从原子论系统到非线性弹性理论的过渡
On the passage from atomistic systems to nonlinear elasticity theory
摘要：
当原子间距离趋于零时，我们严格地导出了非线性弹性理论中原子模型的连续极限。特别是，我们得到了一个积分泛函作用于变形梯度的连续介质理论，这取决于潜在的原子相互作用势和晶格几何。我们的理论适用的相互作用势是一般的有限范围模型的多晶格，特别是也可以考虑多极相互作用和键角依赖的贡献。此外，我们讨论了柯西玻恩规则的适用性。我们的极限能量密度类由一般的拟凸函数组成，线性化的极限能量类与Cauchy-Born规则一致，由不受Cauchy关系限制的一般二次型组成.
英文摘要：
We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models on multilattices which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.
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