Higher-order peridynamic material correspondence model can be developed based on the formulation of higher-order deformation gradient and constitutive correspondence with generalized continuum theories. In this paper, we present formulations of higher-order peridynamic material correspondence models adopting the material constitutive relations from the strain gradient theories. Similar to the formulation of the first-order deformation gradient, the weighted least squares technique is employed to construct the second-order and the third-order deformation gradients. Force density states are then derived as the Frechet derivatives of the free energy density with respect to the deformation states. Connections to the second-order and the third-order strain gradient elasticity theories are established by realizing the relationships between the energy conjugate stresses of the higher-order deformation gradients in peridynamics and the stress measures in strain gradient theories. In addition to the horizon, length-scale parameters from strain gradient theories are explicitly incorporated into the higher-order peridynamic material correspondence models, which enables application of peridynamics theory to materials at micron and sub-micron scales where length-scale effects are significant.

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弹性的高阶近场动力学材料对应模型

高阶近场动力学材料对应模型可以基于高阶变形梯度和广义连续体理论的本构对应的公式来开发。在本文中，我们提出了采用应变梯度理论中的材料本构关系的高阶近场动力学材料对应模型的公式。与一阶变形梯度的公式类似，采用加权最小二乘法构造二阶和三阶变形梯度。然后将力密度状态导出为自由能密度相对于变形状态的 Frechet 导数。通过实现近场动力学中高阶变形梯度的能量共轭应力与应变梯度理论中的应力测度之间的关系，建立了与二阶和三阶应变梯度弹性理论的联系。除了视界，来自应变梯度理论的长度尺度参数被明确地纳入高阶近场动力学材料对应模型，这使得近场动力学理论能够应用于长度尺度效应显着的微米和亚微米尺度的材料。