We review and compare the Born-Huang and the Lemaitre-Maloney theories that lead to analytical expressions for elastic constants, accounting for affine and nonaffine deformations in a lattice (or in a disordered solid). The Born-Huang method is based on Helmholtz free energy while the Lemaitre-Maloney formalism focus on the Gibbs ensemble with the focus on local force. Although starting from different perspectives, in the linear elastic limit, and in equilibrium, material elastic constants must be the same in all these methods. This is explicitly verified on examples of linear chains, and the numerical simulation of a non-centrosymmetric crystal.

我们回顾并比较了Born-Huang和Lemaitre-Maloney理论，这些理论导致了弹性常数的解析表达式，从而说明了晶格（或无序实体）中的仿射和非仿射变形。Born-Huang方法基于亥姆霍兹自由能，而Lemaitre-Maloney形式主义则侧重于Gibbs合奏，而侧重于局部力。尽管从不同的角度出发，但在线性弹性极限和平衡状态下，所有这些方法中的材料弹性常数必须相同。这在线性链的示例以及非中心对称晶体的数值模拟中得到了明确验证。