On a kinetic formulation of elasto-viscoplasticity

Pergamon

Int. J. Engng Sci. Vol. 34, No. 9, pp. 1033-1046, 1996

Published by Elsevier Science Ltd Printed in Great Britain 0020-7225/96 $15.00+ 0.00

PII: S0020-7225(96)00008-0

ON A KINETIC FORMULATION VISCOPLASTICITY

OF ELASTO-

A. I. LEONOV and J. P A D O V A N Department of Polymer Engineering, The University of Akron, Akron, OH 44325-0301, U.S.A. Abstract--A kinetic formulation of elasto-viscoplasticity is proposed. It consists of a non-linear Maxwell-like viscoelastic constitutive equation (CE) that describes the evolution of the recoverable (elastic) part of the total Finger strain tensor, a thermodynamic formulation of stress tensor dependence on the elastic strain tensor, and a kinetic equation for a scalar "bonding" factor. The set of CEs does not contain any mathematical yield criterion but predicts anisotropic yield values for stress as a transition from solid-like deformations to flow through a bifurcation. The set of CEs proposed is shown to be globally Hadamard stable. The dissipative stability is also discussed briefly. In the case when the elastic strains are sufficiently small, the second-order approach is developed. Within this approach, several problems of simple shearing are analyzed to illustrate the model behavior. Published by Elsevier Science Ltd

1. I N T R O D U C T I O N Many concentrated dispersed systems, such as lubricating greases, inks, pastes, water-clay mixtures and polymers filled with small particles, display elasto-viscoplastic behavior. In these two-phase systems, colloidal particles with attractive forces create a particulate network (sometimes incorporating polymer macromolecules, in the case of filled polymers), if the particle concentration exceeds a critical value related to a "percolation threshold". These systems display a solid-like behavior under low stresses, and change it to flow under higher stresses. This occurs because of rupturing of the particulate network into "flocks". After any unloading of these systems, a long process of restoration of flocks into a new network structure happens. This is ~Lccompanied by residual stresses and accumulating irreversible strains. All these phenomena have a striking resemblance to those well known for metals. This is in spite of the differences in structure between metals, filled polymers and dispersed systems. The important features common for these very different materials are: sharp and time-dependent solid-liquid tran~;ition; existence of yield stresses; long time stress-strain relaxation and restoration of the, structure; and the hardening phenomenon. At high stress levels, the yield behavior is not important for these materials, but the viscoelastic effects still are. There have been extensive studies of kinetic and dynamic phenomena in elastoviscoplasticity of metals based on non-linear viscoelastic CEs. Valanis [1, 2] was seemingly the first who used an integral version of non-linear viscoelasticity in his endochronic theory of elasto-viscoplasticity. Godunov and co-workers (see, e.g. [3] and references therein) first developed a Maxwell type non-linear viscoelastic CE to treat high speed phenomena in plasticity of metals. Godunov used and discussed Eckart's concept of "variable elastic strain" [4] extensively, which was developed after Eckart in many publications [5-9]. Bodner and Partom [10] and Rubin [11, 12], among others, seemingly unaware of publications by Godunov, developed similar approaches to elasto-viscoplasticity of metals. The similarities between non-yield elasto-viscoplastic formulations for metals and viscoelastic CEs [13-15] for polymer liquids were discussed recently [16]; but the yield effects in metals and dispersed systems are too important kinetically and dynamically simply to ignore (for discussion, see e.g. [17], Part 6b). Nevertheles,;, using common formulations of algebraic yield criteria (e.g. of von Mises type), one cannol: properly describe the time-dependent transition (thixotropic) phenomena in these materials and, additionally, those formulations create heavy computational difficulties for solving two- and three-dimensional problems. A phenomenological kinetic approach [18, 19] to time-dependent yield phenomena in filled 1033

Kinetic formulationof elasto-viscoplasticity

1035

below. By introducing the C a u c h y - G r e e n elastic strain tensor b, b = c -~, equation (3a) can also be rewritten in the equivalent form: a

i,

0b + fC~)@(b) = O;

b = c-

1;

.

.

(3b) a where b is the "lower-connected" tensor time derivative of b. To complete the set of CEs for elasto-viscoplasticity, we need to formulate a kinetic equation for evolution of the bonding factor ~. This has been done in [18, 19] as follows: + ~/0, = (1 -

b = b + (Vv) b + b (Vv)

~)EO/(zcOl);

E = 2X/tr e 2 .

(4)

Here 01 is a characteristic time of rebonding, a positive scalar generally depending on temperature T, parameter ~:, and invariants Ik; E is the intensity of strain rate tensor e; and the positive constant z.: is the intensity of critical elastic strains in the marginal solid-like state, just before the start of debonding. The second term on the left-hand side of kinetic equation (4) is proportional to the average dissipation Di due to debonding, the right-hand side is proportional to the debonding power. Also, the storage rebonding energy F~ is proportional to the parameter [19]. In order to guarantee proper physical behavior of the model, the structural functions @(c) and f(~) should have some mathematical properties which have been analyzed in [13-15, 21]. The function *(c) has to be a monotonically increasing function of c and have the additional properties: tr[c - I - ~(c)] = 0;

tr[~r(c) • e 1 i~(c)] >

0;

~(c) ~ A(c - ~)(c---~ fi; A > 0).

(5)

The first relation in (5) guarantees that the value of density after complete unloading will coincide with Po, the initial density at rest state. Consequently, this yields the relation well known in the finite elasticity: po/p = X/~3. The second inequality in (5), along with assumed positiveness of the mobility function f(~), guarantees the positive definiteness of the macroscopic dissipation D, i.e. O = 0-1f(~)tr[~r(c) • c - l ° ~(c)].

(6)

The third relation in (5) guarantees an analytical transition to the infinitesimal elastoviscoplasticity with geometrically small elastic deformations. The general form for the term @(c), which satisfies the first condition (5) is: I~(C) " C -I :

OtI(C --

811/3)

-

0~2(c 2 - ~

tr

c2/3)

(7)

where a, and az are some functions of basic invariants Ik, satisfying the second (dissipative) inequality in (5). Consider now the properties of the mobility function f(~). The case f(~) =- 1 corresponds to Maxwell-like CEs for viscoelastic liquids [13-15, 21]. As already mentioned, these viscoelastic CEs were also applied in [3, 16] to modeling elasto-viscoplasticity. In order to incorporate yield effects into this model we need to formulate the proper features of f(~). According to the evolution equation, (3a), the function f(~) can be considered as a scaling factor for the total relaxation parameter 0 = O/f(~). We assume that there is a region of deformations where the material behaves as a deformed solid with no dissipation. This type of behavior is associated with the limit 0-~, ~, with no debonding in the material, i.e. ~:= 0. This means that f(~)-->0 when ~--+ 0. On the other hand, outside the region of solid-like behavior where ~ > 0, the function f(~) should describe the process of medium "fluidization" caused by increasing debonding, when there is an increase in the stress level. To describe this effect properly the

1036

A . I . LEONOV and J. PADOVAN

total relaxation parameter 0 should be a decreasing function of (, or f ( sr) is increasing. Thus the formal properties of function f(~:) relevant to these considerations are: f(~:) = ~ + O(~2),

~

0;

f'(~) > 0

(f(~)--*fo = const, ~

1).

(8)

The asymptotic expression in (8) for f(~:) as sr--~0 guarantees the existence of (generally different) yield values for the components of stress tensor when E--~0 [18, 19]. Equation (8) complemented by the initial condition, ~:(0)= 0, guarantees the inequality: 0 - ~:< 1 [19]. The property of the mobility function f(~:) described in parentheses in (8) is an additional assumption of "complete fluidization" at very high levels of stress. It can be omitted at the first step of modeling. The evolution equation (3a) for tensor e should be complemented by the initial condition, c[,=o = 6, held for any rest state before plastic deformation. Equations (2)-(4) are the closed set of CEs for the general kinetic approach to elasto-viscoplasticity, as soon as all structural functions, elastic potential F, dissipative scalar functions otj and a2, relaxation functions 0 and 0j, and the mobility function f(~) are specified within the above constraints. The function F in equation (2), as well as the functions ak in equation (7) can be simplified, especially in the incompressible case: det c = 1. It was found [23] that a general form of elastic potential, flexible enough for applications, can be represented in the form: poE

3G(T) 2(k + 1) {(1 - fl)[(l~/3) k÷l - 1] + fl[(I2/3) k+~ - 1]}

(9)

where G ( T ) is the Hookean modulus and/3 and k are some numerical parameters. It was also argued in [13, 23] that a,=a2=-a(ll,12)

(a-~l,

when

I~,I2--.3)

(10)

which guarantees the existence of the planar deformations for the model. The specific forms of the function a can be easily found by analyzing experimental data. For example, in several cases it is possible to use the following specification for the function a [23]:

(11)

ot = (I~/I2) m.

The functions 0 and 01 can be specified using the thermo-fluctuation concept, as proposed by Godunov [3]. The one-parametric specification of the mobility function f(~), convenient for simple deformations f(~) = [(1 - ~)-" - 1]/n

(n > 0),

(12)

was suggested in [19] where some problems of shearing of dispersed systems, along with their experimental verification, were also considered. The parameters/3, K, n, m and others should be chosen within the region of stability for the model, as discussed in the next section. These parameters will also reflect the differences between the various media modeled by this approach.

3. STABILITY OF CONSTITUTIVE E Q U A T I O N S FOR ELASTOVISCOPLASTICITY Formulations of CEs has to be subjected to some additional preventing the equations from non-physical instabilities. Recent viscoelastic CEs can be applied to formulate those constraints for viscoplasticity. Two major instabilities, of Hadamard and dissipative to improper formulation of CEs for viscoelastic liquids.

macroscopic constraints results on stability for the equations of elastotypes, were revealed due

K i n e t i c f o r m u l a t i o n of elasto-viscoplasticity

1037

3.1 Hadarnard insurability

We can define the complete set of equations as Hadamard stable (or evolutionary, or well-posed) when the solution of a boundary-value Cauchy problem for the set at any time provides the complete initial conditions for determining the solution at subsequent instants of time. Thus, the Hadamard stability allows the solutions to continue in the positive direction of the time axis. When it is impossible, very quick blow-up instability happens with extremely short wave disturbances. This results in progressive failure in numerical calculations: the finer the mesh, the worse the degaradation of the results will be [20]. In many cases, one can treat Hadamard instability as a blow-up type increase in the amplitude of initially infinitesimal waves of disturbances as the wavelength tends to zero. Important results on the Hadamard instability were obtained in studies of viscoelastic CEs during the 1980s and summarized in a monograph by Joseph [20]. Those studies analyzed the Hadamard stability for particular flows as described by particular types of viscoelastic CEs. Recently the complete results on global Hadamard stability (i.e. for any type of flow and for any value of viscoelastic characteristic parameter, Deborah number) were obtained for a general class of qaasilinear viscoelastic CEs of Maxwell type [21, 22]. Along with constitutive equations, the analysis of Hadamard stability involves the mass and momentum balance equations, with the method of characteristics used as a general mathematical tool. In the case of the most important class of quasilinear partial differential equations, this method is equivalent to the method of "frozen coefficients" (see, e.g. [20]), where the common linear stability analysis is locally used. Employing this method, not only differential but also a large class of integral viscoelastic CEs was completely investigated for Hadamard stability [22]. For a class of dJifferential operators 0

D~c~-c- v(c" e + e-c)

( - 1 - < v = const <-- 1),

(13)

0

widely used in Maxwell-like viscoelastic CEs, where c is the Jaumann time derivative, the necessary and sufficient condition, v = +1, for global Hadamard stability was obtained in [20, 21]. It means that only upper- or lower-connected time derivatives can be used as stable differential operal:ors in the evolution equations of type (3a). It also means that the use of Jaumann derivatives in rate equations will result in Hadamard instability. Therefore the Hadamard instability in Maxwell-like viscoelastic CEs with upper/lower convected time derivatives could originate only due to improper formulation of the elastic potential F. The necessary zmd sufficient criteria of Hadamard stability for the viscoelastic case, f(~) = 1, in CEs (2), (3) were formulated in [21, 22] as the constraint imposed on elastic potential F. The constraint coincides exactly with that well known in the theory of finite elasticity as the "strong ellipticity" condition [24]: B = Bq,,,,,qirjq,,,r, > O,

B i j ..... =

4c,,,q OCq,~,

pCip

•

(14a)

Here q and r are any two three-dimensional vectors, and the rank four tensor Bij..... is symmetrical with respect to two first but to two second indices. The physical sense of the coincidence of the general criteria for Hadamard stability for elastic and viscoelastic cases is evident: viscoelastic liquids respond as elastic solids on very rapid deformations. It should also I:,e mentioned that such a common thermodynamic constraint imposed on the free energy function as convexity of F with respect to the Hencky measure of elastic deformation h (h = 1/2In c), results in a weaker constraint = B~j.,.~j~ ..... > O.

(14b)

1038

A.I. LEONOV and J. PADOVAN

This is true for any arbitrary symmetric second-rank tensor 13. In the case of incompressibility, the vectors q and r in equation (14a) are orthogonal, and the tensor [3 in (14b) is traceless. The necessary and sufficient conditions in the compressible [25] and incompressible [26] cases were established in the form of algebraic criteria imposed on the first and second derivatives of the free energy function F with respect to basic invariants Ik. These criteria represent a set of awkward algebraic inequalities. Therefore the easier, sufficient conditions for Hadamard stability were also formulated for the incompressible case. Leonov [13] suggested a restrictive sufficient criterion for Hadamard stability, where F should be a monotonically increasing and a convex function of/1 and 12 (Criterion H1). Renardy [27] proved a less restrictive sufficient condition for Hadamard stability, namely F should be monotonically increasing and a convex function of ~ and X/~2(Criterion H2). These results can be immediately applied to the set of CEs (2)-(4) of the present paper. Indeed, to study Hadamard stability one should neglect the dissipative terms in equations (3a) and (4) and consider the remaining terms along with mass and momentum balance equations; but without those dissipative terms equations (3a) and (4) are uncoupled and their Hadamard stability is exactly related to the behavior of only elastic terms.

3.2 Dissipative instabilities Studies of dissipative types of instability in viscoelastic CEs, which are caused by an improper formulation of the dissipative term $(e) in Maxwell-like CEs, even if the dissipative function is positive definite, are not as advanced as the Hadamard type. A sufficient condition (close to necessary) for boundedness of elastic strain tensor e in any flow with a given strain history was found [21]. This stability condition also holds for the kinetic elasto-viscoplasticity (2), (3), (8) in the flow region when f(~) > 0. In the latter case, the presence of the mobility function f(~) in equation (3) is a softening factor for applying this condition. It was also shown [28] that when stress history is given, the above boundedness criterion does not prevent some viscoelastic Maxwell-like CEs from severe dissipative instability. Thus an additional (and very evident) necessary condition for the dissipative stability was proposed [22, 28]: the unboundedness and monotonic increase in steady dependencies of stress on strain rate in simple shear and simple extension. The same necesary condition obviously holds for the formulation of dissipative terms in the kinetic approach to elasto-viscoplasticity. It was proved [22] that for the Maxwell type viscoelastic CEs, the satisfaction of Criterion H1 guarantees the satisfaction of the above criteria of dissipative stability (boundedness) for the tensor e. However, for the elastoviscoplastic CEs (2)-(4), there is no such a proof.

3.3 Example Let us consider the specific form of elastic potential given by equation (9). Applying the sufficient condition of Hadamard stability H1 yields the following constraints imposed on the parameters k and/3:

k>O,

0-
(15)

Now let us consider the specification of the mobility function f(~) given by equation (12). Then using the monotonicity of both simple shear and simple elongation flow curves as the necessary condition of dissipative stability, we can find the constraint imposed on parameter n in equation (12): 0 < n < 1.

(16)

Kinetic formulation of elasto-viscoplasticity 4. S E C O N D - O R D E R

APPROACH

IN KINETIC

1039

ELASTO-VISCOPLASTICITY

4.1 General formulation L e t us consider the incompressible case. Using the relation b e t w e e n the Finger and H e n c k y tensors, ¢ = exp(2h), we e m p l o y the expansions h = ehl + e2h2 + . . . .

e = fi + 2ehl + 2 e 2 ( h 2 + h2) + . . . .

c -1 = fi - 2ehl + 222(h 2 + h2) + . . . .

(17)

with a small positive p a r a m e t e r e. T h e n the expansions for invariants Ik due to e q u a t i o n (17) are of the form: Ii = tr c = 3 + 22 tr hi + 2e2[tr(h 2) + tr h2] + " • • 12 = tr c - i = 3 - 22 tr ht + 2e2[tr(h 2) + tr h2] + • • • /3 = 1 + 2e tr hi + 2e2[(tr ht) 2 + tr h2] + . . . .

1.

T h e third f o r m u l a here shows that tr hi = 0,

tr h2 = 0.

(18)

T h e r e f o r e the expressions for It and 12 will take the form: It ~ 12 = 3 + 2e2(tr ht) 2 + . . . .

(19)

E q u a t i o n s (17)-(19) are consistent with evolution e q u a t i o n (3a) and kinetic e q u a t i o n (4), if 0 = o(e),

Zc = O ( e ) .

(20)

This m e a n s that the m e d i u m u n d e r consideration is "rigid" enough. T h e r e f o r e in the following we will simply use e = zc. Substituting e q u a t i o n (19) and (20) into e q u a t i o n (3) and equations (7), (10) and (11) yields for the t e r m s of O(e): 0rhl + f(sC)hn = Oe/Zc

(~ = t/O),

(21)

w h e r e e is the strain rate tensor, and for the t e r m s of O(22): c9h2 + f(¢)h2 = [ - ( v . V)h, + h i " Vv + (Vv)* • h,]O/zc - cgh~ - 2f(~:)h 2.

(22)

T h e last equation can be simplified using e q u a t i o n (21), to the form: 0

0

0r(h2 -- hi) + f(g)h2 + ht = 0

(hi = ht + to. hi - h , . to),

(23)

0

w h e r e hi is the J a u m a n n tensor time derivative of tensor hi and to is the vorticity tensor. With the s a m e precision, the kinetic e q u a t i o n (4) takes the form:

roof + ~ = (1 - ¢)EO/zc

(r = 01/0).

(24)

Finally, using e q u a t i o n (9) with the simplifying assumptions k = 0, /3 = 0, we can rewrite e q u a t i o n (4) holding the first- and s e c o n d - o r d e r terms, as follows: or = - p 8 + 2Gz¢[ht + z~(h~ + h2)].

(25)

E q u a t i o n s ( 2 1 ) - ( 2 5 ) r e p r e s e n t the closed set of equations for the s e c o n d - o r d e r elastoviscoplasticity. T h e initial conditions for the set, related to the initial rest state, are: hl[~=o = 0,

h2l~=o = 0,

¢1~=o = 0.

N o w we consider e x a m p l e s of this f o r m u l a t i o n for two types of simple d e f o r m a t i o n . E$ 34-9-C

(26)

1040

A.I. LEONOV and J. PADOVAN

4.2 Simple extension

In this example, the deformation is a homogeneous extension of a bar with circular cross-section. The kinematic matrices are: Vv = e = ~ • diag{1, - 1 / 2 , -1/2}, h, = h, diag{1, - 1 / 2 , -1/2}

to = 0, (~ ->0).

(27)

Therefore, due to equation (4), E = kX/-3. Because in this type of deformation, (v- V)h, = 0, equation (23) with initial condition (26) gives: h2 = 0. Also, the component h~(T) satisfies the equation: Orh, + f ( ~ ) h , = ~O/z¢.

(28)

The kinetic equation (24) is of the form: r O ~ + ~ = V3 (1 - ~)~O/z~.

(29)

The following expression for the total extensional stress 0- is obtained by using the zero stress condition on the free surface of the bar: (30)

o" = 3Gzch,(1 + zch,/2). 4.3 Simple shear

The deformation in this example is the homogeneous shearing between two infinite parallel plates. The kinematic matrices are of two-dimensional form:

;]

4/

15

4/

=[hi.,,

h,,12],

(31)

Here hi (i = 1, 2) are the traceless first- and second-order elastic Hencky strain tensors whose time evolution is described by equations (21) and (23). Using these equations one can obtain the following evolution equations for non-vanishing components of these tensors: Orh + f ( ~ ) h = r / z ,

(h = h,.,2; h,.,, = 0),

o~/~ +f(£)/~ = F h ,

F = 4/O/z~

(32)

(/~ = h2.,,; h2.,2 =0).

(33)

The kinetic equation (24) takes the form: rO~£ + £ = (1 - £) IFI.

(34)

Due to equations (25), (32) and (33), the components of stress tensor are represented as follows: 0-,2 = 2Gzch,

o',l = - p + 2Gz~(h 2 + fz),

0-22 = - P + 2Gz¢(h z - h ),

0"33 =

-p.

They are rewritten in the more compact dimensionless form: 0-]~ = 2h, 6"12----Gzc

/¢/t =

o'ii - 0-22 2/z, 2Gz 2

/~2 - -

0"22 --

h 2 - nr.

0"33

2Gz2~

(35)

-

Here N, and Nz are the first and second normal stress differences. Equation (35) demonstrates that the occurrence of normal stresses in simple shearing is a second-order effect. The set of equations (32)-(35), along with the specification of the mobility function f ( £ ) given by equation (12), is closed. We now illustrate some basic features of the model by analyzing regimes of simple shear deformation, using equations (32)-(35) and (12). 4.3.1 Start up shearing flow. In this case, r = roll(r)

(ro > o)

(36)

Kinetic formulation of elasto-viscoplasticity

1041

where H ( r ) is the Heaviside step function. To obtain the expressions for stresses (35) we need to solve the set of equations (32)-(34) with the initial conditions: hlt=o = 0,

/~l~=,, = 0,

~1~=o = 0.

(37)

The solution of equation (34) for bonding factor Fo exp[-(1 = 1 + ro {1 -

+ ro)r/r]}

(38)

shows that in this type of loading, the process of debonding occurs which increases progressively from zero to the steady value Fo/(1 + Fo). The solutions of equations (32) and (33) were found numerically for two values of parameter Fo, 0.2 and 5. Parameter r was taken equal to 10. To illustrate the effect of parameter n in equation (12) for the mobility function f(~), two values of n, equal to 0.2 and 0.7, were considered. The results of calculations for shear stress o'12 are represented in dimensionless form in left parts (loading) of Fig. 1 (a,b). It is seen that the stress overshoots occur at high values of Fo with increasing intensity of peaks, shifting them to the smaller times with increasing Fo. It should be mentioned that the stress overshoots in Fig. 1 are too high only because in our demonstration we considered the example 0 = const. It can also be seen that the effect of parameter n is noticeable only for steady plastic shearing flow. Both normal stress components haw; qualitatively similar behavior, but their dimensional values are considerably less than those for shear stress in the second-order approach. 4.3.2 Steady plastic shearing flow. In the steady plastic shearing flow (F > 0),

r

nr/2

= 1 +-----F'

h - (1 + F)" - 1'

nr2/2 h - [(1 + F)" - 1]2

(39)

and the dimensionless stresses are of the form: nF a,2 - (1 + F)" - 1'

nF 2 F¢, = [(1 + F)" - 1]z'

~/2 = a212/4 - F/,/2.

(40a)

The asymptotic formulae for small and large values of F are: ~ , = [1 + (1 - n)F]/n + . . . .

#,2 = 1 + (1 - n)F/2 + . . . .

/~/z = - [ 1 + (1 - n)r](2 - n)/(4n) + ....

(F << 1)

(40b)

and

~12 = n r ' - " ,

R, = n r 2-2",

N2 ~" - n ( 2 - n)/4~/l

([" >> 1).

(40c)

At F--->0, equation (40b) demonstrates the yield values for the shear and normal stresses:

Y12 = 2Gz¢,

111= (2/n)Gz~,

Y2 = - ( 2 - n)/(2n)Gz~.

(41)

It is also remarkable that near the yield, the dependencies of shear and normal stresses on shear rate have linear (Bingham) behavior. Equation (40c) also demonstrates that the parameter ( 1 - n) plays the role of index of pseudoplasticity: the more n grows the more pseudoplasticity (shear thinning) is induced in plastic flow. 4.3.3 Relaxation after cessation o f steady plastic shearing flow. In this case,

r(r) =

roll(to

- r),

(42)

1042

A . I . LEONOV and J. PADOVAN

6

3

1 -\

Loading

~

Relaxation

2

t

I

I

I

10

20

30

40

50

60

t

4

3

---...t'

~,.~RelaxaUon

Loading

S-

2

I

I

I

I

I

10

20

30

40

50

60

Fig. 1. The time dependencies of dimensionless shear stress try2 in start up flows (left-hand side branches of curves) and relaxation after cessation of flow (right-hand side branches of curves) for different values of dimensionless shear rate F: (1) F = 0.2; (2) F = 5. (a) n = 0.2; (b) n = 0.7.

w i t h i n i t i a l v a l u e s so0, h~) a n d /7o d e p e n d i n g o n Fo as d e s c r i b e d b y e q u a t i o n (39). T h e n t h e s o l u t i o n s of e q u a t i o n s ( 3 2 ) - ( 3 4 ) , ~:(r) = ~,, e x p [ - ( r

- ro)/r],

[ /rlf('(t))dt]

{ h ( r ) , / i (r)} = {ho, aro}exp -

,

(43)

s h o w t h a t r e b o n d i n g o c c u r s u n d e r this t y p e o f u n l o a d i n g w i t h t h e t e n d e n c y ~ ~ 0 at r ~ ~ . I n

Kinetic formulation of elasto-viscoplasticity

1043

the case F<< 1, J'(~)~, and formulae for h(z) and h(z) in the relaxation regime have the explicit expressions: h(v) ~ 1/211+ ( 1 - n ) F / 2 ] e x p { - r r [ l - e x p (

v r Vo)]}

~(z)~- l/(2n)[l + (1- n)F]exp{-rF[1-exp(- V- v°)]}

(44)

Equation (44) along with (35) distinctly shows the appearance of residual stresses at z---> oo. The simple analysis also shows that the residual stresses decrease when the value of F is increasing. The numerical examples of relaxation are demonstrated in the right branches of Fig. l(a,b).

4.3.4 Creep in simple shear Being mostly interested in the qualitative demonstrations of kinetic effects in elastoviscoplasticity, we consider in this section only the infinitesimal approach. Hence we ignore the normal stresses. Then equation (32) can be rewritten in terms of the dimensionless shear stress o- (o- = 6q2) represented in equation (35), along with the kinetic equation (34) for active loading (F-> 0), to yield the following:

do-~dr + f(~)o-

= F;

r d~/dz +

~ = (1 - ~)r.

(45)

Let us consider the regime of deformation when a(v) is given. If o-(z) is varied slowly enough, there always exists the quasistatic solution: = 0,

F = 0,

h = o-/2.

(46)

On the other hand, if the value of o- is high enough and o-(v) is still varied slowly, there is a quasistationary solution for the creep plastic shearing flow represented by the first formula in equation (40a). I:a order to clarify this behavior we consider below the typical regime of creep deformation, o-(z) = o-oH(v),

F ( - 0 ) = 0,

~ ( - 0 ) = 0,

(47)

where o-o is constant. Equation (45) immediately yields: F(z) = o-o[8(v) + f(~)],

~o = ~(+0) = 1 - exp(-o-o/r).

(48)

The second expression in equation (48) shows that there is an instantaneous debonding caused by application of instantaneous stress. The evolution ~(z) at z > 0 is described by the second equation (45) represented as: r d~/dz = (1 - £)f(~:)[o-o - N(£)];

N(£) - (1 - s~)f(s~)"

(49)

Here the multiplier ( 1 - £)f(£) is always positive for 0 < £ < 1, and the function N(£) is monotonically increasing. The physical case occurs only when o-o/r<< 1. Then, £o --~ o-o/r and N(£0)~ 1. We now consider the limit case of equation (49) when z ~ +0: r d~/drl+0 ~ (1 - ~0)f(~0)[o-0 - 1].

(50)

If o-o < 1 (o-~2< YJ2 = Gz¢), d~/dv[+o < 0, and due to equation (49), ~(z) decreases monotonically to zero. This means that initial debonding in the material will be "healing" and it returns eventually to a solid-like static equilibrium. It should also be mentioned that because of the initial debonding on the material, the final solid-like state will be accompanied by irreversible deformations. If o-o > 1, dg/dvl+o > 0, and due to equation (49), g(z) increases monotonically to the value described by forraulae (39) and (40) for the steady flow regime.

1044

A . I . L E O N O V and J. P A D O V A N 0.40

4 0.35 0.30 0.25 ~u, 0.20 0.15 0.10

0.05 1,3 0.00

I

I

I

5O

100

150

200

t Fig. 2. The plots of bonding factor ~: vs dimensionless time r in creep for different values of parameter n and constant normalized shear stress fro: (1) ~ro = 0.8, n = 0.7; (2) tro = 1.2, n = 0.7; (3) tro = 0.8, n = 0.7; (4) o-o = 1.2, n = 0.2.

This

behavior

behavior

shows

distinctly

to plastic flow through

Numerical

illustrations

s h e a r s t r a i n ¢/(~) =

that

of the creep

3'(r)/zc

where

the

model

predicts

the

transition

from

a solid-like

a bifurcation. behavior

for bonding

factor

y = f~ F(t) dt are represented

~(z) and

in dimensionless

the normalized form in Figs 2

5 P- 4

3 2

1,3

1 0

I

I

I

I

50

100

150

200

t Fig. 3. The dimensionless time dependencies of normalized shear strain ¢/= y/zc in creep for different values of parameter n and constant normalized shear stress o'o: (1) ~o=0.8, n =0.7; (2) tro= 1.2, n = 0.7; (3) tro = 0.8, n = 0.7; (4) tro = 1.2, n = 0.2.

Kinetic formulation of elasto-viscoplasticity

1045

and 3. Here curves 1 and 3 in both the figures are attributed to o'0 = 0.8, i.e. the value of shear stress below the dimensionless yield value equal to one. One can see that in this case, the curves for n = 0.2 and n = 0.7 are indistinguishable. It is seen that the curves 1 and 3 in Fig. 2 demonstrate the behavior of g(z) corresponding to rebonding (healing) after instantaneous debonding, with the tendency ~:--+0 at z---> ~. The curves 1 and 3 in Fig. 3 demonstrate the plot ~,(z) corresponding to the solid-like behavior. It is not ideal, however, because of accumulating irreversible strain during initial debonding. Curves 2 and 4 in both Figs 2 and 3, which correspond to n = 0.7 and n = 0.2, respectively, are attributed to the value or = 1.2, i.e. the value of shear stress above the dimensionless yield value equal to one. Curves 2 and 4 in Fig. 2 illustrate the behavior of ~(z) which corresponds to the debonding p:rocess in creep. Here function ~(z) increases monotonically, with ~--->~(~) as r--* oo; ~(~) corresponds to the steady plastic shearing flow under the creep condition. One can see that in the case n = 0.2 (curve 4), more rapid debonding kinetic occur in comparison with the case n = 0.7 (curve 2). This produces more rapid kinetics of strain shown in Fig. 3, for the case n = 0.2 (curve 4) as compared with the case n = 0.7 (curve 2). 5. C O N C L U S I O N S

(1) In this paper, a general kinetic Eulerian approach to the formulation of elastoviscoplasticity was proposed. It consists of non-linear Maxwell type viscoelastic constitutive equation for evolution of the recoverable (elastic) part of the total Finger strain tensor and the hyperelastic relation between the elastic strain and stress tensor. The tensorial evolution equation was coupled with a kinetic equation which describes the evolution of a scalar bonding factor. (2) Some properties of several structural functions were analyzed and suggestions on how to specify them are given. (3) The formulation does not employ the mathematical yield criterion but predicts anisotropic yield values for stress as a transition from solid-like deformations to flow through bifurcation. This gives this approach a computational advantage over those which employ the algebraic yield criteria. Thus the formulation represents a general scheme which unifies pure viscoelastic approaches to the elasto-viscoplasticity without description of yield phenomena, and standard viscoplastic models. (4) Two types of stability constraints imposed on the formulation were analyzed. One was related to well-posedness (Hadamard stability) of the formualtion. Another was related to the possible type of constitutive instability caused by improper formulation of dissipative terms in constitutive equations. It was shown that the proposed formulation is globally well-posed. It was also noted that all elasto-viscoplastic formulations which employ Jaumann (co-rotational) time derivatives are ill-posed. Some remarks about dissipative stability for the present formulation are made. (5) In order to reveal some features of the formulation a second-order approach was also developed which has been tested before for some dispersed systems. Several problems were solved for simple shear to illustrate the model behavior. These included start up and steady plastic shearing flow, stress relaxation and creep. The most important effects demonstrated are stress overshoots in start up flow, shear thinning in steady flow, debonding under loading and rebonding after unloading and bifurcation transition from solid-like behavior to flow in creep. Acknowledgement--ldany thanks are due to S. Acharia for numerical illustrations. REFERENCES [1] K. V. VALANIS, Archs. Mech. 23, 517 (1971). [2] K. C. VALANIS, Archs. Mech. 27, 857 (1975).

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A . I . LEONOV and J. PADOVAN

[3] S. K. GODUNOV, Elements o f Continuum Mechanics, pp. 117-174. Nauka, Moscow (1978) (in Russian). [4] C. ECKART, Phys. Rev. 73, 373 (1948). [5] K. KONDO, Non-Riemannian Geometry o f Imperfect Crystals From a Macroscopic Viewpoint. Memoirs o f the Unifying Study o f the Basic Problems in Engineering Sciences by Means o f Geometry (Edited by K. KONDO), p. 1. Gakujutsu Bunken Fukyu-kai, Tokyo (1955). [6] B. A. BILBY, R. BULLOUGH and E. SMITH, Proc. R. Soc. Lond. A231, 263 (1955). [7] E. KRONER, Arch. Rat. Mech. Anal. 4, 273 (1960). [8] J. F. BESSELING, A Thermodynamic Approach to Rheology. Irreversible Aspects o f Continuum Mechanics and Transfers o f Physical Character in Moving Fluids (Edited by H. PARKUS and L. I. SEDOV), IUTAM Symposia, Vienna, 22-28 June (1966). [9] E. H. LEE and D. T. LIU, Finite Strain Elastic-Plastic Theory (Edited by H. PARKUS and L. I. SEDOV), IUTAM Symposia, Vienna, 22-28 June (1966). [10] S. R. BODNER and Y. PARTOM, J. Appl. Mech. 42, 385 (1975). [11] M. B. RUBIN, Int. J. Engng Sci. 24, 1083 (1986). [12] M. B. RUBIN, Int. L Engng Sci. 25, 1175 (1987). [13] A. I. LEONOV, Rheol. Acta 15, 85 (1976). [14] A. I. LEONOV, Ann. New York Acad. Sci. 410, 23 (1983). [15] A. I. LEONOV, J. Non-Newtonian Fluid Mech. 25, 1-59 (1987). [16] M. B. RUBIN and A. L. YARIN, J. Non-Newtonian Fluid Mech. 50, 79 (1993). [17] P. M. NAGHDI, J. Appl. Math. Phys. (ZAMP) 41, 316 (1990). [18] A. |. LEONOV, J. Rheol. 34, 1039 (1990). [19] P. COUSSOT, A. I. LEONOV and J. M. PIAU, J. Non-Newtonian Fluid Mech. 46, 179 (1993). [20] D. D. JOSEPH, Fluid Dynamics o f Viscoelastic Liquids, pp. 127-163. Springer, New York (1990). [21] A. 1. LEONOV, J. Non-Newtonian Fluid Mech. 42, 323 (1992). [22] Y. KWON and A. I. LEONOV, J. Non°Newtonian Fluid Mech. 58, 25 (1995). [23] M. V. SIMHAMBHATIA and A. I. LEONOV, Rheol. Acta 34, 259 (1995). [24] C. TRUESDELL and W. NOLL, The Non-Linear Field Theories o f Mechanics. Springer, New York (1992). [25] J. K. KNOWLES and E. STERNBERG, Arch. Rat. Mech. Anal. 63, 321 (1983). [26] L. ZEE and E. STERNBERG, Arch. Rat. Mech. Anal. 83, 53 (1983). [27] M. RENARDY, Arch. Rat. Mech. Anal. 88, 83 (1985). [28] Y. KWON and A. I. LEONOV, J. Rheol. 36, 1515 (1992). (Received 9 May 1995)