A gradient theory of internal variables - Springer Link

Acta Mechanica 116, 1 - 14 (1996)

ACTA MECHANICA 9 Springer-Verlag 1996

A gradient theory of internal variables K. C. Valanis, Vancouver, Washington (Received August 4, 1995)

Summary. In this paper we develop a gradient theory of internal variables using a variational principle in conjunction with the dissipation inequality. The basic findings are (i), that the internal variables are, non-local in that they obey field equations instead of evolution equations and (ii) they are subject to boundary conditions that are dictated by the applied tractions and displacements as well as the physical structure of the material domain. As a consequence, spatially inhomogeneousstrain fields exist in the presence ofuniforrn boundary tractions and/or displacements. This phenomenon is illustrated in the simple case of one dimension.

1 Introduction The phenomenon of localization of macroscopic deformation has been the motivating force for the development of continuum theories with "higher gradients". In earlier works the goal was the development of elasticity (hyperelasticity) theories with higher deformation gradients (e.g. Toupin [1], [2], Mindlin [3], [4] and Green and Rivlin [5]). It is not our purpose to discuss these theories in detail except to say that the motivation here was "a more precise" formulation of the constitutive response of an elastic material when strong variations in the boundary tractions and/or displacements at the surface of a domain were present. The central issue that we are concerned with here is the experimental observation that macroscopically uniform material domains, under uniform surface tractions, develop localized, i.e., non-uniform deformation fields - contrary to the predictions of "local" theories. There are other issues such as the "regularization" of solutions, i.e., the correction of pathological situations where a local theory gives rise to ill-posed boundary and/or initial value problems, by the introduction of gradients in the constitutive parameters. These "other issues", however, are murky, often due to the inadequacy of the constitutive theories rather than the material behavior itself. While the introduction of higher gradients of constitutive parameters in a constitutive equation goes back as far as Van der Waal, their recent advocacy is due to Aifantis [6], [7] who called for the necessity for including gradients of these variables in a constitutive equation. He used physical arguments for constructing evolution equations for such variables and showed cases where localization did occur under special circumstances [8]. Recently, other contributions in this direction have been forthcoming, notably the work of Vardoulakis and Aifantis [9] and Vardoulakis and Frantziskonis [10] in the area of plasticity where higher order plastic deformation gradients were introduced. Our object in this paper is to develop a cogent gradient theory of internal variables under isothermal conditions. The condition of uniform temperature allows us to develop the theory in a strictly mechanical setting without a need for entropy arguments. The full thermodynamic

2

K.C. Valanis

treatment is given in a seperate paper as a natural consequence. Ofimportane is that the physical nature of the boundary conditions for the internal variables is demonstrated. Specifically, under uniform surface tractions and/or displacements these conditions are null.

2 Physical foundations Local constitutive theories of material behavior are based on the physical propositions that (i) the range of potential interaction among particles is short, (ii) that there is no particle diffusion in the course of deformation and (iii) the idealization that the discrete constitution of matter may be approximated by a continuous distribution such that, given a material microdomain of volume dVand mass din, however small, there exists a density function 0, at least piecewise continuous, such that dm = Qdv. The second proposition means that nearest neighbors of a particle remain such in the course of deformation, or that the map that sends the undeformed to the deformed configuration is anne. To discuss the third proposition we recognize,as is often done in the literature, a characteristic length 1 associated with the underlying material structure. This may be the lattice length in the case of crystalline solids, the "grain size" in the case of granular media, or the largest dimension of a constituent in the case of heterogeneous media. We then note that the third proposition is based on the assumption that the actual mechanical behavior of a domain of discrete matter may represented by that of a "materially averaged" equivalent continuous domain of volume V and surface S, whose dimensions may be made vanishingly small. However, the experimental measurement of the behavior of the actual material is performed on a mesodomain which isfinite. If the mesodomain is a cube of linear dimension d, then give a fixed error of measurement, a thought experiment will demonstrate that d has an irreducible value do, which is a material property that can be defined operationally and can be measured as follows. Observation shows that the mechanical behavior of a material with a discrete structure is a function of the ratio (dfl). Now if a local theory is to be applicable to a material of discrete constitution the following must be true: If P is a property then there must exist a do such that for all d > do: (i)

P(d/l) -- P(do/1) = fi~

(1.1.1)

where 6 is the difference between the measured properties and

(ii)

Lim 6 = &o

(1.1.2)

d--+ oo

(iii) Lim 5~o -+ 0.

(1.1.3)

do~oO

We note that 5o is an upper bound. Thus do is the minimum dimension of the domain used to determine the property P such that the "error" associated with the size of a material domain, whose material behavior is sought, is tolerable. However, if these measurements are to have a meaning in the context of a local theory, the applied tractions must be "locally uniform", i.e., uniform over the faces of a cube of dimension do, thus giving rise to constant stress and strain fields within the cube. The displacement fields over rge faces of the cube are also uniform, varying linearly within the cube. In reality, however, one

A gradient theory of internal variables

3

can envision situations (i) where applied tractions and/or displacement fields suffer strong variations within do, or (ii) where internal material motions give rise to internal displacement fields, with strong nonlinear variations within the cube. Such a surface traction field T~' can, however, always be decomposed, over each "averaging" sub-surface Sa of dimension do 2, into a locally uniform field T~ and a self-equilibrating field Qi such that T~' = T~ + Q,

(1.2.1)

where ~ Q~dS = 0. In the same manner, a surface displacement field ui' can be expressed as the s, sum of a locally uniform field u~ and its complement q~, i.e.,

ui' = ui + ql

(1.2.2)

where S q~dS = O. The fields Qi and q~ we shall call "locally inhomogeneous". Sa

In view of the above definitions,

I T/~ui'dS = I T~6u,dS + I QfiqidS S

S

(1.3)

$

where repeated indices are summed. In the ensuing analysis all repeated indices are summed unless a statement is made to the contrary. We note that even if the tractions are locally uniform over the entire surface, the resulting displacement field in the cube m a y not be linear, because of the complex internal material structure. In this event the free energy of the cube will depend on the displacement gradient uij as well as on an internal displacement field qi, known as an internal variable in irreversible thermodynamics. In this case a local form of the internal variable theory will result. However, if

ql suffers strong variations within the cube, the free energy may then depend not only on qi but also its gradient. A gradient theory of internal variables will then be called for. We thus have the following two possibilities: (i) A local theory in which lIOl]l = 0 on the subsurface ST on which tractions are prescribed, Ilqill = 0 on the subsurface S, on which displacements are prescribed, and the free energy depends on uij and qi but not on its gradient. (ii) A non-locgl, i.e., gradient theory where IIQ~II # 0 on Sr, Ilqzll # 0 on S., in general, and the free energy depends on u~,j on q~ and its gradient qij. Remarks: The physical model discussed above is the motivation for the development of a gradient theory of internal variables. The theory is mathematically self-consistent. Furthermore a framework is provided for a correlation, in quantitative terms, of the physical model with the mathematical theory. In Section 4 a method is given in some detail for calculating the inhomogeneous fields Qi and qi on the surface S, from the applied fields. Since Qi is self-equilibrating, its effects in some cases are likely to be confined to the vicinity of the surface. The same is true of q~. In this event, if the surface fields are moderately homogeneous, in the sense of Section 4, then as a first approximation Q~ or q~ m a y be set equal to zero on the surface. Some extreme cases are worth mentioning. A concentrated load on the surface hardly qualifies as a locally uniform traction field. On the other hand a uniform traction is an exact such field.

4

K.C. Valanis

3 A variational principle We begin with a integral form of a principle which is of purely mechanical character, in that it avoids questions of entropy and temperature under conditions of irreversibility (even though the question of existence of entropy was dealt with by Valanis [11], [12], in earlier work). Furthermore, it is simple and leads to direct results. The principle is in the form of the global statement that, if ~, is the stored energy (free energy in thermodynamics) of an arbitrary domain D of a continuous medium with surface S, a variation bO of ~, is given by Eq. (2.3), i.e.,

bt) = ~ T~bu~dS + S Qibq~dS + ~ fibu, dV - ~ Q/bq~dV S

S

D

(2.1)

D

where 6u~ is a variation in the displacement field ul, 6q, is the variation in the displacement field q,, T~ and Q, are the locally homogeneous and inhomogeneous surface tractions respectively, fi are the body forces (including inertial forces) and Q~' are the internal dissipative force fields, complementary to q~, such that Q/bq, is a variational increment in the dissipation. Equation (2.1) is a statement to the fact that, for all admissible variational increments bu~ and bql, at constant T/, Qi, J) and Qi', the increment of the free energy of a region is equal to the increment in the work done by the external body as well as surface forces, minus the increment in dissipation due to the work done by the internal forces Q/. In regard to the admissibility of the increments bu~ and 6qi we point out that while bu~ are completely arbitrary, bq, must satisfy the dissipation inequality

Q~'bq~ > O; if [Ibq~l[> O; [IQ/II> 0

(2.2)

so that inequality (2.1) may be written in terms of the inequality (2.3.1):

6~ O, with the proviso that the equality sign applies only in the case where IIQi'l[ = 0 and/or ]lbqil] = O. We complete the variational statement by stipulating that for all bUg* 6~, = 0,

116qzll= o,

(2.3.2)

where buff constitute rigid body motion. This is a constitutive statement. The fact that this is also a statement of equilibrium raises philosophical questions as to whether equilibrium is an independent law, or a form of constitutive behavior common to all materials whose constitution is determined by the form of dependence of the free energy density on the displacement gradients and the internal variables and their gradients. We remark that this Variational Principle contains the local theory as a special case and will also serve as the basis of a constitutive theory with higher gradients of internal variables, through the internal gradient fields. We also show that, under isothermal conditions, internal variables may be treated as mechanical quantities, giving rise to constitutive theories and equations without the aid of irreversible thermodynamics. At this juncture, and for the purposes of the analysis in the subsequent Sections, we introduce the free energy density 4t such that

~b = ~ ~dV. D

(2.4)

A gradient theory of internal variables

5

4 The local theory We first show that an isothermal local theory, in the sense of Valanis [13] and Coleman and Gurtin [14], is obtained by setting

O = O(uij; qi)

(2.5.1)

in D, while on S IlOell = Ilq~ll = 0.

(2.6.2)

The functions 0, u~, and q~are presumed to possess differentiability to a degree required by the analysis. Inequality (2.3.2) and Eqs. (2.4) and (2.5.1), in conjunction with the Green-Gauss theorem, give

5 (T~ - O0/c~u~,i)nj6uidS + ~ {(OO/c~ui.j),j + J~} 6u~dV+ I (O0/aq,)6qidV < 0. S

D

(2.6)

D

Quite clearly, since distributions 6ul on c~and in D can be applied independently of each other, then, given any set of admissible 6qi appropriate distributions may be found so as to violate inequality (2.6), unless certain conditions prevail. Typically for instance, 6ui(S), (~ui(V) may be distributions in the form of Dirac delta functions, in which event inequality (2.6) becomes

(Ti - 6q~//~ui,j) OUi + ((cqO/cqui,j).j + fi) (~ui + (OO/cqqi) bqi < 0

(2.7)

where 6U~, 6u, and 6qi are the vector intensities of the distributions. On the basis of the above discussion, given any admissible bqi, inequality (2.7) will not be violated if and only if on S:

T~ = (e~9/Ou,d) nj

(2.8)

(O~/OuJj + f~ -- 0

(2.9)

while in D:

(a~/Sqi) cSq~< 0.

(2.10)

Remark. One may now reverse the process and multiply Eqs. (2.8) and (2.9) by c~ui and integrate over any domain D and use inequality (2.10) to arrive at inequality (2.6). This means that if inequality (2.6) is true for one domain D, then it must be true for all domains. Use of Eqs. (2.5.1), (2.8) and (2.9) in Eq. (2.1) yields the relation

(Qi' + cq~/cqqi) 5qidV = 0

(2.10.1)

D

which, as above, leads to the local form

{Q~' + c~k/c~ql} 6q~ = 0.

(2.10.2)

One may now show that (2.10.2) cannot be satisfied for all admissible 6qi, i.e., those that satisfy inequalities (2.2) and (2.10, unless

Q~' = -c3~/c3q~. For a proof see the Appendix. This last relation completes the analysis.

(2.10.3)

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K.C. Valanis

Remark: In the thermodynamic derivation by Valanis on the one hand [13] and Coleman and Gurtin on the other [14], Qi' do not appear explicitly. The free energy gradients -OO/Oq~ are simply presumed to have the significance of thermodynamic forces. Here, their existence is posited ab initio in Eq. (2.1) and are then shown to be equal to -&~/~q~.

Rigid body motions (i) Rigid body translation. In this case 6u~* is constant in D and 6q~ = 0 in D by virtue of Eq. (2.3.2). Thus, in view of Eq. (2.3.1): r, as + Sa~dV = 0. S

(2.11)

D

Application of Eq. (2.11) to an infinitesimal tetrahedron, gives rise to the known relation: Ton~= Tj

(2.12)

which used in conjunction with Eq. (2.11) and the Green-Gauss theorem yields the equilibrium relation (law of conservation of linear momentum): T~i,~+j)=0

(2.13)

where T~j is the stress tensor. Remark: The law of conservation of linear momentum and the zero variation in the free energy under conditions of rigid body translation are mathematically equivalent. We add however that the dynamic relation

= gl - r

(2.14)

which is Newton's law of motion, where g~ are body forces other than inertia forces, is not derivable from the variational principle. One could introduce the kinetic energy term in this principle ab initio, but this would be possible only if Eq. (2.14) were known ahead of time. (ii) Rigid body rotation. In this case,

,Sui* = eiik6f2jfxk

(2.15)

where ~f2j is the infinitesimal rotation vector. Again ~q~ = 0 in D by virtue of Eq. (2.3.2). Use of Eq. (2.15) and in view of Eq. (2.13),

eokTjk=O

(2.16)

which is also the law of conservation of angular momentum and a statement to the fact that T~ is symmetric. Remark: The principle of conservation of angular momentum, the symmetry of the stress tensor, and the null variation of the free energy under infinitesimal rigid body rotation are mathematically equivalent. We add, however, that the last statement regarding the free energy is a constitutive statement. Therefore, the law of conservation of angular momentum is a constitutive consequence of the nuI1 variation of the free energy under conditions of infinitesimal rigid body rotation.

A gradient theory of internal variables

7

Relation o f the stress tensor to the free energy gradient. At this point we note that Eqs. (2.12) and (2.8) lead to Eq. (2.17): (T~ -- O~/c3uj, i) n~ = 0

(2.17)

for all arbitrary nl and thus r,~ = ~'l'/&,,.

(2.18)

Furthermore the symmetry of T~j demands that a~91au, o - aOIc?u;., = 0

(2.19)

for all i andj. When i = j Eq. (2.19) is satisfied trivially, but when i # j it is a well-known partial differential equation of the form O0/Ox - aO/ay = 0

(2.20)

with the general solution ~/, = O(x + y).

(2.21)

Hence ~l = ~t(ui,j -]- uj.i).

(2.22)

R e m a r k : The conservation of angular momentum (or the zero variation of ~O under infinitesimal rigid body rotation, or the symmetry of the stress tensor) demands that ~ be a function of the symmetric part of the displacement gradient, i.e., the (small) strain tensor ei~. R e m a r k : The Variational Principle, as stated in its totality by inequalities (2.3.1) and (2.3.2), in conjunction with Newton's law of motion (Eq. (2.14)), contains all the general laws of mechanical behavior of a dissipative, locally continuous medium under small strain and isothermal conditions. Specific material behavior is the domain of constitutive laws. Specific constitutive laws

Such laws are obtained by assigning a causal relation between Qe' and OqJ&, or, more generally, c3qdOz where z (dz > 0) is an intrinsic time scale, as discussed by Valanis in previous works [15], [16]. Such laws must satisfy inequality (2.10). Previously, we discussed particular laws of the type Qi' = b~jc3qj/c3t

(2.23.1)

or

Q / = bijOqi/Oz

(2.23.2)

where bij was a positive definite and symmetric material matrix. Equation (2.23.1) leads to viscoelasticity whereas Eq. (2.23.2) leads to (endochronic) plasticity. It is not our intention to elaborate on these equations in this local formulation, as the subject was treated in detail previously in those references. We note, however, that in the works cited above, the internal variables are second order tensors. How vectors can be used to construct tensorial variables of second order will be discussed in the section o Multiple Internal Variables.

8

K.C. Valanis

5 A gradient theory of internal variables Field and constitutive equations in the presence of gradient fields We pursue the Gradient Theory on the basis of the variational principle already stated in Eq. (2.3.1), inequality (2.3.2). The gradient theory is based on the proposition that O = q,(u,j; q~; q~.3.

(3.1)

Higher gradients of the internal variables may be included. This is not the purpose of the present paper. The ensuing analysis follows precisely the steps of the local theory and will not be repeated in detail here. Specifically, inequality (2.3.1) gives rise to the following variational inequality for all 6uz and all admissible 6qz, i.e., those which are arbitrary on S but satisfy the inequality Qi'6ql > 0 in D: {(~tP/Oui,i) n i - Ti} dS + ~ {(OOfi?qi,2) nj - Qi} 6qidS - ~ {(~?O/~ui,j)j S

+fi} ~u~dV

S

+ ~ {c~t~/~qi -- (O0/Oq,d),j } 6q~dV O, r not summed.

(4.3)

The analysis follows precisely the arguments of Section 3, so that all Eqs. (3.3) - (3.16.2) apply absolutely, with the additional proviso that all equations involving ql apply to each individual internal variable separately. Discussion: The need for multiple internal variables lies in the mathematical representation of the memory, i.e., history dependence of a material. For instance, in the local theory of linear viscoelasticity in one dimension, a single internal variable will give rise to the constitutive equation

T = i G(t - z) ~g/~d~

(4.4)

0

where G(t) is of the form

G(t) = Go exp ( - a t ) ,

(4.5)

Go and e being positive constants. The memory kernel given in Eq. (4.5) is inadequate for describing the behavior of actual materials. When multiple, say n, internal variables are used, G(t) has the form given by Eq. (4.6): G(t)= ~ G r e x p ( - ~ , t ) .

(4.6)

r=l

The actual material memory kernel is a positive monotonically decaying function and can be represented accurately by a finite sum of decaying exponential terms.

Multiplicity of internal variables. Boundary conditions There remains the question of origin of multiplicity of the internal variables. Let tractions T~'(S) be prescribed over the entire surface. We shall discuss two asymptotic cases: (i) S _-> do 2 . Case (i) is strict philosophical terms always true, since a domain consisting of costituents of finite size, contiguous or not, must be of infinite extent (do = ~ ) if it is to be represented by a continuum. Thus, in this case, the averaging surface Sa is the entire surface. Consider the situation where tractions T/(S) are given over the entire surface. Now given the vector valued surface function T/(S), the average, "locally uniform" function Ti is found, where T~ = S - t ~ T/dS, in which event Q~ = T / - T~.One may use a set of orthonormal functions ~b{(S) to obtain the series representation

Qi(S) = z A/~ir(S)

(4.7.1)

Qir = Armpit(S), r not summed

(4.7.2)

A gradient theory of internal variables

11

where now Q r are the "internal tractions" appropriate to the internal variables q**. In a similar manner if displacements Ui are prescribed on the entire boundary then ul = S - 1 ~ UidS and qi = Ui - ul q,{S) = Z a,OF(S) q ( = a,c~i'(S),

r not summed.

(4.8.1) (4.8.2)

The functions qi" constitute the displacement boundary values for the internal variables qir(D) of which ~ is a function. Of course if the boundary tractions (displacements) are locally uniform then QF(qi r) are identically zero. Case (ii) arises when the material domain is made of small contiguous constituents. In this event the area do 2 of the averaging sub-surface S, m a y be much less than S, within the tolerable error discussed in Section 2. The procedure described above is now applied over each averaging sub-surface Sa, and the coeff• A, and a, are no longer constant over the entire surface as was the situation in case (i) but vary in value from one averaging sub-surface to the next. The advantage of this approach lies in the possibility that the surface tractions, or displacements, m a y vary slowly over each sub-surface so that the distribution is locally uniform even though the variation over the entire surface m a y be considerable. Thus there exists a class of "slowly" varying surface functions for which Qi ~ and ql ~ are identically zero, to a good approximation. Internal variables as second order tensors in a local theory

Let qi "s (r = 1, 2, 3... n, s -- 1, 2, 3) be 3n linearly independent vectors. Then qi~ where qi~ = ~', qirnqj "s (r ----1, 2... n)

(4.9)

s

is a set of r second order tensors whose principal invariants are all different from zero. Furthermore they are linearly independent. Thus ~(eo; qir~) m a y be assigned the special form

~b = $(~ii; q~J),

(4.10)

and a concomitant evolution equation be given the special form g~/Oq~j + bbuSq~i/~z = O.

(4.11)

For the thermodynamic treatment of this ease see Valanis [15], [16].

7 A worked example To illustrate the ramifications of the non-local theory we present in this Section a worked example in one dimension. Let

= (1/2) Aux 2 + Buxqx + (1/2) Cqx 2

(5.1)

where u x - Ou/~x, qx =- 8q/Sx, i.e., a suffix denotes differentiation. We then consider the quasistatie extension of a semi-infinite rod ( < x < oo) when at x = 0, a stress To(t) is applied. The stress is posited to be uniform in the thickness dimensions so that the applied traction is locally

12

K.C. Valanis AUx/To

/ ~ ~

X=XlX=X2

~S m

i

Fig. 1. Strain vs. time at fixed points on the bar

uniform. The pertinent boundary conditions then are: At x = 0: T = To; Q = 0. At x = oe: all variables are bounded. At t = 0: T = u = Q = q = 0. The equilibrium condition, Eq. (3.8), gives (~0/0u~)x = o

(s.2)

while the evolution equation for the internal variables is given by Eq. (5.3):

(30/Oqx)x = bqt.

(5.3)

In view of Eq. (5.2) the stress T(= OO/Ou~) is uniform in the rod as in the local theory. However this not true of the strain. Equations (5.2) and (5.3) combine to give the following (diffusion) equation for q:

Clqx~ = bqt

(5.4)

where C1 = C - Ba/A. We have solved Eq. (5.4) by the usual Laplace Transform technique and obtained the following expression for the strain e:

= i J(x; t - z) (aTlas) dz.

(5.5)

0

Note that the memory function J(x, t) plays the role of a creep function except that now, in variance with the local theory, it is a function of x as well as t. Equation (5.6) gives the analytical form of J, found from the solution

J(x, t) = A - l ( H ( t ) + (B2/AC1) erfc [x/(2atl/2)]).

(5.6)

In Fig. 1 we show the dependence of strain on time at various x-stations when To has the form of a Heaviside step function step function, in which event e(x, t) = J(x, t). Evidently the strain "diffuses" into the rod as time increases.

Appendix The task is to prove that if

(Qi' -1- ~1i) 6qi = 0

(A. 1)

A gradient theory of internal variables

13

for all admissible 6ql, where ~i = O~/qi, then

(Qi' -4- i//i) = O.

(A. 2)

As discussed in Section 4, 6q~ is admissible if

Qi'gqi > 0; ~'i6qi < 0

(A. 3.1,2)

for all IlQi'll # 0, llq~ll # 0, II0qill # 0. Otherwise 6qi are arbitrary. Proof." If Qi' and Oi are coUinear and either of the same sign or unequal, the proof is trivial. Let Qi' = e~i, where e # - 1. Then in view ofEq. (A. 2) Oifqi = 0. However, the constraints ~i6qi = 0 and O~6q~ < 0 cannot be satisfied simultaneously if It,ll # 0. Thus, in this event, e = - 1 and Qi' "~ ffJi = O.

If Q~' and #~ are not collinear then there exists a vector B~ normal to the plane of the vectors Q~' and r such that the scalar product p = e~kQ~'~'jBk > 0. The three vectors Q~' ff~, and B~ are clearly linearly independent. We now introduce two vectors R~, normal to the plane of ~,~ and Bi, and P~ normal to the plane of Q~' and B~, i.e.,

Ri = eijkl~jBk'~

Pi = eijkQj'Bk.

(A. 4.1,2)

It m a y now be shown that all vectors 6q~ of the form

6qi = ~Ri + flPi + ?Bi

(A. 5)

where ~ and fl are positive scalars and 7 is non-negative, are admissible. In fact

Qi'gq~ = ~p;

~fiq~ = - t i p

(A. 6.1,2)

thus satisfying inequalities (A. 3.1,2). Three vectors 6q{ are then constructed as in Eq. (A. 7):

6q{ = ~'Ri + firPi q- ?'Bi

(A. 7)

where ~r > 0, fir > 0, ? r > 0. These vectors are admissible and, furthermore linearly independent if: 71 = 1, T2 = 0, 73 --- 0, and the determinant condition (A. 8) is satisfied: ;11 ~fl~ # 0 .

(A. 8)

Since one can always find positive scalars such that cdfl 2 - ~2fl~ # 0, condition (A. 1) now demands that the vector (Q~' + ~,~) be orthogonal to three linearly independent vectors. This is not possible if the said vector is different from zero. Thus Qi' = - ~/~.

References [1] [2] [3] [4]

Toupin, R. A.: Elastic materials with couple stress. Arch. Rat. Mech. Anal. 11, 119-132 (1962). Toupin, R. A.: Theories of elasticity with couple stress. Arch. Rat. Mech. Anal. 17, 85-112 (1964). Mindlin, R. D.: Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16, 51-78 (1964). Mindlin, R. D.: Second gradient of strain and surface tension in linear elasticity. Int. J. Solids Struct. 1, 417-438 (1965). [5] Green, A. E., Rivlin, R. S.: Multipolar continuum mechanics. Arch. Rat. Mech. Anal. 17, 113--147 (1965).

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K.C. Valanis: A gradient theory of internal variables

[6] Aifantis, E. C.: Remarks on media with microstructure. Int. J. Eng. Sci. 22, 961-968 (1984). [7] Aifantis, E. C.: On the structure of single slip and its implications for inelasticity. In: Physical basis and modelling of finite deformations of aggregates (Gittus, J., Nemat-Nasser, S., Zarka, J., eds.), pp. 283-324. London: Elsevier 1986. [8] Aifantis, E. C.: On the microstructural origin of certain inelastic models. J. Eng. Mat. Techn. 106, 326-330 (1984). [9] Vardoulakis, I., Aifantis, E. C.: A gradient flow theory of plasticity for granular materials. Acta Mech. 87, 197-214 (1991). [10] Vardoulakis, I., Frantziskonis, G.: Micro-structure in kinematic-hardening plasticity. Eur. J. Mech. 11, 467-486 (1992). [11] Valanis, K. C.: Irreversibility and existence of entropy. J. Nonlinear Mech. 6, 337-360 (1971). [12] Valanis, K. C.: Partial integrability as a basis of existence of entropy in irreversible system. ZAMM 63, 7 3 - 8 0 (1983). [13] Valanis, K. C.: The viscoelastic potential and its thermodynamic foundations. Iowa State University Report 52 (1967); J. Math. Phys. 47, 267-276 (1968). [14] Coleman, B., Gurtin, M.: Thermodynamics of internal state variables. J. Chem. Phys. 47, 599-613 (1967). [15] Valanis, K. C.: A theory of viscoplasticity without a yield surface. Arch. Mech. 23, 517--533 (1971). [16] Valanis, K. C.: Fundamental consequences of a new intrinsic time measure. Plasticity as a limit of the endochronic theory. Arch. Mech. 32, 171-191 (1980).

Author's address: K. C. Valanis, Endochronics, Inc., 8605 Lakecrest Ct., Vancouver, WA 98665, U.S.A.