The paper presents a new procedure to construct micro-mechanical damage models able to describe size effects in solids. The new approach is illustrated in the case of brittle materials. We use homogenization based on two-scale asymptotic developments to describe the overall behavior of a damaged elastic body starting from an explicit description of elementary volumes with micro-cracks. An appropriate micro-mechanical energy analysis is proposed leading to a damage evolution law that incorporates stiffness degradation, material softening, size effects, unilaterality, different fracture behaviors in tension and compression, induced anisotropy. The model also accounts for micro-crack nucleation and growth.Finite element solutions for some numerical tests are presented in order to illustrate the ability of the new approach to describe known behaviors, like the localization of damage and size-dependence of the structural response. Based on a correct micro-mechanical description of the energy dissipation associated with failure, the model avoids significant mesh dependency for the localized damage finite element solutions.

In this work, a mathematical modeling of the elastic properties of cubic crystals with centrosymmetry at small scales by means of the Toupin–Mindlin anisotropic first strain gradient elasticity theory is presented. In this framework, two constitutive tensors are involved, a constitutive tensor of fourth-rank of the elastic constants and a constitutive tensor of sixth-rank of the gradient-elastic constants. First, $$3+11$$ 3 + 11 material parameters (3 elastic and 11 gradient-elastic constants), 3 characteristic lengths and $$1+6$$ 1 + 6 isotropy conditions are derived. The 11 gradient-elastic constants are given in terms of the 11 gradient-elastic constants in Voigt notation. Second, the numerical values of the obtained quantities are computed for four representative cubic materials, namely aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using an interatomic potential (MEAM). The positive definiteness of the strain energy density is examined leading to 3 necessary and sufficient conditions for the elastic constants and 7 ones for the gradient-elastic constants in Voigt notation. Moreover, 5 lattice relations as well as 8 generalized Cauchy relations for the gradient-elastic constants are derived. Furthermore, using the normalized Voigt notation, a tensor equivalent matrix representation of the two constitutive tensors is given. A generalization of the Voigt average toward the sixth-rank constitutive tensor of the gradient-elastic constants is given in order to determine isotropic gradient-elastic constants. In addition, Mindlin’s isotropic first strain gradient elasticity theory is also considered offering through comparisons a deeper understanding of the influence of the anisotropy in a crystal as well as the increased complexity of the mathematical modeling.

In the present paper we investigate conservation and balance laws in the framework of linear elastodynamics considering the strain energy density depending on the gradients of the displacement up to the third order, as originally proposed by Mindlin (Int. J. Solids Struct. 1, 417-438, 1965). The conservation and balance laws that correspond to the symmetries of translation, rotation, scaling and addition of solutions are derived using Noether's theorem. Also, the formulas of the dynamical J , L and M-integrals are presented for the problem under study. Moreover, the balance law of addition of solutions gives rise to explore the dynamical reciprocal theorem as well as the restrictions under which it is valid.

A theoretical framework for dislocation dynamics in quasicrystals is provided according to the continuum theory of dislocations. Firstly, we present the fundamental theory for moving dislocations in quasicrystals giving the dislocation density tensors and introducing the dislocation current tensors for the phonon and phason fields, including the Bianchi identities. Next, we give the equations of motion for the incompatible elastodynamics as well as for the incompatible elasto-hydrodynamics of quasicrystals. We continue with the derivation of the balance law of pseudomomentum thereby obtaining the generalized forms of the Eshelby stress tensor, the pseudomomentum vector, the dynamical Peach-Koehler force density and the Cherepanov force density for quasicrystals. The form of the dynamical Peach-Koehler force for a straight dislocation is obtained as well. Moreover, we deduce the balance law of energy that gives rise to the generalized forms of the field intensity vector and the elastic power density of quasicrystals. The above balance laws are produced for both models. The differences between the two models and their consequences are revealed. *

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