Processes in Masonry Bodies and the Dynamical Significance of Collapse

Abstract

We consider masonry bodies with dissipation of the rate type. Global in time existence and uniqueness of weak solutions for arbitrary loads and initial conditions is established; this includes the loads for which the masonry structure collapses. "Safe" and "collapse" loads are distinguished by different behaviors of solutions (= processes) at large times. Three situations can arise according to the properties of the equilibrium problem. (i) The equilibrium problem has a (typically nonunique) solution in the Sobolev space W^1,2 of displacements; then each process stabilizes inasmuch as the kinetic energy tends to 0 and the L² distance of the displacement from the set of all equilibrium displacements tends to 0. (ii) The equilibrium problem has no solution in W^l,2 but the infimum of the energy functional on the space of admissible displacements is finite. Then the kinetic energy tends to 0 but the W^l,2 norm of the displacement tends to oo. This may correspond either to a collapse or to a situation when the process approaches an equilibrium solution in a larger function space. (iii): The infimum of the energy functional on the space of admissible displacements is -∞. Then the total energy approaches -∞ in any process and the W^l,1 norm of the displacement tends to ∞; the structure collapses.