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A micromorphic computational homogenization framework has recently been developed to deal with materials showing long-range correlated interactions, i.e. displaying patterning modes. Typical examples of such materials are elastomeric mechanical metamaterials, in which patterning emerges from local buckling of the underlying microstructure. Because pattern transformations significantly influence the resulting effective behaviour, it is vital to distinguish them from the overall deformation. To this end, the following kinematic decomposition into three parts was introduced in the micromorphic scheme: (i) a smooth mean displacement field, corresponding to the slowly varying deformation at the macro-scale, (ii) a long-range correlated fluctuation field, related to the buckling pattern at the meso-scale, and (iii) the remaining uncorrelated local microfluctuation field at the micro-scale. The micromorphic framework has proven to be capable of predicting relevant mechanical behaviour, including size effects and spatial as well as temporal mixing of patterns in elastomeric metamaterials, making it a powerful tool to design metamaterials for engineering applications. The long-range correlated fluctuation fields need to be, however, provided a priori as input parameters. The main goal of this study is experimental identification of the decomposed kinematics in cellular metamaterials based on the three-part ansatz. To this end, a full-field micromorphic Integrated Digital Image Correlation (IDIC) technique has been developed. The methodology is formulated for finite-size cellular elastomeric metamaterial specimens deformed in (i) virtually generated images and (ii) experimental images attained during in-situ compression of specimens with millimetre sized microstructure using optical microscopy., Comment: 27 pages, 8 figures, abstract shortened to fulfil 1920 character limit

The structural properties of mechanical metamaterials are typically studied with two-scale methods based on computational homogenization. Because such materials have a complex microstructure, enriched schemes such as second-order computational homogenization are required to fully capture their non-linear behavior, which arises from non-local interactions due to the buckling or patterning of the microstructure. In the two-scale formulation, the effective behavior of the microstructure is captured with a representative volume element (RVE), and a homogenized effective continuum is considered on the macroscale. Although an effective continuum formulation is introduced, solving such two-scale models concurrently is still computationally demanding due to the many repeated solutions for each RVE at the microscale level. In this work, we propose a reduced-order model for the microscopic problem arising in second-order computational homogenization, using proper orthogonal decomposition and a novel hyperreduction method that is specifically tailored for this problem and inspired by the empirical cubature method. Two numerical examples are considered, in which the performance of the reduced-order model is carefully assessed by comparing its solutions with direct numerical simulations (entirely resolving the underlying microstructure) and the full second-order computational homogenization model. The reduced-order model is able to approximate the result of the full computational homogenization well, provided that the training data is representative for the problem at hand. Any remaining errors, when compared with the direct numerical simulation, can be attributed to the inherent approximation errors in the computational homogenization scheme. Regarding run times for one thread, speed-ups on the order of 100 are achieved with the reduced-order model as compared to direct numerical simulations.

The role of heterogeneity in the plastic flow of thin ferrite specimens is investigated in this study. This is done through a recently introduced quasi-2D experimental-numerical framework that allows for a quantitative comparison of the deformation fields of metal microstructures between experiments and simulations at a high level of detail and complexity. The method exploits samples that are locally ultra-thin ("2D") and hence have a practically uniform microstructure through their thickness. This allows testing more complex loading conditions compared to uniaxial micromechanical experiments while avoiding the complexity of an unknown subsurface microstructure, which limits comparisons between experiments and simulations in traditional integrated approaches at the level of the polycrystalline microstructure. The present approach enables to study the effect of microstructural features such as grain boundaries. To study the role of stochastic fluctuations, a constitutive model is employed which introduces random heterogeneity into a crystal plasticity model. A detailed analysis of the simulations is performed at the level of individual slip systems. Since both experimental and numerical results are susceptible to stochastic fluctuations, the outcomes of many simulations are compared to the experimentally obtained result. This comparison allows us to determine how a single experiment relates to an ensemble of simulations. Additionally, results obtained with a conventional crystal plasticity model are considered. The analysis reveals that the heterogeneity in the plasticity model is essential for accurately capturing the deformation mechanisms., Comment: 25 pages, 13 figures, submitted for publication

The slip system activity in microtensile tests of ferrite single crystals is compared with predictions made by the discrete slip plane model proposed by Wijnen et al. (International Journal of Solids and Structures 228, 111094, 2021). This is an extension of conventional crystal plasticity in which the stochastics and physics of dislocation sources are taken into account in a discrete slip band. This results in discrete slip traces and non-deterministic mechanical behavior, similar to what is observed in experiments. A detailed analysis of which slip systems are presumed to be active in experiments is performed. In small-scale mechanical tests on BCC metals and alloys, non-Schmid effects are often needed to explain the observed response. Therefore, these effects are incorporated into the model by extending a non-Schmid framework commonly used to model {110} slip to {112} planes. The slip activity in the simulations is compared to the slip activity in single crystal ferrite microtensile tests. This is done for the discrete slip plane model with and without non-Schmid effects, as well as for a conventional crystal plasticity model. The conventional crystal plasticity model fails to predict the diversity in active slip systems that is observed experimentally. The slip activity obtained with the discrete slip plane model is in convincingly better agreement with the experiments. Including non-Schmid effects only entails minor differences. This suggests that stochastic effects dominate the behavior of ferrite crystals with dimensions in the order of a few micrometers and that non-Schmid effects may not play a large role., Comment: 26 pages, 10 figures, submitted for publication

Elastomeric mechanical metamaterials exhibit unconventional mechanical behaviour owing to their complex microstructures. A clear transition in the effective properties emerges under compressive loading, which is triggered by local instabilities and pattern transformations of the underlying cellular microstructure. Such transformations trigger a non-local mechanical response resulting in strong size effects. For predictive modelling of engineering applications, the effective homogenized material properties are generally of interest. For mechanical metamaterials, these can be obtained in an expensive manner by ensemble averaging of the direct numerical simulations for a series of translated microstructures, applicable especially in the regime of small separation of scales. To circumvent this expensive step, computational homogenization methods are of benefit, employing volume averaging instead. Classical first-order computational homogenization, which relies on the standard separation of scales principle, is unable to capture any size and boundary effects. Second-order computational homogenization has the ability to capture strain gradient effects at the macro-scale, thus accounting for the presence of non-localities. Another alternative is micromorphic computational homogenization scheme, which is tailored to pattern-transforming metamaterials by incorporating prior kinematic knowledge. In this contribution, a systematic study is performed, assessing the predictive ability of computational homogenization schemes in the realm of elastomeric metamaterials. Three representative examples with distinct mechanical loading are employed for this purpose: uniform compression and bending of an infinite specimen, and compression of a finite specimen. Qualitative and quantitative analyses are performed for each of the load cases where the ensemble average solution is set as a reference., Comment: 32 pages, 19 figures, 1 table, abstract shortened to fulfil 1920 character limit

The state-of-the-art in paper mechanics calls for novel experimental data covering the full-field hygro-expansion of inter-fiber bonds in paper, i.e., the 3D morphological changes and inter-fiber interactions. Therefore, a recently developed full-field single fiber hygro-expansion measurement methodology based on global digital height correlation is extended to orthogonally bonded inter-fiber bonds, to investigate their full-field quasi-3D hygroscopic behavior. A sample holder has been developed which enables the quasi-3D characterization of the initial geometry of individual inter-fiber bonds, including the fiber thickness and width along the length of the fibers as well as the degree of wrap around and contact area of the bond, which are vital for understanding the inter-fiber bond hygro-mechanics. Full-field hygroscopic testing revealed novel insights on the inter-fiber interactions: (i) the transverse hygro-expansion of each fiber strongly reduces when approaching the bonded area, due to the significantly lower longitudinal hygro-expansion of the other bonded fiber. (ii) The relatively large transverse strain of one fiber stretches the other crossing fiber in its longitudinal direction, thereby significantly contributing to the sheet scale hygro-expansion. (iii) Out-of-plane bending is observed in the bonded region which is driven by the significant difference in transverse and longitudinal hygro-expansion of, respectively, the top and bottom fiber constituting the bond. A bi-layer laminate model is derived to rationalize the bending deformation and an adequate match is found with the experimental data. Under the assumption of zero bending, which represents constrained inter-fiber bonds inside a paper sheet, the model is able to predict the contribution of the transverse strain in the bonded regions to the sheet-scale hygro-expansion., Comment: 30 pages, 8 figures, 4 tables

Earlier works in literature on the hygro-expansion of paper state that the larger hygro-expansivity of freely compared to restrained dried handsheets is due to structural differences between the fibers inside the handsheet. To unravel this hypothesis, first, the hygro-expansion of freely and restrained dried, hardwood and softwood handsheets has been characterized. Subsequently, the transient full-field hygro-expansion (longitudinal, transverse, and shear strain) of fibers extracted from these handsheets was measured using global digital height correlation, from which the micro-fibril angle was deduced. The hygro-expansivity of each individual fiber was tested before and after a wetting period, during which the fiber's moisture content is maximized, to analyze if a restrained dried fiber can "transform" into a freely dried fiber. It was found that the longitudinal hygro-expansion of the freely dried fibers is significantly larger than the restrained dried fibers, consistent with the sheet-scale differences. The difference in micro-fibril angle between the freely and restrained dried fibers is a possible explanation for this difference, but merely for the hardwood fibers, which are able to "transform" to freely dried fibers after being soaked in water. In contrast, this "transformation" does not happen in softwood fibers, even after full immersion in water for a day. Various mechanisms have been studied to explain the observations on freely and restrained dried hardwood and softwood, fiber and handsheets including analysis of the fibers' lumen and cross-sectional shape. The presented results and discussion deepens the understanding of the differences between freely and restrained dried handsheets., Comment: 43 pages, 15 figures, 2 tables

The transient dimensional changes during \textit{hygro}-expansion and \textit{hydro}-expansion of freely and restrained dried, softwood and hardwood sheets and fibers is monitored, to unravel the governing micro-mechanisms occurring during gradual water saturation. The response of individual fibers is measured using a full-field global digital height correlation method, which has been extended to monitor the transient \textit{hydro}-expansion of fibers from dry to fully saturated. The \textit{hygro}- and \textit{hydro}-expansion is larger for freely versus restrained dried and softwood versus hardwood handsheets. The transient sheet-scale \textit{hydro}-expansion reveals a sudden strain and moisture content step. It is postulated that the driving mechanism is the moisture-induced softening of the so-called "dislocated regions" in the fiber's cellulose micro-fibrils, unlocking further fiber swelling. The strain step is negligible for restrained dried handsheets, which is attributed to the "dislocated cellulose regions" being locked in their stretched configuration during restrained drying, which is supported by the single fiber \textit{hydro}-expansion measurements. Finally, an inter-fiber bond model is exploited and adapted to predict the sheet-scale \textit{hygro}-expansion from the fiber level characteristics. The model correctly predicts the qualitative differences between freely versus restrained dried and softwood versus hardwood handsheets, yet, its simplified geometry does not allow for more quantitative predictions of the sheet-scale \textit{hydro}-expansion., Comment: 37 pages; 12 figures; 5 tables

Micromechanical constitutive parameters are important for many engineering materials, typically in microelectronic applications and material design. Their accurate identification poses a three-fold experimental challenge: (i) deformation of the microstructure is observable only at small scales, requiring SEM or other microscopy techniques; (ii) external loadings are applied at a (larger) engineering or device scale; and (iii) material parameters typically depend on the applied manufacturing process, necessitating measurements on material produced with the same process. In this paper, micromechanical parameter identification in heterogeneous solids is addressed through multiscale experiments combined with Integrated Digital Image Correlation (IDIC) in conjunction with various possible computational homogenization schemes. To this end, some basic concepts underlying multiscale approaches available in the literature are first reviewed, discussing their respective advantages and disadvantages from the computational as well as experimental point of view. A link is made with recently introduced uncoupled methods, which allow for identification of material parameter ratios at the microscale, still lacking a proper normalization. Two multiscale methods are analysed, allowing to bridge the gap between microstructural kinematics and macroscopically measured forces, providing the required normalization. It is shown that an integrated experimental--computational scheme provides relaxed requirements on scale separation. The accuracy and performance of the discussed techniques are analysed by means of virtual experimentation under plane strain and large strain assumptions for unidirectional fibre-reinforced composites. The robustness against image noise is also assessed., Comment: 44 pages, 16 figures, 4 tables, 7 procedures, abstract shortened to fulfil 1920 character limit

In particular types of layer- or lamellar-like microstructures such as pearlite and lath martensite, plastic slip occurs favorably in directions parallel to inter-lamellar boundaries. This may be due to the interplay between morphology and crystallographic orientation or, more generally, due to constraints imposed on the plastic slip due to the lamellar microstructural geometry. This paper proposes a micromechanics based, computationally efficient, scale independent model for particular type of lamellar microstructures containing softer lamellae, which are sufficiently thin to be considered as discrete slip planes embedded in a matrix representing the harder lamellae. Accordingly, the model is constructed as an isotropic visco-plastic model which is enriched with an additional orientation-dependent planar plastic deformation mechanism. This additional mode is activated when the applied load, projected on the direction of the soft films, induces a significant amount of shear stress. Otherwise, the plastic deformation is governed solely by the isotropic part of the model. The response of the proposed model is assessed via a comparison to direct numerical simulations (DNS) of an infinite periodic two-phase laminate. It is shown that the yielding behavior of the model follows the same behavior as the reference model. It is observed that the proposed model is highly anisotropic, and the degree of anisotropy depends on the contrast between the slip resistance (or yield stress) of the planar mode versus that of the isotropic part. The formulation is then applied to model the substructure of lath martensite with inter-layer thin austenite films. It is exploited in a mesoscale simulation of a dual-phase (DP) steel microstructure.The results are compared with those of a standard isotropic model and a full crystal plasticity model.

Experimental findings in the literature reveal that DP steels with identical strength and composition, but different microstructures, exhibit inconsistent trends under either necking-controlled or damage-controlled ductility tests. A special case of this phenomenon is referred to in the forming community as the cut-edge failure or edge cracking issue of DP steels. It is observed that globally ductile microstructures are prone to premature damage at cut edges while a comparatively less ductile DP steel performs better under the locally applied deformation at such edges. In this paper, a systematical statistical study is conducted to analyse this paradox from a micromechanical point of view. The obtained results qualitatively confirm the experimental observations and allowed us to explain/rationalize them, as follows. To reach a higher necking-controlled ductility, i.e. global ductility, the mechanical contrast between ferrite and martensite must be increased. Since martensite reveals a very high initial hardening and fast saturation, activation of martensite plasticity will lead to early necking in the microstructure and should therefore be avoided; a high mechanical phase contrast has this effect. However, at the same time a higher mechanical phase contrast is detrimental for the local ductility. More heterogeneity in the microstructure causes the generation of higher local plastic strains and triaxial stresses which leads to early damages. Therefore, in DP steel, increasing the global ductility may come at the expense of decreasing local ductility, and vice versa. The martensite hardening behavior is the main factor governing this phenomenon, and therefore also constitutes the key to processing improved steels that combine the best of both.

The onset of plasticity in quenched martensitic microstructures is characterized by a low initial yield stress followed by an extremely strong initial hardening response, and then a sudden hardening saturation. Literature attributes this behavior to residual stresses and dislocations inherited from the martensitic transformation, or to microstructural heterogeneities causing strength differences among the grains. Here, we argue that orientation-dependent yielding of lath martensite due to inter-lath sliding, which induces a substructure boundary sliding mechanism, may also contribute significantly to the observed behavior. To demonstrate this, we systematically study its quantitative contribution to the elasto-plastic transition behavior in a numerical microstructural model. In the simulations, we employ an effective laminate model for the martensite packets which takes into account the yielding anisotropy due to sliding along the packet's habit plane orientation. To account for the effect of carbon content, martensitic microstructures with different levels of lath strength are considered. It is shown that the martensite packets with a favorable habit plane orientation start to yield earlier compared to those with an unfavorable orientation, which initially remain elastic. As a consequence, the macro-scale response of the microstructures exhibits a low yield stress, followed by a significant degree of initial hardening which continues until the saturation stress level is approached. The apparent work hardening rate depends on the contrast between the in-habit plane and out-of-habit plane yield strength used in the model.

This work presents a detailed experimental-numerical analysis of a low-carbon dual-phase steel microstructure, revealing evidence of significant anisotropic plastic deformation in lath martensite. A careful determination of the habit plane orientations in the present martensitic variants demonstrates that the observed traces of plastic slip coincide with the directions of the corresponding habit planes. To study the local lath martensite plastic activity, a dedicated substructure-enriched crystal plasticity based model is exploited. Compared with conventional bcc crystal plasticity, the model incorporates an extra crystallographic slip plane containing 3 softer slip systems parallel to the habit planes to capture the role of habit-plane plasticity. Simulations reveal consistent strain localization patterns as those observed in the experiments, with most plasticity localized in the martensite packets with their habit plane oriented favorably with respect to the applied load. Based on these insights, recommendations for future martensite modelling strategies and potential improvements of steels are discussed.

In fibrous paper materials, an exposure to a variation in moisture content causes changes in the geometrical and mechanical properties. Such changes are strongly affected by the inter-fibre bonds, which are responsible for the transfer of the hygro-mechanical response from one fibre to its neighbours in the network, resulting in sheet-scale deformation. Most models developed in literature assume perfect bonding between fibres. In the 3D reality, there is some flexibility in the bond region, even for the perfectly bonded fibres, because of the possibility of deformation gradients through the fibre thickness. In earlier 2D idealizations, perfectly bonded fibres were assumed, implying full kinematic constraint through the entire thickness of the sheet. The purpose of the present study is to assess the effect of this assumption. Using a homogenization approach, a random network of fibres is generated with different coverages and modelled using finite elements. In order to understand the role of bonding between fibres on the hygro-expansive behaviour of a network, a bond model is developed. In this model, the fibres are modelled using 2D regular bulk finite elements and the bonds are represented by interfacial elements of finite stiffness, which are introduced between each pair of fibres bonded in the network. These embedded interfacial elements form a connection between two respective fibres, allowing relative displacements between their mid-planes. The hygro-elastic response of networks obtained with this bond model is investigated by varying the bond stiffness and the network coverage under the application of mechanical loading and changes in moisture content. Furthermore, the bond model is used to analyse the influence of inter-fibre bonds on the anisotropic response of the paper fibre network., Comment: 26 pages, 16 figures

Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour. Such a behaviour is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical macroscale predictions, a micromorphic computational homogenization scheme has recently been developed. Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. To address them, a full Newton method is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge., Comment: 34 pages, 17 figures, 1 table, 1 algorithm, abstract shortened to fulfill 1920 character limit

Materials like paper, consisting of a network of natural fibres, exposed to variations in moisture, undergo changes in geometrical and mechanical properties. This behaviour is particularly important for understanding the hygro-mechanical response of sheets of paper in applications like digital printing. A two-dimensional microstructural model of a fibrous network is therefore developed to upscale the hygro-expansion of individual fibres, through their interaction, to the resulting overall expansion of the network. The fibres are modelled with rectangular shapes and are assumed to be perfectly bonded where they overlap. For realistic networks the number of bonds is large and the network is geometrically so complex that discretizing it by conventional, geometry-conforming, finite elements is cumbersome. The combination of a level-set and XFEM formalism enables the use of regular, structured grids in order to model the complex microstructural geometry. In this approach, the fibres are described implicitly by a level-set function. In order to represent the fibre boundaries in the fibrous network, an XFEM discretization is used together with a Heaviside enrichment function. Numerical results demonstrate that the proposed approach successfully captures the hygro-expansive properties of the network with fewer degrees of freedom compared to classical FEM, preserving desired accuracy., Comment: 27 pages, 22 figures, 4 tables, J. Appl. Mech. June 19, 2020

Cellular elastomeric metamaterials are interesting for various applications, e.g. soft robotics, as they may exhibit multiple microstructural pattern transformations, each with its characteristic mechanical behavior. Numerical literature studies revealed that pattern formation is restricted in (thick) boundary layers causing significant mechanical size effects. This paper aims to experimentally validate these findings on miniaturized specimens, relevant for real applications, and to investigate the effect of increased geometrical and material imperfections resulting from specimen miniaturization. To this end, miniaturized cellular metamaterial specimens are manufactured with different scale ratios, subjected to in-situ micro-compression tests combined with digital image correlation yielding full-field kinematics, and compared to complementary numerical simulations. The specimens' global behavior agrees well with the numerical predictions, in terms of pre-buckling stiffness, buckling strain and post-buckling stress. Their local behavior, i.e. pattern transformation and boundary layer formation, is also consistent between experiments and simulations. Comparison of these results with idealized numerical studies from literature reveals the influence of the boundary conditions in real cellular metamaterial applications, e.g. lateral confinement, on the mechanical response in terms of size effects and boundary layer formation., Comment: 20 pages, 6 figures, Materials & Design, 11 May 2020

Honeycomb-like microstructures have been shown to exhibit local elastic buckling under compression, with three possible geometric buckling modes, or pattern transformations. The individual pattern transformations, and consequently also spatially distributed patterns, can be induced by controlling the applied compression along two orthogonal directions. Exploitation of this property holds great potential in, e.g., soft robotics applications. For fast and optimal design, efficient numerical tools are required, capable of bridging the gap between the microstructural and engineering scale, while capturing all relevant pattern transformations. A micromorphic homogenization framework for materials exhibiting multiple pattern transformations is therefore presented in this paper, which extends the micromorphic scheme of Roko\v{s} et al., J. Mech. Phys. Solids 123, 119-137 (2019), for elastomeric metamaterials exhibiting only a single pattern transformation. The methodology is based on a suitable kinematic ansatz consisting of a smooth part, a set of spatially correlated fluctuating fields, and a remaining, spatially uncorrelated microfluctuation field. Whereas the latter field is neglected or condensed out at the level of each macroscopic material point, the magnitudes of the spatially correlated fluctuating fields emerge at the macroscale as micromorphic fields. We develop the balance equations which these micromorphic fields must satisfy as well as a computational homogenization approach to compute the generalized stresses featuring in these equations. To demonstrate the potential of the methodology, loading cases resulting in mixed modes in both space and time are studied and compared against full-scale simulations. It is shown that the proposed framework is capable of capturing the relevant phenomena, although the inherent multiplicity of solutions entails sensitivity to the initial guess., Comment: 22 pages, 13 figures, Extreme Mechanics Letters, 8 April 2020

The economical lifetime of the divertor is a key concern for realizing nuclear fusion reactors that may solve the world's energy problem. A main risk is thermo-mechanical failure of the plasma-facing tungsten monoblocks, as a consequence of irradiation hardening induced by neutron displacement cascades. Lifetime extensions that could be carried out without prolonged maintenance periods are desired. In this work, the effects of potential treatments for extending the lifetime of an operational reactor are explored. The proposed treatments make use of cyclic recrystallization processes that can occur in neutron-irradiated tungsten. Evolution of the microstructure under non-isothermal conditions is investigated, employing a multi-scale model that includes a physically-based mean-field recrystallization model and a cluster dynamics model for neutron irradiation effects. The model takes into account microstructural properties such as grain size and displacement-induced defect concentrations. The evolution of a hardness indicator under neutron irradiation was studied. The results reveal that, for the given microstructure and under the assumed model behaviour, periodical extra heating can have a significant positive influence on controlling the irradiation hardening. For example, at 800 C, if extra annealing at 1200 C was applied after every 100 hrs for the duration of 1 hr, then the hardness indicator reduces from maximum 140 to below 70.

In nuclear fusion reactors, tungsten will be exposed to high neutron loads at high temper atures (>900 C). The evolution and degradation of the mechanical properties under these conditions is uncertain and therefore constitutes a major risk. Here, the microstructural evolution of tungsten under combined heat and neutron loads is studied, using a multi-scale approach incorporating clusters dynamics and a mean-field recrystallization model. The mean-field recrystallization model contains both nucleation in the bulk and at the grain boundaries. The cluster dynamics model includes the incorporation of loops in the dynamics of the dislocation network as a mechanism. The effects of bulk nucleation on the microstructural evolution are explored. The simulations predict a cyclically occuring neutron-induced recrystallization at all studied temperatures. Furthermore, the evolution of the irradiation hardening during neutron-induced recrystallization is assessed from the simulated microstructures., Comment: Submitted to peer-reviewed journal

The interaction of dislocations with phase boundaries is a complex phenomenon, that is far from being fully understood. A 2D Peierls-Nabarro finite element (PN-FE) model for studying edge dislocation transmission across fully coherent and non-damaging phase boundaries was recently proposed. This paper brings a new dimension to the complexity by extending the PN-FE model with a dedicated cohesive zone model for the phase boundary. With the proposed model, a natural interplay between dislocations, external boundaries and the phase boundary, including decohesion of that boundary, is provided. It allows one to study the competition between dislocation transmission and phase boundary decohesion. Commonly, the interface potentials required for glide plane behaviour and phase boundary decohesion are established through atomistic simulations. They are corresponding to the misfit energy intrinsic to a system of two bulks of atoms that are translated rigidly with respect to each other. It is shown that the blind utilisation of these potentials in zero-thickness interfaces (as used in the proposed model) may lead to a large quantitative error. Accordingly, for physical consistency, the potentials need to be reduced towards zero-thickness potentials. In this paper a linear elastic reduction is adopted. With the reduced potentials for the glide plane and the phase boundary, the competition between dislocation transmission and phase boundary decohesion is studied for an 8-dislocation pile-up system. Results reveal a strong influence of the phase contrast in material properties as well as the phase boundary toughness on the outcome of this competition. In the case of crack nucleation, the crack length shows an equally strong dependency on these properties.

In this paper, the effect of weak nonlinearities in 1D locally resonant metamaterials is investigated via the method of multiple scales. Commonly employed to the investigate the effect of weakly nonlinear interactions on the free wave propagation through a phononic structure or on the dynamic response of a Duffing oscillator, the method of multiple scales is here used to investigate the forced wave propagation through locally resonant metamaterials. The perturbation approach reveals that energy exchange may occur between propagative and evanescent waves induced by quadratic nonlinear local interaction.

Small-scale plasticity problems are often characterised by different patterning behaviours ranging from macroscopic down to the atomistic scale. In successful models of such complex behaviour, its origin lies within non-convexity of the governing free energy functional. A common approach to solve such non-convex problems numerically is by regularisation through a viscous formulation. This, however, may require the system to be overdamped and hence potentially has a strong impact on the obtained results. To avoid this side-effect, this paper addresses the treatment of the full non-convexity by an appropriate numerical solution algorithm -- the truncated Newton method. The presented method is a double iterative approach which successively generates quadratic approximations of the energy landscape and minimises these by an inner iterative scheme, based on the conjugate gradient method. The inner iterations are terminated when either a sufficient energy decrease is achieved or, to incorporate the treatment of non-convexity, a direction of negative curvature is encountered. If the latter never happens, the method reduces to Newton--Raphson iterations, solved by the conjugate gradient method, with a subsequent line search. However, in the case of a non-convex energy it avoids convergence to a saddle point and adds robustness. The stability of the truncated Newton method is demonstrated for the, highly non-convex, Peierls-Nabarro model, solved within a Finite Element framework. A potential drop in efficiency due to an occasional near singular Hessian is remedied by a trust region like extension, which is physically based on the Peierls-Nabarro disregistry profile. The result is an efficient numerical scheme with a high stability that is independent of any regularisation.

To study the nanoscopic interaction between edge dislocations and a phase boundary within a two-phase microstructure the effect of the phase contrast on the internal stress field due to the dislocations needs to be taken into account. For this purpose a 2D semi-discrete model is proposed in this paper. It consists of two distinct phases, each with its specific material properties, separated by a fully coherent and non-damaging phase boundary. Each phase is modelled as a continuum enriched with a Peierls--Nabarro (PN) dislocation region, confining dislocation motion to a discrete plane, the glide plane. In this paper, a single glide plane perpendicular to and continuous across the phase boundary is considered. Along the glide plane bulk induced shear tractions are balanced by glide plane shear tractions based on the classical PN model. The model's ability to capture dislocation obstruction at phase boundaries, dislocation pile-ups and dislocation transmission is studied. Results show that the phase contrast in material properties (e.g. elastic stiffness, glide plane properties) alone creates a barrier to the motion of dislocations from a soft to a hard phase. The proposed model accounts for the interplay between dislocations, external boundaries and phase boundary and thus represents a suitable tool for studying edge dislocation--phase boundary interaction in two-phase microstructures.

This paper presents a homogenization framework for elastomeric metamaterials exhibiting long-range correlated fluctuation fields. Based on full-scale numerical simulations on a class of such materials, an ansatz is proposed that allows to decompose the kinematics into three parts, i.e. a smooth mean displacement field, a long-range correlated fluctuating field, and a local microfluctuation part. With this decomposition, a homogenized solution is defined by ensemble averaging the solutions obtained from a family of translated microstructural realizations. Minimizing the resulting homogenized energy, a micromorphic continuum emerges in terms of the average displacement and the amplitude of the patterning long-range microstructural fluctuation fields. Since full integration of the ensemble averaged global energy (and hence also the corresponding Euler--Lagrange equations) is computationally prohibitive, a more efficient approximative computational framework is developed. The framework relies on local energy density approximations in the neighbourhood of the considered Gauss integration points, while taking into account the smoothness properties of the effective fields and periodicity of the microfluctuation pattern. Finally, the implementation of the proposed methodology is briefly outlined and its performance is demonstrated by comparing its predictions against full scale simulations of a representative example., Comment: 31 pages, 11 figures, 1 table, 1 algorithm, Journal of the Mechanics and Physics of Solids, 30 August 2018

Integrated Digital Image Correlation (IDIC) is nowadays a well established full-field experimental procedure for reliable and accurate identification of material parameters. It is based on the correlation of a series of images captured during a mechanical experiment, that are matched by displacement fields derived from an underlying mechanical model. In recent studies, it has been shown that when the applied boundary conditions lie outside the employed field of view, IDIC suffers from inaccuracies. A typical example is a micromechanical parameter identification inside a Microstructural Volume Element (MVE), whereby images are usually obtained by electron microscopy or other microscopy techniques but the loads are applied at a much larger scale. For any IDIC model, MVE boundary conditions still need to be specified, and any deviation or fluctuation in these boundary conditions may significantly influence the quality of identification. Prescribing proper boundary conditions is generally a challenging task, because the MVE has no free boundary, and the boundary displacements are typically highly heterogeneous due to the underlying microstructure. The aim of this paper is therefore first to quantify the effects of errors in the prescribed boundary conditions on the accuracy of the identification in a systematic way. To this end, three kinds of mechanical tests, each for various levels of material contrast ratios and levels of image noise, are carried out by means of virtual experiments. For simplicity, an elastic compressible Neo-Hookean constitutive model under plane strain assumption is adopted. It is shown that a high level of detail is required in the applied boundary conditions. This motivates an improved boundary condition application approach, which considers constitutive material parameters as well as kinematic variables at the boundary of the entire MVE as degrees of freedom in..., Comment: 37 pages, 25 figures, 2 tables, 2 algorithms

This paper focuses on size effects in periodic mechanical metamaterials driven by reversible pattern transformations due to local elastic buckling instabilities in their microstructure. Two distinct loading cases are studied: compression and bending, in which the material exhibits pattern transformation in the whole structure or only partially. The ratio between the height of the specimen and the size of a unit cell is defined as the scale ratio. A family of shifted microstructures, corresponding to all possible arrangements of the microstructure relative to the external boundary, is considered in order to determine the ensemble averaged solution computed for each scale ratio. In the compression case, the top and the bottom edges of the specimens are fully constrained, which introduces boundary layers with restricted pattern transformation. In the bending case, the top and bottom edges are free boundaries resulting in compliant boundary layers, whereas additional size effects emerge from imposed strain gradient. For comparison, the classical homogenization solution is computed and shown to match well with the ensemble averaged numerical solution only for very large scale ratios. For smaller scale ratios, where a size effect dominates, the classical homogenization no longer applies., Comment: 33 pages, 23 figures

The response of paper to humidity variations is a complex multi-scale problem. The hygroscopic swelling of individual fibres and their interactions within the fibrous network govern the macroscopic response. At this scale, moisture induced instabilities and out-of-plane deformations may occur. This work focuses on several aspects of this issue. A macroscopic phenomenological hygro-mechanical model is first proposed, which aims at predicting moisture induced out-of-plane deformations in paper sheets. The constitutive model is based on the relation between these deformations and typical irreversible phenomena associated to the history of paper manufacturing, i.e. the release of dried-in strains. The model is used to describe bending induced by moisture gradients through the thickness of the sheet as well as buckling due to moisture variation in the presence of mechanical constraints. The results of the model show that the anisotropic sheet-level hygro-expansion has a strong influence on the instability phenomena. Moreover, a comparison with experiments provides adequate semi-quantitative estimates. An additional step is made towards the multi-scale understanding of paper hygro-mechanics. The fundamental mechanisms governing the macroscopic moisture induced response are investigated on the basis of the underlying fibrous network. To this aim, a meso-structural model is developed which consists of a network of fibres randomly positioned in a planar region according to an orientation probability density function. A series of network simulations reveals that upon moisture content variations the expansion of the inter-fibre bonding regions drives the overall deformation. Particularly in the case of anisotropic fibre orientation, this explains the origin of the macro-scale anisotropic hygro-expansion, which is essential for the observed sheet-level instability phenomena.

This paper presents a multi-scale approach to predict the effective hygro-mechanical behaviour of paper sheets based on the properties of the underlying fibrous network. Despite the vast amount of literature on paper hygro-expansion, the functional dependence of the effective material properties on the micro-structural features remains yet unclear. In this work, a micro-structural model of the paper fibrous network is first developed by random deposition of the fibres within a planar region according to an orientation probability density function. Asymptotic homogenization is used to determine its effective properties numerically. Alternatively, two much more idealized micro-structural models are considered, one based on a periodic lattice structure with a regular network of perpendicular fibres and one based on the Voigt average. Despite their simplicity, they reproduce representative micro-structural features, such as the orientation anisotropy and network level hygro-elastic properties. These alternative models can be solved analytically, providing closed-form expressions that explicitly reveal the influence of the individual micro-scale parameters on the effective hygro-mechanical response. The trend predicted by the random network model is captured reasonably well by the two idealized models. The resulting hygro-mechanical properties are finally compared with experimental data reported in the literature, revealing an adequate quantitative agreement.

The hygro-thermo-expansive response of fibrous networks involves deformation phenomena at multiple length scales. The moisture or temperature induced expansion of individual fibres is transmitted in the network through the inter-fibre bonds; particularly in the case of anisotropic fibres, this substantially influences the resulting overall deformation. This paper presents a methodology to predict the effective properties of bonded fibrous networks. The distinctive features of the work are i) the focus on the hygro-thermo-mechanical response, whereas in the literature generally only the mechanical properties are addressed; ii) the adoption of asymptotic homogenization to model fibrous networks. Asymptotic homogenization is an efficient and versatile multi-scale technique that allows to obtain within a rigorous setting the effective material response, even for complex micro-structural geometries. The fibrous networks considered in this investigation are generated by random deposition of the fibres within a planar region according to an orientation probability density function. Most of the available network descriptions model the fibres essentially as uni-axial elements, thereby not explicitly considering the role of the bonds. In this paper, the fibres are treated as two dimensional ribbon-like elements; this allows to naturally include the contribution of the bonding regions to the effective expansion. The efficacy of the proposed study is illustrated by investigating the effective response for several network realizations, incorporating the influence of different micro-scale parameters, such as fibre hygro-thermo-elastic properties, orientation, geometry, areal coverage.

This paper studies a two-phase material with a microstructure composed of a hard brittle reinforcement phase embedded in a soft ductile matrix. It addresses the full three-dimensional nature of the microstructure and macroscopic deformation. A large ensemble of periodic microstructures is used, whereby the individual grains of the two phases are modeled using equi-sized cubes. A particular solution strategy relying on the Fast Fourier Transform is adopted, which has a high computational efficiency both in terms of speed and memory footprint, thus enabling a statistically meaningful analysis. This solution method naturally accompanies the regular microstructural model, as the Fast Fourier Transform relies on a regular grid. Using the many considered microstructures as an ensemble, the average arrangement of phases around fracture initiation sites is objectively identified by the correlation between microstructure and fracture initiation -- in three dimensions. The results show that fracture initiates where regions of the hard phase are interrupted by bands of the soft phase that are aligned with the direction of maximum shear. In such regions, the hard phase is arranged such that the area of the phase boundary perpendicular to the principal strain direction is maximum, leading to high hydrostatic tensile stresses, while not interrupting the shear bands that form in the soft phase. The local incompatibility that is present around the shear bands is responsible for a high plastic strain. By comparing the response to a two-dimensional microstructure it is observed that the response is qualitatively similar (both macroscopically and microscopically). One important difference is that the local strain partitioning between the two phases is over-predicted by the two-dimensional microstructure, leading to an overestimation of damage.

An original experimental approach is presented to automatically determine the average phase distribution around damage sites in multi-phase materials. An objective measure is found to be the average intensity around damage sites, calculated using many images. This method has the following benefits: no phase identification or manual interventions are required, and statistical fluctuations and measurement noise are effectively averaged. The method is demonstrated for dual-phase steel, revealing subtle unexpected differences in the morphology surrounding damage in strongly and weakly banded microstructures.

The precise mechanisms underlying the failure of multi-phase materials may be strongly dependent on the material's microstructural morphology. Micromechanical modeling has provided much insight into this dependence, but uncertainties remain about crucial modeling assumptions. This paper assesses the influence of different grain shapes, damage indicators, and stress states using a structured numerical model. A distinct spatial arrangement of phases around fracture incidents is found, consisting of hard regions in the tensile direction interrupted by soft regions in the directions of shear. These key features are only mildly sensitive to the studied variations.

Multi-phase material are frequently applied in a wide variety of products, as they posses a unique set of properties by combining two or more distinct phases at the level of the microstructure. Although the macroscopic stiffness and hardening are reasonably well understood, questions remain about the dominant failure mechanism(s). We identify the role of the microstructural topology (the distribution of phases) on damage "hot-spot" in the microstructure, by performing a numerical study on a large set of randomly generated topologies. The result identifies a distinct probability distribution of phases around a typical damage "hot-spot". This work is focused on assessing the sensitivity of the result to the assumptions made on the microstructural geometry.

Multi-phase materials are key for modern engineering applications. They are generally characterized by a high strength and ductility. Many of these materials fail by ductile fracture of the, generally softer, matrix phase. In this work we systematically study the influence of the arrangement of the phases by correlating the microstructure of a two-phase material to the onset of ductile failure. A single topological feature is identified in which critical levels of damage are consistently indicated. It consists of a small region of the matrix phase with particles of the hard phase on both sides in a direction that depends on the applied deformation. Due to this configuration, a large tensile hydrostatic stress and plastic strain is observed inside the matrix, indicating high damage. This topological feature has, to some extent, been recognized before for certain multi-phase materials. This study however provides insight in the mechanics involved, including the influence of the loading conditions and the arrangement of the phases in the material surrounding the feature. Furthermore, a parameter study is performed to explore the influence of volume fraction and hardness of the inclusion phase. For the same macroscopic hardening response, the ductility is predicted to increase if the volume fraction of the hard phase increases while at the same time its hardness decreases.

Computational micromechanics and homogenization require the solution of the mechanical equilibrium of a periodic cell that comprises a (generally complex) microstructure. Techniques that apply the Fast Fourier Transform have attracted much attention as they outperform other methods in terms of speed and memory footprint. Moreover, the Fast Fourier Transform is a natural companion of pixel-based digital images which often serve as input. In its original form, one of the biggest challenges for the method is the treatment of (geometrically) non-linear problems, partially due to the need for a uniform linear reference problem. In a geometrically linear setting, the problem has recently been treated in a variational form resulting in an unconditionally stable scheme that combines Newton iterations with an iterative linear solver, and therefore exhibits robust and quadratic convergence behavior. Through this approach, well-known key ingredients were recovered in terms of discretization, numerical quadrature, consistent linearization of the material model, and the iterative solution of the resulting linear system. As a result, the extension to finite strains, using arbitrary constitutive models, is at hand. Because of the application of the Fast Fourier Transform, the implementation is substantially easier than that of other (Finite Element) methods. Both claims are demonstrated in this paper and substantiated with a simple code in Python of just 59 lines (without comments). The aim is to render the method transparent and accessible, whereby researchers that are new to this method should be able to implement it efficiently. The potential of this method is demonstrated using two examples, each with a different material model.

Our goal is to unravel the mechanisms that lead to failure of a ductile two-phase material - that consists of a ductile soft phase and a relatively brittle hard phase. An idealized microstructural model is used to study damage propagation systematically and transparently. The analysis uncovers distinct microstructural features around early voids, whereby regions of the hard phase are aligned with the tensile axis and regions of the soft phase are aligned with the shear directions. These features are consistently found in regions exhibiting damage propagation, whereby the damage remains initiation driven, i.e. voids nucleate independently of each other. Upon localization, damage is controlled on a longer length-scale relying on a critical relative position of 'initiation hot-spots'. The damage rapidly increases in bands of the soft phase wherein several voids are aligned with the shear directions. The relative arrangement of the voids determines whether the microstructure fails early, or at a substantially higher strain. Although much research is needed to refine these findings for real or more realistic microstructures, in particular in three-dimensions, this paper opens a route to a deeper understanding of the ultimate failure of multi-phase materials.

Understanding the microstructural influence on the failure mechanisms in multi-phase materials calls for the identification of the worst-case scenario. This necessitates a statistical approach. By performing simulations directly based on micrographs, such an approach becomes feasible. This is applied here to extract the average microstructure around damage sites.

This paper unravels micromechanical aspects of metallic materials whose microstructure comprises grains of two or more phases. The local plastic response is determined by (i) the relative misorientation of the slip systems of individual grains, and (ii) the different mechanical properties of the phases. The relative importance of these two mechanisms at the meso-scale is unclear: is the plastic response dominated by the grain's anisotropy, or is this effect overwhelmed by the mechanical contrast between the two phases? The answer impacts the modeling of such a material at the meso-scale, but also gives insights in the resulting fracture mechanisms at that length-scale. Until now, this question has been addressed only for particular crystallographies and mechanical properties. In contrast, this paper studies the issue systematically using a large set of phase distributions, crystallographies, and material parameters. It is found that the macroscopic and the mesoscopic (grain-averaged) plastic response of the two extreme modeling choices (crystal plasticity or isotropic plasticity) converge with increasing phase contrast. The effect of the crystallography is completely overwhelmed by the phase contrast when the yield stress of the hard phase is a factor of four higher compared to the soft phase. When this ratio is lower than two, its influence may not be neglected. However, even in this regime, fracture initiation is controlled by the local arrangement of the phases. The latter is quantified in this paper through the average arrangement of the phases around fracture initiation sites.

This paper studies the competition of fracture initiation in the ductile soft phase and in the comparatively brittle hard phase in the microstructure of a two-phase material. A simple microstructural model is used to predict macroscopic fracture initiation. The simplicity of the model ensures highly efficient computations, enabling an comprehensive study: a large range of hard phase volume fractions and yield stress ratios, for wide range of applied stress states. Each combination of these parameters is analyzed using a large set of (random) microstructures. It is observed that only one of the phases dominates macroscopic fracture initiation: at low stress triaxiality the soft phase is dominant, but above a critical triaxiality the hard phase takes over resulting in a strong decrease in ductility. This transition is strongly dependent on microstructural parameters. If the hard phase volume fraction is small, the fracture initiation is dominated by the soft phase even at high phase contrast. At higher hard phase volume fraction, the hard phase dominates already at low phase contrast. This simple model thereby reconciles experimental observations from the literature for a specific combination of parameters, which may have triggered contradictory statements in the past. A microscopic analysis reveals that the average phase distribution around fracture initiation sites is nearly the same for the two failure mechanisms. Along the tensile direction, regions of the hard phase are found directly next to the fracture initiation site. This `band' of hard phase is intersected through the fracture initiation site by `bands' of the soft phase aligned with shear. Clearly, the local mechanical incompatibility is dominant for the initiation of fracture, regardless whether fracture initiates in the soft or in the hard phase.