论文标题
不可压缩弹性材料和应用理论的几何线性化
Geometric linearization of theories for incompressible elastic materials and applications
论文作者
论文摘要
我们从小型位移制度中的非线性弹性理论得出了不可压缩材料的几何线性化理论。我们的非线性存储的能量密度可能在（小）长度尺度上与典型位移变化。这允许应用于多晶体能量，例如，在形状记忆合金和列明弹性体模型的马氏体阶段遇到的应用。在这种密度的渐近行为的自然假设下，我们证明了正确重新缩放的非线性能量功能的伽马连接，以放松有效模型。最终的限制理论是在几何线性上进行了线性性的，因为它作用于无限的位移而不是有限的变形，但通常仍将具有限制的储存能量密度，以非线性的方式取决于无限菌株。我们的结果尤其是为不可压缩的列马体弹性体建立了现有有限和无限理论的严格联系。
We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g., encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.
