Global weak solutions to a phase-field model for motion of grain boundaries

We employ the Galerkin method to prove the global existence of weak solutions to a phase-field model which is suitable to describe a sort of interface motion driven by configurational forces. The higher-order derivative of unknown S exists in the sense of local weak derivatives since it may be not summable over the original open domain. The existence proof is valid in the one-dimensional case.     

WAVE INTERACTIONS IN SULICIU MODEL FOR DYNAMIC PHASE TRANSITIONS

Elementary waves in Suliciu model for dynamic phase transitions are obtained through traveling wave analysis.For any given initial data with two pieces of constant states,the Riemann solutions are constructed as a combination of elementary waves. When the initial profile contains three pieces of constant states,the solution may be constructed from the Riemann solutions,with each two adjacent states connected by elementary waves.A new Riemann problem forms when these two waves collide.Through the exploration of these Riemann problems,the outcome of wave interactions may be classified in a suitable parametric space.     

Constitutive model for third harmonic generation in elastic solids

In this article, we present a new constitutive model for studying ultrasonic third harmonic generation in elastic solids. The model is hyperelastic in nature with two parameters characterizing the linear elastic material response and two other parameters characterizing the nonlinear response. The limiting response of the model as the nonlinearity parameters tend to zero is shown to be the well-known St Venant–Kirchhoff model. Also, the symmetric response of the model in tension and compression and its role in third harmonic generation is shown. Numerical simulations are carried out to study third harmonic generation in materials characterized by the proposed constitutive model. Predicted third harmonic guided wave generation reveals an increasing third harmonic content with increasing nonlinearity. On the other hand, the second harmonics are independent of the nonlinearity parameters and are generated due to the geometric nonlinearity. The feasibility of determining the nonlinearity parameters from third harmonic measurements is qualitatively discussed.     

Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model

A new approach for modeling multivariant martensitic phase transitions (PT) and martensitic microstructure (MM) in elastic materials is proposed. It is based on a thermomechanical model for PT that includes strain softening and the corresponding strain localization during PT. Mesh sensitivity in numerical simulations is avoided by using rate-dependent constitutive equations in the model. Due to strain softening, a microstructure comprised of pure martensitic and austenitic domains separated by narrow transition zones is obtained as the solution of the corresponding boundary value problem. In contrast to Landau-Ginzburg models, which are limited in practice to nanoscale specimens, this new phase field model is valid for scales greater than 100 nm and without upper bound. A finite element algorithm for the solution of elastic problems with multivariant martensitic PT is developed and implemented into the software ABAQUS. Simulated microstructures in elastic single crystals and polycrystals under uniaxial loading are in qualitative agreement with those observed experimentally.     

On a 3D moving load problem for an elastic half space

The paper deals with 3D dynamic response of an elastic half-space loaded by a point force moving at a constant speed along a straight line on the surface. The problem is formulated within the framework of the asymptotic hyperbolic–elliptic model developed earlier by two of the authors. The validity of the model is restricted to the range of speeds close to the Rayleigh wave speed. Steady-state near-field solutions are derived in terms of elementary functions. Transient analysis of surface motion illustrates peculiarities of the resonance associated with the Rayleigh wave.     

The influence of the type of loading on the asymptotic behavior of slender elastic rings

We study the influence of the type of loading on the asymptotic behavior of linearly elastic, isotropic and homogeneous slender circular rings. By using formal asymptotic expansions, we obtain three families of models depending on the properties of the loads. If the loads expend work in inextensional displacements, then we find the classical model where the leading term of the energy corresponds to the bending-torsion energy of inextensional displacements. If the loads do no work in inextensional displacements, the model must be refined and we obtain two other types of models. In these other models, which depend on the type of loading, the leading term of the energy contains additional terms such as, for the second class, an extension energy due to the circumferential stretching of the ring, and even, for the third class, specific load-dependent contributions. This classification is illustrated in several examples.     

A one-dimensional model for incoherent phase changes

We present here a simplified version of the model of incoherent solid-solid transitions with mass diffusion developed by Gurtin and Cermelli in [3]. An incoherent phase change is always associated with some kind of defect production at the interface: we consider here a one-dimensional continuum, so that the resulting equations allow study to be made of the influence of volume (vacancy) production on the evolution of the system.

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Sommario In questo lavoro viene presentato un adattamento del modello di transizione di fase incoerente sviluppato da Gurtin e Cermelli in [3]. Una transizione incoerente è sempre associata alla produzione di un qualche tipo di difetto all'interfaccia: consideriamo qui un modello semplificato di continuo unidimensionale, in modo da poter studiare l'effetto dei difetti di volume (lacune) sull'evoluzione del sistema.

     

The dynamic response of an incompressible non-linearly elastic membrane tube subjected to a dynamic extension

The dynamic response of an isotropic hyperelastic membrane tube, subjected to a dynamic extension at its one end, is studied. In the first part of the paper, an asymptotic expansion technique is used to derive a non-linear membrane theory for finite axially symmetric dynamic deformations of incompressible non-linearly elastic circular cylindrical tubes by starting from the three-dimensional elasticity theory. The equations governing dynamic axially symmetric deformations of the membrane tube are obtained for an arbitrary form of the strain-energy function. In the second part of the paper, finite amplitude wave propagation in an incompressible hyperelastic membrane tube is considered when one end is fixed and the other is subjected to a suddenly applied dynamic extension. A Godunov-type finite volume method is used to solve numerically the corresponding problem. Numerical results are given for the Mooney-Rivlin incompressible material. The question how the present numerical results are related to those obtained in the literature is discussed.     

The role of spinodal region in the kinetics of lattice phase transitions

We consider a one-dimensional chain of phase-transforming springs with harmonic long-range interactions. The nearest-neighbor interactions are assumed to be trilinear, with a spinodal region separating two material phases. We derive the traveling wave solutions governing the motion of an isolated phase boundary through the chain and obtain the functional relation between the driving force and the velocity of a phase boundary which can be used as the closing kinetic relation for the classical continuum theory. We show that a sufficiently wide spinodal region substantially alters the phase boundary kinetics at low velocities and results in a richer solution structure, with phase boundaries emitting short-length lattice waves in both direction. Numerical simulations suggest that solutions of the Riemann problem for the discrete system converge to the obtained traveling waves near the phase boundary.     

A lower bound for a variational model for pattern formation in shape-memory alloys

The Kohn-Müller model for the formation of domain patterns in martensitic shape-memory alloys consists in minimizing the sum of elastic, surface and boundary energy in a simplified scalar setting, with a nonconvex constraint representing the presence of different variants. Precisely, one minimizes

Blowup of solutions for the planar motions of rotating nonlinearly elastic rods

This paper treats the motion of flexible, extensible, shearable nonlinearly elastic rods, described by a geometrically exact theory, when they are confined to a plane rotating about a fixed axis at constant angular speed and when they are confined to a fixed plane with one end rotating at a constant angular speed about an axis perpendicular to the fixed plane. The paper gives restrictions on the constitutive equations and initial conditions that ensure that motions become unbounded at rapid rates as time becomes infinite. The analysis of these constitutive restrictions employs the theory of characteristics for single first-order semilinear partial differential equations.     

Mathematical model of creep for a microinhomogeneous nonlinearly elastic material

A mathematical model is proposed to describe the experimentally observed creep effect on the instantaneous elastic deformation of physically nonlinear elastic microinhomogeneous materials. Using a structural model of the medium, it is shown that, during unloading of a sample after creep at constant stress, the elastic strain can be both larger and smaller than the elastic strain during loading. It is shown that calculation results for a biocomposite material are in good agreement with experimental data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 157–163, May–June, 2008.     

An asymptotic non-linear model for thin-walled rods with strongly curved open cross-section

In this paper, we present a non-linear one-dimensional model for thin-walled rods with open strongly curved cross-section, obtained by asymptotic methods. A dimensional analysis of the non-linear three-dimensional equilibrium equations lets appear dimensionless numbers which reflect the geometry of the structure and the level of applied forces. For a given force level, the order of magnitude of the displacements and the corresponding one-dimensional model are deduced by asymptotic expansions.     

Two-phase flow analysis based on a phase-field model using orthogonal basis bubble function finite element method

In this study, a finite element method based on a phase-field model for gas–liquid two-phase flow is proposed. MINI element based on a bubble function element stabilisation method is employed for the incompressible Navier–Stokes equations. The Cahn–Hilliard equation is employed to estimate the interface of gas and liquid. The orthogonal basis bubble function element is used to solve the Cahn–Hilliard equation. In particular, a detailed explanation for solving the Cahn–Hilliard equation based on a finite element method is given.     

Analytical procedure for determining the linear and nonlinear effective properties of the elastic composite cylinder

In this work we consider a cylindrical structure composed of a nonlinear core (inhomogeneity) surrounded by a different nonlinear shell (matrix). We elaborate a technique for determining its linear elastic moduli (second order elastic constants) and the nonlinear elastic moduli, which are called Landau coefficients (third order elastic constants). Firstly, we develop a nonlinear perturbation method which is able to turn the initial nonlinear elastic problem into a couple of linear problems. Then, we prove that only the solution of the first linear problem is necessary to calculate the linear and nonlinear effective properties of the heterogeneous structure. The following step consists in the exact solution of such a linear problem by means of the complex elastic potentials. As result we obtain the exact closed forms for the linear and nonlinear effective elastic moduli, which are valid for any volume fraction of the core embedded in the external shell.     

On the wedge dispiration in an inhomogeneous isotropic nonlinear elastic solid

In this short note we find the stress field of a wedge dispiration (a combination of a screw dislocation and a wedge dislocation along the same line) in an inhomogeneous incompressible isotropic nonlinear solid. We discuss the effect of the radial inhomogeneity of energy function on both the stress field and the energy per unit length of the dispiration and compare with those in an isotropic linear elastic solid.     

Unsteady one-dimensional water and steam flows through a porous medium with allowance for phase transitions

Fluid flow through a porous medium is considered with allowance for heat conduction and phase transition processes. The one-dimensional problem of the breakdown of an arbitrary discontinuity is solved with reference to the processes of combined nonisothermal water and steam flow through the porous medium. It is assumed that there are two-phase zones of water and steam flow through the porous medium to the left and right of the initial discontinuity. Six qualitatively different discontinuous solutions with internal single-phase water or steam zones are constructed and domains corresponding to each of the solutions are found in the determining parameter space. For the parameters considered a solution of the breakdown problem exists and is unique when the requirements for the existence of a discontinuity structure are satisfied [{xc1}].