint between panel and beam had accordingly

opened up to form a crack. Faber was led to the discovery of

creep by the observation of cracks (c) in partitions which did

partake in the movement of the beams on which they were

standing.

The nature of the creep of concrete was at first not very well

understood. In 1931, Straub [57], speaking of the plastic flow

in concrete arches, proposed a " power law " or what would

now be called a Nutting-Scott Blair equation for the creep of

concrete, of the form

e, =apnT

(1)

238

CREEP

where ec is the deformation due to creep, p is the stress, t is time and a, m and n are constants. The equation did not allow for a yield strength and in the discussion the question was raised : Is concrete a viscous liquid or a plastic solid ? This question could not be answered on the basis of the experimental material existing at the time. In 1930, Glanville [58] had stated that the creep of a 1 : 2 : 4 mix* is approximately twice that of 1 : 1 : 2 mix and it is generally assumed that the creep increases as the quantity of aggregate is increased. This would imply that the aggregates flow as loose sand may flow, and that it is the cement, which binds the aggregates together, that prevents flow.
3. The Creep of Cement and Cement Mortar
Without knowing of Glanville's paper, I co-operated in 1932 with Prof. Bingham in an investigation which aimed at finding out what actually was flowing in reinforced concrete [59]. Reinforced concrete consists offour macroscopic phases: cement, sand, stone, steel, and several microscopic phases in addition, among them water. I t is heterogeneous and seolotropic and cannot be considered even as quasi-homogeneous or quasiisotropic. The seolotropy is introduced through the steel which is inserted into the concrete in the shape of rods, i.e. of bodies one dimension of which exceeds the other two. Concrete, which is an aggregation of the first three material constituents, i.e. cement, sand and stone, can be considered as quasi-isotropic, but is heterogeneous -because of the large size of the stones in the mix. Mortar, which consists of cement and sand only, is quasi-isotropic and can be treated to a first approximation as quasi-homogeneous. But cement itself, after setting, although it will contain water which is not chemically bound, is macroscopically isotropic and homogeneous. We therefore started with the simplest of these mixtures, i.e. hardened neat cement and cement mortar (1 : 3) using the " sagging beam " method. The beams, 2-27 cm. square in section, were placed on two supports, 76*1 cm. apart and allowed to sag under their own weight. The deflections were determined as functions of time. The curves were all of the same shape, viz. starting as parabolas and then proceeding as roughly straight lines for a limited time. The slope of the

* i.e. 1 p a r t cement, 2 parts sand, 4 parts crushed stone.

CREEP OF CEMENT AND CEMENT MORTAR 239
curve is a measure of the Rate of Creep, and the creep curves, therefore, show that the rate of creep first falls off rapidly and then for some time remains nearly constant. The decrease in rate of creep in the parabolic part is mainly due to chemical hardening, i.e. the setting of the cement up to an age of about 60 days when following a short curing. In beams allowed to set or kept wet for curing for a longer period before loading, the parabolic part nearly vanishes, being reduced to a small parabolic start extending over a few days only, which is probably due to an elastic after-effect, a phenomenon, as will be shown, entirely different from creep.
Our investigations brought out two main results :— (i) The rate of creep in its approximately constant stage is the same whatever the previous history of the material as to setting and curing ; (ii) The rate of creep of neat cement stone is about twice or more than that of a 1 : 3 cement mortar stone. However, our investigations were limited in two respects : firstly, there was one load only, i.e. the weight of the beams; secondly, there were two different mixes only (1:0 and 1 : 3). To decide upon the relation between cement and sand in the mechanism of flow, I undertook, about ten years later, a further series of investigations in co-operation with A. Arastein at the Laboratory for Testing Materials at Tel Aviv [60].
4. The Creep Viscosities of Different Mortar Mixes
In these investigations four neat cement beams and two beams each of the mortar mixes 1 : £, 1:1, 1:2 and 1 : 3 were observed under the action of their own weight and of superposed weights of 460 and 920 g. in the centre, producing bending moments approximately twice and three times the moments produced by their own weight. The dimensions of the beams were the same as of those of Bingham and Reiner.
First of all it was found that the rate of creep under the two superposed loads was roughly twice and three times that under its own weight. It had already been observed by GlanviJle [58] that the creep of concrete " can be considered for practical purposes as proportional to the stress." This is a characteristic property of viscous flow, in contradistinction to plastic flow. In the latter case, as there is no flow below the yield point and

AGE IN YEAfcS
w < Oor- oow ooo* Ooo oo—oot*ootnooi oov*
AGE IN PAYS FIG. X I I I , 3. Creep curves of cement and mortars. Deflections
plotted against log of time.

CREEP VISCOSITIES OF MORTAR MIXES 241
therefore zero flow corresponds to a finite stress, there can be no proportionality. This was also Glanville's conclusion " the movement appears to be . . . in the nature of viscous flow " and Glanville and Thomas' [62]: " if there is a yield stress for concrete, its value is negligible ". This justified the calculation of creep viscosities from rates of creep, making use of the formulse of Chapter XII.
The creep curves are shown in Fig. 3 on a semi-log scale. This scale has the advantage that prolonged times can be plotted on convenient lengths of abscissa; it has the disadvantage that a straight line of constant slope appears as curved. The parabolas up to the age of 60 days have not been shown. The creep under superposed loads has been proportionately reduced to creep under the beam weight. The curves confirmed that there exists a roughly straight line portion as shown by Bingham and Reiner. It is followed by a curve of gradually diminishing slope as shown by Glanville.
In Table I (p. 242) the calculated creep viscosities have been entered in columns 13-15. Glanville observed compression in cylinders and deflection in beams of the type to which (XIII, f) refers; Glanville and Thomas observed also extensions in cylinders. The materials were concretes of different mixes, different cements, and mortars. The viscosities change with age and the age of 60 days was selected as the greatest for which observations were available for all different materials, and as one which is past the first parabolic part of the creep curve during which the setting of the cement is proceeding at a high rate.*
In the present case also, as in the case of Bingham and Reiner's beams, the rate of creep of neat cement was over double that of the 1: 3 mortar. This was the more remarkable as the rate of creep of the beams prepared in Easton from the American " Atlas " cement was about four times the rate of creep of the cement beams prepared in Tel Aviv from " Nesher " cement. One conclusion only can be drawn from this as expressed by Thomas [62], viz.: " Concrete (and mortar, M.R.) is considered as comprising two parts (or phases M.R.): (i) the cementitious
* The rate of creep had to be estimated from the slope of the tangent to the creep curves. This procedure does not permit of great accuracy, and in columns 9-11 the whole range of estimated slopes has been entered, and in columns 13-15 corresponding ranges of viscosities.

TABLE. X CALCULATION OF CREEP VISCOSITIES

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