Volume Element Of Microstructure Research Articles

A multi-scale formulation for polycrystalline materials accounting for cohesive micro-cracks: Homogenisation of the traction-separation law

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.

Efficient computational framework for image-based micromechanical analysis of additively manufactured Ti-6Al-4V alloy

The increase in additively manufactured (AM) Ti-6Al-4V alloys in high-performance industrial applications has necessitated the development of robust computational models that can aid in their qualification and certification. Physics-based micromechanical models, relating the AM-processed material microstructure and defect state with the overall material response and life, can play an important role in reducing uncertainty in component behavior and increasing acceptance. Motivated by this need, the present paper develops a novel image-based crystal plasticity finite element model (CPFEM) for efficient micromechanical simulation of the additively manufactured Ti-6Al-4V alloy, whose Widmanstätten microstructure is characterized by 12 HCP α lath variants in the parent β grain. A unique feature of this work is the creation of an efficient crystal plasticity framework for the parent β grain polycrystalline ensembles with parametric representation of the α lath statistics of size, shape, orientation, and crystallography. This statistical representation is expected to significantly enhance its efficiency over models that represent each α lath explicitly in the microstructure. Defects in the form of voids are represented at two scales. The smaller voids in the microstructure are manifested as porosity or void volume fraction distribution in the crystal plasticity model. Larger voids are represented explicitly in the statistically equivalent microstructural volume element (SEMVE) model. The models are built from experimentally acquired electron back scatter diffraction (EBSD) and micro-focus X-ray computed tomography (XCT) images and calibrated and validated with mechanical testing data. This paper extends the developments in Pinz et al. (2022) through the development of a special self-consistent boundary condition in the context of a concurrent model to overcome limitations of periodicity boundary conditions. The concurrent model embeds the SEMVE in a homogenized exterior domain represented by a rate-dependent isotropic plasticity model. Parametric studies are conducted to comprehend the effect of void size, shape and orientation on the overall material response.

A microstructure-based porous crystal plasticity FE model for additively manufactured Ti-6Al-4V alloys

Building from a foundation of microstructural characterization, mechanical testing, 3D statistically equivalent microstructural volume elements (SEMVEs), and image-based microstructural modeling, this paper develops an effective crystal plasticity model with porosity evolution for additively manufactured Ti-6Al-4V alloys. Their microstructure is characterized by a complex Widmanstätten morphology containing 12 α lath variants. In this paper, the morphology is parametrized by statistically equivalent ellipsoids, enabling a parametric representation of the α laths in the models. An effective crystal plasticity framework with the parametric representation of the α lath morphology is achieved by identifying the crystallographic relationship of the α laths in relation to the parent β grains from which they have nucleated and developing methods to incorporate a statistical representation of HCP α laths in β grains. The constitutive model for β grains statistically accounts for the size, shape, orientation, and crystallography of all 12 α lath variants. An important contribution is the integration of porosity evolution with the crystal plasticity model. The model is calibrated and satisfactorily validated with results from experiments on additively manufactured Ti-6Al-4V alloys with and without heat treatment. The model can yield important insights into the underlying physics of this relatively new class of materials.

ON MULTISCALE DAMAGE MODELLING OF HETEROGENEOUS MATERIALS USING NONLOCAL CONTINUUM THEORY

An overview of the modelling of quasi-brittle as well as ductile damage is given. The multiscale procedure employing the nonlocal continuum theory is described in more detail. The softening is introduced at the microlevel in the microstructural volume element and after that the homogenization procedure state variables are mapped at the macrolevel material point via the scale transition approach. In the case of quasi-brittle softening the C1 continuous finite element discretization is applied at both micro- and macrolevel. At the modelling of ductile damage response, the macrolevel is also discretized by the C1 finite element formulation, while the microscale utilizes quadrilateral mixed finite elements employing the nonlocal equivalent plastic strain and gradient-enhanced elastoplasticity. All approaches presented are verified in the standard examples.

On Ductile Damage Modelling of Heterogeneous Material Using Second-Order Homogenization Approach

The paper deals with the numerical modelling of ductile damage responses in heterogeneous materials using the classical second-order homogenization approach. The scale transition methodology in the multiscale framework is described. The structure at the macrolevel is discretized by the triangular C1 finite elements obeying nonlocal continuum theory, while the discretization of microstructural volume element at the microscale is conducted by means of the mixed type quadrilateral finite element with the nonlocal equivalent plastic strain as an additional nodal variable. The ductile damage evolution at the microlevel is modelled by using the gradient enhanced elastoplasticity. The macrolevel softening is governed by two criterions expressed by the increase in homogenized damage variable and the threshold of the local equivalent strain. The softening at each material point at the macrolevel is detected by the critical value of the homogenized damage, where homogenization of the damage variable is performed only within softening area. Due to the nonlocal continuum theory applied, a realistic softening behaviour is demonstrated after the damage initiation, compared to the widely used first-order homogenization approach. All algorithms derived have been embedded into the finite element code ABAQUS by means of the user subroutines and verified on the standard benchmark problems. The damage evolution at both microlevel and macrolevel has been demonstrated.

Multi-dimensional wavelet reduction for the homogenisation of microstructures

One of the recent fields of interest in computational homogenisation is the development of model order reduction frameworks to address the significant computational costs enabling fast and accurate evaluation of the microstructural volume element. Model order reduction techniques are applied to computationally challenging analyses of detailed micro- and or macro-structural problems to reduce both computational time and memory usage. In order to alleviate the costly integration, a wavelet-reduced order model for one-dimensional microstructural problems was presented in van Tuijl et al. (2019). This novel approach addresses both the large number of degrees of freedom and integration costs and provides control on errors in the microstructural fields. In this work, this wavelet reduced order model is extended to a multi-dimensional framework and benchmarked for more realistic multi-scale problems.The Wavelet-Reduced Order Model consists of two reduction steps. First, a Reduced Order Model is constructed to reduce the dimensionality of the microstructural model. Second, a wavelet representation is applied to reduce the integration costs of the microstructural model, whilst maintaining control over the local integration error. The multi-dimensional Wavelet-Reduced Order Model is demonstrated for a set of two-dimensional path-dependent microstructural models, evaluating their accuracy and reduction with respect to the full order models on the microstructural and homogenised fields.

Unsupervised machine learning applied to scanning precession electron diffraction data

Scanning precession electron diffraction involves the acquisition of a two-dimensional precession electron diffraction pattern at every probe position in a two-dimensional scan. The data typically comprise many more diffraction patterns than the number of distinct microstructural volume elements (e.g. crystals) in the region sampled. A dimensionality reduction, ideally to one representative diffraction pattern per distinct element, may then be sought. Further, some diffraction patterns will contain contributions from multiple crystals sampled along the beam path, which may be unmixed by harnessing this oversampling. Here, we report on the application of unsupervised machine learning methods to achieve both dimensionality reduction and signal unmixing. Potential artefacts are discussed and precession electron diffraction is demonstrated to improve results by reducing the impact of bending and dynamical diffraction so that the data better approximate the case in which each crystal yields a given diffraction pattern.

A micro-mechanical model of reinforced polymer failure with length scale effects and predictive capabilities. Validation on carbon fiber reinforced high-crosslinked RTM6 epoxy resin

We propose a micro-mechanical numerical model able to predict the nonlinear behavior and failure of unidirectional fiber reinforced high-crosslinked epoxy subjected to transverse loading conditions. Statistical microstructural volume elements (SMVE) of a realistic composite material are generated from the statistical characterization of the fibers distribution and fiber radius estimated from SEM images of a similar material system. The fibers are assumed to be transversely hyperelastic isotropic and the matrix obeys a hyperelastic viscoelastic–viscoplastic constitutive model enhanced by a multi-mechanism nonlocal damage model. This polymer model captures the pressure dependency and strain rate effects. Besides, it also accounts for size effects through its internal length scales, allowing capturing, with the same unique set of parameters, the behaviors of the epoxy as pure material as well as matrix phase in composites, which are experimentally observed to be different. Additionally, since fiber/matrix interfaces of the considered composite material are categorized as strong ones, the true underlying failure mechanism is located in the matrix close to the fibers, and the interface does not need to be explicitly introduced in the model. The model prediction is found to be in good agreement with experimental results in terms of the global nonlinear stress–strain curves over various strain rates and pressure conditions, on the one hand for pure matrix samples, and on the other hand for the composite coupons, making the proposed framework a predictive virtual testing facility for material design. Finally, using this model, we study the localization behavior in order to characterize the post-failure behavior of the composite material: the cohesive strength is given by the stress–strain curve peak stress while the critical energy release rate is estimated by evaluating the dissipated energy accumulated during the post-peak localization stage. Finally, different SMVE realizations are considered allowing assessing the discrepancy in the failure characteristics of composites.

Crack propagation in TRIP assisted steels modeled by crystal plasticity and cohesive zone method

The influence of transformation induced plasticity (TRIP) on materials mechanical behaviours, as well as failure phenomena including crack propagation and phase boundary debonding in multiphase steels (e.g. dual phase steels, TRIP steels) are studied by using an advanced crystal plasticity finite element method. We have coupled the crystal plasticity model Ma and Hartmaier (2015), which explicitly considers elastic-plastic deformation of ferrite and austenite, austenite-martensite phase, with a cohesive zone model designed for crack propagation, to study the deformations of several representative microstructural volume elements (RVE). Results shows that, the transformation induced plasticity enhances materials strength and ductility, hinders crack propagation and influences interface debonding. Furthermore, the martensitic transformation kinetics in TRIP steels was found depending on the crystallographic orientation and the stress state of a retained austenite grain. The current simulation results helps to investigate and design multiphase steels with improved mechanical properties.

On micromechanical parameter identification with integrated DIC and the role of accuracy in kinematic boundary conditions

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 the IDIC procedure, assuring that both are identified with equal precision and importance. This problem has been studied in the literature with a different method, i.e. Finite Element Method Updating framework.

Role of exterior statistics-based boundary conditions for property-based statistically equivalent representative volume elements of polydispersed elastic composites

The property-based statistically equivalent RVE or P-SERVE has been introduced in the literature as the smallest microstructural volume element in non-uniform microstructures that has effective material properties equivalent to those of the entire microstructure. An important consideration is the application of appropriate boundary conditions for optimal property-based statistically equivalent representative volume element domains. The exterior statistics-based boundary conditions have been developed, accounting for the statistics of fiber distributions and interactions in the domain exterior to the property-based statistically equivalent representative volume element. This paper is intended to validate the efficacy of the exterior statistics-based boundary condition-based property-based statistically equivalent representative volume elements for evaluating homogenized stiffnesses of a unidirectional polymer matrix composite with a polydispersed microstructure characterized by nonuniform dispersion of carbon fibers of varying sizes in an epoxy matrix. Experimental tests and microstructural characterization of the polymer matrix composite are conducted for calibration and validation of the model. Statistically equivalent microstructural volume elements are constructed from experimental micrographs for direct numerical simulations. The performance of the property-based statistically equivalent representative volume element with exterior statistics-based boundary conditions is compared with other boundary conditions, as well as with the statistical volume elements. The tests clearly show the significant advantages of the exterior statistics-based boundary conditions in terms of accuracy of the homogenized stiffness and efficiency.

Integration efficiency for model reduction in micro-mechanical analyses

Micro-structural analyses are an important tool to understand material behavior on a macroscopic scale. The analysis of a microstructure is usually computationally very demanding and there are several reduced order modeling techniques available in literature to limit the computational costs of repetitive analyses of a single representative volume element. These techniques to speed up the integration at the micro-scale can be roughly divided into two classes; methods interpolating the integrand and cubature methods. The empirical interpolation method (high-performance reduced order modeling) and the empirical cubature method are assessed in terms of their accuracy in approximating the full-order result. A micro-structural volume element is therefore considered, subjected to four load-cases, including cyclic and path-dependent loading. The differences in approximating the micro- and macroscopic quantities of interest are highlighted, e.g. micro-fluctuations and stresses. Algorithmic speed-ups for both methods with respect to the full-order micro-structural model are quantified. The pros and cons of both classes are thereby clearly identified.

Microstructural statistics for fatigue crack initiation in polycrystalline nickel-base superalloys

In advanced engineering alloys where inclusions and pores are minimized during processing, the initiation of cracks due to cyclic loading shifts to intrinsic microstructural features. Criteria for the identification of crack initiation sites, defined using elastic-plastic loading parameters and twin boundary length, have been developed and applied to experimental datasets following cyclic loading. The criteria successfully quantify the incidence of experimentally observed cracks. Statistical microstructural volume elements are defined using a convergence approach for two nickel-base superalloys, IN100 and René 88DT. The material element that captures the fatigue crack-initiating features in René 88DT is smaller than IN100 due to a combination of smaller grain size and higher twin density.

Continuum understanding of twin formation near grain boundaries of FCC metals with low stacking fault energy

Deformation twinning from grain boundaries is often observed in face-centered cubic metals with low stacking fault energy. One of the possible factors that contribute to twinning origination from grain boundaries is the intergranular interactions during deformation. Nonetheless, the influence of mechanical interaction among grains on twin evolution has not been fully understood. In spite of extensive experimental and modeling efforts on correlating microstructural features with their twinning behavior, a clear relation among the large aggregate of grains is still lacking. In this work, we characterize the micromechanics of grain-to-grain interactions that contribute to twin evolution by investigating the mechanical twins near grain boundaries using a full-field crystal plasticity simulation of a twinning-induced plasticity steel deformed in uniaxial tension at room temperature. Microstructures are first observed through electron backscatter diffraction technique to obtain data to reconstruct a statistically equivalent microstructure through synthetic microstructure building. Grain-to-grain micromechanical response is analyzed to assess the collective twinning behavior of the microstructural volume element under tensile deformation. Examination of the simulated results reveal that grain interactions are capable of changing the local mechanical behavior near grain boundaries by transferring strain across grain boundary or localizing strain near grain boundary.

A probabilistic mesoscale damage detection in polycrystals using a random matrix approach

This article is concerned with a probabilistic mesoscale damage detection in polycrystals. For this purpose, we make use of a stochastic model describing the linear elasticity matrix of material at the mesoscale. The model is constructed using a maximum entropy principle and random matrix theory and allows one to directly construct a probabilistic model for the system random matrices characterizing the constitutive behavior of the system. First, the theoretical framework and upscale scheme in the construction of the model are briefly reviewed. For each case of healthy and damaged materials, where the damage is introduced in the form of intergranular microcavities, the random matrix model is calibrated by performing simulations on an ensemble of statistical volume elements of microstructure. The calibrated models are then used in a simple coarse-scale simulation in order to explore the sensitivity of the model in detecting the location of mesoscale damages. The result shows that in most cases, one can identify the location of cracks by comparing the probabilistic description of a suitable response quantity of interest predicted for both healthy and damaged systems. Such a probabilistic description is suitable for detecting signature of fine-scale defects where the consequences are reflected at the coarse scale in the form of random fluctuations around the mean behavior. The model can be used as a predictive tool in the context of structural health monitoring and damage prognosis of metallic systems.

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

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

Multi-scale continuous–discontinuous framework for computational-homogenization–localization

In this paper, a multi-scale technique is proposed for the modeling of microstructured materials up to the point of macroscopic failure. A continuous–discontinuous computational homogenization–localization framework is developed, which involves a discontinuity enhanced macroscale description. The underlying microstructural volume element (MVE) enables the incorporation of a band with high strains, i.e. a localization band. For the multi-scale coupling, special scale transition relations are established to handle the underlying material response of both the bulk and the localization zone. At a macroscopic (integration) point, the macroscale displacement jump and the deformation of the surrounding continuum material are used to formulate kinematical boundary constraints for the microscale MVE problem. Upon proper averaging of the MVE response, the macroscopic generalized stress is obtained. Simultaneously, a microscale effective displacement jump is recovered and returned to the macrolevel, based on the MVE deformation field. The equality of the macro- and effective microscale displacement jumps is enforced at the macroscale by Lagrange multipliers, being cohesive tractions at the interface. The applicability of the developed continuous–discontinuous computational homogenization framework is illustrated on two simple benchmark problems involving evolving macroscale discontinuities as a result of microstructural degradation by void growth.

Novel boundary conditions for strain localization analyses in microstructural volume elements

SUMMARYMulti‐scale modeling frequently relies on microstructural representative volume elements (RVEs) on which macroscopic deformation is imposed through kinematical boundary conditions. A particular choice of these boundary conditions may influence the obtained effective properties. For strain localization and damage analyses, the RVE is pushed beyond the limits of its representative character, and the applied boundary conditions have a significant impact on the onset and the type of macroscopic material instability to be predicted.In this article, we propose a new type of boundary conditions for microstructural volume elements, called percolation‐path‐aligned boundary conditions. Intrinsically, these boundary conditions capture the constraining effect of the material surrounding the RVE upon developing localization bands. The alignment with evolving localization bands allows the highly strained band to cross the RVE and fully develop with minimal interference of the applied boundary conditions. For an illustration of the performance of the newly proposed boundary conditions, macroscopic deformation has been imposed on a voided elasto‐plastic RVE using different types of boundary conditions. It is observed that the new RVE boundary conditions provide a good estimate for the effective stiffness, are not susceptible to spurious localization, and permit the development of a full strain localization band up to failure. Copyright © 2011 John Wiley & Sons, Ltd.

Enabling microstructure-based damage and localization analyses and upscaling

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

Microstructural effects on ductile fracture in heterogeneous materials. Part II: Applications to cast aluminum microstructures

Abstract This is the second of a two part paper aimed at investigating the effects of microstructural morphology, material properties and loading on rate-dependent ductile fracture of heterogeneous materials. The locally enhanced Voronoi cell finite element method (LE-VCFEM) is used for micromechanical analyses of deformation and failure in complex microstructural volume elements. The first part of this paper sequence evaluates the sensitivity of strain to failure of computer simulated microstructures to loading rate, microstructural morphology and material properties. In this second part, LE-VCFEM simulations of actual microstructures of a cast aluminum alloy micrograph are used to validate a strain to failure model developed in the first part. A method for identification of critical regions within a heterogeneous microstructure is also developed and validated using in-situ observations of a two-point bending test. The influence of applied strain rates on ductile fracture of micrograph-based complex microstructures is also investigated.