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See detailA large strain hyperelastic viscoelastic-viscoplastic-damage constitutive model based on a multi-mechanism non-local damage continuum for amorphous glassy polymers
Nguyen, Van Dung ULg; Lani, Frédéric; Pardoen, Thomas et al

in International Journal of Solids and Structures (in press)

A large strain hyperelastic phenomenological constitutive model is proposed to model the highly nonlinear, rate-dependent mechanical behavior of amorphous glassy polymers under isothermal conditions. A ... [more ▼]

A large strain hyperelastic phenomenological constitutive model is proposed to model the highly nonlinear, rate-dependent mechanical behavior of amorphous glassy polymers under isothermal conditions. A corotational formulation is used through the total Lagrange formalism. At small strains, the viscoelastic behavior is captured using the generalized Maxwell model. At large strains beyond a viscoelastic limit characterized by a pressure-sensitive yield function, which is extended from the Drucker-Prager one, a viscoplastic region follows. The viscoplastic flow is governed by a non-associated Perzyna-type flow rule incorporating this pressure-sensitive yield function and a quadratic flow potential in order to capture the volumetric deformation during the plastic process. The stress reduction phenomena arising from the post-peak plateau and during the failure stage are considered in the context of a continuum damage mechanics approach. The post-peak softening is modeled by an internal scalar, so-called softening variable, whose evolution is governed by a saturation law. When the softening variable is saturated, the rehardening stage is naturally obtained since the isotropic and kinematic hardening phenomena are still developing. Beyond the onset of failure characterized by a pressure-sensitive failure criterion, the damage process leading to the total failure is controlled by a second internal scalar, so-called failure variable. The final failure occurs when the failure variable reaches its critical value. To avoid the loss of solution uniqueness when dealing with the continuum damage mechanics formalism, a non-local implicit gradient formulation is used for both the softening and failure variables, leading to a multi-mechanism non-local damage continuum. The pressure sensitivity considered in both the yield and failure conditions allows for the distinction under compression and tension loading conditions. It is shown through experimental comparisons that the proposed constitutive model has the ability to capture the complex behavior of amorphous glassy polymers, including their failure. [less ▲]

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See detailChapter 6: Effective Properties, 6.1.1 Review of Homogenization Methods for Heterogeneous Materials
Noels, Ludovic ULg; Wu, Ling ULg; Adam, Laurent et al

in Integrated Computational Materials Engineering (ICME) (in press)

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See detailMulti-scale stochastic study of the grain orientation and roughness effects on polycrystalline thin structures
Lucas, Vincent ULg; Wu, Ling ULg; Golinval, Jean-Claude ULg et al

Conference (2016, June 09)

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 ... [more ▼]

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. [less ▲]

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See detailPrediction of intra- and inter-laminar failure of laminates using non-local damage-enhanced mean-field homogenization simulations
Wu, Ling ULg; Sket, Federico; Adam, Laurent et al

Conference (2016, June 08)

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 ... [more ▼]

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 [less ▲]

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See detailCohesive band model: a triaxiality-dependent cohesive model for damage to crack transition in a non-local implicit discontinuous Galerkin framework
Leclerc, Julien ULg; Wu, Ling ULg; Noels, Ludovic ULg et al

Conference (2016, June 07)

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 ... [more ▼]

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 [less ▲]

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See detailProbabilistic prediction of the quality factor of micro-resonator using a stochastic thermo-mechanical multi-scale approach
Wu, Ling ULg; Lucas, Vincent ULg; Nguyen, Van Dung ULg et al

Scientific conference (2016, May 23)

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 ... [more ▼]

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 to the case of thermoelastic damping. In this method, stochastic volume elements (SVEs) 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. The extracted mesoscopic apparent properties tensors can then be used to define a spatially correlated mesoscale 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 microstructure of the poly-silicon material. [less ▲]

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See detailA Study Of Dry Stiction Phenomenon In MEMS Using A Computational Stochastic Multi-scale Methodology
Hoang Truong, Vinh ULg; Wu, Ling ULg; Paquay, Stéphane et al

in EuroSimE 2016 in Montpellier (2016, April 19)

This work studies the uncertainties of the adhesive contact problems for reduced size structures, e.g. the stiction failure of microelectromechanical systems (MEMS). In MEMS, because of the large surface ... [more ▼]

This work studies the uncertainties of the adhesive contact problems for reduced size structures, e.g. the stiction failure of microelectromechanical systems (MEMS). In MEMS, because of the large surface to volume ratio, the surfaces forces, such as van der Waals forces and capillary forces, are dominant in comparison with the body forces. As these force magnitudes strongly depend on the contact distance, when the two contacting surfaces are rough, the contact distances vary, and the physical contact areas are limited at the highest asperities of the contacting surfaces. Therefore, the adhesive contact forces between two rough surfaces can suffer from a scatter, and the involved structural behaviors can be indeterministic. To numerically predict the probability behaviors of structures involving adhesion in dry environments, in this paper, a computational stochastic model-based multi-scale method developed by the authors is applied. The effects of van der Waals is studied and compared with experimental data as well as with the effects of capillary forces. [less ▲]

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See detailA probabilistic model for predicting the uncertainties of the humid stiction phenomenon on hard materials
Hoang Truong, Vinh ULg; Wu, Ling ULg; Paquay, Stéphane et al

in Journal of Computational & Applied Mathematics (2015), 289

Stiction is a major failure in microelectromechanical system (MEMS) devices in which two contacting surfaces can remain stuck together because of the adhesive forces. Due to the difference between the ... [more ▼]

Stiction is a major failure in microelectromechanical system (MEMS) devices in which two contacting surfaces can remain stuck together because of the adhesive forces. Due to the difference between the surfaces roughness and the adhesive force range, the real contact areas are usually smaller than the apparent one, resulting in a scatter in the adhesive forces. Consequently, the stiction is an uncertain phenomenon. In this work, we develop a probabilistic model to predict the uncertainties of stiction due to the capillary forces acting on stiff materials. This model contains two levels: at the deterministic level, the model can predict the pull-out adhesive contact forces for a given surface topology; at the probabilistic level, the model generates independent identically distributed surfaces on which the deterministic solution can be applied to evaluate the uncertainties related to the stiction phenomenon. [less ▲]

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See detailExperimental and computational micro–mechanical investigations of compressive properties of polypropylene/multi–walled carbon nanotubes nanocomposite foams
Wan, Fangyi; Tran, Minh Phuong; Leblanc, Christophe ULg et al

in Mechanics of Materials (2015), 91(Part 1), 95-118

The compressive behavior of nanocomposite foams is studied by both experimental and computational micro-mechanics approaches with the aim of providing an efficient computational model for this kind of ... [more ▼]

The compressive behavior of nanocomposite foams is studied by both experimental and computational micro-mechanics approaches with the aim of providing an efficient computational model for this kind of material. The nanocomposites based on polypropylene (PP) and different contents of multi-walled carbon nanotubes (CNTs) are prepared by melt mixing method. The nanocomposite samples are foamed using super-critical carbon dioxide (ScCO2) as blowing agent at different soaking temperatures. The influence of this foaming parameter on the morphological characteristics of the foam micro-structure is discussed. Differential Scanning Calorimetry (DSC) measurements are used to quantify the crystallinity degree of both nanocomposites and foams showing that the crystallinity degree is reduced after the foaming process. This modification leads to mechanical properties of the foam cell walls that are different from the raw nanocomposite PP/CNTs material. Three--point bending tests are performed on the latter to measure the flexural modulus in terms of the crystallinity degree. Uniaxial compression tests are then performed on the foamed samples under quasi-static conditions in order to extract the macro-scale compressive response. Next, a two-level multi-scale approach is developed to model the behavior of the foamed nanocomposite material. On the one hand, the micro-mechanical properties of nanocomposite PP/CNTs cell walls are evaluated from a theoretical homogenization model accounting for the micro-structure of the semi-crystalline PP, for the degree of crystallinity, and for the CNT volume fraction. The applicability of this theoretical model is demonstrated via the comparison with experimental data from the described experimental measurements and from literature. On the other hand, the macroscopic behavior of the foamed material is evaluated using a computational micro-mechanics model using tetrakaidecahedron unit cells and periodic boundary conditions to estimate the homogenized properties. The unit cell is combined with several geometrical imperfections in order to capture the elastic collapse of the foamed material. The numerical results are compared to the experimental measurements and it is shown that the proposed unit cell computational micro-mechanics model can be used to estimate the homogenized behavior, including the linear and plateau regimes, of nanocomposite foams. [less ▲]

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See detailA stochastic computational multiscale approach; Application to MEMS resonators
Lucas, Vincent ULg; Golinval, Jean-Claude ULg; Paquay, Stéphane et al

in Computer Methods in Applied Mechanics & Engineering (2015), 294

The aim of this work is to develop a stochastic multiscale model for polycrystalline materials, which accounts for the uncertainties in the micro-structure. At the finest scale, we model the micro ... [more ▼]

The aim of this work is to develop a stochastic multiscale model for polycrystalline materials, which accounts for the uncertainties in the micro-structure. At the finest scale, we model the micro-structure using a random Voronoi tessellation, each grain being assigned a random orientation. Then, we apply a computational homogenization procedure on statistical volume elements to obtain a stochastic characterization of the elasticity tensor at the meso-scale. A random field of the meso-scale elasticity tensor can then be generated based on the information obtained from the SVE simulations. Finally, using a stochastic finite element method, these meso-scale uncertainties are propagated to the coarser scale. As an illustration we study the resonance frequencies of MEMS micro-beams made of poly-silicon materials, and we show that the stochastic multiscale approach predicts results in agreement with a Monte Carlo analysis applied directly on the fine finite-element model, i.e. with an explicit discretization of the grains. [less ▲]

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See detailA study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model
Wu, Ling ULg; Sket, Federico; Molina-Aldareguia, Jon M et al

in Composite Structures (2015), 126

The failure of carbon fiber reinforced epoxy laminates is studied using an anisotropic gradient-enhanced continuum damage model embedded in a mean-field homogenization scheme. In each ply, a homogenized ... [more ▼]

The failure of carbon fiber reinforced epoxy laminates is studied using an anisotropic gradient-enhanced continuum damage model embedded in a mean-field homogenization scheme. In each ply, a homogenized material law is used to capture the intra-laminar failure. The anisotropy of the homogenized material model results from the homogenization method and from the reformulation of the non-local continuum damage theory to account for the material anisotropy. As a result the damage propagation direction in each ply is predicted with accuracy as compared to the experimental results, while the problems of losing uniqueness and strain localization, which occur in classical finite element simulations when strain softening of materials is involved, can be avoided. To model the delamination process, the hybrid discontinuous Galerkin/extrinsic cohesive law method is introduced 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 demonstration, open-hole coupons with different stacking sequences are studied numerically and experimentally. Both the intra- and inter-laminar failure patterns are shown to be well captured by the computational framework. [less ▲]

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See detailA viscoelastic-viscoplastic-damage constitutive model based on a large strain hyperelastic formulation for amorphous glassy polymers
Nguyen, Van Dung ULg; Morelle, Xavier; Lani, Frédéric et al

in proceedings of the 20th International Conference on Composite Materials, Copenhagen, Denmark, 19-24th July 2015 (2015, July 20)

The aim of this work is to develop an efficient large-strain hyperelastic constitutive model for amorphous polymers in the glassy state. These materials exhibit a complex rate- and pressure-sensible ... [more ▼]

The aim of this work is to develop an efficient large-strain hyperelastic constitutive model for amorphous polymers in the glassy state. These materials exhibit a complex rate- and pressure-sensible behavior in both elastic and plastic regimes. After an initial linear elastic region, a nonlinear stage continues until reaching a peak stress, which is followed by a softening stage. At large strains, when the softening is saturated, a re-hardening stage is reached. The viscoelastic effect is captured using the generalized Maxwell model. The viscoplastic effect is considered using a Perzyna-type flow rule incorporating a pressure sensitive yield surface and a non-associated flow potential. This yield surface is extended from the Drucker-Prager one. The saturated softening phenomenon is modelled using an isotropic numerical damage variable progressed by a saturated softening law. With the introduction of the damage parameter, a non-local implicit gradient damage model is used to avoid the loss of the solution uniqueness. Through experimental comparisons, it is shown that the proposed model has the ability to model the complex mechanical responses of amorphous glassy polymers. [less ▲]

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See detailA probabilistic multi-scale model for polycrystalline MEMS resonators
Lucas, Vincent ULg; Wu, Ling ULg; Paquay, Stéphane et al

Conference (2015, July 09)

The size of micro-electro-mechanical systems (MEMS) is only one or two orders of magnitude higher than the size of their micro-structure, i.e. their grain size. As a result, the structural properties ... [more ▼]

The size of micro-electro-mechanical systems (MEMS) is only one or two orders of magnitude higher than the size of their micro-structure, i.e. their grain size. As a result, the structural properties exhibit a scatter. As an example we study the beam resonator illustrated in Fig. 1(a), made of poly-silicon material, in which each grain has a random orientation. Solving the problem with a full direct numerical simulation combined to a Monte-Carlo method allows the probability density function to be computed as illustrated in Fig. 1(b). However this methodology is computationally expensive due to the number of degrees of freedom required to study one sample, motivating the development of a non-deterministic 3-scale approach [3]. In a multiscale approach, at each macro-point of the macro-structure, the resolution of a microscale boundary value problem relates the macro-stress tensor to the macro-strain tensor. At the micro-level, the macro-point is viewed as the center of a Representative Volume Element (RVE). The resolution of the micro-scale boundary problem can be performed using finite-element simulations, as in the computational homogenization framework, e.g. [2]. However, to be representative, the micro-volume-element should have a size much bigger than the microstructure size. In the context of the MEMS resonator, this representativity is lost and Statistical Volume Elements (SVE) are considered. These SVEs are generated under the form of a Voronoi tessellation with a random orientation for each silicon grain. Hence, a Monte-Carlo procedure combined with a homogenization technique allows a distribution of the material tensor at the meso-scale to be estimated. The correlation between the meso-scale material tensors of two SVEs separated by a given distance can also be evaluated. A generator at the meso-scale based on the spectral method [4] is implemented. The generator [3] accounts for a lower bound [1] of the meso-scale material tensor in order to ensure the existence of the second-order moment of the Frobenius norm of the generated material tensor inverse [5]. Using the random meso-scale field obtained with the meso-scale generator, which accounts for the spatial correlation, a Monte-Carlo method can be used at the macro-scale to predict the probabilistic behavior of the MEMS resonator. [less ▲]

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See detailA Non-Local Damage-Enhanced Incremental-Secant Mean-Field-Homogenization For Composite Laminate Failure Predictions
Wu, Ling ULg; Adam, Laurent; Doghri, Issam et al

Conference (2015, July 06)

Recently, the authors have presented an incremental-secant mean-field homogenisation (MFH) process for non-linear composite materials [4]. In this formulation, a virtual elastic unloading is applied to ... [more ▼]

Recently, the authors have presented an incremental-secant mean-field homogenisation (MFH) process for non-linear composite materials [4]. In this formulation, a virtual elastic unloading is applied to evaluate the virtual residual stress and strain states reached in each elasto-plastic phase. These virtual states are then used as a starting point to apply a secant homogenization method. This incremental-secant MFH process can handle non-proportional and nonmonotonic loadings, and naturally possesses an isotropic instantaneous stiffness operator to be used in the Eshelby tensor. This incremental-secant MFH homogenization can account for the first and second statistical moment estimation of the current yield stress in the composite phases during the computation of the plastic flow. When accounting for a second statistical moment estimation, the plastic yield in the composite material phases is captured with a higher accuracy, improving the predictions, mainly in the case of short fiber composite materials [6], see Fig. 1(a). The incremental MFH can handle material softening when extended to include a damage model. Indeed, as the secant formulation is applied from an unloaded state, the inclusion phase can be elastically unloaded during the softening of the matrix phase, contrarily to the case of the incremental-tangent method [3, 5], see Fig. 1(b). Moreover, when formulating the damage model in the composite phases in a non-local way, as with the non-local implicit approach, [1, 2], the MFH scheme can be used to model strain localization in composite structures [5], without suffering from the loss of the solution uniqueness. [less ▲]

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See detailAn XFEM/CZM implementation for massively parallel simulations of composites fracture
Vigueras, Guillermo; Sket, Federico; Samaniego, Cristobal et al

in Composite Structures (2015), 125

Because of their widely spread use in many industries, composites are the subject of many research campaigns. More particularly, the development of both accurate and flexible numerical models able to ... [more ▼]

Because of their widely spread use in many industries, composites are the subject of many research campaigns. More particularly, the development of both accurate and flexible numerical models able to capture their intrinsically multiscale modes of failure is still a challenge. The standard finite element method typically requires intensive remeshing to adequately capture the geometry of the cracks and high accuracy is thus often sacrificed in favor of scalability, and vice versa. In an effort to preserve both properties, we present here an extended finite element method (XFEM) for large scale composite fracture simulations. In this formulation, the standard FEM formulation is partially enriched by use of shifted Heaviside functions with special attention paid to the scalability of the scheme. This enrichment technique offers several benefits, since the interpolation property of the standard shape function still holds at the nodes. Those benefits include (i) no extra boundary condition for the enrichment degree of freedom, and (ii) no need for transition/blending regions; both of which contribute to maintain the scalability of the code. Two different cohesive zone models (CZM) are then adopted to capture the physics of the crack propagation mechanisms. At the intralaminar level, an extrinsic CZM embedded in the XFEM formulation is used. At the interlaminar level, an intrinsic CZM is adopted for predicting the failure. The overall framework is implemented in ALYA, a mechanics code specifically developed for large scale, massively parallel simulations of coupled multi-physics problems. The implementation of both intrinsic and extrinsic CZM models within the code is such that it conserves the extremely efficient scalability of ALYA while providing accurate physical simulations of computationally expensive phenomena. The strong scalability provided by the proposed implementation is demonstrated. The model is ultimately validated against a full experimental campaign of loading tests and X-ray tomography analyses for a chosen very large scale. [less ▲]

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See detailPropagation of uncertainties in the modelling of MEMS resonators (using a 3-scale probabilistic approach)
Lucas, Vincent ULg; Wu, Ling ULg; Golinval, Jean-Claude ULg et al

Conference (2015, May 26)

In order to ensure the accuracy of MEMS vibrometers, the first resonance frequency should be predicted at the design phase. However, this prediction is subjected to randomness: there is a scatter in the ... [more ▼]

In order to ensure the accuracy of MEMS vibrometers, the first resonance frequency should be predicted at the design phase. However, this prediction is subjected to randomness: there is a scatter in the reached value resulting from the uncertainties involved in the manufacturing process. The purpose of this work is to take into account these uncertainties of the microstructure. The objective is a non-deterministic model that can be used since the design stage. The material is the source of uncertainties: the beam resonator is made of a polycrystalline material in which each grain has a random orientation. Solving the problem with a full direct numerical simulation combined to a Monte-Carlo method allows the probability density function of the resonance frequency to be computed. However this methodology is computationally expensive due to the number of degrees of freedom required to study one sample, motivating the development of a computationally efficient method. Towards this end a 3-scales stochastic model for predicting the resonance frequency of a micro-beam made of a polycrystalline linear anisotropic material is described. At the lower scale, we model the micro-structure with micro-volume elements. Due to the small-scale involved, the representativity of these micro-volume elements is not achieved and thus Statistical Volume Elements (SVE) are considered. These SVEs are generated under the form of a Voronoï tessellation, each grain being assigned a random orientation. Computational homogenization is applied over the SVEs, along with a Monte-Carlo procedure, to obtain a stochastic characterization of the elasticity tensor at the second scale of interest, the meso-scale. The spatial correlation between SVEs is also estimated. A generator based on spectral methods is implemented. Afterwards, using a stochastic finite element method, these meso-scale uncertainties are propagated by taking account of the spatial correlation up to the higher scale to predict the probabilistic behavior of the MEMS resonator. [less ▲]

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See detailA stochastic multiscale analysis for MEMS stiction failure
Hoang Truong, Vinh ULg; Wu, Ling ULg; Golinval, Jean-Claude ULg et al

Conference (2015, May 26)

Stiction is a major failure in microelectromechanical system (MEMS) devices in which two contacting surfaces can remain stuck together because of the adhesive forces, such as van der Waals forces and ... [more ▼]

Stiction is a major failure in microelectromechanical system (MEMS) devices in which two contacting surfaces can remain stuck together because of the adhesive forces, such as van der Waals forces and capillary forces. Stiction is a multiscale problem which is characterized by three different lengths: the MEMS device characteristic length, the roughness of the contacting surfaces, and the distance range of the adhesive forces. Because MEMS surfaces roughness and adhesive force distances are of comparable scales, the randomness in the contacting surfaces can result in important uncertainties on the interacting forces, and in turn lead to a scatter in the MEMS structural behavior. The purpose of this work is to quantify the uncertainties on the macro stiction behavior of a MEMS structure due to the randomness in its contacting surfaces. A full analysis, such as the combination of a Monte-Carlo simulation to generate random surfaces combined with finite element (FE) analyses to model the stiction behavior, is expensive in terms of the computational cost due to the difference in the scales between the macro characteristic length and the distance range of the adhesive forces. Thus, in this work, we develop a stochastic multiscale analysis. At the micro scale, the uncertainties in the interacting forces between two rough surfaces are investigated. The power spectral density function of the surface is characterized from experimental topology measurements, and interacting surfaces are then generated as Gaussian random surfaces. For each generated random surface, the interacting adhesive forces are calculated by using a modified Dejarguin-Muller-Toporov (DMT) model. The resulting adhesive contact forces can be integrated using the finite element method at the structural scale by associating to each discretized contacting point a sampled surface. We then use the Monte-Carlo method to quantify the uncertainties in the stiction behavior of the MEMS device. [less ▲]

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See detailPropagation of uncertainties using probabilistic multi-scale models
Lucas, Vincent ULg; Wu, Ling ULg; Paquay, Stéphane et al

Conference (2015, February 25)

When applying a multiscale approach, the material behavior at the macro-scale can be obtained from an homogenization scheme. To this end, at each integration-point of the macro-structure, the macrostress ... [more ▼]

When applying a multiscale approach, the material behavior at the macro-scale can be obtained from an homogenization scheme. To this end, at each integration-point of the macro-structure, the macrostress tensor is related to the macro-strain tensor through 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. When considering structures of reduced sizes, such as micro-electro-mechanical systems (MEMS), as the size of the devices is only one or two orders of magnitude higher than the size of their microstructure, i.e. their grain size, the structural properties exhibit a scatter at the macro-scale. The representativity of the micro-scale volume element is lost and Statistical Volume Elements (SVE) should be considered in order to account for the micro-structural uncertainties. These uncertainties should then be propagated to the macro-scale in order to predict the device properties in a probabilistic way. In this work we propose a non-deterministic multi-scale approach [1] for poly-silicon MEMS resonators. A set of SVEs is first generated under the form of Voronoi tessellations with a random orientation assigned for each silicon grain of each SVE. The resolution of each micro-scale boundary problem is performed by recourse to the computational homogenization framework, e.g. [2], leading to meso-scale material properties under the form of a linear material tensor for each SVE. Applying a Monte-Carlo procedure allows a distribution of this material tensor to be determined at the meso-scale. The correlation between the meso-scale material tensors of two SVEs separated by a given distance can also be evaluated. A generator of the meso-scale material tensor is then implemented using the spectral method [3]. The generator [1] accounts for a lower bound [4] of the meso-scale material tensor in order to ensure the existence of the second-order moment of the Frobenius norm of the tensor inverse [5]. A macro-scale finite element model of the beam resonator can now be achieved using regular finite-element, i.e. not conforming with the grains, and the material tensor at each Gauss point is obtained using the meso-scale generator, which accounts for the spatial correlation. A Monte-Carlo method is then used at the macro-scale to predict the probabilistic behavior of the MEMS resonator. As an example the beam resonator illustrated in Fig. 1(a) is made of poly-silicon, and each grain has a random orientation. Solving the problem with a full direct numerical simulation combined to a Monte-Carlo method allows the probability density function to be computed as illustrated in Fig. 1(b). However this methodology is computationally expensive due to the number of degrees of freedom required to study one sample. The proposed non-deterministic multi-scale strategy allows reducing this computational cost as the Monte-Carlo processes are applied on much smaller finite-element models. The method can also be applied in the context of fracture of thin poly-silicon film [6]. In this case, a set of meso-scopic cohesive laws can be obtained at the meso-scale from the resolution of different SVEs. The meso-scopic cohesive laws are obtained for each RVE from the finite element resolution of the Voronoi tessellations using the method proposed in [7]. The resulting statistical values for the critical energy release rate and for the critical strength can then be used for macro-scale simulations. [less ▲]

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See detailMultiscale modelling framework for the fracture of thin brittle polycrystalline films - Application to polysilicon
Mulay, Shantanu; Becker, Gauthier ULg; Vayrette, Renaud et al

in Computational Mechanics (2015), 55(1), 73-91

Micro-electro-mechanical systems (MEMS) made of polycrystalline silicon are widely used in several engineering fields. The fracture properties of polycrystalline silicon directly affect their reliability ... [more ▼]

Micro-electro-mechanical systems (MEMS) made of polycrystalline silicon are widely used in several engineering fields. The fracture properties of polycrystalline silicon directly affect their reliability. The effect of the orientation of grains on the fracture behaviour of polycrystalline silicon is investigated out of the several factors. This is achieved, firstly, by identifying the statistical variation of the fracture strength and critical strain energy release rate, at the nanoscopic scale, over a thin freestanding polycrystalline silicon film, having mesoscopic scale dimensions. The fracture stress and strain at the mesoscopic level are found to be closely matching with uniaxial tension experimental results. Secondly, the polycrystalline silicon film is considered at the continuum MEMS scale, and its fracture behaviour is studied by incorporating the nanoscopic scale effect of grain orientation. The entire modelling and simulation of the thin film is achieved by combining the discontinuous Galerkin method and extrinsic cohesive law describing the fracture process. [less ▲]

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