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A Viscoplastic Contact Problem with Normal Compliance, Unilateral Constraint and Memory Term

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A Viscoplastic Contact Problem with Normal Compliance, Unilateral Constraint and Memory Term
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  Appl Math Optim (2014) 69:175–198DOI 10.1007/s00245-013-9216-2 A Viscoplastic Contact Problem with NormalCompliance, Unilateral Constraint and Memory Term M. Sofonea  · F. P˘atrulescu  · A. Farca¸s Published online: 30 August 2013© Springer Science+Business Media New York 2013 Abstract  We consider a mathematical model which describes the quasistatic contactbetween a viscoplastic body and a foundation. The material’s behavior is modelledwitharate-typeconstitutivelawwithinternalstatevariable.Thecontactisfrictionlessand is modelled with normal compliance, unilateral constraint and memory term. Wepresent the classical formulation of the problem, list the assumptions on the dataand derive a variational formulation of the model. Then we prove its unique weak solvability. The proof is based on arguments of history-dependent quasivariationalinequalities. We also study the dependence of the solution with respect to the dataand prove a convergence result. Keywords  Viscoplastic material  ·  Frictionless contact  ·  Normal compliance  · Unilateral constraint  ·  Memory term  ·  History-dependent variational inequality  · Weak solution  ·  Fréchet space 1 Introduction The aim of this paper is to study a frictionless contact problem for rate-type vis-coplastic materials within the framework of the Mathematical Theory of Contact M. Sofonea ( B )Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia,52 Avenue de Paul Alduy, 66 860 Perpignan, Francee-mail: sofonea@univ-perp.frF. P˘atrulescuTiberiu Popoviciu Institute of Numerical Analysis, P.O. Box 68-1, 400110 Cluj-Napoca, RomaniaA. Farca¸sFaculty of Mathematics and Computer Science, Babe¸s-Bolyai University, Kog˘alniceanu street, no. 1,400084 Cluj-Napoca, Romania  176 Appl Math Optim (2014) 69:175–198 Mechanics. We model the material’s behavior with a constitutive law of the form ˙ σ  (t) = E  ε  ˙ u (t)  + G   σ  (t), ε  u (t)  , κ (t)  ,  (1.1)where  u  denotes the displacement field,  σ   represents the stress tensor,  ε ( u )  is thelinearized strain tensor and  κ  denotes an internal state variable. Here  E   is a linearoperator which describes the elastic properties of the material and  G   is a nonlinearconstitutive function which describes its viscoplastic behavior. In (1.1) and every-where in this paper the dot above a variable represents the derivative with respect tothe time variable  t  . Following [3, 7], the internal state variable  κ  is a vector-valuedfunction whose evolution is governed by the differential equation ˙ κ (t) = G  σ  (t), ε  u (t)  , κ (t)  ,  (1.2)in which  G  is a nonlinear constitutive function with values in R m ,  m  being a positiveinteger.Various results, examples and mechanical interpretations in the study of viscoplas-tic materials of the form (1.1), (1.2) can be found in [3, 7] and the references therein. Quasistatic contact problems for such materials have been considered in [5, 6] and the references therein. There, the contact was assumed to be frictionless and wasmodelled with normal compliance; the unique weak solvability of the correspondingproblemswasprovedbyusingargumentsofnonlinearequationswithmonotoneoper-ators and fixed point; semi-discrete and fully discrete scheme were considered, errorestimates and convergence results were proved and numerical simulation in the studyof two-dimensional test problems were presented. The normal compliance contactcondition was first introduced in [14] and since then used in many publications, see,e.g., [9–12] and references therein. The term  normal compliance  was first introducedin [10, 11]. In the particular case without internal state variable the constitutive equation (1.1)reads ˙ σ  (t) = E  ε  ˙ u (t)  + G   σ  (t), ε  u (t)  ,  (1.3)and was used in the literature in order to model the behaviour of various materi-als like rubbers, rocks, metals, pastes and polymers. Quasistatic frictionless contactproblems for materials of the form (1.3) have been considered in [1, 6, 15, 17] and the references therein, under various contact conditions. In [6, 15] both the Signorini and the normal compliance condition were used which, recall, describe a contact witha rigid and elastic foundation, respectively. In [1, 17] the contact was modelled with normal compliance and unilateral constraint condition. This condition, introduced forthe first time in [8], models an elastic-rigid behavior of the foundation.With respect to the papers above mentioned, the current paper has three traits of novelties that we describe in what follows. First, the model we consider involves acontact condition with normal compliance, unilateral constraint and memory term.This condition takes into account both the deformability, the rigidity, and the mem-ory effects of the foundation. Second, in contrast with the short note [4], we modelthe behavior of the material with a viscoplastic constitutive law with internal statevariable. And, finally, we study the contact process on an unbounded interval of time  Appl Math Optim (2014) 69:175–198 177 which implies the use of the framework of Fréchet spaces of continuous functions,instead of that of the classical Banach spaces of continuous functions defined on abounded interval of time. The three ingredients above lead to a new and interestingmathematical model. The aim of this work is to prove the unique weak solvability of this model and to study the dependence of the weak solution with respect to the data.The rest of the paper is structured as follows. In Sect. 2 we present the notationwe shall use as well as some preliminary material. In Sect. 3 we describe the modelof the contact process. In Sect. 4 we list the assumptions on the data and derive thevariational formulation of the problem. Then we state and prove our main existenceand uniqueness result, Theorem 4.1. In Sect. 5 we state and prove our converge result, Theorem 5.1. It states the continuous dependence of the solution with respect to thedata. 2 Notations and Preliminaries Everywhere in this paper we use the notation  N ∗  for the set of positive integers and R +  willrepresentthesetofnonnegativerealnumbers,i.e. R +  =[ 0 , +∞ ) .Foragiven r  ∈ R  we denote by  r +  its positive part, i.e.  r +  =  max { r, 0 } . Let  Ω  be a boundeddomain  Ω  ⊂ R d  (d   = 1 , 2 , 3 )  with a Lipschitz continuous boundary  Γ   and let  Γ  1  bea measurable part of   Γ   such that meas (Γ  1 ) >  0. We use the notation  x  =  (x i )  for atypical point in  Ω ∪ Γ   and we denote by  ν  = (ν i )  the outward unit normal at  Γ   . Hereand below the indices  i ,  j  ,  k ,  l  run between 1 and  d   and, unless stated otherwise, thesummation convention over repeated indices is used. An index that follows a commarepresents the partial derivative with respect to the corresponding component of thespatial variable, e.g.  u i,j   = ∂u i /∂x j  .We denote by  S d  the space of second order symmetric tensors on  R d  or, equiva-lently, the space of symmetric matrices of order  d  . The inner product and norm on R d  and S d  are defined by u · v  = u i v i ,   v = ( v  · v ) 12  ∀ u , v  ∈ R d  , σ   · τ   = σ  ij  τ  ij  ,   τ  = ( τ   · τ  ) 12  ∀ σ  , τ   ∈ S d  . Also, we use the notation   κ   for the Euclidean norm of the element  κ  ∈ R m . Inaddition, we use standard notation for the Lebesgue and Sobolev spaces associated to Ω  and  Γ   and, moreover, we consider the spaces V   =  v  = (v i ) ∈ H  1 (Ω) d  : v  = 0  on  Γ  1  ,Q  =  τ   = (τ  ij  ) ∈ L 2 (Ω) d  × d  : τ  ij   = τ  ji  . These are real Hilbert spaces endowed with the inner products ( u , v ) V   =   Ω ε ( u ) · ε ( v )dx, ( σ  , τ  ) Q  =   Ω σ   · τ   dx,  178 Appl Math Optim (2014) 69:175–198 and the associated norms  · V   and  · Q , respectively. Here  ε  represents the defor-mation operator given by ε ( v ) =  ε ij  ( v )  , ε ij  ( v ) = 12 (v i,j   + v j,i )  ∀ v  ∈ H  1 (Ω) d  . Completeness of the space  (V,  ·  V  )  follows from the assumption meas (Γ  1 ) >  0,which allows the use of Korn’s inequality.For an element  v  ∈  V   we still write  v  for the trace of   v  on the boundary andwe denote by  v ν  and  v τ   the normal and tangential components of   v  on  Γ   , given by v ν  =  v  ·  ν ,  v τ   =  v  −  v ν ν . Let  Γ  3  be a measurable part of   Γ   . Then, by the Sobolevtrace theorem, there exists a positive constant  c 0  which depends on  Ω ,  Γ  1  and  Γ  3 such that  v  L 2 (Γ  3 ) d   ≤ c 0  v  V   ∀ v  ∈ V.  (2.1)Also, for a regular function  σ   ∈  Q  we use the notation  σ  ν  and  σ  τ   for the normaland the tangential traces, i.e.  σ  ν  = ( σν ) · ν  and  σ  τ   = σν  − σ  ν ν . Moreover, we recallthat the divergence operator is defined by the equality Div σ   = (σ  ij,j  )  and, finally, thefollowing Green’s formula holds:   Ω σ   · ε ( v )dx  +   Ω Div σ   · v dx  =   Γ  σν  · v da  ∀ v  ∈ V.  (2.2)Finally, we denote by  Q ∞  the space of fourth order tensor fields given by Q ∞  =  E   = ( E  ijkl ) : E  ijkl  = E  jikl  = E  klij   ∈ L ∞ (Ω),  1 ≤ i,j,k,l  ≤ d   , and we recall that  Q ∞  is a real Banach space with the norm  E   Q ∞  =  max 1 ≤ i,j,k,l ≤ d   E  ijkl  L ∞ (Ω) . Moreover, a simple calculation shows that  E  τ   Q  ≤ d   E   Q ∞  τ   Q  ∀ E   ∈ Q ∞ , τ   ∈ Q.  (2.3)For each Banach space  X  we use the notation  C( R + ; X)  for the space of con-tinuous functions defined on  R +  with values in  X . For a subset  K  ⊂  X  we still usethe symbol  C( R + ; K)  for the set of continuous functions defined on  R +  with val-ues in  K . It is well known that  C( R + ; X)  can be organized in a canonical way as aFréchet space, i.e. as a complete metric space in which the corresponding topologyis induced by a countable family of seminorms. Details can be found in [2] and [13], for instance. Here we restrict ourseleves to recall that the convergence of a sequence (x k ) k  to the element  x , in the space  C( R + ; X) , can be described as follows:  x k  → x  in  C( R + ; X)  as  k  →∞  if and only if max r ∈[ 0 ,n ]  x k (r) − x(r)  X  → 0 as  k  →∞ ,  for all  n ∈ N ∗ . (2.4)The following fixed-point result will be used in Sect. 4 of the paper.  Appl Math Optim (2014) 69:175–198 179 Theorem 2.1  Let   (X,  ·  X )  be a real Banach space and let   Λ  :  C( R + ; X)  → C( R + ; X)  be a nonlinear operator with the following property :  there exists  c >  0 such that   Λu(t) − Λv(t)  X  ≤ c    t  0  u(s) − v(s)  X  ds  (2.5)  for all  u, v  ∈ C( R + ; X)  and for all  t   ∈ R + .  Then the operator   Λ  has a unique fixed  point   η ∗  ∈ C( R + ; X) .Theorem 2.1 represents a simplified version of Corollary 2.5 in [18]. We underline that in (2.5) and below, the notation  Λη(t)  represents the value of the function  Λη  atthe point  t  , i.e.  Λη(t) = (Λη)(t) .Consider now a real Hilbert space  X  with inner product  ( · , · ) X  and associatednorm  · X  as well as a normed space  Y   with norm  · Y  . Let  K  be a subset of   X and consider the operators  A  :  K  →  X ,  R :  C( R + ; X)  →  C( R + ; Y)  as well as thefunctions  ϕ  : Y   × K  → R ,  f   : R +  → X  such that: K  is a nonempty closed convex subset of   X.  (2.6)(a) There exists  m >  0 such that (Au 1  − Au 2 ,u 1  − u 2 ) X  ≥ m  u 1  − u 2  2 X  ∀ u 1 , u 2  ∈ K. (b) There exists  M >  0 such that  Au 1  − Au 2  X  ≤ M   u 1  − u 2  X  ∀ u 1 , u 2  ∈ K.  (2.7)For every  n ∈ N ∗  there exists  r n  >  0 such that  R u 1 (t) − R u 2 (t)  Y   ≤ r n    t  0  u 1 (s) − u 2 (s)  X  ds ∀ u 1 , u 2  ∈ C( R + ; X),  ∀ t   ∈[ 0 ,n ] .  (2.8)(a) The function  ϕ(u, · ) : K  → R is convex andlower semicontinuous, for all  u ∈ Y. (b) There exists  α  ≥ 0 such that ϕ(u 1 ,v 2 ) − ϕ(u 1 ,v 1 ) + ϕ(u 2 ,v 1 ) − ϕ(u 2 ,v 2 ) ≤ α  u 1  − u 2  Y   v 1  − v 2  X  ∀ u 1 ,u 2  ∈ Y,  ∀ v 1 ,v 2  ∈ K.  (2.9) f   ∈ C( R + ; X).  (2.10)The following result, proved in [16], will be used in Sect. 4 of this paper. Theorem 2.2  Assume that   (2.6)  –  (2.10)  hold  .  Then there exists a unique function u ∈ C( R + ; K)  such that  ,  for all  t   ∈ R + ,  the inequality below holds :  Au(t),v  − u(t)  X  + ϕ  R u(t),v  − ϕ  R u(t),u(t)  ≥  f(t),v  − u(t)  X  ∀ v  ∈ K.  (2.11)
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