Uniqueness Theorems in Linear Elasticity

Knops, R. J.; Payne, L. E.

doi:10.1007/978-3-642-65101-4

Cited by 174 publications

(138 citation statements)

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“…Usually, the stability ensures the uniqueness of solution, Editor's Choice however the neutral stability does not imply uniqueness of solution. Indeed, for the case of stress boundary and mixed boundary, the solution is non-unique under the condition [25,26].…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, the strong ellipticity condition (3.7) is a necessary and sufficient condition to ensure that the speeds of all plane waves in elastic media filling three-dimensional space are positive. Moreover, for the displacement boundary value problem of linear elastostatics in bounded regions, the strong ellipticity condition (3.7) guarantees the uniqueness of solution [26].…”

Section: Condition Of Strong Ellipticity and Negative Elastic Modulimentioning

confidence: 99%

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Stability of elastic material with negative stiffness and negative Poisson's ratio

Shang

Lakes

2007

Physica Status Solidi (b)

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PACS 46. 05.+b, 62.20.Dc Stability of cuboids and cylinders of an isotropic elastic material with negative stiffness under partial constraint is analyzed using an integral method and Rayleigh quotient. It is not necessary that the material exhibit a positive definite strain energy to be stable. The elastic object under partial constraint may have a negative bulk modulus K and yet be stable. A cylinder of arbitrary cross section with the lateral surface constrained and top and bottom planar surfaces is stable provided the shear modulus G > 0 and -G/3 < K < 0 or K > 0. This corresponds to an extended range of negative Poisson's ratio, -∞ < ν < -1. A cuboid is stable provided each of its surfaces is an aggregate of regions obeying fully or partially constrained boundary conditions.

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Section: Discussionmentioning

confidence: 99%

Section: Condition Of Strong Ellipticity and Negative Elastic Modulimentioning

confidence: 99%

Stability of elastic material with negative stiffness and negative Poisson's ratio

Shang

Lakes

2007

Physica Status Solidi (b)

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“…Summation convention is used throughout this note.) Such an inequality is easily obtained by assuming that the coefficients CiJM are such that there exists a positive constant c0 for which Cijkiix)%i£ki Cf£i£ij (2) for every tensor and every x in D (see, for example, Fichera [2] or Knops and Payne [3]). …”

Section: Introductionmentioning

confidence: 99%

“…A weaker assumption than the inequality (2) is the assumption that C is uniformly strongly elliptic, i.e., that there exists a constant cx for which Ci/3ja1 > c, |a|2 |/3|2 (3) for all vectors a and 0 and all x in D. If the major symmetry condition CiJM -CKUJ is satisfied, then Wheeler [5] has shown that (3) implies uniqueness of solutions of the elastodynamic displacement boundary-value problem. However, Edelstein and Fosdick [1] have shown by example that uniform strong ellipticity alone is not sufficient in general to guarantee uniqueness for the elastostatic displacement boundary-value problem, although uniqueness can be regained in certain circumstances with a few additional assumptions [3].…”

Section: Introductionmentioning

confidence: 99%

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Strong ellipticity and Van Hove’s lemma in inhomogeneous media

Walker¹

1977

Quart. Appl. Math.

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Explanation of locking of four‐node plane element by considering it as elastic Dirichlet‐type boundary value problem

Molenkamp

Sellmeijer

Sharma

et al. 2000

Int. J. Numer. Anal. Meth. Geomech.

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SUMMARYThe aim of this study is to arrive at a better understanding of the phenomenon of locking of low-order compatible displacement type of "nite elements in particular for the hour-glass mode of the plane four-node element and dilative materials. To this end the properties of "nite elements are investigated in an analytical way, where a "nite element is considered as a plane boundary value problem with prescribed boundary displacement (Dirichlet problem). In this paper for the sake of simplicity the simplest possible linear comparison solid, namely isotropic linear elasticity, is applied, although recognizing fully that for a dilative material elasto-plasticity would be more realistic. From the study described in this paper it is concluded that locking of the four-node element is not due to any particular numerical formulation of this compatible "nite element since, even the analytical solution su!ers from this problem. The locking of this element is not related to incompressibility of the material either as the analytical solution shows locking to occur at a parameter set which di!ers signi"cantly from the one in case of incompressibility. It is shown that locking is a consequence of the combination of the dilative material behaviour and the compatible displacement type of boundary conditions, which leads to in"nite isotropic stresses in the element. These in"nite isotropic stresses occur at the limit of uniqueness of the solution, which for this element is shown to occur outside the parameter range of the su$ciency of uniqueness.

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Uniqueness Theorems in Linear Elasticity

Cited by 174 publications

References 0 publications

Stability of elastic material with negative stiffness and negative Poisson's ratio

Stability of elastic material with negative stiffness and negative Poisson's ratio

Strong ellipticity and Van Hove’s lemma in inhomogeneous media

Explanation of locking of four‐node plane element by considering it as elastic Dirichlet‐type boundary value problem

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