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See detailElastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework
Wu, Ling ULg; Becker, Gauthier ULg; Noels, Ludovic ULg

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 ▲]

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See detailMuti-scale methods with strain-softening: damage-enhanced MFH for composite materials and computational homogenization for cellular materials with micro-buckling
Noels, Ludovic ULg; Nguyen, Van Dung ULg; Wu, Ling ULg 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 ▲]

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See detailMultiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous Galerkin formulation
Nguyen, Van Dung ULg; Becker, Gauthier ULg; Noels, Ludovic ULg

in Computer Methods in Applied Mechanics & Engineering (2013), 260

When considering problems of dimensions close to the characteristic length of the material, the size e ects can not be neglected and the classical (so–called first–order) multiscale computational ... [more ▼]

When considering problems of dimensions close to the characteristic length of the material, the size e ects can not be neglected and the classical (so–called first–order) multiscale computational homogenization scheme (FMCH) looses accuracy, motivating the use of a second–order multiscale computational homogenization (SMCH) scheme. This second–order scheme uses the classical continuum at the micro–scale while considering second–order continuum at the macro–scale. Although the theoretical background of the second–order continuum is increasing, the implementation into a finite element code is not straightforward because of the lack of high–order continuity of the shape functions. In this work, we propose a SMCH scheme relying on the discontinuous Galerkin (DG) method at the macro–scale, which simplifies the implementation of the method. Indeed, the DG method is a generalization of weak formulations allowing for inter-element discontinuities either at the C0 level or at the C1 level, and it can thus be used to constrain weakly the C1 continuity at the macro–scale. The C0 continuity can be either weakly constrained by using the DG method or strongly constrained by using usual C0 displacement–based finite elements. Therefore, two formulations can be used at the macro–scale: (i) the full–discontinuous Galerkin formulation (FDG) with weak C0 and C1 continuity enforcements, and (ii) the enriched discontinuous Galerkin formulation (EDG) with high–order term enrichment into the conventional C0 finite element framework. The micro–problem is formulated in terms of standard equilibrium and periodic boundary conditions. A parallel implementation in three dimensions for non–linear finite deformation problems is developed, showing that the proposed method can be integrated into conventional finite element codes in a straightforward and e cient way. [less ▲]

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See detailA micro-meso-model of intra-laminar fracture in fiber-reinforced composites based on a Discontinuous Galerkin/Cohesive Zone Method
Wu, Ling ULg; Tjahjanto, Denny; Becker, Gauthier ULg et al

in Engineering Fracture Mechanics (2013), 104

The recently developed hybrid discontinuous Galerkin/extrinsic cohesive law framework is extended to the study of intra{laminar fracture of composite materials. Toward this end, micro-volumes of di erent ... [more ▼]

The recently developed hybrid discontinuous Galerkin/extrinsic cohesive law framework is extended to the study of intra{laminar fracture of composite materials. Toward this end, micro-volumes of di erent sizes are studied. The method captures the debonding process, which is herein proposed to be assimilated to a damaging process, before the strain softening onset, and the density of dissipated energy resulting from the damage (debonding) remains the same for the di erent studied cell sizes. Finally, during the strain softening phase a micro{crack initiates and propagates in agreement with experimental observations. We thus extract a resulting mesoscale cohesive law, which is independent on the cell sizes, using literature methods. [less ▲]

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See detailA full-discontinuous Galerkin formulation of non-linear Kirchhoff-Love shells: elasto-plastic finite deformations, parallel computation & fracture applications
Becker, Gauthier ULg; Noels, Ludovic ULg

in International Journal for Numerical Methods in Engineering (2013), 93(1), 80-117

Due to its ability to take into account discontinuities, the discontinuous Galerkin (DG) method presents some advantages for modeling crack initiations and propagations. This concept has been recently ... [more ▼]

Due to its ability to take into account discontinuities, the discontinuous Galerkin (DG) method presents some advantages for modeling crack initiations and propagations. This concept has been recently applied to 3D simulations and to elastic thin bodies. In this last case, the assumption of small elastic deformations before crack initiations or propagations reduces drastically the applicability of the framework to a reduced number of materials. To remove this limitation, a full-DG formulation of non-linear Kirchhoff-Love shells is presented and is used in combination with an elasto-plastic finite deformations model. The results obtained by this new formulation are in agreement with other continuum elasto-plastic shell formulations. Then this full-DG formulation of Kirchhoff-Love shells is coupled with the cohesive zone model to perform thin body fracture simulations. As this method allows considering elasto-plastic constitutive laws in combination with the cohesive model, accurate results compared to the experiments are found. In particular, the crack path and propagation rate of a blasted cylinder are shown to match experimental results. One of the main advantages of this framework is its ability to run in parallel with a high speed-up factor, allowing the simulation of ultra fine meshes. [less ▲]

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See detailValidation tests of the full-discontinuous Galerkin / extrinsic cohesive law framework of Kirchhoff-Love shells
Becker, Gauthier ULg; Noels, Ludovic ULg

in International Journal of Fracture (2012), 178(1), 299-322

Due to its ability to account for discontinuities, the discontinuous Galerkin (DG) method presents two main advantages for modeling crack initiations and propagation. On the one hand, it provides an easy ... [more ▼]

Due to its ability to account for discontinuities, the discontinuous Galerkin (DG) method presents two main advantages for modeling crack initiations and propagation. On the one hand, it provides an easy way to insert the cohesive elements during the simulation and therefore avoids the drawbacks inherent to the use of an extrinsic cohesive law. On the other hand, the capture of complex crack path requires very thin meshes and the recourse to a parallel implementation of DG formulations exhibits a high scalability of the resolution scheme. Recently, the authors developed such a DG-fracture framework for Kirchhoff-Love shells in the linear and non-linear ranges. They proved that this framework dissipates, during the fracture process, an amount of energy equal to the fracture energy of the material and that the model is able to propagate the crack with the right speed. In this paper, novel numerical benchmarks are presented to validate the method in various fracture conditions. The two first ones include an initial notch and study the fracture propagation under two different dynamic loadings (impact and blast). The two other ones focus on the fragmentation of initially unbroken specimens due to uniform expansion in order to demonstrate the ability of the new framework to model crack initiations. Results are in all cases in agreement with the ones reported in the literature. [less ▲]

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See detailFull discontinuous Galerkin formulation of shells in large deformations with parallel and fracture mechanics applications
Becker, Gauthier ULg; Noels, Ludovic ULg

Conference (2012, July 11)

Fracture mechanical problems can be solved by coupling the finite elements with a cohesive approach. Unfortunately, the classical cohesive methods suffer from severe limitations. Indeed, on one hand, the ... [more ▼]

Fracture mechanical problems can be solved by coupling the finite elements with a cohesive approach. Unfortunately, the classical cohesive methods suffer from severe limitations. Indeed, on one hand, the intrinsic approach, which inserts the cohesive elements at the beginning, has to model the prefracture stage. This requires an initial slope in the traction separation law that should tend toward infinity to avoid lack of consistency leading to obvious numerical problems. On the other hand, the extrinsic cohesive method inserts the cohesive elements during the simulation when a fracture criterion is reached. This insertion requires topological mesh modifications and therefore a very complicated implementation, especially in a parallel code. To overcome these limitations, new methods were developed and in particular, an approach based on discontinuous Galerkin formulation (DG) has been pioneered by R. Radovitzky (Radovitzky cmame2011). The use of the DG principle allows to formulate the problem with discontinuous elements and the continuity between them is ensured weakly by terms integrated on the elements interface . These interface elements can be easily replaced by a cohesive element during the simulation. We have recently developed this approach for shells (Becker cmame2011) to obtain a full DG method. Moreover, a new cohesive law based on the reduced stresses of the thin bodies formulation is developed to propagate a fracture through the thickness. This cohesive model dissipates the right amount of energy during crack phenomena. These developments are implemented in parallel and validated by the study the blast of a notched cylinder, for which experimental and numerical (by XFEM method) data are reported in the literature by R. Larsson (Larsson ijnme2011). Finally, as thin structures are often made of ductile materials, which show large deformations before fracture, the formulation is extended to the non linear case with hyperelastic material law. This one can take into account the damage and a criterion based on the work of Huespe (Huespe plasticity2009) is developed to localize the damage leading to the apparition and propagation of cracks. [less ▲]

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See detailNumerical simulations of brittle and elasto-plastic fracture for thin structures subjected to dynamic loadings
Becker, Gauthier ULg

Doctoral thesis (2012)

The main purpose of this thesis is the development of a framework to model fracture initiation and propagation in thin bodies. This is achieved by the combination of two original models. On one hand ... [more ▼]

The main purpose of this thesis is the development of a framework to model fracture initiation and propagation in thin bodies. This is achieved by the combination of two original models. On one hand, (full) discontinuous Galerkin formulations of Euler-Bernoulli beams as well as Kirchhoff-Love shells are established. These formulations allow modeling a thin structure with discontinuous elements, the continuity being ensured weakly by addition of interface terms. The first advantage of the recourse to a discontinuous method is an easy insertion of cohesive elements during the simulation without a modification of the mesh topology. In fact with a continuous method, the insertion of the cohesive elements at the beginning of the simulation leads to numerical issues and their insertion at onset of fracture requires a complex implementation to duplicate the nodes. By contrast, as interface elements are naturally present in a discontinuous formulation their substitution at fracture initiation is straightforward. The second advantage of the discontinuous Galerkin formulation is a simple parallel implementation obtained in this work by exploiting, the discontinuity of the mesh in an original manner. Finally, last advantage of the recourse to a discontinuous Galerkin method for thin bodies is to obtain a one field formulation. In fact, the C1 continuity is ensured weakly by interface terms without considering rotational degrees of freedom. On the other hand, the through-the-thickness crack propagation is complicated by the implicit thickness model inherent to thin bodies formulations. Therefore we suggest an original cohesive model based on reduced tresses. Our model combines the different reduced stresses in such a way that the expected amount of energy is released during the crack process leading to a model which respects the energetic balance whatever the applied loadings. The efficiency of the obtained framework is demonstrated through the simulation of several benchmarks whose results are in agreement with numerical and experimental data coming from the literature. Furthermore, the versatility of our framework is shown by simulating 2 very different fracture phenomena: the crack propagation for elastic as well as for elasto-plastic behavior and the fragmentation of brittle materials. This demonstrates that our framework is a powerful tool to study dynamics crack phenomena in thin structure problems involving a large number of degrees of freedom. [less ▲]

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See detailFull discontinuous Galerkin formulation of shell in large deformations with fracture mechanic applications
Becker, Gauthier ULg; Noels, Ludovic ULg

in Hogge, Michel; Van Keer, Roger; Dick, Erik (Eds.) et al Proceedings of the 5th International Conference on Advanded COmputational Methods in Engineering (ACOMEN2011) (2011, November)

Different methods have been developed to model tearing prediction, as e.g., the combination between the cohesive principle and the finite element method. Unfortunately, this method has some well known ... [more ▼]

Different methods have been developed to model tearing prediction, as e.g., the combination between the cohesive principle and the finite element method. Unfortunately, this method has some well known issues that can be fixed by recourse to discontinuous Galerkin formulation. Such a formulation allows to insert very easily an extrinsic cohesive element at onset of fracture without any mesh modification. This promising technique has been recently developed by the authors for linear shell. Although promising numerical results were obtained, it is difficult to compare the method with experiments due to the large plastic deformation present in material before the fracture apparition. Thus, the method is extent herein to elasto-plastic finite deformations. The simulations of some benchmarks prove the ability of this new framework to model accurately the continuum part of the deformation and the crack propagation. [less ▲]

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See detailA one Field Full Discontinuous Galerkin Method for Kirchhoff-Love Shells Applied to Fracture Mechanics
Becker, Gauthier ULg; Geuzaine, Christophe ULg; Noels, Ludovic ULg

in Computer Methods in Applied Mechanics & Engineering (2011), 200(45-46), 3223-3241

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the Finite Element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive ... [more ▼]

In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the Finite Element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture. The purpose of this paper is to develop this formulation for Kirchhoff-Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method. [less ▲]

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See detailA shell fracture framework based on a full discontinuous Galerkin formulation combined with an extrinsic cohesive law
Becker, Gauthier ULg; Noels, Ludovic ULg

Conference (2011, June 06)

The cohesive method can be combined easily with Finite Element method to simulate a fracture problem which can contains fracture initiation and propagation. Nevertheless, the insertion of cohesive ... [more ▼]

The cohesive method can be combined easily with Finite Element method to simulate a fracture problem which can contains fracture initiation and propagation. Nevertheless, the insertion of cohesive elements is not straightforward. Indeed, the two classical approaches suffer from severe limitations. On one hand, in the intrinsic approach, as the cohesive element is inserted at the beginning, this element has to model the continuum stage of deformation before fracture. This is ensured by an initial slope in the cohesive law which leads to a stiffness modification and to an alteration of propagation of wave. On the other hand, the introduction of the cohesive element during the simulation in extrinsic approach requests a dynamic modification of mesh. This operation is very difficult to implement especially in the case of a parallel implementation which is almost mandatory due to the very important number of degrees of freedom inherent to a fine mesh used to track the crack path. A solution to these limitations, pioneered by J. Mergheim and R. Radovitzky is to recourse to a discontinuous Galerkin formulation. Indeed this one used discontinuous test functions and integration at the interface of elements to discretize a structure with discontinuous elements. The integration on the boundary of elements allows ensuring weakly the continuity of displacements in a stable and consistent manner. As interface elements are present they can be easily substituted by cohesive elements when a fracture criterion is reached. The interest of the method has been recently proved by R. Radovitzky etal. for 3D elements and by the authors for Euler-Bernoulli beams. An extension of the formulation to Kirchhoff-Love shell is presented here. A novel extrinsic cohesive law is developed to model a through the thickness fracture. In fact, as in thin bodies formulation the thickness is not “discretized” this operation is not straightforward. Indeed, as the fracture occurs only in tension, in a pure bending case the position of neutral axis has to be move to propagate the fracture. To avoid this complicated step, it is suggested to integrate on the thickness the cohesive law which is then applies on resultant efforts. The coupling between the openings in displacement and rotation is performed in a way which guarantees a proper release of energy for any loading. Furthermore, the combination between fracture modes I and II is realized as suggested by M. Ortiz etal. Some numerical quasi-static and dynamic benchmarks are simulated to show the interest and the good performance of the presented framework. [less ▲]

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See detailA fracture framework for Euler Bernoulli beams based on a full discontinuous Galerkin formulation/extrinsic cohesive law combination
Becker, Gauthier ULg; Noels, Ludovic ULg

in International Journal for Numerical Methods in Engineering (2011), 85(10), 12271251

A new full Discontinuous Galerkin discretization of Euler Bernoulli beam is presented. The main interest of this framework is its ability to simulate fracture problems by inserting a cohesive zone model ... [more ▼]

A new full Discontinuous Galerkin discretization of Euler Bernoulli beam is presented. The main interest of this framework is its ability to simulate fracture problems by inserting a cohesive zone model in the formulation. With a classical Continuous Galerkin method the use of the cohesive zone model is di cult because as insert a cohesive element between bulk elements is not straightforward. On one hand if the cohesive element is inserted at the beginning of the simulation there is a modification of the structure stiffness and on the other hand inserting the cohesive element during the simulation requires modification of the mesh during computation. These drawbacks are avoided with the presented formulation as the structure is discretized in a stable and consistent way with full discontinuous elements and inserting cohesive elements during the simulation becomes straightforward. A new cohesive law based on the resultant stresses (bending moment and membrane) of the thin structure discretization is also presented. This model allows propagating fracture while avoiding through-the-thickness integration of the cohesive law. Tests are performed to show that the proposed model releases, during the fracture process, an energy quantity equal to the fracture energy for any combination of tension-bending loadings. [less ▲]

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See detailA Full Discontinuous Galerkin Formulation Of Euler Bernoulli Beams In Linear Elasticity With Fractured Mechanic Applications
Becker, Gauthier ULg; Noels, Ludovic ULg

Conference (2010, July 21)

A full discontinuous Galerkin method is used to predict the fracture of beams thanks to insertion of an extrinsic cohesive element. In fact, The formulation developed originally by G. Wells etal. to ... [more ▼]

A full discontinuous Galerkin method is used to predict the fracture of beams thanks to insertion of an extrinsic cohesive element. In fact, The formulation developed originally by G. Wells etal. to guarantee weakly the high order derivatives of plates with only displacement field unknown and extended by L. Noels etal. for shells is derived for beam with full discontinuous elements. This new formulation can be advantageously combined, as shown first by J. Mergheim etal. , with an extrinsic cohesive approach as there is no need to modify dynamically the mesh, which is the major drawback of this approach. The pre-fractured stage is modeled by full discontinuous elements in a manner which is proved stable and consistent and the fracture is modeled by a cohesive law applied on stress resultant an stress couple defined by J.C. Simo etal. The suggested study produces two type of results. On one hand, it is shown analytically and verified by numerical examples that the presented framework has got the properties of consistency and convergence expected for a numerical scheme. On the other hand, it is proved by some test cases that the energy released during fracture process is equal to the fracture energy except in the case where the difference of internal energy between not fractured and fractured configurations is bigger than the fracture energy. In this case, the fracture occurs in one time step. The presented work proposed a novel interesting manner to take into account fracture in thin bodies. The verification made on the particularized case of beams suggested great perspectives for plates and shells which allow to simulate more complex problems. [less ▲]

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See detailEtude de modélisation d'impacts sur des structures métalliques
Becker, Gauthier ULg

Master's dissertation (2008)

Dans les cas d'une sollicitation par choc, des phénomènes dynamiques peuvent apparaitre et modifi er le comportement de la structure par rapport à un cas statique. Ce travail se propose d'évaluer ce ... [more ▼]

Dans les cas d'une sollicitation par choc, des phénomènes dynamiques peuvent apparaitre et modifi er le comportement de la structure par rapport à un cas statique. Ce travail se propose d'évaluer ce changement de comportement via l'étude d'un système simplifi é. Le choc sera modélisé de manière analytique en statique et en dynamique a n de mettre en évidence les phénomènes dynamiques en présence. Le même travail sera ensuite réalisé en utilisant la méthode des éléments fi nis. Les modèles seront vérifi és expérimentalement. Une fois véri fiés, les différents modèles seront exploités pour établir une loi décrivant l'effet dynamique. Cette loi postule que les effets dynamiques sont exclusivement fonction du rapport des masses en présence. Plus la masse du poteau sera importante devant celle du mobile impactant et plus les effets dynamiques seront prononcés au niveau du poteau. La loi permettra également d'arriver à la conclusion que les effets dynamiques se matérialisent par une diminution des déformations observées par rapport au cas statique. La vérifi cation menée dans la suite permettra de vérifi er la loi et montrera que l'élasticité du matériau joue également un rôle important au niveau des phénomènes dynamiques. La théorie et les modèles développés seront ensuite appliquées au dimensionnement d'un poteau anti-intrusion. Ce système permet de protéger des bâtiments contre les voitures béliers. La protection ne sera e ficace que si le poteau est correctement dimensionné. A cet effet, un modèle éléments fi nis sera élaboré dans la partie appliquée du travail. La partie théorique ayant permis de démontrer que les effets dynamiques jouent un rôle négligeable dans la déformation du poteau, ce sont les modèles basés une approche quasi-statique qui seront utilisés. [less ▲]

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