Stress regularity in quasi-static perfect plasticity with a pressure dependent yield criterion

0.33 MB·

by  Maria Giovanna Mora

STRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY WITH A
PRESSURE DEPENDENT YIELD CRITERION
JEAN-FRANC¸OISBABADJIANANDMARIAGIOVANNAMORA
7
1
0 Abstract. Thisworkisdevotedtoestablishingaregularityresultforthestresstensorinquasi-
2 static planar isotropic linearly elastic – perfectly plastic materials obeying a Drucker-Prager
or Mohr-Coulomb yield criterion. Under suitable assumptions on the data, it is proved that
n
the stress tensor has a spatial gradient that is locally squared integrable. As a corollary, the
a
usual measure theoretical flow rule is expressed in a strong form using the quasi-continuous
J
representativeofthestress.

6
1

]
P
1. Introduction
A
. Perfect plasticity is a class of models in continuum solid mechanics involving a fixed threshold
h
criterion on the Cauchy stress. When the stress is below a critical value, the underlying mate-
t
a rial behaves elastically, while the saturation of the constraint leads to permanent deformations
m
after unloading back to a stress-free configuration. Elasto-plasticity represents a typical inelastic
[ behavior, whose evolution is described by means of an internal variable, the plastic strain.

1 To formulate more precisely the problem, let us consider a bounded open set Ω Rn (in the
v ⊂
following,onlythedimensionn=2willbeconsidered),whichstandsforthereferenceconfiguration
9
of an elasto-plastic body. In the framework of small strain elasto-plasticity the natural kinematic
0
3 and static variables are the displacement field u : Ω [0,T] Rn and the stress tensor σ :
4 Ω [0,T] Mn×n, where Mn×n is the set of n n×symmetr→ic matrices. In quasi-statics the
× → sym sym ×
0 equilibrium is described by the system of equations
.
1
divσ =f in Ω [0,T],
0 − ×
7 for some given body loads f :Ω [0,T] Rn. Perfect plasticity is characterizedby the existence
1 × →
of a yield zone in which the stress is constrained to remain. The stress tensor must indeed belong
:
v to a given closed and convex subset K of Mn×n with non empty interior:
sym
i
X
σ K.
∈
r
a If σ lies inside the interior of K, the material behaves elastically, so that unloading will bring the
body back to its initial configuration. On the other hand, if σ reaches the boundary of K (called
the yield surface), a plastic flow may develop, so that, after unloading, a non-trivial permanent
plastic strain will remain. The total linearized strain, denoted by Eu := (Du+DuT)/2, is thus
additively decomposed as
Eu=e+p.
The elastic strain e:Ω [0,T] Mn×n is related to the stress through the usual Hooke’s law
× → sym
σ :=Ce,
Date:January17,2017.
Key words and phrases. Elasto-plasticity, Convex analysis, Quasi-static evolution, Regularity, Functions of
boundeddeformation,Capacity.
1
2 J.-F.BABADJIANANDM.G.MORA
where C is the symmetric fourth order elasticity tensor. The evolution of the plastic strain p :
Ω [0,T] Mn×n is described by means of the flow rule
× → sym
p˙ N (σ), (1.1)
K
∈
where N (σ) is the normal cone to K at σ. From convex analysis, N (σ) = ∂I (σ), i.e., it
K K K
coincides with the subdifferential of the indicator function I of the set K (where I (σ) = 0
K K
if σ K, while I (σ) = + otherwise). Hence, from convex duality, the flow rule can be
K
∈ ∞
equivalently written as
σ:p˙ =maxτ:p˙ =:H(p˙), (1.2)
τ∈K
where H : Mn×n [0,+ ] is the support function of K. This last formulation (1.2) is nothing
sym → ∞
but Hill’s principle of maximum plastic work, and H(p˙) denotes the plastic dissipation.
Standard models used for most of metals or alloys are those of Von Mises and Tresca. These
kinds of materials are not sensitive to hydrostatic pressure, and plastic behavior is only generated
through critical shearing stresses. In these models, if σ := σ trσId stands for the deviatoric
D − n
stress, the elasticity set K is of the form
σ Mn×n :κ(σ ) k ,
{ ∈ sym D ≤ }
where k > 0. The Von Mises yield criterion corresponds to κ(σ ) = σ , while that of Tresca
D D
| |
to κ(σ ) = σ σ where σ σ are the ordered principal stresses. The mathematical
D n 1 1 n
− ≤ ··· ≤
analysis of such models has been performed in [28, 30, 5, 12].
Ontheotherhand,inthe contextofsoilmechanics,materialssuchassandorconcreteturnout
to develop permanent volumetric changes due to hydrostatic pressure. Typical models are those
of Drucker-Prager and Mohr-Coulomb (see [18]), which can be seen as generalizations of the Von
Mises and Tresca models, respectively. In these cases, the elasticity set K takes the form
σ Mn×n :κ(σ )+αtrσ k ,
{ ∈ sym D ≤ }
where α,k > 0. The main difference between metals and soils, is that for the latter, there are in
general no directions along which the Cauchy stress is bounded. These models have been studied
in [9] (see also [25]).
A common feature to all the models of perfect plasticity described so far is that they develop
strain concentration leading to discontinuities of the displacement field. This has been a major
difficulty in defining a suitable functional framework for the study of such problems. It has been
overcome by the introduction of the space BD of functions of bounded deformation (see [29, 30])
and through a suitable relaxation procedure (see [7, 24]). Solutions in the energy space must at
least satisfy the following regularity: for all t [0,T],
∈
u(t) BD(Ω), e(t), σ(t) L2(Ω;Mn×n), p(t) (Ω;Mn×n),
∈ ∈ sym ∈M sym
where (Ω;Mn×n) stands for the space of bounded Radon measures in Ω.
M sym
Higher regularity of solutions appears therefore as a natural question. For dynamical problems
ithas(onlyrecently)beenestablishedin[23]thatforanyelasticitysetK thesolutionsaresmooth
inshorttime,providedthe dataaresmoothandcompactlysupportedinspace. Sucharesultdoes
not hold in the static or quasi-static cases (see the examples in [28, Section 2] or [17, Section 10]).
However, some partial regularity results are available for the stress in some particular situations.
Indeed, it has been provedin [10, 26, 17, 14] that for a Von Mises elasticity set, the Cauchy stress
satisfies
σ L∞(0,T;H1 (Ω;Mn×n)).
∈ loc sym
Unfortunately, the proofs of these results are very rigid and do not easily extend to other types of
elasticity sets. A more generalresult has been obtained in [20], where it has been provedthat the
STRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 3
same regularity result holds if K = K RId, where K is a smooth compact convex subset of
D D
⊕
deviatoric symmetric matrices, with positive curvature.
In the footsteps of [9], we address here the question of deriving similar regularity properties for
the stress tensor in the case of Drucker-Pragerand Mohr-Coulomb elasticity sets. A result in this
direction has been obtained in [27] in the static case. The object of this present work is to extend
this resultto the quasi-staticcase. We indeedprovethatin dimensionn=2, the stress tensorhas
the expected regularity σ L∞(0,T;H1 (Ω;M2×2)) (see Theorem 2.7).
∈ loc sym
Our proof rests on a duality approach, analogous to that of [27] and [17]. Since the solutions
to the perfect plasticity model are singular, one needs first to regularize the problem. To this
aim we consider a visco-plastic approximation of Perzyna type (see [28]). In contrast with the
Kelvin-Voigt visco-elastic regularization used in [27, 17], it modifies the dissipation potential H
(see (1.2)) in sucha way that the regularizedflow rule is not givenby a differential inclusion as in
(1.1),butis adifferentialequation. Inotherwords,the regularizedplasticstrainrateisunivocally
determined by the stress (see (3.3)). To the best of our knowledge, this is the first time that this
kind of approximation is used to establish regularity of the stress.
In the following we explain the strategy of our proof and the main difficulties. For simplicity
of exposition we neglect the terms due to the visco-plastic approximation (which are those ε-
dependent), we assume the solution to be smooth, and we show how a uniform estimate on the
H1 (Ω;Mn×n)-norm of σ(t) can be obtained.
loc sym
In the static case [27] and in the absence of external body loads, the equilibrium states (u,e,p)
minimize the energy functional
1
(v,η,q) Cη :ηdx+ H(q)dx
7→ 2ˆ ˆ
Ω Ω
among all triples (v,η,q) satisfying the additive decomposition Ev = η +q and the boundary
condition. Minimizing first with respect to q shows that u is actually a minimizer of
v g(Ev)dx,
7→ˆ
Ω
where g is the convex conjugate of the auxiliary energy τ 1C−1τ : τ +I (τ), and the stress is
7→ 2 K
then given by
σ =Dg(Eu). (1.3)
This formula for the stress is the starting point of the analysis. In order to get estimates on the
spatial gradient of σ, it would be convenient to differentiate (1.3). Unfortunately, the function g
beingonlyofclass 1 withLipschitzcontinuouspartialderivatives,theclassicalchainruleformula
C
forthecompositionofaLipschitzfunctionwithavectorvalued(Sobolev)functiondoesnotapply.
Toovercomethisproblem,wecomputeexplicitelytheexpressionofg (see(3.13))anduseageneral
chain rule formula established in [3] to get a formula of the type ∂ σ =D2g(Ev)∂ Ev.
k k
The study of the quasi-static case introduces further difficulties. In the case of Von Mises plas-
ticity [17], the previous method is applied to the incremental problem where (u ,e ,p ) minimizes
i i i
1
(v,η,q) Cη :ηdx+ H(q p )dx
7→ 2ˆ ˆ − i−1
Ω Ω
among all admissible triples (v,η,q) at time t , or still u minimizes
i i
v g(Ev p )dx.
7→ˆ − i−1
Ω
Itisshownthatthestressσ :=Dg(Eu p )satisfiesanH1 (Ω;Mn×n)estimatethatisuniform
i i− i−1 loc sym
with respect to the viscosity parameter, but may possibly depend on the time step. Afterwards,
a uniform estimate with respect to the time step is established showing the desired regularity
4 J.-F.BABADJIANANDM.G.MORA
result. The main drawback of this approach is that it necessitates to perform twice almost the
same computations.
In contrast with [17], we directly work on a time continuous model, and use the underlying
variational structure to establish a similar formula (see (3.15)) for the stress as
σ =Dg(ξ), ξ :=e+p˙.
Thestrategyconsiststhenindifferentiatingthe equilibriumequation,take(∂ u˙)ψ astestfunction
k
(where ψ is a suitable cut-off function), and deduce an inequality that provides a bound on ∂ σ
k
(seeProposition5.1). Themaindifficultyindoingsoistodealwithatermoftheform(see(5.13))
t
∂ σ ξ ψdxds, (1.4)
ˆ ˆ | k || |
0 Ω
since ξ is onlybounded inL1(Ω (0,T);Mn×n). We thus need to absorbthis termby some ofthe
× sym
coercivity terms of the left hand side
t t
∂ σ :∂ σψdxds+ ∂ σ :∂ ξψdxds.
ˆ ˆ k k ˆ ˆ k k
0 Ω 0 Ω
To this end, we use the special structure of the function g, together with the formula ∂ σ =
k
D2g(ξ)∂ ξ, to show that the integral in (1.4) can be controled by
k
t 1/2
M ∂ σ :∂ ξψdxds ,
ˆ ˆ k k
(cid:18) 0 Ω (cid:19)
where M only depends on various norms of (u,e,p) in the energy space (see (5.23)–(5.24)).
Note that the estimates performed in the proof of Proposition 5.1 should actually hold true in
anyspacedimension. However,sincethefinalestimate(5.27)involvestheL2(Ω (0,T);Rn)-norm
×
of u˙ and this is controled only in dimension n = 2 owing to the continuous embedding of BD(Ω)
intoL2(Ω;R2)1,this estimateturnsouttobeuniformwithrespecttotheviscosityparameteronly
in dimension n =2. This special role played by the dimension n=2 was already observed in the
papers [9, 25, 27], and is a recurrent feature in plasticity models where the elasticity set has no
bounded directions.
A direct consequence of this result is that, by means of the quasi-continuous representative of
thestresswithrespecttotheH1-capacity,onecanexpresstheflowruleinapointwisestrongform.
Indeed,since σ(t) L2(Ω;M2×2)andp˙(t) (Ω;Mn×n),the productbetweenσ(t) andp˙(t)isin
∈ sym ∈M sym
general not well defined. This issue is usually overcome by introducing a distributional notion of
duality σ(t) : p˙(t) as in [22]. In the present situation the precise representative, denoted by σ˜(t),
is defined up to a set of zero capacity and thus it turns out to be p˙(t)-measurable. This enables
| |
oneto givea sensetothe pointwise productofthe (quasi-continuous)stressσ˜(t)withthe measure
p˙(t), and, in particular, to express Hill’s principle of maximal plastic work (1.2) in a strong sense.
The paper is organised as follows. In Section 2 we introduce the precise mathematical setting
to formulate accurately the model of perfect plasticity, and state our main regularity result, The-
orem 2.7. In Section 3 we approximate the perfect plasticity model by means of a visco-plastic
regularization, and establish the convergence of the solutions, as well as some (non uniform) reg-
ularity properties of the approximating solutions. Section 4 is devoted to establish a chain rule
type formula for the stress which, as explained above, is instrumental for the subsequent analysis.
In Section 5 we prove an estimate on the (visco-plastic) stress in L∞(0,T;H1 (Ω;M2×2)), which
loc sym
is uniform with respect to the viscosity parameter. Owing to this estimate we complete the proof
of Theorem 2.7. The last section is devoted to show the validity of the flow rule by means of the
quasi-continuous representative of the stress.
1Indimensionn=3,BD(Ω)onlyembeds intoL3/2(Ω;R3)
STRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 5
2. Mathematical formulation of the problem
2.1. Notation.
2.1.1. Vectors and matrices. If a,b Rn, we write a b for the Euclidean scalar product, and we
∈ ·
denote by a =√a a the associated norm.
| | ·
We write Mn×n for the set of real n n matrices, and Mn×n for that of all real symmetric
× sym
n n matrices. Given two matrices A and B Mn×n, we use the Frobenius scalar product
× ∈
A : B = tr(ATB) (where AT is the transpose of A, and trA is its trace), and we denote
by A = √A:A the associated norm. If A Mn×n, we denote by A := A 1(trA)Id the
dev|iat|oric ppart of A, which is a trace free mat∈rix. We recall that for anyDtwo ve−ctonrs a,b Rn,
a b:=abT Mn×n stands forthe tensorproduct, anda b:=(a b+b a)/2 Mn×n de∈notes
⊗ ∈ ⊙ ⊗ ⊗ ∈ sym
the symmetric tensor product.
2.1.2. Functional spaces. We use standard notation for Lebesgue and Sobolev spaces.
Let X Rn be a locally compact set and Y be an Euclidean space. We write (X;Y) (or
simply ⊂(X)ifY =R)forthespaceofboundedRadonmeasuresinX withvaluesinMY,endowed
M
withthenorm µ(X),where µ (X)isthevariationofthemeasureµ. TheLebesguemeasure
in Rn is denote|d|by n, and|th|e∈(nM 1)-dimensional Hausdorff measure by n−1.
L − H
If U Rn is an open set, BD(U) stands for the space of functions of bounded deformation in
U, i.e.,⊂u BD(U) if u L1(U;Rn) and Eu (U;Mn×n), where Eu := (Du+DuT)/2 and
Du is the ∈distributional d∈erivative of u. We rec∈allMthat, ifsUym Rn is bounded and has a Lipschitz
boundary, BD(U) can be embedded into Ln/(n−1)(U;Rn). W⊂e refer to [30] for general properties
of this space.
2.1.3. Capacity. We finally recall the definition and several facts about capacity (see [1]). Let
Ω Rn be an open set. The capacity of a set A Ω in Ω is defined by
⊂ ⊂
Cap(A):=inf u2dx:u H1(Ω), u 1 n-a.e. in a neighborhood of A .
ˆ |∇ | ∈ 0 ≥ L
(cid:26) Ω (cid:27)
One of the interests of capacity is that it enables one to give an accurate sense to the pointwise
value of Sobolev functions (see [1, Section 6.1]). More precisely, every u H1(Ω) has a quasi-
∈
continuous representative u˜, which is uniquely defined except on a set of capacity zero in Ω. It
meansthatu˜=u n-a.e.inΩ,andthat, foreachε>0,there existsaclosedsetA Ω suchthat
ε
L ⊂
Cap(Ω A )<ε and u˜ is continuous on A . In addition (see [1, Theorem6.2.1]), there exists a
\ ε |Aε ε
Borel set Z Ω with Cap(Z)=0 such that
⊂
1
lim u(y)dy =u˜(x) for all x Ω Z.
r→0+ Ln(Br(x)∩Ω)ˆBr(x)∩Ω ∈ \
2.2. Description of the model.
2.2.1. The reference configuration. We denote by
Ω R2 a bounded connected open set with Lipschitz boundary (A )
1
⊂
the reference configuration of an elasto-plastic material.
2.2.2. Boundary condition. We assume that the body is subjected to a time-dependent boundary
displacement, which is the trace on ∂Ω of a function w(t):Ω R2 with
→
w AC([0,T];H1(Ω;R2)). (A )
2
∈
6 J.-F.BABADJIANANDM.G.MORA
2.2.3. The elastic energy. We consider an isotropic body whose fourth order elasticity tensor C is
given by
Ce=λ(tre)Id+2µe for all e M2×2, (A )
∈ sym 3
where λ andµ are the Lam´ecoefficients satisfying µ>0 and λ+µ>0. Note that there exist two
constants c , c >0 such that
0 1
c e2 Ce:e c e2 for all e M2×2. (2.1)
0| | ≤ ≤ 1| | ∈ sym
Setting K :=λ+µ, the inverse of C can be represented as
0
1 1
C−1σ = (trσ)Id+ σ for all σ M2×2.
4K 2µ D ∈ sym
0
We define the elastic energy, for all e L2(Ω;M2×2), by
∈ sym
1
(e):= Ce(x):e(x)dx.
Q 2ˆ
Ω
2.2.4. Theelasticityset. InthispaperweareinterestedintheDrucker-PragerandMohr-Coulomb2
models, where the elasticity domain is a closed and convex cone with vertex lying on the axis of
hydrostatic stresses given by
K := σ M2×2 : σ +αtrσ κ . (A )
{ ∈ sym | D| ≤ } 4
In the previous formula α > 0 and κ > 0 are positive constants related to the cohesion and the
coefficient of internal friction of the material, respectively.
2.2.5. External forces. We consider a time-dependent body load f(t):Ω R2 satisfying
→
f AC([0,T];L2(Ω;R2)), (A )
5
∈
which satisfies the usual safe-load condition: there exist χ AC([0,T];L2(Ω;M2×2)) and a con-
∈ sym
stant δ (0,κ) such that for every t [0,T]
∈ ∈
divχ(t)=f(t) in Ω,
− (A )
6
(χD(t) +αtrχ(t) κ δ in Ω.
| | ≤ −
2.2.6. The dissipation energy. We define the support function H :M2×2 [0,+ ] of K by
sym → ∞
H(p):= supσ:p for all p M2×2.
∈ sym
σ∈K
Since K is closed and convex, H is convex, lower semicontinuous, and positively 1-homogeneous.
In addition, since 0 belongs to the interior of K, the functions H enjoys the following coercivity
property: there exists c >0
2
H(p) c p for all p M2×2.
≥ 2| | ∈ sym
It is easy to establish the following explicit formula for the function H.
Lemma 2.1. For all p M2×2,
∈ sym
κtrp trp
if p ,
D
2α | |≤ 2α
H(p)=

trp
+ if pD > .
∞ | | 2α

σma2xN−otσemtihna,twihnerdeimσmenaxsio(nresnp.=σm2int)hiestthweolamrgoedsetls(raersep.elqouwievsatl)enetigbenevcaaluuseeooffσtDhe. algebraic identity √2|σD| =
STRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 7
The dissipated energy functional is then defined, for all p L1(Ω;M2×2), by
∈ sym
(p):= H(p(x))dx.
H ˆ
Ω
As a consequence of the previous properties of H, we infer that is sequentially weakly lower
semicontinuous in L1(Ω;M2×2). Since L1(Ω;M2×2) is not reflexivHe (bounded sequences in that
sym sym
space are only weakly* sequentially compact in the space of measures), it will also be useful to
extend the definition of when p (Ω;M2×2). According to [21], we define the non-negative
H ∈ M sym
Borel measure
dp
H(p):=H p,
dp | |
(cid:18) | |(cid:19)
where dp is the Radon-Nikodym derivative of p with respect to its variation p. In general, the
d|p| | |
measure H(p) is not even locally finite. However, if further H(p) has finite mass, i.e., if H(p) is a
bounded Radon measure, we can define the dissipation functional
(p):=H(p)(Ω).
H
Inthatcase,theresultsof[15,16]applyandH(p)canbeexpressedbymeansofadualityformula.
If H(p) (Ω), we get that (see [9])
∈M
ϕdH(p)=sup ϕσ:dp :σ ∞(Ω;K) , (2.2)
ˆ ˆ ∈C
Ω (cid:26) Ω (cid:27)
for any ϕ (Ω) with ϕ 0, and in particular
∈C ≥
(p)=sup σ:dp :σ ∞(Ω;K) . (2.3)
H ˆ ∈C
(cid:26) Ω (cid:27)
Note also that the Reshetnyak Theorem (see [4, Theorem 2.38])applies here, so that is sequen-
tially weakly* lower semicontinuous in (Ω;M2×2). H
M sym
2.2.7. Spaces of admissible fields. Given a prescribed boundary displacement wˆ H1(Ω;R2), we
∈
will consider the following spaces of kinematically admissible fields:
(wˆ):= (v,η,q) H1(Ω;R2) L2(Ω;M2×2) L2(Ω;M2×2):
Ar ∈ × sym × sym
(cid:8) Ev =η+q a.e. in Ω, v =wˆ 1-a.e. on ∂Ω ,
H
and (cid:9)
(wˆ):= (v,η,q) BD(Ω) L2(Ω;M2×2) (Ω;M2×2):
A ∈ × sym ×M sym
n
Ev =η+q in Ω, q =(wˆ v) ν 1 on ∂Ω ,
− ⊙ H
o
where ν is the outer unit normal to ∂Ω.
The space of plastically admissible stresses is defined by
:= τ L2(Ω;M2×2):τ(x) K for a.e. x Ω ,
K { ∈ sym ∈ ∈ }
and the space of statically admissible stresses is given by
:= τ L2(Ω;M2×2):divτ L2(Ω;R2) .
S { ∈ sym ∈ }
8 J.-F.BABADJIANANDM.G.MORA
2.2.8. Stress/strain duality. Thedualitypairingbetweenstressesandplasticstrainsisa priorinot
well defined, since the former are only squared Lebesgue integrable, while the latter are possibly
singular measures. Following [22], we define the following distributional notion of duality.
Definition 2.2. Let σ and (u,e,p) (wˆ) with wˆ H1(Ω;R2). We define the distribution
[σ:p] ′(R2) supporte∈dSin Ω by ∈ A ∈
∈D
[σ:p],ϕ = ϕ(wˆ u) divσdx+ σ :[(wˆ u) ϕ]dx+ σ :(Ewˆ e)ϕdx (2.4)
h i ˆ − · ˆ − ⊙∇ ˆ −
Ω Ω Ω
for every ϕ ∞(R2). The duality product is then defined as
∈Cc
σ,p := [σ:p],1 = (wˆ u) divσdx+ σ :(Ewˆ e)dx.
h i h i ˆ − · ˆ −
Ω Ω
Remark 2.3. Note that the first and second integrals in (2.4) are well defined since BD(Ω) is
embedded into L2(Ω;R2) for n = 2. Moreover, according to the integration by parts formula in
BD(Ω) (see [8, Theorem 3.2]), if σ 1(Ω;M2×2), we have
∈S∩C sym
[σ:p],ϕ = ϕσ :dp for all ϕ ∞(R2). (2.5)
h i ˆ ∈Cc
Ω
Aconvolutionargumentshowsthat(2.5)remainstrueprovidedσ (Ω;M2×2)andϕ ∞(Ω).
∈S∩C sym ∈C
Using this notion of stress/strain duality, the duality formulas (2.2) and (2.3) can be now
extended to less regular statically and plastically admissible stresses. If p (Ω;M2×2) with
∈ M sym
(p)<+ ,
H ∞
ϕdH(p)=sup [σ:p],ϕ :σ ,
ˆ h i ∈K∩S
Ω
n o
for all ϕ ∞(Ω) with ϕ 0, and in particular,
∈C ≥
(p)=sup σ,p :σ .
H h i ∈K∩S
n o
The followingresultestablishesa coercivitypropertyofthe functional p (p) χ(t),p (see
7→H −h i
e.g. [9, Proposition 6.1]).
Proposition 2.4. Let wˆ H1(Ω;R2) and (u,e,p) (wˆ). Then there exists a constant C ,
δ,α
∈ ∈ A
depending on δ and α, such that the following coercivity estimate holds:
H(p)−hχ(t),pi≥Cδ,αkpkM(Ω;M2sy×m2)
for every t [0,T].
∈
2.2.9. Initial condition. We finally consider an initial datum (u ,e ,p ) (w(0)) and σ :=Ce
0 0 0 0 0
∈A
satisfying the stability conditions
σ , divσ =f(0) in Ω. (A )
0 0 7
∈K −
We are now in position to state the existence result obtained in [9].
Theorem 2.5. Assume (A )–(A ). Then there exist
1 7
u AC([0,T];BD(Ω)),
∈
e, σ AC([0,T];L2(Ω;M2×2)),
 ∈ sym
p AC([0,T]; (Ω;M2×2)),
∈ M sym
with 
(u(0),e(0),p(0))=(u ,e ,p )
0 0 0
that satisfy, for all t [0,T]:
∈
STRESS REGULARITY IN QUASI-STATIC PERFECT PLASTICITY 9
(i) the kinematic compatibility
Eu(t)=e(t)+p(t) in Ω,
(p(t)=(w(t) u(t)) ν 1 on ∂Ω,
− ⊙ H
(ii) the static and plastic admissibility
σ(t)=Ce(t),
divσ(t)=f(t) in Ω,

−
σ(t) ,
∈K
(iii) the energy balance 
t t t
(e(t))+ (p˙(s))ds= (e )+ σ :Ew˙ dxds+ f (u˙ w˙)dxds. (2.6)
Q ˆ H Q 0 ˆ ˆ ˆ ˆ · −
0 0 Ω 0 Ω
Moreover, the stress σ is unique, and for a.e. t [0,T] the distribution [σ(t):p˙(t)] is well defined,
∈
and it is a measure in (Ω) satisfying Hill’s principle of maximum plastic work
M
H(p˙(t))=[σ(t):p˙(t)] in (Ω). (2.7)
M
Remark 2.6. According to Lemma 2.1, the flow rule can be equivalently written as
κtrp˙(t) trp˙(t)
=[σ(t):p˙(t)], and p˙ (t) in (Ω).
D
2α | |≤ 2α M
The main result of this work is the following regularity result.
Theorem 2.7. Assume (A )–(A ) and that α=1/√2 in (A ). Under the additional hypotheses
1 7 4
that w H1([0,T];H2(Ω;R2)), χ W1,∞([0,T6 ];L∞(Ω;M2×2)) H1([0,T];H1(Ω;M2×2)), f
∈ ∈ sym ∩ sym ∈
L∞(0,T;H1(Ω;R2)) L2(0,T;H2(Ω;R2)) L∞(Ω (0,T);R2) and e H1 (Ω;M2×2), the stress
∩ ∩ × 0 ∈ loc sym
tensor satisfies
σ L∞(0,T;H1 (Ω;M2×2)).
∈ loc sym
3. Perzyna visco-plastic approximations
InordertoproveTheorem2.7,wewillneedtoconsideraregularizedproblem. Thiswillbedone
bymeansofaso-calledPerzynavisco-plasticapproximation. Thefollowingresult,formulatedhere
in a modern language, has been established in [28].
Since the initial data (u ,e ,p ) givenin (A ) does not belong to the rightenergy space associ-
0 0 0 7
ated to the visco-plasticmodel, we first need to regularizeit. According to [13, Lemma 5.1], there
exists a sequence (u ) H1(Ω;R2) such that u = w(0) 1-a.e. on ∂Ω, u u strongly
0,ε 0,ε 0,ε 0
⊂ H →
in L1(Ω;R2), and Eu ⇀ Eu weakly* in (Ω;M2×2). Setting p = Eu e , we get that
0,ε 0 M sym 0,ε 0,ε− 0
(u ,e ,p ) (w(0)).
0,ε 0 0,ε r
∈A
Proposition 3.1. Assume (A )–(A ). Let ε>0 and let (u ,e ,p ) (w(0)) be constructed
1 7 0,ε 0 0,ε r
∈A
as above. Then there exists a unique triple
(u ,e ,p ) AC([0,T];H1(Ω;R2)) AC([0,T];L2(Ω;M2×2)) AC([0,T];L2(Ω;M2×2))
ε ε ε ∈ × sym × sym
such that (u (0),e (0),p (0))=(u ,e ,p ), for all t [0,T]
ε ε ε 0,ε 0 0,ε
∈
(u (t),e (t),p (t)) (w(t)), σ (t)=Ce (t), divσ (t)=f(t) in Ω,
ε ε ε r ε ε ε
∈A −
and for a.e. t [0,T]
∈
σ (t) εp˙ (t) ∂H(p˙ (t)) in Ω. (3.1)
ε ε ε
− ∈
10 J.-F.BABADJIANANDM.G.MORA
Remark 3.2. For every q M2×2, we define the function
∈ sym
ε
H (q):=H(q)+ q 2.
ε
2| |
The convex conjugate of H is given, for all τ M2×2, by
ε ∈ sym
τ P (τ)2
H∗(τ)= | − K | ,
ε 2ε
where P stands for the orthogonal projection onto the nonempty closed convex set K. The
K
function H∗ turns out to be of class 1 and its differential is given by
ε C
τ P (τ)
DH∗(τ)= − K . (3.2)
ε ε
With these notation, the flow rule (3.1), can be equivalently written, for a.e. t [0,T], as
∈
σ (t) ∂H (p˙ (t)) in Ω,
ε ε ε
∈
or still, by convex analysis,
p˙ (t)=DH∗(σ (t)) in Ω. (3.3)
ε ε ε
We will show that the solution (u ,e ,p ) of the visco-plastic model given by Proposition 3.1
ε ε ε
converges to a solution of the perfectly plastic model, in the sense of Theorem 2.5.
Proposition 3.3. Assume that (A )–(A ) hold, and in addition that w H1([0,T];H1(Ω;R2))
1 7
and χ W1,∞([0,T];L∞(Ω;M2×2)). Then, up to a subsequence (not re∈labeled), (u ,e ,p ) ⇀
∈ sym ε ε ε
(u,e,p) weakly* in H1([0,T];BD(Ω)) H1([0,T];L2(Ω;M2×2)) H1([0,T]; (Ω;M2×2)), where
× sym × M sym
(u,e,p) is a solution of the perfectly plastic model as in Theorem 2.5.
Note that this result was already proven in [28] for different type of elasticity sets that are
bounded in the direction of deviatoric stresses (see also [12]). However we give below a slightly
different and simplified argument since some finer estimates established along the proof will be
useful in that of our regularity result Theorem 2.7.
3.1. A priori estimates. We firstestablishsomeuniforma priori estimateswhichwillenable one
to get weak compactness on the families (u ) , (e ) , and (p ) .
ε ε>0 ε ε>0 ε ε>0
3.1.1. First energy estimates. Standard arguments show that the following energy balance holds:
for all t [0,T],
∈
t t
(e (t))+ (p˙ (s))ds+ε p˙ 2dxds
Q ε ˆ H ε ˆ ˆ | ε|
0 0 Ω
t t
= (e )+ σ :Ew˙ dxds+ f (u˙ w˙)dxds, (3.4)
Q 0 ˆ ˆ ε ˆ ˆ · ε−
0 Ω 0 Ω
or still, using the safe load condition (A ), together with an integration by parts in time,
6
t t t
(e (t))+ (p˙ (s))ds χ:p˙ dxds+ε p˙ 2dxds
Q ε ˆ H ε −ˆ ˆ ε ˆ ˆ | ε|
0 0 Ω 0 Ω
t t
= (e )+ σ :Ew˙ dxds χ˙ :(e Ew)dxds
Q 0 ˆ ˆ ε −ˆ ˆ ε−
0 Ω 0 Ω
+ χ(t):(e (t) Ew(t))dx χ(0):(e Ew(0))dx. (3.5)
ˆ ε − −ˆ 0−
Ω Ω
Therefore, an application of Proposition 2.4 leads to the following first energy estimates:
sεu>p0 keεkL∞(0,T;L2(Ω;M2sy×m2))+kp˙εkL1(0,T;L1(Ω;M2sy×m2)) +√εkp˙εkL2(0,T;L2(Ω;M2sy×m2)) <+∞. (3.6)
(cid:16) (cid:17)