The Armstrong–Frederick cyclic hardening plasticity model with Cosserat effects

27 Pages·2014·0.34 MB·English

by  Krzysztof Chełmiński

Availableonlineatwww.sciencedirect.com
ScienceDirect
J.DifferentialEquations256(2014)3497–3523
www.elsevier.com/locate/jde
The Armstrong–Frederick cyclic hardening plasticity
model with Cosserat effects
Krzysztof Chełmin´skia, Patrizio Neffb, Sebastian Owczareka,∗
aFacultyofMathematicsandInformationScience,WarsawUniversityofTechnology,ul.Koszykowa75,
00-662Warsaw,Poland
bFakultätfürMathematik,UniversitätDuisburg-Essen,LehrstuhlfürNichtlineareAnalysisundModellierung,
Thea-LeymannStrasse9,45141Essen,Germany
Received11May2013;revised25January2014
Availableonline12March2014
Abstract
WeproposeanextensionofthecyclichardeningplasticitymodelformulatedbyArmstrongandFrederick
whichincludesmicropolareffects.Ourmicropolarextensionestablishescoercivityofthemodelwhichis
otherwisenotpresent.Westudythenexistenceofsolutionstothequasistatic,rate-independentArmstrong–
Frederick model with Cosserat effects which is, however,still of non-monotone, non-associated type. In
ordertodothis,weneedtorelaxthepointwisedefinitionoftheflowruleintoasuitableweakenergy-type
inequality.ItisshownthatthelimitintheYosidaapproximationprocesssatisfiesthisnewsolutioncon-
cept.Thelimitfunctionshaveabetterregularitythanpreviouslyknownintheliterature,wheretheoriginal
Armstrong–Frederickmodelhasbeenstudied.
©2014ElsevierInc.Allrightsreserved.
1. Introduction
Oneofthewell-knownmodelstodescribecyclicplasticityisthenon-linearkinematicharden-
ingmodelformulatedbyArmstrongandFrederick[3].Thismodelhasbeenhighlyrated,because
it is based on a physical mechanism of strain hardening and dynamic recovery, and because it
* Correspondingauthor.
E-mailaddresses:[email protected](K.Chełmin´ski),[email protected](P.Neff),
[email protected](S.Owczarek).
http://dx.doi.org/10.1016/j.jde.2014.02.003
0022-0396/©2014ElsevierInc.Allrightsreserved.
3498 K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523
hasthecapabilityofrepresentingreasonablywelltheshapesofstress–strainhysteresisloops,es-
peciallythoseofconstantstrainranges.Therefore,implementationoftheArmstrong–Frederick
modelinfiniteelementmethodshasbeenexaminedinseveralstudiestodate.Thus,thatmodelis
nowavailableasamaterialmodelofcyclichardeningplasticityincommercial,general-purpose
softwareforfiniteelementanalysis.
The Armstrong–Frederick model (AF) is a modification of the Melan–Prager model, which
iswellknownintheliteratureanditcanalsobeseenasanapproximationofthePrandtl–Reuss
model. The key modification of this simple model is the so-called “recall”-term, changing the
evolutionlawforthesymmetricbackstresstensorb fromaclassicallinearkinematichardening
law(Melan–Prager)toanonlinearkinematichardeninglaw,i.e.,
b = cεp − d|εp|b .
t (cid:2)(cid:3)(cid:4)t(cid:5) (cid:2) (cid:3)(cid:4)t (cid:5)
lin.kin.hardening recall-term,nonlinearhardening
Here, εp is the symmetric plastic strain tensor, c and d are positive material constants. The
∞
“recall”-term entails the L -boundedness of the backstress b, a property which is an experi-
mental fact since to the contrary, for high frequency cycles softening and rupture will occur.
Therefore,theAF-modelshowsnonlinearkinematichardening,butonlytowithinacertainex-
tent.ThemorerealisticdescriptionofcyclichardeningplasticityexperimentswiththeAF-model,
however, has a prize to be paid: the model is non-coercive (bounded hardening), it is of non-
monotonetypeandnotofgradienttype(non-associatedflowrule).Thus,theAF-modelisoneof
theprominentsmallstrainplasticitymodelswhichhasyetescapedtheeffortsofmathematicians
toestablishwell-posedness.
The mathematical analysis being quite challenging, there are no encompassing existence
resultsforthismodelintheliterature.Thefirst(partial)mathematicalresultfortheArmstrong–
Frederick model was obtained by the first author in the article [10]. There, the non-monotone,
non-associated AF-model was written as a model of pre-monotone type (for the definition we
refer to [2]). In this work the author used a Yosida approximation to the monotone part of the
flow rule. The obtained a priori estimates are, however, not sufficient to pass to the limit with
such approximations and to get L2-strong solutions (see Section 3 in [10]). It was only shown
that the limit functions satisfy the so-called “reduced energy inequality”. In the article [19] a
regularizationofthe“recall”-termintheequationforthebackstresswasproposed.Theexistence
of a rescaled in time solution to the Armstrong–Frederick model with the regularized equation
forthebackstresscouldthenbeestablished.Therescalingideaisverysimple:anewtimevari-
able τ =ζ(t) is proposed. Then the new system is easier to analyze, because the plastic strain
is now uniformly Lipschitz with respect to the rescaled time. The main problem is to get back
to the original system with the rescaled in time solution. It is, in principle, possible for rate-
independent models, where the flow rule is invariant under scaling of the time variable. The
Armstrong–Frederickmodelisrate-independentbuttheauthorsof[19]arenotabletogetback
to the original system. The rescaling idea has already been applied in the plasticity context in
[4,16,17].
In this paper we want to extend the system of equations proposed by Armstrong and Fred-
erick to include micropolar effects. In the classical metal perfect plasticity models at infinites-
imal strains it has been shown in a series of papers [11–13,24] that a coupling with Cosserat
elasticity may also regularize the ill-posedness of the Prandtl–Reuss plasticity model. This
is possible because the Cosserat coupling leads to coercivity. Perfect plasticity, however, is
yet characterized by a monotone flow rule of gradient type (associated plasticity). Therefore,
K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523 3499
the question arises naturally, whether adding Cosserat microrotations to the AF-model is still
enough to regularize the problem in the way to satisfy the flow rule in a standard point-
wise sense. From a modelling perspective, adding microrotations means to consider a material
made up of individual particles which can rotate and interact with each other [21–23,26]. For
phenomenological polycrystalline plasticity adding Cosserat effects is arguably a physically
motivated regularization: the individual crystal grains are rotating and interacting with each
other.
TheextensionoftheArmstrong–FrederickmodeltoincludeCosserateffectsfollowsthelines
proposed in [11], where the authors added the Cosserat effect to the classical elastoplasticity
modelwithamonotoneflowrule.There,andinourpresentapproach,onlytheelasticityrelation
isaugmentedwithCosserateffects,theplasticconstitutiveequations,andnotablythe“recall”-
term, is left unchanged, contrary to [19]. Regarding the effect of the Cosserat-modification for
classicalplasticitymodels,ithasbeenprovedthatthenewmodelisthermodynamicallyadmis-
sibleandthatthereexistsaunique,globalintimesolutiontoCosseratelasto-plasticity.In[14]
anH1 -regularityresultforthestressesandstrainswasproved,cf.[27].ThedynamicCosserat
loc
plasticity was studied in [12], see also [13,24]. Another way to regularize classical plasticity
models is by introducing gradient plasticity effects [18,25,28–30]. However, a modification of
the AF-model to include higher gradients will be left to future work. Moreover, the coupling
with thermal effects can be treated as another attempt to regularize models from the inelastic
deformationtheory(cf.[5,6]).
Manynon-monotonemodelsfromthetheoryofinelasticdeformationprocessesinmetalsare
also non-coercive (for the definition see [2]) and the existence results for such models is only
veryweaklyexamined.Forexample:thesolutionsobtainedinarticles[10],and[19]hadalow
regularitywithrespecttotimeandspace(seealso[31,32],wherethenon-monotonemodelsof
poroplasticitywasconsidered).Inouropinion,itisexpedienttofirstconsidernon-monotonebut
coercivemodelsandtoobtainbetterregularityresultsforthesolutions.Thisarticlepresentsthe
first mathematicalresult in this respect for our new Armstrong–Frederickmodel with Cosserat
effects,whichisnon-monotone,non-associatedbutcoercive.
2. TheArmstrong–FrederickmodelwithCosserateffect
This section is devotedto the formulation of the Armstrong–Frederickmodel with Cosserat
effects.
FromthemechanicalresultsforCosseratplasticity(seeforexample[11,12])weconcludethat
wedealwiththefollowinginitial–boundaryvalueproblem:wearelookingforthedisplacement
fieldu:Ω×[0,T]→R3,themicrorotationmatrixA:Ω×[0,T]→so(3)(so(3)isthesetof
skew-symmetric3×3matrices)andthevectorofinternalvariablesz=(εp,b):Ω×[0,T]→
S3 ×S3 (εp istheclassicalinfinitesimalsymmetricplasticstraintensor,b isthesymmetric
dev dev
backstresstensorandthespaceS3 denotesthesetofsymmetric3×3-matriceswithvanishing
dev
trace)satisfyingthefollowingsystemofequations
div T =−f,
x
(cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7)
T =2μ ε(u)−εp +2μ skew(∇ u)−A +λtr ε(u)−εp 1,
c x
(cid:6) (cid:7)
−l (cid:7) axl(A)=μ axl skew(∇ u)−A ,
c x c x
εp ∈∂I (T ),
t K(b) E
3500 K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523
(cid:6) (cid:7) (cid:6) (cid:7)
T =2μ ε(u)−εp +λtr ε(u)−εp 1,
E
(cid:8) (cid:8)
b =cεp−d(cid:8)εp(cid:8)b, (2.1)
t t t
where ε(u)=sym(∇ u) denotes the symmetric part of the gradient of the displacement. The
x
aboveequationsarestudiedforx∈Ω⊂R3 andt∈[0,T],whereΩ⊂R3 isaboundeddomain
withsmoothboundary∂Ω andt denotesthetime.
The set of admissible elastic stresses K(b(x,t)) is defined in the form K(b)={T ∈S3:
E
|dev(T )−b|(cid:2)σ },wheredev(T )=T − 1tr(T )·1,σ isamaterialparameter(theyield
E y E E 3 E y
limit)and1denotestheidentitymatrix.ThefunctionI istheindicatorfunctionoftheadmis-
K(b)
siblesetK(b)and∂I isthesubgradientoftheconvex,proper,lowersemicontinuousfunction
K(b)
I . The function f :Ω ×[0,T]→R3 describes the density of the applied body forces, the
K(b)
parametersμ,λarepositiveLaméconstants(theelasticconstitutiveequationcanbegeneralized
in the obvious way to anisotropic case), μ >0 is the Cosserat couple modulus and l >0 is
c c
a material parameter with dimensions [m2], describing a length scale of the model due to the
Cosserat effects. c,d >0 are material constants. The operator skew(T)= 1(T −TT) denotes
2
theskew-symmetricpartofa3×3-tensor.Theoperatoraxl:so(3)→R3 establishestheiden-
tificationofaskew-symmetricmatrixwithvectorsinR3.ThismeansthatifwetakeA∈so(3),
whichisintheformA=((0,α,β),(−α,0,γ),(−β,−γ,0)),thenaxl(A)=(α,β,γ).
Notice again that the expression |εp|b is a perturbation of Melan–Prager model – if d =0
t
thenweobtaintheMelan–Pragerlinearkinematichardeningmodel.
Thesystem(2.1)isconsideredwithDirichletboundaryconditionforthedisplacement:
u(x,t)=g (x,t) forx∈∂Ω andt(cid:3)0 (2.2)
D
andwithDirichletboundaryconditionforthemicrorotation:
A(x,t)=A (x,t) forx∈∂Ω andt(cid:3)0. (2.3)
D
Finally,weconsiderthesystem(2.1)withthefollowinginitialconditions
εp(x,0)=εp,0(x), b(x,0)=b0(x). (2.4)
Thefreeenergyfunctionassociatedwiththesystem(2.1)isgivenbytheformula
(cid:6) (cid:7) (cid:9) (cid:9) (cid:9) (cid:9)
ρψ ε,εp,A,b =μ(cid:9)ε(u)−εp(cid:9)2+μ (cid:9)skew(∇ u)−A(cid:9)2
c x
(cid:6) (cid:6) (cid:7)(cid:7) (cid:9) (cid:9)
+ λ tr ε(u)−εp 2+2l (cid:9)∇ axl(A)(cid:9)2+ 1 (cid:8)b(cid:8)2, (2.5)
c x
2 2c
whereρ isthemassdensitywhichweassumetobeconstantintimeandspace.Thetotalenergy
isoftheform:
(cid:10)
(cid:6) (cid:7) (cid:6) (cid:7)
E ε,εp,A,b (t)= ρψ ε(x,t),εp(x,t),A(x,t),b(x,t) dx.
Ω
FromSection2 ofthearticle[10]weknowthattheinelasticconstitutiveequationoccurringin
(2.1)isofpre-monotonetype.Thismeansthatthereexistsamultifunctiong:D(g)⊂S3×S3→
K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523 3501
2S3×S3 “monotone at the point zero” (∀z∈D(g) g(z)·z(cid:3)0) such that the flow rule can be
writtenintheform
(cid:6) (cid:7) (cid:6) (cid:6) (cid:7)(cid:7)
εp,b ∈g −ρ∇ ψ ε,εp,A,b ,
t t z
where ψ(ε,εp,A,b) is the free energy associated with the model. Additionally, every pre-
monotone system of equations is thermodynamically admissible – see [2] for more details.
Moreover, if initial data (εp,0,b0)∈S3 ×S3 , then any solution (εp(t),b(t)) (if there ex-
dev dev
ists) belongs to S3 ×S3 , because the right hand side of (2.1) is a subset of S3 and the
dev dev 4 dev
followingsystem
(cid:6) (cid:7)
d
trεp =0 with trεp(0)=0,
dt
(cid:8) (cid:8)
d (trb)+d(cid:8)εp(cid:8)trb=0 with trb(0)=0,
dt t
possesses the unique solution (trεp,trb)=(0,0). Inspired by the work [10], we propose to
rewritethesetofadmissiblestressesintheform
(cid:11)(cid:6) (cid:7) (cid:8) (cid:8) (cid:12)
K= dev(T ),b ∈S3 ×S3 : (cid:8)dev(T )+cb(cid:8)(cid:2)σ ,
E dev dev E y
wheretheconstantσ isthesameasintheyieldconditionandtheinelasticconstitutiveequation
y
cannowbewrittenintheequivalentform
(cid:13) (cid:14) (cid:13)(cid:13) (cid:14)(cid:14) (cid:13) (cid:14)
d εp 1 0
dt b ∈∂IK dev(TE),−cb − d|εp|b . (2.6)
t
3. Mainresult
Herewedefineanotionofthedefinitionofthesolutionforthesystem(2.1).Nextweformu-
latethemainresultofthispaper.First,letusstartwiththedefinitionofL2-strongsolutionofthe
system(2.1).
Definition 3.1. Fix T >0. We say that a vector (u,A,T,εp,b)∈W1,∞(0,T;H1(Ω;R3)×
H2(Ω;so(3))×L2(Ω;R9)×L2(Ω;S3 )×L2(Ω;S3 ))isL2-strongsolutionofthesystem
dev dev
div T =−f,
x
(cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7)
T =2μ ε(u)−εp +2μ skew(∇ u)−A +λtr ε(u)−εp 1,
c x
(cid:6) (cid:7)
−l (cid:7) axl(A)=μ axl skew(∇ u)−A ,
c x c x
(cid:13) (cid:14) (cid:13) (cid:14) (cid:13) (cid:14)
d εp 0 dev(2μ(ε(u)−εp))
dt b + d|εp|b ∈∂IK −1b
t c
if
1. |dev(2μ(ε(u(x,t))−εp(x,t)))−b(x,t)|(cid:2)σ foralmostall(x,t)∈Ω×(0,T),
y
2. theequationsandinclusionabovearesatisfiedforalmostall(x,t)∈Ω×(0,T).
3502 K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523
Next, we are going to define a weaker notion of solutions to the system (2.1). We give a
motivationofthisdefinition.Itwillbetheenergyinequalitycombinedwithspecialtestfunctions.
Letusconsideranotherconvexset(whichwillbeusedassetoftestfunctionsfurtheron)
(cid:15)(cid:13) (cid:14) (cid:16)
(cid:8) (cid:8)
K∗= dev(T ),−1b ∈S3 ×S3 : (cid:8)dev(T )−b(cid:8)+ d |b|2(cid:2)σ ,
E c dev dev E 2c y
wheretheconstantσ isthesameasintheyieldconditionandletusassumethatσ > c .This
y y 2d
technicalassumptionimpliesthatthepointswith|b|2= 2σyc (“cornerpoints”ofthesetK∗)lie
d
outside of the set {|b|(cid:2) c} – for details we refer to Fig. 1 of [10]. It is not difficult to see that
d
K∗⊂K and on the set K∗ the constitutive equation associated withe the system (A-F) can be
writtenintheform(therighthandsideoftheflowruleisorthogonaltotheboundaryoftheset
K∗ –seealsoFig.1from[10])
(cid:13) (cid:14) (cid:13)(cid:13) (cid:14)(cid:14)
d εp 1
dt b ∈∂IK∗ dev(TE),−cb .
The set K∗ is convex and closed, thus the subgradient ∂IK∗ generates a maximal monotone
operator hence for all L2-strong solutions the above differential inclusion is equivalent to the
followingpointwisevariationalinequality
(cid:13) (cid:14)
(cid:6) (cid:6) (cid:7) (cid:6) (cid:7)(cid:7)
εp(x,t) dev T (x,t) −dev Tˆ (x,t) +b (x,t) −1b(x,t)+ 1bˆ(x,t) (cid:3)0 (3.1)
t E E t c c
forall(dev(Tˆ ),bˆ)∈K∗ suchthat(Tˆ ,bˆ)∈L2(0,T;L2(Ω;S3)×L2(Ω;S3 )).
E E dev
Proposition3.2.Assumethatavector
(cid:6) (cid:7) (cid:6) (cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7)
u,A,T,εp,b ∈W1,∞ 0,T;H1 Ω;R3 ×H2 Ω;so(3) ×L2 Ω;R9
(cid:6) (cid:7) (cid:6) (cid:7)(cid:7)
×L2 Ω;S3 ×L2 Ω;S3
dev dev
isL2-strongsolutionofthesystem(2.1)withboundaryandinitialconditions(2.2)–(2.4).Then
forallfunctions(Tˆ ,bˆ)∈L2(0,T;L2(Ω;S3)×L2(Ω;S3 ))suchthat
E dev
(cid:6) (cid:7) (cid:6) (cid:6) (cid:7)(cid:7)
dev(Tˆ ),bˆ ∈K∗, divTˆ ∈L2 0,T;L2 Ω,R3 ,
E E
theinequality
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8)
1 C−1T (x,t)T (x,t)dx+μ (cid:8)skew ∇ u(x,t) −A(x,t)(cid:8)2dx
E E c x
2
Ω Ω
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7)(cid:8) (cid:8) (cid:8)
+2l (cid:8)∇axl A(x,t) (cid:8)2dx+ 1 (cid:8)b(x,t)(cid:8)2dx
c
2c
Ω Ω
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8)
(cid:2) 1 C−1T0(x)T0(x)dx+μ (cid:8)skew ∇ u(x,0) −A(x,0)(cid:8)2dx
2 E E c x
Ω Ω
K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523 3503
(cid:10) (cid:10) (cid:10)t (cid:10)
(cid:8) (cid:8) (cid:8) (cid:6) (cid:7)(cid:8)
+ 1 (cid:8)b(x,0)(cid:8)2dx+2l (cid:8)∇axl A(x,0) (cid:8)2dx+ u (x,τ)f(x,τ)dxdτ
c t
2c
Ω Ω 0 Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
(cid:6) (cid:7)
+ u (x,τ)divTˆ (x,τ)dxdτ + g (x,τ) T(x,τ)−Tˆ (x,τ) ·n(x)dSdτ
t E D,t E
0 Ω 0 ∂Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
+ C−1T (x,τ)Tˆ (x,τ)dxdτ + 1 b (x,τ)bˆ(x,τ)dxdτ
E,t E t
c
0 Ω 0 Ω
(cid:10)t (cid:10)
(cid:6) (cid:7) (cid:6) (cid:7)
+4l ∇axl A(x,τ) ·naxl A (x,τ) dSdτ
c D,t
0 ∂Ω
holdsforallt∈(0,T).
Proof. Fromthetheoryofelasticity,weknowthatthereexistsapositivedefiniteoperatorC−1:
S3→S3 suchthatC−1T =ε −εp.Integrating(3.1)overΩ×(0,t)fort(cid:2)T weobtain
E,t t t
(cid:10)t (cid:10) (cid:10)t (cid:10)
1
C−1T (x,τ)T (x,τ)dxdτ + b (x,τ)b(x,τ)dxdτ
E,t E t
c
0 Ω 0 Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
(cid:6) (cid:7)
(cid:2) ε (x,τ) T (x,τ)−Tˆ (x,τ) dxdτ + C−1T (x,τ)Tˆ (x,τ)dxdτ
t E E E,t E
0 Ω 0 Ω
(cid:10)t (cid:10)
+ 1 b (x,τ)bˆ(x,τ)dxdτ
t
c
0 Ω
(cid:10)t (cid:10)
(cid:6) (cid:7)
= ∇u (x,τ) T(x,τ)−Tˆ (x,τ) dxdτ
t E
0 Ω
(cid:10)t (cid:10)
(cid:6) (cid:6) (cid:7) (cid:7) (cid:6) (cid:7)
−2μ skew ∇ u(x,τ) −A(x,τ) skew ∇ u (x,τ) dxdτ
c x x t
0 Ω
(cid:10)t (cid:10)
+ 1 b (x,τ)bˆ(x,τ)dxdτ. (3.2)
t
c
0 Ω
Integrating by parts in the first term on the right hand side of (3.2), using Eq.(2.1) and the
1
boundarydatawehavethefollowinginequality
(cid:10) (cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8) (cid:8) (cid:8)
1 C−1T (x,t)T (x,t)dx+μ (cid:8)skew ∇ u(x,t) −A(x,t)(cid:8)2dx+ 1 (cid:8)b(x,t)(cid:8)2dx
E E c x
2 2c
Ω Ω Ω
3504 K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8)
(cid:2) 1 C−1T (x,0)T (x,0)dx+μ (cid:8)skew ∇ u(x,0) −A(x,0)(cid:8)2dx
E E c x
2
Ω Ω
(cid:10) (cid:10)t (cid:10) (cid:10)t (cid:10)
(cid:8) (cid:8)
+ 1 (cid:8)b(x,0)(cid:8)2dx+ u (x,τ)f(x,τ)dxdτ + u (x,τ)divTˆ (x,τ)dxdτ
t t E
2c
Ω 0 Ω 0 Ω
(cid:10)t (cid:10)
(cid:6) (cid:7)
+ g (x,τ) T(x,τ)−Tˆ (x,τ) ·n(x)dSdτ
D,t E
0 ∂Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
+ C−1T (x,τ)Tˆ (x,τ)dxdτ + 1 b (x,τ)bˆ(x,τ)dxdτ
E,t E t
c
0 Ω 0 Ω
(cid:10)t (cid:10)
(cid:6) (cid:6) (cid:7) (cid:7)
−2μ skew ∇ u(x,τ) −A(x,τ) A (x,τ)dxdτ. (3.3)
c x t
0 Ω
Using(2.1) tothelasttermoftheaboveinequalityandintegratingbypartsinthelasttermon
3
therighthandsideof(3.3)weobtain
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8)
1 C−1T (x,t)T (x,t)dx+μ (cid:8)skew ∇ u(x,t) −A(x,t)(cid:8)2dx
E E c x
2
Ω Ω
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7)(cid:8) (cid:8) (cid:8)
+2l (cid:8)∇axl A(x,t) (cid:8)2dx+ 1 (cid:8)b(x,t)(cid:8)2dx
c
2c
Ω Ω
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8)
(cid:2) 1 C−1T (x,0)T (x,0)dx+μ (cid:8)skew ∇ u(x,0) −A(x,0)(cid:8)2dx
E E c x
2
Ω Ω
(cid:10) (cid:10) (cid:10)t (cid:10)
(cid:8) (cid:8) (cid:8) (cid:6) (cid:7)(cid:8)
+ 1 (cid:8)b(x,0)(cid:8)2dx+2l (cid:8)∇axl A(x,0) (cid:8)2dx+ u (x,τ)f(x,τ)dxdτ
c t
2c
Ω Ω 0 Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
(cid:6) (cid:7)
+ u (x,τ)divTˆ (x,τ)dxdτ + g (x,τ) T(x,τ)−Tˆ (x,τ) ·n(x)dSdτ
t E D,t E
0 Ω 0 ∂Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
+ C−1T (x,τ)Tˆ (x,τ)dxdτ + 1 b (x,τ)bˆ(x,τ)dxdτ
E,t E t
c
0 Ω 0 Ω
(cid:10)t (cid:10)
(cid:6) (cid:7) (cid:6) (cid:7)
+4l ∇axl A(x,τ) ·naxl A (x,τ) dSdτ, (3.4)
c D,t
0 ∂Ω
K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523 3505
wheretheboundaryintegralsaredefinedinthesenseofdualitybetweenthespace
(cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7)
H12 ∂Ω;R3 and H−12 ∂Ω;R3 see[1]fordetails .
Theproofiscompleted. (cid:2)
LetusassumethatforallT >0thegivendataF,g ,A havetheregularity
D D
(cid:6) (cid:6) (cid:7)(cid:7) (cid:6) (cid:6) (cid:7)(cid:7)
f ∈H1 0,T;L2 Ω;R3 , gD∈H1 0,T;H12 ∂Ω;R3 , (3.5)
(cid:6) (cid:6) (cid:7)(cid:7)
AD∈H1 0,T;H32 ∂Ω;so(3) . (3.6)
Additionallyletusassumethattheinitialdata(εp,0,b0)∈L2(Ω;S3 )×L2(Ω;S3 )satisfy
dev dev
(cid:8) (cid:8) (cid:8) (cid:6) (cid:7) (cid:8)
(cid:8)b0(x)(cid:8)(cid:2) c and (cid:8)dev T0(x) −b0(x)(cid:8)(cid:2)σ foralmostallx∈Ω, (3.7)
d E y
where the initial stress T0 =2μ(ε(u(0))−εp,0)+λtr(ε(u(0))−εp,0)1∈L2(Ω;S3) is the
E
uniquesolutionofthefollowinglinearproblem
div T0(x)=−f(x,0),
x
(cid:6) (cid:7) (cid:6) (cid:6) (cid:7) (cid:7)
−l (cid:7) axl A(x,0) =μ axl skew ∇ u(x,0) −A(x,0) ,
c x c x
u(x,0)|∂Ω =gD(x,0) A(x,0)|∂Ω =AD(x,0), (3.8)
with
(cid:6) (cid:6) (cid:7) (cid:7) (cid:6) (cid:6) (cid:7) (cid:7)
T0(x)=2μ ε u(x,0) −εp,0(x) +2μ skew ∇ u(x,0) −A(x,0)
c x
(cid:6) (cid:6) (cid:7) (cid:7)
+λtr ε u(x,0) −εp,0(x) 1.
Remark. The energy inequality from Proposition 3.2 that an L2-strong solution satisfies, is
equivalenttothefollowingintegralversionofinequality(3.1)
(cid:10)t (cid:10) (cid:13) (cid:14)
(cid:6) (cid:6) (cid:7) (cid:6) (cid:7)(cid:7)
εp(x,t) dev T (x,t) −dev Tˆ (x,t) +b (x,t) −1b(x,t)+ 1bˆ(x,t) dxdτ (cid:3)0
t E E t c c
0 Ω
for all (dev(Tˆ ),bˆ)∈K∗ such that (Tˆ ,bˆ)∈L2(0,T;L2(Ω;S3)×L2(Ω;S3 )). From this
E E dev
followsthatthelastinequalitycanalsobeusedasasolutionconcept.Inthisarticlewewantto
work with so-called energetic solution introduced by Suquet in [33] and by Chełmin´ski in [8]
thereforeweintroducethefollowingdefinitionofsolution.
Definition3.3(Solutionconcept-energyinequality).FixT >0.Supposethatthegivendatasat-
isfy(3.5)and(3.6).Wesaythatavector(u,T,A,εp,b)∈L∞(0,T;H1(Ω;R3)×L2(Ω;S3)×
H2(Ω;so(3))×(L∞(Ω;S3 ))2)solvestheproblem(2.1)–(2.4)if
dev
(cid:6) (cid:7) (cid:6) (cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:6) (cid:7)(cid:7) (cid:7)
u ,T ,A ,εp,b ∈L2 0,T;H1 Ω;R3 ×L2 Ω;R9 ×H2 Ω;so(3) × L2 Ω;S3 2 ,
t t t t t dev
3506 K.Chełmin´skietal./J.DifferentialEquations256(2014)3497–3523
Eqs. (2.1) and (2.1) are satisfiedpointwisealmosteverywhereon Ω ×(0,T) and forall test
1 3
functions(Tˆ ,bˆ)∈L2(0,T;L2(Ω;S3)×L2(Ω;S3 ))suchthat
E dev
(cid:6) (cid:7) (cid:6) (cid:6) (cid:7)(cid:7)
dev(Tˆ ),bˆ ∈K∗,divTˆ ∈L2 0,T;L2 Ω,R3 ,
E E
theinequality
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8)
1 C−1T (x,t)T (x,t)dx+μ (cid:8)skew ∇ u(x,t) −A(x,t)(cid:8)2dx
E E c x
2
Ω Ω
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7)(cid:8) (cid:8) (cid:8)
+2l (cid:8)∇axl A(x,t) (cid:8)2dx+ 1 (cid:8)b(x,t)(cid:8)2dx
c
2c
Ω Ω
(cid:10) (cid:10)
(cid:8) (cid:6) (cid:7) (cid:8)
(cid:2) 1 C−1T0(x)T0(x)dx+μ (cid:8)skew ∇ u(x,0) −A(x,0)(cid:8)2dx
2 E E c x
Ω Ω
(cid:10) (cid:10) (cid:10)t (cid:10)
(cid:8) (cid:8) (cid:8) (cid:6) (cid:7)(cid:8)
+ 1 (cid:8)b(x,0)(cid:8)2dx+2l (cid:8)∇axl A(x,0) (cid:8)2dx+ u (x,τ)f(x,τ)dxdτ
c t
2c
Ω Ω 0 Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
(cid:6) (cid:7)
+ u (x,τ)divTˆ (x,τ)dxdτ + g (x,τ) T(x,τ)−Tˆ (x,τ) ·n(x)dSdτ
t E D,t E
0 Ω 0 ∂Ω
(cid:10)t (cid:10) (cid:10)t (cid:10)
+ C−1T (x,τ)Tˆ (x,τ)dxdτ + 1 b (x,τ)bˆ(x,τ)dxdτ
E,t E t
c
0 Ω 0 Ω
(cid:10)t (cid:10)
(cid:6) (cid:7) (cid:6) (cid:7)
+4l ∇axl A(x,τ) ·naxl A (x,τ) dSdτ (3.9)
c D,t
0 ∂Ω
is satisfied for all t ∈ (0,T), where T0 ∈ L2(Ω;S3) and (u(0),A(0)) ∈ H1(Ω;R3) ×
E
H2(Ω;so(3)) are unique solutions of the problem (3.8). Moreover the stress constraint 1 in
Definition3.1issatisfied.
Theorem3.4(Mainexistenceresult).Letusassumethatthegivendataandinitialdatasatisfy
theproperties,whicharespecifiedin(3.5)–(3.8).Thenthereexistsaglobalintimesolution(in
thesenseof Definition3.3)ofthesystem(2.1)withboundaryconditions(2.2),(2.3)andinitial
condition(2.4).
Notice that the solution defined above has a quite nice regularity. We even get that εp ∈
t
L2(L2), which yields |εp|b ∈L2(L2). Unfortunately, this information is still not enough to
t
obtain L2-strong solutions. This paper presents the first existence result for the Armstrong–
FrederickmodelwithCosserateffectsandthenewsolutionconcept.