ISSN:

1937-1632

eISSN:

1937-1179

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## Discrete & Continuous Dynamical Systems - S

April 2013 , Volume 6 , Issue 2

Issue dedicated to Michel Frémond on the occasion of his 70th birthday

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2013, 6(2): i-ii
doi: 10.3934/dcdss.2013.6.2i

*+*[Abstract](2010)*+*[PDF](82.5KB)**Abstract:**

This special volume of Discrete and Continuous Dynamical Systems - Series S is dedicated to Michel Frémond on the occasion of his 70th birthday, for his important contributions to several theoretical and applied problems in Mechanics, Thermodynamics and Engineering.

For more information please click the “Full Text” above.”

2013, 6(2): 277-291
doi: 10.3934/dcdss.2013.6.277

*+*[Abstract](2070)*+*[PDF](443.1KB)**Abstract:**

In this paper we introduce a 3D phenomenological model for shape memory behavior, accounting for: martensite reorientation, asymmetric response of the material to tension/compression, different kinetics between forward and reverse phase transformation. We combine two modeling approaches using scalar and tensorial internal variables. Indeed, we use volume proportions of different configurations of the crystal lattice (austenite and two variants of martensite) as scalar internal variables and the preferred direction of stress-induced martensite as tensorial internal variable. Then, we derive evolution equations by a generalization of the principle of virtual powers, including microforces and micromovements responsible for phase transformations. In addition, we prescribe an evolution law for phase proportions ensuring different kinetics during forward and reverse transformation of the oriented martensite.

2013, 6(2): 293-316
doi: 10.3934/dcdss.2013.6.293

*+*[Abstract](2272)*+*[PDF](729.7KB)**Abstract:**

A one-dimensional model for a shape memory alloy is proposed. It provides a simplified description of the pseudo-elastic regime, where stress-induced transitions from austenitic to oriented martensitic phases occurs. The stress-strain evolution is ruled by a bilinear rate-independent o.d.e. which also accounts for the fine structure of minor hysteresis loops and applies to the case of single crystals only. The temperature enters the model as a parameter through the yield limit $y$.Above the critical temperature $\theta_A^*$, the austenite-martensite phase transformations are described by a Ginzburg-Landau theory involving an order parameter $φ$, which is related to the anelastic deformation. As usual, the basic ingredient is the Gibbs free energy, $\zeta$, which is a function of the order parameter, the stress and the temperature. Unlike other approaches, the expression of this thermodynamic potential is derived rather then assumed, here. The explicit expressions of the minimum and maximum free energies are obtained by exploiting the Clausius-Duhem inequality, which ensures the compatibility with thermodynamics, and the complete controllability of the system. This allows us to highlight the role of the Ginzburg-Landau equation when phase transitions in materials with hysteresis are involved.

2013, 6(2): 317-330
doi: 10.3934/dcdss.2013.6.317

*+*[Abstract](2507)*+*[PDF](503.5KB)**Abstract:**

We propose a model describing the liquid-vapour phase transition according to a phase-field method. A phase variable $φ$ is introduced whose equilibrium values $φ=0$ and $φ=1$ are associated with the liquid and vapour phases. The phase field obeys Ginzburg-Landau equation and enters the constitutive relation of the density, accounting for the sudden density jump occurring at the phase transition. In this paper we concern ourselves especially with the problems arising in the phase field approach due to the existence of the critical point in the coexistence line, which entails the merging of the phases described by $φ$.

2013, 6(2): 331-351
doi: 10.3934/dcdss.2013.6.331

*+*[Abstract](1968)*+*[PDF](467.7KB)**Abstract:**

In this paper, we deal with a PDE system describing a phase transition problem characterized by irreversible evolution and ruled by a nonlinear heat flux law. Its derivation comes from the modelling approach proposed by M. Frémond. Our main result consists in showing the global-in-time existence and the uniqueness of the solution of the related initial and boundary value problem.

2013, 6(2): 353-368
doi: 10.3934/dcdss.2013.6.353

*+*[Abstract](2395)*+*[PDF](455.7KB)**Abstract:**

This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $\rho$ and the chemical potential $\mu$; each equation includes a viscosity term -- respectively, $\varepsilon \,\partial_t\mu$ and $\delta\,\partial_t\rho$ -- with $\varepsilon$ and $\delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(\varepsilon,\delta)-$solutions. Here we discuss the asymptotic limit of the system as $\varepsilon$ tends to $0$. We prove convergence of $(\varepsilon,\delta)-$solutions to the corresponding solutions for the case $\varepsilon =0$, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.

2013, 6(2): 369-386
doi: 10.3934/dcdss.2013.6.369

*+*[Abstract](2757)*+*[PDF](426.1KB)**Abstract:**

We address the thermal control of the quasi-static evolution of a polycrystalline shape memory alloy specimen. The thermomechanical evolution of the body is described by means of the phenomenological SOUZA$-$AURICCHIO model [6,53]. By assuming to be able to control the temperature of the body in time we determine the corresponding quasi-static evolution in the

*energetic*sense. By recovering in this context a result by RINDLER [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals and comment on their possible approximation.

2013, 6(2): 387-400
doi: 10.3934/dcdss.2013.6.387

*+*[Abstract](2586)*+*[PDF](346.8KB)**Abstract:**

Our aim in this paper is to define proper dynamic boundary conditions for a generalization of the Cahn-Hilliard system proposed by M. Gurtin. Such boundary conditions take into account the interactions with the walls in confined systems. We then study the existence and uniqueness of weak solutions.

2013, 6(2): 401-422
doi: 10.3934/dcdss.2013.6.401

*+*[Abstract](2494)*+*[PDF](481.4KB)**Abstract:**

In this article, we give an asymptotic expansion, with respect to the viscosity which is considered here to be small, of the solutions of the $3D$ linearized Primitive Equations (EPs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice of the

*non-local*boundary conditions for the non-viscous PEs.

2013, 6(2): 423-438
doi: 10.3934/dcdss.2013.6.423

*+*[Abstract](2210)*+*[PDF](348.5KB)**Abstract:**

In this paper we consider some mechanical phenomena whose dynamics is described by a class of quasi-variational inequalities of parabolic type. Our system consists of a second-order parabolic variational inequality with gradient constraint depending on the temperature and the heat equation. Since the temperature is unknown in our problem, the constraint function is unknown as well. In this sense, our problem includes the quasi-variational structure, and in the mathamtical analysis one of main difficulties comes from it. Our approach to the problem is based on the abstract theory of quasi-variational inequalities with non-local constraint which has been developed in [6]. However the abstract theory is not directly used in the existence proof of a solution, since the mathematical situation of the problem is much nicer than that in the abstract theory [6]. In this paper we prove the existence of a weak solution of our system.

2013, 6(2): 439-460
doi: 10.3934/dcdss.2013.6.439

*+*[Abstract](2569)*+*[PDF](455.6KB)**Abstract:**

We propose an improved model explaining the occurrence of high stresses due to the difference in specific volumes during phase transitions between water and ice. The unknowns of the resulting evolution problem are the absolute temperature, the volume increment, and the liquid fraction. The main novelty here consists in including the dependence of the specific heat and of the speed of sound upon the phase. These additional nonlinearities bring new mathematical difficulties which require new estimation techniques based on Moser iteration. We establish the existence of a global solution to the corresponding initial-boundary value problem, as well as lower and upper bounds for the absolute temperature. Assuming constant heat conductivity, we also prove uniqueness and continuous data dependence of the solution.

2013, 6(2): 461-478
doi: 10.3934/dcdss.2013.6.461

*+*[Abstract](2557)*+*[PDF](420.8KB)**Abstract:**

In this paper, tensegrity structures are modeled by introducing suitable energy convex functions. These allow to enforce both ideal and non-ideal constraints, gathering compatibility, equilibrium, and stability problems, as well as their duality relationships, in the same functional framework. Arguments of convex analysis allow to recover consistently a number of basic results, as well as to formulate new interpretations and analysis criterions.

2013, 6(2): 479-499
doi: 10.3934/dcdss.2013.6.479

*+*[Abstract](2517)*+*[PDF](308.4KB)**Abstract:**

We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,Φ,K)$ defining the evolution $\dot U=-K(U)D Φ(U)$. Here $Φ$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G =K ^{-1}$ exists, the triple $(X,Φ,G)$ defines a gradient system.

Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi)=1/2\langle \Xi, K(U)\Xi\rangle$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.

2013, 6(2): 501-515
doi: 10.3934/dcdss.2013.6.501

*+*[Abstract](1896)*+*[PDF](411.8KB)**Abstract:**

We consider the inverse problem of determining the possible presence of an inclusion in a thin plate by boundary measurements. The plate is made by non-homogeneous linearly elastic material belonging to a general class of anisotropy. The inclusion is made by different elastic material. Under some a priori assumptions on the unknown inclusion, we prove constructive upper and lower estimates of the area of the unknown defect in terms of an easily expressed quantity related to work, which is given in terms of measurements of a couple field applied at the boundary and of the induced transversal displacement and its normal derivative taken at the boundary of the plate.

2013, 6(2): 517-546
doi: 10.3934/dcdss.2013.6.517

*+*[Abstract](2194)*+*[PDF](469.2KB)**Abstract:**

An initial-boundary-value problem for a class of sixth order viscous Cahn-Hilliard type equations with a nonlinear diffusion is considered. The study is motivated by phase-field modelling of various spatial structures, for example arising in oil-water-surfactant mixtures and in modelling of crystal growth on atomic length, known as phase field crystal model. For such problem we prove the existence and uniqueness of a global in time regular solution. First the finite-time existence is proved by means of the Leray-Schauder fixed point theorem. Then, due to suitable estimates, the finite-time solution is extended step by step on the infinite time interval.

2013, 6(2): 547-565
doi: 10.3934/dcdss.2013.6.547

*+*[Abstract](2534)*+*[PDF](689.7KB)**Abstract:**

The non-smooth view of Michel Frémond has already been proven successful in managing collisions between rigid particles and in this paper, it will be adapted so as to represent pedestrians and their strategy of displacement. The developed discrete approach applies a rigorous thermodynamic framework in which the local interactions between particles are managed by the use of pseudopotentials of dissipation. It handles local interactions such as pedestrian-pedestrian and pedestrian-obstacle in order to reproduce the global and real dynamics of pedestrian traffic. Social forces are introduced and implemented in order to simulate the behavior of pedestrians and subgroups of pedestrians. The numerical implementation allows us to perform simulations in various situations so that the safety and comfort of public spaces can be enhanced.

2013, 6(2): 567-590
doi: 10.3934/dcdss.2013.6.567

*+*[Abstract](1989)*+*[PDF](492.7KB)**Abstract:**

Pseudo-potentials are very useful tools to define thermodynamically admissible constitutive rules. Bipotentials are convenient for numerical purposes, in particular for non-associative rules. Unfortunately, these functionals are not always easy to construct starting from a given constitutive law. This work proposes a procedure to find the pseudo-potentials and the bipotential starting from the usual description of a non-associative constitutive law. This method is applied to different non-associative plasticity models such as the Drucker-Prager model and the non-linear kinematic hardening model. The same procedure allows one to obtain the pseudo-potentials of an endochronic plasticity model. The pseudo-potentials for the contact problem with dissipation are constructed using the same ideas. For all these non-associative constitutive laws a bipotential is then automatically deduced.

2013, 6(2): 591-610
doi: 10.3934/dcdss.2013.6.591

*+*[Abstract](2164)*+*[PDF](520.2KB)**Abstract:**

The quasistatic rate-independent evolution of delamination in the so-called mixed-mode, i.e. distinguishing opening (mode I) from shearing (mode II), devised in [45], is described in detail and rigorously analysed as far as existence of the so-called energetic solutions concerns. The model formulated at small strains uses a delamination parameter of Frémond's type combined with a concept of interface plasticity, and is associative in the sense that the dissipative force driving delamination has a potential which depends in a 1-homogeneous way only on rates of internal parameters. A sample numerical simulation documents that this model can really produce mode-mixity-sensitive delamination.

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