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首页> 外文期刊>European Journal of Mechanics. A, Solids >An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity
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An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

机译：在超弹性的空间和材料设置中椭圆度条件损失的等价图

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摘要

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.
机译：对于连续力学的空间和物质运动问题，详细阐述了由秩一凸性条件表示的椭圆度的损失。虽然空间运动问题是通过根据变形梯度参数化的经典平衡方程来表征的，但材料运动问题却是由反变形梯度驱动的。这样，它处理了与固定空间点处的材料放置变化在能量上共轭的构造力学的材料力。两个问题之间的对偶性在平衡律，线性化（包括一致的切线算符）和张量方面得到了强调。在这两种设置中都讨论了一阶凸性问题。特别地，证明了，如果违反了一阶凸性条件，则控制方程的适定性的损失在空间和物质运动环境中同时发生。因此，材料运动问题，即构形力平衡，不会导致确保椭圆度的附加要求。空间和材料运动方法的这种二元性在超弹性情况下得到了总体说明，并在分析和数值上对新胡克式的超弹性材料进行了举例说明。在这两种设置中，椭圆度条件的几何表示都特别受重视。

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