Effective Potentials and Elastic Properties in the Lattice-Element Method: Isotropy and Transverse Isotropy

Abstract

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., 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., Affes et al.

Disorder-induced stiffness degradation of highly disordered porous materials

“…Much akin to Potential of Mean Force (PMF) approaches used in Soft Matter Physics (see e.g. Masoero et al, 2012 ), these mass points interact with their nearest neighbors through effective interaction potentials ( Laubie et al, 2017a ). In this approach ( Fig.…”

“…Herein l 0 i j = r i j (with r i j = x j − x i = l 0 i j e n ) is the distance between solid mass points i and j in the reference configuration, while the solid's energy parameters n,t i j ∼ a 3 0 E s are calibrated to recover the desired effective (or macroscopic) elastic behavior of the (homogeneous) solid phase ( ϕ = 0 ; with Young's modulus E s and Poisson's ratio ν s ), following the procedure outlined in Laubie et al (2017a ). If only the stretch term is considered (setting t i j = 0 ), the value of the Poisson's ratio of the composite is defined by the geometric limit value of the cubic lattice; that is ν lim = 1 / (D + 1) (with D the space dimension).…”

A potential-of-mean-force approach for fracture mechanics of heterogeneous materials using the lattice element method

“…The starting point of this approach is the realization that LEM can be viewed as a potential-of-mean-force approach (PMF) akin to the one employed by the soft-matter physics community: a number of mass points defined on a regular or irregular lattice interact via effective potentials from which forces and moments derive. This PMF approach to LEM was recently proposed for elastic systems ( Laubie et al, 2017a;2017b ), showing that the elasticity in the PMF context is but an evaluation of the energy content of the system around the equilibrium state defined by the lattice structure, for which most non-harmonic potentials degenerate to harmonic potentials commensurable to the original truss-beam type formulation used in classical LEM approaches. On the other hand, the PMF approach puts the LEM on the same footing as molecular approaches thus permitting to employ the canon of statistical physics, such as thermodynamic ensemble definitions, to extend the LEM approach as a tool of solid mechanics to poromechanics ( Monfared et al, 2017 ).…”

Mesoscale Poroelasticity of Heterogeneous Media

“…(see (Laubie et al 2017b)). Lastly, it is readily recognized that ì e n ⊗ ì e n is the fabric tensor, H p ,…”

“…C is the fourth-order elastic stiffness tensor, b is the second-order tensor of Biot ì ϑ i , degrees of freedom, where ì X i and ì x i denote the position vectors of mass point i in the reference 66 and the deformed configurations, respectively (for a detailed derivation, see (Laubie et al 2017b)):…”