After the wide premise of Part I, where the equations for Cauchy’s continuum were retrieved through the energy minimization and some differential geometric perspectives were specified, the present paper as Part II outlines the variational derivation of the equilibrium equations for second gradient materials and their transformation from the Eulerian to the Lagrangian form. Volume, face and edge contributions to the inner virtual work were provided through integration by parts and by repeated applications of the divergence theorem extended to curved surfaces with border. To sustain double forces over the faces and line forces along the edges, the role of the third rank hyperstress tensor was highlighted. Special attention was devoted to the edge work, and to the evaluation of the variables discontinuous across the edge belonging to the contiguous boundary faces. The detailed expression of the contact pressures was provided, including multiple products of normal vector components, their gradient and a combination of them: in particular, the dependence on the local mean curvature was shown. The transport of the governing equations from the Eulerian to the Lagrangian configuration was developed according to two diverse strategies, exploiting novel differential geometric formulae and revealing a coupling of terms transversely to the involved domains.

在第一部分的广泛前提下，通过能量最小化检索柯西连续体方程并指定了一些微分几何观点，本文作为第二部分概述了第二梯度材料平衡方程的变分推导及其从欧拉到拉格朗日形式。体积、面和边缘对内部虚拟功的贡献是通过零件集成和重复应用扩展到有边界的曲面的散度定理来提供的。为了维持面的双重力和沿边缘的线力，强调了三阶超应力张量的作用。特别注意边缘工作，以及跨属于连续边界面的边缘不连续的变量的评估。提供了接触压力的详细表达式，包括法向量分量的多个乘积、它们的梯度和它们的组合：特别是显示了对局部平均曲率的依赖性。控制方程从欧拉配置到拉格朗日配置的传输是根据两种不同的策略开发的，利用新的微分几何公式并揭示了与相关域横向的项耦合。