Wu, L., Chung, C. N., Major, Z., Adam, L., & Noels, L. (2018). From SEM images to elastic responses: a stochastic multiscale analysis of UD fiber reinforced composites. *Composite Structures, 189C*, 206-227.

Peer reviewed (verified by ORBi)

In this work, the elastic response of unidirectional fiber (UD) reinforced composites is studied in a stochastic multiscale way. First, the micro-structure of UD carbon fiber reinforced composites is statistically studied based on SEM images of its cross-section and an algorithm to generate numerical micro-structures with an equivalent random distribution of fibers is developed. In particular, based on the images spatial analysis, the empirical statistical descriptors are considered as dependent variables and represented using the copula framework, allowing generating micro-structure realizations used as Stochastic Volume Elements (SVEs). Second, a stochastic scale transition is conducted through the homogenization of SVEs. With a view to the use of the resulting meso-scale random field in structural stochastic analyzes, the homogenization is performed in two steps in order to respect the statistical content from the micro-meter-long
SVEs to the millimeter-long structural finite elements. To this end, the computational homogenization is applied in a hierarchy model: i) Micro-structure generator produces Small SVEs (SSVEs) which are homogenized; ii) Big SVEs (BSVEs) are constructed from the SSVEs. Finally, it is shown on simple illustrative examples that the scatter of the (homogenized) stress distribution in a composite ply can be simulated by means of the developed methodology.

Homsi, L., & Noels, L. (2018). A discontinuous Galerkin method for non-linear electro-thermo-mechanical problems; Application to shape memory composite materials. *Meccanica, 53*(6), 1357-1401.

Peer reviewed

A coupled Electro-Thermo-Mechanical Discontinuous Galerkin (DG) method is developed considering the non-linear interactions of electrical, thermal, and mechanical fields. In order to develop a stable discontinuous Galerkin formulation the governing equations are expressed in terms of energetically conjugated fields gradients and fluxes. Moreover, the DG method is formulated in finite deformations and finite fields variations. The multi-physics DG formulation is shown to satisfy the consistency condition, and the uniqueness and optimal convergence rate properties are derived under the assumption of small deformation. First the numerical properties
are verified on a simple numerical example, and then the framework is applied to simulate the response of smart composite materials in which the shape memory effect of the matrix is triggered by the Joule effect.

Leclerc, J., Wu, L., Nguyen, V. D., & Noels, L. (2018). A damage to crack transition model accounting for stress triaxiality formulated in a hybrid non-local implicit discontinuous Galerkin - cohesive band model framework. *International Journal for Numerical Methods in Engineering, 113*(3), 374-410.

Peer reviewed (verified by ORBi)

Modelling the entire ductile fracture process remains a challenge. On the one hand, continuous damage models succeed in capturing the initial diffuse damage stage but are not able to represent discontinuities or cracks. On the other hand, discontinuous methods, as the cohesive zones, which model the crack propagation behaviour, are suited to represent the localised damaging process. However, they are unable to represent diffuse damage. Moreover, most of the cohesive models do not capture triaxiality effect.
In this paper, the advantages of the two approaches are combined in a single damage to crack transition framework. In a small deformation setting, a non-local elastic damage model is associated with a cohesive model in a discontinuous Galerkin finite element framework. A cohesive band model is used to naturally introduce a triaxiality-dependent behaviour inside the cohesive law. Practically, a numerical thickness is introduced to recover a 3D-state, mandatory to incorporate the in-plane stretch effects. This thickness is evaluated to ensure the energy consistency of the method and is not a new numerical parameter. The traction-separation law is then built from the underlying damage model.

Guzman, C. F., Yuan, S., Duchene, L., Flores, E. I. S., & Habraken, A. (2018). Damage prediction in single point incremental forming using an extended Gurson model. *International Journal of Solids and Structures, 151*, 45-56.

Peer reviewed (verified by ORBi)

Single point incremental forming (SPIF) has several advantages over traditional forming, such as the high
formability attainable by the material. Different hypotheses have been proposed to explain this behavior,
but there is still no straightforward relation between the particular stress and strain state induced by SPIF
and the material degradation leading to localization and fracture. A systematic review of the state of the
art about formability and damage in SPIF is presented and an extended Gurson–Tvergaard–Needleman
(GTN) model was applied to predict damage in SPIF through finite element (FE) simulations. The line
test was used to validate the simulations by comparing force and shape predictions with experimental
results. To analyze the failure prediction, several simulations of SPIF cones at different wall angles were
performed. It is concluded that the GTN model underestimates the failure angle on SPIF due to wrong
coalescence modeling. A physically-based Thomason coalescence criterion was then used leading to an
improvement on the results by delaying the onset of coalescence.

Nguyen, V. D., Wu, L., & Noels, L. (2017). Unified treatment of boundary conditions and efficient algorithms for estimating tangent operators of the homogenized behavior in the computational homogenization method. *Computational Mechanics, 59*(3), 483-505.

Peer reviewed (verified by ORBi)

This work provides a unified treatment of arbitrary kinds of microscopic boundary conditions usually considered in the multi-scale computational homogenization method for nonlinear multi-physics problems. An efficient procedure is developed to enforce the multi-point linear constraints arising from the microscopic boundary condition either by the direct constraint elimination or by the Lagrange multiplier elimination methods. The macroscopic tangent operators are computed in an efficient way from a multiple right hand sides linear system whose left hand side matrix is the stiffness matrix of the microscopic linearized system at the converged solution. The number of vectors at the right hand side is equal to the number of the macroscopic kinematic variables used to formulate the microscopic boundary condition. As the resolution of the microscopic linearized system often follows a direct factorization procedure, the computation of the macroscopic tangent operators is then performed using this factorized matrix at a reduced computational time.

Mertens, A., L'Hoest, T., Magnien, J., Carrus, R., & Lecomte-Beckers, J. (2017). On the Elaboration of Metal-Ceramic Composite Coatings by Laser Cladding. *Materials Science Forum, 879*, 1288-1293.

Peer reviewed

This paper reports on a preliminary investigation into the elaboration, by the additive process known as laser cladding, of composite coatings with a matrix of stainless steel 316L reinforced with varying contents of tungsten (WC) or silicon carbides (SiC) particles. Laser cladding is characterised by ultra-fast solidification and cooling rates, thus giving rise to ultra-fine out-of-equilibrium microstructures and potentially enhanced mechanical properties. Both types of composite coatings – i.e. with SiC or WC ‒ are compared in terms of their microstructures and hardness. Special attention is given to the dissolution of the carbides particles and to interfacial reactions taking place between the particles and the metallic matrix.

Noels, L., Wu, L., Adam, L., Seyfarth, J., Soni, G., Segurado, J., Laschet, G., Chen, G., Lesueur, M., Lobos, M., Böhlke, T., Reiter, T., Oberpeilsteiner, S., Sallaberger, D., Weichert, D., & Broekmann, C. (2016). Chapter 6: Effective Properties. In P., Ulrich & G. J., Schmitz (Eds.), *Handbook of Software Solutions for ICME* (pp. 433-441). Weinheim, Germany: Wiley-VCH.

As one of the results of an ambitious project, this handbook provides a well-structured directory
of globally available software tools in the area of Integrated Computational Materials
Engineering (ICME).
The compilation covers models, software tools, and numerical methods allowing describing
electronic, atomistic, and mesoscopic phenomena, which in their combination determine the
microstructure and the properties of materials. It reaches out to simulations of component
manufacture comprising primary shaping, forming, joining, coating, heat treatment, and
machining processes. Models and tools addressing the in-service behavior like fatigue, corrosion,
and eventually recycling complete the compilation.
An introductory overview is provided for each of these different modelling areas highlighting the
relevant phenomena and also discussing the current state for the different simulation approaches.
A must-have for researchers, application engineers, and simulation software providers seeking
a holistic overview about the current state of the art in a huge variety of modelling topics.
This handbook equally serves as a reference manual for academic and commercial software developers
and providers, for industrial users of simulation software, and for decision makers seeking
to optimize their production by simulations. In view of its sound introductions into the different
fields of materials physics, materials chemistry, materials engineering and materials processing
it also serves as a tutorial for students in the emerging discipline of ICME, which requires a broad
view on things and at least a basic education in adjacent fields.

Wu, L., Lucas, V., Nguyen, V. D., Golinval, J.-C., Paquay, S., & Noels, L. (2016). A Stochastic Multi-Scale Approach for the Modeling of Thermo-Elastic Damping in Micro-Resonators. *Computer Methods in Applied Mechanics and Engineering, 310*, 802-839.

Peer reviewed (verified by ORBi)

The aim of this work is to study the thermo-elastic quality factor (Q) of micro-resonators with a stochastic multi-scale approach. In the design of high-Q micro-resonators, thermo-elastic damping is one of the major dissipation mechanisms, which may have detrimental effects on the quality factor, and has to be predicted accurately. Since material uncertainties are inherent to and unavoidable in micro-electromechanical systems (MEMS), the effects of those variations have to be considered in the modeling in order to ensure the required MEMS performance. To this end, a coupled thermo-mechanical stochastic multi-scale approach is developed in this paper. Thermo-mechanical micro-models of polycrystalline materials are used to represent micro-structure realizations. A computational homogenization procedure is then applied on these statistical
volume elements to obtain the stochastic characterizations of the elasticity tensor, thermal expansion, and conductivity tensors at the meso-scale. Spatially correlated meso-scale random fields can thus be generated to represent the stochastic behavior of the homogenized material properties. Finally, the distribution of the thermo-elastic quality factor of MEMS resonators is studied through a stochastic finite element method using as input the generated stochastic random field.

Nguyen, V. D., Wu, L., Homsi, L., & Noels, L. (2016, September 08). *Unified treatment of microscopic boundary conditions in computational homogenization method for multiphysics problems*. Paper presented at 15th edition of the European Mechanics of Materials Conference (EMMC15), Brussels, Belgium.

Computational homogenization (so-called FE2) method is an effective tool to model complex behavior of heterogeneous media allowing direct coupling between the structure response and the evolving microstructure not only in purely mechanical problems but also in multiphysics problems [1]. The basic idea of this method is to obtain the macroscopic constitutive relationships from the resolution of the microscopic boundary value problem (BVP) defined on a representative volume element. This method does not requires any constitutive assumption at the macroscopic level, but an appropriate microscopic boundary condition has to be defined. Our work focuses on the unified treatment of the microscopic boundary condition in a multiphysics microscopic BVP. In particular, an efficient way to compute the tangent operator is developed for an arbitrary kind of boundary conditions.
When considering the FE2method, the homogenized stresses and homogenized tangents at every macroscopic integration points are required. From the energy consistency condition between macroscopic and microscopic problems, the homogenized stresses can be easily computed by the volumetric averaging integrals of the microscopic counterparts. The required homogenized tangents often follows a stiffness condensation from the microscopic stiffness matrix at the equilibrium state [2]. When using the stiffness condensation, the microscopic stiffness matrix needs to be partitioned, and dense matrices based on Schur complements (under a matrix form 𝐊̃ 𝑏𝑏=𝐊𝑏𝑏−𝐊𝑏𝑖𝐊𝑖𝑖−1𝐊𝑖𝑏) have to be estimated. The matrix operations based on Schur complements require a large time consuming and a lot of memory when increasing the number of degrees of freedom of the microscopic BVPs. This work proposes an efficient method allowing to compute the homogenized tangents without significant effort. The microscopic stiffness matrix does not need to be partitioned. The homogenized tangents are computed by solving a linear system, which is based on the linearized system at the converge solution of the microscopic BVP, with multiple right hand sides.
With proposed numerical improvements, the FE2 method is used in a fully thermo-mechanically-coupled simulation. The temperature-dependent elastoplastic behavior, thermal conduction as well as the heat conversion from the mechanical deformation are considered in the hyperelastic large strain framework.
[1]. Geers, M. G. D., Kouznetsova, V. G., Brekelmans, W. A. M., 2010. J. Comput. Appl. Math. 234 (7), 2175-2182.
[2]. Kouznetsova, V., Brekelmans, W. A. M., Baaijens, F. P. T., 2001. Comput. Mech. 27 (1), 37-48.

Wu, L., Lucas, V., Golinval, J.-C., Paquay, S., & Noels, L. (2016, September 07). *Probabilistic prediction of the quality factor of micro-resonator using a stochastic thermo-mechanical multi-scale approach*. Paper presented at 15th edition of the European Mechanics of Materials Conference (EMMC15), Brussels, Belgium.

As the size of the device is only one or two orders of magnitude higher than the size of the grains, the structural properties, such as the thermo-elastic quality factor (Q), of micro-electro-mechanical systems (MEMS) made of poly-crystalline materials exhibit a scatter, due to the existing randomness in the grain size, grain orientation, surface roughness...
In order to predict the probabilistic behavior of micro-resonators, the authors extend herein a previously developed stochastic 3-scale approach [1] to the case of thermoelastic damping [2]. In this method, stochastic volume elements (SVEs) [3] are defined by considering random grain orientations in a tessellation. For each SVE realization, the mesoscopic apparent elasticity tensor, thermal conductivity tensor, and thermal dilatation tensor can be obtained using thermo-mechanical computational homogenization theory [4]. The extracted mesoscopic apparent properties tensors can then be used to define a spatially correlated meso-scale random field, which is in turn used as input for stochastic finite element simulations. As a result, the probabilistic distribution of the quality factor of micro-resonator can be extracted by considering Monte-Carlo simulations of coarse-meshed micro-resonators, accounting implicitly for the random micro-structure of the poly-silicon material.
[1] V. Lucas, J.-C. Golinval, S. Paquay, V.-D. Nguyen, L. Noels, L. Wu, A stochastic computational multiscale approach; Application to MEMS resonators. Computer Methods in Applied Mechanics and Engineering, 294, 141-167, 2015.
[2] L. Wu, V. Lucas, V.-D. Nguyen, J.-C. Golinval, S. Paquay, L. Noels, A Stochastic Multiscale Approach for the Modeling of Thermoelastic Damping in Micro-Resonators. Submitted.
[3] 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, 35--49, 1999.
[4] I. Özdemir, W. A. M. Brekelmans, M. G. D. Geers, Computational homogenization for heat conduction in heterogeneous solids, International Journal for Numerical Methods in Engineering 73, 185-204, 2008.

Wu, L., Adam, L., Bidaine, B., & Noels, L. (2016, September 07). *Simulations of composite laminates inter and intra-laminar failure using on a non-local mean-field damage-enhanced multi-scale method*. Paper presented at 15th edition of the European Mechanics of Materials Conference (EMMC15), Brussels, Belgium.

The failure of carbon fiber reinforced composite laminates is studied using a multiscale method.
A non-local mean-field homogenization (MFH) method accounting for the damage evolution of the matrix phase of the composite material [1] is considered in each ply in order to capture the intra-laminar failure. In that formulation, an incremental-secant MFH approach is used to account for the elastic unloading of the fibers during the strain softening of the matrix. In order to avoid the strain/damage localization caused by the matrix material softening, the damage enhanced MFH was formulated in an implicit non-local way [2]. Accurate predictions of the composite softening behavior and of the different phases response is then achieved. The delamination process is modeled by recourse to a hybrid discontinuous Galerkin (DG)/ extrinsic cohesive law approach.
An open-hole composite laminate with a quasi-isotropic sequence ([90/45/-45/90/0]S) is then studied experimentally and using the multiscale method [3]. The numerical model is found to predict the damage bands along the fiber directions in agreement with the experimental samples inspected by X-ray computed tomography (XCT). Moreover, the predicted delamination pattern is found to match the experimental observations.
Finally, with a view to stochastic analysis, the effect of the volume fraction and orientation variations on the failure is studied by defining them as random variables.
REFERENCES
[1] L. Wu, L. Noels, L. Adam, I. Doghri, An implicit-gradient-enhanced incremental-secant mean- field homogenization scheme for elasto-plastic composites with damage, International Journal of Solids and Structures, 50, 3843-3860, 2013.
[2] R. Peerlings, R. de Borst, W. Brekelmans, S. Ayyapureddi, Gradient-enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering, 39, 3391-3403, 1996.
[3] L. Wu, F. Sket, J.M. Molina-Aldareguia, A. Makradi, L. Adam, I. Doghri, L. Noels, A study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model, Composite Structures, 126, 246–264, 2015.

Mertens, A., Paydas, H., Rigo, O., Carrus, R., & Lecomte-Beckers, J. (2016, August 05). *On the effect of microstructural anisotropy on the mechanical and thermophysical properties of Ti6Al4V processed by Laser Beam Melting*. Paper presented at The 9th Pacific Rim International Conference on Advaced Materials and Processing (PRICM9), Kyoto, Japon.

Laser beam melting (LBM) is a strongly directional process in which a metallic powder is deposited layer by layer in a powder bed and molten locally according to the desired shape. When processing Ti6Al4V, it is well known that the latest layer tends to solidify epitaxially on the previous layers, thus giving rise to elongated columnar primary β(BCC) crystals extending over several successive layers. These primary β grains then transforms into the α(HCP) structure upon cooling. The present work aimed at studying the microstructural anisotropy of LBM Ti6Al4V, as well as its consequences on the mechanical and thermophysical properties (i.e. thermal expansion and thermal conductivity). In order to gain a deeper undestanding of thermal phenomena in the LBM of Ti6Al4V, great care was also taken to characterize the thermophysical properties over a wide temperature range from room temperature.

Mertens, A., & Lecomte-Beckers, J. (2016). On the role of interfacial reactions, dissolution and secondary precipitation during the laser additive manufacturing of metal matrix composites - A Review. In I. V., Shishkovsky (Ed.), *New Trends in 3D Printing* (pp. 187-213). Rijeka, Croatia: Intech.

Peer reviewed

Since current trends in the transportation, energy or mechanical industries impose increasingly demanding service conditions for metallic parts, metal matrix composites (MMC) are the object of a growing interest. Powder-based laser additive manufacturing, that allows to make parts with complex shapes, appears particularly adapted for the production of MMCs. This paper reviews the current state-of-the-art in the production of MMCs by additive processes, with the aim of assessing the potentials and difficulties offered by these techniques. Two main processing routes are envisaged i.e. (1) the processing of ex-situ composites in which the reinforcing phase as a powder – often of ceramic particles − is directly mixed with the powder of the matrix alloy, and both powders are simultaneously processed by the laser. (2) Alternatively, the reinforcing phase can be produced in-situ by a chemical reaction during the fabrication of the composite. For both processing routes, a careful control is needed to overcome challenges brought e.g. by the behaviour of the reinforcement particles in the laser beam, by changes in laser absorptivity or by the dissolution of the reinforcing particles in the molten metal, in order to produce metal matrix composites with enhanced usage properties.

Wu, L., Lucas, V., Adam, L., & Noels, L. (2016, July 12). *Mean-Field-Homogenization-based stochastic multiscale methods for composite materials*. Paper presented at 2nd International Conference on Mechanics of Composites (MechComp 2016), Porto, Portugal.

When considering a homogenization-based multiscale approach, at each integration-point of the macro-structure, the material properties are obtained from the resolution of a micro-scale boundary value problem. At the micro-level, the macro-point is viewed as the center of a Representative Volume Element (RVE). However, to be representative, the micro-volume-element should have a size much bigger than the micro-structure size. For composite materials which suffer from a large property and geometrical dispersion, either this requires RVE of sizes which cannot usually be obtained numerically, or simply the structural properties exhibit a scatter at the macro-scale. In both cases, the representativity of the micro-scale volume element is lost and Statistical Volume Elements (SVE) [1] should be considered in order to account for the micro-structural uncertainties, which should in turn be propagated to the macro-scale in order to predict the structural properties in a probabilistic way.
In this work we propose a non-deterministic multi-scale approach for composite materials following the methodology set in [2].
Uncertainties on the meso-scale properties and their (spatial) correlations are first evaluated through the homogenization of SVEs. This homogenization combines both mean-field method in order to gain efficiency and computational homogenization to evaluate the spatial correlation. A generator of the meso-scale material tensor is then implemented using the spectral method [3]. As a result, a meso-scale random field can be generated, paving the way to the use of stochastic finite elements to study the probabilistic behavior of macro-scale structures.
[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, 35–49, 1999.
[2] V. Lucas, J.-C. Golinval, S. Paquay, V.-D. Nguyen, L. Noels, L. Wu, A stochastic computational multiscale approach; Application to MEMS resonators. Computer Methods in Applied Mechanics and Engineering, 294, 141–167, 2015.
[3] Shinozuka, M., Deodatis, G. Simulation of stochastic processes by spectral representation. Appl. Mech. Rev., 1991: 44(4): 191-204, 1991.

Wu, L., Adam, L., Doghri, I., & Noels, L. (2016, July 12). *Failure multiscale simulations of composite laminates based on a non-local mean-field damage-enhanced homogenization*. Paper presented at 2nd International Conference on Mechanics of Composites (MechComp 2016), Porto, Portugal.

A multiscale method is developed to study the failure of carbon fiber reinforced composites.
In order to capture the intra-laminar failure, a non-local mean-field homogenization (MFH) method accounting for the damage evolution of the matrix phase of the composite material [1] is considered. In that formulation, an incremental-secant MFH approach is used to account for the elastic unloading of the fibers during the strain softening of the matrix. In order to avoid the strain/damage localization caused by the matrix material softening, an implicit non-local method [2] was reformulated to account for the composite material anisotropy. As a result, accurate predictions of the composite softening behavior and of the different phases response is possible, even for volume ratios of inclusions around 60%. In particular it is shown that the damage propagation direction in each ply follows the fiber orientation in agreement with experimental data.
The delamination process is modeled by recourse to a hybrid discontinuous Galerkin (DG)/ extrinsic cohesive law approach. As for the extrinsic cohesive law (ECL), which represents the fracturing response only, and for which cohesive elements are inserted at failure onset, the method does not suffer from a mesh-dependent effect. However, because of the underlying discontinuous Galerkin method, interface elements are present since the very beginning of the simulation avoiding the need to propagate topological changes in the mesh with the propagation of the delamination. Moreover, the pre-failure response is accurately captured by the material law though the DG implementation, by contrast to usual intrinsic cohesive laws.
As a demonstration of the efficiency and accuracy of the method, a composite laminate with a quasi-isotropic sequence ([90/45/-45/90/0]S) and an open-hole geometry is studied using the multiscale method [3] and the results are compared to experimental data. The numerical model is found to predict the damage bands along the fiber directions as observed in the experimental samples inspected by X-ray computed tomography (XCT). Moreover, the predicted delamination pattern is found to match the experimental observations.
REFERENCES
[1] L. Wu, L. Noels, L. Adam, I. Doghri, An implicit-gradient-enhanced incremental-secant mean- field homogenization scheme for elasto-plastic composites with damage, International Journal of Solids and Structures, 50, 3843-3860, 2013.
[2] R. Peerlings, R. de Borst, W. Brekelmans, S. Ayyapureddi, Gradient-enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering, 39, 3391-3403, 1996.
[3] L. Wu, F. Sket, J.M. Molina-Aldareguia, A. Makradi, L. Adam, I. Doghri, L. Noels, A study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model, Composite Structures, 126, 246–264, 2015.

Lucas, V., Wu, L., Golinval, J.-C., Paquay, S., Voicu, R., Baracu, A., & Noels, L. (2016, June 09). *Multi-scale stochastic study of the grain orientation and roughness effects on polycrystalline thin structures*. Paper presented at ECCOMAS Congress 2016 VII European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece.

When studying micro-electro-mechanical systems (MEMS) made of poly-crystalline materials, as the size of the device is only one or two orders of magnitude higher than the size of the the grains, the structural properties exhibit a scatter at the macro-scale due to the existing randomness in the grain size, grain orientation, surface roughness... In order to predict the probabilistic behavior at the structural scale, the authors have recently developed a stochastic 3-scale approach [1]. In this method, stochastic volume elements (SVEs) [2] are defined by considering random grain orientations in a tessellation. For each SVE realization, a meso-scopic apparent material tensor can be obtained using the computational homogenization theory. The extracted meso-scopic apparent material tensors can then be used to defined a spatially correlated meso-scale random field, which is in turn used as input for stochastic finite element
simulations.
In this work we intend to study the effect of different material-related uncertainty sources on the structural behavior of vibrating micro-devices manufactured using low pressure chemical vapor deposition. First, the effect of preferred grain orientation on vibrating micro-structures is assessed. To this end, SVEs are generated so that their grain orientation distributions follow XRD measurements. Second, the effect of the roughness of the vibrating micro-structures is studied. Toward this end, SVEs, whose rough surface statistical properties follow AFM measurements, are generated. A second-order computational homogenization [3] applied on the different SVE realizations allows defining a meso-scale random field of the in-plane and out-of-plane meso-scale shell properties. Stochastic shell finite elements can then be applied to predict the MEMS probabilistic behavior.
[1] V. Lucas, et al., Comp. Meth. in Appl. Mech. and Eng., 294, 141-167, 2015
[2] M. Ostoja-Starzewski, X.Wang, Comp. Meth. in Appl. Mech. and Eng., 168, 35–49, 1999
[3] E.W.C. Coenen, V. Kouznetsova, M.G.D. Geers. Int. J. for Numer. Meth. in Eng., 83, 1180–1205, 2010.

Wu, L., Sket, F., Adam, L., Doghri, I., & Noels, L. (2016, June 08). *Prediction of intra- and inter-laminar failure of laminates using non-local damage-enhanced mean-field homogenization simulations*. Paper presented at ECCOMAS Congress 2016 VII European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece.

The failure of carbon fiber reinforced composites with a quasi-isotropic sequence ([90/45/-45/90/0]S) and open-hole geometry is studied using a multiscale method [1]. On the one hand, the intra-laminar failure is captured using a damage-enhanced mean-field homogenization scheme. To this end, each ply is modeled as a homogenized material whose anisotropic damage behavior is captured from the homogenization method [2]. In order to avoid the problem of loss of solution uniqueness the mean-field homogenization process is formulated in the context of the non-local continuum damage theory [3]. On the other hand, an hybrid discontinuous Galerkin/extrinsic cohesive law method is used to model the delamination process at the ply
interfaces. This hybrid method avoids the need to propagate topological changes in the mesh with the propagation of the delamination while it preserves the consistency and stability in the un-cracked interfaces.
As a result, the multiscale framework allows predicting damage propagation directions in each ply along the fiber directions accordingly to the experimental results as it is demonstrated by considering an openhole [90/45/-45/90/0]S-laminate studied both numerically and experimentally.
[1] L. Wu, et al., Composite Struct., 126, 246–264, 2015.
[2] L. Wu, L. Noels, L. Adam, I. Doghri, Int. J. of Solids and Struct., 50, 3843-3860, 2013.
[3] R. Peerlings, et al., Int. J. for Numer. Meth. in Eng., 39, 3391-3403, 1996

Leclerc, J., Wu, L., Noels, L., & Nguyen, V. D. (2016, June 07). *Cohesive band model: a triaxiality-dependent cohesive model for damage to crack transition in a non-local implicit discontinuous Galerkin framework*. Paper presented at ECCOMAS Congress 2016 VII European Congress on Computational Methods in Applied Sciences and Engineering, Crete Island, Greece.

Numerical modelling of the complete ductile failure process is still a challenge. On the one hand, continuous approaches, described by damage models, succeed in the initial diffuse damage stage but are still unable to represent physical discontinuities. On the other hand, discontinuous approaches, such as the cohesive zone models, are able to represent the crack propagation behaviour. They are suited for local damaging processes as crack initiation and propagation, and so, fail in diffuse damage prediction of ductile materials. Moreover, they do not usually capture triaxiality effects, mandatory for accurate ductile failure simulations. To describe the ductile failure process, the numerical scheme proposed here combines both approaches [1] in order to beneficiate from their respective advantages: a non-local damage model combined with an extrinsic cohesive law in a discontinuous Galerkin finite element framework. An application example of this scheme is shown on the attached figure. The initial diffuse damage stage is modelled by an implicit nonlocal damage model as suggested by [2]. Upon damage to crack transition, a cohesive band [3] is used to introduce in-plane stretch effects inside the cohesive law or in other words, a triaxiality-dependent behaviour. Indeed, these in-plane strains play an important role during the ductile failure process and have to be considered. Concretely, when crack appears in the last failure stage, all the damaging process is assumed to occur inside a thin band ahead of the crack surface. Thanks to the small but finite numerical band thickness, the strains inside this band can be obtained from the in-plane strains and from the cohesive jump.
Then, the stress-state inside the band and the cohesive traction forces on the crack lips are deduced from the underlying continuum damage model. The band thickness is not a new material parameter but is computed to ensure the energetic consistency during the transition.
[1] Wu L, Becker G, Noels L. Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework. Comput. Methods Appl. Mech. Eng. 279 (2014): 379–409
[2] Peerlings R., de Borst R., Brekelmans W., Ayyapureddi S. Gradient-enhanced damage for quasi-brittle materials, Int. J. for Num. Methods in Eng. 39 (1996): 3391-3403
[3] Remmers J. J. C., de Borst R., Verhoosel C. V., Needleman A. The cohesive band model: a cohesive surface formulation with stress triaxiality. Int. J. Fract. 181 (2013): 177–188

Mertens, A., Dedry, O., Reuter, D., Rigo, O., & Lecomte-Beckers, J. (2016, June 01). *Microstructural evolution during the heat treatment of Laser Beam Melted AlSi10Mg*. Paper presented at Thermec 2016 International Conference on Processing and Manufacturing of Advanced Materials, Graz, Autriche.

Al alloy AlSi10Mg processed by Laser Beam Melting (LBM) exhibits a much finer microstructure than its cast counterpart due to the ultra-fast cooling rates imposed in the LBM process. One important consequence of this microstructural refinement is that a classical T6 age hardening heat treatment will not have the same effect on LBM AlSi10Mg when compared with cast AlSi10Mg. Indeed, a previous study by the present authors has shown that heat treating LBM AlSi10Mg at 510°C for 6 hours followed by a second isothermal hold at 170°C for 6 hours brought a marked improvement of the yield stress by 30% and of the elongation at break by 220%. However, this was achieved at the expense of a decrease in both hardness and ultimate tensile strength. A better understanding of the underlying phenomena is needed in order to optimize the heat treatment of LBM AlSi10Mg. The present work hence aims at investigating in more depth the microstructural evolution induced upon heat treating LBM AlSi10Mg. Changes in texture as well as in the distribution of Si-rich precipitates (size, morphology...) have been studied, with a particular attention to those changes taking place during the first step of the heat treatment at the higher temperature of 510°C.

Mertens, A., L'Hoest, T., Lecomte-Beckers, J., Magnien, J., & Carrus, R. (2016, May 27). *On the laser additive manufacturing of metal matrix composites*. Paper presented at Additive Manufacturing Workshop Brussels, Bruxelles, Belgique.

This paper reviews the challenges and opportunities related to the production of of metal matrix composites (MMCs) and functionally graded materials (FGMs) by additive manufacturing laser-based technologies. Special attention is given to issues such as the stability of the reinforcement particles under the laser beam, particles’ dissolution in the metallic melt pool and interfacial reactions between the particles and the metallic matrix. Illustrations are taken from a preliminary investigation, by the MMS Unit, into the laser cladding of stainless steel + carbides composite coatings, as well as from the open scientific literature.