Mechanics is a fascinating field of physics. Its main purpose is to study the interaction of particles in the presence of forces such as gravity. In contrast to the dynamics of particles, the study of continuous media, such as elastic bodies, is challenging and still today not entirely well-understood. This is due to the fact that continuous media behave in numerous and complex ways. For exampl...

We use Hodge-type orthogonal decompositions for studying the compatibility equations of the displacement gradient and the linear strain with prescribed boundary displacements. We show that the displacement gradient is compatible if and only if for any equilibrated virtual first-Piola Kirchhoff stress tensor field, the virtual work done by the displacement gradient is equal to the virtual work d...

In this paper, we study the Kirchhoff equation with Sobolev critical exponent \begin{document}$ -\left(a+b\int_{ {\mathbb{R}}^3}|\nabla u|^2\right)\Delta u = \lambda u+\mu|u|^{q-2}u+|u|^{4}u\ \ {\rm in}\ {\mathbb{R}}^3 $\end{document} under normalized constraint$ \int_{ {\mathbb{R}}^3}u^2 c^2, $where a, \, b, c>0 are constants, \lambda, \mu\in{\mathbb{R}} and 2<q<6 $\end{document}. The...