1,721,096 research outputs found
Continuum thermodynamics and phase-field models
No full textPhase transitions between two phases are modelled as space regions where a phase-field changes smoothly. The two phases are separated by a thin transition layer, the so-called diffuse interface. All thermodynamic quantities are allowed to vary inside this layer, including the pressure and the mass density. A thermodynamic approach is developed by allowing for the nonlocal character of the continuum. It is based on an extra entropy flux which is proved to be non vanishing inside the transition layer, only. The phase-field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation of rate type. Necessary and sufficient restrictions placed by thermodynamics are derived for the constitutive equations and, furthermore, a general form of the evolution equation for the phase-field is obtained within the schemes of both a non-conserved and a conserved phase-field. Based on the thermodynamic restrictions, a phase-field model for the ice-water transition is established which allows for superheating and undercooling. A model ruling the liquid-vapor phase transition is also provided which accounts for both temperature and pressure variations during the evaporation process. The explicit expression of the Gibbs free enthalpy, the Clausius-Clapeyron formula and the customary form of the vapor pressure curve are recovered
On the existence and uniqueness of solutions for problems in Kelvin-Voigt viscoelasticity
No full textUn problema di esistenza ed unicità per un sistema linearizzato di Navier-Stokes a coefficienti non costanti
No full textThe existence and uniqueness of strong solutions of initial-boundary value problems for the linear system
vt−vΔv+Bk(x,t)vxk=−gradp+fdivv=0(x,t)∈Ω×(0,T)
is established for functionsBk(x, t),k=1,2,3, continuous and bounded on Ω×(0,T), where Ω is a domain inR3. Problems of this kind occur examining some uniqueness questions for Navier-Stokes nonlinear systems
Phase-field models for transition phenomena in materials with hysteresis
No full textNon-isothermal phase-field models of transition phenomena in materials with hysteresis are considered within the framework of the Ginzburg-Landau theory. Our attempt is to capture the relation between phase-transition and hysteresis (either mechanical or magnetic). All models are required to be compatible with thermodynamics and to fit well the shape of the major hysteresis loop. Focusing on uniform cyclic processes, numerical simulations at different temperatures are performed
Ordinary differential equations
No full textWe summarize here the main results in the theory of ordinary differential equations (ODEs). After recalling some general mathematical definitions, the Cauchy problem in R^n is first considered. Some results are explained without proofs: the Peano’s Ttheorem and the existence and uniqueness of solutions in a rectangle and in a strip. Then, linear ODEs with continuous coefficients are examined, both in the homogeneous and nonhomogeneous case. The generalized Abel’s identity and the Wronskian Theorem are proved. The special case of linear ODEs with constant coefficients is scrutinized. We show that a general ODE of order n can be reduced to the a first-order system of n ODEs. In particular, we scrutinize linear systems and describe the classical method called “separation of variables". Finally, first-order ODEs in Banach spaces are considered. In this framework, the global existence of the Cauchy problem is proved by virtue of the Banach contraction principle
Sui potenziali elettromagnetici per alcuni modelli di isteresi magnetica
No full textIn this paper we study electromagnetic potentials (free energy and enthalpy) for some simple models of ferromagnetic materials exhibiting hysteresis loops and obeying constitutive equations of rate type. For the sake of simplicity, mechanical and electric effects are ignored and isothermal conditions are assumed. Involving thermodynamic arguments of Coleman and Owen we show the existence of a convex set of lower-potentials whose minimum, and maximum elements are explicitly constructed as functions of state. Finally, some problems on state-space representation and convexity of various lower-potentials are discussed
New variational principles in quasi-static viscoelasticity
No full textA "saddle point" (or maximum-minimum) principle is set up for the quasi-static boundary-value problem in linear viscoelasticity. The appropriate class of convolution-type functionals for it is taken in terms of bilinear forms with a weight function involving Fourier transform. The "minimax" property is shown to hold as a direct consequence of the thermodynamic restrictions on the relaxation function. This approach can be extended to further linear evolution problems where initial data are not prescribed
A new model for rate-independent hysteresis in permanent magnets
No full textA one-dimensional memory-based model of magnetic materials exhibiting hysteresis loops is proposed here. Mainly, we suggest a modification of the Duhem model investigated by Coleman & Hodgdon in connection with isoperms in order to better fit their real behaviour. Many properties fulfilled by the original model are preserved here. For instance, the presence of the major loop, bounding all hysteresis pathes, the existence and uniqueness of asymptotically stable periodic solutions (primitive loops), the existence of a "universal" demagnetization process (a suitable alternating magnetic field, with slowly decreasing amplitude, reproducing the "virgin state"). On the other hand, a few mathematical features are common to the Preisach model and allow the corresponding hysteretic functional to avoid effects, like self-magnetization, which are typical of the Duhem modelization but physically unsound. In particular, equilibrium states are stable with respect to noises of small amplitude (static vibro-stability)
An existence theorem for a nonlinear evolution equation in viscoelasticity
No full textWe establish the existence of weak global solutions of initial-boundary value problems for a partial differential equation
which occours as the equation of motion in nonlinear Kelvin solids with nonlinear stress components. Each functionai is required to be sufficiently smooth and must satisfy the following conditions:a)∣αi(x,t,η)∣⩽K0{∣η∣p-1+1}a)αi(x,t,η)≶K1∣η∣p-2η η≶0c)(∂/∂t)αi(x,t,η)≶K2(t){∣η∣p-2+1}η≶0d)[αi(x,t,η)—αi(x,t,ξ)].(η—ξ⩾0 for somep≥2, some positive constantsK0,K1, some non negative functionK2∈L1(0,T) and for allx∈Ω, t∈[0, T], ξ and ν∈R
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