Thermodynamical theory of elasticity and plasticity

Kluitenberg,

Physica

G. A.

28

2 17-232

1962

THERMODYNAMICAL ELASTICITY

THEORY

AND

OF

PLASTICITY

by G. A. KLUITENBERG Onderafdeling

der Wiskunde

van

de Technische

Hogeschool,

Eindhoven,

Nederland.

Synopsis h theory of elasticity of irreversible principle

and plasticity

processes.

and

degree

plastic

Onsager may

of freedom.

flow,

relations

exist

elasticity

be due

deformations

The entropy The

media,

hardening

of plasticity

The vanishing

In the theory

is derived.

in anisotropic

strain

the inelastic production,

phenomenological between

with

the help of the

of this principle strain tensor which

the temperature

plays

the role of an

is due to heat conduction

equations

are

given

and plastic

Phenomena

as, for instance,

in this theory.

is an approximation of the

is based on the thermodynamics is defined

heat conduction

are discussed. are included

of the rate of the plastic

to a property

which tensor

In the discussion

for the cross effect

and strain

Mises theory to

is given,

inelastic

of relaxability-in-the-small.

is also taken into account. internal

The

It is shown

to the theory

developed

tensor

the yield

strain

phcnomenological

within tensors.

and

the

flow, which thermo-

that the Von in this paper. surface

It is assumed

is seen that

the

are small.

5 1. Introduction. In this paper a theory of elasticity and plasticity is developed using the general theory of thermodynamics of irreversible processes in continuous medial) 2). In 5 2 - $5 the law of conservation of mass and the balance equations for momentum, total energy, kinetic energy and

internal energy are discussed. In 0 6 the tensor of total strain (which consists of two parts: the tensors of elastic and inelastic strain) is introduced. In the following section the inelastic strain tensor is defined with the help of the principle

of relaxability-in-the-small

and it is introduced

as an internal

degree of freedom in the Gibbs relation. In 9 8 and $ 9, according to the usual procedure of the thermodynamics of irreversible processes, the entropy balance, the phenomenological equations and the Onsager relations are given. In the last three sections plasticity is discussed and it is shown that theVonMises theory is an approximation to the theorydevelopedin this paper. We consider a medium consisting $ 2. The law of conservation of mass. of one chemical component and we shall assume that mass is conserved (i.e., such phenomena as nuclear reactions are excluded). Hence, one has aP -=

-

at -

217

div pv,

-

(2-l)

218

G. A. KLUITENBERG

where p is the density of mass, v is the velocity time. We now define the substantial

derivative

of the medium and t is the

with respect

to time by

where vy is the y-component of v and xy is the y-component vector x(y = 1, 2, 3). (A rectangular axis - frame is used.) From the two preceding equations it follows that dP -=

-p

dt

of the position

div v,

and dv pdt = divv, where 21= p-l

(2.5)

is called the specific volume (volume per unit of mass). Finally, if 4 is an arbitrary quantity (for instance a component of a tensor), one has

dq a(Pq) Pdt= ~ at +

(2.6)

Wpqv),

which follows from (2.2) and (2.3). 5 3. The equations of motion (balance equation for momentum). form for the equations of motion reads

’

dv, __ dt

ah

--I&L

__ax

u

The usual

(3.1)

+ pFa,

where F is the force per unit of mass and Pall is the mechanical pressure tensor. Since pv is the density of momentum one may call v the specific momentum (i.e., the momentum per unit of mass). Hence, Plvs may be considered as the tensor of the density of the conductive flow of momentum and pF as the source of momentum (per unit of volume and per unit of time). Using (2.6) one has dv,

Px==

a(pva)

at

+

a(Pv,vs)

Is:,“=, ax--’

(3.2)

u

and hence, one can write for (3.1) (pvavs

+ PM) +

pF,.

(3.3)

THERMODYNAMICAL

From this equation convective

THEORY

OF ELASTICITY

AND

PLASTICITY

it is clear that PV~ZJBis the tensor of the density

flow of momentum,

219 of the

while pvUa2/b + Pa,g is the tensor of the density

of the total flow of momentum. It will be assumed that Pa&9= PO&. This assumption

is equivalent

internal

momenta.

angular

(3.4)

to the assumption

that the medium

has no

5 4. The balance equation for the total energy. It will be assumed that the only source of energy is the newtonian work done by the external forces. (This means that for instance polarization and magnetization phenomena are not taken into consideration.) Hence, one has de pdt=

-divJ(@+pv*F,

(4-l)

where e is the specific total energy (i.e., the sum of the kinetic and internal energies) and J(e) is the density of the conductive flow of the total energy. 3 5. The balance equations for the kinetic and internal energies. On multiplying (3.1) by ua and summing over 6, one obtains

where (3.4) has been used. Since the quantity iv2 is the specific energy, (5.1) is a balance equation for the kinetic energy. On substracting

kinetic

(5.1) from (4.1) one obtains

c,“,,=, pas

pg=-divJtq)-+ where u is the specific internal

energy given by u = e -

and J(q) is the density

(54

9vz 2

of the conductive

flow of the internal

bY J$q’ = $’

-

cl=,

&Pa&

(5.3) energy given

(5.4)

J(q) is usually called the heat flow. The equation (5.2) is a balance equation for the internal energy and is usually called the first law of thermodynamics. From (5.1) and (5.2) it is seen that

220

G. A. KLUITENBEKG

is the amount of energy which transforms of time from kinetic to internal energy.

per unit of volume and per unit

9 6. The tensor of total stmin. Consider an arbitrary point P of the medium. Let the position of this point at the time t = to be given by the vector E,. At an arbitrary time t the point P will have the position x = X(E> t),

(6.1)

where ~(5, t) is a single valued function of g and t such, that x(E,, to) = g. On the other hand, if at the time t the point P has the position x then E,, the position of P at the time to, is uniquely determined by x and t and hence, also 5 = E(XJ t) (6.2) is a single valued function. The displacement vector of P with respect to the position of P at the time to is defined by u = x - 5 = u(C, t) = u(x, t), (6.3) and the velocity

of P at the time t is

v=

tug,

(6.4)

t).

The state of deformation of the medium in the surrounding of P at the time t with respect to the state of the medium in the surrounding of P at the time to is given by the tensor of total strain defined by

(6.5) It will be assumed in this paper that the deformations From the preceding $

equation

are small.

and (6.4) it follows that

I a W(S* t) = g 1% Q5,

t) + $

h

v/3(&t) . 1

(6.6)

From (6.3) one obtains 4

%%(F,t) = $

where day is the Kronecker

tensor.

%x(5‘ t) + Bay,

(6.7)

Using this equation

one gets from (6.6)

Y

THERMODYNAMICAL

If one assumes that

THEORY

OF ELASTICITY

the deformations

are small,

AND

PLASTICITY

the preceding

221 equation

becomes

Using (6.3), (6.4) mentioned above)

and

(2.2)

one has

(even

without

the

approximation

(6.10) From the two preceding

equations

one gets (6.11)

It should be remarked that in the following sections all quantities will be considered as functions of x and t (which has also been done in the preceding sections). From (6.5) one has (6.12) Eai3= Qor. In the following the state of the medium at the time to will be called the reference state. An assumption about this state will be made in the following section. If one defines the quantities &o and E by e = + c;=,

(6.13)

+Jy,

and SC@= El%/3 - Q&X, c;=,

eyy,

(6.14)

one has (6.15)

“CQ= gal3 + &s, and = 0.

(6.16)

zag = E&ga.

(6.17)

c,“=, I,, From

(6.12) and (6.14) it follows that

Using (2.4), (2.5),

(6.1 1) and (6.13) one obtains dv d& v-1 _ = 3 -. dt dt

Hence, v is a function

(6.18)

of E.

5 7. The Gibbs relation and the principle of relaxability-in-the-small. If one considers pure elastic processes in a solid, the Gibbs relation has the form T ds = du + v Z&i where T is the temperature

Pars dea,r,

and s is the specific entropy.

222

G. A. KLUITENBERG

If, however, the medium has internal degrees of freedom, terms must be added to the right hand side of the preceding equation. For instance, in a fluid

consisting

of one

chemical

component,

T ds = du + P dv, where P is the hydrostatic fluid consists

of several chemical

components,

the

Gibbs

pressure. one has

relation If, however,

reads the

T ds = du + P dv - & purdck, where ck is the concentration of the chemical component K. Meixnera) has added terms of this kind to the Gibbs relation for an elastic solid in order to discuss relaxation phenomena. We shall now assume that there exsists an internal degree of freedom characterized by a tensor .scrS. (O (The physical interpretation of this tensor will be discussed in the following.) Still other degrees of freedom may be present, but, as will be seen, for the explanation of plasticity phenomena it is sufficient to assume that there exists only a degree of freedom characterized by a tensor. For the Gibbs relation one then gets

T ds = du t_ v xc,“,+, Pfig dccrg -j- Z’&, Hence,

7:; d&r;.

one has ;

S(E~O, E$, u) = T-1,

a

a q-

(7.3)

s(eor5, E$, u) =

VT-~

7:;.

It follows from (7.1) that ds du T,,=dt+vC:,B=iP”~

dt

In order to give the physical meaning of .&$ we use the principle of relaxability-in-the-small introduced by EC kar t 4). We consider a small bit of matter (surrounding some point P of the medium), which is large from the microscopic point of view (Le., it contains still a great many molecules (or atoms)) and small from the macroscopic point of view (i.e., the variations within the bit of such quantities as the temperature and the strain tensor may be neglected). It will be assumed that the reference state mentioned in $ 6 has a uniform temperature To,Moreover, it will be assumed that if such a small bit is cut out of the medium while the medium is in the reference state, no changes will occur in the bit. If, when the medium is in an arbitrary state, a small bit is cut out, all strains will be relaxed, since on the surface

THERMODYNAMICAL

THEORY

OF ELASTICITY

AND

of the bit no forces work. In the relaxed state, however, deformation

left with respect to the reference

223

PLASTICITY

there may be some

state. For E$ we now take the

tensor describing the strain (with respect to the reference state) of the dissected bit after one has brought the bit to the temperature To*). One may call 8:; the inelastic strain tensor. From this definition and from (6.12) one gets and one may also assume that The elastic part tensor defined by

(i) EC@= &PA,

(7.6)

7:; = $2.

(7.7)

of the deformation

is described

&$ = E&OFrom

by the elastic

strain

Et;.

(7.8)

(6.12) and (7.6) one obtains (7.9)

e:eil = @)W state P a~ = 0 and hence, one has

In the relaxed

P”b(&&eS), E$, T) = 0 for ES = 0 and T = To,

(7.10)

on account of the definitions of E$’ and E$ given above. If one linearizes the function Pao(&$, E$, T) in E$ and T, one obtains with the help of (7.10) Pap = C;,,= i G,WC es) + aas(T -

(7.11)

To),

where the tensors a,g,,c and aao may still depend on E$. It should be remarked that E&B satisfies the compatibility equationss) on account of the definition (6.5). In general, however, neither E$ nor F$ will satisfy these equations. $ 8. The balance equation for the entrofiy. Using (7.5), (2.5), (5.2) and (6.11) one finds p

$ =-

div(T-1 J(4)) + T-1

T-1 J(P). grad T +

-

~~,s=1$

?!$!_. (8.1)

>

Hence, ds Px=

-

div J(8) + a(s),

(8.2)

where J(s) =

T-lJ(@,

(8.3)

and &9 *) Eckart account

=

T-1

-

4), in his discussion

the temperature.

a unique definition

of &$.

T + C&=1

T-VW*grad of the principle

The temperature,

r$j$

of relaxability-in-the-small,

however,

may not be neglected

.

(8.4)

does not take into if one wants to obtain

224

G. A. KLUITENBERG

From

these

equations

it follows

J(S) is the

that

density

of the

conductive

flow of entropy and G(S) is the entropy production per unit of volume and per unit of time. It is seen that the entropy production is due to the irreversible Since

phenomena the entropy

of heat conduction production must J(a) .grad

and inelastic deformation. be positive definite, it follows

that

T (

(8.5)

0,

and

(8.6) Using

(2.6) the equation

(8.2) may

w =__

div

also be written

in the form

(J(S) + ~SV) + CT(')

(8.7)

8t

Hence, psv is the density of the convective is the density of the total flow of entropy. We now give the following definitions ,(i) = 13 x3= -(i) Tab

_ -

(i) Tafl

1’

1

-

4

E(i) = Q c;=, -(i) = E$ &xP From

these

definitions

(0

&)

Further,

T-1

$)

PSV

(8.8)

i’/’

&x&s c;=,

(8.9)

+f;>

(8.10)

&$j, 4 &x/3c,“=, E!$.

(8.11)

= “$

+ T(i) SLY&

(8.12)

(i) = EC; + &W &.@, &3P

(8.13)

c,“=, ?!i) /y = 0 ’

(8.14)

x.3= : r E?31,)= 0 *

(8.15)

the four preceding zc

and J@) +

one finds rafl

Using

flow of entropy

-T-lJ(q)

one has from

equations

one can write

.grad

~~,8=1

(7.6),

T +

for a@) defined

+tjz

(7.7), (8.9) and

+ 3?-(i)$9.

by (8.4) (8.16)

(8.11)

-(i) = +“’ fin’ TUB

(8.17)

&i’ _ aB

(8.18)

and $0 8%’

THERMODYNAMICAL

THEORY

OF ELASTICITY

AND

PLASTICITY

225

We also have (8.19) and 7(6) de(*) __ 20, dt since the entropy

production

(8.20)

must be positive

definite.

9 9. The phenomenological equations and the Onsager relations. On account of the expression (8.4) for the entropy production, we have for the phenomenological equations in an anisotropic medium

(9.2) where

L$)“,“Q’, LDpB’,‘i), ,Tti$‘d

The Onsager

relations

and

L$,$) ,

are phenomenological

tensors.

read L$‘B)(Q)= LIp,“P’,

(9.3)

L$;i’

= pd,

(9.4)

L$$’

= Ly;’

(9.5)

On account of the symmetry of E$ and v$$ one may choose LDpsl’“‘,L$‘JQ) and L$$’ such that the following relations are satisfied6) 7) LC90 = LC$’

(9.6)

L(i)(q) = LpW”“‘: @Y L’W = LCg’ = L$g ‘%SYC Hence, we have from the five preceding LST’

= Ly;‘.

(9.7) (9.8)

equations

= LCQ0 == Lgj$) = L;WC?),

(9.9)

and L$$’

= LC#

= LC$’

= L?Ci&’= L;W;J = L$X$’ = L:iy’j, = LCi&‘. (9.10)

Due to symmetry in the crystal lattice the number of independent components of the phenomenological tensors may further be reduceda). In an isotropic substance one has the requirement that the phenomenological tensors must be invariant with respect to all rotations and to inversion of

226

G. A. KLUITENBERG

the axis-frame.

It can be shown that in this case one obtains

(9.2)

for (9.1) and (9.1 1)

J(Q) = -A grad T,

(9.12) d&(i) = #Z),(i). dt

(9.13)

Hence, in general, there may exist cross effects in anisotropic media between heat conduction and plastic flow (the second term on the right hand side of (9.1) and the first term on the right hand side of (9.2)). In an isotropic substance, however, such cross effects do not exist. From the positive definite character of the entropy production inequalities may be derived for the components of the phenomenological tensors. As an example we consider an isotropic substance. With the help of (9.1 I), (9.12) and (9.13) one obtains for (8.16) o(s) = T-i{LT-i(grad

T)s + ~(1) C&=,

(?$)2

+ 317@)(7(i))s}.

(9.14)

Hence,

If one considers $ 10. Discussion of plasticity. and u one obtains with the help of (7.8) ds = $

du + &+,

~

s as a function

as

as

a&)

(9.15)

Tj@) > - 0.

$1) 2 0,

120,

d&ao +

C:,,+

In the following s will always be considered Hence, one has from (7.1) and (10.1)

1 $J-as a function

a&$)

of F$, E$!

d+.

(10.1)

of e$‘, ~2, and U.

(10.2)

(10.3) 1

aS

T

at4

-=_

where also (2.5) has been used. With the help of (10.3) one gets for (9.2)

’

(10.4)

THERMODYNAMICAL

THEORY

and for (9.12) and (9.13) one obtains

OF ELASTICITY

AND

PLASTICITY

with the help of (10.3),

227

(8.8) and (8.9)

and d&(i) -=

5 P)

dt

PT

(10.7)

C;=,

In general the components of the phenomenological tensors still depend on E$, E$’ and u (or, for instance, on Paa, E$ and T). It is known from experiments that if a medium is in a state with a certain inelastic deformation no changes in E$! will occur if Pa0 is small enough. This means that there ,exists a function @ = @(P,o,

Et;,

T),

(10.8)

such that for @ < 0 the right hand sides of (10.5), (10.6) and (10.7) vanish. We shall now assume that this is due to the fact that L$fZ), L$$, ~(1) and q@) vanish for @ < 0 and get finite values if @ increases. The other possibility would be that 7:; and L$p) vanish, but it will be shown in 4 12 that this is less probable. It should be remarked that one can write for (9.1)

JF’ = -

T-1 I;&

L$)(‘J)

-& + c;,y=l

L$F’

pT

P

?-

aEg

_

PBV , >

(10.9)

where (10.3) has been used. In the case of isotropy this equation reduces to (9.11). Thermodynamical equilibrium is characterized by the vanishing of the entropy production. This is the case if the heat conduction and the plastic flow vanish. Hence, one must have grad T = 0 and CD< 0. It should be stressed, however, that it is not necessary that Pan, E$,’ or E$ vanish. This means that there is no relaxability-in-the-large. If there is mechanical
however,

6P,b is such that @(P,B

+ ~P,B, E$, T) > 0.

228

G.

A. KLUITENBERG

This effect (often called strain hardening)

can be explained

in the following

way. If one changes the pressure the irreversible process described by (9.2) (or by (9.12) and (9.13)) will start when the yield surface (the surface in P,D-space

for which @ = 0) is reached.

If the pressure

final value P&/3+ dP,o, it may occur that the inelastic

is brought deformation

to its tensor

reaches a value E$ + &$j such that @(P@

+ 6P,p, 8)=P+ BE$, T) < 0,

and, hence, the medium is again in a state of thermodynamical 9 11. Some approximations. In this section we mations of which the meaning will become clear I. The assumption 0 = 0. It is known that if plastic deformations occur in a medium, has a vanishing trace i.e.,

equilibrium.

shall discuss two approxiin the following. from experimental data, the inelastic strain tensor

&W = 0. From

(8.10),

(9.2) and the preceding c;=,

(11.1)

equation

it follows that

LZii”;’ = 0,

(11.2)

and c”= a

1

L(M) ‘Iw

and hence, using the Onsager relations

= 0

(11.3)

,

(9.4) and (9.5), one also has

c;=,

L$$

= 0,

(11.4)

c;=,

L$;i’

= 0.

(11.5)

and

With

the help of (8.12)

and the two preceding

equations

one then gets

for (9.1) and (9.2) (11.6)

For the case of isotropy

it follows from (I 1.1) and (9.13) that q(2) = 0.

II.

An assumption

about

the

entropy.

(1 1.8) If one assumes that (1 1.9)

THERMODYNAMICAL

one obtains

THEORY

for the equations

OF ELASTICITY

(10.5),

Jkp) = _ T-l'&

(10.9),

L$)(q)$

AND

PLASTICITY

229

(10.6) and (10.7)

_

c;,,=,

fl

L$!!i’ Pp,,

(11.11)

(11.12) and d&

= -

dt

$2) P,

(11.13)

where P’ns = P&L3-

(11.14)

4 Bap Lx,“=, P,,,

and P = g g_-,

(11.15)

P,,.

The equations (11.12) and (11.13) only hold in the case of isotropy. We remark that one has from (11.14) and (11.15) (11.16)

Pn$ = R%S + P Baa, c:=, and, finally,

P’,,

(11.17)

= 0,

one has (11.18)

Rx, = &,, from (3.4) and (11.14). III.

The

combination

of

the

(10.3), (11.9) and (11.14) one obtains

For the case of isotropy $2) zzz 0.

two

assumptions.

again the equation

We now define the quantities

Using

(8.9),

for (1 1.6) and (11.7)

(11.12)

holds and in (11.13)

&) and E$ by

&) = + z;=,

(11.21)

E$

and z$j = &$? -

g &-Qc;=,

&g!

(11.22)

230

G. A. KLUITENBERG

Hence, E% = E$) + &@)f&g,

(11.23)

c:= 1 SE Z=Z0.

(1 1.24)

and

Using (7.9) and (11.22)

one has g(e) = $a) a@

From

(11.25)

(7.8) it follows with the help of (6.13),

(8.10) and (1 1.21) that

E = &W + c(i) and using (6.14),

(8.1 1) and (11.22)

(1 1.26)

one finds from (7.8)

- = $e.$+ $j &aB

.

(1 1.27)

From (10.4) and (1 1.9) it follows that (11.28) and from (10.2),

(2.5) and (1 1.9) one obtains

+J(+p,, >= r_

(1 1.29)

0.

On account

of the two preceding Pn,c = v-lf$QJ),

equations

one has

u) = z+f$‘(&$),

T).

Since II = ‘U(E) (cf. the end of 4 6) one gets from (11 .I) and (1 1.26) u-1 = = f(3)(&)). Hence, Pas = fap (‘) (Q(e), T ), and if one linearizes this function one obtains Pas = z;,c= 1 %3yt $2 + Ga(T

-

To),

on account of (7.10). The tensors a,gyt and aao are constant. remark at the end of 4 7.) In the case of isotropy one gets Pa0 = asg + {b&) + c(T -

To)) da&

(11.30) (Cf. also the

(11.31)

where the scalars a, b and c are constants. The equation (11.31) is the Duhamel-Neumann law9) for thermoelasticity. If one assumes that c = 0 (or if T = To) (11.31) reduces to the law of Hooke for isotropic media. It should be noted that the theory developed in this paper is based on the thermodynamics of irreversible processes. In order to give a theory for plasticity, Ziegler lo), however, attempts to extend the thermodynamics by introducing non-linear relations among fluxes and affinities.

THERMODYNAMICAL

THEORY

OF ELASTICITY

AND

9 12. The Von Mises theory for the case of isotropy. theoryri) 1s) 13) 14) it is assumed that @ = $ z;,,=, where K is the yield stress in shearli) theory that

(&5)2

-

231

PLASTICITY

In the Von Mises (12.1)

k2,

14). Moreover,

it is assumed

in this

(12.2) where Y_P vanishes for @ < 012) 14). The latter equation is the so called Von Mises flow law. Using (12.1) and (11.14) one can also write for (12.2) de:; -_=dt

(12.3)

Yj(l) P&p,

a formula which has been proposed by Levy in 187015). In those cases in which the assumptions (11 .l) and (11.9) are correct, the theory developed in this paper also gives the flow law (12.3) (cf. (8.13), (1 1.12), (11.13) and (1 1.8)). In 5 10 we made an assumption about the phenomenological tensors in order to explain the plastic behaviour of a solid. It was remarked in that section that another possibility would be to assume that $i vanishes for @ < 0. In the latter case one would have on account of (10.3)

as = PT-S a&$ a&$)

Pa0 = pT -

for

@ < 0:

It seems, however, less probable that as/&$ = as/&&es).Moreover, to obtain the Von Mises theory (which gives a good description of plasticity), it was necessary to make the assumption (11.9). Hence, one would have P,b = 0 for @ < 0, which is absurd. Therefore, the fact that changes in the plastic deformation only occur if @ > 0 must be due to the vanishing of the phenomenological tensors for @ < 0. In a following paper Maxwell and Kelvin bodies will be considered. Received

10-7-61

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