Analysis of a thermomechanical phase transition model

Subject of this work is a macroscopic thermomechanical model of phase transitions in steel. Effects like transformation strain and transformation plasticity induced by the phase transitions are considered and used to formulate a consistent thermomechanical model. The resulting system of state equations consists of a quasistatic momentum balance coupled with a nonlinear stress-strain relation, a nonlinear energy balance equation and a system of ODEs for the phase volume fractions. We prove the existence of a unique weak solution using fixed-point arguments. A key issue is a regularity analysis for the mechanical subsystem to obtain continuity of the stress tensor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical model for impulse resistance welding

We present a mathematical model of impulse resistance welding. It accounts for electrical, thermal and mechanical effects, which are non‐linearly coupled by the balance laws, constitutive equations and boundary conditions. The electrical effects of the weld machine are incorporated by a discrete oscillator circuit which is coupled to the field equations by a boundary condition. We prove the existence of weak solutions for a slightly simplified model which however still covers most of its essential features, e.g. the quadratic Joule heat term and a quadratic term due to non‐elastic energy dissipation. We discuss the numerical implementation in a 2D setting, present some numerical results and conclude with some remarks on future research. Copyright © 2003 John Wiley & Sons, Ltd.     

Global solution to the full one-dimensional Frémond model for shape memory alloys

In this paper, we prove the existence and uniqueness of the solution to the one-dimensional initial-boundary value problem resulting from the Frémond thermomechanical model of structural phase transitions in shape memory materials. In this model, the free energy is assumed to depend on temperature, macroscopic deformation and phase fractions. The resulting equilibrium equations are the balance laws of (linear) momentum and energy, coupled with an evolution variational inequality for the phase fractions. Fourth-order regularizing terms in the quasi-stationary momentum balance equation are not necessary, and, as far as we know for the first time, all the non-linear terms of the energy balance equation are taken into account.     

Simulating resistance switching in amorphous carbon

The resistance switching in amorphous carbon (a-C) is mathematically simulated on the basis of quantum molecular dynamics. The electric conductivity in thin a-C films is related to a phase transition due to an electric field and corresponding Joule heating and cooling. Simulations show that the transition mechanism is related to the clusterization of existing conducting regions of graphite (sp ²) in a nonconducting diamond (sp ³) matrix. Simulations yield the conditions for the phase transition in a-C from one state to another. Excited molecular orbitals and the energy gap at different temperatures are also calculated.     

Existence of solutions for a mathematical model of structural phase transitions in shape memory alloys

In this paper a thermomechanical model for the dynamics of structural phase transitions in the so-called ‘shape memory alloys’ is developed. These materials exhibit rather spectacular hysteresis phenomena. The resulting mathematical model consists of a coupled and highly non-linear system of partial differential equations reflecting the balance laws of linear momentum and energy. For an appropriate weak formulation the local-in-time existence of weak solutions is shown.     

Global existence for a nonlinear system in thermoviscoelasticity with nonconvex energy

A three-dimensional thermoviscoelastic system derived from the balance laws of momentum and energy is considered. To describe structural phase transitions in solids, the stored energy function is not assumed to be convex as a function of the deformation gradient. A novel feature for multi-dimensional, nonconvex, and nonisothermal problems is that no regularizing higher-order terms are introduced. The mechanical dissipation is not linearized. We prove existence global in time. The approach is based on a fixed-point argument using an implicit time discretization and the theory of renormalized solutions for parabolic equations with L¹ data.     

Transient Finite Element Simulation of the Temperature Field during Cryogenic Turning of Metastable Austenitic Steel AISI 347

The process chain in manufacturing often consists of many steps. As part of current researches the possibility of combining two process steps, turning and hardening, is investigated to optimize the manufacturing time and to decrease the energy consumption of the process. For metastable austenitic steels, deformation induced hardening during turning can be used to achieve surface hardening [1] and thus to increase the wear resistance [2] as well as the fatigue strength [3], by applying high passive forces onto the workpiece. This enables an austenite-martensite phase transformation, for which it is necessary to maintain low process temperatures, typically below room temperature. Thus, cryogenic coolants are applied [4]. For a better understanding of the influence of cutting parameters on the process temperatures and thus martensite formation, knowledge of the exact temperature distribution in the workpiece and in the contact zone between workpiece and tool is essential. Since the experimental determination of the temperature field is hardly possible, an inverse determination of the process temperatures via transient finite element simulation is performed. The present finite element approach only takes thermal loads into account. The simulations are performed in the finite element program FEAP (Finite Element Analysis Program) with an Eulerian mesh, which requires special consideration of the rigid body rotation of the workpiece. In order to prevent unphysical oscillations in the solution, introduced by the convective time derivative, a streamline upwind / Petrov–Galerkin stabilization scheme is utilized. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of dynamic mechanical response of a composite plate using multi-field coupling with thermal constraints

We consider the problem of optimizing the dynamic response of a mechanically loaded, rectangular, electrically conductive anisotropic composite plate by applying an electromagnetic field, which exploits the electro-magneto-mechanical field coupling phenomenon. An important aspect of the formulated nonlinear partial differential equation (PDE)-constrained optimization model is the presence of a thermal constraint that prevents polymer matrix degradation in the composite material due to Joule heating produced by the electromagnetic field. A black-box optimization approach based on the active set algorithm is employed. A system of governing PDEs is solved using a series of sequential numerical procedures that includes the method of lines, Newmark time-stepping scheme, quasilinearization, integration of two-point boundary-value problems, and a superposition method followed by orthonormalization. Implementation in hyper-dual arithmetics facilitated automatic differentiation and computation of the gradient. Optimization results show that application of an electromagnetic field with optimal characteristics enables one to significantly reduce the amplitude of the plate vibrations while controlling for Joule heating.     

Existence for the thermoelastic thermistor problem

We study a new model for the thermistor problem that consists of a system of dynamic thermoelasticity equations of displacement and a stationary charge conservation equation of electrical current. The heat source generated by Joule heating is quadratic in the gradient of the electrical potential. This system is nonlinear and degenerate since the electrical conductivity is assumed to be temperature dependent and vanishes at large temperatures. We establish the existence of a very weak solution, the so-called “capacity solution” for the model. The proof is based on regularization and time retarding.     

Non–Isothermal Shear Band Localization in Crystal Plasticity

We analyze the formation of shear bands in plastically deforming single crystals under non-isothermal quasi-static loading conditions. A key feature of the present work is to motivate the constitutive coupled thermomechanical equations for the slip resistance and the slip evolution by micromechanical investigations of defects in crystals. Here, we put particular emphasis on comparative investigations of hardening–softening effects of crystals under non-isothermal conditions, including a quantification of the latent energy storage. Furthermore, we analyze a numerical solution algorithm for the coupled thermomechanical problem and we test the performance on a typical benchmark example of finite-strain single-crystal thermoplasticity: the localization of deformation into a shear band for the case of a rectangular strip. Experimental evidence for this problem is well documented. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence for a quasi-linear evolution equation with a non-convex energy

We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.

     

A model for resistance welding including phase transitions and Joule heating

In this paper, we introduce a new model for solid–liquid phase transitions triggered by Joule heating as they arise in the case of resistance welding of metal parts. The main novelties of the paper are the coupling of the thermistor problem with a phase‐field model and the consideration of phase‐dependent physical parameters through a mixture ansatz. The PDE system resulting from our modeling approach couples a strongly nonlinear heat equation, a non‐smooth equation for the the phase parameter (standing for the local proportion of one of the two phases) with a quasistatic electric charge conservation law. We prove the existence of weak solutions in the three‐dimensional (3D) case, whereas the regularity result and the uniqueness of solution is stated only in the two‐dimensional case. Indeed, uniqueness for the 3D system is still an open problem. Copyright © 2011 John Wiley & Sons, Ltd.     

On Cremona transformations and quadratic complexes

We study the relation between Cremona transformations in space and quadratic line complexes. We show that it is possible to associate a space Cremona transformation to each smooth quadratic line complex once we choose two distinct lines contained in the complex. Such Cremona transformations are cubo-cubic and we classify them in terms of the relative position of the lines chosen. It turns out that the base locus of such a transformation contains a smooth genus two quintic curve. Conversely, we show that given a smooth quintic curve C of genus 2 in ℙ³ every Cremona transformation containing C in its base locus factorizes through a smooth quadratic line complex as before. We consider also some cases where the curve C is singular, and we give examples both when the quadratic line complex is smooth and singular.     

On a viscous Hamilton-Jacobi equation with an unbounded potential term

We present comparison, uniqueness and existence results for unbounded solutions of a viscous Hamilton-Jacobi or eikonal equation. The equation includes an unbounded potential term V(x) subject to a quadratic upper bound. The results are obtained through a tailor-made change of variables in combination with the Hopf-Cole transformation. An integral representation formula for the solution of the Cauchy problem is derived in the case where V(x)=ω²|x|²/2.     

Global existence of BV solutions and relaxation limit for a model of multiphase reactive flow

We consider a strictly hyperbolic system of balance laws in one space variable, that represents a simple model for a fluid flow in the presence of phase transitions. The state variables are specific volume, velocity and mass-density fraction λ of the vapor in the fluid. A reactive source term drives the dynamics of the phase mixtures; such a term depends on a relaxation parameter and involves an equilibrium pressure, allowing for metastable states.First we prove the global existence of weak solutions to the Cauchy problem, where the initial datum for λ is close either to 0 or 1 (the pure phases) and has small total variation, while the initial variations of pressure and velocity are not necessarily small. Then we consider the relaxation limit and prove that the weak solutions of the full system converge to those of the reduced system.     

Kinks in two-phase lipid bilayer membranes

Common models for two-phase lipid bilayer membranes are based on an energy that consists of an elastic term for each lipid phase and a line energy at interfaces. Although such an energy controls only the length of interfaces, the membrane surface is usually assumed to be at least C ¹ across phase boundaries. We consider the spontaneous curvature model for closed rotationally symmetric two-phase membranes without excluding tangent discontinuities at interfaces a priorily. We introduce a family of energies for smooth surfaces and phase fields for the lipid phases and derive a sharp interface limit that coincides with the Γ-limit on all reasonable membranes and extends the classical model by assigning a bending energy also to tangent discontinuities. The theoretical result is illustrated by numerical examples.     

Bounds on Oscillatory Integral Operators Based on Multilinear Estimates

We apply the Bennett–Carbery–Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L ^p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for H?rmander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for H?rmander-type oscillatory integral operators in even dimensions. These last oscillatory estimates are related to improved bounds on the dimensions of curved Kakeya sets in even dimensions.     

Sul criterio dell'energia per la stabilitá di un sistema termomeccanico

Summary The energy criterion for mechanical stability asserts that the stable configurations are those that minimize the potential energy. Recent studies have shown that the energy criterion can be extended to stability of thermomechanical systems under suitable environment conditions, provided that the «stored energy» is interpreted as the equilibrium free-energy at the environmental temperature ^e. The aim of this paper is to provide a contribution to a general theory of thermomechanical stability. Essentially we have restated the theory for general materials introduced by Gurtin with a new framework in the light of recent theories of Noll and Coleman-Owen on simple materials and on thermodynamical potentials. We define a «thermomechanical system» which posseses two main features: i) state space has a «natural topology» depending on the thermodynamical behaviour of system; ii) internal energy E and entropy S are not supposed to exist but are expressely obtained with their smoothness properties.

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Lavoro eseguito nell'ambito del G.N.F.M. del C.N.R,