## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
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- Evolution Equations & Control Theory
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DCDS-S

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.

CPAA

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.

EECT

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.

PROC

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.

DCDS-S

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

Motivated by the

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.

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.

keywords:

DCDS

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.

DCDS-S

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.
NHM

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.

DCDS

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.

DCDS-B

We consider a thermodynamic model for fatigue accumulation in an oscillating elastoplastic Kirchhoff
plate based on the hypothesis that the fatigue accumulation rate is proportional to
the plastic part of the dissipation rate. For the full model with periodic boundary conditions we prove existence of
a solution in the whole time interval.

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