Bipotentials for Non-monotone Multivalued Operators: Fundamental Results and Applications

Buliga, Marius; Saxcé, Géry de; Vallée, Claude

doi:10.1007/s10440-009-9488-3

Cited by 28 publications

(18 citation statements)

References 29 publications

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“…Other interesting results involving the bipotential theory can be found in the overview paper [13] or in the recent paper [14], as well as in the paper [15]. The present paper extends and improves the results in [15]; there the authors illustrate the applicability of a bipotential function to the solution of a displacement-traction model in elastostatics.…”

Section: Introductionsupporting

confidence: 71%

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A variational approach via bipotentials for unilateral contact problems

Matei

2013

Journal of Mathematical Analysis and Applications

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a b s t r a c tWe consider a unilateral contact model for nonlinearly elastic materials, under the small deformation hypothesis, for static processes. The contact is modeled with Signorini's condition with zero gap and the friction is neglected on the potential contact zone. The behavior of the material is modeled by a subdifferential inclusion, the constitutive map being proper, convex, and lower semicontinuous. After describing the model, we give a weak formulation using a bipotential which depends on the constitutive map and its Fenchel conjugate. We arrive to a system of two variational inequalities whose unknown is the pair consisting of the displacement field and the Cauchy stress field. We look for the unknown into a Cartesian product of two nonempty, convex, closed, unbounded subsets of two Hilbert spaces. We prove the existence and the uniqueness of the weak solution based on minimization arguments for appropriate functionals associated with the variational system. How the proposed variational approach is related to previous variational approaches, is discussed too.

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Section: Introductionsupporting

confidence: 71%

“…According to Theorem 2, the functional J has a unique minimizer. Using now (14), (10), (12), (13), the inequality…”

Section: Theorem 3 (An Existence Result) Assumptions 1 and 2 Hold Trmentioning

confidence: 99%

A variational approach via bipotentials for unilateral contact problems

Matei

2013

Journal of Mathematical Analysis and Applications

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“…The construction of several bipotential functions appears in connection with Coulomb's friction law [4] and Cam-Clay models in soil mechanics [14], cyclic plasticity [13,2] and viscoplasticity of metals with nonlinear kinematical hardening rule [8], Lemaitre's damage law [1], the coaxial laws [15,17] etc. See also the overview paper [3]. In the present paper, we illustrate the applicability of bipotentials by providing a new variational formulation for a general model in elastostatics.…”

Section: Introductionmentioning

confidence: 98%

Weak solutions via bipotentials in mechanics of deformable solids

Matei

Niculescu

2011

Journal of Mathematical Analysis and Applications

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We consider a displacement-traction boundary values problem for elastic materials, under the small deformations hypothesis, for static processes. The behavior of the material is modeled by a constitutive law involving the subdifferential of a proper, convex, and lower semicontinuous map. The constitutive map and its Fenchel conjugate allow us to construct a bipotential function. Based on this construction, we propose a weak formulation of our mechanical problem. Furthermore, we prove the existence of at least one weak solution and we investigate the uniqueness of the weak solution. We also comment on the relevance of our variational approach, by considering three significant examples.

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“…A material whose behavior can be described ( [19,20,21]) by one of the following three implicit constitutive laws (iv) y belongs to the subdifferential of b(ξ, y) with respect to ξ at x (v) x belongs to the subdifferential of b(x, η) with respect to η at y (vi) b(x, y) = x, y is referred to as an "Implicit Standard Material" (ISM).…”

Section: Implicit Standard Materialsmentioning

confidence: 99%

A Class of Non-associated Materials: n-Monotone Materials—Hooke’s Law of Elasticity Revisited

Vallée

Lerintiu

Chaoufi

et al. 2012

J Elast

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Generalized Standard Materials are governed by maximal cyclically monotone operators and modeled by convex potentials. Géry de Saxcé's Implicit Standard Materials are modeled by biconvex bipotentials. We analyze the intermediate class of n-monotone materials governed by maximal nmonotone operators and modeled by Fitzpatrick's functions. Revisiting the model of elastic materials initiated by Robert Hooke, and insisting on the linearity, coaxiality and monotonicity properties of the constitutive law, we illustrate that Fitzpatrick's representation of n-monotone operators coming from convex analysis provides a constructive method to discover the best bipotential modeling a n-monotone material. Giving up the symmetry of the linear constitutive laws, we find out that n-monotonicity is a relevant criterion for the materials characterization and classification.2000 Mathematics Subject Classification. Primary: 74D10 and 47H05 and Secondary: 47H04.

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Bipotentials for Non-monotone Multivalued Operators: Fundamental Results and Applications

Cited by 28 publications

References 29 publications

A variational approach via bipotentials for unilateral contact problems

A variational approach via bipotentials for unilateral contact problems

Weak solutions via bipotentials in mechanics of deformable solids

A Class of Non-associated Materials: n-Monotone Materials—Hooke’s Law of Elasticity Revisited

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