Publications of Ludovic Noels
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See detailBayesian inference of multiscale model parameters with artificial neural networks as surrogate
Wu, Ling ULiege; Noels, Ludovic ULiege

Conference (2021, December 13)

In the context of multiscale models, it is not always possible to identify the constituents properties and inverse analysis is a way to identify them from experimental data conducted at the higher scale ... [more ▼]

In the context of multiscale models, it is not always possible to identify the constituents properties and inverse analysis is a way to identify them from experimental data conducted at the higher scale. For example, non-aligned Short Fibers Reinforced Polymer (SFRP) responses can be modelled by Mean-Field Homogenization (MFH) but some geometrical parameters, such as the effective aspect ratio, and some phase material parameters, such as the matrix model parameters, should be inferred from composite experimental responses in order to avoid extensive measurement campaigns at the micro-scale. In practice, because of the increase in the number of parameters in the non-linear models, this identification requires several loading conditions, and a unique set of parameters cannot reproduce all the experimental tests because, on the one hand, of the model limitations and, on the other hand, of the experimental errors [1]. Bayesian Inference (BI) allows circumventing these difficulties, but requires a large amount of the model evaluations during the sampling process. 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. In this work, a Neural-Network (NNW) is first trained using the MFH model, and is then used as a surrogate model during the BI process which is conducted using experimental composite coupon tests as observation data [2]. [less ▲]

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See detailA Recurrent Neural Network-based Surrogate Model for History-Dependent Multi-scale Simulations of Composite Materials
Wu, Ling ULiege; Noels, Ludovic ULiege

Scientific conference (2021, December 08)

In order to make Computational homogenization affordable, pre-off-line finite element simulations are conducted on the mesoscale problem in order to build a synthetic database that can, in turn, be used ... [more ▼]

In order to make Computational homogenization affordable, pre-off-line finite element simulations are conducted on the mesoscale problem in order to build a synthetic database that can, in turn, be used to train surrogate models, which can be used as a constitutive law on a classical finite element simulation, speeding up the multi-scale process by several orders. Artificial neural networks (NNWs) offer the possibility to serve as a surrogate model, but a difficulty arises for elasto-plasticity because of its history-dependency. This difficulty can be solved by considering a Recurrent Neural Network (RNN), which uses sequential information [1]. Nevertheless, in order to be accurate under multi-dimensional non-proportional loading conditions, a sufficiently wide database is required in order to perform the training. To this end, a sequential training synthetic database is obtained from finite element simulations on an elasto-plastic RVE subjected to random loading paths. The RNN predictions are thus found to be in agreement with the FE2 simulations, while reducing the computational cost by 4 orders. Nevertheless, such a paradigm is essentially used as a mapping between the macro-stress and macro-strain tensors of the micro-scale boundary value response and the micro-structure information could not be recovered in a so-called localization step. We thus also develop Recurrent Neural Networks (RNNs)-based surrogate of the local micro-structure state variables for complex loading scenarios [2]. In order to address the curse of dimensionality arising because of the large amount of internal state variables in the micro-structure, we enrich the RNN with PCA dimensionality reduction and dimensionality break down, i.e. the use of several RNNs instead of a single one. The sequential training strategy is optimized to allow for GPU usage. [less ▲]

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See detailPer-phase spatial correlated damage models of UD fibre reinforced composites using mean-field homogenisation; applications to notched laminate failure and yarn failure
Wu, Ling ULiege; Zhang, Tianyu ULiege; Maillard, Etienne et al

in Computers and Structures (2021), 257

micro-mechanical model for fibre bundle failure is formulated following a phase-field approach and is embedded in a semi-analytical homogenisation scheme. In particular mesh-independence and consistency ... [more ▼]

micro-mechanical model for fibre bundle failure is formulated following a phase-field approach and is embedded in a semi-analytical homogenisation scheme. In particular mesh-independence and consistency of energy release rate for fibre bundles embedded in a matrix phase are ensured for fibre dominated failure. Besides, the matrix cracking and fibre-matrix interface debonding are modelled through the evolution of the matrix damage variable framed in an implicit non-local form. Considering the material parameters of both fibre and epoxy matrix phases identified from manufacturer data sheets, it is shown that the failure strength of a ply loaded along the longitudinal direction is in agreement with the reported values. Finally, the multi-damage homogenisation framework is applied to model, on the one hand, the failure of a notched laminate, in which case the failure modes are observed to be in good agreement with experiments, and, on the other hand, the failure of yarns in a plain woven composite unit-cell under uni-axial tension [less ▲]

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See detailA Recurrent Neural Network-based Surrogate Model for History-Dependent Multi-scale Simulations
Wu, Ling ULiege; Noels, Ludovic ULiege

Scientific conference (2021, November 04)

Homogenization-based multi-scale analyses are widely used to account for the effect of material heterogeneity at a structural material point. Among the existing different homogenization methods ... [more ▼]

Homogenization-based multi-scale analyses are widely used to account for the effect of material heterogeneity at a structural material point. Among the existing different homogenization methods, computational homogenization solves the meso-scale heterogeneous problems using a full field discretization of the micro-structure. When embedded in a multi-scale analyses, computational homogenization results in the so-called FE2 method, which is an accurate methodology but which yields prohibitive computational time. A more efficient approach is to conduct pre-off-line finite element simulations on the meso-scale problem in order to build a surrogate model by means of constructing mapping functions. Once this so-called training step is completed, the surrogate model can be used as the constitutive law of a single-scale simulation, leading to highly efficient simulations. Artificial neural networks (NNWs) offer the possibility to build such a mapping. However, one difficulty arises for history-dependent material behaviours, such as elasto-plasticity, since state variables are needed to account for the loading history. This difficulty can be solved by considering a Recurrent Neural Network (RNN), which uses sequential information. In [1] a RNN was designed using a Gated Recurrent Unit (GRU). In order to achieve accuracy under multi-dimensional non-proportional loading conditions, the sequential training data were obtained from finite element simulations on an elastoplastic composite RVE subjected to random loading paths. The RNN predictions were found to be in agreement with the finite elements simulations. [less ▲]

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See detailA Non-Local Ductile Failure Model Accounting for Void Growth and Coalescence at Low and High Stress Triaxiality
Nguyen, Van Dung ULiege; Pardoen, Thomas; Noels, Ludovic ULiege

Conference (2021, September 09)

Ductile failure is controlled by the nucleation, growth, and coalescence of voids combined with an extensive plastic dissipation accumulating before failure. Although the Gurson- Tvergaard- Needleman [1 ... [more ▼]

Ductile failure is controlled by the nucleation, growth, and coalescence of voids combined with an extensive plastic dissipation accumulating before failure. Although the Gurson- Tvergaard- Needleman [1] model, which is the most popular model of the ductile failure, gives a complete computational methodology for all stages of void evolution, the framework remains phenomenological and does not provide a realistic description of the physics of the void coalescence. In this work, a hyperelastic finite strain multi-surface constitutive model with multiple nonlocal variables is developed for predicting the failure of ductile materials [2]. This model is based on the combination of the three distinct nonlocal solutions of expansion of voids embedded in an elastoplastic matrix. The void growth phase governed by the GTN model considers the diffusion of the plastic deformation around voids. The first coalescence mode considered is by void necking and is governed by a heuristic extension of the Thomason model based on the maximal principle stress. The second coalescence mode considered is by void shearing triggered by the maximal shear stress. In order to avoid the loss of solution uniqueness when material softening occurs whatever the localization mechanism is, an implicit nonlocal formulation with multiple nonlocal variables, including the volumetric and deviatoric parts of the plastic deformation, and the mean plastic deformation of the matrix, regularises the problem. Macro-scale numerical tests show that the current approach allows capturing complex failure modes such as slant failure for specimens in plane strain and cup-cone failure for cylindrical. [less ▲]

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See detailA Recurrent Neural Network-based Surrogate Model for History-Dependent Multi-scale Simulations
Wu, Ling ULiege; Cobian, Lucia; Hössinger-Kalteis, Anna et al

Conference (2021, September)

Homogenization-based multi-scale analyses are widely used to account for the effect of material heterogeneity at a structural material point. Among the existing different homogenization methods ... [more ▼]

Homogenization-based multi-scale analyses are widely used to account for the effect of material heterogeneity at a structural material point. Among the existing different homogenization methods, computational homogenization solves the meso-scale heterogeneous problems using a full field discretization of the micro-structure. When embedded in a multi-scale analyses, computational homogenization results in the so-called FE2 method, which is an accurate methodology but which yields prohibitive computational time. A more efficient approach is to conduct pre-off-line finite element simulations on the meso-scale problem in order to build a surrogate model by means of constructing mapping functions. Once this so-called training step is completed, the surrogate model can be used as the constitutive law of a single-scale simulation, leading to highly efficient simulations. Artificial neural networks (NNWs) offer the possibility to build such a mapping. However, one difficulty arises for history-dependent material behaviours, such as elasto-plasticity, since state variables are needed to account for the loading history. This difficulty can be solved by considering a Recurrent Neural Network (RNN), which uses sequential information. In [1] a RNN was designed using a Gated Recurrent Unit (GRU). In order to achieve accuracy under multi-dimensional non-proportional loading conditions, the sequential training data were obtained from finite element simulations on an elastoplastic composite RVE subjected to random loading paths. The RNN predictions were found to be in agreement with the finite elements simulations. In the current work, we are applying the method to metamaterials. The RNN can be trained for different cell geometries, like BCC metamaterials. [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 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 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 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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