Elasticity equations for incompressible and nearly incompressible materials by a variational theorem.

Herrmann, Leonard R.

doi:10.2514/3.3277

Cited by 345 publications

(133 citation statements)

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“…The approximate fields (15) and (16) are all continuous over the entire body, and therefore, if there are no prescribed jumps within the considered continuum, the integrals corresponding to these jumps would not occur in functional (3). Upon substitution from (15), (16), and (17), into (3), we obtain 9 = f -PoFoU:**] dv where we have assumed that the approximate displacement field (15) is a priori adjusted so as to satisfy the prescribed displacement boundary data.…”

Section: On Variational Methods In Finite and Incremental Elastic Defmentioning

confidence: 99%

“…Upon substitution from (15), (16), and (17), into (3), we obtain 9 = f -PoFoU:**] dv where we have assumed that the approximate displacement field (15) is a priori adjusted so as to satisfy the prescribed displacement boundary data. It should be observed that, since the unit field ^ is completely defined as soon as the coordinates of all the nodes are known,5 all the integrals in (18) can be calculated explicitly, and hence 9 becomes an explicit function of the nodal values of the displacement field , stress field S^B , and the deformation tensor CAB .…”

Section: On Variational Methods In Finite and Incremental Elastic Defmentioning

confidence: 99%

“…The use of variational theorems for the numerical solution of elasticity problems has recently been stressed by a number of authors (see for example, [12], [16][17][18][19][20][21][22]). …”

Section: On Variational Methods In Finite and Incremental Elastic Defmentioning

confidence: 99%

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On variational methods in finite and incremental elastic deformation problems with discontinuous fields

Nemat-Nasser¹

1972

Quart. Appl. Math.

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Abstract.With a view toward a numerical solution by means of the finite-element method, we give here a variational statement for large elastic deformations at finite strains which involves independent variation of the displacement, the (nonsymmetric first Piola-Kirclihoff) stress, and the deformation-gradient fields, and which includes both the boundary and the jump conditions. Then we present, for small deformations superimposed on the large, three variational statements, each involving three independent fields and each including both the boundary and the jump conditions. These statements are such that the first variation of the corresponding functional yields the field equations which characterize the equilibrium of the finitely-deformed state considered and also the field equations that pertain to the incremental deformations. Several specializations of these results are discussed. By way of illustration, finally, we present a finite-element formulation of the large deformation problem, using three independent fields, where each field is approximated by a piecewise-linear function within each element.I. Introduction. It appears that a variational statement in which both the displacement and the stress fields are given independent variation was first developed by Hellinger [1, Sec. 7e, Eqs. (21) and (22a, b)], for finite elastic deformation problems. Hellinger uses a nonsymmetric stress tensor which is now commonly referred to as the first Piola-Kirchhoff (or Lagrangian) stress tensor (denoted in the present work by Trja), together with the deformation gradient (denoted here by xa,A). In the first two chapters of his encyclopedic article he discusses the virtual work theorem for the statics and dynamics of one-, two-, and three-dimensional continua, including the case of polar (oriented) media, and presenting results in terms of both what is now commonly referred to as the Eulerian and the Lagrangian formulation. In the third chapter of his paper, Hellinger introduces the assumption of potential loads and the strain-energy function, developing the above-mentioned variational theorem in Sec. 7e. However, he does not include explicitly in his variational functional the boundary conditions. Hellinger's results, with further generalization and clarification, are presented in Sees. 231-238 of [2] in a more modern notation. A more general statement of a variational theorem for finite elasticity was given independently by Reissner [3] who formulates his results in terms of what is now commonly referred to as the symmetric Piola-Kirchhoff stress tensor (denoted here by SAB) and the Lagrangian strain tensor (denoted here by EAB). In this regard, therefore, Reissner's formulation is essentially different from that of Hellinger. Moreover, he includes explicit boundary data in his formulation by permitting * Received February 10, 1971.

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Section: On Variational Methods In Finite and Incremental Elastic Defmentioning

confidence: 99%

Section: On Variational Methods In Finite and Incremental Elastic Defmentioning

confidence: 99%

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On variational methods in finite and incremental elastic deformation problems with discontinuous fields

Nemat-Nasser¹

1972

Quart. Appl. Math.

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“…This phenomenon, which leads to an erroneous stiffening of the load deformation response, is widely known as`locking', and has been reported in the literature by many researchers (e.g. Herrmann, 1965;Christian, 1968;Zienkiewicz et al, 1971;Naylor, 1974;Nagtegaal et al, 1974;Sloan, 1981;Sloan & Randolph, 1982;de Borst & Vermeer, 1984;Burd & Houlsby, 1990;Yu et al, 1993;Yu & Netherton, 2000).…”

Section: Authors' Replymentioning

confidence: 99%

Discussion: Rigorous plasticity solutions for the bearing capacity of two-layered clays

Goss¹,

Griffiths²

2001

Géotechnique

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“…After the pioneering work of Herrmann et al [3] who first developed a mixed FEM formulation for compressible elasticity capable of considering incompressibility, a plethora of special FEM formulations working in the incompressible and nearly incompressible regime have appeared in the literature. Mixed, Lagrange-multiplier, penalty, augmented, enhanced strain, conforming and non-conforming, field consistent and orthogonal projection methods are some of the proposed techniques.…”

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confidence: 99%

A MLPG(LBIE) numerical method for solving 2D incompressible and nearly incompressible elastostatic problems

Vavourakis

Polyzos

2006

Commun. Numer. Meth. Engng.

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SUMMARYA new meshless local Petrov-Galerkin (MLPG) method, based on local boundary integral equation (LBIE) considerations, is proposed here for the solution of 2D, incompressible and nearly incompressible elastostatic problems. The method utilizes, for its meshless implementation, nodal points spread over the analysed domain and employs the moving least squares (MLS) approximation for the interpolation of the interior and boundary variables. On the local and global boundaries, traction vectors are treated in a way so that no derivatives of the utilized MLS shape functions are required. Both displacement and hydrostatic pressure, at all the considered nodal points, are evaluated with the aid of local integral representations valid for incompressible and nearly incompressible solids. Since displacements and stresses are treated as independed variables, the boundary conditions are imposed directly without any problem via the integrals defined on the global boundary of the analysed body. Singular and hypersingular integrals are evaluated directly with high accuracy through advanced integration techniques. Three numerical examples illustrate the proposed methodology and demonstrates its accuracy.

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Elasticity equations for incompressible and nearly incompressible materials by a variational theorem.

Cited by 345 publications

References 6 publications

On variational methods in finite and incremental elastic deformation problems with discontinuous fields

On variational methods in finite and incremental elastic deformation problems with discontinuous fields

Discussion: Rigorous plasticity solutions for the bearing capacity of two-layered clays

A MLPG(LBIE) numerical method for solving 2D incompressible and nearly incompressible elastostatic problems

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