Finite strain theory (original) (raw)
نظرية الإجهاد المنتهي وتسمى أيضا نظرية الإجهاد الكبير هي نظرية تتعامل مع التشوهات التي تكون فيها السلالات و / أو الدورات كبيرة بما يكفي لإبطال الافتراضات المتأصلة في نظرية الإجهاد المتناهية الصغر. حيث تختلف التكوينات غير المشوهة والمشوهة في السلسلة بشكل كبير مما يتطلب تمييزًا واضحًا بينهما. وهذا هو الحال عادة مع اللدائن المرنة ، ومواد تشويه اللدونة وغيرها من السوائل والأنسجة اللينة البيولوجية.
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| dbo:abstract | نظرية الإجهاد المنتهي وتسمى أيضا نظرية الإجهاد الكبير هي نظرية تتعامل مع التشوهات التي تكون فيها السلالات و / أو الدورات كبيرة بما يكفي لإبطال الافتراضات المتأصلة في نظرية الإجهاد المتناهية الصغر. حيث تختلف التكوينات غير المشوهة والمشوهة في السلسلة بشكل كبير مما يتطلب تمييزًا واضحًا بينهما. وهذا هو الحال عادة مع اللدائن المرنة ، ومواد تشويه اللدونة وغيرها من السوائل والأنسجة اللينة البيولوجية. (ar) Der Deformationsgradient (Formelzeichen: ) ist in der Kontinuumsmechanik ein Mittel zur Beschreibung der lokalen Verformung an einem materiellen Punkt eines Körpers. Zur Veranschaulichung kann man sich einen Körper (in Abbildung 1, gelb) vorstellen auf den eine kurze Linie (weil nur lokale Änderungen beschrieben werden, im Bild fett rot) eingeritzt wird. Wird der Körper deformiert (rechts im Bild), wird die eingeritzte Linie nicht nur ihre Lage im Raum ändern, sondern auch gedehnt (oder gestaucht) und verdreht werden. Die Dehnung und Verdrehung beschreibt der Deformationsgradient und ist so ein Maß für die Deformation, daher der Name. Der Anhang Gradient verweist auf die Tatsache, dass lokale Änderungen beschrieben werden. Aus dem Deformationsgradient lassen sich Maße für die lokale Streckung, Verzerrung, Flächen- und Volumenänderung ableiten. Im allgemeinen Fall ist der Deformationsgradient sowohl vom Ort als auch von der Zeit abhängig. Die zeitliche Änderung des Deformationsgradienten gibt Maße für die Änderungsraten der Streckung, Verdrehung, Verzerrung, Flächen- und Volumenänderung. Der Deformationsgradient ist einheitenfrei. Bei den angesprochenen kurzen Linien handelt es sich mathematisch um Vektoren, die vom Deformationsgradient transformiert werden, wobei die Vektoren im Allgemeinen gedreht und gestreckt werden. Abbildungen von Vektoren leisten Tensoren, siehe Abbildung 2, weswegen der Deformationsgradient ein Tensor ist. Wenn es klar ist, auf welches Koordinatensystem sich der Deformationsgradient bezieht, berechnet er sich wie eine Jacobimatrix und kann dann auch als Matrix notiert werden. Oft bildet der Deformationsgradient die (infinitesimal kleinen) materiellen Linienelemente in der Ausgangs- oder Referenzkonfiguration in die aktuelle oder Momentankonfiguration ab. Ganz allgemein kann eine solche Abbildung aber auch zwischen beliebig anderen zu definierenden Konfigurationen stattfinden. (de) In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. (en) Em mecânica do contínuo, a teoria das deformações finitas — também chamada teoria das grandes deformações — lida com deformações nas quais rotações e deformações são arbitrariamente grandes, invalidando as hipóteses inerentes à teoria das deformações infinitesimais. Neste caso, as configurações indeformada e deformada do contínuo são significativamente diferentes e uma distinção clara tem de ser feita entre ambas. Este é normalmente o caso de elastômeros, materiais com deformação plástica o outros fluidos e tecidos moles biológicos. (pt) 有限应变理论（finite strain theory）也稱為大應變理論或大形變理論，是连续介质力学中處理有較大應變或轉動的形變，已不符合无限小应变理论假設下的理論。此情形下，物體在未形變的組態及已形變的組態有明顯的不同。有限应变理论常用於弹性体、塑性變形材料、流体及生物軟組織。 (zh) |
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| dbp:proof | To see how this formula is derived, we start with the oriented area elements in the reference and current configurations: The reference and current volumes of an element are where . Therefore, or, so, So we get or, Q.E.D. (en) A measure of deformation is the difference between the squares of the differential line element , in the undeformed configuration, and , in the deformed configuration . Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred. Thus we have, In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is Then we have, where are the components of the right Cauchy–Green deformation tensor, . Then, replacing this equation into the first equation we have, or where , are the components of a second-order tensor called the Green – St-Venant strain tensor or the Lagrangian finite strain tensor, In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is where are the components of the spatial deformation gradient tensor, . Thus we have where the second order tensor is called Cauchy's deformation tensor, . Then we have, or where , are the components of a second-order tensor called the Eulerian-Almansi finite strain tensor, Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. For the Lagrangian strain tensor, first we differentiate the displacement vector with respect to the material coordinates to obtain the material displacement gradient tensor, Replacing this equation into the expression for the Lagrangian finite strain tensor we have or Similarly, the Eulerian-Almansi finite strain tensor can be expressed as (en) The stretch ratio for the differential element in the direction of the unit vector at the material point , in the undeformed configuration, is defined as where is the deformed magnitude of the differential element . Similarly, the stretch ratio for the differential element , in the direction of the unit vector at the material point , in the deformed configuration, is defined as The square of the stretch ratio is defined as Knowing that we have where and are unit vectors. The normal strain or engineering strain in any direction can be expressed as a function of the stretch ratio, Thus, the normal strain in the direction at the material point may be expressed in terms of the stretch ratio as solving for we have The shear strain, or change in angle between two line elements and initially perpendicular, and oriented in the principal directions and , respectively, can also be expressed as a function of the stretch ratio. From the dot product between the deformed lines and we have where is the angle between the lines and in the deformed configuration. Defining as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have thus, then or (en) |
| dbp:title | Derivation of Nanson's relation (en) Derivation of the Lagrangian and Eulerian finite strain tensors (en) Derivation of the physical interpretation of the Lagrangian and Eulerian finite strain tensors (en) |
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| rdfs:comment | نظرية الإجهاد المنتهي وتسمى أيضا نظرية الإجهاد الكبير هي نظرية تتعامل مع التشوهات التي تكون فيها السلالات و / أو الدورات كبيرة بما يكفي لإبطال الافتراضات المتأصلة في نظرية الإجهاد المتناهية الصغر. حيث تختلف التكوينات غير المشوهة والمشوهة في السلسلة بشكل كبير مما يتطلب تمييزًا واضحًا بينهما. وهذا هو الحال عادة مع اللدائن المرنة ، ومواد تشويه اللدونة وغيرها من السوائل والأنسجة اللينة البيولوجية. (ar) In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. (en) Em mecânica do contínuo, a teoria das deformações finitas — também chamada teoria das grandes deformações — lida com deformações nas quais rotações e deformações são arbitrariamente grandes, invalidando as hipóteses inerentes à teoria das deformações infinitesimais. Neste caso, as configurações indeformada e deformada do contínuo são significativamente diferentes e uma distinção clara tem de ser feita entre ambas. Este é normalmente o caso de elastômeros, materiais com deformação plástica o outros fluidos e tecidos moles biológicos. (pt) 有限应变理论（finite strain theory）也稱為大應變理論或大形變理論，是连续介质力学中處理有較大應變或轉動的形變，已不符合无限小应变理论假設下的理論。此情形下，物體在未形變的組態及已形變的組態有明顯的不同。有限应变理论常用於弹性体、塑性變形材料、流体及生物軟組織。 (zh) Der Deformationsgradient (Formelzeichen: ) ist in der Kontinuumsmechanik ein Mittel zur Beschreibung der lokalen Verformung an einem materiellen Punkt eines Körpers. Zur Veranschaulichung kann man sich einen Körper (in Abbildung 1, gelb) vorstellen auf den eine kurze Linie (weil nur lokale Änderungen beschrieben werden, im Bild fett rot) eingeritzt wird. Wird der Körper deformiert (rechts im Bild), wird die eingeritzte Linie nicht nur ihre Lage im Raum ändern, sondern auch gedehnt (oder gestaucht) und verdreht werden. Die Dehnung und Verdrehung beschreibt der Deformationsgradient und ist so ein Maß für die Deformation, daher der Name. Der Anhang Gradient verweist auf die Tatsache, dass lokale Änderungen beschrieben werden. Aus dem Deformationsgradient lassen sich Maße für die lokale Streck (de) |
| rdfs:label | نظرية الإجهاد المنتهي (ar) Deformationsgradient (de) Finite strain theory (en) 有限変形理論 (ja) Teoria das deformações finitas (pt) 有限应变理论 (zh) |
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