M. Campo

Unilateral Contact and Damage Analysis in Masonry Arches

Two unilateral models, the first with contact interfaces and the second with continuous damage material (Fremond’s model), are applied on the nonlinear analysis and collapse of masonry arches. The results are compared with the predictions of the classical Heyman theory.

Analysis of a multidimensional thermoviscoelastic contact problem under the Green–Lindsay theory

Abstract In this paper, we investigate the existence, the stability and the numerical approximation of a multidimensional dynamic contact problem modeling the evolution of displacement and temperature in a viscoelastic body that may come into contact with a deformable foundation. The viscoelastic body is assumed to behave according to Kelvin–Voigt constitutive law with added thermal effects under the Green–Lindsay theory. We prove that the presence of viscoelastic terms in the equations provides additional regularity and then an existence and uniqueness result is obtained using the Faedo–Galerkin method. An energy decay property is also shown under the assumption of radial symmetry. Then, a numerical approximation based on the finite element method is proposed. A stability result is proved from which the decay of the discrete energy is deduced. A priori error estimates are shown from which the linear convergence is derived under suitable additional regularity conditions. Finally, some numerical experiments are described to support our results.

Numerical analysis of a dynamic problem involving bulk-surface surfactants

In this paper, a dynamic problem which models the evolution of the concentration of surfactants is analyzed from the numerical point of view. Both bulk and surface diffusions are taken into account into the model, and the relationship between both concentrations, in the bulk and at the surface, is considered by using the well-known Langmuir–Hinshelwood equation. Two convective terms are also included. The variational formulation is then written as a coupled system of parabolic partial differential equations, for which an existence and uniqueness result is stated in an earlier paper (Fernández et al. in SIAM J Math Anal 48(5):3065–3089, 2016). Then, fully discrete approximations are introduced by using the classical finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. An a priori error estimates result is proved, from which the linear convergence of the approximation is derived under suitable additional regularity conditions. Finally, some numerical simulations are presented in order to show the accuracy of the algorithm and the behaviour of the solution in real situations.

Analysis of a dynamic viscoelastic-viscoplastic piezoelectric contact problem

In this paper, we study, from both variational and numerical points of view, a dynamic contact problem between a viscoelastic-viscoplastic piezoelectric body and a deformable obstacle. The contact is modelled using the classical normal compliance contact condition. The variational formulation is written as a nonlinear ordinary differential equation for the stress field, a nonlinear hyperbolic variational equation for the displacement field and a linear variational equation for the electric potential field. An existence and uniqueness result is proved using Gronwall’s lemma, adequate auxiliary problems and fixed-point arguments. Then, fully discrete approximations are introduced using an Euler scheme and the finite element method, for which some a priori error estimates are derived, leading to the linear convergence of the algorithm under suitable additional regularity conditions. Finally, some two-dimensional numerical simulations are presented to show the accuracy of the algorithm and the behaviour of the solution.

Numerical analysis of a dynamic viscoelastic contact problem with damage

In this work we numerically study a model for the process of unilateral frictionless contact between a viscoelastic body and a deformable foundation. Material damage due to the opening and growth of micro-cracks and micro-cavities caused by compression or tension is taken into account. Reliable prediction of the development of material damage in engineering systems which undergo cyclic loadings is of considerable importance, and the effective functioning, reliability and safety of the system may be affected by the decrease in its load carrying capability, as the material damage evolves. Because of the importance of the subject, an increasing number of mathematical and engineering publications

Numerical analysis of a dynamic frictional viscoelastic contact problem with damage

In this talk we present the numerical analysis of a dynamic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage, which measures the density of the microcracks in the material and results from tension or compression ([1]), is then involved in the constitutive law (see [2] for details), and modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary inequalities for which we state the existence of a unique weak solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the numerical scheme is deduced. Finally, numerical results are presented for some two-dimensional examples in order to show the accuracy of the algorithm.