Abstract
We derive the Green tensor of Mindlin’s anisotropic first strain gradient elasticity. The Green tensor is valid for arbitrary anisotropic materials, with up to 21 elastic constants and 171 gradient elastic constants in the general case of triclinic media. In contrast to its classical counterpart, the Green tensor is non-singular at the origin, and it converges to the classical tensor a few characteristic lengths away from the origin. Therefore, the Green tensor of Mindlin’s first strain gradient elasticity can be regarded as a physical regularization of the classical anisotropic Green tensor. The isotropic Green tensor and other special cases are recovered as particular instances of the general anisotropic result. The Green tensor is implemented numerically and applied to the Kelvin problem with elastic constants determined from interatomic potentials. Results are compared to molecular statics calculations carried out with the same potentials.
Highlights
Green functions are objects of fundamental importance in field theories, since they represent the fundamental solution of linear inhomogeneous partial differential equations (PDEs) from which any particular solution can be obtained via convolution with the source term (Green 1828)
The Green tensor is found in terms of a matrix kernel integrated over the unit sphere in Fourier space
Because the Green tensor regularizes its classical counterpart without unphysical singularities, it offers a more realistic description of near-core elastic fields of defects in micro-mechanics, and it provides more accurate boundary conditions for atomistic and ab-initio energy-minimization calculations
Summary
Green functions are objects of fundamental importance in field theories, since they represent the fundamental solution of linear inhomogeneous partial differential equations (PDEs) from which any particular solution can be obtained via convolution with the source term (Green 1828). While simple expressions of the Green tensor exist for the isotropic case (Rogula 1973; Lazar and Po 2018), and for simplified anisotropic theories (Lazar and Po 2015a, b), the Green tensor of the fully anisotropic theory of Mindlin’s strain gradient elasticity has remained so far a rather elusive object. Rogula (1973) provided an expression for the Green tensor in gradient elasticity of arbitrary order, which involves a sum of terms associated with the roots of a certain characteristic polynomial Such representation renders its numerical implementation rather impractical, and it conceals the mathematical structure of the Green tensor in relationship to its classical counterpart. The objective of this paper is to derive a simple representation of the Green tensor of Mindlin’s anisotropic first strain gradient elasticity, whose integral kernel involves only matrix operations suitable for efficient numerical implementation. In the linearized theory of Mindlin’s form-II first strain gradient elasticity (Mindlin 1964; 1968b; Mindlin and Eshel 1968a; Mindlin 1972), the strain energy density of an homogeneous and centrosymmetric[2] material is given by (2019) 3:3
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
