Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity

Abstract

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.

On the elastic wedge problem within simplified and incomplete strain gradient elasticity theories

“…It is known that typical values of the length scale parameters of single crystals have the order of interatomic distance, i.e. within SSGET we have 𝑙 = 𝑙 1 = 𝑙 2 ≈ 1 Å (Lazar et al, 2021). Therefore, if the loading is applied at the atomically sharp edge of single crystal and provides the maximum remote tensile stress equals to 𝜏 11 = 1 MPa at distance 𝐿 = 1000𝑙 = 100 nm from the edge tip.…”

“…Components of the constitutive tensors 𝐂 and 𝐀 within Mindlin Form II can be defined as follows (dell'Isola et al, 2009;Lazar et al, 2021):…”

“…where 𝜆, 𝜇 are the classical Lame constants and 𝑎 𝑖 (i=1...5) are the additional material constants of gradient theory, that can be found, e.g. based on the interatomic potentials or via ab initio calculations (Lazar et al, 2021); 𝛿 𝑖𝑗 is the Kronecker delta.…”

On well‐posedness of the first boundary‐value problem within linear isotropic Toupin–Mindlin strain gradient elasticity and constraints for elastic moduli

“…In what follows we restrict ourselves to an isotropic behavior, so $$\mathbb {E}$$ is zero, whereas $$\mathbb {C}$$ and $$\mathbb {D}$$ have the following representation [3, 25] $$$$\begin{equation} \mathbb {C} \eqcellsep =\eqcellsep \mathbb {C}_{ijkl}\mathbf {i}_i\otimes \mathbf {i}_j\otimes \mathbf {i}_k\otimes \mathbf {i}_l,\quad \mathbb {D} =\mathbb {D}_{ijmkln}\mathbf {i}_i\otimes \mathbf {i}_j\otimes \mathbf {i}_m\otimes \mathbf {i}_k\otimes \mathbf {i}_l\otimes \mathbf {i}_n, \eqbreak \end{equation}$$$$ $$$$\begin{equation} \mathbb {C}_{ijkl}\eqcellsep =\eqcellsep\lambda \delta _{ij}\delta _{kl} +\mu (\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}), \end{equation}$$$$ …”

Self-consistent assessments for the effective properties of two-phase composites within strain gradient elasticity

“…Let us assume that the gradient moduli of crystallites are very small (G gr ≈ 0) and has spherical shape. The first assumption is realistic and it bases on the known results of atomistic calculations for the gradient moduli of monocrystals, for which it was found that the the length scale parameters of SGET has sub-nanometer order [38,39]. The assumption about spherical shape of grains is just an approximation of the model that is widely used within classical micromechanics [2,33].…”