ARTICLE IN PRESS
Journal of the Mechanics and Physics of Solids 57 (2009) 1045–1057
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Journal of the Mechanics and Physics of Solids
journal homepage: www.elsevier.com/locate/jmps
A mathematical basis for strain-gradient plasticity theory. Part II:
Tensorial plastic multiplier
N.A. Fleck a,, J.R. Willis b
a
b
Cambridge University, Engineering Department, Trumpington St., Cambridge CB2 1PZ, UK
Department of Applied Mathematics and Theoretical Physics, Wilberforce Rd, Cambridge CB3 0WA, UK
a r t i c l e in fo
abstract
Article history:
Received 21 October 2008
Accepted 27 March 2009
A phenomenological, flow theory version of gradient plasticity for isotropic and
anisotropic solids is constructed along the lines of Gudmundson [Gudmundson, P., 2004.
A unified treatment of strain-gradient plasticity. J. Mech. Phys. Solids 52, 1379–1406].
Both energetic and dissipative stresses are considered in order to develop a kinematic
hardening theory, which in the absence of gradient terms reduces to conventional J2
flow theory with kinematic hardening. The dissipative stress measures, work-conjugate
to plastic strain and its gradient, satisfy a yield condition with associated plastic flow.
The theory includes interfacial terms: elastic energy is stored and plastic work is
dissipated at internal interfaces, and a yield surface is postulated for the work-conjugate
stress quantities at the interface. Uniqueness and extremum principles are constructed
for the solution of boundary value problems, for both the rate-dependent and the rateindependent cases. In the absence of strain gradient and interface effects, the minimum
principles reduce to the classical extremum principles for a kinematically hardening
elasto-plastic solid. A rigid-hardening version of the theory is also stated and the
resulting theory gives rise to an extension to the classical limit load theorems. This has
particular appeal as previous trial fields for limit load analysis can be used to generate
immediately size-dependent bounds on limit loads.
& 2009 Elsevier Ltd. All rights reserved.
Keywords:
Strain-gradient plasticity
Variational methods
Size effects
Plastic collapse
Strengthening and mechanisms
1. Introduction
Strain-gradient theories of plasticity have emerged in recent years, based on the underlying notion that strain gradients
enhance dislocation density and thereby enhance material strength (see for example, Fleck and Hutchinson, 1997, 2001;
Gao et al., 1999; Gurtin, 2000; Gudmundson, 2004). Both phenomenological and crystal versions have been developed, and
have been implemented within a finite element context in order to predict a number of size effects: the Hall–Petch size
effect, the bending and torsional strengths of micron-scale specimens and the indentation size effect (see for example,
Aifantis and Willis, 2005, 2006; Hwang et al., 2002; Wei and Hutchinson, 2003; Kuroda et al., 2007; Engelen et al., 2006;
Evans and Hutchinson, 2009). These higher order theories require higher order boundary conditions in order to satisfy the
governing higher order partial differential equations. In broad terms, the mechanics literature has focussed more on
prediction than on experimental evidence, and aspects of the underlying physics remain unresolved. For example, the
relationship between the material length scale(s) entering the theories and the salient features of the dislocated state
requires further investigation.
Corresponding author. Tel.: +44 1223 748240; fax: +44 1223 332662.
E-mail address: [email protected] (N.A. Fleck).
0022-5096/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jmps.2009.03.007
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Recently, Fleck and Willis (2009) (hereafter referred to as Part I) have synthesised a phenomenological
strain-gradient theory which involves the scalar plastic multiplier _ P as the primary kinematic unknown in
addition to the displacement rate u_ i . Plastic work is taken to be purely dissipative in nature, and stress measures Q and
ti are work-conjugate to _ P and its spatial gradient _ P;i , respectively. They postulate a quadratic yield surface in (Q, ti)
space, and assume that ð_ P ; _ P;i Þ is normal to it. This incremental plasticity theory assumes associated plastic flow and
possesses a convex yield surface; consequently, the theory ensures positive plastic work. Uniqueness and extremum
principles are stated and thereby give the theoretical underpinning for robust numerical solutions and homogenisation
methods.
We emphasise that in Part I, and in Fleck and Hutchinson (2001), the plastic strain increment dPL
ij is taken to
be co-directional with the deviatoric Cauchy stress s0ij sij 13dij skk . Here, we follow the strategy of Gudmundson
(2004) and treat dPL
ij as a kinematically free but incompressible quantity and derive it from the constitutive law as
specified below. In this manner, the current study extends Part I by considering the more general case of a
strain-gradient theory in terms of a tensorial plastic strain rate _ Pij . This more general theory is built upon the
formulation of Gudmundson (2004) and establishes an explicit kinematic hardening theory, involving both dissipative and
energetic stresses within the solid and at internal interfaces. The inclusion of interface terms is similar in concept to that
outlined by Gudmundson (2004) and Aifantis and Willis (2005, 2006), but differs in its implementation through
concentrating on flow theory and on plastic dissipation at interfaces. (Recently, Fredriksson and Gudmundson (2007) have
explored the size effect associated with gradient plasticity within a thin film resting upon an elastic, visco-plastic
interface.)
The higher order theory presented by Gudmundson (2004) is of similar type to the single crystal theories for single
crystal plasticity and visco-plasticity of Gurtin (2000, 2002). A major thrust of the current paper is to outline the variational
structure that accompanies this type of formulation, particularly in the rate-independent limit.
The structure of this paper is as follows. First, the virtual work statement and accompanying field equations are stated
for the strain-gradient solid. The rate-dependent version and accompanying minimum principles are then given. The
central body of the paper deals with the rate-independent solid, and uniqueness and extremum principles are stated. The
rate independent, rigid-hardening case is also considered: the velocity field is the primary kinematic variable, and
uniqueness theorems and extremum principles are stated. Size-dependent limit loads follow immediately, and in the
absence of gradient terms, the classical limit load theorems are recovered.
2. Précis of the Gudmundson (2004) gradient theory
The thermodynamic framework and constitutive law are essentially those given in the work of Gudmundson (2004),
which in turn draws heavily upon Fleck and Hutchinson (1997). The present treatment provides mathematical structure
that accompanies this constitutive law, including uniqueness and minimum principles for rate-dependent and rateindependent versions. Connections are also made with conventional J2 kinematic hardening theory, and rigid-hardening
theory. First, we briefly outline the framework and we then extend the ideas for both rate-dependent behaviour and rateindependent plasticity.
The concepts have already been elaborated in Part I (Fleck and Willis, 2009) in the context of a scalar plastic strain eP and
its gradient. The present work requires a similar development but employs the tensorial plastic strain measure PL
ij as a free
kinematic quantity in addition to the displacement ui. The internal work statement involves increments in elastic strain
PL
PL
dEL
ij , plastic strain dij and plastic strain gradient dij;k . Internal work is also expended at internal interfaces G such as grain
boundaries, such that the displacement rate u_ i is continuous across these interfaces but the plastic strain rate _ PL
ij can jump
in value. We adopt the sign convention that the normal to the interface ni points the direction from the ve side to the +ve
side. The physical arguments given in the introduction of Part I suggest that _ PL
ij on each side of the interface is best written
PLþ
PL
_
_
_ PLþ þ _ PL
across
a
point
of
G
,
along
with the mean value h_ PL
in terms of the jump in value ½_ PL
ij
ij
ij
ij i ðij
ij Þ=2.
(Gudmundson (2004) did not make use of such a decomposition.)
In order to write the principle of virtual work, we introduce the symmetric Cauchy stress sij and the general stress
PL
PL
measures Qij and tijk as work-conjugates to the increments dEL
ij , dij and dij;k , respectively, within the bulk. Likewise, pij
PL
and qij are work-conjugate to [dPL
]
and
/
d
S,
respectively,
on
internal
interfaces.
Then, for a solid of volume V and
ij
ij
external surface S, the principle of virtual work reads
Z n
Z
Z
o
PL
PL
PL
sij dEL
fpij ½dPL
fT i dui þ t ij dPL
(2.1)
ij þ Q ij dij þ tijk dij;k dV þ
ij þ qij hdij ig dG ¼
ij g dS
V
G
S
from which follow the governing equilibrium relations
in V :
on G :
sij;j ¼ 0 and Q ij tijk;k s0ij ¼ 0
pij ¼ nk htijk i and qij ¼ nk ½tijk and on S :
T i ¼ sij nj and t ij ¼ tijk nk
(2.2)
(2.3)
(2.4)
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PL PL
Following Gudmundson (2004), take the internal energy to be of the form U ¼ U(EL
ij ; ij ; ij;k ) and thereby define the
energetic stresses:
@U
;
@EL
ij
sEij ¼
Q Eij ¼
@U
@U
and tEijk ¼ PL
@PL
@ij;k
ij
(2.5)
The natural decomposition for U is
PL PL PL
PL
U ¼ 12Lijkl ðij PL
ij Þðkl kl Þ þ U ðij ; ij;k Þ
(2.6)
in terms of a positive definite elastic modulus Lijkl, giving the usual elasticity relation
sEij ¼ Lijkl ðkl PL
kl Þ
(2.7)
PL
This form will be adopted in the sequel. It should perhaps be remarked that U could, more generally, be taken as
depending on the history of plastic strain and its gradient. It would be necessary only that, during an
E
PL
increment, dUPL ¼ Q Eij dPL
ij þ tijk dij;k . As in Part I, our main concern is to develop a theory that can be applied with relative
ease.
PL
Likewise, a surface energy US ¼ US([PL
ij ],/ij S) is introduced at internal interfaces G, giving rise to the energetic stress
measures
pEij ¼
@U S
@U S
and qEij ¼
@½PL
@h
PL
i
ij
ij
(2.8)
S
PL
PL
PL
We will not elaborate particular choices for the functions UPL(PL
ij ; ij;k ) and U ([ij ],/ij S) in the present paper, and leave
this to future work. One possibility is to write these contributions to the free energy in terms of scalar invariants of the
arguments, as has been done for a deformation theory of plasticity by Fleck and Willis (2004), and Aifantis and Willis
(2005, 2006).
Dissipative stresses follow immediately
in V :
sDij sij sEij ;
on G :
E
pD
ij ¼ pij pij ;
E
QD
ij Q ij Q ij ;
tDijk tijk tEijk
(2.9)
E
qD
ij ¼ qij qij
(2.10)
By the second law, for any physically allowed increment, the change in internal energy is less than or equal to the internal
work, and consequently
D
PL
D
PL
sDij dEL
ij þ Q ij dij þ tijk dij;k X0 in V
(2.11)
PL
D
PL
pD
ij ½dij þ qij hdij iX0 on G
(2.12)
and
D
E
Since (2.11) holds for any dEL
ij it follows that sij ¼ 0, giving sij ¼ sij . The constitutive laws stated below satisfy (2.11) and
D
D
D
_ PL _ PL
(2.12) if (sD
ij ; tijk ) are related to a convex dissipation potential in ðij ; ij;k Þ, and if (pij ; qij ) are related to a convex dissipation
_ PL
potential in ð½_ PL
ij ; hij iÞ. A wide choice of dissipation potentials is possible. Here, we follow Gudmundson (2004) and Fleck
and Hutchinson (1997), and assume that the plastic dissipation depends upon effective strain rates derived from quadratic
forms within V and on G. The next step is to stipulate these effective strain rates.
3. The effective strain rates
P
_ PL
Introduce an overall effective strain rate measure E_ in terms of ð_ PL
ij ; ij;k Þ within V. The uniqueness and extremum
P
principles given in the following sections can be derived for the case where E_ is a convex function, homogeneous and of
degree one, in the strain rate and strain-gradient rate. However, to make the formulation more explicit, and to provide a
P
prototypical working theory, we limit attention to a general quadratic form for ðE_ Þ2 in terms of ð_ PL ; _ PL Þ. First, recall that
ij
ij;k
_ PL
_ PL
ij is incompressible and symmetric and so it has five independent components. Consequently, its gradient ij;k has 15
P
_ 2
_ PL
independent components. Together, they can be expressed as a 20-dimensional vector e_ ¼ ðe_ I Þ ¼ ð_ PL
ij ; ij;k Þ. Then, ðE Þ can be
written symbolically as
P
ðE_ Þ2 AIJ e_ I e_ J
ðI; JÞ sum over 1; 2; . . . ; 20
(3.1)
in terms of the 20 20 symmetric and positive definite matrix AIJ. This definition is sufficiently general that it can describe
an anisotropic solid in addition to the isotropic case. The formulation simplifies considerably for the isotropic solid, as
P
outlined by Fleck and Hutchinson (2001) and Smyshlyaev and Fleck (1996): in this case, E_ can be written in terms of the
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_ PL
usual von Mises scalar invariant _ P of the plastic strain rate _ PL
ij and of the three invariants of ij;k . The interested reader
should refer to Smyshlyaev and Fleck (1996), Fleck and Hutchinson (2001), Gudmundson (2004) or Part I for the details.
_ PL
Likewise, write d_ ¼ ðd_ a Þ ¼ ð½_ PL
ij ; hij iÞ as a 10-dimensional vector, defined at internal interfaces, and construct
_ 2 ¼ aa b d_ a d_ b
D
ða; bÞ sum over 1; 2; . . . ; 10
(3.2)
in terms of the 10 10 symmetric and positive definite matrix (aab). We assume that the plastic dissipation depends
P
_ on G. A visco-plastic formulation is now given, followed by the rate-independent case.
directly upon E_ within V and upon D
The development closely parallels that of Part I and Gudmundson (2004), and so we minimize the discussion here.
4. Visco-plastic version
4.1. Creep potential within V
The 20-dimensional vector r ¼ (rI) ¼ (Qij, tijk) is work-conjugate to e_ I within V, and is the sum of an energetic part
D
rE ¼ (rIE) ¼ (Q Eij ; tEijk ) and a dissipative part rD ¼ (rID) ¼ (Q D
ij ; tijk ). (More precisely, {(Qij,tijk)} has 24 dimensions but the
incompressible constraint permits restriction to the 20-dimensional subspace {(Qij,tijk):Qjj ¼ 0,tjjk ¼ 0}. This point will not
P
be laboured in the sequel.) A power-law creep theory is constructed from a creep potential fðE_ Þ within V as
Z
f¼
_
rD
I deI P
E_
N þ 1 _ 0
s0 _ 0
!Nþ1
(4.1)
P
where ðs0 ; _ 0 ; NÞ are material constants in the usual notation for power-law creep and E_ has already been specified by (3.1).
P
Define the scalar S as the work-conjugate to E_ , such that
P
@f
E_
¼
s
0
P
_ 0
@E_
S¼
!N
(4.2)
Then,
rD
I ¼
@f
S
¼ AIJ e_ J
@e_ I E_ P
(4.3)
and substitution into (3.1) provides the connection
S2 ¼ DIJ r DI rDJ
(4.4)
where DIJ(A1)IJ. The plastic work-rate reads
_
rD
I eI
P
E_
P
¼ SE_ ¼ s0 _ 0
_ 0
!Nþ1
P
and is non-negative as demanded by (2.11). The constitutive dual f*(S) of fðE_ Þ is
ðNþ1Þ=N
N
S
P
P
f ðSÞ ¼ supðSE_ fðE_ ÞÞ ¼
s0 _ 0
s0
Nþ1
_P
(4.5)
(4.6)
E
and e_ I is related to rID according to
e_ I ¼
P
@f
E_
¼
D rD
S IJ J
@r D
I
(4.7)
4.2. Creep potential on G
_ As for the bulk stresses, s is
Introduce the 10-dimensional vector s ¼ (sa) ¼ (pij,qij) as the work-conjugate to d.
D
D
decomposed additively into an energetic part sE ¼ sEa ¼ pEij ; qEijk and a dissipative part sD ¼ sD
a ¼ pij ; qij . The creep
potential on each internal interface is also taken to be power-law in form:
!Mþ1
Z
s00 _ 00 D_
D _
o ¼ s dd (4.8)
M þ 1 _ 00
_ is given by (3.2). Now introduce S as the work-conjugate to D:
_
where ðs00 ; _ 00 ; MÞ are additional creep constants and D
!M
_
@o
D
(4.9)
S¼
¼ s00 0
_
_ 0
@D
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and (4.8) thereby gives
sD
a ¼
@o
@d_ a
¼
S
a d_
_ ab b
D
(4.10)
Substitution of (4.10) into (3.2) provides the direct relation between S and sD
a as
D
S2 ¼ bab sD
a sb
(4.11)
1
)ab. The plastic work-rate at any point on G is non-negative:
!
_ Mþ1
_ a ¼ SD
_ ¼ s0 _ 0 D
sD
d
X0
0
a
0 0 _
0
where bab(a
(4.12)
_ reads
The constitutive dual o*(S) of oðDÞ
o ðSÞ ¼ sup SD_ o D_
_
D
¼
ðMþ1Þ=M
M
S
s00 _ 00 0
s0
Mþ1
(4.13)
and the generalized strain rate d_ a is related to sa according to
_
@o D
d_ a ¼
¼ bab sD
b
@sa
S
(4.14)
The above formulation is a generalization of Fredriksson and Gudmundson (2007) whose potential o* is reproduced by
the choice b12 ¼ b21 ¼ 0 and b11 ¼ 4b22.
4.3. Minimum principle for the rate-dependent constitutive law
The above rate-dependent problem can be formulated via the successive use of two minimum principles.
4.3.1. Minimum Principle I and its dual
Assume that the displacement field ui(x) and the plastic strain PL
ij ðxÞ are known everywhere in the current state, and the
Cauchy stress sij ¼ sEij is given by the elasticity relation (2.7). The energetic stresses (rE, sE) are known from (2.5) and (2.8).
Suppose that the boundary of the solid is subjected to prescribed traction t 0ij over a portion ST and to a prescribed plastic
_ _
strain rate ð_ PL0
ij Þ over the complementary portion SU. Consider any kinematically admissible trial fields ðe ; d Þ, and note that
D
D
_
_
(r *, s *) follow immediately from (4.3) and (4.10). Then, the actual solution for ðe; d; r; sÞ satisfies the following minimum
statement,
Z
Z
Z
P
_ Þ þ sE d_ g dG g
dV
þ
f
o
ð
D
ft 0ij _ PL
(4.15)
H ¼ inf ffðE_ Þ þ r EI e_ I s0ij _ PL
ij
ij g dS
a a
_ PL
ij
V
ST
G
Proof. Setting to zero the first variation of the r.h.s. of (4.15) with respect to _ PL
ij delivers the equilibrium relation (2.2ii) and
P
_ Þ are
the traction relations (2.3) and (2.4ii), along with the constitutive statements (4.3) and (4.10). Since fðE_ Þ and oðD
PL
_
strictly convex in their arguments, the functional possesses a unique minimum, and so the solution for ij is unique. The
above proof is similar to that outlined by Hill (1956a) for a rigid, visco-plastic solid.
A dual principle also exists, as follows. Consider trial equilibrium fields (r*, s*). Then, with the actual solution written as
_ r; sÞ we find that
ðe_ ; d;
Z
Z
Z
fnk tijk _ PL0
ff ðSÞg dV fo ðSÞg dG
I¼
ij g dS SU
V
G
Z
Z
Z
¼ sup
fnk tijk _ PL0
g
dS
f
f
ð
S
Þg
dV
fo ðS Þg dG
(4.16)
ij
ðr ;s Þ
SU
V
G
with equality attained when (r*, s*) is the actual solution. The proof follows immediately from the principle of virtual work
(2.1), together with the strict convexity of f*(S) and of o*(S). &
4.4. Minimum Principle II and its dual
_
Assume that _ PL
ij is known upon making use of the above Minimum Principle I. Then, the velocity field ui is obtained from
a principle similar to the classical minimum principle for an elastic solid. By minimizing the functional:
Z Z
1
0
_
_ PL
fT_ i u_ i g dS
(4.17)
Jðu_ i Þ ¼
Lijkl ð_ ij _ PL
ij Þðkl kl Þ dV V 2
ST
over all u_ i , such that u_ i ¼ u_ 0i on SU, we obtain the unique actual solution u_ i , along with the equilibrium relation (2.2i) in rate
_ ij follows immediately using the rate form of (2.7).
form. The stress rate s
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_ ij uniquely. Write Mijkl(L1)ijkl as the elastic compliance, and
An alternative dual formulation can be used to obtain s
introduce the functional
Z
Z 1
_ ij s
_ kl þ s
_ ij Þ _ ij _ PL
dV Fðs
fT_ i u_ 0i g dS
(4.18)
M ijkl s
ij
V 2
SU
0
_ ij nj ¼ T_ i on ST in order to deliver the actual solution. The
_ ij Þ is minimized over all trial equilibrium fields with s
Then, Fðs
proof is again standard.
5. Rate-independent version
An elasto-plastic, rate-independent version can be derived by taking the limit N-0 and M-0. Strain hardening is
R P
R
_ dt.
admitted explicitly by replacing s0 by SY(EP) and s00 by SY(D), where EP ¼ E_ dt and D ¼ D
5.1. Yield surface and hardening rule in V
Upon taking the rate-independent limit, the flow rule (4.7) reduces to the normality rule
e_ I ¼
@f
P @S
¼ E_
;
@r D
@rD
I
I
SðrD Þ ¼ SY ðEP Þ
(5.1)
P
provided E_ 40. Thus, the plastic strain rate vector e_ lies along the outward normal to the yield surface S(rD) ¼ SY(EP).
P
When S(rD) is less than SY(EP) the response is elastic and E_ ¼ e_ I ¼ 0; then, rD is indeterminate, but must still satisfy the
P
D
balance law (2.2ii). Alternatively, E_ is non-vanishing and r is given in terms of e_ via (4.3), with S ¼ SY(EP). We further
note that the yield surface is shifted from the origin by the energetic stress rE. The fact that rD is indeterminate in the elastic
zone has a major effect upon the mathematical structure of the incremental plasticity law. For example, it is not possible to
use the relation S(rD)pSY(EP) in order to check on the attainment of yield. This is analogous to the case of a conventional,
rigid-plastic solid where the Cauchy stress is not defined within a rigid region, and cannot be used in order to test for
the onset of yield. Hill (1951, 1956b) gave the solution to the paradox for the rigid-plastic solid by outlining the
variational structure and associated uniqueness and extremum principles. We follow a similar strategy here for the straingradient solid.
_ ¼ hðEP ÞE_ P where hdSY/dEP. Consequently, (5.1) can
The hardening rule follows directly from the consistency relation S
be written as
1 @S @S D
_
(5.2)
e_ ¼
r
h @r D @r D
_
_E
Upon making use of the identity r_ D
I ¼ r I r I we can re-write (5.2) in the usual form for associated plastic flow with
kinematic hardening,
1 @S @S
e_ ¼
r_
(5.3)
D
D
H @r
@r
E
with H determined as follows. First, note that ðQ_ ij ; t_ Eijk Þ follows directly from (2.5) as
E
Q_ ij ¼
@2 U
@2 U
_ PL
_ PL
kl þ
kl;m
PL
PL
PL
@ij @kl
@ij @PL
kl;m
(5.4)
and
t_ Eijk ¼
@2 U
PL @ PL
mn
ij;k
@
_ PL
mn þ
@2 U
@
PL @ PL
ij;k
lm;n
_ PL
lm;n
(5.5)
and these linear relations can be written symbolically as
r_ EI ¼ BIJ e_ J
(5.6)
in terms of the 20 20 matrix of coefficients BIJ. This expression gives the evolution law for the centre of the yield surface.
_
_E
Substitution of (5.6) into r_ D
I ¼ r I r I and suitable rearrangement of (5.2) leads to the result (5.3), with
H hþ
1
S2
D
rD
P DPJ BJK DKQ r Q
(5.7)
D
Given e_ , the component of r_ along the normal to the yield surface is known from the constitutive law, but not its
_ ij ¼ s
_ Eij is known from the rate form of the usual elasticity
tangential component. But given e_ and _ ij , the stress rate s
relation (2.7).
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It should perhaps be noted that, while the present structure provides kinematic hardening, this does not preclude the
possibility of constructing an alternative theory in which the centre of the yield surface is described wholly or partly by a
dissipative stress.
5.2. Yield surface and hardening rule on G
In the rate-independent limit, the flow rule (4.14) reduces to the normality statement
@o
_ @S ;
d_ a ¼ D ¼ D
@sa
@sD
a
SðsD Þ ¼ SY ðDÞ
(5.8)
_
provided D40.
Thus, the plastic strain rate vector d_ lies along the outward normal to the yield surface S(sD) ¼ SY(D). When
_ ¼ d_ a ¼ 0: sD is indeterminate, but must still satisfy the balance law
S(sD) is less than SY(D) the response is elastic and D
D
_
(2.3). Dually, for non-vanishing D, s is given in terms of d_ via (4.10), with S ¼ SY(D).
_ where gdSY/dD. Consequently, (5.8) can be written in the
Plastic loading implies the consistency relation S_ ¼ gðDÞD
usual form for associated plastic flow,
1 @S @S
D
(5.9)
d_ ¼
s_
D
D
g @s
@s
_
_E
Now make use of the identity s_ D
I ¼ sI sI to re-write (5.9) as
1 @S @S
d_ ¼
s_
G @sD @sD
(5.10)
The hardening coefficient G is directly linked to g since
p_ Eij ¼
@2 U S
@2 U S
½_ PL þ PL
h_ PL
kl i
PL kl
PL
@½PL
@½
@½
ij
ij @hkl i
kl
(5.11)
q_ Eij ¼
@2 U S
@2 U S
½_ PL
h_ PL
kl þ
kl i
PL
PL
PL
@hij i@½kl @hij i@hPL
i
kl
(5.12)
and
and these linear relations can be written symbolically as
s_ Ea ¼ C ab d_ b
(5.13)
_
_E
Substitution of (5.13) into s_ D
I ¼ sI sI and suitable rearrangement of (5.9) leads to the result (5.10), with
Ggþ
1
S2
D
sD
a bbw C wZ bZl sl
(5.14)
In the remainder of this paper we shall focus on the rate-independent case. A unique solution to a boundary problem at
any instant requires knowledge of both the current surface traction and its rate. The solution process follows the same
sequence as was outlined in Part I and involves the application of two minimum principles. These minimum principles are
variants of the corresponding principles presented in Part I, now admitting variations with respect to the tensor _ PL
ij rather
than the scalar _ P. The mathematical structure of these principles (and the attendant uniqueness statements) have close
parallels with the development of Hill (1951, 1956b) for a conventional rigid-hardening solid.
6. Minimum Principle I to obtain the stress quantities (r, s)
Suppose that the boundary of the elasto-plastic solid is subjected to prescribed traction fields (T 0i ; t 0ij ) over
a portion ST and to the displacement-type loading (u0i ; PL0
ij ) over the complementary portion SU. Assume that the stress
_ r; sÞ as the actual solution, and recall that (r, s) is given by
sij(x) and the plastic strain PL
are
known
everywhere.
Write
ðe_ ; d;
ij
(4.3) and (4.10) in the active plastic zone and is indeterminate in the elastic zone. Consider any kinematically admissible
trial fields ðe_ ; d_ Þ, and note that r* follows immediately from (4.3) provided e_ a0. If e_ ¼ 0 then r* is indeterminate but is
within or on the yield surface. Likewise, s* follows immediately from (4.10) provided d_ a0. If d_ ¼ 0 then s* is
_ r; sÞ satisfies the following minimum
indeterminate but is within or on the yield surface. Then, the actual solution for ðe_ ; d;
statement:
Z
ft ij _ PL0
H¼
ij g dS
SU
Z
Z
Z
P
_ þ sE d_ g dG ¼ inf fSY E_ þ r EI e_ I s0ij _ PL
fSY D
ft 0ij _ PL
(6.1)
ij g dV þ
ij g dS
a a
_ PL
ij
V
G
ST
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_ are normal to their respective yield surfaces. Also, the yield surfaces
Proof. The relations (5.1) and (5.8) state that ðe_ ; dÞ
S(r) ¼ SY and S(s) ¼ SY are convex, and so the ‘maximum plastic work principle for a material element’ reads
ðr I r I Þe_ I X0 and ðr I r I Þe_ I X0
in V :
(6.2)
ðsa sa Þd_ a X0 and ðsa sa Þd_ a X0
on G :
(6.3)
Now make use of the balance laws (2.2ii) and (2.3) to obtain
Z
Z
Z
Z
fr I e_ I g dV þ fsa d_ a g dGX fr I e_ I g dV þ fsa d_ a g dG
V
G
V
G
Z
Z
¼ fs0ij _ PL
fnk tijk _ PL
ij g dV þ
ij g dS
V
(6.4)
S
P
_ þ sE d_ the above inequality can be re-arranged to the form
Upon recalling that r I e_ I ¼ SY E_ þ r EI e_ I and that sa d_ a ¼ SY D
a a
of (6.1). This Minimum Principle delivers unique values for (r, s) in the active plastic zone (see Appendix A for a discussion
_ are also unique up to a multiplicative factor L
_ which is constant in any active
of uniqueness). The plastic strain rates ðe_ ; dÞ
_ unambiguously,
.
In
order
to define L
plastic zone since the functional in (6.1) is homogeneous and of degree one in _ PL
ij
introduce a strain rate field of ‘unit magnitude’ ^ ij ðxÞ within each active plastic zone Va, such that
Z
1
f^ ^ g dV a ¼ 1
(6.5)
V a V a ij ij
_ as the scaling factor for the plastic strain rate such that
Then, write L
_^
_ PL
ij ðxÞ ¼ Lij ðxÞ &
(6.6)
6.1. Dual formulation
Consider trial equilibrium fields (r*, s*) that do not violate the yield condition at any point. Then, with the actual
_ r; sÞ we find that
solution written as ðe_ ; d;
Z
Z
fni tijk _ PL0
g
dS
¼
sup
fnk tijk _ PL0
(6.7)
I¼
jk
ij g dS
ðr ;s Þ
SU
SU
with equality attained when (r*, s*) is the actual solution. The proof follows immediately from the principle of virtual work
(2.1), together with (6.2) and (6.3). Relation (6.7) provides a lower bound restriction on tij while (6.1) provides an upper
bound, in direct correspondence to the classical theorems of limit load analysis Hill (1948, 1951).
7. Minimum Principle II to obtain the plastic multipliers and the velocity field
Assume that the stress sij(x) is known everywhere, and (r,s) is known in the active plastic zone upon making use of
_
^
Minimum Principle I. Write _ PL
ij ðxÞ in terms of the plastic multiplier L and the known ‘unit’ field ij ðxÞ as given by relation
_ is constant within each active plastic zone, but is as yet undetermined. Then, minimizing the functional
(6.5). Recall that L
Z
_ Þ ¼ 1 fL ð_ _ PL Þð_ _ PL Þ þ HE_ P2 g dV
Jðu_ i ; L
ij
kl
kl
2 V ijkl ij
Z
Z
1
0
0
_ 2 g dG fGD
fT_ i u_ i þ t_ij _ P
(7.1)
þ
ij g dS
2 G
ST
_ X0Þ delivers the actual solution ðu_ i ; L
_ Þ. In (7.1), _ PL ¼ L
_ ^ ij .
over all ðu_ i ; L
ij
P
P
_ Þ in addition to the actual solution ðu_ i ; E_ ; DÞ
_ and write Du_ i u_ u_ i ,
Proof. Consider the trial solution ðu_ i ; E_ ; D
i
P
P
P
_
_
_
_
_
_
_
_
DE E E , DD D D and DJ Jðu_ i ; L Þ Jðu_ i ; LÞ. It follows that
Z
_ EL
_ _
DJX fLijkl _ EL
kl Dij þ HAIJ eJ DeI gdV
V
Z
Z
0
0
þ fGaab d_ b Dd_ a gdG fT_ Du_ þ t_ D_ PL gdS
(7.2)
G
ST
i
i
ij
ij
and this may be re-written as
Z
Z
DJX fðHAIJ e_ J r_ I ÞDe_ I g dV þ fðGaab d_ b s_ a ÞDd_ a g dG
V
(7.3)
G
via the principle of virtual work (2.1). We can demonstrate that the right-hand side of (7.3) is non-negative by proving that
r_ I De_ I pHAIJ e_ J De_ I and s_ a Dd_ a pGaab d_ b Dd_ a at each point. The detailed argument is omitted here, as it routinely follows that
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1053
P
of Hill (1956b), and of Part I. In brief, we show that r_ I De_ I pHAIJ e_ J De_ I by considering in turn the four possibilities: (i) E_ 40,
P
P
P
P
P
P
P
E_ 40; (ii) E_ 40, E_ ¼ 0; (iii) E_ ¼ 0, E_ 40; (iv) E_ ¼ 0, E_ ¼ 0. Likewise, to show that s_ a Dd_ a pGaab d_ b Dd_ a , we consider
the analogous four possibilities. Uniqueness of the solution is demonstrated in Appendix A.
&
Further manipulation of (7.1) using (5.3) and (5.10) allows DJ in (7.3) to be re-expressed as
Z
Z
P
_ DDg
_ ÞDE_ P g dV þ fðGD
_ SÞ
_ dG
DJX fðHE_ S
V
(7.4)
G
Straightforward manipulations can be used to show that at the solution,
Z
Z
0
0
fT_ i u_ 0i þ t_ ij _ PL0
fT_ i u_ i þ t_ij _ PL
2J min ¼
ij g dS ij g dS
SU
(7.5)
ST
7.1. Dual formulation
_ uniquely. Write Mijkl(L1)ijkl as the elastic compliance,
_ ; SÞ
_ ij ; S
An alternative dual formulation can be used to obtain ðs
and introduce the functional
8
8
! 9
! 9
Z <
Z
D 2
D 2=
1 <
1 DIJ r J r_ I
1 bab sb s_ a =
_ _
_ s
_ þ
_ ij ; r I ; sa Þ dV þ
dG
M s
Fðs
;
;
S
S
2 V : ijkl ij kl H
G :2G
Z
fT_ i u_ 0i þ t_ij _ PL0
(7.6)
ij g dS
SU
Fð _ ij ; r_ I ; s_ a Þ
is minimized over all equilibrium fields in order to deliver the actual solution.
Then, s
The proof is based upon the principle of virtual work (2.1) written in the form
Z
Z
Z
_ _
_ ij _ EL
fDT_ i u_ i þ Dt_ ij _ PL
fDs
fDs_ a d_ a g dG
ij g dS ¼
ij þ Dr I eI g dV þ
S
V
(7.7)
G
where the symbol D again denotes the difference between the trial equilibrium field solution (asterisked) and the actual
solution (no asterisk). Substitute the following inequalities into (7.7):
1
2
1
2
_ _
_ _
Ds_ ij _ EL
ij p M ijkl sij skl M ijkl sij skl
P
for E_ 40 :
Dr_ I e_ I p
_
and for D40
:
1
2H
_
DIJ r D
J rI
!2
S
(7.8)
_
DIJ r D
J rI
1
2H
D 1 ba b sb s_ a
Ds_ a d_ a p
S
2G
!2
!2
(7.9)
S
D
1 ba b sb s_ a
S
2G
!2
(7.10)
to obtain
_ ij ; r_ I ; s_ a ÞpFðs
_ ij ; r_ I ; s_ a Þ
Fðs
(7.11)
_ ij ; r_ I ; s_ a Þ can be written in the form
Finally, we note that Fðs
Z
Z
0
0
_ ij ; r_ I ; s_ a Þ ¼
2Fðs
fT_ i u_ 0i þ t_ij _ PL0
fT_ i u_ i þ t_ ij _ PL
ij g dS ij g dS
SU
ST
and so the values of 2J in the extremum of (7.1) and of 2F in the extremum of (7.12) approximate the quantity
R
_0 _
_ 0 _ PL
t_ij _ PL0
ij g dS S fT i ui þ t ij ij g dS from above and below.
(7.12)
R
_ _0
SU fT i ui
þ
T
8. The rigid-hardening solid
The above rate-independent theory can be reduced to a rigid-hardening version absent elasticity.1 This modified
constitutive law is the natural extension to classical, rigid-hardening theory as expounded by Hill (1956b). Uniqueness and
extremum principles persist and limit theorems follow immediately from them. In the absence of elastic straining, the
incompressible plastic strain field is derived directly from the displacement field:
1 _
_
_ PL
ij 2ðui;j þ uj;i Þ;
1
_ PL
kk ¼ 0
(8.1)
This contrasts with the scalar version of Part I. A rigid-hardening version was not developed for the scalar version, for which only the effective stress
se is determined by the scalar analogue of (2.2ii).
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The elastic term in the principle of virtual work (2.1) is now dropped and it becomes
Z
Z 1
PL
PL
PL
Q ij dPL
fpij ½dPL
ij þ sii dkk þ tijk dij;k dV þ
ij þ qij hdij ig dG
3
V
G
Z
¼ fT i dui þ t ij dPL
ij g dS
(8.2)
S
along with the governing equilibrium relations (2.2)–(2.4). In (8.2), the hydrostatic component of Cauchy stress 13sii is the
Lagrange multiplier associated with the material incompressibility, _ PL
kk ¼ 0. The deviatoric part of Cauchy stress is now
defined by (2.2ii). The virtual work statement (8.2) can be simplified further upon exploiting (8.1). Introduce the notation qi
for the gradient operator, and on the surface S with unit normal nj, decompose qi into a normal operator Dniqi and a
surface gradient operator Di such that
@i ¼ Di þ ni D
(8.3)
Upon making use of Stokes’ divergence theorem (see for example, Mindlin (1964) or Fleck and Hutchinson (1997)) the
work principle (8.2) reduces to
Z n
Z o
1
PL
PL
PL
Q ij dPL
pij ½dPL
ij þ sii dkk þ tijk dij;k dV þ
ij þ qij hdij i dG
3
V
G
Z
¼ fT̄ i dui þ Ri ðDdui Þg dS
(8.4)
S
where
T̄ i ¼ T i Dj t ij þ t ij nj ðDp np Þ ¼ sij nj Dj ðtijk nk Þ þ tijk nj nk ðDp np Þ
(8.5)
Ri ¼ t ij nj ¼ tijk nj nk
(8.6)
and
Incompressibility dictates a kinematic restriction on the normal component of Ddui at the surface of the body:
ni Ddui ¼ Dj duj
(8.7)
Consequently, the number of independent higher order boundary conditions is reduced from 3 to 2, and the virtual work
statement is altered. After some manipulation of (8.4), and upon making use of (8.7) we obtain
Z
Z 1
PL
PL
PL
Q ij dPL
fpij ½dPL
ij þ sii dkk þ tijk dij;k dV þ
ij þ qij hdij ig dG
3
V
G
Z
¼ fT^ i dui þ R^ i ðDdui Þg dS
(8.8)
S
where the two independent higher order tractions R^ i tangential to S are
R^ i ¼ Ri ni Rj nj ¼ tijk nj nk ni tpjk np nj nk
(8.9)
and
T^ i ¼ T̄ i ni Rj nj ðDp np Þ ¼ sij nj Dj ðtijk nk Þ þ ðtijk nj nk ni tpjk np nj nk ÞðDp np Þ
(8.10)
The yield surface and the flow laws for the rigid-hardening solid remain as before, as expressed by the relations
(5.1)–(5.14).
Minimum principles (and the attendant uniqueness principles) can be developed for the rigid-hardening solid in a
similar manner to that described above for the elastic–plastic solid. These principles along with their proofs are detailed in
Appendix B.
9. Concluding remarks
The constitutive framework elaborated above for both rate-dependent and rate-independent solids can be implemented
numerically using standard finite element methods and minimization procedures. This has been illustrated by Idiart et al.
(2009) for the prototypical problem of a thin foil in bending.
Acknowledgements
The authors are grateful to Dr. M. Idiart and Prof. J.W. Hutchinson for insightful discussions, and to the EPSRC for
financial support in the form of contract no. EP/C52392X/1. NAF wishes to thank the Department of Applied Mathematics
and Theoretical Physics of Cambridge University for providing hospitality during his sabbatical leave.
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1055
Appendix A. Uniqueness principles
A.1. Uniqueness Principle I for the stress quantities (r, s)
In the current state, the stress sij(x), displacement ui(x) and plastic strain PL
ij ðxÞ are known everywhere. We shall now
show that the generalized stress (r, s) is unique within each active plastic zone, while the plastic strain rate _ PL
ij ðxÞ is known
_ , as defined by (6.6). For the case where _ PL ðxÞ is specified at some point, the
uniquely up to an arbitrary plastic multiplier L
ij
_ is determined.
associated multiplier L
Consider two solutions for (r, s) and for ðu_ i ; _ P Þ, and denote one solution by an asterisk and the other without an asterisk.
The difference between the two solutions is denoted by the symbol D. Then, the principle of virtual work (2.1) gives
Z
Z
Z
_
fDsij D_ EL
fDsa Dd_ a g dG
(A.1)
0 ¼ fDT i Du_ i þ Dt ij D_ PL
ij g dS ¼
ij þ Dr I DeI g dV þ
S
V
G
We assert that Dsij ¼ 0. Now, (6.2) and (6.3) give Dr I De_ I X0 unless (rI* ¼ rI) and Dsa Dd_ a X0 unless (sa* ¼ sa). (A.1) implies
that (r, s) are unique in the active plastic zone while De_ I and Dd_ a may be non-zero. Within any given plastic zone, _ PL
ij ðxÞ is
_.
known up to the value of the arbitrary plastic multiplier L
_ is known for any active plastic zone when a non-zero _ PL ðxÞ is prescribed on a portion of the
Further, the value of L
ij
boundary of the plastic zone. This can occur when a portion SU of the surface of the body is subjected to prescribed nonzero _ PL
ij ðxÞ.
A.2. Uniqueness Principle II for the plastic multiplier
Assume that the stress sij(x) is known everywhere, and (r,s) are known in the active plastic zone. We shall now show
_ within each active plastic zone and
that a stipulation of the traction rate data is sufficient to ensure a unique solution for L
ðxÞ.
thereby _ PL
ij
Consider two solutions for ðr_ ; s_ Þ and for ðu_ i ; _ PL
ij Þ, and denote the first solution by a superscript asterisk (*), and the second
solution by the asterisk absent. The first solution minus the second solution is labeled by the symbol D. Then, the principle
of virtual work (2.1) implies
Z
Z
Z
EL
_
_
_
_
g
dS
¼
f
D
s
D
þ
D
D
g
dV
þ
fDs_ a Dd_ a g dG
(A.2)
0 ¼ fDT_ i Du_ i þ Dt_ ij D_ PL
r
e
I
I
ij
ij
ij
S
V
G
Consider the integrand on the right-hand side of (A.2). We shall show that it is positive unless the two solutions coincide.
Assume that the elastic modulus Lijkl is positive definite. Then,
_ EL _ EL
Ds_ ij D_ EL
ij ¼ Lijkl Dij Dkl X0
(A.3)
Second, note that (5.1) and (5.8) deliver
P
_ E_ and Ds_ a Dd_ a ¼ DS_ DD
_
Dr_ I De_ I ¼ DSD
(A.4)
P
P
Now, follow the argument of Hill (1956b), and of Part I, by considering in turn the four possibilities: (i) E_ 40, E_ 40; (ii)
P
P
P
P
P
_ E_ P XHðDE_ P Þ2 X0 provided H40,
E_ 40, E_ ¼ 0; (iii) E_ ¼ 0, E_ 40; and (iv) E_ ¼ 0, E_ ¼ 0. These lead to the result that DSD
2
_
_
_
and likewise DSDDXGðDDÞ X0, provided G40.
_P _
The work statement (A.2) implies that the elastic strain rate _ EL
kl and the plastic multipliers ðE ; DÞ are unique.
_
_ and ðe_ ; dÞ are unique.
_ ij , L
Consequently, s
P
Appendix B. Minimum principles for the rigid-hardening solid
B.1. Minimum Principle I for the rigid-hardening solid
The Uniqueness Principle as given in Appendix A.1 also applies to the rigid-hardening solid. The principle reveals
that the generalized stress (r, s) is unique within each active plastic zone, while the plastic strain rate _ PL
ij ðxÞ is
_ , as defined by (6.6). For the case where a non-zero _ PL ðxÞ is specified
known uniquely up to an arbitrary plastic multiplier L
ij
_ is determined. The Cauchy stress deviator s0 ij is related to (r, s) via the balance
at some point, the associated multiplier L
law (2.2ii).
The minimum principle (6.1) simplifies considerably upon making use of the constraint (8.1) for the rigid-hardening
_ _
solid. Assume that the stress sij(x) and the plastic strain PL
ij are known everywhere. Write ðe; d; r; sÞ as the actual solution,
and recall that (r, s) is given by (4.3) and (4.10) in the active plastic zone and is indeterminate in the elastic zone. Consider
any kinematically admissible trial fields ðe_ ; d_ Þ, and note that r* follows immediately from (4.3) provided e_ a0. If e_ ¼ 0
_
then r* is indeterminate but is within or on the yield surface. Likewise, s* follows immediately from (4.10) provided d a0.
_
_
If d ¼ 0 then s* is indeterminate but is within or on the yield surface. Then, the actual solution for ðe_ ; d; r; sÞ satisfies the
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following minimum statement:
Z
fT̄ i u_ 0i þ Ri ðDu_ 0i Þg dS
H¼
SU
Z
Z
Z
P
0
_ þ sE d_ g dG fSY E_ þ rEI e_ I g dV þ fSY D
fT̄ i u_ i þ R0i ðDu_ i Þg dS
¼ inf
a a
u_ i
V
(B.1)
ST
G
The proof follows directly upon substitution of (6.2i) and (6.3i) into the principle of virtual work (8.4).
Upon admitting incompressibility, the principle of virtual work (8.8) can be used as the starting point, and a slightly
R
0
0
modified version of the minimum principle will arise, such that the surface term in (B.1) now reads ST fT^ i u_ i þ R^ i ðDu_ i Þg dS.
Similar modifications to the minimum principles stated below can be made in a straightforward manner (omitted for the
sake of brevity).
B.1.1. Dual formulation
A dual formulation exists to (B.1). Instead of considering trial kinematic fields we now consider trial equilibrium fields
_ r; sÞ. Then, we find that
(r*, s*) that do not violate the yield condition at any point. Write the actual solution as ðe_ ; d;
Z
Z
fT̄ i u_ 0i þ Ri ðDu_ 0i Þg dS ¼ sup
fT̄ i u_ 0i þ Ri ðDu_ 0i Þg dS
H¼
SU
ðr ;s Þ
Z
¼ sup
ðr ;s Þ
fQ ij _ PL
ij
_ PL
ijk ij;k g dV
þt V
Z
SU
þ
G
fsa d_ a g dG Z
ST
fT̄ i u_ i þ Ri ðDu_ i Þg dS
(B.2)
with equality attained when (r*, s*) is the actual solution. This statement is analogous to Hill’s (1948) maximum plastic
work principle. The proof is a direct consequence of (6.2ii) and (6.3ii), along with the principle of virtual work (8.4). We
conclude that (B.2) provides a lower bound restriction on H while (B.1) provides an upper bound, in direct correspondence
to the classical theorems of limit load analysis. In fact, the trial fields from upper and lower bound analysis can be used
immediately in order to obtain bounds on size effects. But now a word of caution: the augmented upper bound on collapse
load may be infinite when the upper bound solution for the conventional solid has singular strain fields. A prototypical
example is the circular fan in plane strain slip-line field analysis at the tip of a crack or beneath an indenter. In this case, the
strain rate is of order 1/r and so the volume integral on the right-hand side of (B.1) is unbounded.
B.2. Minimum Principle II for the rigid-hardening solid
Assume that the stress sij(x) and (r,s) are known in the active plastic zone upon making use of Minimum Principle I.
_
^
_ PL
Write _ PL
ij ðxÞ in terms of the plastic multiplier L and the known ‘unit’ field ij ðxÞ as given by relation (6.6). The field ij ðxÞ is
_
derivable from a continuous velocity field u_ i ðxÞ according to (8.1). We emphasise that L is constant within each active
plastic zone, but is as yet undetermined. Consider the functional:
Z
Z
Z
0
0
_ Þ ¼ 1 fHE_ P2 g dV þ 1 fGD
_ 2 g dG JðL
fT̄_ i u_ i þ R_ i ðDu_ i Þg dS
(B.3)
2 V
2 G
ST
_ X0 to deliver the actual solution L
_.
We shall show that this can be minimized over all L
P
_ in addition to the actual solution ðu_ ; E_ P ; DÞ
_ Þ which scales with L
_ which scales
Proof. Consider the trial solution ðu_ i ; E_ ; D
i
P
P
P
_ Þ JðL
_ Þ. It follows that
_ , and write Du_ u_ u_ , DE_ E_ E_ , DD
_ and DJ JðL
_ D
_ D
with L
i
i
i
Z
Z
Z
0
0
DJX fHAIJ e_ J De_ I g dV þ fGaab d_ b Dd_ a g dG fT_ i Du_ i þ t_ij D_ PL
(B.4)
ij g dS
V
ST
G
and, upon making use of the principle of virtual work (8.4) in rate form, this reduces to
Z
Z
DJX fðHAIJ e_ J r_ I ÞDe_ I g dV þ fðGaab d_ b s_ a Þ Dd_ a g dG
V
(B.5)
G
The argument of Section 7 is now repeated to show that the right-hand side of (B.5) is non-negative and thereby the
minimum principle is established. At the solution we find
Z
Z
0
0
2J min ¼
fT̄_ i u_ 0i þ R_ i ðDu_ 0i Þg dS fT̄_ i u_ i þ R_ i ðDu_ i Þg dS &
(B.6)
SU
ST
B.2.1. Dual formulation
_ uniquely. Introduce the functional
_ ; SÞ
An alternative dual formulation can be used to obtain ðS
9
9
8
8
!
!
Z <
Z <
Z
D 2
D 2=
1 DIJ r J r_ I
1 bab sb s_ a =
dV þ
dG Fðr_ I ; s_ a Þ fT̄_ i u_ 0i þ R_ i ðDu_ 0i Þg dS
;
;
:
:
S
S
2H
2G
V
G
SU
(B.7)
ARTICLE IN PRESS
N.A. Fleck, J.R. Willis / J. Mech. Phys. Solids 57 (2009) 1045–1057
1057
Then, Fðr_ I ; s_ a Þ is minimized over all equilibrium fields in order to deliver the actual solution. The proof begins with the
principle of virtual work (8.4) written in the form
Z
Z
Z
_ _ PL
_ _ PL
fDQ_ ij _ PL
fDp_ ij ½_ PL
fDT̄_ i u_ i þ DR_ i ðDu_ i Þg dS
(B.8)
ij þ Dtijk ij;k g dV þ
ij þ Dqij hij ig dG ¼
V
S
G
where the symbol D again denotes the difference between the trial equilibrium field solution (asterisked) and the actual
P
_
we find that (7.10) is satisfied.
solution (no asterisk). For E_ 40 we again find that (7.9) is satisfied, while for D40
Consequently we obtain Fðr_ I ; s_ a ÞpFðr_ I ; s_ a Þ. This completes the proof.
Finally, we note that Fðr_ I ; s_ a Þ can be written in the form
Z
Z
0
0
2Fðr_ I ; s_ a Þ ¼
fT̄_ i u_ 0i þ R_ i ðDu_ 0i Þg dS fT̄_ i u_ i þ R_ i ðDu_ i Þg dS
(B.9)
SU
ST
and so we conclude that the values of
2J in the extremum of (B.3) and of 2F in the extremum of (B.9) approximate the
0
R
R
0
quantity SU fT̄_ i u_ 0i þ R_ i ðDu_ 0i Þg dS ST fT̄_ i u_ i þ R_ i ðDu_ i Þg from above and below.
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