Lecture Notes in Applied and Computational Mechanics, 2005
In these notes we are concerned with heat conduction described by using the entropy balance. The ... more In these notes we are concerned with heat conduction described by using the entropy balance. The main advantage of this approach consists in recovering the positivity of the absolute temperature, necessary to prove thermodynamical consistency, directly by solving the equation. We introduce the model and discuss its thermomechanical consistency. Then, we investigate from the analytical and mechanical point of view,
ABSTRACT In this work, we analytically investigate a multi-component system for describing phase ... more ABSTRACT In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.
ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction be... more ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction between a thermoviscoelastic body and a rigid support.A PDE system, consisting of the evolution equations for the temperatures in the bulk domain and on the contact surface, of the momentum balance, and of the equation for the internal variable describing the state of the adhesion, is derived on the basis of a surface damage theory by M. Frémond.The existence of global-in-time solutions to the associated initial–boundary value problem is proved by passing to the limit in a carefully tailored time-discretization scheme.
We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid... more We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid support, encompassing thermal and frictional effects. Following Frémond's approach, adhesion is described in terms of a surface damage parameter χ . The related equations are the momentum balance for the vector of small displacements, and parabolic-type evolution equations for χ and for the absolute temperatures of the body and of the adhesive substance on the contact surface. All of the constraints on the internal variables, as well as the contact and the friction conditions, are rendered by means of subdifferential operators. Furthermore, the temperature equations, derived from an entropy balance law, feature singular functions. Therefore, the resulting PDE system has a highly nonlinear character.
ABSTRACT The present paper deals with the experimental analysis, constitutive modeling and numeri... more ABSTRACT The present paper deals with the experimental analysis, constitutive modeling and numerical simulation of a class of polymers, exhibiting shape memory effects. We first present and discuss the results of an experimental traction-shrinkage campaign on semi-crystalline shape memory polymers, particularly, on low-density and high-density polyethylene-based polymers. Then, we develop a new one-dimensional phenomenological constitutive model, based on the so-called phase transition approach and formulated in a finite strain framework, in order to reproduce experimental observations. The model is treated through a numerical procedure, consisting in the replacement of the classical set of Kuhn-Tucker inequality conditions by the Fischer-Burmeister complementarity function. Numerical predictions reveal that the model is able to describe qualitative aspects of material behavior, involving both orientation and thermal retraction, as well as to predict experimental orientation processes for semi-crystalline polyethylene-based polymers with different densities.
This work is concerned with the mathematical analysis of a system of partial differential equatio... more This work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin/lead solders. The system combines the (quasistationary) balance of linear momentum with a fourth-order evolution equation of Cahn-Hilliard type for the phase separation, and it is highly nonlinearly coupled. Existence and uniqueness results are shown.
We consider a model describing the evolution of damage in visco-elastic materials, where both the... more We consider a model describing the evolution of damage in visco-elastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinear parabolic variational inclusion, due to the presence of two maximal monotone graphs involving the phase parameter and its time derivative. Existence of a solution is proved in some subinterval of time in which the damage process is not complete. Uniqueness is established in the case when one of the two monotone graphs is assumed to be Lipschitz continuous.
Discrete and Continuous Dynamical Systems - Series B, 2014
Dedicated to Professor Mauro Fabrizio, nice friend and valuable collaborator, wishing him the ver... more Dedicated to Professor Mauro Fabrizio, nice friend and valuable collaborator, wishing him the very best "ad multos annos" Abstract A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Converge results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.
We build a predictive theory for the evolution of mixture of helium and supercooled helium at low... more We build a predictive theory for the evolution of mixture of helium and supercooled helium at low temperature. The absolute temperature θ and the volume fraction β of helium, which is dominant at temperature larger than the phase change temperature, are the state quantities. The predictive theory accounts for local interactions at the microscopic level, involving the gradient of β. The nonlinear heat flux in the supercooled phase results from a Norton-Hoff potential. We prove that the resulting set of partial differential equations has solutions within a convenient analytical frame.
ABSTRACT On étudie en dimension 3 le mouvement dʼun solide en grandes déformations. Ce solide peu... more ABSTRACT On étudie en dimension 3 le mouvement dʼun solide en grandes déformations. Ce solide peut être chargé par des pointes, des fils, des poutres, des coques… Cela nous conduit à retenir une théorie du troisième gradient dans le solide. La matrice dʼélongation qui apparaît dans la décomposition polaire doit être symétrique. Cette liaison interne introduit une contrainte de réaction qui contribue à la contrainte de Piola–Kirchhoff–Boussinesq. On montre alors quʼil existe un mouvement qui satisfait toutes les équations de la Mécanique dans un cadre variationnel convenable. Ce mouvement est local en temps, car il peut être interrompu par un écrasement provoquant une discontinuité de vitesse par rapport au temps, cʼest-à-dire une collision interne.
Modelling shape memory alloys is based on choosing state quantities E which involve the phase vol... more Modelling shape memory alloys is based on choosing state quantities E which involve the phase volume fractions, on defining the free energy Psi(E) and the pseudo-potential of dissipation Phi(E,dE/dt,grad T) depending on the thermal heterogeneity characterised by the temperature gradient grad T. The derivatives of Psi and Phi give the constitutive laws which are used in the momentum and energy
Lecture Notes in Applied and Computational Mechanics, 2005
In these notes we are concerned with heat conduction described by using the entropy balance. The ... more In these notes we are concerned with heat conduction described by using the entropy balance. The main advantage of this approach consists in recovering the positivity of the absolute temperature, necessary to prove thermodynamical consistency, directly by solving the equation. We introduce the model and discuss its thermomechanical consistency. Then, we investigate from the analytical and mechanical point of view,
ABSTRACT In this work, we analytically investigate a multi-component system for describing phase ... more ABSTRACT In this work, we analytically investigate a multi-component system for describing phase separation and damage processes in solids. The model consists of a parabolic diffusion equation of fourth order for the concentration coupled with an elliptic system with material dependent coefficients for the strain tensor and a doubly nonlinear differential inclusion for the damage function. The main aim of this paper is to show existence of weak solutions for the introduced model, where, in contrast to existing damage models in the literature, different elastic properties of damaged and undamaged material are regarded. To prove existence of weak solutions for the introduced model, we start with an approximation system. Then, by passing to the limit, existence results of weak solutions for the proposed model are obtained via suitable variational techniques.
ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction be... more ABSTRACT In this paper we introduce and investigate a model for adhesive contact with friction between a thermoviscoelastic body and a rigid support.A PDE system, consisting of the evolution equations for the temperatures in the bulk domain and on the contact surface, of the momentum balance, and of the equation for the internal variable describing the state of the adhesion, is derived on the basis of a surface damage theory by M. Frémond.The existence of global-in-time solutions to the associated initial–boundary value problem is proved by passing to the limit in a carefully tailored time-discretization scheme.
We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid... more We propose a model for (unilateral) contact with adhesion between a viscoelastic body and a rigid support, encompassing thermal and frictional effects. Following Frémond's approach, adhesion is described in terms of a surface damage parameter χ . The related equations are the momentum balance for the vector of small displacements, and parabolic-type evolution equations for χ and for the absolute temperatures of the body and of the adhesive substance on the contact surface. All of the constraints on the internal variables, as well as the contact and the friction conditions, are rendered by means of subdifferential operators. Furthermore, the temperature equations, derived from an entropy balance law, feature singular functions. Therefore, the resulting PDE system has a highly nonlinear character.
ABSTRACT The present paper deals with the experimental analysis, constitutive modeling and numeri... more ABSTRACT The present paper deals with the experimental analysis, constitutive modeling and numerical simulation of a class of polymers, exhibiting shape memory effects. We first present and discuss the results of an experimental traction-shrinkage campaign on semi-crystalline shape memory polymers, particularly, on low-density and high-density polyethylene-based polymers. Then, we develop a new one-dimensional phenomenological constitutive model, based on the so-called phase transition approach and formulated in a finite strain framework, in order to reproduce experimental observations. The model is treated through a numerical procedure, consisting in the replacement of the classical set of Kuhn-Tucker inequality conditions by the Fischer-Burmeister complementarity function. Numerical predictions reveal that the model is able to describe qualitative aspects of material behavior, involving both orientation and thermal retraction, as well as to predict experimental orientation processes for semi-crystalline polyethylene-based polymers with different densities.
This work is concerned with the mathematical analysis of a system of partial differential equatio... more This work is concerned with the mathematical analysis of a system of partial differential equations modeling the effect of phase separation driven by mechanical actions in binary alloys like tin/lead solders. The system combines the (quasistationary) balance of linear momentum with a fourth-order evolution equation of Cahn-Hilliard type for the phase separation, and it is highly nonlinearly coupled. Existence and uniqueness results are shown.
We consider a model describing the evolution of damage in visco-elastic materials, where both the... more We consider a model describing the evolution of damage in visco-elastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinear parabolic variational inclusion, due to the presence of two maximal monotone graphs involving the phase parameter and its time derivative. Existence of a solution is proved in some subinterval of time in which the damage process is not complete. Uniqueness is established in the case when one of the two monotone graphs is assumed to be Lipschitz continuous.
Discrete and Continuous Dynamical Systems - Series B, 2014
Dedicated to Professor Mauro Fabrizio, nice friend and valuable collaborator, wishing him the ver... more Dedicated to Professor Mauro Fabrizio, nice friend and valuable collaborator, wishing him the very best "ad multos annos" Abstract A thermodynamic model describing phase transitions with thermal memory, in terms of an entropy equation and a momentum balance for the microforces, is adressed. Converge results and error estimates are proved for the related integrodifferential system of PDE as the sequence of memory kernels converges to a multiple of a Dirac delta, in a suitable sense.
We build a predictive theory for the evolution of mixture of helium and supercooled helium at low... more We build a predictive theory for the evolution of mixture of helium and supercooled helium at low temperature. The absolute temperature θ and the volume fraction β of helium, which is dominant at temperature larger than the phase change temperature, are the state quantities. The predictive theory accounts for local interactions at the microscopic level, involving the gradient of β. The nonlinear heat flux in the supercooled phase results from a Norton-Hoff potential. We prove that the resulting set of partial differential equations has solutions within a convenient analytical frame.
ABSTRACT On étudie en dimension 3 le mouvement dʼun solide en grandes déformations. Ce solide peu... more ABSTRACT On étudie en dimension 3 le mouvement dʼun solide en grandes déformations. Ce solide peut être chargé par des pointes, des fils, des poutres, des coques… Cela nous conduit à retenir une théorie du troisième gradient dans le solide. La matrice dʼélongation qui apparaît dans la décomposition polaire doit être symétrique. Cette liaison interne introduit une contrainte de réaction qui contribue à la contrainte de Piola–Kirchhoff–Boussinesq. On montre alors quʼil existe un mouvement qui satisfait toutes les équations de la Mécanique dans un cadre variationnel convenable. Ce mouvement est local en temps, car il peut être interrompu par un écrasement provoquant une discontinuité de vitesse par rapport au temps, cʼest-à-dire une collision interne.
Modelling shape memory alloys is based on choosing state quantities E which involve the phase vol... more Modelling shape memory alloys is based on choosing state quantities E which involve the phase volume fractions, on defining the free energy Psi(E) and the pseudo-potential of dissipation Phi(E,dE/dt,grad T) depending on the thermal heterogeneity characterised by the temperature gradient grad T. The derivatives of Psi and Phi give the constitutive laws which are used in the momentum and energy
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