Discrete Free Energy Functionals for Elastic Materials with Phase Change (original) (raw)
Related papers
On the Variational Limits of Lattice Energies on Prestrained Elastic Bodies
Springer Proceedings in Mathematics & Statistics, 2015
We study the asymptotic behaviour of the discrete elastic energies in presence of the prestrain metric G, assigned on the continuum reference configuration Ω. When the mesh size of the discrete lattice in Ω goes to zero, we obtain the variational bounds on the limiting (in the sense of Γ-limit) energy. In case of the nearest-neighbour and next-to-nearest-neighbour interactions, we derive a precise asymptotic formula, and compare it with the non-Euclidean model energy relative to G.
A General Theory for Elastic Phase Transitions in Crystals
Analysis, Modeling and Simulation of Multiscale Problems
We derive a general theory for elastic phase transitions in solids subject to diffusion under possibly large deformations. After stating the physical model, we derive an existence result for measure-valued solutions that relies on a new approximation result for cylinder functions in infinite settings.
A derivation of linear elastic energies from pair-interaction atomistic systems
2007
Pair-interaction atomistic energies may give rise, in the framework of the passage from discrete systems to continuous variational problems, to nonlinear energies with genuinely quasiconvex integrands. This phenomenon takes place even for simple harmonic interactions as shown by an example by Friesecke and Theil . On the other hand, a rigorous derivation of linearly elastic energies from energies with quasiconvex integrands can be obtained by Γ-convergence following the method by Dal Maso, Negri and Percivale . We show that the derivation of linear theories by Γ-convergence can be obtained directly from lattice interactions in the regime of small deformations. Our proof relies on a lower bound by comparison with the continuous result, and on a direct Taylor expansion for the upper bound. The computation is carried over for a family of lattice energies comprising interactions on the triangular lattice in dimension two.
European Journal of Physics, 2011
The requirement of rotational invariance for lattice potential energies is investigated. Starting from this condition, it is shown that the Cauchy relations for the elastic constants are fulfilled if the lattice potential is built from pair interactions or when the first-neighbour approximation is adopted. This is seldom recognized in widely used solid-state textbooks. Frequently, pair interaction is even considered to be the most general situation. In addition, it is shown that the demand of rotational invariance in an infinite crystal leads to inconsistencies in the symmetry of the elastic tensor. However, for finite crystals, no problems arise, and the Huang conditions are deduced using exclusively a microscopic approach for the elasticity theory, without making any reference to macroscopic parameters. This work may be useful in both undergraduate and graduate level courses to point out the crudeness of the pair-potential interaction and to explore the limits of the infinite-crystal approximation.
Linear elastic deformation of the two-dimensional triangular lattice with multiple vacancies is considered. Closed-form analytical expressions for displacement field in the lattice with doubly periodic system of vacancies are derived. Effective elastic moduli are calculated. The results are compared with the ones obtained by molecular dynamics simulations of a lattice with random distribution of vacancies. At low vacancy concentrations, less than 4%, random and periodic distributions of vacancies produce the same effect on elastic moduli. One of the main goals is to examine the possibilities and limitations of modelling of the lattice with vacancies by an elastic continuum with holes. It is found that the effective elastic properties are modelled adequately, provided the shape of the holes is chosen appropriately. On the contrary, the strain field, in particular, strain concentration differs significantly.
Finite-scale microstructures and metastability in one-dimensional elasticity
Meccanica, 1995
This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy [1]. Following some previous work in this area, we suitably regularize the problem in order to investigate this degenerancy. Our approach gives an explicit framework for the the study of the rich variety of finite-scale equilibrium microstructures for the bar in a hard loading device, and their stability properties. In this way we clarify the role of interracial energy in creating finitescale microstructures, by considering the combined effect of the oscillation-inducing and oscillation-inhibiting terms in the energy functional. Sommario. 11 lavoro riguarda la non unicit~ messa in luce da J.L. Ericksen nella sua analisi dell'equilibrio di barre elastiche con energia non convessa. Seguendo le linee di precedenti lavori, per investigate questa degenerazione si ricorre ad una regolarizzazione del problema e si d~t un esplicito quadro di riferimento per lo studio della ricca varieth delle microstrutture di scala finita e della loro stabilitY. Si chiarisce in particolare il ruolo dell'energia di interfaccia nella creazione di microstrutture di scala finita considerando l'effetto combinato di termini inibitori e favorevoli all'insorgere di oscillazioni nel funzionale energia.
Journal of Nanomechanics and Micromechanics
Lattice approaches have emerged as a powerful tool to capture the effective mechanical behavior of heterogeneous materials using harmonic interactions inspired from beam-type stretch and rotational interactions between a discrete number of mass points. In this paper, the lattice element method (LEM) is reformulated within the conceptual framework of empirical force fields employed at the lattice scale. Within this framework, because classical harmonic formulations are but a Taylor expansion of nonharmonic potential expressions, they can be used to model both the linear and the nonlinear response of discretized material systems. Specifically, closed-form calibration procedures for such interaction potentials are derived for both the isotropic and the transverse isotropic elastic cases on cubic lattices, in the form of linear relations between effective elasticity properties and energy parameters that define the interactions. The relevance of the approach is shown by an application to the classical Griffith crack problem. In particular, it is shown that continuum-scale quantities of linear-elastic fracture mechanics, such as stress intensity factors (SIFs), are well captured by the method, which by its very discrete nature removes geometric discontinuities that provoke stress singularities in the continuum case. With its strengths and limitations thus defined, the proposed LEM is well suited for the study of multiphase materials whose microtextural information is obtained by, e.g., X-ray micro-computed tomography. with the current understanding of the link between texture (here lattice) and the deformation behavior of materials (Greaves et al. 2011). In order to overcome this limitation, several authors suggested the addition of beam-type interactions between mass points in 2D (e.g., Schlangen and Garboczi 1996, 1997; Bolander and Saito 1998) and 3D with or without rotational degrees of freedom (Zhao et al. 2011), with up to 178 interactions for each node in the (random lattice) system (Lilliu et al. 1999; Lilliu and van Mier 2003). While the preceding approaches allowed removing some of the earlier limitations of the central-force model, a search of the relevant literature was not conclusive in finding a rational framework that clearly defines the different elements of the method, from the local interactions that link the lattice's mass points to the macroscopic properties of the assembly of links, which is, in short, the focus of this paper. Such a framework is needed though not only for elastic (i.e., reversible) phenomena, but also for extending the method to poroelasticity (Monfared et al. 2016) or dissipative phenomena, related to plastic deformation, fracture, and so on, for which the method is frequently applied (e.g.,