Finite element approximation of a strain-limiting elastic model

Bonito, Andrea; Girault, Vivette; Süli, Endre

doi:10.1093/imanum/dry065

Cited by 8 publications

(6 citation statements)

References 17 publications

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“…Hence these two problems are equivalent. Furthermore, we can eliminate the unknown u by proceeding as follows; see [5]. First, we introduce the decomposition…”

Section: The Stokes Systemmentioning

confidence: 99%

“…To compute the solution to problem (6.1), we propose a decoupled algorithm based on a Lions-Mercier splitting algorithm [25] (alternating-direction method of the Peaceman-Rachford type [31]) applied to the unknown T h . Following the discussion in [5,Section 7], the algorithm reads, for a pseudo-time step τ > 0:…”

Section: Lions-mercier Decoupled Iterative Algorithmmentioning

confidence: 99%

“…It can be posed within a Hilbert space setting owing to the presence of the coefficient α in (1.10), but nevertheless, it is a challenging problem as it involves two nonlinearities: the monotone part in the constitutive relation and the inertial (convective) term. The problem without the inertial term, see Subsection 2.2 below, has already been analysed in [5], while the analysis of the steady-state incompressible Navier-Stokes equations is well-established, see for instance [41,20]. With both nonlinearities present in the model, proving the existence of a weak solution, for instance, to the best of our knowledge cannot be done by simply coupling the techniques used for these two problems, namely the Browder-Minty theorem and the Galerkin method combined with Brouwer's fixed point theorem and a weak compactness argument.…”

Section: Introductionmentioning

confidence: 99%

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Finite element approximation of steady flows of colloidal solutions

et al. 2021

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We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

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“…Hence these two problems are equivalent. Furthermore, we can eliminate the unknown u by proceeding as follows; see [5]. First, we introduce the decomposition…”

Section: The Stokes Systemmentioning

confidence: 99%

Section: Lions-mercier Decoupled Iterative Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Finite element approximation of steady flows of colloidal solutions

et al. 2021

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“…Hence these two problems are equivalent. Furthermore, we can eliminate the unknown u by proceeding as follows; see [5]. First, we introduce the decomposition M = M ⊕ M ⊥ with…”

Section: The Stokes Systemmentioning

confidence: 99%

Finite Element Approximation of Steady Flows of Colloidal Solutions

Bonito¹,

Girault²,

Guignard³

et al. 2021

Preprint

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We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a unique weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed finite element approximations of the system using both conforming and nonconforming finite element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We propose first a Lions-Mercier type iterative method and next a classical fixed-point algorithm to solve the finite-dimensional problems resulting from the finite element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method.

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“…Using the strain-limiting models, several studies have attempted to revisit the classical problems in elasticity such as two-dimensional V-notches [30][31][32][33], elliptical holes [34][35][36], unsteady problems [37][38][39][40][41], and nonlinear viscoelastic deformations [42][43][44][45]. A rigorous mathematical analysis has been carried out in [31,[46][47][48][49] concerning the existence and uniqueness of weak solutions for the large class of strain-limiting models.…”

Section: Introductionmentioning

confidence: 99%

A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

Yoon

Mallikarjunaiah

2021

Mathematics and Mechanics of Solids

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It is well known that the linearized theory of elasticity admits the logically inconsistent solution of singular strains when applied to certain naive models of fracture while the theory is a first-order approximation to finite elasticity in the asymptotic limit of infinitesimal displacement gradient. Meanwhile, the strain-limiting models, a special subclass of nonlinear implicit constitutive relations, predict uniformly bounded strain in the whole material body including at the strain-concentrator such as a crack tip or reentrant corner. Such a nonlinear approximation cannot be possible within the standard linearization procedure of either Cauchy or Green elasticity. In this work, we examine a finite-element discretization for several boundary value problems to study the state of stress–strain in the solid body of which response is described by a nonlinear strain-limiting theory of elasticity. The problems of notches, oriented cracks, and an interface crack in anti-plane shear are analyzed. The numerical results indicate that the linearized strain remains below a value that can be fixed a priori, therefore, ensuring the validity of the nonlinear model. In addition, we find high stress values in the neighborhood of the crack tip in every example, thereby suggesting that the crack tip acts as a singular energy sink for a stationary crack. We also calculate the stress intensity factor (SIF) in this study. The computed value of SIF in the nonlinear strain-limiting model is corresponding to that of the classical linear model, and thereby providing a tenet for a possible local criterion for fracture. The framework of strain-limiting theories, within which the linearized strain bears a nonlinear relationship with the stress, can provide a rational basis for developing physically meaningful models to study a crack evolution in elastic solids.

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Finite element approximation of a strain-limiting elastic model

Cited by 8 publications

References 17 publications

Finite element approximation of steady flows of colloidal solutions

Finite element approximation of steady flows of colloidal solutions

Finite Element Approximation of Steady Flows of Colloidal Solutions

A finite-element discretization of some boundary value problems for nonlinear strain-limiting elastic bodies

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