. . . "Material hiperel\u00E1stico"@es . . "Proof 2"@en . . . . "Hyper\u00E9lasticit\u00E9"@fr . . . "\u8D85\u5F39\u6027"@zh . . "\u8D85\u5F3E\u6027\uFF08\u3061\u3087\u3046\u3060\u3093\u305B\u3044\u3001Hyperelasticity\uFF09\u3068\u306F\u3001\u7269\u4F53\u3092\u69CB\u6210\u3059\u308B\u7269\u8CEA\u306E\u529B\u5B66\u7684\u7279\u6027\u306E\u6570\u7406\u7684\u8868\u73FE\u306E\u3072\u3068\u3064\u3067\u3042\u308A\u3001\u3072\u305A\u307F\u30A8\u30CD\u30EB\u30AE\u30FC\u5BC6\u5EA6\u95A2\u6570\uFF08\u5358\u4F4D\u4F53\u7A4D\u3042\u305F\u308A\u306E\u3072\u305A\u307F\u30A8\u30CD\u30EB\u30AE\u30FC\u3092\u8868\u3059\u5F3E\u6027\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\uFF09\u3092\u6709\u3059\u308B\u3053\u3068\u304C\u7279\u5FB4\u3067\u3042\u308B\u3002\u8D85\u5F3E\u6027\u3092\u6709\u3059\u308B\u7269\u8CEA\u3092\u8D85\u5F3E\u6027\u4F53\u3068\u3088\u3073\u3001\u30B4\u30E0\u306E\u6700\u3082\u7C21\u6613\u306A\u30E2\u30C7\u30EB\u3068\u3057\u3066\u767B\u5834\u3057\u305F\u3053\u3068\u306B\u7531\u6765\u3057\u3066\u3001\u6570\u5341\uFF05\uFF5E\u6570\u767E\uFF05\u306E\u5927\u3072\u305A\u307F\u72B6\u614B\u3092\u60F3\u5B9A\u3057\u3066\u3044\u308B\u3002"@ja . . . . . . . . "4472938"^^ . . "Hyperelastizit\u00E4t oder Green\u2019sche Elastizit\u00E4t (von griechisch \u1F51\u03C0\u03AD\u03C1 hyper \u201E\u00FCber\u201C, \u03B5\u03BB\u03B1\u03C3\u03C4\u03B9\u03BA\u03CC\u03C2 elastikos \u201Eanpassungsf\u00E4hig\u201C und George Green) ist ein Materialmodell der Elastizit\u00E4t. Elastizit\u00E4t ist die Eigenschaft eines K\u00F6rpers, unter Krafteinwirkung seine Form zu ver\u00E4ndern und bei Wegfall der einwirkenden Kraft in die Ursprungsform zur\u00FCckzukehren (Beispiel: Sprungfeder). Als Ursache der Elastizit\u00E4t kommen Verzerrungen des Atomgitters (bei Metallen), das Dehnen von Molek\u00FClketten (Gummi und Kunststoffe) oder die \u00C4nderung des mittleren Atomabstandes (Fl\u00FCssigkeiten und Gase) in Frage. F\u00FCr viele Materialien beschreibt die lineare Elastizit\u00E4t das beobachtete Materialverhalten nicht genau. Das bekannteste Beispiel mit nichtlinear elastischem Verhalten ist Gummi, das gro\u00DFen Verformungen standh\u00E4lt und dessen Reaktionen in guter N\u00E4herung mit Hyperelastizit\u00E4t nachgebildet werden k\u00F6nnen. Auch biologische Gewebe werden mit Hyperelastizit\u00E4t modelliert. Alle barotropen reibungsfreien Fl\u00FCssigkeiten und Gase sind gleichsam Cauchy-elastisch und hyperelastisch, worauf in der Cauchy-Elastizit\u00E4t eingegangen wird. Der vorliegende Artikel befasst sich mit Feststoffmodellen. Hier ist die Hyperelastizit\u00E4t derjenige Spezialfall der Cauchy-Elastizit\u00E4t, in dem das Materialverhalten konservativ ist. Ronald Rivlin und Melvin Mooney entwickelten die ersten Feststoffmodelle der Hyperelastizit\u00E4t, das Neo-Hooke- bzw. das Mooney-Rivlin-Modell. Andere oft benutzte Materialmodelle sind das Ogden- und Arruda-Boyce-Modell."@de . . "To express the Cauchy stress in terms of the stretches recall that\n\nThe chain rule gives\n\nThe Cauchy stress is given by\n\nPlugging in the expression for the derivative of leads to\n\nUsing the spectral decomposition of we have\n\nAlso note that\n\nTherefore, the expression for the Cauchy stress can be written as\n\nFor an incompressible material and hence . Following Ogden p. 485, we may write\n\nSome care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fr\u00E9chet differentiable. A rigorous tensor derivative can only be found by solving another eigenvalue problem.\n\nIf we express the stress in terms of differences between components,\n\nIf in addition to incompressibility we have then a possible solution to the problem\nrequires and we can write the stress differences as"@en . . . "The second Piola\u2013Kirchhoff stress tensor for a hyperelastic material is given by\n\nwhere is the right Cauchy\u2013Green deformation tensor and is the deformation gradient. The Cauchy stress is given by\n\nwhere . Let be the three principal invariants of . Then\n\nThe derivatives of the invariants of the symmetric tensor are\n\nTherefore, we can write\n\nPlugging into the expression for the Cauchy stress gives\n\nUsing the left Cauchy\u2013Green deformation tensor and noting that , we can write\n\nFor an incompressible material and hence .Then\n\nTherefore, the Cauchy stress is given by\n\nwhere is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.\n\nIf, in addition, , we have and hence\n\nIn that case the Cauchy stress can be expressed as"@en . . . . . "Iperelasticit\u00E0"@it . . . . "Proof 1"@en . . . "Un material hiperel\u00E1stico o material el\u00E1stico de Green\u200B es un tipo de material el\u00E1stico para el cual la ecuaci\u00F3n constitutiva que relaciona tensiones y deformaciones puede obtenerse a partir de una potencial el\u00E1stico o energ\u00EDa el\u00E1stica de deformaci\u00F3n que sea funci\u00F3n de estado. En un material el\u00E1stico el tensor de tensiones (2\u00BA tensor de Piola-Kirchhof) puede relacionarse con el tensor de deformaci\u00F3n de Green-Cauchy mediante la relaci\u00F3n: en componentes Los materiales hiperel\u00E1sticos son un caso particular de material el\u00E1stico de Cauchy."@es . "A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress\u2013strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material. Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney\u2013Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the Ogden model and the Arruda\u2013Boyce model."@en . . . . . "Nella scienza delle costruzioni un materiale \u00E8 definito \"linearmente iperelastico\" quando: \n* \u00C8 linearmente elastico \n* Alla relazione costitutiva di lineare elasticit\u00E0 \u00E8 possibile associare una funzione scalare definita energia specifica (o potenziale elastico di deformazione, misurabile in J/m\u00B3 o F*L/L\u00B3) che \u00E8 appunto l'energia che bisogna spendere per deformare un determinato materiale d'una quantit\u00E0 unitaria. Quindi se questa energia esiste, essa \u00E8 unica."@it . "35659"^^ . . . "\u8D85\u5F39\u6027\u6750\u6599\u6A21\u578B\u53EF\u7528\u4E8E\u4E3A\u7C7B\u6A61\u80F6\u6750\u6599\u5EFA\u6A21\uFF0C\u5176\u4E2D\u7684\u89E3\u4F1A\u6D89\u53CA\u5927\u53D8\u5F62\u3002\u5047\u8BBE\u6750\u6599\u4E3A\u975E\u7EBF\u6027\u5F39\u6027\u3001\u540C\u5411\u6027\u4E14\u4E0D\u53EF\u538B\u7F29\u3002\u5E38\u89C1\u7684\u8D85\u5F39\u6027\u6A21\u578B\u6709\uFF1A \n* Mooney - Rivlin \u8D85\u5F39\u6027\u6A21\u578B\u53EF\u4EE5\u7528\u4E8E\u5B9E\u4F53\u5355\u5143\u548C\u539A\u58F3\u4F53\u3002 \n* \n* \u8D85\u5F39\u6027 Blatz - Ko \u6A21\u578B\u7528\u4E8E\u53EF\u538B\u7F29\u805A\u6C28\u916F\uFF08PU\uFF09\u6CE1\u6CAB\u7C7B\u578B\u6A61\u80F6\u7684\u6A21\u578B\u3002"@zh . . . . "Hyperelastizit\u00E4t oder Green\u2019sche Elastizit\u00E4t (von griechisch \u1F51\u03C0\u03AD\u03C1 hyper \u201E\u00FCber\u201C, \u03B5\u03BB\u03B1\u03C3\u03C4\u03B9\u03BA\u03CC\u03C2 elastikos \u201Eanpassungsf\u00E4hig\u201C und George Green) ist ein Materialmodell der Elastizit\u00E4t. Elastizit\u00E4t ist die Eigenschaft eines K\u00F6rpers, unter Krafteinwirkung seine Form zu ver\u00E4ndern und bei Wegfall der einwirkenden Kraft in die Ursprungsform zur\u00FCckzukehren (Beispiel: Sprungfeder). Als Ursache der Elastizit\u00E4t kommen Verzerrungen des Atomgitters (bei Metallen), das Dehnen von Molek\u00FClketten (Gummi und Kunststoffe) oder die \u00C4nderung des mittleren Atomabstandes (Fl\u00FCssigkeiten und Gase) in Frage."@de . . . "Nella scienza delle costruzioni un materiale \u00E8 definito \"linearmente iperelastico\" quando: \n* \u00C8 linearmente elastico \n* Alla relazione costitutiva di lineare elasticit\u00E0 \u00E8 possibile associare una funzione scalare definita energia specifica (o potenziale elastico di deformazione, misurabile in J/m\u00B3 o F*L/L\u00B3) che \u00E8 appunto l'energia che bisogna spendere per deformare un determinato materiale d'una quantit\u00E0 unitaria. Si dimostra (anche se \u00E8 facilmente immaginabile) che il lavoro per deformare di una certa quantit\u00E0 un solido linearmente iperelastico (lavoro in un cammino deformativo) \u00E8 proprio pari alla variazione di potenziale elastico. Quindi se questa energia esiste, essa \u00E8 unica. Nel caso di sforzo monoassiale per un corpo iperelastico, applicando una forza al corpo lungo una direzione (monoassiale), si ottiene una deformazione. Rilasciando tale forza, non rimangono deformazioni residue e il corpo ritorna nello stato iniziale, senza dissipazioni."@it . "Un material hiperel\u00E1stico o material el\u00E1stico de Green\u200B es un tipo de material el\u00E1stico para el cual la ecuaci\u00F3n constitutiva que relaciona tensiones y deformaciones puede obtenerse a partir de una potencial el\u00E1stico o energ\u00EDa el\u00E1stica de deformaci\u00F3n que sea funci\u00F3n de estado. En un material el\u00E1stico el tensor de tensiones (2\u00BA tensor de Piola-Kirchhof) puede relacionarse con el tensor de deformaci\u00F3n de Green-Cauchy mediante la relaci\u00F3n: en componentes Los materiales hiperel\u00E1sticos son un caso particular de material el\u00E1stico de Cauchy."@es . . . . . . "The isochoric deformation gradient is defined as , resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free. Using this one can subsequently define the isochoric left Cauchy\u2013Green deformation tensor .\nThe invariants of are\n\nThe set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy\u2013Green deformation tensor tensor, , and add into the fray to describe the volumetric behaviour.\n\nTo express the Cauchy stress in terms of the invariants recall that\n\nThe chain rule of differentiation gives us\n\nRecall that the Cauchy stress is given by\n\nIn terms of the invariants we have\n\nPlugging in the expressions for the derivatives of in terms of , we have\n\nor,\n\nIn terms of the deviatoric part of , we can write\n\nFor an incompressible material and hence .Then\nthe Cauchy stress is given by\n\nwhere is an undetermined pressure-like Lagrange multiplier term. In addition, if , we have and hence\nthe Cauchy stress can be expressed as"@en . . . . . . . . . "Hyperelastic material"@en . "\u8D85\u5F3E\u6027"@ja . . . . . . "A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress\u2013strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material. For many materials, linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic and incompressible. Hyperelasticity provides a means of modeling the stress\u2013strain behavior of such materials. The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues are also often modeled via the hyperelastic idealization. Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney\u2013Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the Ogden model and the Arruda\u2013Boyce model."@en . . . . "\u8D85\u5F39\u6027\u6750\u6599\u6A21\u578B\u53EF\u7528\u4E8E\u4E3A\u7C7B\u6A61\u80F6\u6750\u6599\u5EFA\u6A21\uFF0C\u5176\u4E2D\u7684\u89E3\u4F1A\u6D89\u53CA\u5927\u53D8\u5F62\u3002\u5047\u8BBE\u6750\u6599\u4E3A\u975E\u7EBF\u6027\u5F39\u6027\u3001\u540C\u5411\u6027\u4E14\u4E0D\u53EF\u538B\u7F29\u3002\u5E38\u89C1\u7684\u8D85\u5F39\u6027\u6A21\u578B\u6709\uFF1A \n* Mooney - Rivlin \u8D85\u5F39\u6027\u6A21\u578B\u53EF\u4EE5\u7528\u4E8E\u5B9E\u4F53\u5355\u5143\u548C\u539A\u58F3\u4F53\u3002 \n* \n* \u8D85\u5F39\u6027 Blatz - Ko \u6A21\u578B\u7528\u4E8E\u53EF\u538B\u7F29\u805A\u6C28\u916F\uFF08PU\uFF09\u6CE1\u6CAB\u7C7B\u578B\u6A61\u80F6\u7684\u6A21\u578B\u3002"@zh . "Hyperelastizit\u00E4t"@de . "Proof 3"@en . . . . "L'hyper\u00E9lasticit\u00E9 est un formalisme math\u00E9matique utilis\u00E9 en r\u00E9sistance des mat\u00E9riaux pour d\u00E9crire la relation contrainte-d\u00E9formation de certains mat\u00E9riaux grandement d\u00E9formables (polym\u00E8res thermoplastiques, polym\u00E8res thermodurcissables, \u00E9lastom\u00E8res, tissus biologiques). Contrairement \u00E0 l'\u00E9lasticit\u00E9 lin\u00E9aire d\u00E9finie explicitement par la loi de Hooke pour les petites d\u00E9formations, en hyper\u00E9lasticit\u00E9, on postule l'existence d\u2019une densit\u00E9 d\u2019\u00E9nergie de d\u00E9formation not\u00E9e W dont les d\u00E9riv\u00E9es par rapport \u00E0 la d\u00E9formation dans une direction donn\u00E9e donnent l'\u00E9tat de contrainte au sein du mat\u00E9riau dans cette m\u00EAme direction. Physiquement, W repr\u00E9sente la quantit\u00E9 d\u2019\u00E9nergie \u00E9lastique que le mat\u00E9riau emmagasine en fonction de l\u2019\u00E9tirement impos\u00E9."@fr . . "1122158305"^^ . . . . . . . . . . . . . . . . . . . . "\u8D85\u5F3E\u6027\uFF08\u3061\u3087\u3046\u3060\u3093\u305B\u3044\u3001Hyperelasticity\uFF09\u3068\u306F\u3001\u7269\u4F53\u3092\u69CB\u6210\u3059\u308B\u7269\u8CEA\u306E\u529B\u5B66\u7684\u7279\u6027\u306E\u6570\u7406\u7684\u8868\u73FE\u306E\u3072\u3068\u3064\u3067\u3042\u308A\u3001\u3072\u305A\u307F\u30A8\u30CD\u30EB\u30AE\u30FC\u5BC6\u5EA6\u95A2\u6570\uFF08\u5358\u4F4D\u4F53\u7A4D\u3042\u305F\u308A\u306E\u3072\u305A\u307F\u30A8\u30CD\u30EB\u30AE\u30FC\u3092\u8868\u3059\u5F3E\u6027\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\uFF09\u3092\u6709\u3059\u308B\u3053\u3068\u304C\u7279\u5FB4\u3067\u3042\u308B\u3002\u8D85\u5F3E\u6027\u3092\u6709\u3059\u308B\u7269\u8CEA\u3092\u8D85\u5F3E\u6027\u4F53\u3068\u3088\u3073\u3001\u30B4\u30E0\u306E\u6700\u3082\u7C21\u6613\u306A\u30E2\u30C7\u30EB\u3068\u3057\u3066\u767B\u5834\u3057\u305F\u3053\u3068\u306B\u7531\u6765\u3057\u3066\u3001\u6570\u5341\uFF05\uFF5E\u6570\u767E\uFF05\u306E\u5927\u3072\u305A\u307F\u72B6\u614B\u3092\u60F3\u5B9A\u3057\u3066\u3044\u308B\u3002"@ja . . . "L'hyper\u00E9lasticit\u00E9 est un formalisme math\u00E9matique utilis\u00E9 en r\u00E9sistance des mat\u00E9riaux pour d\u00E9crire la relation contrainte-d\u00E9formation de certains mat\u00E9riaux grandement d\u00E9formables (polym\u00E8res thermoplastiques, polym\u00E8res thermodurcissables, \u00E9lastom\u00E8res, tissus biologiques). Contrairement \u00E0 l'\u00E9lasticit\u00E9 lin\u00E9aire d\u00E9finie explicitement par la loi de Hooke pour les petites d\u00E9formations, en hyper\u00E9lasticit\u00E9, on postule l'existence d\u2019une densit\u00E9 d\u2019\u00E9nergie de d\u00E9formation not\u00E9e W dont les d\u00E9riv\u00E9es par rapport \u00E0 la d\u00E9formation dans une direction donn\u00E9e donnent l'\u00E9tat de contrainte au sein du mat\u00E9riau dans cette m\u00EAme direction. Physiquement, W repr\u00E9sente la quantit\u00E9 d\u2019\u00E9nergie \u00E9lastique que le mat\u00E9riau emmagasine en fonction de l\u2019\u00E9tirement impos\u00E9. Tous les polym\u00E8res ne pr\u00E9sentent pas le m\u00EAme comportement m\u00E9canique en fonction de leur microstructure (longueur des macromol\u00E9cule, nature des composants chimiques, degr\u00E9 de r\u00E9ticulation, possibilit\u00E9 de cristallisation sous-tension, temp\u00E9rature d'utilisation, etc.). Une loi de comportement hyper\u00E9lastique est donc une expression de W permettant de d\u00E9crire le comportement particulier d'un mat\u00E9riau. Il en existe une multitude pouvant \u00EAtre s\u00E9par\u00E9es en deux cat\u00E9gories : les mod\u00E8les ph\u00E9nom\u00E9nologiques et les mod\u00E8les statistiques."@fr . .