Computational homogenization of nano‐materials accounting for size effects via surface elasticity

Abstract

The objective of this contribution is to establish a first-order computational homogenization framework for micro-to-macro transitions of porous media that accounts for the size effects through the consideration of surface elasticity at the microscale. Although the classical (firstorder) homogenization schemes are well established, they are not capable of capturing the well-known size effects in nano-porous materials. In this contribution we introduce surface elasticity as a remedy to account for size effects within a first-order homogenization scheme. This proposition is based on the fact that surfaces are no longer negligible at small scales.Following a standard first-order homogenization ansatz on the microscopic motion in terms of the macroscopic motion, a Hill-type averaging condition is used to link the two scales. The averaging theorems are revisited and generalized to account for surfaces. In the absence of surface energy this generalized framework reduces to classical homogenization. The influence of the length scale is elucidated via a series of numerical examples performed using the finite element method. The numerical results are compared against the analytical ones at small strains for tetragonal and hexagonal microstructures. Furthermore, numerical results at small strains are compared with those at finite strains for both microstructures. Finally, it is shown that there exists an upper bound for the material response of nano-porous media. This finding surprisingly restricts the notion of "smaller is stronger".

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

“…This approach is briefly discussed in Appendix B. Strain-driven homogenization is, however, more common as it captures the softening response of the material similar to a displacement-control algorithm. Some of the contents presented in this section bear certain similarities to those proposed in our recent contribution [252], however, for computational homogenization accounting for surface energies. Admittedly, some relations such as macrodeformation gradient M F assume the same format whether or not surface contributions are considered.…”

“…For instance, the format of the macro Piola stress M P as well as the Hill-Mandel condition given in Ref. [252] differ from their classical definitions here. In this section, fundamental definitions and concepts of the theory of computational homogenization are briefly addressed in order for this paper to be self-contained.…”

“…Furthermore, second-order computational homogenization introduces a physical length to the microscale that is missing in the first-order homogenization. It is also possible to formulate a first-order computational homogenization accounting for size effects by taking surface energies at the microscale into account [251,252]. 3 This is justified by the fact that due to the large surface-to-volume ratio at smaller scales, surface contributions to the overall response of the material are no longer negligible at the microscale.…”

“…The extension of the relations and proofs presented in this section to account for surface energies is given in Ref. [252].…”

Bounds on size-dependent behaviour of composites

“…The influence of elastic surfaces using PBC on the response of the material in the context of the computational homogenisation and large deformations has been recently studied [73,74]. Utilising the periodic boundary conditions is commonly justified by the fact that they often produce the most intermediate results as compared to other boundary conditions.…”