In this paper, we compare a biomechanics empirical model of the heart fibrous structure to two models obtained by a non-periodic homogenization process. To this end, the two homogenized models are simplified using the small amplitude homogenization procedure of Tartar, both in conduction and in elasticity. A new small amplitude homogenization expansion formula for a mixture of anisotropic elastic materials is also derived and allows us to obtain a third simplified model.

We present a general numerical method for calculating effective elastic properties of periodic structures based on the homogenization method. Some concrete numerical examples are presented.

We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with natoms where $y\in {ℝ}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy asn tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: ${\min }_y{E}^{\left(n\right)}\left(y\right)=n{E}_{\mathrm{bulk}}+\sqrt{n}{E}_{\mathrm{surface}}+o\left(\sqrt{n}\right),n\to \infty .$ The bulk energy densityEbulk is given by an explicit expression...

We study an atomistic pair potential-energy E(n)(y) that describes the elastic behavior of two-dimensional crystals with n atoms where $y\in {ℝ}^{2\times n}$ characterizes the particle positions. The main focus is the asymptotic analysis of the ground state energy as n tends to infinity. We show in a suitable scaling regime where the energy is essentially quadratic that the energy minimum of E(n) admits an asymptotic expansion involving fractional powers of n: ${\min }_y{E}^{\left(n\right)}\left(y\right)=n{E}_{\mathrm{bulk}}+\sqrt{n}{E}_{\mathrm{surface}}+o\left(\sqrt{n}\right),n\to \infty .$ The bulk energy density Ebulk is given by an explicit expression...