Mathematics of the Masonry–Like model and Limit Analysis

These notes present a brief introduction to the mathematics of equilibrium of no–tension (masonry–like) materials. We first review the constitutive equations using the idea that the stress of the no–tension material must be always negative semidefinite. The strain tensor is naturally split into the sum of the elastic strain and fracture strain. The stress depends linearly on the elastic strain via the fourth–order tensor of elasticities. Then we consider a body made of a no–tension material, introduce the loads and the total energy of the deformation with is the sum of the internal energy and the energy of the loads. Then we examine the question whether the total energy is bounded from below. That brings us to the important notion of the strong compatibility of loads. The loads are strongly compatible if they can be equilibrated (in the sense of the principle of virtual work) by a square integrable negative semidefinite stress field. The total energy is bounded from below if and only if the loads are strongly compatible. The notion of strong compatibility of loads is central in the limit analysis and in a strengthened form in the theory of existence of equilibrium states. Roughly speaking, if the loads are strongly compatible, then the body is safe, while otherwise strongly incompatible loads lead to the collapse of the body. To determine whether the loads are strongly compatible, it is not necessary to solve the full displacement problem, it suffices to find the negative semidefinite square integrable stress field, which is independent of the constitutive theory.