Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part II: Higher-order constitutive properties and application cases

Abstract

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i.) is positive definite only when the discrepancy tensor is negative defined; (ii.) the non-local material symmetries are the same of the discrepancy tensor, and (iii.) the nonlocal effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthortropic matrix.

Identification of second-gradient elastic materials from planar hexagonal lattices. Part II: Mechanical characteristics and model validation

“…showing that only the first three eigenvectors have non-null components in both the strain and the curvature, while t * [4] and t * [5] have non-null components only in the curvature, and t * [6] and t * [7] only in the strain. Note that the non-null curvature components of the vectors t * [4] and t * [5] coincide with the components of vectors q * [2] and q * [4] , respectively. The seven eigenvectors (45) are rotated of an angle θ ∈ [0, 2π] to analyze the directional properties of the energy density (for a material equivalent to a lattice with stiffness ratios, k/k = 10, k/k = 100), plotted as polar diagrams in Fig.…”

A contact problem in couple stress thermoelasticity: The indentation by a hot flat punch

“…The need for such generalized continuum models has also been verified through experimental (Lakes et al, 1985;Beveridge et al, 2013) and theoretical (Smishlayev and Fleck, 1995;Bigoni and Drugan, 2007;Bacca et al, 2013a;2013b;Bacigalupo, 2013) approaches.…”

Identification of second-gradient elastic materials from planar hexagonal lattices. Part I: Analytical derivation of equivalent constitutive tensors

“…The key for the identification procedure performed in the next Section is the imposition to the infinite lattice of a linear and a quadratic nodal displacement fields (as in [3], [4], [7], [10]), together with an 'additional field' ∆u (m,n|i) , namely,…”

“…Therefore, from Eq. (18), the elastic energy of the lattice can be expressed as U (m,n) lat (a, b * ) and therefore can be represented as the following quadratic form in a and b * U (m,n) lat (a, b * ) = 2 a · H [1] k, k, k a + 2 a · mH [2] k, k, k + nH [3] k, k, k + H [4] k, k, k b * + 2 b * · m 2 H [5] k, k, k + n 2 H [6] k, k, k + m nH [7] k, k, k + mH [8] k, k, k + +nH [9] k, k, k + H [10]…”

“…81I [3] 2I [1] I [2] − 9I [3] 32I [1] I 2 [2] , H [3] 33 = 27 √ 3I [3] 2I [1] I [2] − 9I [3] 32I [1] I 2 [2] , H 3 k − k I [1] I [2] − 9I [3] k k − 2k k + k 16I [1] I 2 [2] , H [4] 14 = − 9 k − k I [1] I [2] + 3I [3] k k − 2k k + k 16I [1] I 2 [2] , H 3 k − k I [1] I [2] − 9I [3] k k − 2k k + k 16I [1] I 2 [2] , H [4] 24 = − 9 k − k I [1] I [2] + 3I [3] 81I [3] 2I [1] I [2] + 9I [3] 8I [1] I [2] + 9I [3] 16I 2 [1] I 3 [2] , H [5] 12 = 2187I 2 [3] 2I [1] I [2] + 3I [3] 16I 2 [1] I 3 [2] , H 81I [3] 2I [1] I …”