On the derivative of the stress–strain relation in a no-tension material

Abstract

The stress-strain relation of a no-tension material, used to model masonry structures, is determined by the nonlinear projection of the strain tensor onto the image of the convex cone of negative-semidefinite stresses under the fourthorder tensor of elastic compliances. We prove that the stress-strain relation is indefinitely differentiable on an open dense subset O of the set of all strains. The set O consists of four open connected regions determined by the rank k = 0, 1, 2, 3 of the resulting stress. Further, an equation for the derivative of the stress-strain relation is derived. This equation cannot be solved explicitly in the case of a material of general symmetry, but it is shown that for an isotropic material this leads to the derivative established earlier by Lucchesi et al. (Int J Solid Struct 1996; 33: 1961-1994 and Masonry constructions: Mechanical models and numerical applications. Berlin: Springer, 2008) by different means. For a material of general symmetry, when the tensor of elasticities does not have the representation known in the isotropic case, only general steps leading to the evaluation of the derivative are described.

Modal analysis of masonry structures

“…which is analogous to (30), though here the elastic stiffness matrix K, calculated using the elastic tensor C, has been replaced by the damaged stiffness matrix K, calculated using D E T( E), which takes into account the presence of cracks in body B. Thus, a new numerical method for the modal analysis of masonry constructions has been implemented in an updated version 1.2 of the NOSA-ITACA code.…”

“…For W k the interior of V k , function T turns out to be differentiable in [30]. Thus, for E ∈ W denoting by D E T(E) the derivative of T(E) with respect to E, we have…”

Propagation of waves in masonry-like solids

“…Denoting by ‫ސ‬ W Sym ! Sym the metric projection onto the closed convex cone ‫ރ‬ 1 Sym with respect to the energetic scalar product, it is possible to prove that E e D ‫.ސ‬E / and T D ‫ރ‬OE‫.ސ‬E / [Padovani and Šilhavý 2015]. The stress function ‫ޔ‬ W Sym !…”

“…For W k , the interior of V k , function ‫ޔ‬ turns out to be differentiable on W D S 3 iD0 W i [Lucchesi et al 2008;Padovani and Šilhavý 2015]. The derivative D E ‫.ޔ‬E / of ‫.ޔ‬E / with respect to E in the regions W i has been calculated in [Lucchesi et al 2008].…”

“…Regions V 1 and V 2 exhibit mixed behavior: the stress tensor has respectively two and one eigenvectors corresponding to the zero eigenvalue, and the material can fracture orthogonally to these directions. As demonstrated in [Lucchesi et al 2008;Padovani and Šilhavý 2015], the function ‫ޔ‬ is differentiable on W D S 3 iD0 W i , with W i being the interior of set V i . The derivative D E ‫.ޔ‬E / of ‫ޔ‬ with respect to E is a symmetric fourth-order tensor from Sym with values in Sym, whose spectral representation has been calculated in [Lucchesi et al 2008] and is recalled here.…”