Hyperelasticity as a $$Gamma $$Γ-limit of peridynamics when the horizon goes to zero
Hyperelasticity as a $$Gamma $$Γ-limit of peridynamics when the horizon goes to zero
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当地平线趋于零时，超弹性作为近场动力学的 $$Gamma $$Γ 极限
DOI：
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发表时间：
2015
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通讯作者：
P. Pedregal
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作者：
J. C. Bellido;C. Mora;P. Pedregal
近场动力学是连续介质力学中的非局部模型，特别是弹性模型，由 Silling (2000) 引入。非定域性反映在有限距离处的点彼此施加力的事实。然而，如果这些点比称为地平线的特征长度更远，则通常假设它们不相互作用。我们采用与时间无关的变形的变分方法，根据该方法，它们的能量表示为不涉及梯度的二重积分。我们证明，当地平线趋于零时，该模型的 $$Gamma $$Γ 极限是超弹性的经典模型。我们特别关注从非局部模型的密度到其局部模型的密度是如何发生的。
Peridynamics is a nonlocal model in Continuum Mechanics, and in particular Elasticity, introduced by Silling (2000). The nonlocality is reflected in the fact that points at a finite distance exert a force upon each other. If, however, those points are more distant than a characteristic length called horizon, it is customary to assume that they do not interact. We work in the variational approach of time-independent deformations, according to which, their energy is expressed as a double integral that does not involve gradients. We prove that the $$Gamma $$Γ-limit of this model, as the horizon tends to zero, is the classical model of hyperelasticity. We pay special attention to how the passage from the density of the non-local model to its local counterpart takes place.