International Journal of Plasticity, Vol. 13, Nos. 6-7, pp.

669-696, 1997
Pergamon © 1997ElsevierScienceLtd
Printed in Great Britain. All rights reserved
0749-6419/97$17.00+0.00
PII: S0749-6419(97)00033-8

A TEIERMODYNAMICALLY CONSISTENT THEORY OF

G R A D I E N T - R E G U L A R I Z E D PLASTICITY COUPLED TO
DAMAGE

T. Svedberg and K. R u n e s s o n

Division of Solid Mechanics, Chalmers University of Technology, S-41296 Gfteborg, Sweden

(Received in final revisedform 6 June 1997)

Abstract--A thermodynamically consistent theory of gradient-regularized plasticity with coupling

to isotropic damage is presented. If the internal variables are gradient-enhanced, this leads to a
nonconwmtional definition of the corresponding dissipative stresses (with a separate definition
associated with external boundaries of the body). The main purpose is to use this theory for
describing the successive development of the localization zone that is pertinent to the softening
regime due to excessive damage close to failure. Localization characteristics of the linearized
deformation field are assessed for the simple homogeneous tension bar as well as for the general
stress state. In particular, it is predicted that the localization zone narrows with accumulation of
damage. This prediction is indeed verified by a numerical solution in the case of uniaxial stress.
© 1997 Elsevier Science Ltd

I. INTRODUCTION

Conventional local c o n t i n u u m theory is deficient when the stress (and strain) states satisfy
the conditions for onset o f localization o f the d e f o r m a t i o n field. In a general multiaxial
stress state, 1localization will orien t become possible first after the stress-strain behavior
enters the softening range. Softening must be modelled as "negative h a r d e n i n g " within the
context o f plasticity theory. A m o r e rational constitutive f r a m e w o r k is based on the cou-
pling o f c o n t i n u u m d a m a g e to plastic deformation (which is the a p p r o a c h taken in the
present work:).
A well-known characteristic o f a local c o n t i n u u m theory is that the localization zone
can be infinitely thin, i.e. a displacement discontinuity can develop. This means that dif-
fuse localization phenomena, such as the development o f a s m o o t h neck in a tension-bar,
can n o t be accurately described within classical c o n t i n u u m theory (at least n o t without
suitable regularization in the postlocalized range). It is well-known that a displacement
discontinuity leads to numerical difficulties when conventional F E - m e t h o d s , which
employ continuous displacement approximation, are used to capture localization phe-
n o m e n a . In particular, it appears that the calculated energy dissipation will tend to zero

*Localization in the hardening regime (for local continuum theory of plasticity) requires nonassociative plastic
flow, cf. Ottosen and Runesson (1991).

669
670 T. Svedbergand K. Runesson

when the FE-mesh becomes infinitely refined, which is translated into the physically
unrealistic result that the global postlocalized response becomes infinitely brittle. Since
this is the characteristic result of refining the FE-mesh, it is known as "pathological"
mesh-dependence (as opposed to the smooth dependence of the mesh for regular solu-
tions).
As was already alluded to above, one possibility to ensure that the correct amount of
energy is dissipated is to employ a "local regularization" strategy such that a non-stan-
dard interface is introduced along the localization zone. Such dissipation-objective regu-
larization of the local theory has a long tradition in the modeling of semi-brittle material
response, cf. William et al. (1984) and Ba~ant (1986), and has relied heavily on the "ficti-
tious crack" concept launched by Hillerborg et al. (1976). In its turn, the fictitious crack
concept has deep roots in fracture mechanics, cf. Dugdale (1960) and Barrenblat (1962).
Recent contributions within FE-technology are those of Larsson et al. (1993), Simo et al.
(1993), Larsson and Runesson (1996) and Steinmann and Stein (1996).
Another approach to the regularization issue is to introduce nonlocal effects. This is
accomplished by defining suitably weighted averages (nonlocal formulation) or gradients
(gradient formulation) of a selection of the thermodynamic variables. This approach has
won widespread popularity in recent years, although the concept goes back to Eringen
(1981). Contributions to the development of nonlocal theory, mainly from a numerical
viewpoint, are those of Ba~ant and Lin (1988), Pijaudier-Cabot and Huerta (1991),
Leblond et al. (1994) and Str6mberg and Ristinmaa (1996). As to gradient theory, Dillon
and Kratochvil (1970) developed a theory based on strain gradients within a thermody-
namic framework. In terms of gradient hardening (which has been supported by micro-
mechanics), gradient theory has been advanced by Aifantis and coworkers in a series of
papers for metals as well as geological materials, e.g. Zbib and Aifantis (1988, 1992),
Vardoulakis and Aifantis (1991) and Miihlaus and Aifantis (1991). Large strain effects
have been included, although within the classical hypoelastic setting (which does not have
a thermodynamic basis), e.g. Zbib and Aifantis (1992), Fleck and Hutchinson (1993),
Zbib (1994). Numerical algorithms have been proposed by Sluys et al. (1993) and Pamin
(1994). However, the coupling between plasticity and damage has not been considered in
this context.
The aim of the present paper is to present a thermodynamically consistent theory of
gradient-regularized plasticity coupled to damage. Only those internal variables control-
ling hardening are augmented with gradient terms, thus resembling much of the above-
referenced work (in this sense). In particular, it is shown that this approach leads to the
introduction of "dissipative stresses", cf. Halphen and Son (1975), on the boundary of a
given body. The CDI then imposes restrictions, in a rational fashion, on possible bound-
ary conditions for the plastic multiplier. This is believed to be a novel feature, that con-
tributes in establishing a self-consistent constitutive theory. We remark that, so far in the
literature, only heuristic arguments have been used for choosing the (homogeneous)
boundary conditions. For convenience, restriction is made to small strain kinematics. The
extension to the hyperelastic-plastic format (based on the multiplicative split of the
deformation gradient) is the subject of a forthcoming papert.

*As already mentioned, the hypoelastic-plasticformat does not derive from thermodynamics, and it is thus
unsuitable for developmentsin the spirit of this paper.
 Thermodynamically consistent theory of gradient-regularized plasticity 671

The paper is organized as follows: Section II presents the basic thermodynamic rela-
tions for the gradient theory of a rate-independent dissipative material. This is followed in
Section III by constitutive relations for models including gradients and explicitly for a
model problem comprising scalar damage coupled to von Mises plasticity with isotropic
hardening. In Section IV the localization characteristics for gradient models are investi-
gated. Finally, in Section V the theoretical predictions in the previous sections are verified
from a numerical investigation of the simple problem of a tension-bar (in uniaxial stress).

H. THERMODYNAMICS

II. 1. Energy equation for gradient theory

The energy equation (first law of thermodynamics) states that the energy is conserved,
which for tht.~ body occupying the domain fl with boundary Off can be written as

/~+K=P+Q (1)

with

E=
I pedg2, K=~ ii P l d 12 dr2 (2)

P= J,b udn+ I(or. m)-d d(Of~) (3)

fl 0[2

Q = J pr dft - l h . m d( O~) (4)

[2 0[2

Here, E and K are the internal and the kinetic energy, respectively, of the body, whereas P
and Q represent mechanical and thermal power supply. Moreover, e is the internal energy
density (per ~anit mass), b is a body force, or is the stress, r is a heat source and h is the heat
flux. We have also introduced the displacement u, the (outward) normal t m on 0~2 and the
mass density p.
Assuming that (1) holds for an arbitrarily chosen part of fl, while using the equilibrium
equation, one obtains the locale form of the energy equation:

pb = or : ~ + p r - V - h (5)

which is the usual formulation (for local continuum theory), with ~ being the total strain.
Hence, the developments leading to the local form of the energy equation, given in (5), do
not explicitly involve the arguments of e, and we conclude that (5) is valid for local as well
as gradient theories.

tNotation n is (]gater) used for the normal to a singular surface.

¢Spatially local; should not be confused with local continuum theory.
672 T. Svedberg and K. Runesson

11.2. Entropy inequalityfor gradient theory

The entropy inequality (second law of thermodynamics) states that the entropy (disorder)
of a thermodynamic system can not decrease. This law can be stated as:
- Qo >_0 (6)
with

S= I psdf2, ao = pr d[2- I --0---

I -~- h'm d(Of2) (7)

Here, S is the entropy of the body, whereas Qo is the "entropy flux", Valanis (1971).
Moreover, s is the entropy density (per unit mass) and 0 is the absolute temperature.
Invoking (7) into (6), and eliminating r by combining with (5), we obtain the global form
of the Clausius-Duhem-Inequality (CDI):

cr.b.+pOi_pb_h. 0 dn>O (8)

Following Simo and Miehe (1992), we assume that the state is completely determined by
the elastic strain ee = e _ rp, the "elastic" entropy se = s - sP and the internal variables
q, (which are treated here as scalar quantities, for simplicity). Here, eP is the plastic strain
whereas sP is the plastic configurational entropy. In order to simplify the subsequent
developments, it will be assumed henceforth that only the internal variables q~ are of
nonlocal character. Hence, both q~ and Vq, will appear as arguments in e, such that
e = e(e e, : , q~, Vq,). Upon differentiation and inserting into (8), we obtain

I0[(-P0-~e)
1 Or :~+p(0- Oe\. + p-~-~:
~-~)s Oe : ~p + p -Oe
~ ~p
(9)
ae. ae h. ]
_ ~p_~q q _ ~ p o(vq,~). VOa JO d[2>O

Now, integrating the gradient term by parts, we may rewrite (9) as

II[((r 0 e ) ( Oe)k 0e ~p Oe.

-Pb-~: :~+P ° - W +Pb-~:: + P ~ :
[2
(io)
+ a.o. h._ o an + Q )Oo d(an) >_ 0
~f
8[2

where we introduced the dissipative stresses

Q~ = -p~-~q~ + o r . a( (11)
 Thermodynamicallyconsistenttheory of gradient-regularized plasticity 673

ae
Q~)=-m.pa(Vq~ ) on 0f~ (12)

Here, super~',ript "b" denotes "boundary". In standard fashion (as for local continuum
theory), we postulate that the inequality (10) must hold for any choice of domain [2 and
for any independent thermodynamic process: (1) a reversible process (q~ = 0) under uni-
form temperature (V0 = 0), (2) a reversible process (q. = 0) under non-uniform tem-
perature (V0 # 0). As a result, Coleman's equations are obtained formally like for the
local continu,~m theory:
ae ae
(r = p~-~, 0 = 0-~ (13)

def
D = (r: ~ P + p 0 s P + E Q a q a >_0, -h. V0>0 in [2 (14)
0t

D(b) def
= E Q~) q ~ > 0 on 0[2 (15)
~t

where D is the is the "local" dissipation. It is concluded that the difference, as compared
to local theory, is the additional gradient term in Q~, and the boundary dissipation term
in (15).
Remark: while the global inequality in (10) is necessary in order to satisfy the CDI, the
local inequalities in (14) and (15) are only sufficient conditions. It is also remarked that the
expressions in (11) and (12) resemble those obtained by Valanis (1996) for a generic
situation involving internal variables.
Upon introducing the Helmholtz' free energy density q~ via the Legendre transform
e = qJ + Ose, we obtain from (13)

0* 0*
se = - - - (16)
~ = Pa-~' a0

whereas (11) and (12) are now given as

Q~ = - P a - ~ + 0 v . a(v-qq~) in 0[2 (17)

Q(b) = _ m . p a ( v q ~ ) on 0[2 (18)

m A GRADIENT THEORY FOR PLASTICITY COUPLED WITH DAMAGE

III.1. Constitutive relations from thermodynamics

A thermodynamically consistent theory of gradient-regularized plasticity, whose devel-
opment is coupled to that of damage, is defined next. To this end, ~Paccounts for isotropic
damage (denoted or) but allows for a conceptually quite general format of plasticity.
674 T. Svedberg and K. Runesson

Moreover, restriction is made to the isothermal situation, i.e. 0 is treated as a (known)

parameter only. This means that s ~ and sP become irrelevant quantities. Hence, we adopt
the additive expression for ko:

tIl(g.,e,/¢', VK, 0/) = ~f(f.. e, 0/) "['- tI/P'l°c(/¢") + ~IiP'grad(v/¢") (19)

where tI~ is the elastic energy defined as

ele = (1 - ot)~b~(ee) with p @ ( e e) = } e e" e : e e (20)

whereas qjp,loc and kop,grad are local and gradient contributions due to hardening (which
are left unspecified for the moment), which are expressed in the scalar hardening variable
r. t Gradient effects are restricted to hardening via the inclusion of VK. Moreover, e e is the
tensor of constant elastic moduli, that represent linear (but generally anisotropic) elastic
response.
F r o m the Coleman's relations (16), the following constitutive relations are obtained

0ko
cr=p-~-=(1-c0b, 8=e e:e e (21)

D = o" : ~P --I.-Kk + A6~ > 0, x ~ ~ (22)

D (b) = K(b)k > 0, x 6 a~ (23)

where 8 is the effective stress (like in local theory). The dissipative stresses are defined as:
K = K I°c + Kg~ad with

K t°c GqtlIp'l°c K grad = V - ( O~pp,grad~

=-P ~ ' \P--~V~J' x~ (24)

Oq~I/P,gtad
K(b) = - m . p O(Vr) ' x 6 0f2 (25)

0qf 1A
A = _p__~_=p~e ----~o': (ee) -I :fi" (26)

Like in local theory, the "effective stress principle" is adopted, i.e. 8 (instead o f or) is used
in the yield function for the undamaged material behavior. The convex set B(-) of plas-
tically admissible states is then defined as

tOnly a single scalar hardening variable (that represents isotropic hardening) is considered in order to simplify
the notation (and analysis). The extension to more general hardening is straightforward, although significantly
more technical.
 Thermodynamically consistent theory of gradient-regularized plasticity 675

B(a) = {((r, K) I a)(8, K) _< 0} (27/

where • is tkc convex yield function.

111.2. Constii!utive rate equations

Rate equations for the internal variables are introduced in exactly the same way as for
local theory. Hence, for general nonassociative plastic flow and hardening rules, we
introduce the: dissipative potential ~* such that (for assumed smooth potential surfaces)

.... X 00"
t p- ---- (28)
,.. 1 - a 08

".. t~ = ~. 0.* (29)

OK

whereby ~* Jis chosen in such a fashion that CDI is satisfied in g2. An example of such a
class of thermodynamically admissible rules, that includes nonlinear hardening develop-
ment for metals, is defined by the choice
1 2
• * = • + ~gxK (30)

where the constants gx > O. Moreover, we choose the damage rule as

• 0T
a = ~.-- (31)
0A

where T(A, (~) is a positive, scalar function that is increasing monotonically in both its
arguments.
In order to complete the problem formulation in [2, we adopt the Kuhn-Tucker com-
plementary conditions
>0, ~(8,K)<0, ~(8,K)=0 (32)
Remark: like in local theory, the conditions (32) follow automatically only when the
Maximum Dissipation Principle is adopted, which is equivalent to completely associative
dissipation rules.
By combining (21), (24) with (28), (29), we obtain the following constitutive equations
(on rate foma) that are valid in fl:

~'=b e--~ r e : -0~* with ~ . e = ~ e : ~ (33)

1 - ot 08"

~oc • 8~* (34)

= -~.H OK

/~ad=/2V'( Hgrad'v~0~*~0g] +t2v. m ( a.

(0g)'/
(35)
676 T. Svedberg and K. Runesson

We also obtain on 0~2:

• O~* " 2 a2~*
/(<b) = --tim- H grad • V;~ - ~ - - ~1 m • H grad- VK(aK) 2 (36)

In these expressions we have introduced the "local" hardening modulus H and the (new)
"nonlocal" hardening tensor I-Wad defined as

02qjpJ°c 1 ~qjp,~d
H grad (37)
H = p (at) 2 , = P ~ a(Vr) ® a(vr)

We shall henceforth assume that H > 0, whereas H grad is a positive definite 2nd order
tensor. It is emphasized that, within local theory, H > 0 is not a sufficient condition to
avoid apparent softening (in the sense of a negative tangent modulus), since the damage ot
will induce a destabilizing effect. The desired stabilization is caused by H ~ d , which will be
shown more explicitly below.
The internal length I in (35) may be seen (1 °) as a convenient dimensional parameter, in
order that H and H ~ad will get the same dimension, or (2 °) as a physical entity that is a
characteristic measure of the microstructure. Yet another interpretation is that (3 °) I is a
mere artifact that brings about numerical stabilization to the intrinsically local theory. In
fact, the introduction of H grad formally admits anisotropic nonlocal hardening, which
obviously has little justification from a physical viewpoint in conjunction with isotropic
local hardening•
From the aforesaid, the conclusion is that the role of H ~ad as a purely numerical means
of regularization is quite significant. This is particularly true when that state has been
reached that localization is encountered. In such a situation the spectral properties of
H grad a r e conveniently related to the orientation of the localization zone (defined by its
incipient normal n). A feasible choice is

H grad = n g r a d [ ( l -- k)n ® n + kr], 0< k < 1 (38)

where H grad > 0 is the scalar nonlocal hardening activated across the localization zone.
(In this case//grad is identical to/-/~l tad, which is the largest principle value of H g~ad, cor-
responding to the eigenvector n. All other eigenvalues are M-P ad corresponding to eigen-
vectors that are orthogonal to n.) The requirement k > 0 is necessary in order to ensure
that H grad is nonsingular, whereas k = 1 gives the trivial isotropic case defined by
Hgraa = Hgradr. In this fashion it is possible to "enhance" gradient effects across the
localization zone (with a steep strain gradient), whereas the perturbation of the local
hardening along this zone (where the field variables vary smoothly) is suppressed)
In order to complete the theory, we must choose a constitutive assumption for K~b) as
to ensure that (23) is satisfied. While the technique to satisfy (22) is standard, it is less
obvious (at least from a physical standpoint) h o ~ t o choose the constitutive law for K (b)
or, equivalently, how to define the boundary condition for ~ on ag. The relevant choice of

tThis feature resembles the shock-capturing feature in fluid dynamics, cf. Brooks and Hughes (1982), Johnson et
al. (1987).
 Thermodynamicallyconsistenttheory of gradient-regularizedplasticity 677

/~b) is dependent on the actual model. An explicit expression is proposed for the von
Mises-based model in Section III.4.

III.3. Tangent relation--linearized differential equation for plastic multiplier

It is of inte~rest to establish the linearized behavior in the special case that the material
is in a plastic state, i.e. when 4~ = O. Like in local theory, we have the consistency
conditions:
__0, + ( b , K ) _ < 0 , ~6(b,K)=0 (39)

which are the linearized correspondents of those in (32). The distinct situations of plastic
loading (+ = 0, ~ > 0) and elastic unloading (+ _< 0, 2 = 0) are still identified locally
within fl (like for local theory).
Upon using the constitutive relations (33) to (35), we may derive the pertinent consti-
Ygrad
tutive differential equation for ~. when * = 0 as follows: first we rewrite I~ in (35) more
explicitly, by using the chain rule, as

i~grad : l 2 -0"*
~ [Hgrad : V2~ + (V. Hgrad) • V~] + l 2 02** [VK- H grad V~.

+ XH : v2K+, (v. VX:+ VX. VK] (40)

+ 1-2 (~- *~*. V K" . H grad • VK

Next, we combine
+ = ~ :o'+ ~K (41)

with (33)-(35) and (40) to obtain the linear differential equation in ~ (for given +e)

_+grad + (h + hgrad)i = +e _ + (42)

where we have introduced the "local" loading function +e and the local generalized
plastic modulus h as
+e 0 " ~e
= ~ : • /. (43)

1_ _ 0" ~¢ 0"* --
h-l_ot0b: :~ +H with H=H0*0K0**0K (44)

Moreover, we have introduced the "gradient" augmentation hgrad to the hardening h, and
the "gradient"' loading function +grad as

hgrad .,,a* [ 0 2 " * [Hgrad : V2K+ (V. Hgrad) " V K ] + V K . H grad- V (45)
= 2
678 T. Svedbergand K. Runesson

t~grad = l 2 ~aO (--~-

aO* I n grad : V2~ + (V" H grad )" V~.]
.. + 2 (020*
- ~ VK. H grad • V)~) (46)

In the special case of isotropic hardening, i.e. H grad = Hgradr, where H grad is a scalar, then
the expressions in (45) and (46) are simplified as

h grad = - I 2 ~-g
2 0(020*
k(aK)
0 [ngradV2K + vngrad . VK] + (0K)
3030* ngrad I VK 12) (47)

(
¢bgr,d=12OOo_g _~..
[/dgr~aO*d V2"~.+ VHgrad . V~] + 2 V *"
02O*/,tgradVK.
~~ ( 0 K ) ) (48)

A few other special cases of the general situation, for which (45) and (46) are the pertinent
relations, are considered next.
Homogeneous state: when all state-variables are spatially homogeneous, then we obtain
h grad = 0 and
t~grad = l 200 00* Hgrad . V2)~ (49)
OK OK

Constant noniocal hardening moduli: when H grad is constant, then we obtain

hgrad = - I 200
~ - ~\(ag)
z/'020*
| - - - - x H grad : V2K + ~--~-~VK.H grad VK) (50)
(oh-')
~)grad 2 00 /00* -rad o*
2 020* VK : H grad • V~
= ! ~ - ~ - ~ -H~ . V 2 ~ + (a/0~
) (51)

This case corresponds to the choice

#tltp,grad = _112Vx " ngra d . Vtc (52)

2

111.4. Constitutive relations for a model problem

As a prototype model, we may choose the von Mises yield criterion with linear isotropic
local hardening (but with orthotropic gradient hardening). The corresponding expressions
for the inelastic part of the free energy density then becomes

pqlp,loc = ~Hx2,
1 p~pp,grad= ~12Vx • H grad Vx (53)

where H and H grad are constant moduli. Hence, we obtain

K = - H x + 12V • (H grad. Vat) = - H x + 12H grad : V2t¢ (54)

 Thermodynamically consistent theory o f gradient-regularized plasticity 679

The yield criterion reads, as for local theory

• (0, K) = ~e - ay - K with ~e = V~ I Odev I (55)

where ~e is the equivalent effective stress and try is the (initial) yield stress:
The flow and hardening rules are of the associative type, whereas the damage rule is
defined by the following dissipative potential:

A2
T(A, c~) -- 2S(1 - of)m (56)

where S is a (constant) damage modulus, and m is an exponent that governs the rate of
damage evolution. For m = 1, the damage law proposed by Lemaitre and Chaboche
(1990) is retlieved. Hence, we conclude that the resulting model is thermodynamically
consistent as far as the CDI in ~2 is concerned.
Subsequently, we shall also assume isotropic elasticity (defined by the shear modulus G
and the bulk modulus Kb). Taken together, these assumptions result in the following
constitutive equations (on time differentiated form) that are valid in g2:
)~ 3G^
lYdev = 2G~:dev 1 - - Of °'e lYdev' Orm = Kb~v (57)

/~ = H ~ - 1 2 V • (H grad- V)~), K(0) = 0 (58)

d • R~
& - S ( I -a)m-X3GS(1 - a ) m' a(0)=0 (59)

where
3G (~m'~ 2
Sv = + (60)

which are subjected to the constraints

A ' ^

>__0, ~(trdev, K) ___0, ~,~0raev, K) = 0 (61)

These relations are obtained from (33), (34), (35) and (31) when (55) and (56) are intro-
duced.
Remark: when specialized to isotropic gradient hardening, i.e. Hgrad =/-/graa6, the
expression in (58) shows strong resemblance with the "first order" gradient term of Zbib
and Aifantis (1988) and subsequent papers.
It remain,~ to define a plausible boundary condition for )~ so that CDI will be satisfied.
A simple constitutive assumption is that K (b) = ~K, where ~ is a scalar, whose structure is

tThe deviator stress is defined as O'dev ~ Or - - O ' m 6 , where a m = 16 : or is the m e a n stress.

680 T. Svedbergand K. Runesson

similar to linear local hardening, K--- -Htc in ~2, as apparant from (54). It then appears
that the condition (23) can be written as
~tct~ >_ 0, x ~ 0~2 (62)

To show that there exists a constant -~ > 0 such that the condition (62) is satisfied, we first
conclude that to- - ~ _< 0, which with ~c(0) = 0 means that t¢ < 0. Hence, the result ~ _> 0
follows.
Upon using (25), we first conclude that
/~b) = 12m " Hgrad . V)~ (63)

which may be combined with the r e l a t i o n / 6 ~) = - ~ to give

C/./grad
m- H grad. V)~ = - ~.. 1 "~ ~' (64)

which is the adopted natural boundary condition (of Robin-type) for ~ on af2. In order to
obtain this expression, we introduced the relation ~ = I~ladlc, where/-~rl ad > 0 is the lar-
gest principal value of H Srad, whereas c > 0 is a non-dimensional scalar constant.
In the literature, cf. Pamin (1994), it is common to choose c = 0 ad hoc, whereas the
condition (64) has a therrnodynamical motivation. Clearly, when l = 0 (local theory), the
boundary condition will be irrelevant. As to the physical motivation for the choice of
c = 0, we remark that, if c > 0, then no bifurcation problem would be obtained in the
ductile regime, i.e. after plastic deformation has occured (to > 0). The only possibility for
bifurcation would be in a semibrittle deformation mode (at the onset of yielding when
tc = 0). Hence, it is difficult to justify any other choice than c = 0 for ductile materials.
However, c > 0 would be possible for cementitious materials, where gradient theory is
used essentially to define the length scale of the process zone in conjunction within the
"fictitious crack" framework.
The pertinent linearized differential equation that determines ~ in this particular case, is
obtained directly from (42) to (46) as

_+grad + h~ = +e _ + (65)

where we used that h grad = 0, whereas

~e = -x-3GO'dev": ~., ~)grad : 12Hgrad " V2L, h - --3G d- H (66)

ere 1-or

Equation (65) can be solved together with the boundary condition (64) and the consis-
tency conditions
~>0, ~<0, ~O=0 (67)

IV. LOCALIZATION CHARACTERISTICS

IV. 1. Linearized solution for homogeneous tension bar

IV. 1.1. Preliminaries. As a preliminary to a bifurcation analysis for the multiaxial states
o f stress and strain, we analyze the homogenous tension bar, as shown in Fig. 1, subjected
 Thermodynamically consistent theory of gradient-regularized plasticity 681

6=[0-/2
X
I I
0
1.12 _1_ t./2
F"

Fig. 1. Homogenous tension bar subjected to prescribed end displacement.

to prescribed end displacement (extension). For the particular case of uniaxial stress, the
material model described in Section Ill.4 can be simplified upon replacement of the gen-
eral expression for • in (55) with the "uniaxial" expression

=1 8 1 -Cry - K (68)

This leads to the specialized form of the rate eqns (57) to (59) as follows:

a= Ek-~.E-- (69)
181

'2 ra° 2 (70,

. 1812
a = x 2 E g d -,~)m (71)

The corresponding linearized differential eqn (65) for the determination of L is given as

- 4 , g~ad+ h~ = +° - + (72)

where
~e = 8 • @grad= 12/_/~add2X h- E t- H (73)
F~-i ~ ' dx2' 1~

We shall now assume that the state has been achieved at which the onset of localization is
possible, i.e. that a secondary (inhomogeneous) solution for ~ is possible in addition to the
primary (homogeneous) solution. However, from equilibrium considerations then follows
that ~ must be homogeneous along the whole bar for both the primary and secondary
solutions. It is then convenient to eliminate ~ from ~e as data in (72) in favor of d. t As a
result of thi,; elimination, we finally obtain the following equation, that replaces (72):

~ ~ (74)
_~d + ~O~ = 18 1 1 - o~

tThis is formaL~ equivalent to choosing ~ as the control variable (rather than ~), although it must be born in
mind that the real control variable is the end displacement of the bar. This boundary condition is used to deter-
mine &
682 T. Svedberg and K. Runesson

where/~ is the "effective" hardening modulus defined by

/t = H 1¢~13 (75)
2ES(1 - - o/) m + l

Once again, it is assumed that the state is completely homogeneous up to the state where
onset of localization can happen. We also note that the tangent stiffness E ~p for plastic
loading in the prelocalized regime is defined by

8 = ffl,~ with ~ P - - (1 - t~,___ (76)

h

The critical value of H for which E ep = 0 ( a n d / 1 = 0) is denoted Her. From (75) follows
that

I:I=H-Hcr with Her - l~13 (77)

2ES(1 - a)m+!

We may envision three principally different situations as illustrated in Fig. 2.

Localization situation 1: H = 0 is achieved fight at the onset of yielding, i.e. when u = 0.
We then obtain from (77) that this situation is encountered if H is chosen as

HmHcr,O= ~ (78)
2ES
Localization situation 2: H < Her' 0, which means that / t < 0 fight after traversing the
initial yield point, which defines a "kink" in the stress-strain relation. (The situation
/ t < 0 is denoted "effective softening" in what follows.)

O Of"~'ffcr, l::ep_o

I
~+t~. - . . . . . . . . . . . . . .

' >Her

Fig. 2. Typical (local) stress-strain response curves depending on the amount of hardening H.
 Thermodynamically consistent theory of gradient-regularized plasticity 683

Localizatioa situation 3: H > Her,0, which means that/~ = 0 will be achieved along the
path of loading at ot = aer, K = K¢~ defined by the relation

[(~, +/~)~]~
0tcr--~ 1 - L 2 E ~ - J (79)

It is noted that the value Kcr depends on the history of damage development (that must in
practice be calculated while the constitutive relations are integrated numerically). More-
over, the stress-strain relation is assumed to be smooth at H = 0.
At the onset of localization, we assume that continued plastic loading will be confined
to the zone qP = [- ~, ~, where 8 < L and where 8 is the width of the localization zone.
Since + = 0 within this zone, we obtain the pertinent constitutive differential equation
from (74) as
• 2 - ~ d d2)~ ~ b (80)
-~ n~'- ~x2+gr;~_ i~ I l - o r
In order to obtain a completely homogeneous solution in the prelocalized regime it is
necessary to set c = 0 in the boundary condition (64). Hence, the pertinent boundary
conditions are (due to symmetry at x = 0):

dX 8
~=0 at x = O and x--- (81)
dx 2

which must be satisfied in addition to the constraint

;~>0 for 0 < x < ~ 8 and ; ~ = 0 for ~8 < x < ~ L (82)

Clearly, the response is elastic outside ~'~P, which is indicated by the condition (82)2.

IV.1.2. Homogeneous solution. In the case of (local) "effective hardening", i.e. H > 0,
the general solution of (80) becomes

[(x)=Alexp +A2exp - q lal(1-ot)H with f l = >0 (83)

where the integration constants AI and A2 are determined by the boundary conditions
(81). It appears that the only possibility is Ai = A2 -----0 regardless of the value of 8, and
the solution reduces to the usual spatially homogeneous solution pertinent to the local
theory:

~.(x) = I~1
~ (1 - a ) ~ (84)

which gives rise to the tangent stiffness relation in (76). F r o m / ~ > 0 and the condition
> 0 follow that sign (b) = sign (o).
684 T. Svedberg and K. Runesson

IV.l.3. Localized solution. In the case of (local) "effective softening", i.e. /~ < 0 the
general solution of (80) becomes

d //grad
~(x) = A1 san + A2 COS -'[ with fl = ~ > 0 (85)
I~1 (1 - o 0 H
A nontrivial solution in now obtained for 0 < x < ~/2 as

d [1 [2~rx~]
- +cos[--~---)J with 7=27rfl (86)
i(x) - I,~I (1 --~)/~

which is a valid solution when 8 _< L. This condition is satisfied if l is sufficiently small, i.e.
if l/L < (2~r~) -l. Whenever this condition on l is not satisfied, then the only solution is the
homogeneous solution given already in (83). (The solution (86) is illustrated in Fig. 3(a)
for the case or > 0.)
Remark: From H < 0 and the condition )~ E 0 it now follows that sign (d) = -sign (or).
It also appears that X takes the maximum value at x = 0. Moreover, the "local" value of
~, given by (84), is obtained for x = 8/4.
Upon inserting (86) into (69), we obtain the tangent stiffness relation for 0 < x < 8/2 as

t~ = EvP'grad(x)E(X) with Eep'grad(x) -- (1 - cOEk(x)fiI

h + [k(x) - 1]/I (87)

where we have introduced the scalar

/,2JrX, .]_1
k(x)= 1 +cos T)j (88)

and it follows readily that ½ < k < ~ .

The following special cases are of interest:

k = ½ (x = 0), k = l (x=~)
(89)
1 k=oo (x=~)

(a) ~~L(1,,a)l'~o' (b) 0

2--

1--

I I "x "$uperlocal* behavior ~ "sublocal* behavior

(a) (b) (d)
Fig. 3. (a) Localized solution k(x), (b) characteristic stress-strain behavior due to gradient effects.
 Thermodynamically consistent theory of gradient-regularized plasticity 685

where the position ~ is defined by E~, grad = --OO and is given by

X= arccos with 0 < ^< 1 (90)

h h-H-
The various situations are depicted in Fig. 3(b). The response is termed "sublocal" if
k < 1, whereas it is termed "superlocal" when k > 1. The obvious motivation for this
terminology is that the "nonlocal softening modulus" k(x) I H I may be smaller or larger
than the "local softening modulus" I H I . Moreover, in the "sublocal" and "superlocal"
regimes, the plastic strain rate ~ is smaller and larger, respectively, than that encountered
for local theory. It is also noted that E ~p, grad > 0 for ~ < x < ~/2, which corresponds to
apparent "snapback" behavior. Finally, the situations (b) and (d) represent "local load-
ing" and "local unloading", respectively in terms of the strain-controlled loading criteria.
Next, we evaluate the proper loading criterion (for plastic loading). This is given for-
mally as

Se,~ad > 0 -~, $ = O, ~ _> 0 for x e ~'~P (91)

where we imroduced

Se,grad= Se + ~grad (92)

Upon combining (73)! and (87)1, we obtain

8. . 8" E
the(x) = - ~ ] Ee(x) = ] 8. ] ~p,grad(x)6" (93)

whereas combining (73)3 with (86) gives

Sgrad(x )
=
d
12/_/grad )
_ _
8.
18.[
cos{2rtx~
,T) 1
-
(r
.
(94)

Since sign (~) = -sign (~), it follows from (93) that

Se(x)>0 when 0 < x < ~ ( L i )

8 (95)
~e(x)<0 when ~ < x < (Lii)

where the position ~ was given in (90). Hence, it is possible that Se < 0 in ~'2 p, although
(~e, grad > 0. In particular, for x = ~ we obtain ~e(~) = 0 and

¢~e,grad(~) = ¢~grad(~) __ 6 h 6
> 0 (96)
I 8. I h - / ~ 1-or

where (90) was used. It is also readily concluded that Se. grad(~$/2) = 0, which follows upon
observing that k = o~ gives E ~p, grad = (1 -- ot)E at x = 8/2.
686 T. Svedberg and K. Runesson

IV.2. Bifurcation--onset of localization for linearized problem

IV.2.1. Localization condition• Next, we shall investigate the conditions for existence of
a bifurcation of the incremental (or rate) fields in the general situation of multiaxial stress
and strain fields. It is then assumed that the state is completely homogeneous just before
the onset of localization. Assuming plastic loading (~ = 0), we use the results of Section
111.3 to conclude that the pertinent differential equation for ~ is

°qtI:~
- 1 2 H grad : V2~ + h~ = ~ - " £e . I~ w i t h ~grad ~--- Hgrad 0t:I~
OK oqO*
OK (97)

which-is, indeed, completely equivalent to (80) for the simple tension bar. We shall also
need the equilibrium equation, which together with the constitutive relation (33) gives

V.cr=0
04:
with 6 = ( 1 - o t ) E e ' b . - ~ r e" 0--~- (98)

Here, we have introduced the "effective flow potential"

¢*(~, K, a) - ¢*(b, K) + T(A(~), tr) (99)

which gives
0~* -- 0~* 0T
Ob ~ . ~ ( f e ) - I :~. (100)
06"
Remark: since we shall only be concerned with the possibility of bifurcations in the
incremental solution, the difference between two possible solutions of ~r must satisfy the
homogeneous equilibrium eqn (98)1. Hence, subsequently we assume that all incremental
variables represent the difference between the primary and secondary solutions (without
new notation).
An infinite domain is considered and the solutions for the displacement rate 6 and for
are expressed in terms of plane waves, i.e.

• /2~r [i27r • x )
ti(x, t ) = U ( t ) e x p ( ~ - n - x), )~(x, t) =/~(t) exp~-~- n (lO1)

where x is the position vector (in Cartesian coordinates), n is the normal direction of the
wave, and 8 is the wave length, Moreover I9 and A are spatially homogeneous amplitudes
of the wave solutions•
Upon introducing the assumed solutions, given in (101), into the two eqns (97) and (98),
we conclude that these equations are, indeed, satisfied for each x if

(1-ot)Q e h+~---~a ®a .l~l=O (102)

where hgrad(n) is the "gradient" counterpart of h, and is given as

h grad ~-~n • a grad . n/-~--f~ 2 (103)

 Thermodynamically consistent theory of gradient-regularized plasticity 687

and where Qe(n), i*(n) and a(n) are given as

a~ a6*
Qe = n. e e • n, a = ~ : e e • n, i* = O~ : e e .n (104)

It appears from (102) that

Qep = ( ~ e 1 (~e
h + h grad fi* ® a with = (1 - oOQe 005)

is the acoustic tensor for gradient-regularized plasticity coupled to damage, and it differs
from the "local" counterpart only by the additional term h grad. It is clear that bifurcations
are possible, i.e. lJ ~ 0 in (102), only when d e t Q ep = 0 precisely as in the case of local
theory. We may, therefore, put forward arguments in close agreement with those of
Ottosen and Runesson (1991) for the investigation of the spectral properties of Qep. First,
it is concluded that the smallest eigenvalue of Qep, with respect to the metric defined by
the "undam.'lged" elastic tensor Qe, is given as

X0)(n)=l-a y(n) with h = z +H (106)

h + hgrad(n) 1 -- tx

where we have introduced the abbreviated notation

y(n) = a(n). PC(n). i*(n), z = ~0~" ee : 0~*

0--8-' pe = (Qe)-l (107)

At any given thermodynamic state s = { e e, r, ce}, we define the localization direction fi(s)
such that
fi = arg(min kl(n)~ = arg(max y(n)'~ (108)
\lnl=l ,,/ \lnl=l h -~ hgrad(li)J

where the relation (106) was used. Once again, it is emphasized that fi is a function of the
state.
Remark: in the special case when (38) is adopted (with gradient isotropy as the simplest
~--~rail ad
special case), then we obtain n- H • n = ]~]gr . This means that (108) is simplified as

fi = arg(max y(n)~ (109)

\lul=l /

and the explLicit results obtained by Ottosen and Runesson (1991) can be used directly.
The critical state &r, corresponding to the critical localization direction nor = n(&r),
must satisfy the critical state relation k(1)(fi) = 0, i.e.

H(1-~) -y(fi)+z:-(1 -~)hg~d(fi) (110)

m

where it is realized that H, y and z may also be functions of the state. Hence, in general we
should write H = H(s), y = y(s, fi) and z = z(s). Clearly, eqn (110) is not sufficient to
688 T. Svedbergand K. Runesson

determine the critical state Scr. Rather this state is achieved along a (given) loading path in
stress, strain or mixed stress-strain space. After integration of the pertinent constitutive
relations, we may check whether the relation (110) is satisfied. If so, we have found the
critical values of interest, in particular ncr.

IV.2.2. Special localization situations. Depending on the actual constitutive properties,

three principally different situations may be encountered at the localization state in prac-
tice. (These situations allude to the localization situations in subsection IV.I.1 for the
tension bar.)
Localization situation 1: bifurcation is possible at the very onset of yielding (and
development of damage), i.e. when o~= 0. Since h grad > 0, which follows from (103), it is
clear that the maximum value of Hcr is obtained with the choice hgrad= O, i.e. when
8 = o¢. This value of Her is denoted Her, 0 and is obtained directly from eqn (110) when
the stress state is known at initial yielding. It also appears that the critical value Her, 0 is
identical to that which is pertinent to local theory. For uniaxial stress state, we may
readily retrieve Her,0 in (78).
Localization s i t u a t i o n 2 : H 0 < Her, 0, where we have used the notation H0 for the initial
value of H (at the onset of yielding). This means that the critical state is now traversed
already at the yield point. We may then solve for the pertinent localization width (wave
length) 8 from (110) as

]8 = 2zr nor" ~grad,, .her with Ho = 11o - Hcr,O (111)

H

where ner is obtained directly from the stress state, t In other words, we obtain a truly
localized solution right from the bifurcation point. Clearly, eqn (111) represents the gen-
eralization of (86)2 for the tension bar, in which case Her, 0 is given by (78).
Since the onset of localization at initial yielding characterizes "semi-brittle" response,
i.e. localization represents an incipient macroscopic crack, the post-localized (post-peak)
stress-strain relation should not be considered as a constitutive relation. It should rather
be determined from arguments in Nonlinear Fracture Mechanics, involving the localiza-
tion width 8 and the fracture energy release Wf.
As to the width of the localization zone, it is noted from (111) that 8 becomes infinitely
small when H grad vanishes. This pathological situation (pertinent to local theory) causes
the notorious mesh-dependence in FE-analysis. In fact, this situation can be interpreted as
the evolution of a displacement discontinuity, which is the key assumption in recent work
on theoretical evaluations and numerical procedures for capturing localized behavior, cf.
Simo et al. (1993), Larsson and Runesson (1996).
Localization situation 3:H0 > Her. 0 and H is a continuously decreasing function of the
state along the considered loading path. (In the extreme case H is constant, like for linear
isotropie hardening of the yon Mises yield criterion.) In this ease it is possible to obtain
real solutions Her and Otcr only after some damage has developed, i.e. oter > 0, w h i c h
is typical for ductile response. It appears that the earliest possibility for bifurcation

tit is a simple matter to calculate ncr since the critical state sc~ is known a priori, i.e. no integration of the con-
stitutive relation is necessary.
 Thermodynamically consistent theory of gradient-regularized plasticity 689

(represented by the smallest value of acr) occurs when h gr~ = 0 which corresponds to
8 = oc. Hence, the critical state and orientation of the "localization zone" will be the same
as for local theory.
At continued loading beyond the onset of localization, when H _< Her and ot > act, it
may be conjectured that the secondary solution is characterized by a finite width 8.
Moreover, ~: will successively decrease towards zero as the state approaches complete
material failure defined by a = 1. In this extreme case a true displacement discontinuity
has emerged. For the tension bar this scenario means that a diffuse neck starts to develop
at the peak (nominal) stress (as predicted by local theory), and it evolves in the post-peak
regime (who:re H < 0) while becoming more localized.
Let us return to the multiaxial state. It is realized that the state becomes non-homoge-
neous on the secondary loading path (after bifurcation). However, if it is assumed that the
quantities y and z, defined in (107), do not depend significantly on the state, then we may
give an approximate expression for 8 from (110) as follows:

~ " 2Jr.I (-l---- (X)ncr "~I-][--grad - ~---~- (112)

l -- ~(1 -- ot)H - (1 - Otcr)Hcr

where H and a represent the current state. The anticipated behavior is shown in Fig. 4. In
practice, of course, it is necessary to solve a full-fledged boundary value problem in order
to obtain the current value of &
Remark: in the special case that H is constant, we have Her = H, which may be inserted
directly into (112).
Remark: after the critical state has been reached, any point along the primary loading
path is a bifurcation point, as shown schematically in Fig. 4. However, for given material
parameters ,,'ach such secondary loading path is well-defined (as opposed to the situation
for local continuum theory, as discussed previously).

,0

15=o~ (z>txcr

Primary
loading
/path
[ l o a d i n g
] p a t h s _ _
[ H<Hcr

Fig. 4. Characteristics along primary stress-strain response curve in uniaxial stress (corresponding to prevailing
homogeneous deformation).
690 T. Svedberg and K. Runesson

V. NUMERICAL INVESTIGATION OF TENSION BAR

V. 1. Problem definition and material data

In order to numerically confirm (and extend) the analytical results for uniaxial stress (as
discussed in Section IV. 1), we analyze the h o m o g e n e o u s tension b a r in Fig. 5. T h e b a r is
subjected to the prescribed end displacement u, and the average strain ~ in the bar is,

E = 103ay

H = 0, 50<7y or 100ay
i u,
T Vl
/./grad = 50Gy/(41r 2)
'x
S = 10-sexy
4 L i

m=l

l = 0 or l = LI4
Fig. 5. Homogeneous tension bar (subjected to prescribed end displacemen0 with used material parameters.

1 , . , , ,
:: . t~.., ::Elernont lonoth # bar !onOth

i .i..., , i.. ......... i ............ i ....... I .... oa ii............

0,8 ............
,/,,,..,
i .......... I'"'~ ...... i" :: i i
, I--o.o,
" i
I,
!o.
0.7 ............ i/ .... ,...... :i! .......... .::i............ ! ....... l--o.o~s I! ..........
i/ ....... ', , i b, i ' : 'i :
.......... :"..........
.:/ i ........ "f! ............ i " "::. .... ! ............ i ............ ! ...........

: i' ~l ! "'.

j / li ::, ::,. i :: '....i ::

°-~1- .... 1~i ............ i', .......... i-"; ....... i............ i ............ :'--:.- ........ i ...........
/ / :i i, i "l i i i ".. i
0,21-"/ . . . . . !............ ~" .......... ~ ...... i ..... ~............ i ............ ::...... :".:'"! ............

o., .................................... i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /: ..........

2.5 3 3.5 4
Mean strain / yield strain

Fig. 6. Influence of FEomesh on global stress-strain relation for localtheory (1 = 0) when H = 0 < H~, 0.
 Thermodynamically consistent theory of gradient-p~,ularized plasticity 691

therefore, ~ := u/L. Localization was triggered (at peak nominal stress) at the midsection
(x = L/2) via a slight reduction of the yield stress oy (in a finite element). The used material
parameters are shown in Fig. 5.

............ i . . . . . . . . . . . i"''.: ....... ::

................. o.~ ...........
i i ".. i -- o.os

"5;, I
i

'~;0.5["......................... !............ i i.............. i............ :.~ ............. !.............

I
• "I" ....... ! ! i! ! i "'.. !

r 'i! .............. '.......................... '-7! .............

0.2 ....................................... ............................. :- ............. ::; ...........

o. 1 } ~ ~ . . . . - - i--~........
0
0 0.5 1 1.5 2 2.5 3 3.5
M l m n lltrain /yteld strain

Fig. 7. I n f l u e n c e o f F E - m e s h on global stress-strain relation for gradient-regularized t h e o r y (l = L/4) when

H = 0 (8 = L/4).

2.5 x 10 -4 w

2 ............................................ i ......... ......

1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ .......................... :. . . . . . . . . : ........

-
.Z !.
t .......... " .................
...... .................... .......... ?.........

0 . 5 . . . . . . . . . . . . . . . . . . , . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

[ :

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Coordinate along bar

]Fig. 8. P l a s t i c z o n e d e v e l o p m e n t for 1 = L/4 a n d H = 0 w h e n ~/ey = 1 : 0 . 0 5 : 1.35.

692 T. Svedberg and K. Runesson

V.2. Results
Results are first s h o w n for the choice H = 0. Since Her, 0 = 50~ry, as given from (77)2, we
conclude that / t = (-Her. 0 = ) - 50*ry in this case and (86)2 yields that 8 = 1 with the
possibility for localization right at the onset o f yielding. Figures 6 and 7 s h o w the global

x 10~

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Coordinate along bar

Fig, 9. Strain field development for l -- LI4 and H = 0 w h e n ~/ey = 0 : 0 . 0 5 : 1.35.

: i I .... 1 .................. ~?,,...! ..................

l ! ! I ! !'.

/ i/ ! iii i! ....

i
,o.~r

/
.........../
...........

/
~ ~

~lenlient lenglh I~lr ~

i
i.

i i
i;; ']ii
"

i i

°'~lt ................. il~ i]7:="

..... o~ II............... !ii.................... ~i .........
o~ . . . . . . . . . . . . . . . . . . .................... i ............ i .....

I
0.5 1 1.5 2 2.5
Mean s h i n / yield slmin

Fig. I0. Influence o f F E - m e s h o n global stress-strain relation for local theory (1 = 0) w h e n H = 50~y = H~r. 0.
 Thermodynamically consistent theory of gradient-regularized plasticity 693

force versus displacement characteristics (in terms of average stress ~ versus average strain
~) for different uniform element meshes, where Axe denotes the element length. The well-
known pathological mesh-dependence in the case of local theory (l = 0) is clearly

0.9

0.8

0.7

--~0.6

~0.5

J O.3

0.2

0.1 - - °

0
0 0.5 1.5 2 2.5
Mean strain / yield strain

Fig. 11. Influence of FE-mesh on global stress-strain relation for gradient-regularizedtheory (1 = L/4) when
H = 50~y(8 = ~).

3 x 10"4

2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -', ...................................................

2 ...............................................

~ 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i .... i ........................................

0.1 0.2 0.3 0.4 0.5 0~6 0.7 0.8 0.9

Co~d~atl alongbar

Fig. 12. Plastic zone development for 1 = L/4 and H = 50% when g/ey = 1:0.05:1.75.
694 T. Svedberg and K. Runesson

7 x 10"~

4
i

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I
Coordinate along bar

Fig. 13. Strain field development for l = L/4 and H = 50try when E/Ey = 0:0.05: 1.75.

demonstrated in Fig. 6. That this deficiency is eliminated upon invoking the gradient-
regularized theory (1 = L/4) is evident in Fig. 7. It appears that a converged result has
been obtained when Axe < L/20 = !/5. Figure 8 shows how the plastic multiplier incre-
ment AX =l AeP ] is distributed (along the bar) for increasing end displacement. In parti-
cular, it is demonstrated that the current plastic zone narrows as the damage in the bar
increases, which is in agreement with the conjectured behavior in subsection IV.2. At the
end of the deformation process, "infinite localization" is detected, which may be associ-
ated with the occurrence of a macroscopic crack. Finally, Fig. 9 shows the development of
total strain.

The next series of calculations were carried out for H = 50Oy, which gives/~ = 0. We
thus conclude that localization can (also in this case) occur at the onset of plastic yielding
but in a diffuse manner with ~ = oo initially. The corresponding response curves are
shown in Figs 10-13. Even in this case, an infinitely thin localization zone is obtained at
failure.

VI. CONCLUSIONS

As part of the thermodynamics consideration, the CDI was established for the gradient-
enhanced continuum. This results in the definition of an unorthodox dissipative stress on
the boundary of the body, which is used in establishing thermodynamical constraints on
the boundary conditions for the plastic multiplier increment. To the author's knowledge,
this is a novel feature that sheds some light on the issue of choosing this boundary con-
dition. In fact, previous suggestions to choose homogeneous conditions, that were chosen
ad hoe by Mfihlhaus and Aifantis (1991), are confirmed as admissible.
 Thermodynamically consistent theory of gradient-regularized plasticity 695

Constitutive relations were established for a class of plasticity models with coupling to
damage but with the gradient-regularization restricted to the hardening variables. As part
of the bifurcation analysis of the corresponding incremental deformation field, it was
concluded tlhat the critical orientation of the localization zone is the same as for local
theory for a class of gradient hardening models including the situation that it is isotropic.
(To investigate the more general situation is the subject of future research.) Finally,
numerical results for the tension bar were shown to support the theoretical findings.

Acknowledgem~mts---We wish to acknowledge support from the Swedish National Research Program in
Aeronautics, tl~e Swedish National Institute of Applied Mathematics, and the Swedish Council for Technological
Sciences.

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