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Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity
This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under polynomial growth assumptions, we prove that the energy functionals stored in the deformation of an -scaling of a stochastic lattice Γ -converge to a continuous energy functio...
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| Published in: | Archive for rational mechanics and analysis 2011-06, Vol.200 (3), p.881-943 |
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| cites | cdi_FETCH-LOGICAL-c381t-8a0981b44e06007891044fa72725035657d11fda6ac964b1437d7b8dd360a3af3 |
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| container_title | Archive for rational mechanics and analysis |
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| creator | Alicandro, Roberto Cicalese, Marco Gloria, Antoine |
| description | This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under polynomial growth assumptions, we prove that the energy functionals
stored in the deformation of an
-scaling of a stochastic lattice
Γ
-converge to a continuous energy functional when
goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic. We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance, isotropy and natural states. |
| doi_str_mv | 10.1007/s00205-010-0378-7 |
| format | article |
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stored in the deformation of an
-scaling of a stochastic lattice
Γ
-converge to a continuous energy functional when
goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic. We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance, isotropy and natural states.</description><identifier>ISSN: 0003-9527</identifier><identifier>EISSN: 1432-0673</identifier><identifier>DOI: 10.1007/s00205-010-0378-7</identifier><identifier>CODEN: AVRMAW</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer-Verlag</publisher><subject>Classical Mechanics ; Complex Systems ; Exact sciences and technology ; Fluid- and Aerodynamics ; Fundamental areas of phenomenology (including applications) ; Mathematical and Computational Physics ; Mathematics ; Numerical Analysis ; Physics ; Physics and Astronomy ; Solid mechanics ; Static elasticity (thermoelasticity...) ; Structural and continuum mechanics ; Theoretical</subject><ispartof>Archive for rational mechanics and analysis, 2011-06, Vol.200 (3), p.881-943</ispartof><rights>Springer-Verlag 2010</rights><rights>2015 INIST-CNRS</rights><rights>Springer-Verlag 2011</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c381t-8a0981b44e06007891044fa72725035657d11fda6ac964b1437d7b8dd360a3af3</citedby><cites>FETCH-LOGICAL-c381t-8a0981b44e06007891044fa72725035657d11fda6ac964b1437d7b8dd360a3af3</cites><orcidid>0000-0001-7502-606X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,314,776,780,881,27901,27902</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=24203602$$DView record in Pascal Francis$$Hfree_for_read</backlink><backlink>$$Uhttps://inria.hal.science/inria-00437765$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Alicandro, Roberto</creatorcontrib><creatorcontrib>Cicalese, Marco</creatorcontrib><creatorcontrib>Gloria, Antoine</creatorcontrib><title>Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity</title><title>Archive for rational mechanics and analysis</title><addtitle>Arch Rational Mech Anal</addtitle><description>This article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices. Under polynomial growth assumptions, we prove that the energy functionals
stored in the deformation of an
-scaling of a stochastic lattice
Γ
-converge to a continuous energy functional when
goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic. We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics. 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stored in the deformation of an
-scaling of a stochastic lattice
Γ
-converge to a continuous energy functional when
goes to zero. In particular, the limiting energy functional is of integral type, and deterministic if the lattice is ergodic. We also generalize, to systems and nonlinear settings, well-known results on stochastic homogenization of discrete elliptic equations. As an application of the main result, we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics. We finally address some mechanical properties of the limiting models, such as frame-invariance, isotropy and natural states.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer-Verlag</pub><doi>10.1007/s00205-010-0378-7</doi><tpages>63</tpages><orcidid>https://orcid.org/0000-0001-7502-606X</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Classical Mechanics Complex Systems Exact sciences and technology Fluid- and Aerodynamics Fundamental areas of phenomenology (including applications) Mathematical and Computational Physics Mathematics Numerical Analysis Physics Physics and Astronomy Solid mechanics Static elasticity (thermoelasticity...) Structural and continuum mechanics Theoretical |
| title | Integral Representation Results for Energies Defined on Stochastic Lattices and Application to Nonlinear Elasticity |
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