SUMMARY The occurrence of earthquakes can be regarded as shear fracture releasing the elastic potential energy stored in the Earth. The potential energy density of linear elastic forces is generally represented in the quadratic form of strain tensor components with the fourth-order coefficient tensor of elastic stiffness. When the material is isotropic, since the stiffness tensor is expressible as a linear combination of two independent symmetric tensors, we can decompose the elastic potential energy density into two independent parts, namely the volumetric part and the shearing part. By definition, the partial derivatives of the elastic potential energy density with respect to volumetric and shearing deformations give the corresponding generalized forces in the sense of Lagrangian mechanics: specifically, one-third of the first invariant of stress tensor (equivalent to the mean stress) to volumetric deformation and the square root of the second invariant of deviatoric stress tensor (equivalent to $\sqrt {3/2} $ times the octahedral shear stress). With these generalized forces instead of the normal and tangential stresses on a specific fault plane, we correctly represented the original concept of Coulomb's failure criterion (shear failure occurs when shearing stress is equal to shearing strength) and defined energetics-based failure stress (EFS). The change in EFS associated with the occurrence of a main fracture (ΔEFS) gives a rational metric for aftershock generation, which can be reduced to previously proposed various metrics in special cases. For example, when the level of background deviatoric stress is much higher than the magnitude of coseismic stress changes, the expression of ΔEFS is reduced to a similar form to the well-known Coulomb failure stress change (ΔCFS). Even in the energetics-based metric, the effects of pore-fluid pressure changes are essential. We theoretically examined the mechanical effects of induced and enforced pore-fluid pressure changes and elucidated that the difference between them is reflected in the focal mechanisms of aftershocks.

中文翻译：

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弹性势能的分解和余震产生的合理度量

总结 地震的发生可以看作是剪切断裂释放了储存在地球中的弹性势能。线弹性力的势能密度一般以应变张量分量的二次形式表示，弹性刚度的四阶系数张量。当材料为各向同性时，由于刚度张量可以表示为两个独立的对称张量的线性组合，我们可以将弹性势能密度分解为两个独立的部分，即体积部分和剪切部分。根据定义，弹性势能密度关于体积变形和剪切变形的偏导数给出了拉格朗日力学意义上的相应广义力：具体而言，应力张量第一不变量（相当于平均应力）的三分之一到体积变形和偏应力张量第二不变量的平方根（相当于 $\sqrt {3/2} $ 乘以八面体剪应力） . 使用这些广义力而不是特定断层面上的法向和切向应力，我们正确地表示了库仑失效准则的原始概念（当剪切应力等于剪切强度时发生剪切破坏）并定义了基于能量学的破坏应力 (EFS) . 与主裂缝发生相关的 EFS 变化（ΔEFS）为余震产生提供了一个合理的度量，在特殊情况下可以简化为先前提出的各种度量。例如，当背景偏应力水平远高于同震应力变化的幅度时，ΔEFS 的表达减少到与众所周知的库仑失效应力变化（ΔCFS）相似的形式。即使在基于能量学的度量中，孔隙流体压力变化的影响也是必不可少的。我们从理论上研究了诱发和强制孔隙流体压力变化的机械效应，并阐明它们之间的差异反映在余震的震源机制中。