The present paper presents a theoretical formulation of different self-healing variables. Healing variables based on the recovery in elastic modulus, shear modulus, Poisson's ratio, and bulk modulus are defined. The formulation is presented in both scalar and tensorial cases. A new healing variable based on elastic stiffness recovery in proposed, which is consistent with the continuum damage-healing mechanics. The evolution of the healing variable calculated based on cross-section as function of the healing variable calculated based on elastic stiffness is presented in both hypotheses of elastic strain equivalence and elastic energy equivalence. The components of the fourth-rank healing tensor are also obtained in the case of isotropic elasticity, plane stress, and plane strain. It is found that the healing variable calculated based on elastic stiffness reduction is greater than the one calculated based on cross-section reduction in the case of the hypothesis of elastic energy equivalence. It is also shown that the healing tensor fits the boundary conditions of the healing variable in the case of scalar formulation.

中文翻译：

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平面应力、平面应变和各向同性弹性中的损伤和愈合力学

本论文提出了不同自愈变量的理论公式。定义了基于弹性模量、剪切模量、泊松比和体积模量恢复的愈合变量。该公式以标量和张量情况呈现。提出了一种基于弹性刚度恢复的新愈合变量，这与连续损伤-愈合力学一致。在弹性应变等价和弹性能量等价的两个假设中都提出了基于横截面计算的愈合变量作为基于弹性刚度计算的愈合变量的函数的演变。在各向同性弹性、平面应力和平面应变的情况下，也得到了四阶愈合张量的分量。发现在弹性能量等价假设的情况下，基于弹性刚度折减计算的愈合变量大于基于横截面折减计算的愈合变量。还表明，在标量公式的情况下，愈合张量符合愈合变量的边界条件。