In this study, some pioneering contributions, envisaged in the works of Gabrio Piola, were developed through tools of the modern differential geometry and applied to the second gradient continua. Part I introduced the variational approach for the equilibrium problem according to the first gradient theory and exploited differential geometric perspectives for the present scenario. By prescribing the stationarity of the Lagrangian energy functional, the virtual work equations for a Cauchy’s medium were recovered. The focus was on the deformation process regarded as a diffeomorphism between Riemannian embedded submanifolds, emphasizing the roles of the pullback metrics and of the covariant differentiation. Novel transport formulae were provided for normal and tangent vectors in the neighborhood of a boundary edge. The divergence theorem for curved surfaces with border was revisited, providing remarkable relationships between Lagrangian and Eulerian expressions involving projectors.

在这项研究中，Gabrio Piola 的作品中设想的一些开创性贡献是通过现代微分几何工具开发的，并应用于第二梯度连续体。第一部分根据第一个梯度理论介绍了平衡问题的变分方法，并为当前场景利用了微分几何视角。通过规定拉格朗日能量泛函的平稳性，恢复了柯西介质的虚功方程。重点是变形过程被视为黎曼嵌入子流形之间的微分同胚，强调回拉度量和协变微分的作用。为边界边缘附近的法线和切线向量提供了新的传输公式。