V Varvara Kouznetsova

Multi-scale computational homogenization: Trends and challenges

In the past decades, considerable progress had been made in bridging the mechanics of materials to other disciplines, e.g. downscaling to the field of materials science or upscaling to the field of structural engineering. Within this wide context, this paper reviews the state-of-the-art of a particular, yet powerful, method, i.e. computational homogenization. The paper discusses the main trends since the early developments of this approach up to the ongoing contributions and upcoming challenges in the field.

Multi-scale computational homogenization of structured thin sheets

Structured and layered thin sheets are used in a variety of innovative applications, e.g. flexible displays, rollable solar cells or flexible electronics. Stacks of different materials, with often highly complex interconnects between layers, are thereby used, which are typically loaded in bending combined with intrinsic thermo-mechanical mismatches. As a result, different failure mechanisms at the level of the layered substructure occur, which constitutes a serious reliability concern.This paper deals with the two-scale homogenization of structured thin sheets, whereby a higher-order through-thickness representative volume element (RVE) is used. The methodology relies on the computational homogenization of the mechanics of microstructures, for which first-order and second-order solution strategies have been developed in the past decade. The upscaling of the deformation of structured thin sheets towards a shell-type continuum is second-order in nature. The higher-order kinematics is defined on the basis of a microstructural RVE, which represents the full thickness of the macroscopic structure and a periodic in-plane cell (e.g. a single pixel in a flexible display). The elaboration of the boundary conditions and the solution of the micro-scale boundary value problem are discussed. The obtained micro-scale stress state is homogenized towards a 3D macroscopic shell structure, for which detailed aspects will be emphasized. The coupled numerical solution strategy is briefly outlined. Finally, an example is given and the application to a number of practical problems is highlighted, where the solution provides direct information on each scale. The incorporation of failure events at the substructure level is thereby naturally at hand.

A multi-scale approach to bridge microscale damage and macroscale failure : a nested computational homogenization-localization framework

This paper presents a multi-scale modelling approach for bridging the microscale damage and macroscale failure. The proposed scheme evolves from a classical computational homogenization scheme (FE2) towards a discontinuity enriched framework. The classical homogenization approaches typically rely on the separation of scales principle, which is violated as soon as a strain localization band develops within a microstructural volume element (MVE). The newly developed scheme resolves this limitation by considering the bifurcation of the microscale deformation into a continuum ‘bulk’ part and a localization related part. The most distinct feature of the proposed framework is that both, the local macroscale traction-opening response of the cohesive crack and the stress-strain response of the surrounding ‘bulk’, are obtained from a single MVE analysis. The discontinuity enriched macroscale description is formulated to accommodate for the micro-macro coupling. The macroscale boundary value problem and the corresponding implementation are detailed for the use within the embedded discontinuities approach. The presented multi-scale method is demonstrated on a numerical example of a cohesive crack propagation in a macroscopic double notch specimen, with underlying voided microstructure.

Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum

This contribution presents a novel homogenization technique for modeling heterogeneous materials with micro-inertia effects such as locally resonant acoustic metamaterials. Linear elastodynamics is used to model the micro and macro scale problems and an extended first order Computational Homogenization framework is used to establish the coupling. Craig Bampton Mode Synthesis is then applied to solve and eliminate the microscale problem, resulting in a compact closed form description of the microdynamics that accurately captures the Local Resonance phenomena. The resulting equations represent an enriched continuum in which additional kinematic degrees of freedom emerge to account for Local Resonance effects which would otherwise be absent in a classical continuum. Such an approach retains the accuracy and robustness offered by a standard Computational Homogenization implementation, whereby the problem and the computational time are reduced to the on-line solution of one scale only.

Enabling microstructure-based damage and localization analyses and upscaling

This paper presents a framework that enables microstructural modelling of complex microstructures involving damage, localization and fracture. Classical computational homogenization schemes hinge on the separation of scales and the existence of representative volume elements. Due to the accumulation of micro-damage, the microstructural volume elements gradually loose their representative character and evolve towards a unique volume with a developing strain localization band, rendering classical homogenization approaches inapplicable.The assumption that the representative nature of the microstructure along the strain localization band is preserved enables the definition of advance scale transition relations for both imposing the overall macroscale load and coarse graining the cohesive behaviour of the strain localization band. Newly developed strain percolation path aligned boundary conditions have been used for this purpose. It is shown that this enables the development and progressive evolution of a strain localization band with minimal interference of the imposed boundary conditions. In addition to classical homogenization of the stress–strain response, a coarse graining scheme for the effective cohesive behaviour of the strain localization band is proposed. This enables the assessment of the microstructural evolution within the strain localization band and simultaneously provides the effective cohesive response useful for macroscale failure models.

Multi-layered inclusions in locally resonant metamaterials: two-dimensional vs. three-dimensional modeling

Locally resonant metamaterials (LRMs) controlling low-frequency waves due to resonant scattering are usually characterized by narrow band gaps (BGs) and a poor wave filtering performance. To remedy this shortcoming, multiresonant metamaterial structures with closely located BGs have been proposed and widely studied. However, the analysis is generally limited to two-dimensional (2D) structures neglecting the finite height of any real resonator. The aim of this paper is the comparison of the wave dispersion for two- and three-dimensional (3D) metamaterial models and evaluation of the applicability ranges of 2D results. Numerical study reveals that dual-resonant structures with cylindrical inclusions possess only a single (compared to two in the 2D case) BG for certain height-to-width ratios. In contrast, the wave dispersion in metamaterials with multiple spherical resonators can be accurately evaluated using a 2D approximation, enabling a significant simplification of resource-consuming 3D models.

Retardation of plastic instability via damage-enabled microstrain delocalization

AbstractMulti-phase microstructures with high mechanical contrast phases are prone to microscopic damage mechanisms. For ferrite–martensite dual-phase steel, for example, damage mechanisms such as martensite cracking or martensite–ferrite decohesion are activated with deformation, and discussed often in literature in relation to their detrimental role in triggering early failure in specific dual-phase steel grades. However, both the micromechanical processes involved and their direct influence on the macroscopic behavior are quite complex, and a deeper understanding thereof requires systematic analyses. To this end, an experimental–theoretical approach is employed here, focusing on three model dual-phase steel microstructures each deformed in three different strain paths. The micromechanical role of the observed damage mechanisms is investigated in detail by in-situ scanning electron microscopy tests, quantitative damage analyses, and finite element simulations. The comparative analysis reveals the unforeseen conclusion that damage nucleation may have a beneficial mechanical effect in ideally designed dual-phase steel microstructures (with effective crack-arrest mechanisms) through microscopic strain delocalization.

Multi-scale mechanics in microelectronics: a paradigm in miniaturization

This paper reviews the inherent change in the observed mechanical behavior of electronic components, structures, and multimaterials as a result of the ongoing miniaturization. In general, the size of microstructures is no longer negligible with respect to the component size in micro and submicron applications. Additionally, surface layers start to play a more prominent role in the mechanical response. Microstructural effects, macroscopically triggered gradient effects, and surface effects jointly appear and constitute the various size effects that can be observed. Classical continuum mechanics theories fail to describe these phenomena, and higher-order multiscale theories are required to arrive at an appropriate prediction of the mechanical behavior of miniaturized structures.

Predictive modeling of interfacial damage in substructured steels: application to martensitic microstructures

Metallic composite phases, like martensite present in conventional steels and new generation high strength steels exhibit microscale, locally lamellar microstructures characterized by alternating layers of phases or crystallographic variants. The layers can be sub-micron down to a few nanometers thick, and they are often characterized by high contrasts in plastic properties. As a consequence, fracture in these lamellar microstructures generally occurs along the layer interfaces or within one of the layers, typically parallel to the interface. This paper presents a computational framework that addresses the lamellar nature of these microstructures, by homogenizing the plastic deformation at the mesoscale by using the microscale response of the laminates. Failure is accounted for by introducing a family of damaging planes that are parallel to the layer interface. Mode I, mode II and mixed-mode opening are incorporated. The planes along which failure occurs are captured using a smeared damage approach. Coupling of damage with isotropic or anisotropic plasticity models, like crystal plasticity, is straightforward. The damaging planes and directions do not need to correspond to crystalline slip planes, and normal opening is also included. Focus is given on rate-dependent formulations of plasticity and damage, i.e. converged results can be obtained without further regularization techniques. The validation of the model using experimental observations in martensite-austenite lamellar microstructures in steels reveals that the model correctly predicts the main features of the onset of failure, e.g. the necking point, the failure initiation region and the failure mode. Finally, based on the qualitative results obtained, some material design guidelines are provided for martensitic and multi-phase steels.

A multi-scale model of martensitic transformation induced plasticity at finite strains

A physically-based multi-scale model for martensitic transformation induced plasticity is presented. At the fine scale, a model for one transforming martensitic variant is established based on the concept of a lamellae composed of a martensitic plate and an austenitic layer. Next, the behaviour of 24 potentially transforming variants is homogenized towards the behaviour of an austenitic grain. As a simple example, the model is applied to deformation and transformation of a single austenitic grain under different deformation modes.