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.

中文翻译：

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不可压缩弹性材料和应用理论的几何线性化

我们从小位移状态下的非线性弹性理论推导出不可压缩材料的几何线性化理论。我们的非线性储能密度可能在与典型位移相同（小）的长度范围内变化。这允许应用于多孔能量，例如在形状记忆合金的马氏体相和向列弹性体模型中遇到。在对这种密度的渐近行为的自然假设下，我们证明了适当重新缩放的非线性能量泛函对有效模型的松弛的 Gamma 收敛。由此产生的极限理论是几何线性化的，它作用于无穷小位移而不是有限变形，但通常仍具有有限的存储能量密度，它以非线性方式取决于无穷小的应变。特别是，我们的结果为不可压缩向列弹性体的现有有限和无穷小理论建立了严格的联系。