Computer Methods in Applied Mechanics and Engineering, Dec 1, 2008
The phenomena of crack initiation, propagation and ultimate fracture are studied here under the f... more The phenomena of crack initiation, propagation and ultimate fracture are studied here under the following assumptions: (i) the crack law is modelled by means of a cohesive zone model and (ii) the crack paths are postulated a priori. In this context, a variational formulation is proposed which relies on an augmented Lagrangian. A mixed interface finite element is introduced to discretise the crack paths, the degrees of freedom of which consist in the displacement on both crack lips and the density of cohesive forces. This enables an exact treatment of multi-valued cohesive laws (e.g. initial adhesion, contact conditions, possible rigid unloading, etc.), without penalty regularisation. A special attention is paid to the convergence with mesh-refinement, i.e. the well-posedness of the problem, on the basis of theoretical results of contact mechanics and some complementary numerical investigations. Fulfilment of the LBB condition is the key factor to gain the desired properties. Moreover, it is shown that the integration of the constitutive law admits a unique solution as soon as some condition on the augmented Lagrangian is enforced. Finally, a 3D simulation shows the applicability to practical engineer problems, including in particular the robustness of the formulation and its compatibility with classical solution algorithms (Newton method, line-search, path-following techniques, …).
A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete... more A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete under tensile loading. Here, the target scale is the individual crack. In order to deal with localised damage, the model is inherently nonlocal: the gradient of the damage field is explicitly involved in the constitutive equations; it is parameterised by a nonlocal length scale which is interpreted as the width of the process zone. The model is defined so that its predictions are close to those of a cohesive law for vanishing nonlocal length scales. Therefore, the current model is plainly consistent with cohesive zone model analyses: the nonlocal length scale appears as a small parameter which does not need any specific identification. And four parameters -among which the tensile strength and the fracture energy -enable to adjust the softening cohesive response. Besides, a special attention has been paid to the shape of the initial damage surface and to the relation between damage and stiffness. The damage surface takes into account not only the contrast between tensile and compressive strengths but also experimental evidences regarding its shape in multiaxial tension. And the damage -stiffness relation is defined so as to describe important phenomena such as the stiffness recovery with crack closure and the sustainability of compressive loads by damaged structures. Finally, several comparisons with experimental data (global force/opening responses, size dependency, curved crack paths, crack opening profiles) enable to validate qualitatively and quantitatively the pertinence of the constitutive law in 2D and 3D.
HAL (Le Centre pour la Communication Scientifique Directe), May 9, 2011
La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'... more La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'elle est employée en mécanique non linéaire avec des algorithmes de résolution itératifs. Ce critère de convergence vise à mesurer la proximité du champ de contrainte obtenu avec l'équilibre. Les critères usuels ne fournissent pas une mesure objective de la distance à l'équilibre : la taille des mailles et les dimensions physiques qui composent le résidu influe sur la valeur du critère. Un nouveau critère est proposé qui répond à cette double critique. Mots clefs -méthode éléments finis, critère de convergence, Code_Aster.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 24, 2009
fatigue dans une structure bi-ou tri-dimensionnelle soumise à un chargement cyclique d'amplitude ... more fatigue dans une structure bi-ou tri-dimensionnelle soumise à un chargement cyclique d'amplitude quelconque. Le modèle présenté est fondé sur un principe de minimisation d'énergie. Les résultats numériques corroborent des conclusions théoriques, et mettent en évidence un certain nombre de phénomènes liés à la propagation par fatigue, notamment l'identification d'une loi de Paris pour un chargement d'amplitude constante et l'effet retard dû à une surcharge dans un matériau ductile.
HAL (Le Centre pour la Communication Scientifique Directe), May 15, 2017
On décrit un modèle simplifié de restauration d'écrouissage utilisé dans la simulation numérique ... more On décrit un modèle simplifié de restauration d'écrouissage utilisé dans la simulation numérique du soudage (SNS). Ce modèle, écrit sous forme d'équation différentielle, permet de prendre en compte l'effet de restauration des propriétés plastiques de l'acier lors de la simulation du soudage multi-passes. Les paramètres de ce modèle sont déterminés par recalage à partir d'essais expérimentaux développés et réalisés par l'EPRI. L'étape de validation du modèle, au cours de laquelle les résultats numériques sont confrontés à des résultats expérimentaux, est réalisée en deux temps : d'abord en simulant les essais EPRI puis en simulant des essais Satoh.
Most structures exposed to severe loadings may suffer damage, possibly resulting in safety risk o... more Most structures exposed to severe loadings may suffer damage, possibly resulting in safety risk or economic loss. In complement to experimental studies and design codes and standards, advanced numerical simulation of damage appears as a promising approach to describe such critical regimes. However, damage modelling still faces several difficulties. In the case of ductile damage, characterized by a large amount of plastic strain, spurious strain localization and plastic volumetric locking are observed and should be dealt with.
This study aims at investigating the properties of a non-local locking-free GTN ductile damage mo... more This study aims at investigating the properties of a non-local locking-free GTN ductile damage model [1] at finite strain within the framework of small-scale yielding. This model solves the problems of spurious localization and volumetric locking. The former is achieved by introducing the gradient of the hardening variable into the Helmholtz free energy (gradient plasticity). On a numerical ground, this results in spatial gradients of state variables within the constitutive relations: a decomposition-coordination techniques [2] is used to treat the corresponding term. Regarding the volumetric locking resulting from plastic incompressibility prior to damage, the Hu-Washizu mixed variational principle [3] is put in practice. An additional penalty term is also introduced into the corresponding Lagrangian in order to ensure coercivity. Finally, a new 5-field finite element is derived from the non-local locking-free variational formulation, in combination with the set of constitutive rel...
Dans notre travail de modelisation de l'endommagement ductile, une formulation de grandes def... more Dans notre travail de modelisation de l'endommagement ductile, une formulation de grandes deformations logarithmiques est utilisee. Afin de traiter la dependance au maillage, nous regularisons la loi de comportement avec un modele non local a gradient d'une variable interne. Ce modele non local est ensuite coupleavec des elements mixtes pour eviter le verrouillage volumique. Des simulations sur eprouvettes sont enfin confrontees a des resultats d'essais.
For some concrete structures submitted to mechanical loadings, the safety assessment requires to ... more For some concrete structures submitted to mechanical loadings, the safety assessment requires to predict potential crack propagations. To this end, the fictitious crack approach introduced by Hilleborg thirty years ago and based on cohesive laws brings acceptable results regarding the crack length and the overall structural response [1], thanks to a proper energy balance between strain energy and dissipation through micro and macrocracks. However, the prediction of crack paths may remain difficult and/or cumbersome on a finite element level when dealing with 3D simulations despite constant progresses reported in the literature. An alternative relies on the smeared crack approach where Continuum Damage Mechanics provides the setting to predict damage levels (i.e. crack lengths) as well as damage patterns (i.e. crack paths). But the inherent illposedness of softening laws – resulting namely in spurious mesh dependency and wrong dissipation assessment – has to be given a proper answer ...
This paper presents an approach to predict ductile fracture of real-life structures. It relies on... more This paper presents an approach to predict ductile fracture of real-life structures. It relies on Rousselier’s constitutive model to describe plastic void growth, a specific finite strain formulation that preserves energetic properties and a non local theory to deal with strain localisation. It is finally applied to the computation of a notched specimen.
International Journal for Numerical Methods in Engineering, 2018
The major goal of this work is to develop a robust modelling strategy for the simulation of ducti... more The major goal of this work is to develop a robust modelling strategy for the simulation of ductile damage development including crack initiation and subsequent propagation. For that purpose, a Gurson-type model is used. This model class, as many other damage models, leads to significant material softening and must be used within a large deformation framework due to the ductile character of the materials. This leads to 2 main difficulties that should be dealt with carefully: mesh dependency and volumetric locking. In this work, a logarithmic finite strain framework is adopted in which the Gurson-Tvergaard-Needleman constitutive law is reformulated. Then a nonlocal formulation with regularisation of hardening variable is applied so as to solve mesh dependency and strain localization problem. In addition, the nonlocal model is combined with mixed "displacement-pressure-volume variation" elements to avoid volumetric locking. Thereby, a mesh-independent and locking-free finite strain framework suitable for the modelling of ductile rupture is established. Attention is paid to mathematical properties and numerical performance of the model. Finally, the model parameters are identified on an experimental database for a nuclear piping steel. Simulations of standard test specimens (notched tensile bars and compact tension and single edge notched tensile cracked specimens) are performed and compared to experimental results.
A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete... more A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete under tensile loading. Here, the target scale is the individual crack. In order to deal with localised damage, the model is inherently nonlocal: the gradient of the damage field is explicitly involved in the constitutive equations; it is parameterised by a nonlocal length scale which is interpreted as the width of the process zone. The model is defined so that its predictions are close to those of a cohesive law for vanishing nonlocal length scales. Therefore, the current model is plainly consistent with cohesive zone model analyses: the nonlocal length scale appears as a small parameter which does not need any specific identification. And four parameters -among which the tensile strength and the fracture energy -enable to adjust the softening cohesive response. Besides, a special attention has been paid to the shape of the initial damage surface and to the relation between damage and stiffness. The damage surface takes into account not only the contrast between tensile and compressive strengths but also experimental evidences regarding its shape in multiaxial tension. And the damage -stiffness relation is defined so as to describe important phenomena such as the stiffness recovery with crack closure and the sustainability of compressive loads by damaged structures. Finally, several comparisons with experimental data (global force/opening responses, size dependency, curved crack paths, crack opening profiles) enable to validate qualitatively and quantitatively the pertinence of the constitutive law in 2D and 3D.
En fait, l'étendue du voisinage représentatif devrait être fixée par l'échelle à laquelle on peut... more En fait, l'étendue du voisinage représentatif devrait être fixée par l'échelle à laquelle on peut supposer une distribution affine de la variable α . Elle dépend donc du choix de paramétrage du 1 Pour s'en persuader, deux options : une solide vision dans l'espace ou bien une construction à l'aide de balles de pingpong, très convaincante.
By using a principle of least energy and a Barenblatt-type surface energy assumed as a smooth fun... more By using a principle of least energy and a Barenblatt-type surface energy assumed as a smooth function of the jump of the displacements, we prove that it is possible to predict the onset of cracking by searching relative minima of the total energy of a body, the yield criterion depending on the choice of the energies. These types of brittle fracture models are then tested numerically by using cohesive elements.
On propose une méthode permettant d'assurer la transition et l'équivalence énergétique entre un m... more On propose une méthode permettant d'assurer la transition et l'équivalence énergétique entre un modèle non local à gradient d'endommagement et un modèle de zones cohésives. Dans un premier temps, on identifie analytiquement (sur un problème unidimensionnel) une famille de lois cohésives permettant de basculer du modèle non local vers le modèle cohésif, et ce quelque soit le niveau de dégradation atteint par le matériau. Cette stratégie est ensuite étendue au cadre bidimensionnel éléments finis, en couplant les représentations continues et discrètes dans le cas de la propagation de fissures rectilignes en mode I. Mots clefs -endommagement non local, modèles de zones cohésives, équivalence énergétique. Le matériau considéré est élastique fragile, son état est caractérisé par le tenseur des déformations ε et par une variable scalaire d'endommagement a qui dégrade progressivement sa rigidité initiale.
The modeling of damage processes at the material point scale through a specific constitutive law ... more The modeling of damage processes at the material point scale through a specific constitutive law addresses a wide variety of constitutive behaviors, for instance brittle elasticity where the evolution of damage degrades the elastic stiffness, or ductile plasticity where both dissipative mechanisms (plasticity and damage) are intimately coupled. In spite of their obvious differences, all these laws possess a common feature: beyond some load level, the domain of reversibility expressed in the stress space shrinks. This phenomenon is called strain-softening. This essential property, characteristic of damage mechanisms, is responsible for the localization of the mechanical fields (strain, damage, plastic strain, . . . ) inside narrow bands, an experimental fact observed on many degrading materials such as soils, concrete, metals, etc., see for instance . Despite twenty years of intensive research, numerical modeling of the localization phenomenon remains a challenging objective which has not yet been given a fully satisfactory answer. This is in contrast with the great progresses that have been achieved regarding the local description of damage mechanisms for various materials, on the basis of micro-macro derivations as well as phenomenological intuitions. Unfortunately, localization partially hinders predictive computations of propagating damaged zones in real-life structures submitted to a given load history. The difficulty originates from the fact that the equations that describe the structural behavior (including not only the constitutive law but also the equilibrium and compatibility equations) result in an ill-posed problem which manifests itself by a spurious mesh-dependency when making use of finite element solutions. After a first insight into the localization phenomena, the problem of loss of uniqueness and loss of ellipticity is addressed for elastoplastic solids. General (diffuse) and localized bifurcation modes are distinguished. The existence of jumps of the velocity gradient is explored in the dynamic (acceleration waves) and static cases (stationary waves). Critical values of hardening moduli for loss of uniqueness or loss of ellipticity are derived. A link between the notion of material stability and acceleration waves is also presented. The collected strain localization criteria are summarized in section 1.2.6. Note that the derivations and formula are presented in the case of elastoplastic materials. However the results can be extended to damaging materials in a straightforward
La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'... more La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'elle est employée en mécanique non linéaire avec des algorithmes de résolution itératifs. Ce critère de convergence vise à mesurer la proximité du champ de contrainte obtenu avec l'équilibre. Les critères usuels ne fournissent pas une mesure objective de la distance à l'équilibre : la taille des mailles et les dimensions physiques qui composent le résidu influe sur la valeur du critère. Un nouveau critère est proposé qui répond à cette double critique. Mots clefs -méthode éléments finis, critère de convergence, Code_Aster.
A general framework for deriving and using a class of constitutive laws incorporating spatial gra... more A general framework for deriving and using a class of constitutive laws incorporating spatial gradients of internal variables is presented. It user two basic ingredients : a derivation of such models by homogenization techniques and a reformulation of the et~olution eyuation at the scale of the whole structure.
Computer Methods in Applied Mechanics and Engineering, Dec 1, 2008
The phenomena of crack initiation, propagation and ultimate fracture are studied here under the f... more The phenomena of crack initiation, propagation and ultimate fracture are studied here under the following assumptions: (i) the crack law is modelled by means of a cohesive zone model and (ii) the crack paths are postulated a priori. In this context, a variational formulation is proposed which relies on an augmented Lagrangian. A mixed interface finite element is introduced to discretise the crack paths, the degrees of freedom of which consist in the displacement on both crack lips and the density of cohesive forces. This enables an exact treatment of multi-valued cohesive laws (e.g. initial adhesion, contact conditions, possible rigid unloading, etc.), without penalty regularisation. A special attention is paid to the convergence with mesh-refinement, i.e. the well-posedness of the problem, on the basis of theoretical results of contact mechanics and some complementary numerical investigations. Fulfilment of the LBB condition is the key factor to gain the desired properties. Moreover, it is shown that the integration of the constitutive law admits a unique solution as soon as some condition on the augmented Lagrangian is enforced. Finally, a 3D simulation shows the applicability to practical engineer problems, including in particular the robustness of the formulation and its compatibility with classical solution algorithms (Newton method, line-search, path-following techniques, …).
A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete... more A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete under tensile loading. Here, the target scale is the individual crack. In order to deal with localised damage, the model is inherently nonlocal: the gradient of the damage field is explicitly involved in the constitutive equations; it is parameterised by a nonlocal length scale which is interpreted as the width of the process zone. The model is defined so that its predictions are close to those of a cohesive law for vanishing nonlocal length scales. Therefore, the current model is plainly consistent with cohesive zone model analyses: the nonlocal length scale appears as a small parameter which does not need any specific identification. And four parameters -among which the tensile strength and the fracture energy -enable to adjust the softening cohesive response. Besides, a special attention has been paid to the shape of the initial damage surface and to the relation between damage and stiffness. The damage surface takes into account not only the contrast between tensile and compressive strengths but also experimental evidences regarding its shape in multiaxial tension. And the damage -stiffness relation is defined so as to describe important phenomena such as the stiffness recovery with crack closure and the sustainability of compressive loads by damaged structures. Finally, several comparisons with experimental data (global force/opening responses, size dependency, curved crack paths, crack opening profiles) enable to validate qualitatively and quantitatively the pertinence of the constitutive law in 2D and 3D.
HAL (Le Centre pour la Communication Scientifique Directe), May 9, 2011
La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'... more La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'elle est employée en mécanique non linéaire avec des algorithmes de résolution itératifs. Ce critère de convergence vise à mesurer la proximité du champ de contrainte obtenu avec l'équilibre. Les critères usuels ne fournissent pas une mesure objective de la distance à l'équilibre : la taille des mailles et les dimensions physiques qui composent le résidu influe sur la valeur du critère. Un nouveau critère est proposé qui répond à cette double critique. Mots clefs -méthode éléments finis, critère de convergence, Code_Aster.
HAL (Le Centre pour la Communication Scientifique Directe), Aug 24, 2009
fatigue dans une structure bi-ou tri-dimensionnelle soumise à un chargement cyclique d'amplitude ... more fatigue dans une structure bi-ou tri-dimensionnelle soumise à un chargement cyclique d'amplitude quelconque. Le modèle présenté est fondé sur un principe de minimisation d'énergie. Les résultats numériques corroborent des conclusions théoriques, et mettent en évidence un certain nombre de phénomènes liés à la propagation par fatigue, notamment l'identification d'une loi de Paris pour un chargement d'amplitude constante et l'effet retard dû à une surcharge dans un matériau ductile.
HAL (Le Centre pour la Communication Scientifique Directe), May 15, 2017
On décrit un modèle simplifié de restauration d'écrouissage utilisé dans la simulation numérique ... more On décrit un modèle simplifié de restauration d'écrouissage utilisé dans la simulation numérique du soudage (SNS). Ce modèle, écrit sous forme d'équation différentielle, permet de prendre en compte l'effet de restauration des propriétés plastiques de l'acier lors de la simulation du soudage multi-passes. Les paramètres de ce modèle sont déterminés par recalage à partir d'essais expérimentaux développés et réalisés par l'EPRI. L'étape de validation du modèle, au cours de laquelle les résultats numériques sont confrontés à des résultats expérimentaux, est réalisée en deux temps : d'abord en simulant les essais EPRI puis en simulant des essais Satoh.
Most structures exposed to severe loadings may suffer damage, possibly resulting in safety risk o... more Most structures exposed to severe loadings may suffer damage, possibly resulting in safety risk or economic loss. In complement to experimental studies and design codes and standards, advanced numerical simulation of damage appears as a promising approach to describe such critical regimes. However, damage modelling still faces several difficulties. In the case of ductile damage, characterized by a large amount of plastic strain, spurious strain localization and plastic volumetric locking are observed and should be dealt with.
This study aims at investigating the properties of a non-local locking-free GTN ductile damage mo... more This study aims at investigating the properties of a non-local locking-free GTN ductile damage model [1] at finite strain within the framework of small-scale yielding. This model solves the problems of spurious localization and volumetric locking. The former is achieved by introducing the gradient of the hardening variable into the Helmholtz free energy (gradient plasticity). On a numerical ground, this results in spatial gradients of state variables within the constitutive relations: a decomposition-coordination techniques [2] is used to treat the corresponding term. Regarding the volumetric locking resulting from plastic incompressibility prior to damage, the Hu-Washizu mixed variational principle [3] is put in practice. An additional penalty term is also introduced into the corresponding Lagrangian in order to ensure coercivity. Finally, a new 5-field finite element is derived from the non-local locking-free variational formulation, in combination with the set of constitutive rel...
Dans notre travail de modelisation de l'endommagement ductile, une formulation de grandes def... more Dans notre travail de modelisation de l'endommagement ductile, une formulation de grandes deformations logarithmiques est utilisee. Afin de traiter la dependance au maillage, nous regularisons la loi de comportement avec un modele non local a gradient d'une variable interne. Ce modele non local est ensuite coupleavec des elements mixtes pour eviter le verrouillage volumique. Des simulations sur eprouvettes sont enfin confrontees a des resultats d'essais.
For some concrete structures submitted to mechanical loadings, the safety assessment requires to ... more For some concrete structures submitted to mechanical loadings, the safety assessment requires to predict potential crack propagations. To this end, the fictitious crack approach introduced by Hilleborg thirty years ago and based on cohesive laws brings acceptable results regarding the crack length and the overall structural response [1], thanks to a proper energy balance between strain energy and dissipation through micro and macrocracks. However, the prediction of crack paths may remain difficult and/or cumbersome on a finite element level when dealing with 3D simulations despite constant progresses reported in the literature. An alternative relies on the smeared crack approach where Continuum Damage Mechanics provides the setting to predict damage levels (i.e. crack lengths) as well as damage patterns (i.e. crack paths). But the inherent illposedness of softening laws – resulting namely in spurious mesh dependency and wrong dissipation assessment – has to be given a proper answer ...
This paper presents an approach to predict ductile fracture of real-life structures. It relies on... more This paper presents an approach to predict ductile fracture of real-life structures. It relies on Rousselier’s constitutive model to describe plastic void growth, a specific finite strain formulation that preserves energetic properties and a non local theory to deal with strain localisation. It is finally applied to the computation of a notched specimen.
International Journal for Numerical Methods in Engineering, 2018
The major goal of this work is to develop a robust modelling strategy for the simulation of ducti... more The major goal of this work is to develop a robust modelling strategy for the simulation of ductile damage development including crack initiation and subsequent propagation. For that purpose, a Gurson-type model is used. This model class, as many other damage models, leads to significant material softening and must be used within a large deformation framework due to the ductile character of the materials. This leads to 2 main difficulties that should be dealt with carefully: mesh dependency and volumetric locking. In this work, a logarithmic finite strain framework is adopted in which the Gurson-Tvergaard-Needleman constitutive law is reformulated. Then a nonlocal formulation with regularisation of hardening variable is applied so as to solve mesh dependency and strain localization problem. In addition, the nonlocal model is combined with mixed "displacement-pressure-volume variation" elements to avoid volumetric locking. Thereby, a mesh-independent and locking-free finite strain framework suitable for the modelling of ductile rupture is established. Attention is paid to mathematical properties and numerical performance of the model. Finally, the model parameters are identified on an experimental database for a nuclear piping steel. Simulations of standard test specimens (notched tensile bars and compact tension and single edge notched tensile cracked specimens) are performed and compared to experimental results.
A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete... more A rate-independent damage constitutive law is proposed to describe the fracture of plain concrete under tensile loading. Here, the target scale is the individual crack. In order to deal with localised damage, the model is inherently nonlocal: the gradient of the damage field is explicitly involved in the constitutive equations; it is parameterised by a nonlocal length scale which is interpreted as the width of the process zone. The model is defined so that its predictions are close to those of a cohesive law for vanishing nonlocal length scales. Therefore, the current model is plainly consistent with cohesive zone model analyses: the nonlocal length scale appears as a small parameter which does not need any specific identification. And four parameters -among which the tensile strength and the fracture energy -enable to adjust the softening cohesive response. Besides, a special attention has been paid to the shape of the initial damage surface and to the relation between damage and stiffness. The damage surface takes into account not only the contrast between tensile and compressive strengths but also experimental evidences regarding its shape in multiaxial tension. And the damage -stiffness relation is defined so as to describe important phenomena such as the stiffness recovery with crack closure and the sustainability of compressive loads by damaged structures. Finally, several comparisons with experimental data (global force/opening responses, size dependency, curved crack paths, crack opening profiles) enable to validate qualitatively and quantitatively the pertinence of the constitutive law in 2D and 3D.
En fait, l'étendue du voisinage représentatif devrait être fixée par l'échelle à laquelle on peut... more En fait, l'étendue du voisinage représentatif devrait être fixée par l'échelle à laquelle on peut supposer une distribution affine de la variable α . Elle dépend donc du choix de paramétrage du 1 Pour s'en persuader, deux options : une solide vision dans l'espace ou bien une construction à l'aide de balles de pingpong, très convaincante.
By using a principle of least energy and a Barenblatt-type surface energy assumed as a smooth fun... more By using a principle of least energy and a Barenblatt-type surface energy assumed as a smooth function of the jump of the displacements, we prove that it is possible to predict the onset of cracking by searching relative minima of the total energy of a body, the yield criterion depending on the choice of the energies. These types of brittle fracture models are then tested numerically by using cohesive elements.
On propose une méthode permettant d'assurer la transition et l'équivalence énergétique entre un m... more On propose une méthode permettant d'assurer la transition et l'équivalence énergétique entre un modèle non local à gradient d'endommagement et un modèle de zones cohésives. Dans un premier temps, on identifie analytiquement (sur un problème unidimensionnel) une famille de lois cohésives permettant de basculer du modèle non local vers le modèle cohésif, et ce quelque soit le niveau de dégradation atteint par le matériau. Cette stratégie est ensuite étendue au cadre bidimensionnel éléments finis, en couplant les représentations continues et discrètes dans le cas de la propagation de fissures rectilignes en mode I. Mots clefs -endommagement non local, modèles de zones cohésives, équivalence énergétique. Le matériau considéré est élastique fragile, son état est caractérisé par le tenseur des déformations ε et par une variable scalaire d'endommagement a qui dégrade progressivement sa rigidité initiale.
The modeling of damage processes at the material point scale through a specific constitutive law ... more The modeling of damage processes at the material point scale through a specific constitutive law addresses a wide variety of constitutive behaviors, for instance brittle elasticity where the evolution of damage degrades the elastic stiffness, or ductile plasticity where both dissipative mechanisms (plasticity and damage) are intimately coupled. In spite of their obvious differences, all these laws possess a common feature: beyond some load level, the domain of reversibility expressed in the stress space shrinks. This phenomenon is called strain-softening. This essential property, characteristic of damage mechanisms, is responsible for the localization of the mechanical fields (strain, damage, plastic strain, . . . ) inside narrow bands, an experimental fact observed on many degrading materials such as soils, concrete, metals, etc., see for instance . Despite twenty years of intensive research, numerical modeling of the localization phenomenon remains a challenging objective which has not yet been given a fully satisfactory answer. This is in contrast with the great progresses that have been achieved regarding the local description of damage mechanisms for various materials, on the basis of micro-macro derivations as well as phenomenological intuitions. Unfortunately, localization partially hinders predictive computations of propagating damaged zones in real-life structures submitted to a given load history. The difficulty originates from the fact that the equations that describe the structural behavior (including not only the constitutive law but also the equilibrium and compatibility equations) result in an ill-posed problem which manifests itself by a spurious mesh-dependency when making use of finite element solutions. After a first insight into the localization phenomena, the problem of loss of uniqueness and loss of ellipticity is addressed for elastoplastic solids. General (diffuse) and localized bifurcation modes are distinguished. The existence of jumps of the velocity gradient is explored in the dynamic (acceleration waves) and static cases (stationary waves). Critical values of hardening moduli for loss of uniqueness or loss of ellipticity are derived. A link between the notion of material stability and acceleration waves is also presented. The collected strain localization criteria are summarized in section 1.2.6. Note that the derivations and formula are presented in the case of elastoplastic materials. However the results can be extended to damaging materials in a straightforward
La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'... more La méthode des éléments finis formulée en déplacement nécessite un critère de convergence lorsqu'elle est employée en mécanique non linéaire avec des algorithmes de résolution itératifs. Ce critère de convergence vise à mesurer la proximité du champ de contrainte obtenu avec l'équilibre. Les critères usuels ne fournissent pas une mesure objective de la distance à l'équilibre : la taille des mailles et les dimensions physiques qui composent le résidu influe sur la valeur du critère. Un nouveau critère est proposé qui répond à cette double critique. Mots clefs -méthode éléments finis, critère de convergence, Code_Aster.
A general framework for deriving and using a class of constitutive laws incorporating spatial gra... more A general framework for deriving and using a class of constitutive laws incorporating spatial gradients of internal variables is presented. It user two basic ingredients : a derivation of such models by homogenization techniques and a reformulation of the et~olution eyuation at the scale of the whole structure.
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Papers by Eric Lorentz