Analysis of non-local models through energetic formulations

Lorentz, Éric; Andrieux, Stéphane

doi:10.1016/s0020-7683(03)00110-0

Cited by 85 publications

(66 citation statements)

References 31 publications

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“…The bulk equations and the boundary conditions (17) have been proposed and discussed by several authors, cf. [10,12,19,20,22,26].…”

Section: Gsm-formalism: Energy and Dissipation Analysismentioning

confidence: 99%

On standard dissipative gradient models

Nguyen

2010

Ann. Solid Struct. Mech.

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International audienceA general presentation on gradient models for solids in thermo- mechanics is given in this paper. Firstly, it is shown that the introduction of the gradients of the internal variables can be conveniently done under the formalism of the generalized standard materials. The derivation of the governing equations and the associated boundary conditions of a solid can be derived in a straightforward manner from the basic assumption of existence of an energy potential and a dissipation potential. Secondly, the attention is focussed on the introduction of the temperature gradient. Two descriptions, respectively by Forest & al. and by Nguyen & Andrieux are discussed. The last one is based upon an interesting definition of the entropy and of the internal energy by Legendre transform of the free energy with respect to the temperature and the temperature gradient. The obtained results are justified in a simple example of micro-macro modeling of thermal conduction in a rigid composite. In particular, it is shown that the associated thermal equation removes the paradox of instantaneous propagation

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“…The bulk equations and the boundary conditions (17) have been proposed and discussed by several authors, cf. [10,12,19,20,22,26].…”

Section: Gsm-formalism: Energy and Dissipation Analysismentioning

confidence: 99%

On standard dissipative gradient models

Nguyen

2010

Ann. Solid Struct. Mech.

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“…It should be pointed out that this computation is local. The localization of damage and strains due to strain softening would require the use of a regularization technique (Pijaudier-Cabot and Bazant, 1987;Peerlings et al, 1996;Lorentz and Andrieux, 2003) to avoid mesh dependency due to strain softening (Crisfield, 1982). Nevertheless, a former study (Jason et al, 2004b) has shown that using such a technique would be numerically too expensive.…”

Section: Mechanical Problemmentioning

confidence: 99%

Hydraulic behaviour of a representative structural volume for containment buildings

Jason

Pijaudier‐Cabot

Ghavamian

et al. 2007

Nuclear Engineering and Design

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For particular structures like containment buildings of nuclear power plants, the study of the hydraulic behaviour is of great concern. These structures are indeed the third barrier used to protect the environment in case of accidents. The evolution of the leaking rate through the porous medium is closely related to the changes in the permeability during the ageing process of the structure. It is thus essential to know the relation between concrete degradation and the transfer property when the consequences of a mechanical loading on the hydraulic behaviour have to be evaluated. A chained approach is designed for this purpose. The mechanical behaviour is described by an elastic plastic damage formulation, where damage is responsible for the softening evolution while plasticity accounts for the development of irreversible strains. The drying process is evaluated according to a non-linear equation of diffusion. From the knowledge of the damage and the degree of saturation, a relation is proposed to calculate the permeability of concrete. Finally, the non-homogeneous distribution of the hydraulic conductivity is included in the hydraulic problem which is in fact the association of the mass balance equation for gas phase and Darcy law. From this methodology, it is shown how an indicator for the hydraulic flows can be deduced.

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“…A consequence is the necessary introduction of an additional material parameter, the internal length. Despite of the large literature on the subject, the discussion on the effectiveness of the several possible regularization techniques is still open (Lorentz and Andrieux, 2003). The main issue is that even the regularized non-local strain-softening models continue to be ill-posed, showing complex phenomena, such as infinite number of linearly independent solutions and continuous bifurcations (Benallal and Marigo, 2007;Pijaudier-Cabot and Benallal, 1993), which strongly depend on the specific constitute model under consideration.…”

Section: Introductionmentioning

confidence: 99%

Gradient Damage Models and Their Use to Approximate Brittle Fracture

Pham

Amor

Marigo

et al. 2010

International Journal of Damage Mechanics

530

435

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In its numerical implementation, the variational approach to brittle fracture approximates the crack evolution in an elastic solid through the use of gradient damage models. In the present paper, we first formulate the quasi-static evolution problem for a general class of such damage models. Then, we introduce a stability criterion in terms of the positivity of the second derivative of the total energy under the unilateral constraint induced by the irreversibility of damage. These concepts are applied in the particular setting of a one-dimensional traction test. That allows us to construct homogeneous as well as localized damage solutions in a closed form and to illustrate the concepts of loss of stability, of scale effects, of damage localization, and of structural failure. Considering several specific constitutive models, stress-displacement curves, stability diagrams, and energy dissipation provide identification criteria for the relevant material parameters, such as limit stress and internal length. Finally, the one-dimensional analytical results are compared with the numerical solution of the evolution problem in a two dimensional setting.

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Analysis of non-local models through energetic formulations

Cited by 85 publications

References 31 publications

On standard dissipative gradient models

On standard dissipative gradient models

Hydraulic behaviour of a representative structural volume for containment buildings

Gradient Damage Models and Their Use to Approximate Brittle Fracture

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