Hyperelastic material (original) (raw)

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Un material hiperelástico o material elástico de Green​ es un tipo de material elástico para el cual la ecuación constitutiva que relaciona tensiones y deformaciones puede obtenerse a partir de una potencial elástico o energía elástica de deformación que sea función de estado. En un material elástico el tensor de tensiones (2º tensor de Piola-Kirchhof) puede relacionarse con el tensor de deformación de Green-Cauchy mediante la relación: en componentes Los materiales hiperelásticos son un caso particular de material elástico de Cauchy.

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dbo:abstract Hyperelastizität oder Green’sche Elastizität (von griechisch ὑπέρ hyper „über“, ελαστικός elastikos „anpassungsfähig“ und George Green) ist ein Materialmodell der Elastizität. Elastizität ist die Eigenschaft eines Körpers, unter Krafteinwirkung seine Form zu verändern und bei Wegfall der einwirkenden Kraft in die Ursprungsform zurückzukehren (Beispiel: Sprungfeder). Als Ursache der Elastizität kommen Verzerrungen des Atomgitters (bei Metallen), das Dehnen von Molekülketten (Gummi und Kunststoffe) oder die Änderung des mittleren Atomabstandes (Flüssigkeiten und Gase) in Frage. Für viele Materialien beschreibt die lineare Elastizität das beobachtete Materialverhalten nicht genau. Das bekannteste Beispiel mit nichtlinear elastischem Verhalten ist Gummi, das großen Verformungen standhält und dessen Reaktionen in guter Näherung mit Hyperelastizität nachgebildet werden können. Auch biologische Gewebe werden mit Hyperelastizität modelliert. Alle barotropen reibungsfreien Flüssigkeiten und Gase sind gleichsam Cauchy-elastisch und hyperelastisch, worauf in der Cauchy-Elastizität eingegangen wird. Der vorliegende Artikel befasst sich mit Feststoffmodellen. Hier ist die Hyperelastizität derjenige Spezialfall der Cauchy-Elastizität, in dem das Materialverhalten konservativ ist. Ronald Rivlin und Melvin Mooney entwickelten die ersten Feststoffmodelle der Hyperelastizität, das Neo-Hooke- bzw. das Mooney-Rivlin-Modell. Andere oft benutzte Materialmodelle sind das Ogden- und Arruda-Boyce-Modell. (de) A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress–strain 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–strain 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–Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the Ogden model and the Arruda–Boyce model. (en) Un material hiperelástico o material elástico de Green​ es un tipo de material elástico para el cual la ecuación constitutiva que relaciona tensiones y deformaciones puede obtenerse a partir de una potencial elástico o energía elástica de deformación que sea función de estado. En un material elástico el tensor de tensiones (2º tensor de Piola-Kirchhof) puede relacionarse con el tensor de deformación de Green-Cauchy mediante la relación: en componentes Los materiales hiperelásticos son un caso particular de material elástico de Cauchy. (es) L'hyperélasticité est un formalisme mathématique utilisé en résistance des matériaux pour décrire la relation contrainte-déformation de certains matériaux grandement déformables (polymères thermoplastiques, polymères thermodurcissables, élastomères, tissus biologiques). Contrairement à l'élasticité linéaire définie explicitement par la loi de Hooke pour les petites déformations, en hyperélasticité, on postule l'existence d’une densité d’énergie de déformation notée W dont les dérivées par rapport à la déformation dans une direction donnée donnent l'état de contrainte au sein du matériau dans cette même direction. Physiquement, W représente la quantité d’énergie élastique que le matériau emmagasine en fonction de l’étirement imposé. Tous les polymères ne présentent pas le même comportement mécanique en fonction de leur microstructure (longueur des macromolécule, nature des composants chimiques, degré de réticulation, possibilité de cristallisation sous-tension, température d'utilisation, etc.). Une loi de comportement hyperélastique est donc une expression de W permettant de décrire le comportement particulier d'un matériau. Il en existe une multitude pouvant être séparées en deux catégories : les modèles phénoménologiques et les modèles statistiques. (fr) Nella scienza delle costruzioni un materiale è definito "linearmente iperelastico" quando: * È linearmente elastico * Alla relazione costitutiva di lineare elasticità è possibile associare una funzione scalare definita energia specifica (o potenziale elastico di deformazione, misurabile in J/m³ o F*L/L³) che è appunto l'energia che bisogna spendere per deformare un determinato materiale d'una quantità unitaria. Si dimostra (anche se è facilmente immaginabile) che il lavoro per deformare di una certa quantità un solido linearmente iperelastico (lavoro in un cammino deformativo) è proprio pari alla variazione di potenziale elastico. Quindi se questa energia esiste, essa è 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) 超弾性(ちょうだんせい、Hyperelasticity)とは、物体を構成する物質の力学的特性の数理的表現のひとつであり、ひずみエネルギー密度関数(単位体積あたりのひずみエネルギーを表す弾性ポテンシャル)を有することが特徴である。超弾性を有する物質を超弾性体とよび、ゴムの最も簡易なモデルとして登場したことに由来して、数十%~数百%の大ひずみ状態を想定している。 (ja) 超弹性材料模型可用于为类橡胶材料建模,其中的解会涉及大变形。假设材料为非线性弹性、同向性且不可压缩。常见的超弹性模型有: * Mooney - Rivlin 超弹性模型可以用于实体单元和厚壳体。 * * 超弹性 Blatz - Ko 模型用于可压缩聚氨酯(PU)泡沫类型橡胶的模型。 (zh)
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dbp:proof To express the Cauchy stress in terms of the stretches recall that The chain rule gives The Cauchy stress is given by Plugging in the expression for the derivative of leads to Using the spectral decomposition of we have Also note that Therefore, the expression for the Cauchy stress can be written as For an incompressible material and hence . Following Ogden p. 485, we may write Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable. A rigorous tensor derivative can only be found by solving another eigenvalue problem. If we express the stress in terms of differences between components, If in addition to incompressibility we have then a possible solution to the problem requires and we can write the stress differences as (en) The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by where is the right Cauchy–Green deformation tensor and is the deformation gradient. The Cauchy stress is given by where . Let be the three principal invariants of . Then The derivatives of the invariants of the symmetric tensor are Therefore, we can write Plugging into the expression for the Cauchy stress gives Using the left Cauchy–Green deformation tensor and noting that , we can write For an incompressible material and hence .Then Therefore, the Cauchy stress is given by where is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint. If, in addition, , we have and hence In that case the Cauchy stress can be expressed as (en) 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–Green deformation tensor . The invariants of are The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, , and add into the fray to describe the volumetric behaviour. To express the Cauchy stress in terms of the invariants recall that The chain rule of differentiation gives us Recall that the Cauchy stress is given by In terms of the invariants we have Plugging in the expressions for the derivatives of in terms of , we have or, In terms of the deviatoric part of , we can write For an incompressible material and hence .Then the Cauchy stress is given by where is an undetermined pressure-like Lagrange multiplier term. In addition, if , we have and hence the Cauchy stress can be expressed as (en)
dbp:title Proof 1 (en) Proof 2 (en) Proof 3 (en)
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rdfs:comment Un material hiperelástico o material elástico de Green​ es un tipo de material elástico para el cual la ecuación constitutiva que relaciona tensiones y deformaciones puede obtenerse a partir de una potencial elástico o energía elástica de deformación que sea función de estado. En un material elástico el tensor de tensiones (2º tensor de Piola-Kirchhof) puede relacionarse con el tensor de deformación de Green-Cauchy mediante la relación: en componentes Los materiales hiperelásticos son un caso particular de material elástico de Cauchy. (es) 超弾性(ちょうだんせい、Hyperelasticity)とは、物体を構成する物質の力学的特性の数理的表現のひとつであり、ひずみエネルギー密度関数(単位体積あたりのひずみエネルギーを表す弾性ポテンシャル)を有することが特徴である。超弾性を有する物質を超弾性体とよび、ゴムの最も簡易なモデルとして登場したことに由来して、数十%~数百%の大ひずみ状態を想定している。 (ja) 超弹性材料模型可用于为类橡胶材料建模,其中的解会涉及大变形。假设材料为非线性弹性、同向性且不可压缩。常见的超弹性模型有: * Mooney - Rivlin 超弹性模型可以用于实体单元和厚壳体。 * * 超弹性 Blatz - Ko 模型用于可压缩聚氨酯(PU)泡沫类型橡胶的模型。 (zh) Hyperelastizität oder Green’sche Elastizität (von griechisch ὑπέρ hyper „über“, ελαστικός elastikos „anpassungsfähig“ und George Green) ist ein Materialmodell der Elastizität. Elastizität ist die Eigenschaft eines Körpers, unter Krafteinwirkung seine Form zu verändern und bei Wegfall der einwirkenden Kraft in die Ursprungsform zurückzukehren (Beispiel: Sprungfeder). Als Ursache der Elastizität kommen Verzerrungen des Atomgitters (bei Metallen), das Dehnen von Molekülketten (Gummi und Kunststoffe) oder die Änderung des mittleren Atomabstandes (Flüssigkeiten und Gase) in Frage. (de) A hyperelastic or Green elastic material is a type of constitutive model for ideally elastic material for which the stress–strain 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–Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the Ogden model and the Arruda–Boyce model. (en) L'hyperélasticité est un formalisme mathématique utilisé en résistance des matériaux pour décrire la relation contrainte-déformation de certains matériaux grandement déformables (polymères thermoplastiques, polymères thermodurcissables, élastomères, tissus biologiques). Contrairement à l'élasticité linéaire définie explicitement par la loi de Hooke pour les petites déformations, en hyperélasticité, on postule l'existence d’une densité d’énergie de déformation notée W dont les dérivées par rapport à la déformation dans une direction donnée donnent l'état de contrainte au sein du matériau dans cette même direction. Physiquement, W représente la quantité d’énergie élastique que le matériau emmagasine en fonction de l’étirement imposé. (fr) Nella scienza delle costruzioni un materiale è definito "linearmente iperelastico" quando: * È linearmente elastico * Alla relazione costitutiva di lineare elasticità è possibile associare una funzione scalare definita energia specifica (o potenziale elastico di deformazione, misurabile in J/m³ o F*L/L³) che è appunto l'energia che bisogna spendere per deformare un determinato materiale d'una quantità unitaria. Quindi se questa energia esiste, essa è unica. (it)
rdfs:label Hyperelastizität (de) Material hiperelástico (es) Iperelasticità (it) Hyperelastic material (en) Hyperélasticité (fr) 超弾性 (ja) 超弹性 (zh)
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