In this work we consider a solid body $\Omega \subset {ℝ}^3$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f$ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda$ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by $\overline{\lambda }$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri, RAIRO: Modél. Math. Anal. Numér. 29 (1995)...

In this work we consider a solid body $\Omega \subset {ℝ}^3$ constituted by a nonhomogeneous elastoplastic material, submitted to a density of body forces $\lambda f$ and a density of forces $\lambda g$ acting on the boundary where the real $\lambda$ is the loading parameter. The problem is to determine, in the case of an unbounded convex of elasticity, the Limit load denoted by $\overline{\lambda }$ beyond which there is a break of the structure. The case of a bounded convex of elasticity is done in [El-Fekih and Hadhri, RAIRO: Modél. Math. Anal. Numér. 29 (1995)...

We investigate collisions (assumed to be instantaneous) and fractures of three-dimensional solids. Equations of motion and constitutive laws provide a set of partial differential equations, whose corresponding variational problem may be solved in the space of special functions with bounded deformations ($SBD$), exploiting the direct method of calculus of variations.

Let $F^{\mathrm{p}}\in \mathrm{GL}\left(3\right)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form $\mathrm{Curl}\left[F^{\mathrm{p}}\right]·{\left(F^{\mathrm{p}}\right)}^T$ applied to rotations controls the gradient in the sense that pointwise $\forall R\in C^1({ℝ}^3,\mathrm{SO}\left(3\right)):\parallel \mathrm{Curl}\left[R\right]·R^T{\parallel }_{{𝕄}^{3\times 3}}^2\ge \frac{1}{2}{\parallel \mathrm{D}R\parallel }_{{ℝ}^{27}}^2$. This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461–1506; John, Comme Pure Appl. Math. 14 (1961) 391–413; Reshetnyak, Siberian Math. J. 8 (1967) 631–653)] as well as an associated linearized theorem...

Let $F^{\mathrm{p}}\in \mathrm{GL}\left(3\right)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form $\mathrm{Curl}\left[F^{\mathrm{p}}\right]·{\left(F^{\mathrm{p}}\right)}^T$ applied to rotations controls the gradient in the sense that pointwise $\forall R\in C^1({ℝ}^3,\mathrm{SO}\left(3\right)):\parallel \mathrm{Curl}\left[R\right]·R^T{\parallel }_{{𝕄}^{3\times 3}}^2\ge \frac{1}{2}{\parallel \mathrm{D}R\parallel }_{{ℝ}^{27}}^2$. This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying...