Bounds on size-dependent behaviour of composites

Abstract

Computational homogenisation is a powerful strategy to predict the effective behaviour of heterogeneous materials. While computational homogenisation cannot exactly compute the effective parameters, it can provide bounds on the overall material response. Thus, central to computational homogenisation is the existence of bounds. Classical firstorder computational homogenisation cannot capture size effects. Recently, it has been shown that size effects can be retrieved via accounting for elastic coherent interfaces in the microstructure. The primary objective of this contribution is to present a systematic study to attain computational bounds on the sizedependent response of composites. We show rigorously that interface-enhanced computational homogenisation introduces two relative length scales into the problem and investigate the interplay between them. To enforce the equivalence of the virtual power between the scales, a generalised version of the Hill-Mandel condition is employed, and accordingly, suitable boundary conditions are derived. Macroscopic quantities are related to their microscopic counterparts via extended average theorems. Periodic boundary conditions provide an effective behaviour bounded by traction and displacement boundary conditions. Apart from the bounds due to boundary conditions for a given size, the size-dependent response of a composite is bounded, too. The lower bound coincides with that of a composite with no interface. Surprisingly, there also exists an upper bound on the size-dependent response beyond which the expected 'smaller is stronger' trend is no longer observed. Finally, we show an excellent agreement between our numerical results and the corresponding analytical solution for linear isotropic materials which highlights the accuracy and broad applicability of the presented scheme.

Stochastic local FEM for computational homogenization of heterogeneous materials exhibiting large plastic deformations

“…The recently introduced novel weak PBC [46] and other advanced techniques still belong to one of the three classical types of BC from this perspective. It is well-known that the DBC overestimate stresses, while the TBC underestimate them [6,44,45,48,55,56]. The most reliable results even in case of random materials are obtained using the PBC, which is however often criticized in literature since random samples are not periodic.…”

“…Firstly, we need to design a suitable model of the microstructure and transfer macroscopic loads and deformations to the microscale. This is done using the Hill-Mandel [14,23,44,45,54] condition which states that the virtual work performed by microscopic stresses at microscopic deformation gradients must be equal to the virtual work performed by macroscopic (homogenized) stresses at macroscopic (homogenized) deformation gradients. The second step is to simulate the response of the micromodel.…”

Fuzzy‐stochastic FEM–based homogenization framework for materials with polymorphic uncertainties in the microstructure

“…The use of periodic boundary conditions for the statistically similar (substitute) model is motivated by a number of studies demonstrating that the periodic boundary conditions are the most reliable and converge faster than Dirichlet and Neumann boundary conditions . They are often used even if the model is not periodic, because Dirichlet and Neumann boundary conditions always give an overestimation and an underestimation for the stress in computational homogenization. Moreover, the simple comparison of the homogenized stresses in periodic composites modeled with only one inclusion in the unit cell and composites with a random microstructure, performed in the work of Zabihyan et al, demonstrated close homogenized stress values in both models.…”

Understanding the role of general interfaces in the overall behavior of composites and size effects

“…A major shortcoming of the classical computational homogenization is that it fails to account for size-dependent material behavior, often referred to as size effects. On the other hand, the size effects in composites are essentially attributed to surface [31] and interface effects [32,33] leading to significant properties of nano-materials due to their pronounced area-to-volume ratio, see [34][35][36][37][38] among others. Therefore, it is important to extend the computational homogenization method to account for the interfaces between the constituents of a micro-structure so as to capture the size effects.…”