Publications of Ludovic Noels
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See detailTowards stochastic multi-scale methods in continuum solid mechanics
Noels, Ludovic ULiege

in Advances in Applied Mechanics (in press), 55

The scientific community has realised that non-determinism is a major issue that affects structural and material performance and reliability. Because experimental characterisation alone cannot reliably ... [more ▼]

The scientific community has realised that non-determinism is a major issue that affects structural and material performance and reliability. Because experimental characterisation alone cannot reliably sample the tails of distributions, virtual stochastic testing has thus become a research field of growing interest. Since the uncertainties at the structural level also result from the variability of the micro-structure, there is a need to develop computationally efficient stochastic multi-scale methods. The purpose of this work is to provide a summary of the different methods that have been developed in the context of micro-structure characterisation and reconstruction, of stochastic homogenisation and of uncertainties up-scaling. [less ▲]

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See detailTensile failure model of carbon fibre in unidirectionally reinforced epoxy composites with mean-field homogenisation
Wu, Ling ULiege; Maillard, Etienne; Noels, Ludovic ULiege

in Composite Structures (2021), 273

This paper presents an extension of the so-called incremental-secant mean-field homogenisation (MFH) formulation accounting for fibre bundle failure and matrix cracking in Unidirectional (UD) composites ... [more ▼]

This paper presents an extension of the so-called incremental-secant mean-field homogenisation (MFH) formulation accounting for fibre bundle failure and matrix cracking in Unidirectional (UD) composites. First a model for fibre bundle failure is developed bth failure probability of the carbon fibre described by a Weibull distribution. This fibre bundle failure model is then framed in a damage model of embedded bundles in a matrix by considering an exponential relation to describe the longitudinal stress build-up profile experimentally observed during failure of embedded fibre bundles. Cracking of the matrix in UD composites is accounted for through an anisotropic non-local damage model, which allows capturing the so-called 0∘ splits experimentally observed during the longitudinal tension of UD plies. A Mean Field Homogenisation (MFH) model is then extended to account for these damage models as component behaviours of the 2-phase composite material. A finite element multi-scale simulation of a notched laminate shows that the intra-laminar failure modes observed by an in situ experiment reported in the literature are well captured by the damage variables related to the matrix and fibre bundle failure processes. Inter-laminar failure is also captured by an extrinsic cohesive law introduced between the plies. [less ▲]

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See detailMicro-mechanics and data-driven based reduced order models for multi-scale analyses of woven composites
Wu, Ling ULiege; Adam, Laurent; Noels, Ludovic ULiege

in Composite Structures (2021), 270

Order reduction of woven composite materials is based on the definition of short fibres reinforced matrix material pseudo-grains completed by pure matrix parts. The former ones model the curved yarns ... [more ▼]

Order reduction of woven composite materials is based on the definition of short fibres reinforced matrix material pseudo-grains completed by pure matrix parts. The former ones model the curved yarns, which are assimilated to continuous fibre reinforced matrix materials, in woven composites, and the latter ones model the matrix response. The homogenisation is achieved by recursively using micro-mechanics models, such as mean-field homogenisation and Voigt’s rule of mixture, and on the laminate theory. The pseudo-grains number and micro-structural features such as orientation, aspect ratio and volume fraction are considered as the Reduced Order Model (ROM) parameters and are identified following the approach of Deep Material Network (DMN): a set of homogenised elasticity tensors evaluated by computational homogenisation of woven unit-cells is used as training data in order to identify the topological parameters of the ROM. Once the topological parameters are identified, the proposed ROM can be used to conduct nonlinear analyses of woven composites. The accuracy and efficiency of the proposed ROM have been verified by comparing the predictions with direct numerical simulations on two different woven unit cells. [less ▲]

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See detailMultiscale Modeling of Composites – Piecewise-Uniform Model Order Reduction
Spilker, Kevin ULiege; Noels, Ludovic ULiege

Conference (2021, June 14)

Two-scale simulations for multiscale modeling purposes require the solution of boundary value problems for each macroscopic material point. Each macroscopic point contains a representative volume element ... [more ▼]

Two-scale simulations for multiscale modeling purposes require the solution of boundary value problems for each macroscopic material point. Each macroscopic point contains a representative volume element (RVE) that exhibits the micro-structure of the material, constituted by microscopic points.When dealing with complex heterogeneous micro-structures, the computational effort to solve the boundary problems for all macroscopic points is immense. In order to make multiscale simulations utilizable for a wider range of purposes, a reduction of the computational complexity is indispensable.A reduction of the systems internal variables can be achieved by a decomposition of the full RVE into several subdomains, constituted by clusters of microscopic material points. Constitutive equations need to be solved for all subdomains instead of for all microscopic points in the so-called “online”stage, using quantitities pre-computed in the “offline” stage. In this work, the Transformation Field Analysis (TFA) strategy is implemented, assuming uniform stress and strain fields within the subdomains(Dvorak, 1992). The division of all microscopic points into the clusters depends on their mechanical behavior, represented by the strain concentration tensors at all microscopic locations. The subdivision based on the mechanical behavior is expected to improve results in the elasto-plastic range due to the enhanced ability to account for strain concentrations, a known shortcoming of the original TFA method.The constitutive equations for the TFA model are coupling relations between the macroscopic internal variables and the internal variables in the single subdomains. The coupling equations rely on the “offline”and once for all computed strain concentration tensors of the subdomains, representing the distribution of the applied macroscopic strain in the single subdomains. After the onset of plasticity in one or more subdomains, interactions between the occurring plastic strain, treated as present eigenstrains in the corresponding subdomains, and the strain in the other clusters need to be taken into account to compute the overall RVE response. These influences rely on eigenstrain – strain interaction tensors,which are also determined once for all and numerically. The numerical instead of analytical determination of the interaction tensors allows to account for eigenstrain influences in highly heterogeneous and anisotropic geometries.For the solution of the TFA equations in the “online” stage, an increasing number of clusters provides more accurate results due to a better capability to represent plastic strain effects.The incremental tangent stiffness is commonly utilized to account for the inelastic deformation in the single subdomains. However, this approach can lead to over-stiff results according to various numerical findings. In order to recuperate this shortcoming, a different approach was tested: the use of the incremental secant stiffness instead of the incremental tangent stiffness for the subdomains. The incremental secant stiffness is determined by a virtual elastic unloading step to a vanishing stress of the homogenized material and computation of the new internal variables of the subdomains from the total unloaded state. The use of the incremental secant stiffness instead of the incremental tangent stiffness is expected to provide more accurate results and an improved way for the modeling of the material behavior under non-proportional loading conditions. The research has been funded by the Walloon Region under the agreement no.7911-VISCOS in the context of the 21st SKYWIN call [less ▲]

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See detailDuctile fracture of high strength steels with morphological anisotropy. Part II: Nonlocal micromechanics-based modeling
Leclerc, Julien ULiege; Martelleur, Matthieu; Colla, Marie Stéphane et al

in Engineering Fracture Mechanics (2021), (248), 107716

The ductile fracture behavior of a high strength steel is addressed in this two-part study using a micromechanics-based approach. The objective of Part II is to propose, identify, and validate a numerical ... [more ▼]

The ductile fracture behavior of a high strength steel is addressed in this two-part study using a micromechanics-based approach. The objective of Part II is to propose, identify, and validate a numerical model of ductile fracture based on the Gurson-Tvergaard-Needleman model. This model is enhanced by the Nahshon-Hutchinson shear modification in combination with the Thomason coalescence criterion within a fully nonlocal form and relying on a damage-to-crack transition technique. The material model involves parameters of different nature either related to the micro-mechanics of porous materials or to semi-empirical formalisms. The void nucleation model and elastoplastic behavior have been developed and identified in Part I. The other parameters are identified in this part using inverse modeling based on both the numerical results of void cell simulations and the experimental measurements. The model is shown to adequately predict the effect of stress triaxiality and Lode parameter on the fracture strain as well as the fracture anisotropy. While the cup-cone and slant fracture paths in the round bars and in the plane strain specimens, respectively, cannot be captured using the pure continuum approach, the damage-to-crack transition framework reproduces these experimental observations. [less ▲]

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See detailDuctile fracture of high strength steels with morphological anisotropy. Part I: Characterization, testing, and void nucleation law
Marteleur, Matthieu; Leclerc, Julien ULiege; Colla, Marie Stéphane et al

in Engineering Fracture Mechanics (2021), 244

The ductile fracture behavior of a high strength steel is investigated using a micromechanics-based approach with the objective to build a predictive framework for the fracture strain and crack ... [more ▼]

The ductile fracture behavior of a high strength steel is investigated using a micromechanics-based approach with the objective to build a predictive framework for the fracture strain and crack propagation under different loading conditions. Part I of this study describes the experimental results and the determination of the elastoplastic behavior and damage nucleation under different stress triaxiality and Lode parameter. The damage mechanism starts early void nucleation from elongated inclusions, either by particle cracking under loading oriented along the major axis, or by matrix decohesion when the main loading is transverse. Void nucleation is followed by plastic growth and coalescence. The long inclusion axis is preferentially aligned in one direction leading to signi cant failure anisotropy with the fracture strain in the transverse direction being almost 50% lower compared to the longitudinal one, even though the plastic behavior is isotropic. The experimental data are first used to calibrate the elastoplastic model. An enhanced anisotropic nucleation model is then developed and integrated into the Gurson-Tvergaard-Needleman scheme. The parameters identification of the anisotropic nucleation model is finally performed and validated towards the experimental results. All these elements are subsequently used in Part II to simulate the full failure behavior of all testing specimens in the entire spectrum of stress states, from nucleation to final failure. [less ▲]

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See detailNeural network-based surrogate model for multi-scale analyses
Noels, Ludovic ULiege; Wu, Ling ULiege

Scientific conference (2021, February 08)

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See detailUNSUPERVISED LEARNING BASED MODEL ORDER REDUCTION FOR HYPERELASTOPLASTICITY
Vijayaraghavan, Soumianarayanan ULiege; Beex, Lars; Noels, Ludovic ULiege et al

Scientific conference (2021, January)

Many model order reduction approaches use solutions of a few ‘offline’ training simulations to reduce the number of degrees of freedom of the many ‘online’ simulations of interest. In proper orthogonal ... [more ▼]

Many model order reduction approaches use solutions of a few ‘offline’ training simulations to reduce the number of degrees of freedom of the many ‘online’ simulations of interest. In proper orthogonal decomposition, singular value decomposition is applied to a matrix with the training solutions in order to capture the most essential characteristics in the first few modes - which are used as global interpolation bases. Proper orthogonal decomposition has proven itself as an accurate reduced order model approach for elliptical partial differential equations. In the field of solid mechanics, this means that it is accurate for (hyper)elastic material models, but not for (hyper)elastoplasticity. Based on the study of [1], the current contribution investigates how clustering of the training solutions and extracting global modes from each cluster can improve the accuracy of proper orthogonal decomposition for hyperelastoplasticity. Both centroid-based clustering (i.e. k-means clustering) and connectivity-based clustering (based on chinese whispers) are investigated. The approach is applied to a hyperelastoplastic representative volume element exposed to monotonic loading, quasi-monotonic loading and quasi-random loading. In case of monotonic and quasi-montonic loading, the components of the macroscopic deformation tensor are the variables to which clustering is applied. In case of quasi-random loading however, not only the components of the macroscopic deformation tensor and the incremental changes of these components are the variables to which clustering is applied, but also all history variables of all integration points. [1] David Amsallem1, Matthew J. Zahr2 and Charbel Farhat Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Meth. Engng. VOL 92 IS-10 SN-0029-5981 [less ▲]

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See detailA recurrent neural network-accelerated multi-scale model for elasto-plastic heterogeneous materials subjected to random cyclic and non-proportional loading paths
Wu, Ling ULiege; Nguyen, Van Dung ULiege; Kilingar, Nanda Gopala ULiege et al

in Computer Methods in Applied Mechanics and Engineering (2020), 369

An artificial Neural Network (NNW) is designed to serve as a surrogate model of micro-scale simulations in the context of multi-scale analyzes in solid mechanics. The design and training methodologies of ... [more ▼]

An artificial Neural Network (NNW) is designed to serve as a surrogate model of micro-scale simulations in the context of multi-scale analyzes in solid mechanics. The design and training methodologies of the NNW are developed in order to allow accounting for history-dependent material behaviors. On the one hand, a Recurrent Neural Network (RNN) using a Gated Recurrent Unit (GRU) is constructed, which allows mimicking the internal variables required to account for history-dependent behaviors since the RNN is self-equipped with hidden variables that have the ability of tracking loading history. On the other hand, in order to achieve accuracy under multi-dimensional non-proportional loading conditions, training of the RNN is achieved using sequential data. In particular the sequential training data are collected from finite element simulations on an elasto-plastic composite RVE subjected to random loading paths. The random loading paths are generated in a way similar to a random walking in stochastic process and allows generating data for a wide range of strain-stress states and state evolution. The accuracy and efficiency of the RNN-based surrogate model is tested on the structural analysis of an open-hole sample subjected to several loading/unloading cycles. It is shown that a similar accuracy as with a FE2 multi-scale simulation can be reached with the RNN-based surrogate model as long as the local strain state remains in the training range, while the computational time is reduced by four orders of magnitude. [less ▲]

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See detailA tutorial on Bayesian inference to identify material parameters in solid mechanics
Rappel, Hussein; Beex, Lars A A; Hale, Jake S et al

in Archives of Computational Methods in Engineering (2020), 27(2), 361385

The aim of this contribution is to explain in a straightforward manner how Bayesian inference can be used to identify material parameters of material models for solids. Bayesian approaches have already ... [more ▼]

The aim of this contribution is to explain in a straightforward manner how Bayesian inference can be used to identify material parameters of material models for solids. Bayesian approaches have already been used for this purpose, but most of the literature is not necessarily easy to understand for those new to the field. The reason for this is that most literature focuses either on complex statistical and machine learning concepts and/or on relatively complex mechanical models. In order to introduce the approach as gently as possible, we only focus on stress-strain measurements coming from uniaxial tensile tests and we only treat elastic and elastoplastic material models. Furthermore, the stress-strain measurements are created artificially in order to allow a one-to-one comparison between the true parameter values and the identified parameter distributions. [less ▲]

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See detailA nonlocal approach of ductile failure incorporating void growth, internal necking, and shear dominated coalescence mechanisms
Nguyen, Van Dung ULiege; Pardoen, Thomas; Noels, Ludovic ULiege

in Journal of the Mechanics and Physics of Solids (2020), 137

An advanced modeling framework is developed for predicting the failure of ductile materials relying on micromechanics, physical ingredients, and robust numerical methods. The approach is based on a ... [more ▼]

An advanced modeling framework is developed for predicting the failure of ductile materials relying on micromechanics, physical ingredients, and robust numerical methods. The approach is based on a hyperelastic finite strain multi-surface constitutive model with multiple nonlocal variables. The three distinct nonlocal solutions for the expansion of voids embedded in an elastoplastic matrix are considered: a void growth phase governed by the Gurson-Tvergaard-Needleman yield surface, a void necking coalescence phase governed by a heuristic extension of the Thomason yield surface based on the maximum principal stress, and a competing void shearing coalescence phase triggered by the maximum shear stress. The first solution considers the diffused plastic deformation around the voids while the last two solutions correspond to a state of plastic localization between neighboring voids. This combination captures the Lode variable and shear effects, which play important roles in dictating the damage evolution rates. The implicit nonlocal formulation with multiple nonlocal variables, including the volumetric and deviatoric parts of the plastic strain, and the mean equivalent plastic strain of the matrix, regularizes the problem of the loss of solution uniqueness when material softening occurs whatever the localization mechanism. The predictive capability of the proposed model is demonstrated through different numerical simulations in which complex failure patterns such as slant and cup-cone of respectively plane strain and axisymmetric samples under tensile loading conditions develop. [less ▲]

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See detailA micromechanics-based non-local damage to crack transition framework for porous elastoplastic solids
Leclerc, Julien ULiege; Nguyen, Van Dung ULiege; Pardoen, Thomas et al

in International Journal of Plasticity (2020), 127

The failure process of ductile porous materials is simulated by representing the damage nucleation, growth and coalescence stages up to crack initiation and propagation using a physically-based ... [more ▼]

The failure process of ductile porous materials is simulated by representing the damage nucleation, growth and coalescence stages up to crack initiation and propagation using a physically-based constitutive model. In particular, a non-local damage to crack transition framework is developed to predict the fracture under various loading conditions while minimising case-dependent calibration process. The formulation is based on a discontinuous Galerkin method, making it computationally efficient and scalable. The initial stable damage process is simulated using an implicit non-local damage model ensuring solution uniqueness beyond the onset of softening relying on the Gurson-Tvergaard-Needleman (GTN) model. Once the coalescence criterion is satisfied, which can physically arise before or during the softening stage, a cohesive band is introduced. Within the cohesive band, a void coalescence-based governing law is solved, accounting for the stress triaxiality state and material history, in order to capture the near crack tip failure process in a micro-mechanically sound way. Two coalescence models are then successively considered and compared. First, with a view to model verification towards literature results, a numerical coalescence model detects crack initiation at loss of ellipticity of a local model, and the crack opening is governed by ad-hoc parameters of the GTN model. Alternatively, the Thomason criterion is used to detect crack nucleation during the softening stage while the Thomason coalescence model governs the crack opening process. This latter model is able to reproduce slant and cup-cone failure modes in plane-strain and axisymmetric specimens, respectively. [less ▲]

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See detailBayesian inference of non-linear multiscale model parameters accelerated by a Deep Neural Network
Wu, Ling ULiege; Zulueta Uriondo, Kepa; Major, Zoltan et al

in Computer Methods in Applied Mechanics and Engineering (2020), 360

We develop a Bayesian Inference (BI) of a non-linear multiscale model and material parameters using experimental composite coupons tests as observation data. In particular we consider non-aligned Short ... [more ▼]

We develop a Bayesian Inference (BI) of a non-linear multiscale model and material parameters using experimental composite coupons tests as observation data. In particular we consider non-aligned Short Fibers Reinforced Polymer (SFRP) as a composite material system and Mean-Field Homogenization (MFH) as a multiscale model. Although MFH is computationally efficient, when considering non-aligned inclusions, the evaluation cost of a non-linear response for a given set of model and material parameters remains too prohibitive to be coupled with the sampling process required by the BI. Therefore, a Neural-Network-type (NNW) is first trained using the MFH model, and is then used as a surrogate model during the BI process, making the identification process affordable. [less ▲]

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See detailA stochastic Mean-Field-Homogenization-based micro-mechanical model of unidirectional composites failure
Wu, Ling ULiege; Calleja, Juan Manuel ULiege; Nguyen, Van Dung ULiege et al

Conference (2019, December 20)

Homogenization approaches are commonly developed in order to account for micro-structural geometrical and material properties in the framework of multiscale analyses. Although most of the approaches ... [more ▼]

Homogenization approaches are commonly developed in order to account for micro-structural geometrical and material properties in the framework of multiscale analyses. Although most of the approaches postulate the existence of a statistically Representative Volume Element (RVE), such representativity is not always ensured, in particular when studying the failure of composite materials, because of the existing micro-structural uncertainties. In this work we develop a stochastic multi-scale approach for unidirectional composite materials in order to predict the scatter existing at the structural behaviour. Statistical characteristics of the micro-structure are first extracted from SEM images in order to build a Stochastic Volume Elements (SVE) generator [1], allowing the extraction of probabilistic meso-scale stochastic behaviours from direct numerical simulations. Finally, a probabilistic Mean-Field-Homogenization (MFH) method is developed [2,3] such that the phase parameters of the MFH are defined as random fields identified from the stochastic homogenized behaviours obtained through the direct simulations of the SVEs. As a result, non-deterministic macro-scale behaviours can be studied, allowing to predict composite failure in a probabilistic way. [1] L. Wu, C. N. Chung, Z. Major, L. Adam, and L. Noels. "From SEM images to elastic responses: a stochastic multiscale analysis of UD fiber reinforced composites." Composite Structures 189C (2018): 206-227. [2] L. Wu, L. Adam, and L. Noels. "A micro-mechanics-based inverse study for stochastic order reduction of elastic UD-fiber reinforced composites analyzes." International Journal for Numerical Methods in Engineering 115, no. 12 (2018): 1430-1456. [3] L. Wu, V. D. Nguyen, L. Adam, and L. Noels. "An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites." Computer Methods in Applied Mechanics and Engineering (2019). [less ▲]

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See detailA multi-mechanism non-local porosity model for highly-ductile materials; application to high entropy alloys
Nguyen, Van Dung ULiege; Harik, Philippe; Hilhorst, Antoine et al

Conference (2019, December 18)

High ductility materials are characterized by high failure strains and high toughness properties. As a result, modelling their response up to failure requires the development of robust constitutive models ... [more ▼]

High ductility materials are characterized by high failure strains and high toughness properties. As a result, modelling their response up to failure requires the development of robust constitutive models able to represent both the hardening phase during which large deformation gradients of several tens of percent arise in combination with nucleation and growth of micro-voids, as well as the softening phase characterized by high critical energy release rate and during which coalescence of micro-voids develops. The most popular model of the ductile failure is the Gurson- Tvergaard- Needleman (so-called GTN) model, which provides a complete computational methodology for all stages of void evolution with a limited number of material parameters that can be identified based on macroscopic mechanical tests. However, the underlying phenomenological concept of void coalescence does not provide a realistic description of the void coalescence physics. Instead, the micro-mechanical-based coalescence model pioneered by Thomason provides a more physical basis under the assumption that the coalescence starts when the localization of the plastic deformation occurs in the ligaments between neighbouring voids. In this work a coupled finite-strain Gurson Thomason model is completed by a set of appropriate evolution laws governing the internal variables. The void growth phase is governed by the GTN plasticity solution and the Thomason model is used as a closed form of the plasticity problem during the coalescence stage. This provides a physically based numerical framework to represent the hardening, damage diffusion and localization stages of ductile materials. In order to avoid the loss of solution uniqueness, the damage model is formulated within an implicit gradient enhancement in which length scale effects are considered to take into account the influence of the neighbouring material points. Since the combined Gurson/Thomason model developed herein is driven by multiple softening mechanisms, it is formulated in a nonlocal setting using multiple nonlocal variables. It is shown that this approach allows recovering complex failure patterns such as slant and cup-cone of respectively plane strain and axisymmetric samples tests. Besides, the formulation is calibrated considering experimental tests performed on High Entropy Alloys (HEAs). HEAs form a new material family characterized by a combination of high strength and high toughness properties. Because of these exceptional properties, modelling their response up to failure requires the development of robust constitutive models and it is shown that the developed multi-mechanism nonlocal Gurson Thomason model provides such a framework able to reproduce the failure of HEA samples of different geometries. [less ▲]

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See detailA Bayesian framework to identify random parameter fields based on the copula theorem and Gaussian fields: Application to polycrystalline materials
Hussein, Rappel; Wu, Ling ULiege; Noels, Ludovic ULiege et al

in Journal of Applied Mechanics (2019), 86(12), 121009

For many models of solids, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application ... [more ▼]

For many models of solids, we frequently assume that the material parameters do not vary in space, nor that they vary from one product realization to another. If the length scale of the application approaches the length scale of the micro-structure however, spatially fluctuating parameter fields (which vary from one realization of the field to another) can be incorporated to make the model capture the stochasticity of the underlying micro-structure. Randomly fluctuating parameter fields are often described as Gaussian fields. Gaussian fields however assume that the probability density function of a material parameter at a given location is a univariate Gaussian distribution. This entails for instance that negative parameter values can be realized, whereas most material parameters have physical bounds (e.g. the Young’s modulus cannot be negative). In this contribution, randomly fluctuating parameter fields are therefore described using the copula theorem and Gaussian fields, which allow different types of univariate marginal distributions to be incorporated, but with the same correlation structure as Gaussian fields. It is convenient to keep the Gaussian correlation structure, as it allows us to draw samples from Gaussian fields and transform them into the new random fields. The benefit of this approach is that any type of univariate marginal distribution can be incorporated. If the selected univariate marginal distribution has bounds, unphysical material parameter values will never be realized. We then use Bayesian inference to identify the distribution parameters (which govern the random field). Bayesian inference regards the parameters that are to be identified as random variables and requires a user- defined prior distribution of the parameters to which the observations are inferred. For the homogenized Young’s modulus of a columnar polycrystalline material of interest in this study, the results show that with a relatively wide prior (i.e. a prior distribution without strong assumptions), a single specimen is sufficient to accurately recover the distribution parameter values. [less ▲]

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See detailComputational generation of open-foam representative volume elements with morphological control using distance fields
Kilingar, Nanda Gopala ULiege; Ehab Moustafa Kamel, Karim; Sonon, Bernard et al

in European Journal of Mechanics. A, Solids (2019), 78

This paper presents an automated approach to build computationally Representative Volume Elements (RVE) of open-foam cellular materials, enabling the study of the effects of the microstructural features ... [more ▼]

This paper presents an automated approach to build computationally Representative Volume Elements (RVE) of open-foam cellular materials, enabling the study of the effects of the microstructural features on their macroscopic behavior. The approach strongly relies on the use of distance and level set functions. The methodology is based on the extraction of random tessellations from inclusion packings following predetermined statistical packing distribution criteria. With the help of simple recombination operations on the distance fields, the tessellations are made to degenerate in Laguerre tessellations. Predetermined morphological characteristics like strut cross-section variation based on commercially available materials are applied on the RVE to ensure the extraction of closely matching models using simple surface extraction tools, and a detailed morphology quantification of the resulting RVEs is provided by comparing them with experimental observations. The extracted RVE surface is then treated with smoothening criteria before obtaining a 3D tetrahedralized model. This model can then be exported for multi-scale simulations to assess the effects of microstructural features by an upscaling methodology. The approach is illustrated by the simulation of a compression test on an RVE incorporating plasticity with geometrically non-linear behavior. [less ▲]

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See detailAn inverse Mean-Field-Homogenization-based micro-mechanical model for stochastic multiscale simulations of unidirectional composites
Wu, Ling ULiege; Calleja, Juan Manuel ULiege; Nguyen, Van Dung ULiege et al

Conference (2019, October 03)

Homogenization approaches have been widely developed in order to account for micro-structural geometrical and material properties in the framework of multiscale analyses. Most of the approaches postulate ... [more ▼]

Homogenization approaches have been widely developed in order to account for micro-structural geometrical and material properties in the framework of multiscale analyses. Most of the approaches postulate the existence of a statistically Representative Volume Element (RVE). However, such representativity is not always ensured, in particular when studying the failure of composite materials, because of the existing micro-structural uncertainties. In this work we develop a stochastic multi-scale approach for unidirectional composite materials in order to predict the scatter existing at the structural behaviour. Statistical characteristics of the micro-structure are first extracted from SEM images in order to build a Stochastic Volume Elements (SVE) [1] generator [2]. Probabilistic meso-scale stochastic behaviours are then extracted from direct numerical simulations of the generated SVEs. Finally, in order to provide an efficient way of exploiting the meso-scale random fields, while keeping information such as stress/strain history at the micro-scale during the resolution of macro-scale stochastic finite element, a probabilistic Mean-Field-Homogenization (MFH) method is developed [3,4]. To this end, the phase parameters of the MFH are defined as random fields, which are identified from the stochastic homogenized behaviours obtained through the stochastic direct simulations of the SVEs. As a result, non-deterministic macro-scale behaviours can be studied while having access to the micro-scale different phase stress-strain evolution, allowing to predict composite failure in a probabilistic way. [1] M. Ostoja-Starzewski, X. Wang, Stochastic finite elements as a bridge between random material microstructure and global response, Computer Methods in Applied Mechanics and Engineering 168 (14) (1999) 35 - 49, [2] L. Wu, C. N. Chung, Z. Major, L. Adam, and L. Noels. "From SEM images to elastic responses: a stochastic multiscale analysis of UD fiber reinforced composites." Composite Structures 189C (2018): 206-227. [3] L. Wu, L. Adam, and L. Noels. "A micro-mechanics-based inverse study for stochastic order reduction of elastic UD-fiber reinforced composites analyzes." International Journal for Numerical Methods in Engineering 115, no. 12 (2018): 1430-1456. [4] L. Wu, V. D. Nguyen, L. Adam, and L. Noels. "An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites." Computer Methods in Applied Mechanics and Engineering 348 (2019): 97-138. [less ▲]

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See detailNumerical Evaluation of Interaction Tensors in Heterogeneous Materials
Spilker, Kevin ULiege; Noels, Ludovic ULiege; Wu, Ling ULiege

Conference (2019, September 16)

Two-scale simulations for multiscale modeling purposes require the solution of boundary value problems for each macroscopic material point. Each macroscopic point contains a representative volume element ... [more ▼]

Two-scale simulations for multiscale modeling purposes require the solution of boundary value problems for each macroscopic material point. Each macroscopic point contains a representative volume element (RVE) that exhibits the micro-structure of the material, constituted by microscopic points. When dealing with complex heterogeneous micro-structures, the computational effort to solve the boundary problems for all macroscopic points is immense. In order to make multiscale simulations utilizable for a wider range of purposes, a reduction of the computational complexity is indispensable. A reduction of the systems internal variables can be achieved by a decomposition of the full RVE into several subdomains, where constitutive equations need to be solved for all subdomains instead of for all microscopic points. In this work, the Transformation Field Analysis (TFA) strategy [1] will be implemented, assuming uniform stress and strain fields within the subdomains. The strain inside the subdomains is affected by the present eigenstrains in all other subdomains. This requires the determination of strain concentration tensors and eigenstrain – strain interaction tensors. The computation of these quantities and the domain decomposition of the RVE can be performed once for all by FE simulations in the so-called “off-line” stage. In order to achieve a reasonable decomposition into subdomains, strain concentration tensors of all microscopic points inside the RVE, representing their mechanical behavior, are computed by the application of various boundary conditions on the RVE. Subsequently, the microscopic points are decomposed into subdomains by a clustering method based on the similarity of their mechanical behavior. The applied clustering approach for the domain decomposition may allow both for a high reduction of computational costs for the simulations and settle shortcomings due to not well captured plastic strain fields of the original TFA method. The constitutive relations for the single clusters rely on interaction effects between the clusters. Interaction tensors can be evaluated in the “off-line” stage by analytical or numerical approaches. Analytical approaches include homogenized overall properties of the RVE, being not representative in cases of the presence of dominant heterogeneous microstructures. In this work, the eigenstrain – strain interaction tensors for the TFA approach are determined numerically by off-line FE simulations. Eigenstrains are applied on each single cluster, and a comparison with the resulting strain in all clusters allows for the complete characterization of the interaction tensors. References [1] Dvorak J. Transformation Field Analysis of Inelastic Composite Materials. Proceedings: Mathematical and Physical Sciences 1992; 437:311–327. [less ▲]

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See detailA Damage to Crack Transition Framework for Ductile Failure
Leclerc, Julien ULiege; Nguyen, Van Dung ULiege; Noels, Ludovic ULiege

Conference (2019, September 03)

Accurate numerical predictions of the entire ductile failure process is still challenging. Indeed, it combines a diffuse damage stage followed by a damage localisation and crack initiation and propagation ... [more ▼]

Accurate numerical predictions of the entire ductile failure process is still challenging. Indeed, it combines a diffuse damage stage followed by a damage localisation and crack initiation and propagation. On the one hand, continuous damage models are suited for the diffuse damage stage but are inadequate for the description of physical discontinuities. On the other hand, discontinuous approaches, as cohesive zone models, are able to reproduce crack initiation and propagation, but not the damage diffusion. In this work, the presented numerical scheme joins both approaches in a discontinuous Galerkin finite element framework. A non-local implicit damage model computes the initial diffuse damage stage beyond the softening point without mesh-dependency. Then, a crack is introduced using a cohesive band model [1, 2]. Contrarily to classical cohesive models, a 3D state is recreated at the crack interface by considering a small, but finite, fictitious cohesive thickness. As a result, a strain tensor can be recomposed from the cohesive jump and the neighbouring bulk deformation gradient. A stress tensor at the interface, from which the cohesive forces are deduced, is computed using an appropriate local damage law. The framework is applied to ductile failure, modelled by a combination of the Gurson and the Thomason model [3]. The initial diffuse void growth phase is modelled by the (non-local) Gurson model [4] accounting for shear effects [5]. Then, a crack is introduced when the coalescence is reached and the behaviour of the cohesive law is computed from the Thomason model [3]. The framework capabilities are demonstrated by reproducing the slanted and the cup-cone failure respectively of a plane strain specimen and a round bar. REFERENCES [1] J.J.C. Remmers, R. de Borst., C.V. Verhoosel and A. Needleman. The cohesive band model: a cohesive surface formulation with stress triaxiality. Int. J. Fract. (2013). [2] J. Leclerc, L. Wu, V.D. Nguyen and L. Noels. Cohesive band model: a cohesive model with triaxiality for crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework. Int. J. for Num. Methods in Eng. (2018). [3] A.A. Benzerga, J.-B. Leblond, A. Needleman, V. Tvergaard. Ductile failure modelling. Int. J. Fract. (2016). [4] F. Reusch, B. Svendsen and D. Klingbeil. A non-local extension of Gurson-based ductile damage modelling. Comp. Mat. Sci. (2003). [5] K. Nahshon and J.W. Hutchinson. Modi cation of the Gurson Model for shear failure. Eur. J. of Mech. A/Sol. (2008). [less ▲]

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