A continuum-molecular model for anisotropic electrically conductive materials

Anisotropic Continuum-Molecular Models: A Unified Framework Based on Pair Potentials for Elasticity, Fracture and Diffusion-Type Problems

“…where, under the assumed conditions, the pairwise deformations s( ) and ( ) of the generic virtual fiber can be expressed as linear functions of the homogeneous macro strain components ij and quadratic functions of the direction cosines of the orthonormal unit vectors = n i i and = t i i (as previously defined), by which are obtained through the Cauchy-Born rule [87,88], particularized to our case 4 , and according to ε = 𝜺 , where is given by Eq. 44 [72,76]. Hence, one obtains Substituting Eq.…”

“…In non-local models of this kind, the representative cell is the horizon region H over the integral in Eq. 11 is defined, while the micro-macro moduli correspondence principle is based on a micro-macro energy equivalence, which allows macroscopic properties to be directly associated to microscale-defined constitutive laws [72]. Since in-plane Cauchy anisotropic linear elasticity is defined by six independent elastic moduli, it follows that k n ( ) and k t ( ) must be functions of at least six independent microelastic constants.…”

“…Due to the central force assumption and the directional dependency laws adopted, two of the four elastic moduli which define orthotropic in-plane elasticity have to be preset, whereas square-symmetric materials [70] cannot be modeled. A mechanical continuum-molecular formulation based on pair potentials without restrictions in the number of material constants was proposed later by Diana et al [40,71,72].…”

“…The latter analyzed the problem of anisotropic heat transfer focusing on the micro-conductivity function definition without making particular assumptions about the nature of the interactions. Recently, Diana and Carvelli have derived an analytical peridynamic formulation for anisotropic materials accounting for their anisotropic overall conductivity and fracture properties, together with a general procedure for the identification of the model parameters [72]. For the best of the author's knowledge, other anisotropic models based on pair potentials with similar features have not been proposed so far.…”

“…The proposed method is validated against analytical solutions and experimental data, when available. By summarizing the work that we have been doing on this topic [40,71,72,76], the manucript introduces a general approach to anisotropic problems that can be used for further developments of continuum-molecular formulations to include other mechanical behaviors and coupled phenomena involving different physics. The proposed model couples the intuitive simplicity of the concept of mutual interaction in molecular models based on pair potentials (removing Cauchy relations restrictions), with the (modern) mathematical formalism of a peridynamic continuum formulation for anisotropic materials.…”