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This paper presents a methodology where a macroscopic linear material response incorporates microscopic variations, such as transient interactions and micro-inertia effects. This is achieved by implementing the temporal coupling between macro and microstructures, along with the spatial coupling, within a dynamic computational homogenisation framework. In the context of dynamic multiscale modelling, the temporal coupling method offers significant advantages by effectively reducing deviations emerging from micro-inertia effects and transient phenomena. The effectiveness of the developed procedure is validated by a comparison of the macroscopic results with the solutions of direct numerical simulation for a one-dimensional periodic laminate bar with different contrast levels. The homogenised results obtained using the developed procedure indicate that a better prediction of the macroscopic requires a larger Representative Volume Element (RVE) which improves the estimation of multiscale strain energy and a larger time window which improves the estimation of multiscale kinetic energy. The simultaneous increase in the RVE size and the time averaging window yields the best results in predicting the macroscopic response. [ABSTRACT FROM AUTHOR]

THE PAPER PRESENTS THE FORMULATION OF A GRANULAR COMPUTATIONAL HOMOGENISATION PROBLEM and the proposition of a method to solve it, which enables multiscale analysis of materials with uncertain microstructure parameters. The material parameters and the geometry, represented by the interval and fuzzy numbers, are assumed to be unprecise. An α-cut representation of fuzzy numbers allows the use of interval arithmetic for epistemic uncertainties. Directed interval arithmetic is used to reduce the effect of interval widening during arithmetic operations. Response surfaces of diverse types, including Artificial Neural Networks, are used as model reduction methods. The finite element method is employed to solve the boundary value problem on a micro scale. Numerical examples are provided to demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

Purpose: The present paper aims to explore a computational homogenisation procedure to investigate the full geometric representation of yield surfaces for isotropic porous ductile media. The effects of cell morphology and imposed boundary conditions are assessed. The sensitivity of the yield surfaces to the Lode angle is also investigated in detail. Design/methodology/approach: The microscale of the material is modelled by the concept of Representative Volume Element (RVE) or unit cell, which is numerically simulated through three-dimensional finite element analyses. Numerous loading conditions are considered to create complete yield surfaces encompassing high, intermediate and low triaxialities. The influence of cell morphology on the yield surfaces is assessed considering a spherical cell with spherical void and a cubic RVE with spherical void, both under uniform strain boundary condition. The use of spherical cell is interesting as preferential directions in the effective behaviour are avoided. The periodic boundary condition, which favours strain localization, is imposed on the cubic RVE to compare the results. Small strains are assumed and the cell matrix is considered as a perfect elasto-plastic material following the von Mises yield criterion. Findings: Different morphologies for the cell imply in different yield conditions for the same load situations. The yield surfaces in correspondence to periodic boundary condition show significant differences compared to those obtained by imposing uniform strain boundary condition. The stress Lode angle has a strong influence on the geometry of the yield surfaces considering low and intermediate triaxialities. Originality/value: The exhaustive computational study of the effects of cell morphologies and imposed boundary conditions fills a gap in the full representation of the flow surfaces. The homogenisation-based strategy allows us to further investigate the influence of the Lode angle on the yield surfaces. [ABSTRACT FROM AUTHOR]

The interaction between deformable bodies and rigid foundations undergoing finite strains is explored in this work with the Method of Multiscale Virtual Power, unlocking novel insights into contact homogenisation theories. The focus lies in establishing the foundational kinematical links across scales and achieving seamless homogenisation of the traction vector through rigorous duality arguments. Two distinct families of contact homogenisation models are formulated: surface–surface and surface–volume models. These families emerge from the virtual power balance established between the target macroscopic surface and the corresponding microscopic contact surface or volume. They not only represent fundamental kinematical entities at each scale but also enable the replacement of the heterogeneous contact traction distribution with a smooth, homogenised version. Microscale equilibrium problems and homogenisation relations, along with the relevant macro- and microscale quantities, are derived by means of straightforward variational arguments. Furthermore, potential finite size effects are addressed in the homogenised response, enhancing the reliability and applicability of the strategy across diverse scenarios. Promising submodels have been identified within both families, broadening the understanding of multiscale contact phenomena and including existing models as particular cases. The proposed continuum-based variational families of models naturally lead to generic finite element-based frameworks for contact homogenisation of heterogeneous solids, as described in a forthcoming paper. [ABSTRACT FROM AUTHOR]

Computational multiscale methods are highly sophisticated numerical approaches to predict the constitutive response of heterogeneous materials from their underlying microstructures. However, the quality of the prediction intrinsically relies on an accurate representation of the microscale morphology and its individual constituents, which makes these formulations computationally demanding. Against this background, the applicability of an adaptive wavelet-based collocation approach is studied in this contribution. It is shown that the Hill–Mandel energy equivalence condition can naturally be accounted for in the wavelet basis, (discrete) wavelet-based scale-bridging relations are derived, and a wavelet-based mapping algorithm for internal variables is proposed. The characteristic properties of the formulation are then discussed by an in-depth analysis of elementary one-dimensional problems in multiscale mechanics. In particular, the microscale fields and their macroscopic analogues are studied for microstructures that feature material interfaces and material interphases. Analytical solutions are provided to assess the accuracy of the simulation results. [ABSTRACT FROM AUTHOR]

The effective material response as observed at a macro level is a manifestation of the material microstructure and lower scale processes. Due to their distinct atomic arrangement, compared to bulk material, grain boundaries significantly affect the electrical properties of metals. However, a scale-bridging understanding of the associated microstructure–property relation remains elusive so that phenomenological approaches such as the Andrews method are typically applied. In the present contribution we revisit Andrews method from a computational multiscale perspective to analyse its limits and drive concepts to go beyond. By making use of homogenisation techniques we provide a solid theoretical foundation to the Andrews method, discuss its applicability and tacit assumptions involved, and resolve its core limitations. To this end, simplistic analytical examples are discussed in a one-dimensional setting to show the fundamental relation between the Andrews method and homogenisation approaches. Building on this knowledge the importance of the underlying microscale morphology and associated morphology-induced anisotropies are in the focus of investigations based on simplified microstructures. Concluding the analysis, scaling laws for isotropic microstructures are derived and the transferability of the results to realistic, (quasi-)isotropic polycrystals is shown. • Computational homogenisation-based theoretical foundation to the Andrews method. • Scaling laws for isotropic microstructures. • Transferability to realistic (quasi-)isotropic polycrystals. [ABSTRACT FROM AUTHOR]

The numerical assessment of fracture properties of cortical bone is important in providing suggestions on patient-specific clinical treatments. We present a generic finite element modelling framework incorporating computational fracture approaches and computational homogenisation techniques. Finite element computations for statistical volume elements (SVEs) at the microscale are performed for different sizes with random osteon packing with a fixed volume fraction. These SVEs are loaded in the transverse direction under tension. The minimal SVE size in terms of ensuring a representative effective cohesive law is suggested to be 0.6 mm. Since cement lines as weak interfaces play a key role in bone fracture, the effects of their fracture properties on the effective fracture strength and toughness are investigated. The extracted effective fracture properties can be used as homogenised inputs to a discrete crack simulation at macroscopic or structural scale. The extrinsic toughening mechanisms observed in the SVE models are discussed with a comparison against experimental observations from the literature, giving beneficial insights to cortical bone failure. [ABSTRACT FROM AUTHOR]

Motivated by advances in flexible electronic technologies and by the endeavour to develop non-destructive testing methods, this article analyses the capability of computational multiscale formulations to predict the influence of microscale cracks on effective macroscopic electrical and mechanical material properties. To this end, thin metal films under mechanical load are experimentally analysed by using in-situ confocal laser scanning microscopy (CLSM) and in-situ four point probe resistance measurements. Image processing techniques are then used to generate representative volume elements from the laser intensity images. These discrete representations of the crack pattern at the microscale serve as the basis for the calculation of effective macroscopic electrical conductivity and mechanical stiffness tensors by means of computational homogenisation approaches. A comparison of simulation results with experimental electrical resistance measurements and a detailed study of fundamental numerical properties demonstrates the applicability of the proposed approach. In particular, the (numerical) errors that are induced by the representative volume element size and by the finite element discretisation are studied, and the influence of the filter that is used in the generation process of the representative volume element is analysed. [ABSTRACT FROM AUTHOR]

This article addresses numerical simulation of the mechanical behaviour of materials comprising heterogeneous ductile micro-structures in the presence of voids using a multi-scale approach considering plasticity processes. This kind of material has been used in several applications as structural solutions. Therefore, for safety reasons, studies about stress analysis of porous ductile materials are essential to understand their mechanical behaviour. Numerical modelling is performed in micro-structures using the concept of representative volume element (RVE) where the matrix is considered as an ideally plastic material governed by the von Mises model with isotropic hardening, while inclusions are adopted as very stiff elastic materials. Also, fracture and contact finite elements are used to model the phase debonding. Different distributions of voids and inclusions are assumed in the RVE domain to investigate their influences on the proposed analyses. We conclude, for instance, that the concentration of voids in the RVE decreases its loading capacity. On the other hand, we show in the numerical examples that the RVEs containing random distributions of voids present loading capacity improved when compared to the RVEs containing symmetric distributions of voids. Moreover, the results show that the insertion of reinforcements into porous ductile media has limited efficiency when dealing with high values of loading. [ABSTRACT FROM AUTHOR]

• The effect of pre-strain history on macro yielding has been shown for a periodic unit cell with spherical void at its centre. • Yield surfaces are defined as iso-surface of constant macro plastic dissipation rate. • Yield surfaces of a prestrained heterogeneous ductile solid, corresponds to a low plastic dissipation rate, exhibit kinematic hardening in a manner that carries signatures of its pre-straining. • At high plastic dissipation rate, the subsequent yield surface shows isotropic hardening. • Distortion of the subsequent yield surface is concentrated at locations marked by the prestrained Lode angle and triaxiality level. Subsequent yield surfaces of ductile solids, pre-strained in shear and/or tension, exhibit various characteristic features that depend on the proof strain used to detect yield. These features include kinematic hardening when the proof strain denoting yield is low, isotropic hardening when it is high, formation of a 'nose' in the loading direction and flattening of the rear part of the yield surface. With the aid of a computational homogenisation scheme that allows application of arbitrary macro stress or macro deformation gradient to a unit cell, we show that essential features of the subsequent yield surfaces owe their origin to the extent of plasticity accumulated within the unit cell. For instance, a simple unit cell containing a spherical void, where the void serves to create a heterogenous distribution of plastic strain, suffices to computationally reproduce the experimentally observed features. Moreover, in contrast to experiments which generally allow plotting of the yield surface in a normal stress – shear stress plane, the computational scheme allows fuller exploration of the subsequent macro yield surface in the macro stress space. We find that what seems to be a translation and distortion of the yield surface in experiments, is actually most pronounced near some critical octahedral planes characterised by levels of triaxiality induced by the pre-straining process. The evolution of the yield surface in regions where the triaxiality is higher is considerably different. Moreover, as homogeneous yielding is approached with loading of the unit cell, the material seems to forget the nature of the pre-straining process and undergoes isotropic hardening. [ABSTRACT FROM AUTHOR]

A generic and straightforward numerical strategy for the determination of the size of Representative Contact Elements (RCEs), which are employed in finite element contact homogenization procedures, is proposed in this contribution. The approach enables the determination of the length, height and mesh discretisation of RCEs that provide a good statistical representation of the contact interface, described by a given set of topography properties. The case of Gaussian self-affine and elastic rough profiles under normal, frictionless and non-adhesive contact is analysed in detail. A collection of conditions has been derived for two-dimensional problems, based on a trade-off between the convergence of the real contact area and the computational cost. With the increasing width of the roughness power spectrum, the restrictions imposed on the length and mesh size can be relaxed, allowing to reduce the size of numerical models. The corresponding class of problems in three-dimensions has also been studied. In this case, the influence of the numerical scheme adopted for the evaluation of the contact area has been analysed leading to the identification of two bounds, which converge for the same value with progressively finer meshes. State-of-the-art numerical results fall within the bounded region, and the application of the area correction technique proposed by Yastrebov et al. to the upper node-based bound, accelerates the convergence of the mesh and renders a good agreement with reference data. [ABSTRACT FROM AUTHOR]

Two-scale models for reinforced concrete, where the large-scale problems are defined in terms of Euler–Bernoulli beam and Kirchhoff–Love plate models, are constructed. The subscale problem on the Representative Volume Element (RVE) is correspondingly outlined as finding the response of the three-dimensional RVE comprising plain concrete continuum, reinforcement bars and the bond between them. The boundary region of the periodic mesh is modelled with special solid elements, which allow for prescribing the macroscopic input via strongly periodic boundary conditions in an effective way. The effective response of the reinforced concrete RVEs of different sizes subjected to tension and pure bending is investigated for both effective beam and plate models. A series of experiments on reinforced concrete panels subjected to bending and membrane loads is simulated, and the effective moment–curvature response is studied. Within the developed framework, an arbitrary macroscopic loading in terms of membrane strains and curvatures can be prescribed on the RVE, and the corresponding effective response is obtained, making the proposed formulation feasible for future use in an FE2 scheme. [ABSTRACT FROM AUTHOR]

SUMMARY: A two‐scale model for reinforced concrete, in which the large‐scale problem formulation is enriched by an effective reinforcement slip variable, is derived from the single‐scale model describing the response of plain concrete, reinforcement steel, and the bond between them. The subscale problem on the representative volume element (RVE) is correspondingly defined as finding the response of the RVE subjected to effective variables (strain, slip, and slip gradient) imposed from the large scale. A novel volumetric definition of effective reinforcement slip and its gradient is devised, and the corresponding subscale problem is formulated. The newly defined effective variables are imposed on the RVE in a weak sense using Lagrange multipliers. The response of the RVEs of different sizes was investigated by means of pull‐through tests, and the novel boundary condition type was used in FE2 analyses of a deep beam. Locally, prescribing the macroscopic reinforcement slip and its gradient in the proposed manner resulted in reduced RVE‐size dependency of effective work conjugates, which allows for more objective description of reinforcement slip in two‐scale modeling of reinforced concrete. Globally, this formulation produced more consistent amplitudes of effective slip fluctuations and more consistent maximum crack width predictions. [ABSTRACT FROM AUTHOR]

In the contribution at hand, a new material modelling approach is introduced. This formulation is based upon the Principle of Multiscale Virtual Power and consideration of micromechanically motivated assumptions. Consequently, the evolution of dissipative phenomena depends on the chosen microstructure. Therefore, a strong anisotropy, which is induced by damage, is represented even with isotropic material formulations. This phenomenon is present in concrete. Furthermore, the modelling approach is validated by different material tests. Tensile-tensile and compression-tensile tests are used for validation of the proposed description. Some material tests are taken from the existing literature, while others are presented in the contribution at hand. Furthermore, the capabilities of the proposed formulation to capture different amounts of textile reinforcement in concrete are shown by additional experiments from the literature. Subsequently, the consistent linearisation of the proposed model is verified based on numerical analyses. • Consistent formulation of a multiscale analysis applied to concrete. • Novel microlayer framework for concrete. • Validation for concrete. • Application to concrete under biaxial loads. [ABSTRACT FROM AUTHOR]

This paper proposes a novel method to accurately and efficiently reduce a microstructural mechanical model using a wavelet based discretisation. The model enriches a standard reduced order modelling (ROM) approach with a wavelet representation. Although the ROM approach reduces the dimensionality of the system of equations, the computational complexity of the integration of the weak form remains problematic. Using a sparse wavelet representation of the required integrands, the computational cost of the assembly of the system of equations is reduced significantly. This wavelet-reduced order model (W-ROM) is applied to the mechanical equilibrium of a microstructural volume as used in a computational homogenisation framework. The reduction technique however is not limited to micro-scale models and can also be applied to macroscopic problems to reduce the computational costs of the integration. For the sake of clarity, the W-ROM will be demonstrated using a one-dimensional example, providing full insight in the underlying steps taken. [ABSTRACT FROM AUTHOR]

The present contribution focuses on the analysis of diverse deformation mechanisms that impact the behaviour of PC/ABS blends using computational homogenisation. This includes analysing internal particle cavitation, PC/ABS interface debonding, and PC matrix shear-yielding. The goal is to investigate the optimal composition for specific applications and create tailored materials. The work involves establishing a microstructure Representative Volume Element, defining the constitutive description of both material phases, and explicitly modelling PC/ABS interfaces and matrix damage to achieve accurate predictions. A Python programme is devised to efficiently integrate zero-thickness cohesive interface elements around ABS particles, incorporating the PPR potential-based cohesive model to characterise the interface. Additionally, the finite strain visco-elastic visco-plastic constitutive model of the PC matrix is extended to incorporate a damage variable, addressing the shear-yielding failure mechanism. The PC/ABS blend's thermomechanical response is homogenised using first-order hierarchical multi-scale analyses. The impact of considering the interface phase in the microstructure is assessed through various numerical analyses. The synergy between the constitutive models effectively captures the blend's behaviour. These findings lay the foundation for broader applicability beyond PC/ABS blends, paving the way for future studies in the field. • Development of a homogenisation framework to study the behaviour of PC/ABS blends. • Examining the individual influence and synergistic effects of the deformation mechanisms. • Determining the optimal size and mesh discretisation for Representative Volume Elements. • Analysing how interfacial and bulk material parameters impact the behaviour. • Evaluating which loading conditions are most affected by considering realistic interfaces. [ABSTRACT FROM AUTHOR]

This contribution investigates the modelling of failure in polycrystalline materials due to the nucleation and propagation of cohesive cracks at the micro-scale. First, a two-scale formulation that enables the analysis of failure, even when the evolution and propagation of cohesive micro-cracks induce material instability, is proposed. The development of this macro-continuous/micro-discontinuous formulation is based on the Method of Multi-Scale Virtual Power, from which dynamic equilibrium and homogenisation equations are derived, ensuring a variationally consistent scale transition. The respective minimal kinematical constraint is retrieved and enforced with the Lagrange multiplier method. The formulation is also extended for periodic boundary conditions. The discretisation of the resulting equilibrium equations, with the finite element method and the generalised- α method, is described in detail. Second, a fracture-based computational homogenisation procedure is developed to obtain homogenised traction-separation laws and fracture properties from microstructural volume elements. It is derived from a crack-based Hill-Mandel principle and employs a novel energetic-based damage variable, within the Park-Paulino-Roesler cohesive model, proposed to define the crack domain and compute the crack homogenised quantities. A strategy for accurately computing the homogenised unit normal vector of the equivalent macroscopic crack is also suggested. Numerical examples demonstrate the formulation's ability to analyse fracture problems, including a study on the impact of different microscopic boundary conditions. This work also reveals the convergence of the traction-separation law with increasing microstructure model size when using the fracture-based computational homogenisation. Furthermore, it investigates how advanced deformation mechanisms, like slip plasticity and martensitic transformation, affect intergranular crack propagation mechanisms. • A new two-scale model is formulated for the analysis of polycrystalline materials. • Failure due to cohesive micro-cracks and dynamic effects are modelled. • Lagrange multipliers are used to enforce the microscopic constraints. • A fracture-based computational homogenisation procedure is developed. • Homogenised traction-separation laws and fracture properties are obtained. [ABSTRACT FROM AUTHOR]

A single-scale model for reinforced concrete, comprising the plain concrete continuum, reinforcement bars and the bond between them, is used as a basis for deriving a two-scale model. The large-scale problem, representing the "effective" reinforced concrete solid, is enriched by an effective reinforcement slip variable. The subscale problem on a Representative Volume Element (RVE) is defined by Dirichlet boundary conditions. The response of the RVEs of different sizes was investigated by means of pull-out tests. The resulting two-scale formulation was used in an FE2 analysis of a deep beam. Load-deflection relations, crack widths, and strain fields were compared to those obtained from a single-scale analysis. Incorporating the independent macroscopic reinforcement slip variable resulted in a more pronounced localisation of the effective strain field. This produced a more accurate estimation of the crack widths than the two-scale formulation neglecting the effective reinforcement slip variable. [ABSTRACT FROM AUTHOR]

Summary: A two‐scale model is derived from a fully resolved model where the response of concrete, steel reinforcement, and bond between them are considered. The pertinent “effective” large‐scale problem is derived from selective homogenisation in terms of the equilibrium of reinforced concrete considered as a single‐phase solid. Variational formulations of the representative volume element problem are established in terms of the subscale displacement fields for the plain concrete continuum and the reinforcement bars. Dirichlet and Neumann boundary conditions (BCs) are imposed on the concrete (pertaining to uniform boundary displacement and constant boundary traction, respectively) and on the reinforcement bars (pertaining to prescribed boundary displacement and vanishing sectional forces, respectively). Different representative volume element sizes and combinations of BCs were used in FE2 analyses of a deep beam subjected to four‐point bending. Results were compared with those of full resolution (single‐scale). The most reliable response was obtained for the case of Dirichlet‐Dirichlet BCs, with a good match between the models in terms of the deformed shape, force‐deflection relation, and average strain. Even though the maximum crack widths were underestimated, the Dirichlet‐Dirichlet combination provided an approximate upper bound on the structural stiffness. [ABSTRACT FROM AUTHOR]

There is a wide range of industrially-relevant problems where mechanical stresses directly affect kinetics of chemical reactions. For example, this includes formation of oxide layers on parts of micro-electro-mechanical systems (MEMS) and lithiation of Si in Li-ion batteries. Detailed understanding of these processes requires thermodynamically-consistent theories describing the coupled thermo-chemo-mechanical behaviour of those systems. Furthermore, as the majority of materials used in those systems have complex microstructures, multiscale modelling techniques are required for efficient simulation of their behaviour. Hence, the purpose of the present paper is two-fold: (1) to derive a thermodynamically-consistent thermo-chemo-mechanical theory; and (2) to propose a two-scale modelling approach based on the concept of computational homogenisation for the considered theory. The theory and the two-scale computational approach are implemented and tested using a number of computational examples, including the case of the reaction locking due to mechanical stresses. • A general thermodynamically-consistent thermo-chemo-mechanical theory is proposed. • Computational homogenisation framework is proposed for the theory. • Chemical reaction locking due to mechanical stresses is modelled. [ABSTRACT FROM AUTHOR]

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 capturesize effects. Recently, it has been shown thatsize effectscan 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. [ABSTRACT FROM PUBLISHER]

In this paper, the influence of microscopic parameters on the macroscopic behaviour of polycrystalline materials under different loading configuration is investigated. Linear elastic grains with zero-thickness cohesive interfaces are considered at the microscale with in depth introduction of effective parameters. A multiscale method based on homogenisation technique is employed to bridge the scales. In order to minimize the homogenisation error, a representative volume element (RVE) of the microscopic structure is statistically determined to be used in the numerical analysis. For each loading condition of the RVE, several numerical examinations are conducted to illustrate the relationship between the microscopic parameters. Finally, the effects of microscopic critical fracture energies, maximum tensile and shear strengths of grain interfaces on the mechanical properties, i.e. stress-strain curve and yield surface at the macroscale are discussed in details. It is shown that macroscopic yield surface and stress-strain curves can be used to characterise the microscopic properties. [ABSTRACT FROM AUTHOR]

Computing the macroscopic material response of a continuum body commonly involves the formulation of a phenomenological constitutive model. However, the response is mainly influenced by the heterogeneous microstructure. Computational homogenisation can be used to determine the constitutive behaviour on the macro-scale by solving a boundary value problem at the micro-scale for every so-called macroscopic material point within a nested solution scheme. Hence, this procedure requires the repeated solution of similar microscopic boundary value problems. To reduce the computational cost, model order reduction techniques can be applied. An important aspect thereby is the robustness of the obtained reduced model. Within this study reduced-order modelling (ROM) for the geometrically nonlinear case using hyperelastic materials is applied for the boundary value problem on the micro-scale. This involves the Proper Orthogonal Decomposition (POD) for the primary unknown and hyper-reduction methods for the arising nonlinearity. Therein three methods for hyper-reduction, differing in how the nonlinearity is approximated and the subsequent projection, are compared in terms of accuracy and robustness. Introducing interpolation or Gappy-POD based approximations may not preserve the symmetry of the system tangent, rendering the widely used Galerkin projection sub-optimal. Hence, a different projection related to a Gauss-Newton scheme (Gauss-Newton with Approximated Tensors- GNAT) is favoured to obtain an optimal projection and a robust reduced model. [ABSTRACT FROM AUTHOR]

A multiscale approach to analyse historical masonry buildings is presented and the numerical results deriving from its implementation are discussed. The modelling of this kind of heterogeneous material composed by irregular, stones and mortar joints is performed at the level of the microstructure by also describing the nonlinear behaviour of stones and mortar joints through an elasto-plastic constitutive law. At the macro-level, a finite element description based on a generic anisotropic material is implemented. This micro-macro model aims to assess the structural behaviour and the safety condition of historic masonries. [ABSTRACT FROM AUTHOR]

Due to their low Young's Modulus, high strength and suitability for additive manufacturing, non-toxic beta-type titanium alloys are emerging as next-generation biomaterials. We present novel experimental results that demonstrate significant variation of Young's Modulus with direction for selective laser melted (SLM) biocompatible Ti-24Nb-4Zr-8Sn (Ti2448). Grain orientation data for SLM-processed Ti2448 is measured using electron backscatter diffraction. By assuming the grain orientations are fixed relative to the axes of the SLM build machine, the measured grain orientation data is used to generate a detailed microstructural finite element model of the polycrystalline SLM-processed material. The computational model provides excellent predictions of the anisotropic properties of SLM-processed Ti2448, indicating that preferential grain orientations that form during SLM processing of Ti2448 cause the experimentally measured variation of the Young's Modulus. The results show that computational models are able to accurately predict the anisotropic Young's Modulus of polycrystalline materials, and, in the context of biocompatible Ti2448 show how to tailor the modulus of SLM components by choosing the build orientation. [Display omitted] [ABSTRACT FROM AUTHOR]

The simulation of complex engineering structures built from magneto-rheological elastomers is a computationally challenging task. Using the FE 2 method, which is based on computational homogenisation, leads to the repetitive solution of micro-scale FE problems, causing excessive computational effort. In this paper, the micro-scale FE problems are replaced by POD reduced models of comparable accuracy. As these models do not deliver the required reductions in computational effort, they are combined with hyper-reduction methods like the Discrete Empirical Interpolation Method (DEIM), Gappy POD, Gauss⁻Newton Approximated Tensors (GNAT), Empirical Cubature (EC) and Reduced Integration Domain (RID). The goal of this work is the comparison of the aforementioned hyper-reduction techniques focusing on accuracy and robustness. For the application in the FE 2 framework, EC and RID are favourable due to their robustness, whereas Gappy POD rendered both the most accurate and efficient reduced models. The well-known DEIM is discarded for this application as it suffers from serious robustness deficiencies.

This contribution focuses on the development of an adaptive hierarchical FFT-based approach for the efficient solution of microscale boundary value problems. To this end, the classic Moulinec–Suquet scheme is revisited and enhanced by making use of wavelet analysis. Governing fields are represented in a wavelet basis and higher level stress approximations in a nested set of approximation spaces are successively derived by making use of wavelet transforms. By adaptively refining the computational grid based on the solution profile, localised features can be resolved accurately while the overall number of material model evaluations is significantly reduced. The performance is demonstrated by a detailed study of representative boundary value problems in one- and two-dimensional domains, whereby a reduction in the number of material model evaluations of up to 95% has been achieved. • State of the art wavelet-enhanced FFT-based solution approach. • Adaptive grid refinement. • Significant reduction in number of material model evaluations. • Eshelby–Green operator for wavelet-based discretisations. [ABSTRACT FROM AUTHOR]

Effective properties of materials with random heterogeneous structures are typically determined by homogenising the mechanical quantity of interest in a window of observation. The entire problem setting encompasses the solution of a local PDE and some averaging formula for the quantity of interest in such domain. There are relatively standard methods in the literature to completely determine the formulation except for two choices: i) the local domain itself and the ii) boundary conditions. Hence, the modelling errors are governed by the quality of these two choices. The choice i) relates to the degree of representativeness of a microscale sample, i.e., it is essentially a statistical characteristic. Naturally, its reliability is higher as the size of the observation window becomes larger and/or the number of samples increases. On the other hand, excepting few special cases there is no automatic guideline to handle ii). Although it is known that the overall effect of boundary condition becomes less important with the size of the microscale domain, the computational cost to simulate such large problem several times might be prohibitive even for relatively small accuracy requirements. Here we introduce a machine learning procedure to select most suitable boundary conditions for multiscale problems, particularly those arising in solid mechanics. We propose the combination Reduced-Order Models and Deep Neural Networks in an offline phase, whilst the online phase consists in the very same homogenisation procedure plus one (cheap) evaluation of the trained model for boundary conditions. Hence, the method allows an implementation with minimal changes in existing codes and the use of relatively small domains without losing accuracy, which reduces the computational cost by several orders of magnitude. A few test cases accounting for random circular and elliptical inclusions are reported aiming at proving the potentials of the DeepBND method. • Physically aware Neural Networks for numerical homogenisation in multiscale mechanics. • Data-driven enhanced boundary conditions for cell problems by reduced-order modelling. • NN architecture tailored to exploit symmetries and a priori mechanical knowledge. • The novel boundary condition allows considerable speed-ups with enhanced accuracy. [ABSTRACT FROM AUTHOR]

Although "classical" multi-scale methods can capture the behaviour of cellular, including lattice, materials, when considering lattices or metamaterial local instabilities, corresponding to a change of the micro-structure morphology, classical computational homogenisation methods fail. On the one hand, first order computational homogenisation, which considers a classical continuum at the macro-scale cannot capture localisation bands inherent to cell buckling propagation. On the other hand, second-order computational homogenisation, which considers a higher order continuum at the macro-scale, introduces a size effect with respect to the Representative Volume Element (RVE) size, which is problematic when the RVE has to consider several cells to recover periodicity during local instability. In this paper we reformulate in a finite-strain setting the second-order computational homogenisation using the idea of equivalent homogenised volume. From this equivalence, arises at the micro-scale a non-uniform body force that acts as a supplementary volume term over the RVE. In the presented method, this non-uniform body-force term arises from the equivalence of energy, i.e. the Hill–Mandel condition, between the micro- and macroscopic volumes and depends mainly on the relation between the micro-scale and macro-scale deformation gradient. We show by considering elastic and elasto-plastic metamaterials and cellular materials that this approach reduces the RVE size dependency on the homogenised response. • Second order homogenisation is reformulated through equivalent homogenised volume. • A non-uniform body force arises as a volume term on the RVE definition. • The expression of this non-uniform body-force arises from the equivalence of energy. • This approach reduces the RVE size dependency on the homogenised response. • Instabilities in non-linear metamaterials and cellular materials can be captured. [ABSTRACT FROM AUTHOR]

The convergence behaviour of representative volume elements (RVEs) with an increasing number of crystalline grains, which can undergo crystallographic slip and mechanically induced martensitic formation, is analysed in the present contribution. Instead of analysing a single micromechanical model subjected to relevant loading scenarios, the representativeness of the RVEs is assessed by running distinct macroscopic simulations that promote different deformation modes with several random realisations of the polycrystalline microstructure. To avoid the computational cost of FE2 approaches and make this convergence analysis feasible, a computationally efficient Taylor's condition (FE-T) is assumed. Then, an FE2 simulation is performed with periodic boundary conditions for a converged RVE size to illustrate the impact of Taylor's assumption on the macroscopic response. The introduction of martensitic transformation slightly increases the dispersion of the results. Nevertheless, the macroscopic response converges whether this phenomenon is considered or not. • The representativeness of polycrystalline volume elements is assessed for an increasing number of grains. • Crystal plasticity and phase transformation mechanisms are taken into account. • Macroscopic specimens are employed to perform the convergence analysis with several realisations. • A FE-T technique that employs the Taylor constraint is developed to speedup simulations. • The FE-T response is compared against a FE2 simulation with periodic boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, we present new reliable model order reduction strategies for computational micromechanics. The difficulties rely mainly upon the high dimensionality of the parameter space represented by any load path applied onto the representative volume element. We take special care of the challenge of selecting an exhaustive snapshot set. This is treated by first using a random sampling of energy dissipating load paths and then in a more advanced way using Bayesian optimization associated with an interlocked division of the parameter space. Results show that we can insure the selection of an exhaustive snapshot set from which a reliable reduced-order model can be built. [ABSTRACT FROM AUTHOR]

Asphalt concrete (AC) is a composite material consisting of bituminous binders, mineral aggregate, and voids. Experimental and computational efforts to assess the bulk mechanical properties of AC ubiquitously raise the question of representative sample size. It is well known that the dimension of a sample has to be larger than the largest morphological entity. However, it has been shown that the required size is a function of the morphological and physical properties under consideration, e. g. the difference between the properties of the constituents at the microscale, and their volume fractions. In the present contribution, representative sample sizes are determined numerically for a variation of parameters in terms of a tolerable amount of statistical scatter by conducting virtual experiments. If the scatter between distinct heterogeneous samples falls below a certain threshold, the corresponding volume is called a representative volume element (RVE). Finite element (FE) discretisations of AC volume samples are generated by means of a shrunk Poisson Voronoi tessellation. In order to provide estimates for the evolution of the RVE size under changing material properties of the mortar, the stiffness ratio between rocks and mortar is varied by two orders of magnitude. Furthermore, the influence of a variation in volume fraction is investigated. [ABSTRACT FROM AUTHOR]

Motivated by the significant influence of the underlying microstructure on the effective electrical properties of a material system and the desire to monitor defect evolution through non-destructive electrical characterisation, this contribution is concerned with a detailed study of conductivity changes caused by the presence of sub-microscale pores. Reducing the complexity of the material system, geometrically well-defined pore arrays are created by focused ion beam (FIB) milling in Cu thin films and characterised by 4-point probe electrical measurements. The experiment is designed such that it reduces to a (quasi-)one-dimensional electrical problem which is amenable to analytical techniques when invoking a computational homogenisation scheme to approximate the effective electrical properties of a given microstructure. The applicability of the proposed approach is shown in a first step by comparing simulation results for different pore volume fractions and pore shapes against their experimental counterparts. In a second step, a sensitivity analysis of the experimental data is carried out and the usefulness of the proposed modelling approach in interpreting the experimental data is demonstrated. In particular, the findings suggest that the proposed experimental method allows (at best) the determination of pore volume fractions with an accuracy of ± 0. 5 %. • Geometrically well-defined nanoscale unit cell arrays are created by FIB milling. • 4-point probe measurements are used to electrically characterise the samples. • Computational homogenisation is used to relate porosity and electrical conductivity. • Experimental limits under idealised conditions are revealed. [ABSTRACT FROM AUTHOR]

This work is concerned with particulate composites filled with hollow spherical inclusions, i.e., syntactic foams . We aim at the micromechanical evaluation of the effective uniaxial compressive strength for the most relevant case of glass inclusions of wall thickness of few micrometers ( microballoons ) filling a thermoset matrix. We develop a three-dimensional Finite Element (FE) modelling which extends and improves that recently proposed by our group. Different microstructures are described by cubic unit cells containing fifty hollow spheres accounting for different filler polydispersions and filler volume fraction f up to 60%. Each microballoon is assumed to undergo brittle failure according to a structural criterion. Here, we account for the matrix nonlinear behaviour and, in a phenomenological way, for the detriment of its mechanical properties, proportional to its defectiveness, which increases with the filler content and becomes extremely relevant at f larger than 50%. Our findings agree with experimental observations from the literature and reveal room for improvement in the effective mechanical properties by acting on the manufacturing process. [ABSTRACT FROM AUTHOR]

A mixed finite element (FE) procedure of the gradient Cosserat continuum for the second-order computational homogenisation of granular materials is presented. The proposed mixed FE is developed based on the Hu-Washizu variational principle. Translational displacements, microrotations, and displacement gradients with Lagrange multipliers are taken as the independent nodal variables. The tangent stiffness matrix of the mixed FE is formulated. The advantage of the gradient Cosserat continuum model in capturing the meso-structural size effect is numerically demonstrated. Patch tests are specially designed and performed to validate the mixed FE formulations. A numerical example is presented to demonstrate the performance of the mixed FE procedure in the simulation of strain softening and localisation phenomena, while without the need to specify the macroscopic phenomenological constitutive relationship and material failure model. The meso-structural mechanisms of the macroscopic failure of granular materials are detected, i.e. significant development of dissipative sliding and rolling frictions among particles in contacts, resulting in the loss of contacts. [ABSTRACT FROM AUTHOR]

A computational homogenisation technique is used to investigate the role of fine-scale dilatancy on the stress-induced permeability evolution in a granitic material. A representative volume element incorporating the heterogeneous fabric of the material is combined with a fine-scale interfacial decohesion model to account for microcracking. A material model that incorporates dilatancy is used to assess the influence of dilatant processes at the fine scale on the averaged mechanical behaviour and on permeability evolution, based on the evolving opening of microcracks. The influence of the stress states on the evolution of the spatially averaged permeability obtained from simulations is examined and compared with experimental results available in the literature. It is shown that the dilatancy-dependent permeability evolution can be successfully modelled by the averaging approach. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, we propose upper and lower error bounding techniques for reduced order modelling applied to the computational homogenisation of random composites. The upper bound relies on the construction of a reduced model for the stress field. Upon ensuring that the reduced stress satisfies the equilibrium in the finite element sense, the desired bounding property is obtained. The lower bound is obtained by defining a hierarchical enriched reduced model for the displacement. We show that the sharpness of both error estimates can be seamlessly controlled by adapting the parameters of the corresponding reduced order model. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a computational homogenisation-based technique for flexural effects in textile reinforced composite planar shells. An homogenisation procedure is used for the in-plane and the out-of-plane behaviour of three-dimensional woven composite shells, taking the in-plane periodicity of the material into account while relaxing any periodicity tying in the thickness direction. Several types of damage (matrix or reinforcement cracking, delamination, …) can appear in a composite material. In this paper, material non-linear computations are used to assess the importance of bending on the risk for delamination at the reinforcement/matrix interface. The normal and tangential stresses at the interface are computed and a simplified criterion for delamination is used for this purpose. The effect of flexural loading on the stress components responsible for a potential delamination failure mode at the interface is analysed. The values of interface stresses obtained by means of flexural homogenisation are compared with 3D homogenisation results using periodicity constraints along the thickness direction, and compared qualitatively with experimental facts available from the literature. The importance for taking flexural effects into account properly is emphasised. [Copyright &y& Elsevier]

This paper reviews multi-scale computational homogenisation frameworks for the non-linear behaviour of heterogeneous thin planar shells. Based on a review of some of the currently available methods, a computational homogenisation scheme for shells is applied on to representative volume elements for plain weave composites. The effect of flexural loading on the potential failure modes of such materials is analysed, focusing on the reinforcement-matrix delamination mechanism. The attention is next shifted toward failure localisation in masonry unit cells. Subsequently, a recently developed computational FE² solution scheme accounting for damage localisation at structural scales based on RVE computations is applied. [ABSTRACT FROM AUTHOR]