Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound - Saeb, Saba; Steinmann, Paul; Javili, Ali (Applied Mechanics Reviews, 2016)
The objective of this contribution is to present a unifying review on strain-driven computational homogenization at finite strains, thereby elaborating on computational aspects of the finite element method. The underlying assumption of computational homogenization is separation of length scales, and hence, computing the material response at the macroscopic scale from averaging the microscopic behavior. In doing so, the energetic equivalence between the two scales, the Hill–Mandel condition, is guaranteed via imposing proper boundary conditions such as linear displacement, periodic displacement and antiperiodic traction, and constant traction boundary conditions. Focus is given on the finite element implementation of these boundary conditions and their influence on the overall response of the material. Computational frameworks for all canonical boundary conditions are briefly formulated in order to demonstrate similarities and differences among the various boundary conditions. Furthermore, we detail on the computational aspects of the classical Reuss' and Voigt's bounds and their extensions to finite strains. A concise and clear formulation for computing the macroscopic tangent necessary for FE2 calculations is presented. The performances of the proposed schemes are illustrated via a series of two- and three-dimensional numerical examples. The numerical examples provide enough details to serve as benchmarks.
EAM Import::2019-03-13

Aspects of implementing constant traction boundary conditions in computational homogenization via semi-Dirichlet boundary conditions - Javili, Ali; Saeb, Saba; Steinmann, Paul (Computational Mechanics, 2016)
In the past decades computational homogenization has proven to be a powerful strategy to compute the overall response of continua. Central to computational homogenization is the Hill–Mandel condition. The Hill–Mandel condition is fulfilled via imposing displacement boundary conditions (DBC), periodic boundary conditions (PBC) or traction boundary conditions (TBC) collectively referred to as canonical boundary conditions. While DBC and PBC are widely implemented, TBC remains poorly understood, with a few exceptions. The main issue with TBC is the singularity of the stiffness matrix due to rigid body motions. The objective of this manuscript is to propose a generic strategy to implement TBC in the context of computational homogenization at finite strains. To eliminate rigid body motions, we introduce the concept of semi-Dirichlet boundary conditions. Semi-Dirichlet boundary conditions are non-homogeneous Dirichlet-type constraints that simultaneously satisfy the Neumann-type conditions. A key feature of the proposed methodology is its applicability for both strain-driven as well as stress-driven homogenization. The performance of the proposed scheme is demonstrated via a series of numerical examples.
EAM Import::2019-03-13

Bounds on size-dependent behavior of composites - Saeb, Saba; Steinmann, Paul; Javili, A. (Philosophical Magazine, 2017)
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 first-order 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 size-dependent 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.
EAM Import::2019-03-13

Designing tunable composites with general interfaces - Saeb, Saba; Steinmann, Paul; Javili, A. (International Journal of Solids and Structures, 2019)
In this manuscript, we employ interface enhanced computational homogenization to explore and detail on a number of unfamiliar characteristics that composites can exhibit at different length scales. Here, the interface between the constituents is general in the sense that both displacement and traction jumps across the interface are admissible. We carry out numerous computational investigations using the finite element method for a broad range of various material parameters. Our numerical results reveal that the effective response of a microstructure embedding general interfaces is intuitively unpredictable and highly complex. In particular, for certain ranges of material parameters the overall response shows insensitivity with respect to either microstructure size or stiffness-ratio between inclusion and matrix. This unique behavior is observed likewise for two- and three-dimensional unit-cells. Our findings provide a valuable guideline to design tunable composites utilizing interfaces.
CRIS-Team Scopus Importer:2019-05-02
Keywords: Composites; Computational modeling; Interface/interphase; Micro-mechanics; Size-effect; Smart materials

Generalized interfaces via weighted averages for application to graded interphases at large deformations - Saeb, Saba; Firooz, Soheil; Steinmann, Paul; Javili, A. (Journal of the Mechanics and Physics of Solids, 2021)
Finite-thickness interphases between different constituents in heterogeneous materials are often replaced by a zero-thickness interface model. Commonly accepted interface models intuitively assume that the interface layer is situated exactly in the middle of its associated interphase. Furthermore, it has been reported in the literature that this assumption is necessary to guarantee the balance of angular momentum on the interface. While the interface coincides with the mid-layer of a uniform interphase, we argue that this assumption fails to sufficiently capture the behavior of graded or inhomogeneous interphases. This contribution extends the formulation of the general interface model to account for arbitrary interface positions. The issue of angular momentum balance on general interfaces is critically revisited. It is proven that the interface position does not necessarily have to coincide with the mid-layer in order to satisfy the angular momentum balance. The analysis here leads to a unique definition of the controversially discussed interface configuration. The presented general interface model is essentially based upon the weighted average operator instead of the commonly accepted classical average operator. The framework is geometrically exact and suitable for finite deformations. The significance of the interface position is demonstrated via a series of examples where the interface position is identified based on a full resolution interphase.
CRIS-Team Scopus Importer:2021-02-05
Keywords: Cohesive interface; General interface model; Interface elasticity; Interphase; Weighted average

On effective behavior of microstructures embedding general interfaces with damage - Saeb, Saba; Steinmann, Paul; Javili, A. (Computational Mechanics, 2019)
The interface between constituents of a multiphase material exhibits properties different from those of the bulk and can lead to major alternation of the material response. Interface effects are particularly important for multiphase nano-materials where the area-to-volume ratio is significantly large. In this contribution, we study the influence of a degrading general interface. That is, we allow for the initiation and accumulation of damage on a generalized interface accounting for both jumps of the displacement and the traction across the interface. The applicability of the proposed framework is demonstrated through several numerical examples. We present a parametric study on the influence of a broad range of interface material parameters on the overall behavior of various microstructures subject to volumetric loading and unloading. The numerical results illustrate that the resistance along the interface plays a key role in the resulting damage mechanism and could potentially prevent the detachment of the inclusion from the matrix regardless of the resistance across the interface or bulk material parameters. This behavior is observed and shown for both two- and three-dimensional examples. Moreover, the size-effect due to the general interface model is examined and compared against other interface models. Finally, the influence of the boundary conditions on the effective response and damage initiation of several microstructures is studied.
CRIS-Team Scopus Importer:2019-06-04
Keywords: Cohesive interface; Computational homogenization; Elastic interface; Finite deformation; General interface

Systematic study of homogenization and the utility of circular simplified representative volume element - Firooz, Soheil; Saeb, Saba; Chatzigeorgiou, George; Meraghni, Fodil; Steinmann, Paul; Javili, Ali (Mathematics and Mechanics of Solids, 2019)
Although both computational and analytical homogenization are well-established today, a thorough and systematic study to compare them is missing in the literature. This manuscript aims to provide an exhaustive comparison of numerical computations and analytical estimates, such as Voigt, Reuss, Hashin–Shtrikman, and composite cylinder assemblage. The numerical computations are associated with canonical boundary conditions imposed on either tetragonal, hexagonal, or circular representative volume elements using the finite-element method. The circular representative volume element is employed to capture an effective isotropic material response suitable for comparison with associated analytical estimates. The analytical results from composite cylinder assemblage are in excellent agreement with the numerical results obtained from a circular representative volume element. We observe that the circular representative volume element renders identical responses for both linear displacement and periodic boundary conditions. In addition, the behaviors of periodic and random microstructures with different inclusion distributions are examined under various boundary conditions. Strikingly, for some specific microstructures, the effective shear modulus does not lie within the Hashin–Shtrikman bounds. Finally, numerical simulations are carried out at finite deformations to compare different representative volume element types in the nonlinear regime. Unlike other canonical boundary conditions, the uniform traction boundary conditions result in nearly identical effective responses for all types of representative volume element, indicating that they are less sensitive with respect to the underlying microstructure. The numerical examples furnish adequate information to serve as benchmarks.
CRIS-Team Scopus Importer:2019-06-04
Keywords: circular representative volume element; composite cylinder assemblage; composites; Hashin–Shtrikman bounds; hexagonal representative volume element; Homogenization