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Chapter 6: Effective Properties, 6.1.1 Review of Homogenization Methods for Heterogeneous Materials Noels, Ludovic ; Wu, Ling ; et al in Integrated Computational Materials Engineering (ICME) (in press) Detailed reference viewed: 33 (9 ULg)Classifying simulated wheat yield responses to changes in temperature and precipitation across a european transect ; ; et al Conference (2016, March) Detailed reference viewed: 15 (2 ULg)A probabilistic model for predicting the uncertainties of the humid stiction phenomenon on hard materials Hoang Truong, Vinh ; Wu, Ling ; 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 ▲] Detailed reference viewed: 211 (87 ULg)A Non-Local Damage-Enhanced Incremental-Secant Mean-Field-Homogenization For Composite Laminate Failure Predictions Wu, Ling Conference (2015, October 07) Detailed reference viewed: 25 (7 ULg)A stochastic computational multiscale approach; Application to MEMS resonators Lucas, Vincent ; Golinval, Jean-Claude ; 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 ▲] Detailed reference viewed: 179 (113 ULg)A study of composite laminates failure using an anisotropic gradient-enhanced damage mean-field homogenization model Wu, Ling ; ; 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 ▲] Detailed reference viewed: 144 (42 ULg)A probabilistic multi-scale model for polycrystalline MEMS resonators Lucas, Vincent ; Wu, Ling ; 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 ▲] Detailed reference viewed: 43 (8 ULg)A Non-Local Damage-Enhanced Incremental-Secant Mean-Field-Homogenization For Composite Laminate Failure Predictions Wu, Ling ; ; 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 ▲] Detailed reference viewed: 33 (6 ULg)An XFEM/CZM implementation for massively parallel simulations of composites fracture ; ; 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 ▲] Detailed reference viewed: 125 (20 ULg)Propagation of uncertainties in the modelling of MEMS resonators (using a 3-scale probabilistic approach) Lucas, Vincent ; Wu, Ling ; Golinval, Jean-Claude 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 ▲] Detailed reference viewed: 27 (6 ULg)A stochastic multiscale analysis for MEMS stiction failure Hoang Truong, Vinh ; Wu, Ling ; Golinval, Jean-Claude 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 ▲] Detailed reference viewed: 50 (18 ULg)Propagation of uncertainties using probabilistic multi-scale models Lucas, Vincent ; Wu, Ling ; 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 ▲] Detailed reference viewed: 109 (11 ULg)An incremental-secant mean-field homogenisation method with second statistical moments for elasto-plastic composite materials Wu, Ling ; ; Noels, Ludovic in Philosophical Magazine (2015), 95(28-30), 3348-3384 In this paper, the incremental-secant mean-field homogenisation (MFH) scheme recently developed by the authors is extended to account for second statistical moments. The incremental-secant MFH method ... [more ▼] In this paper, the incremental-secant mean-field homogenisation (MFH) scheme recently developed by the authors is extended to account for second statistical moments. The incremental-secant MFH method possesses several advantages compared to other MFH methods. Indeed the method can handle non-proportional and non-monotonic loadings, while the instantaneous stiffness operators used in the Eshelby tensor are naturally isotropic, avoiding the isotropisation approximation required by the affne and incremental-tangent methods. Moreover, the incremental-secant MFH formalism was shown to be able to account for material softening when extended to include a non-local damage model in the matrix phase, thus enabling an accurate simulation of the onset and evolution of damage across the scales. In this work, by accounting for a second statistical moment estimation of the current yield stress in the composite phases, the plastic flow computation allows capturing with a better accuracy the plastic yield in the composite material phases, which in turn improves the accuracy of the predictions, mainly in the case of short fibre composite materials. The incremental-secant mean-field-homogenisation (MFH) can thus be used to model a wide variety of composite material systems with a good accuracy. [less ▲] Detailed reference viewed: 87 (33 ULg)Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework Wu, Ling ; Becker, Gauthier ; Noels, Ludovic in Computer Methods in Applied Mechanics & Engineering (2014), 279 One current challenge related to computational fracture mechanics is the modeling of ductile fracture and in particular the damage to crack transition. On the one hand, continuum damage models, especially ... [more ▼] One current challenge related to computational fracture mechanics is the modeling of ductile fracture and in particular the damage to crack transition. On the one hand, continuum damage models, especially in their non-local formulation which avoids the loss of solution uniqueness, can capture the material degradation process up to the localization of the damage, but are unable to represent a discontinuity in the structure. On the other hand cohesive zone methods can represent the process zone at the crack tip governing the crack propagation, but cannot account for the diffuse material damaging process. In this paper we propose to combine, in a small deformations setting, a non-local elastic damage model with a cohesive zone model. This combination is formulated within a discontinuous Galerkin nite element discretization. Indeed this DG weak formulation can easily be developed in a non-local implicit form and naturally embeds interface elements that can be used to integrate the traction separation law of the cohesive zone model. The method remains thus consistent and computationally e cient as compared to other cohesive element approaches. The effects of the damage to crack transition and of the mesh discretization are respectively studied on the compact tension specimen and on the double-notched specimen, demonstrating the efficiency and accuracy of the method. [less ▲] Detailed reference viewed: 113 (24 ULg)Prediction of meso-scale mechanical properties of poly-silicon materials Lucas, Vincent ; Wu, Ling ; Arnst, Maarten et al Conference (2014, August 27) The miniature sizes of micro–electro–mechanical systems (MEMS) as well as the nature of their manufacturing processes, such as etching, material layer deposition, or embossing, are responsible for the ... [more ▼] The miniature sizes of micro–electro–mechanical systems (MEMS) as well as the nature of their manufacturing processes, such as etching, material layer deposition, or embossing, are responsible for the existence of a scatter in the final dimensions, material properties ... of manufactured micro–sensors. This scatter is potentially threatening the behavior and reliability of samples from a batch fabrication process, motivating the development of non-deterministic computational approaches to predict the MEMS properties. In this work we extract the meso-scale properties of the poly-silicon material under the form of a probabilistic distribution. To this end, Statistical Volume Elements (SVE) of the micro-structure are generated under the form of a Voronoï 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. As the finite element method is used to discretize the SVE and to solve the micro-scale boundary value problem, the homogenization technique used to extract the material tensor relies on the computational homogenization theory. In a future work, we will investigate, in the context of MEMS vibrometers, the propagation to the macro–scale of the meso-scale distribution of the homogenized elasticity tensor, with the final aim of predicting the uncertainty on their resonance frequencies. [less ▲] Detailed reference viewed: 73 (23 ULg)A probabilistic model of the adhesive contact forces between rough surfaces in the MEMS stiction context Hoang Truong, Vinh ; Wu, Ling ; Arnst, Maarten et al Conference (2014, June 26) Stiction is a common failure mechanism in microelectromechanical systems (MEMS) in which two interacting bodies permanently adhere together. This problem is due to the comparability of adhesive surface ... [more ▼] Stiction is a common failure mechanism in microelectromechanical systems (MEMS) in which two interacting bodies permanently adhere together. This problem is due to the comparability of adhesive surface forces (e.g. Van der Waals forces, capillary forces) and body forces in the MEMS context. To predict the adhesive contact forces coupled with stiction phenomenon, the combination of the Nayak statistical approach with the asperity-based theories is a common solution. However, this method contains some limitations due to the underlying assumptions: infinite size of the interacting rough surfaces and negligibility of asperity interactions. Furthermore, the Nayak solution suffers from a considerable dependency on the choice of the minimum wave length considered in the surface representation, which remains diXcult to select. The main goal of our research is to propose an improved method (i) which accounts for the Vnite size of the interacting surfaces, (ii) accounts for the uncertainties related to these surface topologies, (iii) in which the minimum wave length selection is physically based, and (iv) in which the validity of the asperity-based theories is demonstrated. From the topology measurements of MEMS samples, an analysis of the power spectral density function is carried out to determine the minimum relevant wave length for the problem of interest (here capillary stiction). We also show that at this scale of interest the asperity-based theories remain valid for polysilicon materials. Moreover, instead of considering inVnite surfaces as in the Nayak approach, a set of surfaces, whose sizes are representative of the MEMS structure, is generated based on the approximated power spectral density analysis and using the Monte Carlo method. From this description of the contacting surfaces, the adhesive contact forces can be evaluated by applying the asperity contact theories, leading to a probabilistic distribution of the adhesive contact forces. In addition, as the contact forces are rooted from the micro-scale adhesive forces, while their consequence, stiction, happens at the macro-scale of the considered device, the multi-scale nature of the phenomenon is accounted for. To predict this macro-scale behavior in a probabilistic form, the uncertainty quantiVcation process is coupled with a multiscale analysis. [less ▲] Detailed reference viewed: 141 (20 ULg)Muti-scale methods with strain-softening: damage-enhanced MFH for composite materials and computational homogenization for cellular materials with micro-buckling Wu, Ling ; Nguyen, Van Dung ; et al Conference (2014, June 06) In this work, multi-scale methods with strain softening are developed in the contexts of damage modeling for composite laminates and of buckling analyses in cellular materials. First, an anisotropic ... [more ▼] In this work, multi-scale methods with strain softening are developed in the contexts of damage modeling for composite laminates and of buckling analyses in cellular materials. First, an anisotropic gradient–enhanced continuum damage model is embedded in a mean–field homogenization (MFH) process for elasto-plastic composites. The homogenization procedure is based on the newly developed incremental secant mean-field homogenization formulation, for which the residual stress and strain states reached in the phases upon a fictitious elastic unloading are considered as starting point to apply the secant method. The mean stress fields in the phases are then computed using isotropic secant tensors, which are naturally used to define the Linear Comparison–Composite The resulting multi– scale model is then applied to study the damage process at the meso–scale of laminates, and in particular the damaging of plies in a composite stack. By using the gradient–enhanced continuum damage model, the problem of losing uniqueness upon strain softening is avoided. Second, an efficient multi–scale finite element framework capturing the buckling instabilities in cellular materials is developed. As a classical multi–scale computational homogenization scheme loses accuracy with the apparition of the macroscopic localizations resulting from the micro–buckling, the second order multi–scale computational homogenization scheme is considered. This second–order computational framework is enhanced with the following novelties so that it can be used for cellular materials. At the microscopic scale, the periodic boundary condition is used because of its efficiency. As the meshes generated from cellular materials exhibit a large void part on the boundaries and are not conforming in general, the classical enforcement based on the matching nodes cannot be applied. A new method based on the polynomial interpolation2 without the requirement of the matching mesh condition on opposite boundaries of the representative volume element (RVE) is developed. Next, in order to solve the underlying macroscopic Mindlin strain gradient continuum of this second–order scheme by the displacement–based finite element framework, the treatment of high order terms is based on the discontinuous Galerkin (DG) method to weakly impose the C1-continuity. Finally, as the instability phenomena are considered at both scales of the cellular materials, the path following technique is adopted to solve both the macroscopic and microscopic problems. [less ▲] Detailed reference viewed: 57 (18 ULg)Prediction of macroscopic mechanical properties of a polycrystalline microbeam subjected to material uncertainties Lucas, Vincent ; Wu, Ling ; Arnst, Maarten et al in Cunha, Álvaro; Caetano, Elsa; Ribeiro, Pedro (Eds.) et al Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 (2014, June) The first resonance frequency is a key performance characteristic of MEMS vibrometers. In batch fabrication, this first resonance frequency can exhibit scatter owing to various sources of manufacturing ... [more ▼] The first resonance frequency is a key performance characteristic of MEMS vibrometers. In batch fabrication, this first resonance frequency can exhibit scatter owing to various sources of manufacturing variability involved in the fabrication process. The aim of this work is to develop a stochastic multiscale model for predicting the first resonance frequency of MEMS microbeams constituted of polycrystals while accounting for the uncertainties in the microstructure due to the grain orientations. At the finest scale, we model the microstructure of polycrystaline materials using a random Voronoï 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 second scale of interest, the meso-scale. In the future, using a stochastic finite element method, we will propagate these meso-scale uncertainties to the first resonance frequency at the coarser scale. [less ▲] Detailed reference viewed: 121 (46 ULg)Muti-scale methods with strain-softening: damage-enhanced MFH for composite materials and computational homogenization for cellular materials with micro-buckling Wu, Ling ; Nguyen, Van Dung ; et al Scientific conference (2014, April 28) Materials used in the aerospace industry, as composite or foamed materials are multiscale in nature. To predict the macroscopic behaviour of structures made of such materials, the micro-scopic responses ... [more ▼] Materials used in the aerospace industry, as composite or foamed materials are multiscale in nature. To predict the macroscopic behaviour of structures made of such materials, the micro-scopic responses should also be computed within a nested scheme. This is particularly true when non-linear behaviours are modelled, or when the failure and post failure analyses are sought. In this work, multi-scale methods with strain softening are developed in the contexts of damage modelling for composite laminates and of buckling analyses in cellular materials. First, an anisotropic gradient–enhanced continuum damage model is embedded in a mean–field homogenization (MFH) process for elasto-plastic composites. The homogenization procedure is based on the newly developed incremental secant mean-field homogenization formulation, for which the residual stress and strain states reached in the phases upon a fictitious elastic unloading are considered as starting point to apply the secant method. The mean stress fields in the phases are then computed using isotropic secant tensors, which are naturally used to define the Linear Comparison–Composite The resulting multi– scale model is then applied to study the damage process at the meso–scale of laminates, and in particular the damaging of plies in a composite stack. By using the gradient–enhanced continuum damage model, the problem of losing uniqueness upon strain softening is avoided. Second, an efficient multi–scale finite element framework capturing the buckling instabilities in cellular materials is developed. As a classical multi–scale computational homogenization scheme loses accuracy with the apparition of the macroscopic localizations resulting from the micro–buckling, the second order multi–scale computational homogenization scheme is considered. This second–order computational framework is enhanced with the following novelties so that it can be used for cellular materials. At the microscopic scale, the periodic boundary condition is used because of its efficiency. As the meshes generated from cellular materials exhibit a large void part on the boundaries and are not conforming in general, the classical enforcement based on the matching nodes cannot be applied. A new method based on the polynomial interpolation2 without the requirement of the matching mesh condition on opposite boundaries of the representative volume element (RVE) is developed. Next, in order to solve the underlying macroscopic Mindlin strain gradient continuum of this second–order scheme by the displacement–based finite element framework, the treatment of high order terms is based on the discontinuous Galerkin (DG) method to weakly impose the C1-continuity. Finally, as the instability phenomena are considered at both scales of the cellular materials, the path following technique is adopted to solve both the macroscopic and microscopic problems. [less ▲] Detailed reference viewed: 85 (14 ULg)Muti-scale methods with strain-softening: damage-enhanced MFH for composite materials and computational homogenization for cellular materials with micro-buckling Noels, Ludovic ; Nguyen, Van Dung ; Wu, Ling et al Scientific conference (2014, April 14) Materials used in the aerospace industry, as composite or foamed materials are multiscale in nature. To predict the macroscopic behaviour of structures made of such materials, the micro-scopic responses ... [more ▼] Materials used in the aerospace industry, as composite or foamed materials are multiscale in nature. To predict the macroscopic behaviour of structures made of such materials, the micro-scopic responses should also be computed within a nested scheme. This is particularly true when non-linear behaviours are modelled, or when the failure and post failure analyses are sought. In this work, multi-scale methods with strain softening are developed in the contexts of damage modelling for composite laminates and of buckling analyses in cellular materials. First, an anisotropic gradient–enhanced continuum damage model is embedded in a mean–field homogenization (MFH) process for elasto-plastic composites. The homogenization procedure is based on the newly developed incremental secant mean-field homogenization formulation, for which the residual stress and strain states reached in the phases upon a fictitious elastic unloading are considered as starting point to apply the secant method. The mean stress fields in the phases are then computed using isotropic secant tensors, which are naturally used to define the Linear Comparison–Composite The resulting multi– scale model is then applied to study the damage process at the meso–scale of laminates, and in particular the damaging of plies in a composite stack. By using the gradient–enhanced continuum damage model, the problem of losing uniqueness upon strain softening is avoided. Second, an efficient multi–scale finite element framework capturing the buckling instabilities in cellular materials is developed. As a classical multi–scale computational homogenization scheme loses accuracy with the apparition of the macroscopic localizations resulting from the micro–buckling, the second order multi–scale computational homogenization scheme is considered. This second–order computational framework is enhanced with the following novelties so that it can be used for cellular materials. At the microscopic scale, the periodic boundary condition is used because of its efficiency. As the meshes generated from cellular materials exhibit a large void part on the boundaries and are not conforming in general, the classical enforcement based on the matching nodes cannot be applied. A new method based on the polynomial interpolation2 without the requirement of the matching mesh condition on opposite boundaries of the representative volume element (RVE) is developed. Next, in order to solve the underlying macroscopic Mindlin strain gradient continuum of this second–order scheme by the displacement–based finite element framework, the treatment of high order terms is based on the discontinuous Galerkin (DG) method to weakly impose the C1-continuity. Finally, as the instability phenomena are considered at both scales of the cellular materials, the path following technique is adopted to solve both the macroscopic and microscopic problems. [less ▲] Detailed reference viewed: 61 (5 ULg) |
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