Effective medium theory for second-gradient nonlinear elasticity with chirality

Abstract

We derive effective, parsimonious models from a heterogeneous second-gradient nonlinear elastic material taking into account chiral scale-size effects. Our classification of the effective equations depends on the hierarchy of four characteristic lengths: The size of the heterogeneities ℓ, the intrinsic lengths of the constituents ℓ SG and ℓ chiral , and the overall characteristic length of the domain L. Depending on the different scale interactions between ℓ SG , ℓ chiral , ℓ, and L we obtain either an effective Cauchy continuum or an effective second-gradient continuum. The working technique combines scaling arguments with the periodic homogenization asymptotic procedure. Both the passage to the homogenization limit and the unveiling of the correctors' structure rely on a suitable use of the periodic unfolding and related operators.

On a hierarchy of effective models for the biomechanics of human compact bone tissue

“…It is still not clear how the two approximations are related. We simply point out in the Cosserat context we have two local problems (5.10) and (5.11), while in the second-gradient context there in only one (see (4.11) in [29]).…”

“…As a last remark, we would like to draw the readers attention to another way of approximating a linear elastic material using a different theory of generalized continua, namely, second-gradient elasticity (see [29]). It is still not clear how the two approximations are related.…”

A gradient system for a higher-gradient generalization of Fourier’s law of heat conduction

“…We will not address such a case here. We refer the reader to [12] for different type of scalings in the context of non-simple materials. Therefore, dropping the star notation for expediency, and introducing the parameter ε on the temperature u to show the dependence of the the solution on ε, equation (4.3) becomes,…”