Jose R. Fernández

University of Vigo | UVIGO · Department of Applied Mathematics I

In this article we study, from the numerical point of view, a problem involving an extensible thermoelastic beam with microtemperatures derived recently by Aouadi. The corresponding variational problem leads to a coupled system which is written in terms of the mechanical extensibility, the temperature, and the microtemperatures. An existence and un...

The aim of this work is to obtain an alternative of the Phragmén-Lindelöf type for homogeneous elastic materials when the elastic tensor is not positive definite. Indeed, it is necessary to impose some conditions to this tensor in order to prove the estimates. We propose several examples of elastic tensors which are not positive definite but satisf...

The aim of this work is to provide spatial-time decay estimates for three backward-in-time nonlinear parabolic problems. To this end, we will impose several conditions on the nonlinear terms in order to obtain our estimates. It is worth noting that this type of conditions is satisfied for a large number of nonlinear problems. We will study three pr...

In this work, we study, from both analytical and numerical points of view, a heat conduction model which is based on the Moore-Gibson-Thompson equation. The second gradient effects are also included. First, the existence of a unique solution is proved by using the theory of linear semigroups, and the exponential energy decay is also shown when the...

In this work, we study some qualitative properties arising in the solution of a thermoelastic problem with heat radiation. The so-called Moore–Gibson–Thompson equation is used to model the heat conduction. By using the logarithmic convexity argument, the uniqueness and instability of solutions are proved without imposing any condition on the elasti...

In this paper, a thermoelastic problem involving a viscous strain gradient beam is considered from the analytical and numerical points of view. The so-called type Ⅱ thermal law is used to model the heat conduction and two possible dissipation mechanisms are introduced in the mechanical part, which is considered for the first time within strain grad...

The comprehension and modeling of the mechanical behavior of soft biological tissues are essential due to their clinical applications. This knowledge is essential for predicting tissue responses accurately and enhancing our ability to compute the behavior of biological structures and bio-prosthetic devices under specific loading conditions. The cur...

In this paper, we study a porous thermoelastic problem with microtemperatures assuming parabolic higher order in time derivatives for the thermal variables. The model is derived and written as a coupled linear system. Then, a uniqueness result is proved by using the logarithmic convexity method in the case that we do not assume that the mechanical...

In this paper, we propose a new thermal model based on the so‐called Moore‐Gibson‐Thompson equation for heat conduction, assuming that the material is not centrosymmetric. The existence of a unique solution is proved, although only the main steps of its proof are provided for the sake of simplicity in the presentation. A sufficient condition is pro...

In this paper, we consider, from both analytical and numerical viewpoints, a thermoelastic problem. The so-called MGT model, with two different relaxation parameters, is used for both the displacements and the thermal displacement, leading to a linear coupled system made by two third-order in time partial differential equations. Then, using the the...

In this paper, a thermomechanical problem involving a viscoelastic Timoshenko beam is analyzed from a numerical point of view. The so-called thermodiffusion effects are also included in the model. The problem is written as a linear system composed of two second-order-in-time partial differential equations for the transverse displacement and the rot...

In this paper, we analyze, from the numerical point of view, two thermo-elastic problems involving the Green–Lindsay theory. The coupling term is different for each case, involving second order or first order spatial derivatives, respectively. The variational formulation leads to a linear coupled system which is written in terms of the velocity and...

In this paper, we consider the time decay of the solutions to some problems arising in strain gradient thermoelasticity. We restrict to the two-dimensional case, and we assume that two dissipative mechanisms are introduced, the temperature and the mass dissipation. First, we show that this problem is well-posed proving that the operator defining it...

In this paper, we study the three-dimensional porous elastic problem in the case that three dissipative mechanisms act on the three porosity structures (one in each component). It is important to remark that we consider the case when the material is not centrosymmetric, and therefore, some coupling, not previously considered in the literature conce...

In this note, two problems arisen in micropolar elasticity are considered from the analytical point of view. Following the Kelvin–Voigt theory of micropolar viscoelasticity, two dissipative mechanisms are imposed: in the first problem, it is defined on the microscopic structure and, for the second problem, on the macroscopic structure. Then, an exi...

In this work, we study a two-dimensional problem involving a thermoelastic body with four dissipative mechanisms. The well-known theory proposed by Lord and Shulman is used. The existence and uniqueness of solution is proved by using theory of linear semigroups. Then, introducing some assumptions of the coupling coefficients, we prove that the ener...

In this paper, we analyze, from the numerical point of view, a swelling porous thermo-elastic problem. The so-called second-sound effect is introduced and modeled by using the simplest Maxwell–Cattaneo law. This problem leads to a coupled system which is written by using the displacements of the fluid and the solid, the temperature and the heat flu...

Knowledge on the dynamics of Xylella fastidiosa infection is an essential element for the effective management of new foci. In this study, we propose an Eco-epidemiological Model (XEM) describing the infection dynamics of X. fastidiosa outbreaks. XEM can be applied to design disease management strategies and compare their level of efficacy. XEM is...

It is accepted that we cannot obtain the exponential decay of the solutions to the type III thermoelasticity when the dimension of the domain is greater than one. In this short note, we prove that we can show the exponential decay if we consider n2$$ {n}^2 $$ type III dissipative mechanisms whenever the coupling terms satisfy a certain condition.

We study the one-dimensional problem for the linear strain gradient porous elasticity. Our aim is to analyze the behavior of the solutions with respect to the time variable when a dissipative structural mechanism is introduced in the system. We consider five different scenarios: hyperviscosity and viscosity for the displacement component and hyperv...

In this work, we numerically study a problem including several dissipative mechanisms. A particular case involving the symmetry of the coupling matrix and three mechanisms is considered, leading to the exponential decay of the corresponding solutions. Then, a fully discrete approximation of the general case in two dimensions is introduced by using...

In this short note, we want to describe the logarithmic convexity argument for third order in time partial differential equations. As a consequence, we first prove a uniqueness result whenever certain conditions on the parameters are satisfied. Later, we show the instability of the solutions if the initial energy is less or equal than zero.

In this work, we study, from the numerical point of view, a type III thermoelastic model with double porosity. The thermomechanical problem is written as a linear system composed of hyperbolic partial differential equations for the displacements and the two porosities, and a parabolic partial differential equation for the thermal displacement. An e...

Treating specific tissues without affecting other regions is a difficult task. It is desirable to target the particular tissue where the chemical has its biological effect. To study this phenomenon computationally, in this work we numerically study a mathematical model which is written as a nonlinear system composed by three parabolic partial diffe...

In this paper, we study, from both analytical and numerical points of view, a problem involving a mixture of two viscoelastic solids. An existence and uniqueness result is proved using the theory of linear semigroups. Exponential decay is shown for the one-dimensional case. Then, fully discrete approximations are introduced using the finite element...

In this work, we study a new two-temperatures thermoelastic model. Both thermodynamic and conductive temperatures are included, being related by means of an elliptic or parabolic equation. Then two problems are considered assuming the dependence or not on the rate of conductivity temperature. Existence of solutions for the three-dimensional setting...

We study a one-dimensional problem arising in strain gradient porous-elasticity. Three different Moore–Gibson–Thompson dissipation mechanisms are considered: viscosity and hyperviscosity on the displacements, and weak viscoporosity. The existence and uniqueness of solutions are proved. The energy decay is also shown, being polynomial for the two fi...

In this work, we study a high order derivative in time problem. First, we show that there exists a sequence of elements of the spectrum which tends to infinity and therefore, it is ill posed. Then, we prove the uniqueness of solutions for this problem by adapting the logarithmic arguments to this situation. Finally, the results are applied to the b...

In this paper, we prove that the solutions to the problem determined by an elastic material with $$n^2$$ n 2 coupling dissipative mechanisms decay in an exponential way for every (bounded) geometry of the body, where n is the dimension of the domain, and whenever the coupling coefficients satisfy a suitable condition. We also give several examples...

In this short note, we consider some issues regarding the instability of some elastodynamical problems when the elasticity tensor is not positive definite. By using the so-called logarithmic convexity argument, we prove the instability of solutions when the time derivative of the elasticity tensor is semi-definite negative or it satisfies another r...

In this short paper we analyze, from both analytical and numerical points of view, a dynamic problem arising in micropolar viscoelasticity. An existence and uniqueness result is proved by using the theory of linear semigroups. The exponential decay of the solution is also shown. Then, by using the finite element method and the implicit Euler scheme...

In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The...

In this paper, we consider the energy decay of some problems involving domains with radial symmetry. Three different settings are studied: a strong porous dissipation and heat conduction, a weak porous dissipation and heat conduction and poro-thermoelasticity with microtemperatures. In all the three problems, the exponential energy decay is shown....

In this paper, we numerically study a thermoelastic problem arising in the Moore–Gibson–Thompson theory. Dielectrics effects are also included within the model. The corresponding problem is written in terms of the displacement field, the temperature and the electric potential. A viscous term is added in the heat equation to provide the numerical an...

In this work we study from the mathematical and numerical point of view a problem arising in vector-borne plant diseases. The model is written as a nonlinear system composed of a parabolic partial differential equation for the vector abundance function and a first-order ordinary differential equation for the plant health function. An existence and...

In this work, we analyze, from the numerical point of view, a problem including a mixture made of a MGT viscoelastic solid and an elastic solid. The corresponding variational problem is a linear system composed of two coupled hyperbolic equations written in terms of the acceleration of the first constituent and the velocity of the second one. Then,...

In this work, we consider a multi-dimensional dual-phase-lag problem arising in porous-thermoelasticity with microtemperatures. An existence and uniqueness result is proved by applying the semigroup of linear operators theory. Then, by using the finite element method and the Euler scheme, a fully discrete approximation is numerically studied, provi...

In this work, we numerically analyze a porous elastic problem including several dissipation mechanisms of MGT type. The resulting variational problem is written in terms of the acceleration and the porosity speed. An existence and uniqueness result is recalled. Then, fully discrete approximations are introduced by using the classical finite element...

In this work, we study, from the numerical point of view, a dynamic thermoviscoelastic problem involving micropolar materials. The model leads to a linear system composed of parabolic partial differential equations for the displacements, the microrotation and the temperature. Its weak form is written as a linear system made of first-order variation...

A lot of attention has been paid recently to the study of mixtures and also to the Moore–Gibson–Thompson (MGT) type equations or systems. In fact, the MGT proposition can be used to describe viscoelastic materials. In this paper, we analyze a problem involving a mixture composed by a MGT viscoelastic type material and an elastic solid. To this end,...

In this work we study three different dissipation mechanisms arising in the so-called Moore-Gibson-Thompson porosity. The three cases correspond to the MGT-porous hyperviscosity (fourth-order term), the MGT-porous viscosity (second-order term) and the MGT-porous weak viscosity (zeroth-order term). For all the cases, we prove that there exists a uni...

In this paper, we numerically study porosity problems with three different dissipation mechanisms. The root behavior is analyzed for each case. Then, by using the finite element method and the Newmark-β scheme, fully discrete approximations are introduced and some numerical results are described to show the energy evolution depending on the viscosi...

In this paper, we consider several problems arising in the theory of thermoelastic bodies with voids. Four particular cases are considered depending on the choice of the constitutive tensors, assuming different dissipation mechanisms determined by Moore–Gibson–Thompson-type viscosity. For all of them, the existence and uniqueness of solutions are s...

In this paper we consider the Lord–Shulman thermoelastic theory with porosity and microtemperatures. The new aspect we propose here is to introduce a relaxation parameter in the microtemperatures. Then we obtain an existence theorem for the solutions. In the case that a certain symmetry is satisfied by the constitutive tensors, we prove that the se...

In this work, we consider, from the numerical point of view, a boundary-initial value problem for non-simple porous elastic materials. The mechanical problem is written as a coupled hyperbolic linear system in terms of the displacement and porosity fields. The resulting variational formulation is used to approximate the solution by the finite eleme...

In this work we study, from the numerical point of view, a one-dimensional thermoelastic problem where the thermal law is of type III. Quasi-static microvoids are also assumed within the model. The variational formulation leads to a coupled linear system made of variational equations and it is written in terms of the velocity, the volume fraction a...

In this paper, we deal with the numerical approximation of some porous-thermoelastic problems. Since the inertial effects are assumed to be negligible, the resulting motion equations are quasistatic. Then, by using the finite element method and the implicit Euler scheme, a fully discrete approximation is introduced. We prove a discrete stability pr...

In this work, we study a thermoelastic Bresse system from both mathematical and numerical points of view. The dual-phase-lag heat conduction theory is used to model the heat transfer. An existence and uniqueness result is obtained by using the theory of linear semigroups. Then, fully discrete approximations are introduced by using the finite elemen...

In this work we study a contact problem between a thermoelastic body with dual-phase-lag and a deformable obstacle. The contact is modelled using a modification of the well-known normal compliance contact condition. An existence and uniqueness result is proved applying the Faedo–Galerkin method and Gronwall’s inequality. The exponential stability i...

In this paper, we consider a contact problem between a viscoelastic Bresse beam and a deformable obstacle. The well-known normal compliance contact condition is used to model the contact. The existence of a unique solution to the continuous problem is proved using the Faedo-Galerkin method. An exponential decay property is also obtained defining an...

In this work we consider the temperature-rate dependent two temperatures thermoelastic theory. It has been proposed very recently. We study the case in which the elasticity tensor may not be positive definite. Thus, the problem can be ill posed in the sense of Hadamard. We adapt the logarithmic convexity argument to the specific situation proposed...

In the last twenty years, the analysis of problems involving dual-phase-lag models has received an increasing attention. In this work, we consider the coupling between one of these models and the microtemperatures effects. In order to overcome the infinite speed paradox, two relaxation parameters are introduced for each evolution equation related t...

In this work, we study an approximate problem arising in the incremental thermoelasticity. The existence of a unique solution is proved applying the theory of linear semigroups. The exponential energy decay is also considered. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discret...

In this paper, we consider the Moore–Gibson–Thompson thermoelastic theory. We restrict our attention to radially symmetric solutions and we prove the exponential decay with respect to the time variable. We demonstrate this fact with the help of energy arguments. Later, we give some numerical simulations to illustrate this behaviour.

We consider the system of equations determining the linear thermoelastic deformations of dielectrics within the recently called Moore-Gibson-Thompson (MGT) theory. First, we obtain the system of equations for such a case. Second, we consider the case of a rigid solid and show the existence and the exponential decay of solutions. Third, we consider...

In this work we study from both variational and numerical points of view a thermoelastic problem which appears in the dual-phase-lag theory with two temperatures. An existence and uniqueness result is proved in the general case of different Taylor approximations for the heat flux and the inductive temperature. Then, in order to provide the numerica...

There are many works dealing with the dynamics of bone remodeling, proposing increasingly complex and complete models. In the recent years, the efforts started to focus on developing models that not only reproduce the temporal evolution, but that include the spatial aspects of this phenomenon. In this work, we propose the spatial extension of an ex...

In this work we study a thermoelastic problem involving binary mixtures. Type III thermal theory is considered for the modeling of the heat conduction. Existence, uniqueness and continuous dependence of solutions are proved by using the semigroup theory. Then, the numerical analysis of the resulting variational problem is considered, by using the f...

In this paper, we deal with the numerical analysis of the Lord–Shulman thermoelastic problem with porosity and microtemperatures. The thermomechanical problem leads to a coupled system composed of linear hyperbolic partial differential equations written in terms of transformations of the displacement field and the volume fraction, the temperature a...

In this paper, we analyze a linear problem describing the vibrations of a coupled suspension bridge. The single-span roadbed is modeled as an extensible thermoelastic beam of Timoshenko type. The main cable is modeled as an elastic string and is connected to the roadbed by a distributed system of elastic springs. For this linear model, we obtain th...

In this work, we consider, from the numerical point of view, a poro-thermoelastic problem. The thermal law is the so-called of type III and the microtemperatures are also included into the model. The variational formulation of the problem is written as a linear system of coupled first-order variational equations. Then, fully discrete approximations...

In this paper, we study, from the numerical point of view, a dynamic one-dimensional problem arising in thermoelasticity and thermoviscoelasticity of types II and III. Porosity is also included into the models. The generic variational formulation leads to a coupled system written in terms of the velocity, the volume fraction speed and the temperatu...

Bone tissue is a material with a complex structure and mechanical properties. Repetitive loads or diseases can cause microfractures to appear in the bone tissue, which results in a deterioration of its mechanical properties. On the other hand, bone is a constantly evolving tissue, adapting its density to the loading conditions it is subjected to. I...

In this work, we study from the mathematical and numerical points of view a poro-thermoelastic problem. A long-term memory is assumed on the heat equation. Under some assumptions on the constitutive tensors, the resulting linear system is composed of hyperbolic partial differential equations with a dissipative mechanism in the temperature equation....

In this note, we study a linear system of partial differential equations modelling a one-dimensional two-temperatures thermo-porous-elastic problem with microtemperatures. A new system of conditions is proposed to guarantee the existence, uniqueness and exponential decay of solutions. Our arguments are based on the theory of semigroups of linear op...

This paper investigates several aspects of the linear type III thermoelastic theory. First, we consider the most general system of equations for this theory in the case that the conductivity rate is not definite and we prove an existence theorem by means of the semigroups theory. In fact, we show that the solutions of the problem generate a quasi-c...

In this work, we numerically consider a thermoelastic problem where the thermal law is modeled using the so-called Moore-Gibson-Thompson equation. This thermomechanical problem is written as a coupled system composed of a hyperbolic partial differential equation for a transformation of the displacement field and a parabolic partial differential equ...

In this work we study, from the numerical point of view, a thermoelastic problem involving a microstretch plate. The variational problem is written as a linear system composed of parabolic equations written in terms of the velocity field, the microrotations speed, the microstretch speed and the temperature. Then, a fully discrete approximation is i...

Passive safety systems of cars include parts on the structure that, in the event of an impact, can absorb a large amount of the kinetic energy by deforming and crushing in a design-controlled way. One such energy absorber part, located in the front structure of a Formula Student car, was measured under impact in a test bench. The test is modeled wi...

In this work we study a bone remodeling model for the evolution of the myeloma disease. The biological problem is written as a coupled nonlinear system consisting of parabolic partial differential equations. They are written in terms of the concentrations of osteoblasts and osteoclasts, the density of the relative bone and the concentration of the...

In this paper, we characterized the hyperelastic and damage behavior of the Extensor Digitorum Longus (EDL) human tendon under loading conditions. The study was conducted in both categories of models, phenomenological and physically motivated, to allow the prediction and the macroscopic response of the tendon under specific loading conditions, assu...

In this work, we study a bone remodeling model used to reproduce the phenomenon of osseointegration around endosseous implants. The biological problem is written in terms of the densities of platelets, osteogenic cells, and osteoblasts and the concentrations of two growth factors. Its variational formulation leads to a strongly coupled nonlinear sy...

In this paper, we numerically analyse a phase‐lag model with two temperatures which arises in the heat conduction theory. The model is written as a linear partial differential equation of third order in time. The variational formulation, written in terms of the thermal acceleration, leads to a linear variational equation, for which we recall an exi...

In this paper, we study, from the numerical point of view, a dynamic problem involving a mixture of two viscoelastic solids. The mechanical problem is written as a system of two coupled partial differential equations. Its variational formulation is derived and an existence and uniqueness result, and an energy decay property, are recalled. Then, ful...

In this work, we numerically study a thermo-mechanical problem arising in poro-viscoelasticity with the type III thermal law. The thermomechanical model leads to a linear system of three coupled hyperbolic partial differential equations, and its weak formulation as three coupled parabolic linear variational equations. Then, using the finite element...