Global Existence for Rate-Independent Gradient Plasticity at Finite Strain

Mainik, Andreas; Mielke, Alexander

doi:10.1007/s00332-008-9033-y

Cited by 96 publications

(159 citation statements)

References 45 publications

(46 reference statements)

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“…The reader will note the presence of two gradient terms in the expression for the energy, one for P and one for Q. This is to be expected, if recalling the setting of [26], because our decomposition of the deformation gradient involves both P and Q, as already discussed earlier.…”

Section: Hence In Particular (See Eg [34 Remark 22 and Propositiomentioning

confidence: 65%

“…The energy W hard is often referred to as a hardening energy while the plastic gradient term is called just that, a plastic gradient. An existence theorem can then be proved under adequate assumptions on the energy density W and on the induced dissipation distance D of Remark 4.9; see [26,Theorem 3.1]. In the next section we quickly show that adding regularization terms which enforce compactness of the plastic strains and of the rotations produces a similar result, at least in the setting of Remarks 3.4, 4.5, 4.7.…”

Section: Definition 46 a Variational Evolution (ϕ(T) E(t) P (T) mentioning

confidence: 81%

“…[30,31,32] was later regularized in [26] through the introduction of both a plasticity term and a gradient plasticity term in the energy that, together, enforce compactness of the plastic strains associated with the minimizing sequences. Schematically, in the simplest setting, the free energy becomes W(E) + W hard (P ) + |∇P | r , r > 1.…”

Section: Definition 46 a Variational Evolution (ϕ(T) E(t) P (T) mentioning

confidence: 99%

“…The theory of variational rate independent evolutions has in the past 20 years or so proved to be a handy tool in the mathematical understanding of a variety of rate independent evolutions, be it brittle fracture [18], [12], [17], [11], various approaches to damage [19], [15], [28], or still small strain elasto-plasticity [9], etc..... To our knowledge the only mature attempt in the direction of finite strain elasto-plasticity is that of A. Mielke, together with several co-workers; among the many contributions see in particular [30], [31], [26]. The resulting model will be commented upon and distinguished from ours in Remark 4.9 below.…”

Section: Introductionmentioning

confidence: 99%

“…An adequate regularization of the model, similar in spirit to that performed in [26], alleviates all obstacles and results in a well-posed variational evolution. To this effect, one should introduce an additional energy which depends super-linearly on both the plastic strain and its gradient; in essence this amounts to introducing both hardening and gradient plasticity into the model.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A Critical Revisiting of Finite Elasto-Plasticity

Davoli¹,

Francfort²

2015

SIAM J. Math. Anal.

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Abstract. We propose a new evolution model for large strain elastoplasticity. In a quasi-static setting the suggested model can be formulated as a variational evolution. We establish existence of a closely related evolution in a regularized context. We then discuss how the proposed model performs in both a rigid-plastic, and a one-dimensional settings and compare the results with those that could be achieved using other formulations.

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Section: Hence In Particular (See Eg [34 Remark 22 and Propositiomentioning

confidence: 65%

Section: Definition 46 a Variational Evolution (ϕ(T) E(t) P (T) mentioning

confidence: 81%

Section: Definition 46 a Variational Evolution (ϕ(T) E(t) P (T) mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Critical Revisiting of Finite Elasto-Plasticity

Davoli¹,

Francfort²

2015

SIAM J. Math. Anal.

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Testing a thermo‐chemo‐hydro‐geomechanical model for gas hydrate‐bearing sediments using triaxial compression laboratory experiments

Gupta

Deusner

Haeckel

et al. 2017

Geochem Geophys Geosyst

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Natural gas hydrates are considered a potential resource for gas production on industrial scales. Gas hydrates contribute to the strength and stiffness of the hydrate‐bearing sediments. During gas production, the geomechanical stability of the sediment is compromised. Due to the potential geotechnical risks and process management issues, the mechanical behavior of the gas hydrate‐bearing sediments needs to be carefully considered. In this study, we describe a coupling concept that simplifies the mathematical description of the complex interactions occurring during gas production by isolating the effects of sediment deformation and hydrate phase changes. Central to this coupling concept is the assumption that the soil grains form the load‐bearing solid skeleton, while the gas hydrate enhances the mechanical properties of this skeleton. We focus on testing this coupling concept in capturing the overall impact of geomechanics on gas production behavior though numerical simulation of a high‐pressure isotropic compression experiment combined with methane hydrate formation and dissociation. We consider a linear‐elastic stress‐strain relationship because it is uniquely defined and easy to calibrate. Since, in reality, the geomechanical response of the hydrate‐bearing sediment is typically inelastic and is characterized by a significant shear‐volumetric coupling, we control the experiment very carefully in order to keep the sample deformations small and well within the assumptions of poroelasticity. The closely coordinated experimental and numerical procedures enable us to validate the proposed simplified geomechanics‐to‐flow coupling, and set an important precursor toward enhancing our coupled hydro‐geomechanical hydrate reservoir simulator with more suitable elastoplastic constitutive models.

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Mathematical Foundations of Elastoplastic Deformations in Solids

Reddy

2017

Encyclopedia of Computational Mechanics Second Edition

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This chapter presents a treatment of the mathematical aspects of continuum elastoplasticity. The focus is on rate‐independent material behavior, with the flow relations constructed within the framework of convex analysis. The basis for variational treatments of the problem is the versions of the flow relation that use the dissipation function or, alternatively, the yield surface with a normality law: these are referred to, respectively, as the primal and dual formulations. Both formulations are discussed, with emphasis on the former. Some results on the convergence of fully discrete approximations, using finite elements in space, are presented. The associated predictor–corrector algorithms are described, and their convergence behavior is discussed. A discussion of the large‐deformation theory includes an overview of the derivative‐free energetic formulation, due to Mielke and coworkers, and of which the primal formulation discussed in the small‐strain theory is a special case. Relevant results from functional analysis and function spaces are summarized at appropriate places in the presentation.

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Global Existence for Rate-Independent Gradient Plasticity at Finite Strain

Cited by 96 publications

References 45 publications

A Critical Revisiting of Finite Elasto-Plasticity

A Critical Revisiting of Finite Elasto-Plasticity

Testing a thermo‐chemo‐hydro‐geomechanical model for gas hydrate‐bearing sediments using triaxial compression laboratory experiments

Mathematical Foundations of Elastoplastic Deformations in Solids

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