The Lavrentiev Gap Phenomenon in Nonlinear Elasticity

Foss, Mikil; Hrusa, William J.; Mizel, Victor J.

doi:10.1007/s00205-003-0249-6

Cited by 44 publications

(41 citation statements)

References 11 publications

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“…1/2 (cos(θ/2), sin(θ/2)) gives zero energy but the large power makes approximation difficult and it can be shown that the problem exhibits the Lavrentiev phenomenon. Further, the deformation y * has finite energy for ν > 0 and hence, for ν sufficiently small the Lavrentiev effect remains [16].…”

Section: Maniá-type Examplesmentioning

confidence: 99%

“…In this section, we present numerical results for a modification of the example provided by Foss, Hrusa and Mizel [16]. In their original example a semi-circle Ω is transformed into a quarter-circle y(Ω) with stored energy…”

Section: A Convex Example In 2dmentioning

confidence: 99%

“…Other well-known examples are the one-dimensional examples of Mania [23] and of Ball and Mizel [7,6], or the convex example of Foss, Hrusa and Mizel [16]. In nonlinear elasticity, the Lavrentiev phenomenon is closely related to the occurence of cavitation [4].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of a Class of Penalty Methods for Computing Singular Minimizers

Carstensen

Ortner

2010

Comput. Methods Appl. Math.

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-Amongst the more exciting phenomena in the field of nonlinear partial differential equations is the Lavrentiev phenomenon which occurs in the calculus of variations. We prove that a conforming finite element method fails if and only if the Lavrentiev phenomenon is present. Consequently, nonstandard finite element methods have to be designed for the detection of the Lavrentiev phenomenon in the computational calculus of variations. We formulate and analyze a general strategy for solving variational problems in the presence of the Lavrentiev phenomenon based on a splitting and penalization strategy. We establish convergence results under mild conditions on the stored energy function. Moreover, we present practical strategies for the solution of the discretized problems and for the choice of the penalty parameter.2000 Mathematics Subject Classification: 65N12; 65N25; 65N30; 65N50.Keywords: Lavrentiev phenomenon, finite element method, computational calculus of variations.

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Section: Maniá-type Examplesmentioning

confidence: 99%

Section: A Convex Example In 2dmentioning

confidence: 99%

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Analysis of a Class of Penalty Methods for Computing Singular Minimizers

Carstensen

Ortner

2010

Comput. Methods Appl. Math.

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“…Examples of special cases of the one treated here are known from the scientific literature. For instance, in two-dimensional ambient space there are examples of non-linear elastic simple materials admitting a gap between the infimum of the energy over admissible continuous deformations belonging to a Sobolev space W 1,r and the analogous infimum over admissible continuous deformations belonging to a Sobolev space W 1,s with s < r [20]. A gap phenomenon driven this time by the topology of the substructural manifold M can also arise.…”

Section: Remarks About the Possible Presence Of A Lavrentiev Gap Phenmentioning

confidence: 99%

Ground states in complex bodies

Mariano

Modica

2008

ESAIM: COCV

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Abstract.A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and Cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and Cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance of configurational actions follows. After describing a list of possible applications of the general results collected here, a concrete discussion of the existence of ground states in thermodynamically stable quasicrystals is presented at the end.

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“…In two-dimensional elasticity there are examples of its existence [4]. A gap phenomenon driven by the topology of M is also possible when M has non-trivial homology (less trivial of the one of S 2 ) Remark 2.3 When M coincides with S 2 or has non-trivial homology, minimizers can be found when ν is intended in terms of Cartesian currents (see [3]).…”

Section: Theorem 21 ( [3]) the Functional E Achieves The Minimum Valmentioning

confidence: 99%