stability of flow processes for dissipative solids with internal imperfections

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/84/006848-20 $ 5.50/0 Vol. 35, November 1984 �9 Birkh/iuser Verlag Basel, 1984

Stability of flow processes for dissipative solids with internal imperfections

By Piotr Perzyna, Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland

1. Introduction

The main objective of the paper is the description of the postcritical behaviour of inelastic solids.

To describe the postcritical behaviour for dissipative solids, we need very precise constitutive modelling and we must take into consideration important cooperative effects such as internal imperfections, strain rate sensitivity and thermal effects.

In this paper the formulation of a quasi-static, isothermal flow process for a bounded body is given. Criteria of localization of plastic deformations are considered.

The constitutive modelling is developed within the framework of the rate type material structure with internal state variables.

The evolution of internal imperfections generated by the nucleation and growth of voids is treated as the main phenomenon intervening with plastic deformation process.

A simple model of an elastic-viscoplastic solid with internal imperfections is proposed. This model is justified by physical mechanisms of polycrystalline matter flow in some regions of temperature and strain rate changes. The thermally-activated mechanism causes elastic-viscoplastic flow, and the mechanism of void nucleation and growth plus its diffusion field lead to the evolution of internal imperfections in the solid considered.

The model proposed has some important features. First, by introducing the control function (control vector) the model can describe the properties of a material in a range of strain rates near the static value. This model satisfies also the requirement that during the deformation process in which the effective strain rate is equal to the static value the response of a material becomes elastic-plastic. Second, it can describe the evolution of imperfections during the deformation process by taking account of the nucleation of voids as well as the growth of voids. Third, it includes in the description the diffusional accommodated flow mechanism.

Vol. 35, 1984. Stability of flow processes for dissipative solids 849

The identification procedure for all material functions and constants has been based on available experimental data. Two kinds of experimental tests have been used. First, the mechanical test data for a broad range of strain rate changes are utilized to determine material functions and constants in the evolution equations for the inelastic deformation tensor. Second, the physical, metallurgical observations are assumed as a basis for determining material functions and constants in the evolution equation for the concentration of imperfections.

As an example of a quasi-static, isothermal flow process the boundary-initial- value problem describing the necking phenomenon has been considered. The problem is formulated in such a way that enables discussion of influence of the strain rate effect, as well as of imperfections and diffusion effects on the onset of localization. Comparison of theoretical predictions with available experimental results is given.

2. Quasi-static, isothermal flow process for dissipative solids

To describe the flow process for dissipative solids with internal imperfections induced by the nucleation and growth of voids in the material, we utilize the modified material structure with internal state variables. We shall postulate that the evolution equations for some of the internal state variables have the form of partial differential equations. This material structure is sufficently general to include such effects like strain rate sensitivity and diffusion processes for kinetics of voids (imperfections).

So, the quasi-static, isothermal flow process for dissipative solids with internal imperfections will be determined in the material description by

(i) the constitutive equation for the Piola-Kirchhoff stress tensor

7-= 7'(G)

where o- denotes the intrinsic state which is given by the pair - the strain tensor field E and the field of the internal state vector ~, i.e.

cr = (E, a) e Z (2.2)

and X denotes the intrinsic state space. Basing on the previous results (of. Rfs. [22-25]) we can write

T = 2 00 ~E 50 (o-*) (2.3)

where 50 denotes the free energy constitutive function and ~o 0 is the mass density in the reference configuration:

The intrinsic state a* = (E, ~1) does depend on the internal state variables �9 1 which are described by ordinary differential evolution equations and it is independent of the internal state variables ~2 which are described by partial

850 P. Perzyna ZAMP

differential evolution equations. This result follows from a thermodynamic re- striction (cf. Ref. [7]) and has very important consequences on the constitutive modelling;

(ii) the evolution equation for the internal state variable vector ~ in the form (cf. Ref. [22, 23])

S t ~ (X, t) = Aa a (X, t) + f(o'), (2.4)

where 5e is a linear spatial differential operator, f is a nonlinear function of cr and 8~ denotes the differentiation with respect to time;

(iii) the equation for the strain tensor E, i.e.

E = �89 + H r + n H r ) , (2.5)

where H = Vx u denotes the displacement gradient;

(iv) the equilibrium equation

Div (F T) = 0, (2.6)

where F is the deformaion gradient field;

(v) the initial values

u (X, 0) = 0 , v (X, 0) = v ~ (X), �9 (X, 0) = ~o (X) (2.7)

for X ~ ~ , where v = a t u is the velocity vector field;

(vi) the boundary conditions

u (X, t) -- u l (X, t), a ~ . �9 (X, t) + b �9 (X, t) = 0 (2.8)

for (X, t) 6 ~ • [0, de], where n is the unit outward normal vector on ~ , ul, a and b are bounded functions on ~ x [0, de].

A solution q~ = {u, ~} of the problem of a quasi-static, isothermal flow process is a set of functions u and ~ which satisfy the equations (2.1)-(2.6) with the initial-boundary-value conditions (2.7)-(2.8).

3. Criteria o f local ization o f plastic deformat ions

The criteria of the onset of instability of the flow process for solids can be considered from two point of views. The first is purely mathematical in nature and can be achieved within by investigating discontinuous 1) solutions or bifurcation 2) (branching) of the solution. The second is a more engineering

1) The criteria of discontinuous solution for a flow process have been discussed basing on the first and second Liapunov methods in Ref. [24]. z) The criteria of bifurcation have been broadly investigated by Hill [9 11], cf. also the review papers by Hill [12], Hill and Hutchinson [13], Miles [16] and Asaro [2].

Vol. 35, 1984 Stability of flow processes for dissipative solids 851

approach and is based on the investigation of the load-extension curve during a flow process.

During plastic flow processes for dissipative solids the onset of instability is usually connected with the localization of plastic deformations (cf. Rice [28] and Ref. [23]).

The most widely recognized mode of localization of plastic deformations is that of necking. A necessary condition for the occurrence of necking are the fluctuations in the cross-sectional area. This leads to a maximum load criterion for the instability condition (cf. Chakrabarti and Spretnak [4]).

A second basic mode of localization is the phenomenon of plastic instability in direction of pure shear. A necessary condition for the localization of plastic deformations in the form of the plastic shear band is a maximum true flow stress criterion 3).

In this paper we would like to study the necking phenomenon for dissipative solids with internal imperfections and to use the criterion of maximum load as a fundamental condition for the onset of instability.

It is noteworthy that experimental as well as theoretical studies of necking phenomena showed that some additional effects can influence the onset of instability by shifting the point of initiation of localization from the maximum load point on the load-extension curve4).

Needleman and Rice [18] have proved that the onset of localization does depend critically on the assumed constitutive law. This means that the onset of instability is influenced not only by the geometry and boundary conditions of the body considered but also by the properties of the material.

In this study we focus mostly on the description of the postcritical behaviour of solids, so that the influence of different effects and material properties on the criterion of localization needs further investigation.

4. Elastic-viscoplastic solids with internal imperfections

4.1 Physical motivation for the concept of internal imperfections

In the constitutive modelling for the postcritical behaviour of solids we have to take into consideration the internal imperfections generated by the nucleation and growth of voids. As it has been pointed out by Le Roy, Embury, Edward and Ashby [14] ductile fracture is the end of a sequence of three processes:

3) Experimental observations of this criterion were conducted by Chakrabarti and Spretnak [4]. For recent investigations in this subject see the review papers by Peirce, Asaro and Needleman [19] and Asaro [21. 4) For discussion of these effects see the review papers by Peirce, Asaro and Needleman [19] and Asaro [21.

852 P. Perzyna ZAMP

(i) Nucleation of voids at second phase particles; (ii) Growth of voids mostly due to plastic deformations and yield stress state and due to the diffusional accommodated flow; (iii) Linkage of voids. It is assumed that voids coalesce when the void length reaches some multiple of its distance from the neighbour- ing void.

From physical metallurgical observations during tensile tests, it is clear that the postcritical behaviour of ductile solids is controlled by several phenomena. We can recognize some characteristic features for all of them.

The most essential are the nucleation and growth of voids. The behaviour of the bulk specimen in tensile test is influenced by evolution of voids during the deformation process.

Temperature and strain rate sensitivity of the material is also observed as very important (cf. Wray [30]). For very low strain rate the thermally-activated process can be assumed as responsible for dislocation creep flow.

For intergranular creep-controlled flow and for pure diffusional mechanisms the transport phenomena contribute to void growth. The transport phenomenon is usually approximated by the diffusion mechanism s).

In void nucleation processes particles and inclusions (inhomogeneities in polycrystalline materials) are very important. Particle fracture and interracial decohesion result as a consequence of the local state of deformation which exists in the vicinity of void inhomogeneities. Existing models for void nucleation are generally grouped into one of three categories based upon either an energy criterion, a local stress criterion or a local strain criterion is postulated (cf. Fisher [6]).

Once nucleated, the voids grow mainly in the tensile direction and only in the late stages of necking.

Basing on careful examination of the void nucleation, the void growth and the accumulation of damage during tensile loading and the subsequent necking process and taking advantage of physical suggestions, we can propose a proper evolution equation for the scalar measure of the concentration of imperfections.

4.2 Rate type material structure

We shall now use a different method than that proposed in Section 2 to formulate the equations which describe the elastic-viscoplastic response of solids with internal imperfections. We would like to make use of the rate type material structure in Eulerian description. The advantages of the rate type formulation are sound when the constitutive structure is applied to the solution of the initial-boundary-value problem.

s) See the informations given by Ashby and Verrall [3].


Let us denote the symmetric rate of deformation tensor by D and postulate (cf. Refs. [20, 21]).

D ~ = D - D P,

v 1 D e = _ ~ tr~}l (4.1) 2G l + v '

D e - 7o I ~ I f ( J 1 , J l z , J ~ , ~ )

where ~ denotes the symmetric Zaremba-Jaumann rate of change of the Cauchy stress tensor a, G is the shear modulus, v is the Poisson ratio, 7o is the viscosity constant, q~ is interpreted as a control function and is assumed to depend on (I2/I~) - 1, I 2 = (I lo) 1/2 is the second invariant of the rate of deformation tensor, I~ = (I Ios) 1/2, D ~ is the static rate of deformation tensor, ~ denotes the overstress viscoplastic function, f is the quasi-static yield function and is assumed to have the form

f = f 1 1 (J1, J2, J3, ~), (4.2)

by J1 we denote the first invariant of the Cauchy stress tensor ~r, J~ and j1 the second and third invariants of the stress deviator, respectively, z plays the role of a material function and is postulated as

= ( 4 . 3 )

denotes the imperfection internal state variable which is interpreted as the void volume fraction parameter, and the symbol ([]) is understood according to the definition

{~ if f < z , (4.4) ([1) = if f > z.

It is noteworthy that the material function ~ (~) describes the softening effect induced by internal imperfections.

As a result of Eqs. (4.1) we have the evolution equation for the Cauchy stress tensor o in the form

- - - t r # ! = D - - - ~ ' - 1 ~ o f . (4 .5 ) 2G l + v ~0

In addition, we need the evolution equation for the imperfection internal state variable r

For the interpretation of the internal state variable ~ as the void volume fraction parameter we have to consider the voided solids (porous solids), cf. Gurson [8], Needleman and Rice [18] and Saje, Pan and Needleman [29].

854 P. Perzyna ZAMP

Our aim is to describe the evolution of imperfections generated by the nucleation and growth of voids and to take into consideration the transport phenomena suggested by the physical mechanisms. To do this, we postulate the evolution equation for the imperfection parameter ~ as follows

at ~ = (at ~)diffusion + (at ~)nucleation + (at ~)growth" (4.6)

The diffusional term is assumed in the form

(at r = Do W 2 r (x, t), (4.7)

where D O is the diffusion constant. To fix precisely the diffusion constant D O for a particular material, we can use the expression

D(~) = constexp - ~ ,

where Up is the activation energy of the diffusion process assumed, N denotes the gas constant and 0 temperature. By proper selection of the diffusion process 6) we can determine the activation energy U D as a function of temperature. For an isothermal process we can write

D O = D (0)f~=ao- (4.9)

The nucleation term is postulated in the form (cf. Needleman and Rice [18] and Saje, Pan and Needleman [29])

(at ~)nucleation = A ~ + I J1, (4.10)

where A and 1 are the material functions and ~ denotes the tensile flow strength of the matrix material. Calling further ge the matrix equivalent plastic tensile strain any h-= d6/dg e the equivalent tensile hardening rate of the matrix material and using the equivalence of the plastic power (cf. Refs. [18, 291)

(1 - ~ ) ~ e = tr(aDe), (4.11)

we obtain

- - - tr (a De). (4.12) (1 - 4 )

Taking advantage of the last result in Eq. (4.10) we have

h (at ~)nucleation = 1 ~ t r (o 'D e) + l i t , (4.13)

where h = A h/tY denotes the nucleation material function.

6) For a broad discussion of diffusion process in solids see Flynn [5].


The growth term is postulated

(0t ~)growth = (1 -- ~) Z tr (De), (4.14)

where E is the growth material function. Eq. (4.14) suggests that the growth of voids is mostly due to inelastic defor-

mations. The evolution equation for the void volume fraction parameter can be

written in the following form

h ~t~ = D o V 2 ~ + i ~ t r (aDe) + I J1 + (1 - ~)Ztr(DP). (4.15)

We shall consider the following particular form of the yield function for voided material 7)

1 J ~ (4.16) f(.) = J~ + n ~ j~j ,

where n denotes the material constant. The material softening function 2 is postulated

z = • 2 (1 - ~llZ), (4.17)

where z o denotes the yield constant of the material. The last two assumptions yield to the following evolution equation for the

inelastic deformation tensor

D e='~~ [-Jl+n~J2 l l ) S+2n~J~ l (4.18)

where S denotes the deviator of the Cauchy stress tensor ~.

4.3 Determination of the material functions and constants

To determine the material functions and constants appearing in the evolution equation for the inelastic deformation tensor, we shall use the available experimental data obtained in dynamical tests. As it has been shown in Ref. [21] the determination of the overstress material function O, the control function q0 and the material constants can be based on both combined and one-dimensional loading tests.

We shall use the procedure of identification developed for the elastic- perfectly viscoplastic material in Ref. [21] and generalized to the elastic-work hardening viscoplastic material in Ref. [27].

Let us show this procedure for the dynamical tests under combined loading. We shall use experimental data obtained by Lindhotm [15] for mild steel.

v) A thorough discussion of different particular forms of the yield function for voided solids is given in Ref. [26].

856 P. Perzyna ZAMP

Figure 1

1

(Ii~) ~ (KSI)

6O

5O

4O

3O

2O

10

MILD STEEL (1018)

�9 PURE TENSION ) EXPERIMENTAL DATA

PURE TORSION | LINDHOLM (1965) !

COMBINED LOADING)

~/o : 25.82374 ~-~

i i i �9 i i I l 1 i ~

10 4 10-s 10-4 10-3 10-2 10-1 100 101 102(1101 {- (s-l)

From these data we have the experimental curves

(I Is) ~/2 = ff (Iz)[(Sl,),/2 = oonst, (4.19)

where e denotes the Cauchy strain tensor. Using as an opt imum criterion for the best fitting of experimental data by

theoretical results the functional

J = max II J~ (t) - ~ (I2) 1(s~)1/2 : c o n s t II, (4.20) te[O,d/~l

where

J~(t) xo{ l+cP-aV (tr(D~(t))z)l/2 ( 12 ) 1 } = 9 - 1 (4.21) L 70 ~ '

we obtain results for mild steel (plotted in Fig. 1). To determine the material functions in the evolution equation for the imper-

fection parameter ~ (see Eq. (4.15)), we shall use Fisher's metallurgical obser- vation data [6]. The material functions h, S and I are obtained by using the best curve - fitting method (as it is shown in Fig. 2).

5. E l a s t i c - p l a s t i c r e s p o n s e

The elastic-plastic response of a material is reached in the limit case, when 12 = I s. Then we have the evolution equations

D p = AO~,f(.), h

- tr(o 'D') + IJl + (1 - O~t r (DP) , (5.1) 1 - 4


o B1 B2

e, B3 �9 W1 A �9 W2 ~ % 0.03 .W3 t ~ /

o#, &k Igl

0.01[ �9 ,~'m

I �9 la i~

Figure 2 0.~ 0.~ 0.8 ~ 1.2 1.4 G

for

f(-) = x and tr(8~f#) > 0. (5.2)

It is noteworthy that the diffusion effect has been neglected. The parameter A can be determined from the condition

f = ~ (5.3)

The evolution equation for the plastic deformation in index notation can be written in the form (cf. Refs. [18, 29])

v

D~ = ~ PU Qk, akl, (5.4)

where

ef Pu - ~%, l ~ f

�9 4- I bkt,

1[ ~f H- 4J~ ( 1 - ~ ) ~ 6 u + - 1 - ~ i A \ e ~ ~ "

(5.5)

F o r

f(.) = J~ + n ~ j ~ j and x = zg(1 - ~1/2), (5.6)

we have

Sq fl Ski # P~j - ~ = - - + 6kl, 2 T e 4- ~ij, Qkl 2 z~ (5.7)

858 P. Perzyna ZAMP

Figure 3

1,00

0,90

0,80

0,70 O,6O

0,49,

0,4C

0,30

0,20

0,10

~=o

138 1,87 2,67 3,46 4,52 5,31 6,11 9,40 J~ ~o

PRESENT PROPOSITION ) n = 2.10

- - - - - - 6URSON'S SOLUTION

where

"Ce = N / / ~

3n~Jl

1 2 l(nJ 2 + 2Zo~ -1/2) it

2 x/Xo 2 (1 - 41/2) - n ~ j 2 '

H - 4"c 2 ( 1 - ~ ) S 5 u + ~u

(5.8)

It is noteworthy that the normality does not apply because # 4=//. If I = 0 then/ t =/~ and normality applies.

The comparison of the Gurson solution (cf. Ref. [8]) with the present proposition is given in Fig. 3.

6. Necking process

6.1 Formulation of the initial-boundary-value problem

Let us study the tensile deformation of a circular cylindrical bar of initial length 2 L 0 and initial radius Ro, cf. Fig. 4. The problem is described in the cylindrical coordinates r, 0, z. It has been assumed that the problem is axisym- metric and additionally that the deformations are symmetric about the mid plane z = 0. The ends of the specimen are assumed to remain shear free.

Vol. 35, t984 Stability of flow processes for dissipative solids 859

Iz

, r -- F- AO LO Ro

r free U end s

INITIAL-BOUNDARY VALUE PROBLEM

ELASTIC-VISCOPLASTIC FLOW PROCESS}

r

: L ~ 0

U

A=A(z,t)

_ ~ R=R(z,t]

Figure 4 INITIAL STATE ACTUAL STATE

We use the Eulerian spatial description and the rate type constitutive modelling.

For the yield function (4.16) and the material softening function (4.17) we have the constitutive evolution equations

2G L & +v~-r +v~ Sz l + v

_ Svz 70 ( ~ [ ]) S~ + 2n~J1, Sz ~p Xo

_ vr Yo Qb[])Soo + 2n{Jt, (6.I) r ~o ~o

111 Srr+2n~Jlxo

I [~o-~ ~o'~ ~o',.~ v tr(~)l 2OL_at +Vr~r +v~ az t + v

a~r 7o / F J~ + n ~ J~

1 [So-00 ~ &r00 v tr(~)] 2G L St +v~ +vz Sz 1 + v

860 P. Perzyna ZAMP

The evolution equation for the imperfection parameter is independent of the assumed yield condition and has the form

Ot+~-rvr+~-zvz=D~ 8z2J (6.2t

h + (1 - ~ )E t rD ~' + ~ t r ( a D P) + l J1.

The equations of motion are as follows

OVr @ ~Vr

8v~ @ 8v~ O ~ + v, 8r

8r J + v~ Oz

6a,~'] (O 8v~ Or J + v~ 8z

az / ? ('~" - ,~oo) = o ;

The kinematical expressions have the form

Oo'~z" ~ 1

a z / - 7 % : ~

(6.3)

OU r ~u r OU r v" = f f[ + V" ~r + V~ Oz '

~u z ~u z ~u z v~ = - ~ + v" ~r + vz 8z

(6.4)

The initial values for the problem considered are determined as follows

t = 0, r e [0,Ro], z6[0,L o]

u (r, z, 0) = 0, a (r, z, 0) = 0 ,

(r, z, O) = 4 ~ (r, z).

The boundary conditions are postulated in the following form:

(i)

(6.5)

the lateral surface of the specimen is required to remain stress flee during the straining process

jr= ~ + V z ~ z arr+arZ~Z +f;(z't)( ~t +Vr~-r +Vz 8Z

+ ~ - - Z ~ t ~ r z + ~ k T i - + V , ~ # + V z e z / 3 = ~

+ ~ G = + ~ z \ at +v '~-r +v=T;-z ) J = ~

(6.6)

[0 ~ 0,


for t e [0, dp], r = R (z, t), z e [0, L(t)] where

~R n~ = fi (z, t), n~ = ~zz fi (z, t), (6.7)

and - t /2

f i (z , t ) = [ ( O R ~ 2 _ 1 �9 L\az)

the specimen has shear-free ends

/'0 = 0,

(ii)

(6.8)

(6.9)

for t e [0, d p ] , r ~ [0, R (z, t)] 1~ = L(t);

(iii) TO the ends of the specimen is prescribed constant velocity ~) in direction of z axis, i.e.

v~(r, L(t), t) = U, (6.10)

for t e [0, de], r e [13, R (z, t)] [~ = L(t);

(iv) the assumed symmetry about z = 0 requires

#(r ,0 , t) = 0, ] ~ vz(r, 0, t) = 0, (6.11)

for t ~ [0, dp], r ~ [0, R (0, t)];

(v) the imperfection parameter ~ has to satisfy the boundary condit ion

a ~ . ~(r, z, t) + b ~ (r,z, t) = 0, (6.12)

for t ~ [0, dp], r = R (z, t), z ~ [0, L(t)],

and

for t E [0, dp], r ~ [0, R (z, t)], z = L(t ) ,

i.e. for the lateral surface as well as for the end surfaces. The necking problem is described by the evolution equations for the compo-

nents of the Cauchy stress tensor (6.1), the evolution equat ion for the imperfection parameter (6.2), the equations of mot ion (6.3), the kinematic equations (6.4), the initial values (6.5) and the boundary conditions (6.6)-(6.12).

The solution q~ of the ini t ial-boundary-value problem is represented in the cylindrical coordinates system by

q) = {ur,u~,vr, vz,ar~,~oo, a z z , ~ z , ~ } ( r , z , t ) . (6.13)

The problem formulated is very complex. So, it is worthwhile to consider some particular cases which can be treated as reasonable simplifications of the general problem.

862 P. Perzyna ZAMP

6.2 Influence o f the strain rate effect on the onset o f localization

Let us assume that there are no internal imperfections in the material of the specimen and let us consider a sequence of the initial-boundary-value problems by postulating

v z (r, L(t), t) = gl, ~f2, ~33,..., [3n, (6.14)

instead of the condition (6.11). The spectrum of velocities C a, C 2, C3,. . . , U, has to be assumed such that

it causes changes of the mean strain rate in the specimen in a sufficiently large range.

To simplify the numerical procedure as far as possible, let us neglect the inertial terms in the equations of motion (6.4). Thus, we shall treat problem as quasi-static (cf. Ref. [21]).

Since the problem has been simplified to a quasi-static one, we have to restrict our considerations to values of the velocities ~ (i = 1,2, . . . , n) for which our quasi-static approximation is valid.

The advantage of the problem discussed in Sec. 6.1 is the unified formulation for the elastic-viscoplastic range as well as for the elastic-plastic response of a material. Thus, we can obtain the Needleman results as a limit case of our processes under the assumption that the velocity C 1 = Cstat ie = [-) (assumed by Needleman [17]).

The numerical results obtained are plotted in Figs. 5 and 6. These theoretical results can be compared with experimental data obtained for 316 L stainless steel by Albertini and Montagnani [2].

It can be said that the theoretical predictions are consistent with experimental observations and that the load at the instability point is an increasing function of the strain rate while the strain at the same point is a decreasing function of the strain rate.

LOA D

. . . . ~ On

U2

0,

0 STRAIN 9

Figure 5 {5}

LOAD "~ AT iNSTABiLiTY STRAIN J POINT

I " ~ / LOAD

I \ STRAIN '~"

I STRAI N RATE

�9 0s

Figure 6


6.3 Influence of imperfections on the elastic-plastic solution

We shall now consider the elastic-plastic response of a material with internal imperfections. In the evolution equation (6.3) for the imperfection parameter 4, we neglect the diffusional term (as it has been postulated for the elastic-plastic range, cf. Sec. 5).

The numerical (finite-difference) procedure may be developed in such a way that it can be applicable for the yield function postulated by Eq. (4.16). However, since we are interested in the comparison of our results for voided solids (material with imperfections) with those obtained by Needleman [17], we intend to focus our attention on the J~ flow theory, i.e. we assume the yield condition in the form

(3"1)1/2 = 34, (6.15)

where x is the material softening function. The engineering criterion of the onset of necking in the form of a maximum

load condition (cf. Sec. 3) is postulated. We postulate also that the process of deformation of the specimen is the

same as that considered by Needleman [17] until the maximum load point is reached. Starting from this point we superpose internal imperfections by postulating the following distribution of the initial imperfections

(6.16)

This linear approximation is based on the experimental observations presented by Fisher [13].

The material function ~ = ~ (4) is assumed in two particular forms, giving the parabolic and linear approxiinations, cf. Fig. 7.

Figure 7

% o J I

I ",4."--

I \ I \

i i I , i , \

05 10

smaLL

=0,005 +q3

864

Figure 8

1.2

1.Q

0,8

~ - ~ 0,6

0,4

0,2

0

P. Perzyna ZAMP

BifurCation

Moximum / / Fundament~ .~_ ~Sotution

%%

------------ I NEEDLEMAN'S SOLUTION

PRESENT SOLUTION

...... ~=XoI~- ~�89

. . . . . . . . . . . ~L =~to(1- ~ )

0,04 0.08 0.12 uz/L o

0,16 0.20

The results for voided elastic-plastic solid are compared with those obtained by Needleman [17] in Fig. 8.

6.4 Influence of diffusion effects

To investigate the difffusional effects on the onset of necking as well as on the postcritical behaviour of the specimen we have to solve the initial-boundary- value problem (with inertial terms neglected) formulated in Sec. 6.1 for an elastic-viscoplastic material with internal imperfections. The evolution equation for the imperfection parameter ( may be assumed in a general form (6.3) with the diffusional term included.

This problem enables us to show the importance of the diffusional effects on the solution and to discuss further effects by taking into consideration the strain rate sensitivity of the material as well as the internal imperfections.

Preliminary numerical results obtained for the problem described proved the importance of the diffusional effects. However, the solution of the problem is very difficult numerically and it needs a refinement of the numerical procedure. This will be the subject of further investigations.

7. Conclusions and comments

The main objective of the paper was to describe some phenomena generated by nucleation, growth and diffusion of voids during a deformation process for postcritical behaviour of dissipative solids.

Studies (cf. Fisher [13]) dealing with the deformation of solids involving dispersion of hard second phases or inclusions have shown that the nucleation of


microcracks or voids is associated with these particles when the material of a body is subjected to various types of deformation. These voids appear either as cracks in the particles or as failures of the particle - matrix interfacial bonding. The actual void morphology depends upon the interrelation of various micro- structural parameters as well as the local deformation state. In very high purity materials, the absence of particles acting as void sites requires that other mechanisms of void nucleation be operative. It has been observed that in high purity silver and stainless steel voids are nucleated in the areas characterized by high dislocation densities. During high temperature creep deformation void forma- tion may be associated with diffusion controlled processes in which grain bound- aries act as sinks and sources for vacancies with directional diffusion occuring as a result of the local stresses.

The theory proposed can describe such phenomena as influence of strain rate effects and imperfections and transport effects on main inelastic deformation process. However, this theory can not describe the final mechanism of fracture. The mechanism of fracture is initiated by linking of voids and by forming a long and narrow cavity within a body. This final stage of deformation process is of great practical importance and needs further study.

References

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[2] R. J. Asaro, Micromechanics of crystals andpolycrystals. Brown University Report, April 1982. [3] M. F. Ashby and R. A. Verrall, Diffusion-accommodatedflow and superplasticity, Acta Metall.

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Brown University, 1980. [7] K. Frischmuth and P. Perzyna, Thermodynamics of modified material structure with internal

state variables. Arch. Mechanics 35, 279-286, 1983. [8] A. L. Gurson, Continuum theory of ductile rupture by void nucleation and growth. J. Eng.

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tinuum Mechanics", N. I. Muskhelishvili Volume, pp. 155-164, Soc. Ind. Appl. Math. Phila- delphia 1961.

[10] R. Hil}~, Uniqueness in general boundary-value problems for elastic and inelastic solids. J. Mech. Phys. Solids 9, 114-130, 1961.

[11] R. Hill, Uniqueness and extremum principles in self-adjoint boundary-value problems in continuum mechanics. J. Mech. Phys. Solids, 10, 185-194, 1962.

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[19] D. Peirce, R. J. Asaro and A. Needleman, An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 30, 1087-1119, 1982.

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[22] P. Perzyna, Stability phenomena of dissipative solids with internal defects and imperfections. XVth IUTAM Congress, Toronto, August 1980; in "Theoretical and Applied Mechanics", Proc., eds. F. P. J. Rimrott and B. Tabarrok, North-Holland, Amsterdam 1980, pp. 369 374.

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Abstract

In the paper the description of the postcritical behaviour of dissipative solids is presented. A simple model of an elastic-viscoplastic material with internal imperfections is proposed. This model is justified by physical mechanisms of polycrystalline matter flow in some regions of temperature and strain rate changes.

A model proposed satisfies the requirement that during the deformation process in which the effective strain rate is equal to the assumed static value the response of a material becomes elastic- plastic.

The identification procedure for all material functions and constants are based on available experimental data. Both the mechanical test data and physical, metallurgical observations are used. As an example of a quasi-static, isothermal flow process the boundary-initial-value problem describing the necking phenomenon is considered. The problem is formulated in such a way that enables discussion of the influence of strain rate effects, as well as of imperfection and diffusion effects on the onset of localization. Comparison of theoretical predictions with available experimental results is given.

Zusammenfassung

In der vorliegenden Arbeit wird das nachkritische Verhalten yon Festk6rpern mit innerer Reibung beschrieben. Ein einfaches Modell cines elastisch-viskoplastischen Materials mit Struk- turfehlern wird vorgeschlagen. Dieses Modell ist gerechtfertigt durch die FlieBmechanismen in


polykristallinem Material, die in gewissen Bereichen yon Temperatur und Dehnungsgeschwindig- keiten auftreten.

Das vorgeschlagene Modell erfiillt die Forderung, dab sieh ein Material wfihrend eines Defor- mationsvorganges, bei dem die effektive Dehnungsgeschwindigkeit gleich dem statistischen Wert angenommen wird, elastiseh-plastisch verh/ilt.

Fiir die Materialfunktionen und -konstanten werden sowohl mechanische Experimente wie auch metallurgische Beobachtungen beriicksichtigt. Als Beispiel fiir einen quasistatischen, isother- men FlieBvorgang wird das Rand- und Anfangswertproblem benfitzt, welches die Einschniirung bei einem Zugversuch beschreibt. Das Problem ist in einer Art und Weise formuliert, die es erlaubt, den Einflul3 von Dehnungsgeschwindigkeiten, Strukturfehlern und Diffusion auf den Beginn der Lokali- sierung zu diskutieren. Die theoretischen Voraussagen werden mit vorhandenen Resultaten aus Experimenten verglichen.

(Received: September 19, 1983)

stability of flow processes for dissipative solids with internal imperfections

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