Juan M. Viaño

A three-dimensional numerical simulation of mandible fracture reduction with screwed miniplates.

A three-dimensional finite element model of a fractured human mandible treated with plating technique was developed to simulate and to study the biomechanical loads and the stress field distribution. Biomechanical properties of bone have been thoroughly investigated experimentally. In this work, using the finite element method, complete clinical conditions (after surgical reduction, post-operatory period, complete healing period) were simulated. The mandible fracture was located in the symphysis region and one or two titanium miniplates, fixed with monocortical screws, were evaluated. The behaviour of a reduced human mandible with screwed miniplates, as well as its complete healing, is investigated and described.

A class of integro-differential variational inequalities with applications to viscoelastic contact

We consider a class of abstract evolutionary variational inequalities arising in the study of frictional contact problems for linear viscoelastic materials with long-term memory. First, we prove an abstract existence and uniqueness result, by using arguments of evolutionary variational inequalities and Banachs fixed-point theorem. Next, we study the dependence of the solution on the memory term and derive a convergence result. Then, we consider a contact problem to which the abstract results apply. The problem models a quasistatic process, the contact is bilateral and the friction is modeled with Trescas law. We prove the existence of a unique weak solution to the model and we provide the mechanical interpretation of the corresponding convergence result. Finally, we extend these results to the study of a number of quasistatic frictional problems for linear viscoelastic materials with long-term memory.

Asymptotic derivation of a general linear model for thin-walled elastic rods☆

In this work we derive from three-dimensional elasticity a general mathematically justified theory for thin-walled elastic rods. This theory is obtained as an asymptotic approximation of the three-dimensional linear model as the area and the thickness of the cross-section becomes successively small. It constitutes an extension of the classical Vlassovs theory for thin-walled beams (the most complete among models currently used in engineering).

Numerical solving of frictionless contact problems in perfectly plastic bodies

Abstract We consider the frictionless unilateral contact problems of a perfectly plastic solid with a rigid foundation and two perfectly plastic solids, under an analogous variational formulation. For each one we introduce a discrete approximation via the finite element method and in due course prove convergence. Also, we introduce a converging iterative algorithm to the solution of the discrete problems and present some numerical experiences.

Numerical analysis of a frictional contact problem for viscoelastic materials with long-term memory

We consider a mathematical model which describes the bilateral contact between a deformable body and an obstacle. The process is quasistatic, the material is assumed to be viscoelastic with long memory and the friction is modeled with Tresca’s law. The problem has a unique weak solution. Here we study spatially semi-discrete and fully discrete schemes using finite differences and finite elements. We show the convergence of the schemes under the basic solution regularity and we derive order error estimates. Finally, we present an algorithm for the numerical realization and simulations for a two-dimensional test problem.

Asymptotic justification of an evolution linear thermoelastic model for rods

Abstract The justification of a dynamic thermoelastic model for a linearized beam is presented using asymptotic analysis methods. We prove that the three-dimensional thermoelastic model converges weakly to a limit model introduced here and we deduce a compatibility condition in order to obtain strong convergence.

Mathematical justification of a one‐dimensional model for general elastic shallow arches

We present a bending model for a shallow arch, namely the type of curved rod where the curvature is of the order of the diameter of the cross section. The model is deduced in a rigorous mathematical way from classical tridimensional linear elasticity theory via asymptotic techniques, by taking the limit on a suitable re-scaled formulation of that problem as the diameter of the cross section tends to zero. This model is valid for general cases of applied forces and material, and it allows us to calculate displacements, axial stresses, bending moments and shear forces. The equations present a more general form than in the classical Bernoulli–Navier bending theory for straight slender rods, so that flexures and extensions are proved to be coupled in the most general case.

A Signorini frictionless contact problem for viscoelastic materials with long-term memory

We consider a quasistatic problem which models the contact between a deformable body and an obstacle, the so-called foundation. The material is assumed to have a viscoelastic behavior that we model with a constitutive law with long-term memory; thus, at each moment of time, the stress tensor depends not only on the present strain tensor, but also on its whole history. The contact is frictionless and is modeled by the well-known Signorini conditions. We derive a weak formulation of the problem and, under appropriate regularity hypotheses, we prove its unique solvability. We also discuss a fully discrete scheme for which we obtain error estimates and convergence results. Finally, in order to verify the accuracy of the numerical method, we present numerical simulations for a one-dimensional test problem.

Derivation of an evolution model for nonlinearly elastic beams by asymptotic expansion methods

Abstract The asymptotic expansion method has been shown to be a powerful tool in order to mathematically justify well-known mechanical models and obtain new ones. In this paper we justify the classical semi-linear evolution equation for the bending of an elastic beam submitted to axial forces at both ends. In addition to a theoretical study of this limit model, in the last section, we obtain a generalization of this model for more complete situations.

Numerical Approximation of the Elastic-Viscoplastic Contact Problem with Non-matching Meshes

Summary. This work deals with the approximation of a time dependent variational inequality modelling the unilateral contact problem of elastic-viscoplastic bodies in a bidimensional context. The problem is approximated in the space variable with nonconforming finite element methods which allow the handling of nonmatching meshes on the contact zone. Several error estimates are established and the corresponding numerical experiments are achieved.