On anisotropic elasticity and questions concerning its Finite Element implementation
On anisotropic elasticity and questions concerning its Finite Element implementation
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关于各向异性弹性及其有限元实现的问题
作者：
L. Vergori;M. Destrade;P. McGarry;R. Ogden
我们给出了非线性各向异性超弹性材料的应变能函数的条件，确保与各向异性弹性力学的经典线性理论的兼容性。我们发现了与应变能的体积-偏量分离相关的局限性，例如，在许多有限元（FE）代码中，它不能完全代表线性区域中各向异性材料的行为。这种限制具有重要的后果。我们表明，在小变形制度，有限元代码的基础上的体积偏分离假设预测，一个球体的可压缩各向异性材料制成的流体静压载荷下变形成另一个球体，而不是预期的椭球体。对于有限变形，通常采用的假设，纤维不能支持压缩是不正确的，在目前的有限元代码，并导致非物理的结果，在静水压力下的可压缩各向异性材料的球体变形成一个更大的球。
We give conditions on the strain–energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the volumetric–deviatoric separation of the strain–energy used, for example, in many Finite Element (FE) codes in that it does not fully represent the behavior of anisotropic materials in the linear regime. This limitation has important consequences. We show that, in the small deformation regime, a FE code based on the volumetric–deviatoric separation assumption predicts that a sphere made of a compressible anisotropic material deforms into another sphere under hydrostatic pressure loading, instead of the expected ellipsoid. For finite deformations, the commonly adopted assumption that fibres cannot support compression is incorrectly implemented in current FE codes and leads to the unphysical result that under hydrostatic tension a sphere of compressible anisotropic material deforms into a larger sphere.