A thermodynamic derivation of the stress-strain relations for burgers media and related substances

Kluitenberg, 1968

Physica 38 5 13-548

G. A.

A THERMODYNAMIC DERIVATION OF THE STRESS-STRAIN RELATIONS FOR BURGERS MEDIA AND RELATED SUBSTANCES by G. A. KLUITENBERG Onderafdeling der Wiskunde van de Technische Hogeschool, Eindhoven, Nederland

synopsis A generalization is given of the author’s thermodynamic theory for mechanical phenomena in continuous media. The developments are based on the general methods of non-equilibrium thermodynamics. Temperature effects are fully taken into account. It is assumed that several microscopic phenomena occur which give rise to inelastic strains (for instance, slip, dislocations, etc.). The contributions of these phenomena to the inelastic strain tensor are introduced as internal degrees of freedom in the Gibbs relation. Moreover, it is assumed that a viscous flow phenomenon occurs which is analogous to the viscous flow of ordinary fluids. An explicit form for the entropy production is derived. The phenomenological equations (Fourier’s law and generalizations of Levy’s law and of Newton’s law for viscous fluid flow) are given, and the Onsager-Casimir reciprocity relations are formulated. It follows from the theory that several types of (macroscopic) stress fields may occur in a medium: A stress field $$) cui) which is analogous to the viscous which is of a thermoelastic nature, a stress field rss stresses in ordinary fluids, and stress fields ~1~)~ which are probably connected with the microscopic stress fields surrounding imperfections in the medium. The stress field ~2) + r$) is the mechanical stress field which occurs in the equations of motion and in the first law of thermodynamics, and stress fields of the type 7%) + T[$,~ play the role of thermodynamic affinities in the phenomenological equations which are generalizations of Levy’s law. If the equations of state may be linearized (for example, Hooke’s law and the Duhamel-Neumann law), and if the phenomenological coefficients may be regarded as constants, an explicit form for the stress-strain relation may be derived. In this case the relation for distortional phenomena in isotropic media has the form of a linear relation among the deviators of the mechanical stress tensor, the first w derivatives with respect to time of this tensor, the tensor of total strain (the sum of the elastic and inelastic strains), and the first n + 1 derivatives with respect to time of the tensor of total strain, where n is the number of phenomena that give rise to inelastic deformations. The well-known Burgers equation is a special case of this relation if n = 2. Moreover, the stress-strain relations for ordinary viscous fluids, for thermoelastic media, and for Maxwell, Kelvin, Jeffreys, and Poynting-Thomson media are also special cases of the more general relation mentioned above. In case the equations of state may be linearized explicit expressions are given for the free energy, the internal energy, and the entropy, both for isotropic and anisotropic media. If it is not permissible to linearize the equations of state and/or to regard the phenomenological coefficients as constants, the stress-strain relation is of a very complicated nature. Plasticity phenomena are left out of consideration.

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G. A. KLUITENBERG

5 1. Introd~tiort. In some previous papers (which will be quoted as 11)) IIS), IIIS), IV4), V5), VI6), and VII 7)) the author developed a theory for mechanical phenomena, which is based on the thermodynamics of irreversible processes s) a) 10) 11) 12). In th e p resent paper a generalization of this theory is developed. It is assumed that several

microscopic

phenomena

occur which give rise

to inelastic strains and that the total inelastic deformation is additively composed of contributions of these phenomena. These contributions are introduced as internal degrees of freedom in the Gibbs relation. Moreover, it is assumed that a viscous flow phenomenon occurs which is analogous to the viscous flow of ordinary fluids. In 5 2 we introduce several types of deformations, the Gibbs relation, and the notion of memory. From the first law of thermodynamics and the Gibbs relation we derive an explicit form for the entropy production in $3. It is seen that the entropy production is due to heat conduction, to a viscous flow phenomenon which is analogous to the viscous flow phenomena occurring in fluids, and to changes in the inelastic deformations (inelastic flow). The heat conduction and the flow phenomena are irreversible processes and the phenomenological equations for these irreversible processes are given in 3 4. The phenomenological equations are Fourier’s law for heat conduction and generalizations of Levy’s law for inelastic flow and Newton’s law for viscous fluid flow. The coefficients which occur in the phenomenological equations satisfy symmetry relations and these relations (the OnsagerCasimir reciprocity relations and some others) are formulated in $5. In $6 the phenomenological equations and the Onsager-Casimir relations are discussed for isotropic media. The free energy, the thermodynamic potential, and the enthalpy are introduced in $7. In 9 8 some inequalities are derived from the positive definite character of the entropy production. A discussion of thermodynamic equilibrium is given in 3 9. It follows from the theory that several types of stress fields may occur in a medium. In § 10 the physical meaning of these stress fields is discussed. Explicit stress-strain relations may be obtained if one assumes that the equations of state may be linearized (for instance Hooke’s law and the Duhamel-Neumann law). In $ 11-S 20 the results are investigated to which the theory leads if these linearizations are introduced. Explicit expressions for the thermodynamic functions and the forms of the linear equations of state are given in 9 11 for anisotropic media and in $ 12 for isotropic media. In 9 13-s 20 the stress-strain relations are discussed. In $ 15 we obtain for shear phenomena in isotropic media the stress-strain relation

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OF BURGERS

where ?a~ is the deviator deviator

of the tensor

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AND

of the mechanical

RELATED

stress

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tensor

515

and lap is the

of total

strain. The coefficients R&im (m = 0, 1, . . . . n - 1) and Rgjm (m = 0, 1, . . . . n + 1) are algebraic functions of the coefficients occurring in the phenomenological equations and in the equations of state, and n is the number of phenomena that give rise to inelastic strains. In 5 17 it it shown that the coefficient R[&+l vanishes if the viscous phenomenon (analogous to the viscosity of fluids) does not occur. It is also shown in this section that if one of the phenomena which give rise to inelastic strains is not associated with changes in the microscopic structure of the medium, the coefficient R#,, vanishes. In 5 18-s 20 some special cases of the relation (1.1) are discussed. Particularly in 3 20 it is shown that the Burgers stress-strain relation is a special case of (1.1). 5 2. Elastic deformations, inelastic deformations, the Gibbs relation, memory. If both elastic and inelastic deformations occur, we have &a@= &$) + &I*;‘,

and

(2.1)

where E&S is the tensor of the total strain and E$’ and E$) are tensors describing the elastic and inelastic strains, respectively. In contradistinction to the elastic strains, the inelastic deformations are due to lattice defects and related phenomena (slip, dislocations, etc.). It is known, however, that there are several types of such defects which may occur simultaneously. Let us suppose that there occur n different types of microscopic phenomena giving rise to inelastic strains and let us further assume that &$“’ = z;=,

&$‘,

(2.2)

where E$ is the contribution to the inelastic strain of the K-th microscopic phenomenon. We shall call E$, E$‘, . . . . E$’ partial inelastic strain tensors and .$rs”’the total inelastic

strain tensor. From E&b= E$l’ + x:;=i

(2.1) and (2.2) we have &$).

(2.3)

Since the n microscopic phenomena are assumed to be different, it is reasonable to suppose that in general the entropy will depend on all the tensors E$ (k = 1, . . . . n). Moreover, the entropy will depend on the internal energy and on the elastic strain tensor E$‘. Hence, s = s(zc, &$‘, &$‘, E$, . . .) ES)) ) where s is the specific entropy of (2.3) we may also write

and zc the specific internal

s = s(u, E&y,&$, E$‘, *. . ) E$)). The temperature,

(2.4) energy. By virtue

(2.5)

T, is given by

(2.6)

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G. A. KLUITENBERG

Moreover, we define the equilibrium-stress tensor, $$), by

P-7) and the tensors T$‘, T$‘, . . ., 7%’ by

P-8) where p is the mass density. We shah call r$) the affinity-stress tensor conjugate to E$ (k = 1, . . ., ut). With the aid of (2.6), (2.7), and (2.8) we obtain from (2.5) for the differential ds of s

T ds = dzc - v x&r where

r$J’ d&as+ v x;= 1 Z:,,= 1 7:;’ d@, v = p-l

(2.9) (2.10)

is the specific volume (volume per unit of mass). A relation of the type (2.9) is called Gibbs relation. Using (2.3) we may also write for (2.9)

T ds = du - v x;,s= r $r’ de$) + v x;= r I;&= r +&z d&$‘,

(2.11)

where 7g)@ = 7:;’ - +‘.

(2.12)

We shall call ~12,~ the memory-stress tensor conjugate to E$‘. The tensors (n)2T(m)ap (1) $$q3> ***aT(m)@3 (n) may be termed thermodynamic stress $$Jt; +)0;s> . **>Tap tensors. We shall call a medium of which the entropy has the form (2.4) a viscoanelastic medium of order n. We shall say that such a medium has no memory if one of the memory-stress tensors, say 7gm, vanishes identically. The Gibbs relation for a viscoanelastic medium of order 1zwithout memory may be written either in the form

T ds = dzl - v z;, P=r r$) ds$’ + v x;:;

x&z r TK,~~d@),

(2.13)

(cf. (2.11))) or in the form

T ds = dzl - v &=,

T$’ d&r - ~2’) + v x;::

x&=,

T$’ de’,

(2.14)

(cf. (2.9) and (2.12)), wh ere the third terms on the right hand sides of (2.13) and (2.14) vanish if n = 1. The deviator &a and the scalar part A of an arbitrary tensor field A&,T are defined by (2.15) J&x, = &I3 - &k/rYE:“,=,A,,, A = * Ix;=, A,,.

(2.16)

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517

Hence, AaB = &¶ + A&O?

(2.17)

and &,

(2.18)

Ayy = 0.

We shall call a substance a medium without shape memory if the deviator of one of the memory-stress tensors, say $$,r, vanishes identically. From (2.11) it is seen that the Gibbs relation for a viscoanelastic medium of order 12without shape memory may be written in the form Tds=dzc-vC~B=1~~~)d~~)+v~~=:~~,8=17~)~~d~~~)+3~~~~d~(lt),

(2.19)

where rej and e(n) are the scalar parts of $&s and E$‘, respectively. Thus, the entropy is a function of U, E$‘, E$,),. . . . .z$-l), and e(n) if the medium has no shape memory. Finally, if the scalar part of one of the memory-stress tensors, say 7$, vanishes, we may say that a medium has no volume memory. In this case the entropy is a function of zc, E$), E$‘, . . . . &l), and ~2). The tensor of total strain is the symmetric part of the gradient of the displacement field. Hence, this tensor is symmetric. We shall also assume that the elastic strain tensor and the partial inelastic strain tensors are symmetric. Hence, &a/3 = QCO Je$) = &l) Pa’ (in) = ,(in) Ba ’ %4J

z*

= E&,

(2.20)

-(el) = g(eZ) Ba ’

(2.21)

-(in) _ -(in)

(2.22)

&ols %

-

%a 9

Moreover, we shall assume that (en)

=

T(w)

-kw)

=

e.4)

Tas Bar ’ *as T/Ta ’ (k) -Uc) T(m)@= $$?Lz, ~(?n)crS = q$%X~ (k) = (k) -(k) = -(k) rsczJ T&z* Tas T@

(2.24) (2.25) (2.26)

Hence, the thermodynamic stress tensors are also symmetric. f 3. Entropy balance. The first law of thermodynamics reads (see (II. 1.3) *)) ,,$-=

-div.J@)

+ ~&=rT~~dsdS, dt

(3.1)

where J(g) is the heat flux and rarpis the mechanical stress tensor. The *) Equation (1.3) of paper 11s) will be indicated as (11.1.3) and in an analogous way we denote the other equations of the papers I-VII.

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G. A. KLUITENBERG

latter tensor is related to the mechanical pressure tensor PLvsby 7as = --Paa. The tensor 7ap also occurs in the equations of motion. The tensor $‘, which is defined by (90 = 7aB _ +) Tars B J

(3.2)

is called the viscous stress tensor. It follows from (2.9) that one may also write for the Gibbs relation

Multiplying both sides of the preceding equation by pT-1, and using the first law (3.1), the definition (3.2) of the viscous stress tensor, and the relation (2. lo), we obtain ds

Pz=

J(q) T + CT@),

-div

(3.4)

where a@) is given by ~(8) = T-1

--T-i

J(q)-grad T +

The equation (3.4) is the balance equation for the entropy, J(@/T is the entropy flow, and ~(8) is the entropy production. It is seen that the entropy production is due to three types of phenomena. The first term on the right hand side of (3.5) gives the contribution of heat conduction, the second term represents the entropy production due to viscous flow phenomena, and the last sum is the contribution of changes in the inelastic deformations (anelastic flow) to the entropy production. Using (2.17) and (2.18) one may also write for (3.5) u(S) =

T-1

--T-l

J(q) .grad T + C~,p=l #‘$)

!$!? + d&(k) dt . >

This form for the entropy production media are considered (see 5 8). We shall assume that rag = Tfla, and hence, (cf. (2.24),

.$“Bi = $) (3.2), and (3.7)).

is in particular

?a@ = +j% -(vi) = $vi) J Tas Bar’

(3.6)

useful if isotropic

(3.7) (3.8)

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519

5 4. Phenome~ological eqzlations. According to the usual procedure of nonequilibrium thermodynamics we have for anisotropic media the following phenomenological equations by virtue of the form (3.5) for the entropy production

The tensors I,$$’ (j, k = 0, 1, . . ., n), L$y’ (i = 0, 1, . . ., a), L$vf’ (i = 0, 1, . ..) n), and L$(“) are called phenomenological tensors. The first of these equations is the phenomenological equation for the irreversible process of anelastic flow. This equation may be regarded as a generalization of Levy’s law. The equation (4.2) describes the viscous flow phenomenon and this relation may be considered to be a generalization of Newton’s law for ordinary viscous fluids. Finally, (4.3) may be regarded as a generalization of Fourier’s law for heat conduction. The second and third sums on the right hand side of (4.1), the first and third sums on the right hand side of (4.2), and the first and second sums on the right hand side of (4.3) represent possible cross effects among the irreversible phenomena mentioned above. Moreover, there are n different anelastic flows and among these flows cross effects may also occur. These cross effects are represented by the terms on the right hand side of (4.1) which contain the phenomenological tensors L$$) (j, K = 1, . . . , n ;j # k). 5 5. Symmetry relations. Since the tensors cab, E$, T$‘, and 7:;) are symmetric, one may choose the phenomenological tensors so that L$dWF) = Lj$)

= L$;;’

= L$;U&),

Q&!l’ = L;$q’ Ix 1 Q$3’ = LW$

(1’,k = 0, 1) . . .) 72)

(5.1)

(j = 0, 1, . ..) n)

(5.2)

(j = 0, 1, . ..) 12)

(5.3)

Since the time derivatives ds,o/dt and de$/dt and the heat flow J(q) are odd functions of the microscopic particle velocities and the stresses T$’ and T$ and the temperature gradient aTlax, are even functions of these velocities, the Onsager-Casimir reciprocity relations reads) L$;$’ = L$$‘, Q#’

= _L$W&)>

(j,k=

l,...,n)

(i = 1, . . . . 12)

(5.4) (5.5)

520

G. A. KLUITENBERG

p;y’

=

L$F)

L’O”0’ YW

= L$$!)

(5.6)

’

,

(i =

1, . ..) n)

(5.7)

LW&‘P’= _L$FO)

(5.8)

Q$‘,“P’ = LJ$‘“‘.

(5.9)

several equations may be derived analogously to (1.9.9) From (5.1)-(5.8) and (1.9.10). The equations (5.1)-(5.9) reduce the number of independent components of the phenomenological tensors. Owing to symmetry of, for instance, the crystal lattice, this number may be further reduced. The special case of isotropy will be discussed in the next section. 3 6. Phenomenological equations and symmetry relations for isotro$ic media. The phenomenological tensors are determined by the physical properties of the medium. If the medium is isotropic these tensors must be invariant with respect to all rotations and to inversion of the axis-frame. It can be showns) 13) that these tensors have the following forms L$;;’

= $$‘k’(6a,6~~

L?J@

= LLW&.’= 0,

+ &ySa~) + g(rjp)

-

?p’)

G,j&,

(6.1)

(j, R = 0, 1, . . . . n) (i = 0, 1, . ..) n)

(6.2)

L;‘$)p,“q’ = RTc&~,

(6.3)

where the scalar quantities 1, offs’, and qfyk’ (i, k = 0, 1, . . ., a) are called phenomenological coefficients, and da0 is the Kronecker tensor (also termed the unit tensor). It is seen that the expressions (6.1) and (6.2) for the phenomenological coefficients satisfy the symmetry relations (5. l), (5.2), and (5.3). The reciprocity relations (5.4)-(5.9) are satisfied if @‘“’

= @‘i’,

(i, K = 1, . . . . n)

(6.4)

$0)

= _BLo’i’,

(i =

(6.5)

$,k’

= &%A,

(i, k =

$‘O’

= _-r$o’i’*

(j =

1, . . . . n) 1, . ..) n)

1, . ..) W)

(6.6) (6.7)

These four relations are also called Onsager-Casimir reciprocity relations. Introducing (6.1), (6.2), and (6.3) in (4.1), (4.2), and (4.3), we obtain the following phenomenological equations for isotropic media

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AND

.(vO = ;clt=, q,(OvwT(k) + $LO’

RELATED

SUBSTANCES

JAY,

521

(6.11)

J(q) = ---Agrad T,

(6.12)

where we also used the symmetry of ea,r and To@ (k) and the definitions of scalar part and deviator of a tensor. The phenomenological coefficients with subscript s occur in the equations for irreversible shear phenomena (cf. (6.8) and (6.9)) and the coefficients with subscript z( occur in the volumetric equations (cf. (6.10) and (6.11)). In particular Amoco’ and q$“” may be called the shear viscosity and the volume viscosity, respectively. These coefficients also occur in the theory of ordinary (Newtonian) viscous fluids. The coefficients qFk’ and qzWk’ (i, k = 1, . . ., n) are fluidities and $“, qioS3), q$*“, and qi”‘i’ (j = 1, . . . . n) are dimensionless coefficients connected with possible cross effects among viscous flow and anelastic flows. The coefficients qciPk, and $Sk’, (j,k = 1, . . . , n; j # k) represent possible cross effects among tie n anelastic flows. Finally, A is the coefficient of heat conductivity. $7. The thermodynamic functions. In the usual way we define the specific free energy f, the specific thermodynamic potential g, and the specific enthalpy h by f =u-Ts,

(7.1)

g=u-Tsh = u Moreover,

v cl:,&+, @&3, v ~;,s=,

we define the thermodynamic f* =

(7.4

+‘ea,q.

(7.3)

functions

f*, g*, and h* by

f + 7Jlz:=1 T&z=, $?‘@,

g* = g + v x;=,

&=i

h* = h + v x;= 1 ~,“,+,

(7.4)

T$‘@>

(7.5)

T~~‘E$‘.

(7.6)

From these definitions we obtain, with the aid of the Gibbs relation for the differentials of f, g, h, f’, g*, and h* df

= -s

dg = -s

dT + v &=, dT -

dh = T ds df’

x&=,

= T ds -

x&=,

&xi

- V x;zl

~a6 d(v+‘) &a/rd(V@)

= --s dT + v Z:+,

dg’ = --s dT a*

xf&,

T$’ dw

&?_I

(k)de(k) TEp as ’

(7.7)

d@,

(7.8)

- v X:=1 ~,“,s=,

7;;’

v x;=i

x&=i

7;;’

T$’ deaa + E:bi

X&z1

e:;’ d(v$),

E&Bd(vT$@) + x;=,

&=,

~a,gd(v@‘)

-

+ I;;=r

x&i

(2.9),

ds$,

E$) d(vT$‘), E$) d(v$).

(7.9) (7.10) (7.11) (7.12)

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G. A.

KLUITENBERG

One may also write for (7.7), (7.1 l), and (7.12) df

= --s dT + v I;&=

dg* = -s

dT -

dk* = T ds -

i T$’ de$) -

x:,+i

s&i

caP (ez)d(v+)) E$’ d(v$?)

v x;=i

x&=

i $=$as d&A;‘,

+ x;= i x$= + C;= i -&=

i E$’ d(v+&,J, i E$ d(v$&.),

(7.13) (7.14) (7.15)

where we used the relation (2.3) and the expression (2.12) for the memorystress tensors. If a medium has no memory (cf. 3 2) one of the memory-stress tensors, say @jab, vanishes and, hence, in this case (7.13), (7.14), and (7.15) reduce to df

= -s

dT + v x&=i

dg* = --s dT dh* = T ds -

~;,+,

z&=,

d&

7%’

-

e$’ d(v+))

v z;=; + C;:;

&a~d(vT$@) + x;:;

C;,,=,

+,$018 de;;),

(7.16)

I;,3,+i

E$’ d(v.T~~,&

(7.17)

x;,+i

E$’ d(v$‘&B).

(7.18)

Thus, for a viscoanelastic medium of order n without memory f is a function of T E$? E$‘, .P e$-l). Analogous remarks hold for g* and h*. Fiom (7.7) it Oljb;lo& that a

(efl)Tc@ -P

(1) &,,g, (2) . . . . @,, .~a, &x8,

(7.19)

f (T, Tao,E$‘,E$‘,.. ., @),

(7.20)

-f(T,

s = -

klT

&

(7.21) and from (7.13) we obtain

f(T, ES), EL;‘,EL;‘,. . ., E$)),

(7.22)

f(T, E$? E$‘, &:;‘a. ..> E$?),

(7.24

s = - &

(W) Tap -

p&

UC) T(m)aS = -P -a& Analogous

f(T, @,

results may be obtained

E;;‘, E;;‘, . .., @).

from (7.8)-(7.12)

(7.24)

and (7.14)-(7.18).

3 8. The entrofiy fwodution artd some inequalities. Introducing the expressions (4. l), (4.2), and (4.3) for ds$/dt, Ts’, and I$‘, respectively, in the

THERMODYNAMICS

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(3.5), we obtain for the entropy production

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523

the form

(8.1) By virtue of the Onsager-Casimir reciprocity relations (5.5) and (5.8), the last two sums in (8.1) vanish. If, moreover, we use the reciprocity relation (5.7), (8.1) reduces to

It is seen from (8.2) that the entropy production is a quadratic expression in the components of the gradient of the temperature, the components of the time derivative of the tensor of total strain, and the components of the affinity-stress tensors. Moreover-g),

cm 2

(8.3)

0,

where the equality sign holds only if the above mentioned components vanish, i.e. the entropy production is a positive definite quadratic form. From this positive definite character of the entropy production several inequalities may be derived for the components of the phenomenological tensorsQ). For instance, we have

L%(Q) > 0,

L$_$$f,kk’ 3 0.

(k = 0, 1, . ..) n)

(8.4)

Next, we consider isotropic media. By substituting (6.8)-(6.12) in (3.6) we obtain with the aid of the Onsager-Casimir reciprocity relations (6.5) and (6.7) a@) = T-1

Again,

IT-r(grad

from

T)s + qL”*o)C&= r

the positive

definite

s (‘>

character

+

of the entropy

production

524

G. A.

KLUITENBERG

several inequalities for the phenomenological coefficients may be derived. For example, we have 2 20, Tp’

(8.6)

(j = 0, 1, . ..) n)

> 0.

(8.7)

More detailed discussions concerning these matters may be found in reference 9. 5 9. The refereme state and thermodynamic eqadibrium. Let us choose a temperature TO, Furthermore, let us consider a state of the medium, with the uniform temperature To, in which the equilibrium-stress tensor and all affinity-stress tensors vanish. Hence, (fi)) = 0, T$‘(To, E&B,E:;‘, . . ., cap (n)) = 0. . . . , eaS

T$‘( To> KM, &

(k = 1, . . . . 1z)

(9.1)

(9.2)

Since the tensors T$) and 7:;’ are symmetric, (9.1) and (9.2) form a set of 6(rc + 1) equations from which the components of the n + 1 symmetric tensors eLXfl, E$, . . . , E$) may be determined. The state (with temperature To) characterized by (9.1) and (9.2) will be called the reference state. All strains will be measured with respect to this state, i.e. we choose the tensors ea,9, s(l) Cm) so that they vanish in the reference state. Hence, afl> ***>Em@ 7::) = 0 for T = To 7:;’ = 0 for T = To

and

(n) = 0, ea,r = E$) = . . . = E,@

(9.3)

and

(1) -~~0 = &aS

(9.4)

(n) = 0. . . . = caP

This implies that .Q @‘) also vanishes in the reference state (see (2.3)). If the medium has no memory one of the memory-stress tensors vanishes identically (i.e. one of these tensors vanishes for all values of cap, E$), . . ., @, and T), and one of the affinity-stress tensors equals the equilibriumstress tensor. In this case (9.1) and (9.2) form a set of 612 equations and an arbitrary state with temperature To, satisfying the conditions (9.1) and (9.2), may be chosen as reference state. Analogous considerations may be given if a medium has no shape memory. A medium is in a state of thermodynamic equilibrium if the entropy production vanishes. It follows from the positive definite character of this quantity that it vanishes if (see (8.2)) grad T = 0,

$

= 0,

and $(T,

&a,~,@,

. ..) &$‘) = 0.

(k = 1, . . . . PZ)

(9.6)

By the latter condition the partial inelastic strain tensors EL;‘, . .., 8%’ are

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!jz

determined if T and E&Bare given. It follows from (9.6) that the reference state is a state of thermodynamic equilibrium, provided the tensor of total strain, E&B,(determined by (9.1) and (9.2)) is kept constant. We note that in the reference state the medium has the uniform temperature To and, hence, grad T vanishes in this state. By virtue of (2.12) the condition (9.6) is equivalent to T$“aQ + 7$)@ = 0,

(k = 1, . . . . n)

(9.7)

i.e. in thermodynamic equilibrium each of the memory-stress tensors is balanced by the equilibrium-stress tensor. It follows from the conditions (9.5) and (9.6) and the phenomenological equation (4.2) that in thermodynamic equilibrium the viscous stress tensor T$’ vanishes, and hence, because of (3.2), we have bx) . Tag = T@

(9.8)

If a medium has no memory one of the memory-stress tensors vanishes identically and it then follows from (9.7) that in thermodynamic equilibrium all the memory-stress tensors and the equilibrium-stress tensor must vanish. In view of (9.8) the latter condition implies that the mechanical stress tensor must vanish if a medium without memory is in thermodynamic equilibrium. In an analogous manner one may discuss thermodynamic equilibrium of substances without shape memory (see $2). $ 10. The thermodynamic stress tensors. We give an example of a medium without memory. In fig. I a crystal lattice is represented schematically. Let us suppose that there occurs only one microscopic phenomenon that gives rise to inelastic deformations (i.e. n = 1 in (2.2)) and that this phe-

Fig.

1

Fig. 2

nomenon is slip along the horizontal and vertical lines in fig. 1. In fig. 2 a possible result of such a slip phenomenon is shown. In this case the inelastic deformations are not associated with changes in the microscopic structure of the medium, and if the medium in the two states (shown in fig. 1 and fig. 2) undergoes the same elastic deformations and changes in temperature, the thermodynamic state variables will have the same values

526

G. A. KLUITENBERG

in both cases (i.e. all thermodynamic state variables may be considered to be functions of T and E$)). This means that the medium has no memory (cf. (2.13)) and hence r{z,, = 0. We note that in this case no lattice defects are associated with the inelastic deformations (see fig. 2) and, hence, there are no microscopic stress fields surrounding imperfections in the medium. Moreover, neither the components of ~~0 nor the components of E$) are thermodynamic state variables. More generally, let us suppose that in a medium a microscopic phenomenon occurs that gives rise to inelastic deformations. If this phenomenon does not change the microscopic structure of the medium (independent of the degree of advancement of possible other microscopic phenomena that may give rise to inelastic strains) the medium has no memory and the phenomenon is not associated with microscopic stress fields -which may surround imperfections in the medium. Let us assume that a medium is nonviscous, i.e. the viscous stress tensor 7$) vanishes, and, hence, the mechanical stress tensor rap equals the equilibrium-stress tensor 72) (cf. (3.2)). Furthermore, let us consider strain relaxation at vanishing mechanical stresses (i.e. ran = r$) = 0). During this irreversible process T$’ = $&t)as (see (2.12)) and, hence, in (4. l), the phenomenological equation for inelastic flow, the affinity-stress tensors r$) may be replaced by the memory-stress tensors rgbfi. Hence, in the absence of (macroscopic) mechanical stresses, the driving forces for changes in the inelastic strains are the memory-stress tensors ~$2)~. The process of strain relaxation will go on until a state of thermodynamic equilibrium is reached, and in such a state the memory-stress tensors vanish since 7::’ = $!&P (cf. also (9.6)). If, moreover, the temperature is brought to the temperature To, the medium will return to the reference state (i.e. the medium has memory). If the medium has no memory, one of the memory-stress tensors, say rgNfl, vanishes, and complete recovery does not occur. The remarks in the three preceding paragraphs suggest that the memorystress tensors $&GIs (k = 1, . . . . n) give a macroscopic description of the influence of microscopic stress fields, surrounding imperfections in the medium, on the mechanical and thermodynamic behaviour of the medium. We have chosen the reference state in such a manner that the equilibriumstress tensor, 7$@, vanishes in this state (see the first paragraph of the preceding section). We shall now introduce the physical assumption that the equilibrium-stress tensor vanishes not only in the reference state but in any state with the temperature of the reference state and with vanishing elastic strains i.e. T$$ = 0 for T = To

and

E$? = 0.

(10.1)

This assumption is closely connected with Eckart’s principle of relaxability in the small (see $7 of I and 5 9 of IV). Moreover, this assumption leads

THERMODYNAMICS

OF BURGERS

MEDIA.

AND

RELATED

SUBSTANCES

527

(at least in a first approximation) to a decoupling of elastic and inelastic strains (see the next section) which is in agreement with experimental findings. If one linearizes (for small values of T - To and E$?) the function on the right hand side of (7.23) one gets by virtue of (10.1) (eq)= &= TuP

i a$$%F)

+ a$‘(T -

To).

(10.2)

Since this law is closely connected with Hooke’s law and with the DuhamelNeumann law, and since 7::’ is a part of the mechanical stress tensor (cf. (3.2)), the equilibrium-stress tensor, 7$?), may be looked upon as the stress tensor connected with thermoelastic phenomena. Hence, the affinity-stress tensor, 7$‘, (which occurs in the flow law (4.1)) consists of two parts (k) _ (eq) (10.3) Tap - TclB + ~l~)c@ (cf. (2.12)). The first part is the stress due to thermoelastic strains, while the second part, the memory-stress field, is probably due to stress fields surrounding (microscopically small) imperfections in the medium. In thermodynamic equilibrium the memory-stress field is balanced by the thermoelastic stress field (see § 9). b, which occurs in the equations of The mechanical stress tensor, 7Ly motion and in the first law of thermodynamics (3.1), also consists of two parts T&g= 7$) + 721, (10.4) (cf. (3.2)), where the viscous stress tensor 72’ is analogous to the viscous stress tensor which occurs during flow in ordinary viscous liquids and gases. 5 11. Linear eqzlations of state for anisotropic media. In order to obtain some special stress-strain relations illustrating the type of results to which the theory leads, we shall assume that in a first approximation the thermodynamic stress tensors 7$), T$‘, and $$, are linear functions of the temperature and of the elastic and inelastic strains (linear equations of state). More precisely, we suppose that the thermodynamic stress tensors are linear functions of the temperature and of the strains provided T - To and the strains are sufficiently small (i.e. the deviations from the reference state are sufficiently small). We postulate the following form for the specific free energy of an anisotropic medium f = M.4

Z,B,v,C=l

+ 4 c;,= + (T -

(O)(O)(et) (eU %g3 Ert +

%qT~C

1 Iz:,3,@,v,t= 1 (4gf) To){Z&l

- ~$$y’, &$b$’ +

a:;‘~$-) + IZ,“=,l&=1

(~:;‘-a$Q

E:;}] -Y(T),

(11.1)

where ~0 is the specific volume in the reference state, F(T) is some function

528

G. A. KLUITENBERG

of the temperature and the tensors a$$?), a$), a$$), and a($ (i, k = 1, . . . . n) are constant (i.e. they do not depend on the temperature and the strains). It follows from (11.1) that these tensors may be chosen so that they satisfy the symmetry relations &:“’

= @$) (1

= &g’

u;gw;’ = (p$ a

= u~Oy$J) = u (O)(O)= @$W = @$‘ao’ = +W$‘p”’ , (11.2) /WY

= &y

= q:y”’ c(

= uy&’

= u;m# = u~;wf’ = &~’ o! (1’,k =

&& = @

>

(11.3)

1, . . ..n)

(j = 0, 1, ..*,?z)

(11.4)

Furthermore, these tensors are determined by the physical properties of the medium in the reference state. The expression (11.1) for the free energy has been chosen so that it leads to linear equations of state, where the thermodynamic stress tensors satisfy the conditions (9.4) and (10.1). By virtue of (7.22) we obtain from (11.1) for the specific entropy s = -Vo{gJ=i Because

a$)&$) + x:i”=, g,=,

u$) E(S) + g.

(US’ -

of (7.1) one has from this relation

(11.5)

and (11.1) for the specific internal

energy u = VO[$ gJ+i

u$;$%$)&g)

+ a qc=l -

Z,&v,C=

To{x:,+1

@$qy

+ 1 (4$y +

-

xi”=,

(OW) %@yC g&4=1

$& (Q

_ -

a:;,

4$}1

+

dY +Tm-!P. Finally,

by virtue

librium-stress

(11.6)

of (7.23)

and (7.24) we obtain

from (11 .l) for the equi-

tensor ~2’ +’

= &=,

and for the memory-stress

a$$“~~’

+ u$(T

-

(11.7)

To),

tensors $AjaB

~~~,,S=~~~1~y3,~~1(u~~~‘-u~~~~)~~~+(u~~-u~~’)(T-To).

(j=l,...,@

(11.8)

In the derivation of the two preceding relations we used the symmetry relations (11.2) and (11.3). Moreover, we replaced in (7.23) and (7.24) the mass density p by the mass density pe (= I/Q) of the reference state. From (10.3), (11.7), and (11.8) we obtain for the affinity-stress tensors T$ 7:; = &=

1 a $::O’%C -

ZL,

g&l

where we also used the relation (2.3). generalization of the Duhamel-Neumann

a$;;),$)

+

a$(T -

To),

(11.9)

The equation of state (11.7) is a law for thermoelastic phenomena.

THERMODYNAMICS

OF BURGERS

MEDIA

AND

RELATED

529

SUBSTANCES

If a$’ vanishes or if one considers isothermal processes at the temperature To, (11.7) is analogous to Hooke’s law. The equations of state (11.8) and (11.9) are specific for the theory developed in this paper. It should be noted that in the quadratic expressions (11.1) for the free energy, (11.5) for the entropy, and (11.6) for the internal energy no cross terms occur between the elastic and inelastic parts of the strain. Furthermore, the equilibrium-stress tensor depends only on the elastic strains and the temperature, while the memory-stress tensors depend only on the inelastic strains and the temperature. This expresses that there is a decoupling between elastic and inelastic strains which is in agreement with experimental findings. The specific free energy, the specific entropy, and the specific internal energy may also be expressed as functions of the total strain, the inelastic strains, and the temperature. With the aid of (2.2) and (2.3) one may also write for (11.1) f = vo{J zz:,&Q,C=1 @$%&JS c( B , y , c=l X3,

+ & XL1 +

(T

-

To)(ZE,~_~

- 2&Y

+

a$$%~+$) a$%w

-

+ E;,“=,

I;i,s=,

a$h$)}

-

K

(11.10)

where we also used the symmetry relation (11.2). By virtue of (7.19) one obtains from the expression (11.10) for the free energy s = --VO(CX+~ a$& and since u =

f +

-

Zc,“=,E$=,

d?P a($$$!) + do,

(11.11)

Ts the two preceding relations yield

ZJ= vo{&z,&v,c=1 ~$$%9(~y~ - 2&$?)) + +

WW)E(i)E(k) PZk=, Z,B,W %j+t cq9 yt

-

dY

(11.12)

+Tm-Y.

The specific heat at constant deformations, cle), may be defined by a c(8) = aT zc(T, Em@, Es), . . . . @).

(11.13)

From the two preceding relations we get d?P ‘C(E)= T---, dT2

(11.14)

and hence,

T p = c(t)T log k -j- SOT- ct8)fT - To) - uo if c(e) is constant,

(11.15)

530

G. A.

where SO and 210 are integration

KLUITENBERG

constants.

It is seen from (11.1 I), (11.12),

and the preceding relation that SO and 210 are the specific entropy and the specific internal energy in the reference state, respectively. If a medium has no memory, one of the n memory-stress tensors, say TE),~, vanishes,

and, hence, one has from (11.8) @;;!r”’ = &OiW = ($;;(j:”

(k = 1, . ..) n)

,

(11.16)

and (11.17)

aS ’ a$’ = u(O) where we also used the symmetry

relations

(11.2) and (11.3).

$ 12. Linear equations of state for isotropic media. As we noted in the preceding section, the tensors a$::), a$$’ (j, k = 1, . . ., n), and a$ (j = = 0, 1, . ..) n) are determined by the physical properties of the medium in the reference state. If the medium is isotropic in the reference state these tensors have the following formss) 1s)

&a(j* k)(c&Sgg +

dj&g)

+ g(w k) - df* “‘) daj&,

(12.2)

(j, k = 1, . . . . n) (i = 0, 1, . ..) 12)

a$ = c(JM,e,

(12.3)

where the quantities a( 09O), W O), a(jv “), b(Jpk), and c(j) are scalar constants, and Bag is the unit tensor (or Kronecker tensor). By virtue of (11.3) the coefficients a(Jp k) and b(jr w satisfy the symmetry relations &k)

= &f),

(j,k=

bV,k) = bW,l). Using

the symmetry

properties

1, . . ..N)

(12.4)

(j, k = 1, . . . . n)

(12.5)

of the strain

tensors

and the relations

(2.15)-(2.18) for the deviators and scalar parts of tensors, we obtain from the expression (11.1) for the free energy and the expressions (12. l), (12.2), and (12.3) for the tensors a$;/‘, a$$), and atj the following form for the specific free energy of an isotropic medium f = v,[&+O) + z;,=, +

(T

-

x;,+l {&(a”* k, -

TO){3c(e)&J)

In an analogous s =

(zy)2

+ gb(O*O)(&(e~y+

a(O*O’)x&, + 3 z;=i

manner

(11.5)-(

-vo{3c(0)&r)

g_$@ (c(e) -

3 b&k) + 2(

c(j)) N)]

-

-

bCO,O,)

&W&W)}

Y.

+

(12.6)

11.12) reduce to

+ 3 &

(~(0)

-

c(j))

E(J)}

+

d?P

dT,

(12.7)

THERMODYNAMICS

24 =

vo[&z(O, 0)

OF BURGERS

~~,s=,($32

{*(u”P k) -

+

xrkzl

-

To{3c(OMe~) + 3

MEDIA

+ @CO,O)(&(eO)2

av-h 0’) -&_ ~gl(dO)

-

#&)

AND

+

*pu,w dY T dT

= x;=i

(a@, 0) -

+ {x;=,

(b(“rO) -

7:; = a@, 0),&&g-

&

b(% 0’) &U)&c)} -

-

$JesQ’= a@, O),$) + (bC0,W.&U + ~(0)(T 7&&

531

SUBSTANCES

+

+

c(3)) .&)}I

RELATED

Y,

(12.8)

To)} ,&,

(12.9)

“1) $$’ +

b(Js “‘) .N

+ (c(J) -

c’O’)(T -

To)} Bara,

(12.10)

To)} dab

(12.11)

c;= 1 UU,kQ$) + + {b(O10)&-

f = vo{$z~o~a) g,=,

&@(Eafl -

x;=, 2E$‘)

+ QUO*O)&(&+ (T -

Wtk)&

+ cU)(T -

+ 4 ET&i

a(J* “)(X&r

2&@nl) + # -&=i

To)(3cW

-

Egg;))

+

b(A W&W&W)+

3 zI”=, c(W))}

-

(12.12)

Y,

d!P

To(3c%

-

Furthermore,

-

3 zi”=, c(W))}

+ T dT

-

Y.

(12.14)

we note that it follows from (12.9), (12. lo), and (12.11) that

--(WI= u(O,O)~@z) 74 M’ &Q) = b(e,e)&)

(12.15)

+ c@)(T -

To),

(12.16) (12.17)

= z;=r

(a (O,O)-

a&k)) g$‘,

+A, = 2t-r

(bCO,O)-

b(3,N) e(N + (c(3) -

$&@

“$ = a@, 0)&q T(3) = b(O,O,, -

c”J’)(T -

I;;= 1 UU,k)@,

x;=,

To),

(12.18) (12.19)

b(f,k)&W) + cW(T -

To)_

(12.20)

The equations of state ( 12.15) and (12.16) are generalizations of the DuhamelNeumann law for thermoelastic phenomena in isotropic media. If c(a) vanishes or if one considers isothermal processes at the temperature To, the relations (12.15) and (12.16) are analogous to Hooke’s law for isotropic media. The equations of state (12.17)-( 12.20) are specific for the theory developed in this paper. Finally,

if a medium

has no shape memory

the deviator

of one of the

532

G. A. KLUITENBERG

memory-stress tensors, say ?gjM, vanishes, and it is seen from (12.17) that in this case &N zzza(“, k) = a(& O), (K = 1, . ..) T8) (12.21) where we also used the symmetry relation (12.4). In the same way we have for a medium without volume memory b(k,n) = b&k) = bCO,O), and

c(n) zzzc(O).

(k = 1, . . . . 12)

(12.22) (12.23)

If a medium has no memory, it has neither shape nor volume memory. Hence, for a medium without memory one has the conditions (12.21), (12.22), and (12.23). 3 13. Stress-strain relations. We shall now discuss the stress-strain relations which follow from the theory if one assumes linear equations of state. With the aid of the phenomenological equation (4.2) one obtains from (3.2)

Using the linear equations of state (11.7) and (11.9) and the relation (2.3), (13.2) (13.3)

(13.4)

Hence, Q{iiow depends on the total strain, the mechanical stresses, the temperature, and the gradient of the temperature. With the aid of the linear equation of state (11.9) the phenomenological equation (4.1) may be written in the form

THERMODYNAMICS

OF BURGERS MEDIA AND RELATED SUBSTANCES

533

Hence,Q(t, o)as depends

on the total strain and the temperature. With the aid of the expression (11.12) for the internal energy and the phenomenological equation (4.3) for the heat flow the first law of thermodynamics (3.1) becomes

-

To X;l+,

+

l&4=1&a

u$’ %

T-lL+$k'

+ pow -

aT Zi,p=lTaS7,d&aB >

7-j-g

+

dT dt

=

(13.8)

where we also used (2.2), the symmetry property (11.3) of the tensors a$$‘, the definition (11.14) of cc,), and the equation of state (11.9) for $‘, Moreover, in the first law we replaced p by pe = 1/VO. The equations (13.2), (13.5), and (13.8) form a set of 6(n + 1) + 1 equations for the 6 independent components of the mechanical stress tensor Tao, the 6n independent components of the partial inelastic strain tensors &$ (i = 1, . . . . n), and the temperature if the tensor of total strain &a@is given as a function of position and time. On the other hand, if Tas is known (13.2), (13.5), and (13.8) form a set of equations for the temperature, the total strain, and the partial inelastic strains. Since v = du/dt, where v is the velocity field and u the displacement field, the equations of motion may be written in the form (13.9)

534

G. A. KLUITENBERG

(cf. (1.3. l)), where F, is the volume force per unit of mass. As ~~6 = = 4(&,/&z, + &S/&X,), (13.2), (13.5), (13.8), and (13.9) form a set of 6(rt + 1) + 4 equations for the 6 independent components of the mechanical stress tensor, the 6% independent components tensors, the three components of the displacement ture. All other thermodynamic the quantities just mentioned

of the inelastic strain field, and the tempera-

quantities may be regarded as functions ot and hence, (13.2), (13.5), (13.8), and (13.9)

form a complete set of equations describing the mechanical and thermodynamical behaviour of the medium. It is seen that, though we introduced linear equations of state, this set of equations is very complicated. In the following sections we shall discuss some simplifying assumptions, and it will be shown that, if these assumptions hold, explicit relations between the mechanical stress tensor and the tensor of total strain may be derived. On the other hand, if the equations of state are not to be linearized, the equations to which the theory leads are so extremely complicated that it is not very likely that the stress-strain relations can be given in a simple explicit form (except for some special cases of which examples are discussed in 5 7 of IV). 3 14. Volumetric phenomena. In the three preceding sections we assumed that the strains are so small that the equations of state may be linearized. Such strains are called small from a physical point of view. Moreover, in this paper we confine ourselves to strains which are small from a geometrical point of view. For such strains one has the relation 2, = &)(I + 3&),

(14.1)

(see (111.4.1 l)), w h ere ZIis the specific volume, 110is the specific volume in the reference state, and E is the scalar part of the tensor of total strain. By virtue of (2.3) it follows from (14.1) that d&U dv ~ = 3ve d;-. dt

+ c;=r

d&(k) 7 *

(14.2)

Hence, 3ve(d.@/dt) and 3z10(C~=r ds(k)/dt) are the contributions to the time rate of the specific volume due to the elastic and inelastic phenomena, respectively. However, in general the inelastic phenomena are not associated with volume changes, i.e. d&)/dt = 0 (no volume anelasticity). Thus, for anisotropic media one obtains from (4.1)

-&_I Lf$’ -&, where

LgJ’

= ~c,3=1L$~~)=0, = gxl

Lgy’

we also used the

= 0,

reciprocity

(j=

l,...,

(j =

1, . ..) 9%)

relations

n;k=O,l,...,n)

(5.4),

(14.3) (14.4)

(5.5),

and

(5.7).

For

THERMODYNAMICS

isotropic

OF BURGERS

media one obtains +k)

= $N) 2)

MEDIA

AND

RELATED

SUBSTANCES

535

from (6.10)

= 0

(j = 1, . ..) n; k = 0, 1, . . . . ?z)

>

(14.5)

where we used the reciprocity relations (6.6) and (6.7). and T(M) are the scalar Because of (3.2), T = A@) + T(v~), where T, +), parts of the mechanical stress tensor, the viscous stress tensor, and the equilibrium-stress tensor, respectively (cf. 3 2). For isotropic media without volume anelasticity it follows from the relation 7 = T(@) + T(VQ that T

=

b(‘,‘)&

+

7j;“‘)

d&

dt

+

d’)(T

-

(14.6)

T(J),

where we used (12.16), (6.11), and (14.5). This equation is of the Kelvin type. We note that (14.6) is not simplified if the medium has no memory or no volume memory. 5 15. Distortional fikenomena in isotropic media. Since +a~ = +$’ + ?$’ (cf. (3.2)), we obtain, with the help of the phenomenological equation (6.9), for the deviator of the viscous stress tensor (15.1) By substituting the expressions (12.15) for @“a” and (12.19) for ?$ in (15.1) and in the flow law (6.8) for the deviators of the partial inelastic strain tensors, we get with the aid of (2.3) the equations Iz;=,

@$J)

(15.2)

= Q[&,@,

and dE$ -$-

+ CL1

J%&?) = Q&O)@

(i = 1, . . . . n)

(15.3)

1, . . . . 12)

(15.4)

where cil) = a(O,e) + zy=r ~~“~i)a(j,k), hjk = ~~=I ~~~~i)aB,k),

(i,k=

(k = 1, . . ..n)

(1) QCO,OjaS = a(o~e)(l + xz=r rjA”*k’)EIafl+ qL”,O)-d$ Q (i, Ok@--

d&r a(% a)(~~= 1 qz~~)) gas + l;lF”) dt.

(15.5) -

TUB,

(15.6)

(j = 1, . ..) n)

(15.7)

The equations (15.2) and (15.3) are analogous to (13.2) and (13.5), respectively. We shah now show that the partial inelastic strain tensors may be eliminated from the differential equations (15.3) and the constraint (15.2) pro-

536

G. A. KLUITENBERG

vided the phenomenological coefficients are constants and the stresses and strains have time derivatives of a sufficiently high order. We shall use the notations

d”+ao

__

=

_

?-a@,

-

do&

dO$ =

Eao, -

dto

dto

dt”

g(i) a@’

(15.8)

d0Ql&)@9 dOQ(i,ok@ (1) =Q (i,0k@’ dte = &(O,O)ap’ dte Furthermore, we define the following quantities @+I) = ~~=I c:;i’&, (1) Q(O,m)@ =

dm@o,as

(i,K=

l,...,lz)

(15.9)

=

&m

dm& = Cz(O~O)(l + s;,“=, ?j;O’k))dtm

dm?,,j

dm+rEaa

+ r!O*O)

dtm+l

-7’

(m = 0, 1, . ..) ?z)

= a(o,o)(~;=r #k))

dm& -d.

+ #O’

(15.10)

dm+r& d@+l

’

(j = 1, . ..* n; m = 0, 1, . . . . n -

1)

(15.11)

and

Q$t:& = I&

C:;iY&,m)aP - Q{t,m+lh9~

(i = 1, . ..) n; m = 0, 1, . . . . 12 - i)

(15.12)

The latter equalities in (15. IO) and (15.11) hold because of the definitions (15.6) and (15.7). Moreover, we used the assumptions that the phenomenological coefficients are constant and that the stresses and strains have time derivatives of a sufficiently high order. Next, we consider for each i, where i = 1, . . . . n, the following set of equations cg=r cjjk$’ = Qllo’,o,lxs,

(i = 1, . ..) i)

(j = 1, .*., lz; m = 0, 1, . . . . n - i)

(15.13)

(15.15)

THERMODYNAMICS

These three relations = 5{n2 + n(2 -

OF BURGERS

MEDIA

AND

form a set of 5i + 5(12 -

RELATED

i +

537

SUBSTANCES

1) + 512(n -

i +

1) =

l} 1inear equations for the 5{& + ~(2 - i)) independent components of the tensors .$$, d.$/dt, . . . , dn-~+i@/dtn-$+I. If i = 1, (15.13) reduces to (15.2), and (15.14) and (15.15) may be obtained from (15.2) and (15.3), respectively, by differentiation with respect to time (see also the definitions (15.10) and (15.11)). This shows that the set of equations (15.13), (15.14), and (15.15) holds for i = 1. Next, we show that if these equations hold for i = i’, they also hold for i = i’ + 1 (i’ = 1, . .., n - 1). It is obvious that (15.15) holds for all values of i. Let us multiply bothsidesof (15.15) bycp’andsumover j. With theaidof (15.9) and (15.14) this gives

Q’

{&z+l)c@

+

Using (15.12)

c;=l

i) +

(i+l)

'k

dnaE”k’ A

m=

xi"=, qQ(j,nz)crp

and (15.8), we obtain from this relation x;=i

andif

=

dt7n

c,+i)@

(m = 0, 1, . . . . n -

i)

if m = 0

= Q{iofb:@,

l,...,~--_

These two relations show that if i = i’, these equations also hold that the set of equations (15.13), For i = n this set of equations

the equations (15.13) and (15.14) hold for for i = i’ + 1. This completes the proof (15.14), and (15.15) holds for i = 1, . . . . n. reduces to

(15.17)

Again, we multiply both sides of (15.18) by cj”) and sum over j. With aid of (15.9), (15.17), and (15.12) this gives x;=i Finally,

we may combine z;=i

c,+%$)

(15.16)

= &flf,$;$.

and (15.19)

c$%“;’ = &~~~o,c(s.

the

(15.19)

into the set

(j =

1, . ..) ?z +

1)

(15.20)

This is a set of 5(~ + 1) equations for the 512 independent components the deviators of the n partial inelastic strain tensors @, Es’, . . ., @.

of

538

G.

If the rank of the matrix

A. KLUITENBERG

C, defined by

(1)

(I)

Cl

c2

(2)

C2)

Cl

C=

c2

(n)

Cn)

...

($2)

m A

,$+l)

Cl

and sufficient

Cl

c2

(l)

...

(2) Cl

(2) c2

...

@)

71

p+ 1)

...

condition

(15.21)

>

.. .

c2

(n+l)

(1)

c(1)

....................................... .......................................

Cl is n, a necessary (15.20) reads

...

Tl

for the solvability c(1) n

(1) & (0, ObS

p

(2) Q (O,Ok4

1L

of the equations

... ................................................

...... ... ........................... ............... (n)

(n)

Cl

c2

@+l)

,p+u

The Cayley-Hamilton characteristic equation.

(+)

*** ...

(15.22)

0.

Q I;lO)aP

11

c(n+ 1L

=

(n+l) Q (O,OMB

1)

theorem states that a square matrix satisfies Hence, using matrix notation, we have

Hn + hIHn-l

its

(15.23)

+ . . . + hn-1H + h,E = 0,

where H is the n x n matrix with elements hi, E is the n x n unit matrix, and the coefficients Izr, . . ., 12, are algebraic functions of the elements ha, (i, i = 1, . ..) n) of H. We note that hc is a homogeneous polynomial of degree i in these elements. From (15.9) and (15.23) we obtain ,(c”+l) = -_Iz&)

_ hzcr-1’

By virtue of this relation

-

... -

h,_&’

the rank of the matrix (1)

Cl

(2) Cl

(1)

c2

...

cp

...

-

&&‘.

(15.24)

C is n if and only if

p) ~ C(2)

. . . . . . . . . . . . . . . . . . . . . . . . . . . T:.

(15.25)

#O.

........................... ... (n)

Cl

Furthermore, to From

(n)

c2

...

c(“) n

with the aid of (15.24) and (15.25) the relation

&:Z,:&

+ hr&{:;oj~B + .e. + hn-r@$,,~8

(15.12)

we obtain

+ J&#,)gP

(15.22) reduces = 0.

(15.26)

THERMODYNAMICS

By substituting I;L

(-

OF BURGERS

MEDIA

this result in (15.26) + I;z,

1)” L&fl?n,.s

AND

RELATED

SUBSTANCES

539

we get

~n-tL&:lJ

(-1P

(C~=“=,c:,i-m)Q~j,?@aSN=O1 (15.28)

where ho= By changing

the order of summation

left hand side of (15.28) ZL,

(-

(-

over m and i in the latter

sum on the

we get

1)” hn-@&zp + c::rlI

(15.29)

1.

+ 1)m [E&+r

With the aid of the definitions

k-r{&

C~-nL)&($,na)a&J= 0.

_(15.10) and (15.11)

this equation

(15.30) becomes

(15.31) Next, we multiply both sides of the preceding rearrangement of terms we then obtain

Hence, we have obtained

the stress-strain

equation

by (-

l)“+l. After a

relation

(15.33)

540

G. A.

KLUITENBERG

where R&;m = (- l)“+” hn-, R~~,=(--l)~a~~~~~{h,(l

(m = 0, 1, . ..) vz-

+ ~;=l~~o,k’ ) + C&&-c

R&;% = &‘, a’( 1 + x;=,

+‘,k’)

R’“’

(d)n+l

=

hl17fP’o’ -

1)

(15.34)

(z;k=r Cf+f’k))},

( 15*35)

x:i”=, c;‘+f”‘,

(15.36)

(0,O)

73.3

(15.37)

’

and, if n > 2, R&i% = (- l)m+“ {a(“J’)h,,(l

+ J$= i ~r~‘~‘)- hnm+l~~O*o)+

i T>L+ 1 h,_a(x;k= 1 c~-~~)T$~)) - xF=“=,hn.&3$= 1 ~j++%$~))>. + dolo) C”= (~83 2; m = 1, . . . . n -

1)

(15.38)

Thus, the result is that if there are n phenomena which give rise to inelastic strains and if, moreover, a viscous effect occurs which is analogous to the viscosity of an ordinary fluid, the relation between the deviators fa,v and Corb of the mechanical stress tensor and the tensor of total strain, respectively, has the form (15.33) for an isotropic substance, provided the equations of state may be linearized, the phenomenological coefficients may be regarded as constants, the stresses and strains have time derivatives of sufficiently high order, and the rank of the matrix C, defined by (15.21), is n. $ 16. Farther discussion of distortiortul pkenomenu ire isotropic media. As may be seen from the examples discussed in $ 18-g 20, there is no general physical argument from which one may infer that the rank of the matrix C, defined by (15.21), must be less than n. It is not excluded that the values of the phenomenological coefficients and the coefficients which occur in the equations of state are exactly so that the rank of the matrix C is less than 12, but such exceptional cases will be left out of consideration in this paper. In 5 15 we have shown that the rank of C is n if and only if the inequality (15.25) holds. Furthermore, it may be shown that if (15.25) holds, (15.22) is a sufficient condition for the solvability of the differential equations (15.3) and the constraint (15.2), i.e. the stress-strain relation (15.33), which is equivalent to (15.22), is a necessary and sufficient condition for the solvability of (15.2) and (15.3). We also note that if (15.25) holds, the deviators of the partial inelastic strain tensors may be determined from (15.16). Hence, because of the definitions (15.10), (15.1 l), and (15.12) the deviators of the partial inelastic strain tensors may be expressed as functions of the deviators of the tensor of total strain, the mechanical stress tensor, and time derivatives of these quantities. In an analogous way it may be shown that the scalar parts of the partial inelastic strain tensors may be expressed as functions of the temperature, and of the scalar parts of the tensor of total strain and of the mechanical stress tensor, and of time derivatives of

THERMODYNAMICS

OF BURGERS

MEDIA

AND

RELATED

SUBSTANCES

541

these quantities. Examples of such relations are given in $4 of IV. Hence, it is seen from (12.12)-(12.14) that the free energy, the entropy, and the internal energy may be regarded as functions of the temperature, of the components of the tensor of total strain and of the mechanical stress tensor, and of time derivatives of these quantities. Analogous remarks hold for the thermodynamic stress tensors, the viscous stress tensor, the entropy production, and the heat dissipation function. We shall assume that

det a(tBf) =

au, 1)

a(l.2)

.. .

~,(l, n)

at29 1)

u(2,2)

...

&n)

............ ............... ............

# 0.

(16.1)

. . . . . . ..*...........*.................. a(“, 1)

u(na2)

...

a(% n)

As may also be seen from the examples discussed in 5 18-$20 this inequality holds with the exception of very special cases which probably do not occur in nature. Finally, it follows from the positive definite character of the entropy production that

&t

q?f) =

~)1(1,1) 8

q’L2’ 8

...

p,n)

ay’

#‘2’

...

,y

8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . > 0.

(16.2)

...... ..................... ...... ... q”t’1) 8

$0) 8

...

p,n) 8

One has det qs(‘J) = 0 if and only if one or more of the irreversible phenomena which give rise to inelastic strains do not occur. If, for instance, the first of these phenomena does not occur, one has #‘” = 0 (j = 1, . . ., PZ) (see (6.8)). The quantity ho is defined by (see (15.29)) ho=

1.

(16.3)

Furthermore, it follows from (15.23) that 121= -

(16.4)

C;zi hrt,

and that hall h, = (-I)%

det hg, = (-1)”

h12

...

hln

hz2 ... ha hzzn ...... ........................ . ... ... ........................ h nl

hn2

...

h nn

(16.5)

542

G. A.

KLUITENBERG

Using (15.5), one may also write for the two preceding lzl = -

X;zI=i ~~%&,I),

k, = (-

l)n (det qpj))(det

relations (16.6)

&Jl).

(16.7)

Hence, by virtue of (16.1) and (I 6.2) we have h, = (-1)”

det &I # 0.

(16.8)

We shall now show that the expression (15.35) for R{& may be simplified. Let us denote by A(33 k) the (TZ- 1) x (m -’ 1) subdeterminant of det a(i~j) which is conjugate to the element a(l~k). Hence, we have CE=i (-

1)3+k act, k)A V, Jc) = (det a(r,J)) d(i; i),

zE=i

l)j+k a(k,QA(k,l)

(16.9)

and (-

= (det aV,J)) S(i; i).

(16.10)

Using (15.9) and (15.24) we obtain

h,&‘hjr

ET,=,

= -Iz~c~~),

(16.11)

or, with the aid of (15.5) x&nz=l

J.zn_tc~+~,~)&,

k) = -_Iz,C~i’.

(16.12)

We multiply both sides of this relation by (- l)l+k Aczpk) and sum over I and K. We then get with the help of (16.9), (16.7), and (16.1) C$

m=l hn_,,!i’@“) 7 8 ,3

= - (-I)%

(det T$~)) x;,,_,

Using this result and the definition R&,

(-

l)l+k ci’)A(l~k).

(15.4) of ci”, the expression

(16.13)

(15.35)

for

becomes R&,

= a@,@((det

a(“tj)) -

~(%a) ~~,=,

(-

l)j+k Au,*)) det T$“),

(16.14)

where we also used (16.7) and (16.9). Again, with the aid of (16.9) one may also write for (16.14)

R& = u(o*o){~in,k=l (u(s,f) -~(O,O))(-_l)l+kA(I,k)}detr~~). We also note that (15.34)

(16.15)

and (16.7) yield

R&,, = (-1)”

Jz, = (det &f))(det

Hopi’).

(16.16)

If the medium is in a state of equilibrium, where the time derivatives the stresses and strains vanish, we have, because of (15.33), R”’ (&a,9 or, with the aid of (16.14),

(16.16), &,(O,0)

det a(i*j)

of

= R&$&/3, (16.1), and (16.2), XFk=i

(-1)‘f”

A(f*kl

(16.17)

THERMODYNAMICS

OF BURGERS

MEDIA

We note that u(a*e) is the coefficient

AND

in (16.17) is connected

$ 17. Media

witho&

viscosity

SUBSTANCES

which occurs in the equation

(12.15) for the elastic part of the deformation. brackets

RELATED

of state

The second term between the

with the inelastic and media

543

strains.

witho&

memory.

In $ 15 we de-

rived the stress-strain relations for a viscoanelastic medium of order n with memory (see the end of $2). If in such a medium no viscous effects occur which are analogous to the viscosity of ordinary fluids (i.e. if 72) vanishes), we shall call such a medium an anelastic medium of order n with memory. If $) vanishes, it follows from (6.9) that q!‘*‘) and qiol”’ (k = 1, . . . . n) vanish, and by virtue of the Onsager-Casimir relations (6.5) the coefficients ?$‘O) (k = 1, . ..) n) also vanish. It is seen from (15.37) that in this case the stress-strain relation (15.33) reduces to R”’

(d)o+a~+ R{$,

ZzzRl”dok

d+aB

- dt

+ . . . + RI;; n_ ’

d&S

+ R{&

-

where the expressions reduce to

dt

+

(15.34),

R{$m = (-

&n-l

. . . + R’“’ (d)n_ 1 (16.15),

+

dnqara -= dtn

dn-r& dtn-1

(15.36),

and (15.38) for the coefficients

(m = 0, 1, . . . . n -

l)m+n hn-,

R&i0 = a(a,e){~~k=l

dn-rf,8

(a(i,k) -

a(&@)(-

1)

(17.2)

l)lfk AU,k)} det qc*“,

Rl”dm= a(ate) ,

(17.3) (17.4)

and, if n 2 2, R{f& = (-

I)m+n a(O~O){h,,

+ x:t,+,

Iz~__z(C~~=~ ~~-~)qf,~))}.

(n > 2; m = We also note that the expression

1, . . . . n -

1)

(17.5)

(15.4) for cp) reduces to

cp) = a(&@.

(17.6)

Finally, we note that in this type of media an instantaneous elastic response occurs if the medium is suddenly loaded. This may be seen from (15.1) and (12.15). If a viscoanelastic medium of order n has no shape memory, the deviator of one of the memory-stress tensors, say ?gjXP, vanishes (see 9 2). In this case the relations (12.21) hold. If we take i = n in (16.15), it is seen that Rl”d,o vanishes if a medium has no shape memory. Hence, by virtue of (15.33) the stress-strain relation of a viscoanelastic medium of order 12without shape

544

G. A.

KLUITENBERG

memory reads

and the stress-strain relation for an anelastic medium of order m without shape memory becomes

d+arl

R&,+% + R$\, ~

dt

+ . . . + R”’(d)n _ 1 LR$‘~~~

d’+r?,O &n-l

d&e

dndaS +

F

=

+ . . . + R&z.

(17.8)

Finally, we note that for n = 1, (17.1) reduces to the Poynting-Thomson relation (standard linear solid), (17.7) re duces to the Jeffreys relation, and (17.8) reduces to the Maxwell relation. 3 18. Isotrofiic viscoanelastic media of order one. Let us suppose that only one microscopic phenomenon occurs which gives rise to inelastic strains (i.e. n = 1). In this case the expressions (15.4) and (15.5) for cil) and hir read

cl’) = a(O,O)+ ~~“~l)a(i,i),

(18.1)

jgii = ~~l~*)aU,r),

(18.2)

and the relation (15.23) reduces to kir + hi = 0.

(18.3)

Hence, (131)a(l,l). lzl = --hii = --_rlS

(18.4)

The condition (15.25) reduces to cy) = a(e,o) + aeonl)a(l* 1) # 0.

(18.5)

As has been shown in 5 4 of VII the stress-strain relation reduces t_o the Kelvin equation if cr(l) = 0. However, it is not very likely that the relation a@, 0) + ~6”,l)a(r, 1) = 0 connects the phenomenological coefficient ~i’*~), which occurs in the description of irreversible phenomena, with the coefficients a@* 0) and a(i,l), which occur in the equations of state, and, as noted in $ 16, such exceptional cases wiIl be left out of consideration. As has been shown in 5 5 of VII, the coefficients a(lpl) and a@~@ satisfy the inequality a(i.1) > a@, 0) > 0. (18.6)

THERMODYNAMICS

OF BURGERS

MEDIA

AND

RELATED

SUBSTANCES

545

Hence, if &l) vanishes, we have a( 191) = 0~0) = 0, and in this case the deviators of all thermodynamic stress tensors would vanish (see (12.15), (12.17), and (12.19)). This exceptional case may also be excluded. Therefore, the inequality (16.1) certainly holds if n = 1. We also note that the inequality (16.2) reduces to

#J)

(18.7)

> 0.

If n = 1 (15.33) reads (18.8) For the coefficients in this relation we obtain from (15.34), (16.14), ( 15.36), and (15.37) with the aid of (18.1) and (18.4) R{;;,, = & l)+”8 l) * RI:)0 =

R{;\r = &0,(1

a@,0) (&

1) -

a@,0’) vi” I),

+ si”“’ - #PO)) + &r)(rl;o~o)#~l) R’“’ _ (d)2 -

%

(18.9)

- ~~“+!r’o)),

(0~0)

(18.10) (18.1 1) (18.12)

*

It should be noted that if one takes A( 1~1) = 1, (16.14) holds for 12= 1 (see also (16.9) and (16.10)). With the aid of the Onsager-Casimir relations (6.5), one may also write for (18.11) Rl”d,r= #,e’(l

+ 2r];091’)+ &l){rl;o~o)r;l~l) + (r]t”*l))s}.

(18.13)

As has been shown in IV and VII the stress-strain relations for ordinary viscous fluids, for elastic media, and for Maxwell, Kelvin, Jeffreys, and Poynting-Thomson bodies may be regarded as special cases of (18.8). (Plasticity phenomena are left out of consideration in the present paper.) 3 19. Isotrofiic anelastic media of order two with memory. Let us now consider the case where two microscopic phenomena occur that give rise to inelastic strains (i.e. n = 2). Furthermore, we shall assume that the medium is nonviscous (see the first paragraph of $ 17). Hence, qi”“‘, aeon‘), qi”‘2’, #,O), and 7i2p‘) vanish. In this case we obtain from (15.5), (15.4), and (15.9) & = x:=1 #ni’&, fil,

(i, k = 1, 2)

cl” = cir) = a(O,O), ci2’ = a(% 0) -& ci2) = &e)

h,l = &X0) x&=r $&(“,‘),

x,“=, h,2 = &e)

x:,“,=, #+N.s).

(19.1) (19.2) (19.3) (19.4)

546 The relation

G.

(15.23)

A.

KLUITENBERG

reduces to z:i”=r ~&gc +

WZik

+

h2 qi;

k) =

(19.5)

0,

where and hs = (det $yi’)(det (see (16.6) and (16.7)). The condition (15.25)

a@~j)),

(19.7)

becomes

= (&3,0))2

yp’(&2)

g..,

-

&l))

# 0.

(19.8)

Again, there is no general argument from which one may infer that the inequality in (19.8) does not hold. The same remark holds for the inequality (16.1). By virtue of (17.1) the stress-strain relation medium of order two with memory reads

For the coefficients

in this expression

R{& R&j),, = ~(0, e){&

l)&,

2) -

= &=r

~(1, s),(s, 1) -

= &,a)

-&,i

$g3)),

(19.10)

&~)lljli*i),

R&i

(19.11)

a@, a)(& ~(1,s) -

@W

-

1) + &, 2) a(s,l))}

(19.6),

det qpi),

&%e)) Bf,i),

R$\s = &&a), where we used (19.2),

anelastic

we obtain from (17.2)-(17.5)

= (det u@J)(det

R&,

for an isotropic

(19.12) (19.13) (19.14)

and (19.7).

$20. Isotropic anelastic media of order two without shufie memory (Burgers media). If an anelastic medium of order two has no shape memory, the deviator of one of the memory-stress tensors, say ?[zbP, vanishes. We then have from (12.21) a(% 2) = &

2) =

@,(2,1)

z.cz a@,

0).

(20.1)

THERMODYNAMICS

OF BURGERS

MEDIA

AND RELATED

SUBSTANCES

547

In this case the condition (19.8) reduces to (ace, 0) - &l))(+i)

+ 1;1i2”‘)# 0.

(20.2)

From (19.12) and (20.1) it is seen that Rj& vanishes. Hence, the stressstrain relation (19.9) reduces to

From (19.10), (19.1 I), (19.13), and (19.14), we obtain with the aid of (20.1) for the coefficients which occur in this equation R&,, = a(a,e)(~(r,r) - a(e,e)) det qc~i),

(20.4)

R{& = aWV~l.r’

(20.5)

R[& = a(e,a)(&r)

+ a(0,a)(r1~i~2)+ 71;s’1)+ @s’), - &,a)) #r),

R&\2 = &,a).

(20.6) (20.7)

Using the Onsager relation Q(Lo) = qL2,1) (20.5) may also be written in the form R{& = &r)#,r’ + &W(1;1(2~2) + 2r1;r,s’)S (20.8) S The relation (20.3) is the well-known stress-strain relation for Burgers media l4) l5). Hence, in Burgers media two phenomena occur which give rise to inelastic strains. One of these phenomena is associated with changes in the microscopic structure of the medium (see 5 10) and is responsible for the elastic after-effect. The other phenomenon is not associated with changes in the microscopic structure of the medium and is responsible for the flow phenomenon in Burgers media. Moreover, since there is no viscosity analogous to the viscosity of fluids, an instantaneous elastic response occurs if the medium is suddenly loaded (see (12.15) and (15.1)), In some other papers plasticity phenomena and large elastic and inelastic strains will be discussed. Moreover, a generalization of a more fundamental nature will be developed. Received 25-10-67

REFERENCES 1) 2) 3) 4) 5) 6)

Kluitenberg, Kluitenberg, Kluitenberg, Kluitenberg, Kluitenberg, Kluitenberg,

G. G. G. G. G. G.

A., A., A., A., A., A.,

Physica 28 Physica 28 Physica 28 Physica 29 Physica 30 Application

(1962) 217. (1962) 561. (1962) 1173. (1963) 633. (1964) 1945. of the Thermodynamics

of Irreversible Processes to Conti-

548 -

7) 8) 9) 10) 11) 12) ‘3) 14)

15)

THERMODYNAMICS

OF BURGERS

MEDIA

AND RELATED

SUBSTANCES

-

nuum Mechanics, g 6 of Non-Equilibrium Thermodynamics, Variational Techniques, and Stability, The University of Chicago Press (Chicago, Illinois, 1966) 91. Kluitenberg, G. A., Physica 36 (1967) 177. De Groot, S. R., Thermodynamics of Irreversible Processes, North-Holland Publishing Company, Amsterdam and Interscience Publishers Inc., New York (1951). De Groot, S. R. and Mazur, P., Non-Equilibrium Thermodynamics, North-Holland Publishing Company, Amsterdam and Interscience Publishers Inc., New York (1962). Meixner, J. and Reik, H. G., Thermodynamik der irreversiblen Prozesse, Handbuch der Physik, Band III/Z, Springer-Verlag (Berlin, 1959). Prigogine, I., Etude Thermodynamique des Phenomenes irreversibles, Dunod, Paris et Editions Desoer (Liege, 1947). Prigogine, I., Introduction to Thermodynamics of Irreversible Processes, Interscience Publishers - John Wiley and Sons (New York - London, 1961). Jeffreys, H., Cartesian Tensors, Cambridge University Press (Cambridge, 1957). Burgers, J. M., Mechanical Considerations - Model Systems - Phenomenological Theories of Relaxation and of Viscosity, Chapter I of First Report on Viscosity and Plasticity, Royal Academy, Amsterdam (1935) 5. Reiner, M., Rheology, Handbuch der Physik, Band VI, Springer-Verlag (Berlin, 19581 434.