Fatigue accumulation in an oscillating plate
Michela Eleuteri Jana Kopfov Pavel Krej?
Discrete & Continuous Dynamical Systems - S 2013, 6(4): 909-923 doi: 10.3934/dcdss.2013.6.909
A thermodynamic model for fatigue accumulation in an oscillating elastoplastic Kirchhoffplate based on the hypothesis that the fatigue accumulation rate is proportional tothe dissipation rate, is derived for the case that both the elastic and the plasticmaterial characteristics change with increasing fatigue. We prove the existence ofa unique solution in the whole time interval before a singularity (material failure) occursunder the simplifying hypothesis that the temperature history is a priori given.
keywords: Elastoplastic plate Prandtl-Ishlinskii operator material fatigue.
An asymptotic convergence result for a system of partial differential equations with hysteresis
Michela Eleuteri Pavel Krejčí
Communications on Pure & Applied Analysis 2007, 6(4): 1131-1143 doi: 10.3934/cpaa.2007.6.1131
A partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.
keywords: hysteresis Partial differential equations asymptotic convergence Preisach operator.
Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials
Michela Eleuteri Luca Lussardi
Evolution Equations & Control Theory 2014, 3(3): 411-427 doi: 10.3934/eect.2014.3.411
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 an extension of the phenomenological Souza-Auricchio model [6,7,8,57] accounting also for permanent inelastic effects [9,11,27]. 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. In a similar way as in [28], using results by Rindler [49,50] we prove the existence of optimal controls for a suitably large class of cost functionals.
keywords: permanent inelasticity. rate-independent Shape memory alloys optimal control quasi-static evolution
An existence result for a P.D.E. with hysteresis, convection and a nonlinear boundary condition
Michela Eleuteri
Conference Publications 2007, 2007(Special): 344-353 doi: 10.3934/proc.2007.2007.344
In this paper a partial differential equation containing a continuous hysteresis operator and a convective term is considered. This model equation, which appears in the context of magnetohydrodynamics, is coupled with a nonlinear boundary condition containing a memory operator. Under suitable assumptions, an existence result is achieved using an implicit time discretization scheme.
keywords: hysteresis Partial differential equations convection nonlinear boundary condition.
Preface: Special issue on rate-independent evolutions and hysteresis modelling
Stefano Bosia Michela Eleuteri Elisabetta Rocca Enrico Valdinoci
Discrete & Continuous Dynamical Systems - S 2015, 8(4): i-i doi: 10.3934/dcdss.2015.8.4i
The interest in hysteresis and rate-independent phenomena is shared by scientists with a great variety of different backgrounds. We can encounter these processes in several situations of common life: for instance in elasto-plasticity, ferromagnetism, shape-memory alloys, phase transitions. Beyond physics, hysteresis and rate-independent phenomena appear also in engineering, biology, economics as well as in many other settings, playing an important role in many applications. The complexity arising in these fields necessarily requires a joint contribution of experts with different backgrounds and skills. Therefore, only synergy and cooperation among these several people can lead to concrete advances in the technological capabilities of our society.
    This special issue of Discrete and Continuous Dynamical Systems is devoted to the latest advances and trends in the modelling and in the analysis of this family of complex phenomena. In particular, we gathered contributions from different fields of science (mathematical analysis, mathematical physics, engineering) with the intent of presenting an updated picture of current research directions, offering a new and interdisciplinary perspective in the study of these processes.
    Motivated by the Spring School on Rate-independent Evolutions and Hysteresis Modelling, held at the Politecnico di Milano and University of Milano on May 27-31, 2013, this special issue contains different kinds of original contributions: some of them originate from the courses held in that occasion and from the discussions they stimulated, but are here presented in a new perspective; some others instead are original contributions in related topics. All the papers are written in the clearest possible language, accessible also to students and non-experts of the field, with the intent to attract and introduce them to this topic.
    Final acceptance of all the papers in this volume was made by the normal referee procedure and standard practices of AIMS journals.
    We wish to thanks all the referees, who kindly agreed to devote their time and effort to read and check all the papers carefully, providing useful comments and recommendations. We are also grateful to all the authors for their great job and the high quality of their contributions. We finally wish to express our gratitude to AIMS and in particular to Prof. Alain Miranville for the opportunity to publish this special issue and for the technical support.
On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids
Michela Eleuteri Elisabetta Rocca Giulio Schimperna
Discrete & Continuous Dynamical Systems - A 2015, 35(6): 2497-2522 doi: 10.3934/dcds.2015.35.2497
We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity $u$ is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter $\psi$ representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a priori estimates and showing weak sequential stability of families of approximating solutions.
keywords: weak solutions. Cahn-Hilliard Navier-Stokes incompressible non-isothermal binary fluid global-in-time existence
Thermal control of the Souza-Auricchio model for shape memory alloys
Michela Eleuteri Luca Lussardi Ulisse Stefanelli
Discrete & Continuous Dynamical Systems - S 2013, 6(2): 369-386 doi: 10.3934/dcdss.2013.6.369
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 energeticsense. 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.
keywords: optimal control Shape memory alloys variational evolutions. rate independent energetic solutions
Local Lipschitz continuity of minimizers with mild assumptions on the $x$-dependence
Michela Eleuteri Paolo Marcellini Elvira Mascolo
Discrete & Continuous Dynamical Systems - S 2019, 12(2): 251-265 doi: 10.3934/dcdss.2019018

We are interested in the regularity of local minimizers of energy integrals of the Calculus of Variations. Precisely, let $Ω $ be an open subset of $\mathbb{R}^{n}$. Let $f≤\left( {x, \xi } \right) $ be a real function defined in $Ω × \mathbb{R}^{n}$ satisfying the growth condition $|{f_{\xi x}}\left( {x, \xi } \right)| \le h\left( x \right)|\xi {{\rm{|}}^{p - 1}}$, for $x∈ Ω $ and $\xi ∈ \mathbb{R}^{n}$ with $|\xi {\rm{|}} \ge {M_0}$ for some $M_{0}≥ 0$, with $h \in L_{{\rm{loc}}}^r\left( \Omega \right) $ for some $r>n$. This growth condition is more general than those considered in the mathematical literature and allows us to handle some cases recently studied in similar contexts. We associate to $f\left( {x, \xi } \right) $ the so-called natural $p-$growth conditions on the second derivatives ${f_{\xi \xi }}\left( {x, \xi } \right)$; i.e., $\left( {p - 2} \right) - $growth for $|{f_{\xi \xi }}\left( {x, \xi } \right)| $ from above and $\left( {p - 2} \right) - $growth from below for the quadratic form $({f_{\xi \xi }}\left( {x, \xi } \right)\lambda , \lambda {\rm{ }})$; for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer $u \in W_{{\rm{loc}}}^{1, p}\left( \Omega \right) $ of the energy integral $\int_\Omega {f(x, Du\left( x \right)){\mkern 1mu} dx} $.

keywords: Local minimizers local Lipschitz continuity standard growth
A rate-independent model for permanent inelastic effects in shape memory materials
Michela Eleuteri Luca Lussardi Ulisse Stefanelli
Networks & Heterogeneous Media 2011, 6(1): 145-165 doi: 10.3934/nhm.2011.6.145
This paper addresses a three-dimensional model for isothermal stress-induced transformation in shape memory polycrystalline materials in presence of permanent inelastic effects. The basic features of the model are recalled and the constitutive and the three-dimensional quasi-static evolution problem are proved to be well-posed. Finally, we discuss the convergence of the model to reduced/former ones by means of a rigorous $\Gamma$-convergence analysis.
keywords: Shape memory materials $\Gamma$-convergence. well-posedness permanent inelastic effects
A new phase field model for material fatigue in an oscillating elastoplastic beam
Michela Eleuteri Jana Kopfová Pavel Krejčí
Discrete & Continuous Dynamical Systems - A 2015, 35(6): 2465-2495 doi: 10.3934/dcds.2015.35.2465
We pursue the study of fatigue accumulation in an oscillating elastoplastic beam under the additional hypothesis that the material can partially recover by the effect of melting. The full system consists of the momentum and energy balance equations, an evolution equation for the fatigue rate, and a differential inclusion for the phase dynamics. The main result consists in proving the existence and uniqueness of a strong solution.
keywords: Prandtl-Ishlinskii operator material fatigue thermo-elasto-plasticity Elastoplastic beam phase transition.

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