On the stress function approach in three-dimensional elasticity

Bertóti, E.

doi:10.1007/s00707-006-0432-6

Cited by 3 publications

(3 citation statements)

References 6 publications

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“…The Lagrange multiplier term ensuring the symmetry of the stress tensor in functional (12) for cylindrical shells using (11) in terms of the expanded variables (31), (32), and (25) reads…”

Section: Dimensional Reductionmentioning

confidence: 99%

“…It should also be mentioned that the possible structure of the first-order stress functions giving zero complementary strain energy has to be found to obtain correct finite element solutions. This problem was investigated in two-dimensional elasticity in [17], and a possible structure of the zero-energy stress functions for the three-dimensional case was given in [11]. In addition, to avoid redundant approximation of the stress function components, the indeterminacy property of the stress function tensor should be taken into account.…”

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confidence: 99%

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Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells

Kocsán

2011

Arch Appl Mech

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The two-field dual-mixed Fraeijs de Veubeke variational formulation of three-dimensional elasticity serves as the starting point of the derivation of a dimensionally reduced shell model presented in this paper. The fundamental variables of this complementary energy-based variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium, while the rotation tensor plays the role of a Lagrange multiplier to ensure rotational equilibrium. The volumetric locking-free shell model uses unmodified three-dimensional constitutive equations, and no classical kinematical hypotheses are employed during the derivation. The numerical performance of the related low-order h-, and higher-order p-version finite elements developed for axisymmetrically loaded cylindrical shells is investigated by two representative model problems. It is numerically proven that no negative effect can be experienced when the thickness is small and tends to zero.

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“…The Lagrange multiplier term ensuring the symmetry of the stress tensor in functional (12) for cylindrical shells using (11) in terms of the expanded variables (31), (32), and (25) reads…”

Section: Dimensional Reductionmentioning

confidence: 99%

mentioning

confidence: 99%

Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells

Kocsán

2011

Arch Appl Mech

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“…The stress function approach and the related complementary energy-based variational principles have been successfully applied in the finite element analysis of twodimensional elasticity problems, see, e.g., [2,[7][8][9]. For three-dimensional problems, however, no efficient finite element model exists, which is due mainly to the C 1continuity requirement for the second-order stress functions on the one hand, and to the structure of the zero-energy modes and the difficulties with their suppression for the case of first-order stress functions, on the other hand [10].…”

Section: Introductionmentioning

confidence: 99%

Equilibrated stress space for nonlinear dimensionally reduced shell models in terms of first-order stress functions

Bertóti¹

2020

JCAM

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Considering the power series expansion of the three-dimensional variables with respect to the shell thickness coordinate, an equilibrated stress space for the first Piola-Kirchhoff stress vectors is derived in convective curvilinear coordinate system. The infinite series of the two-dimensional translational equilibrium equations are satisfied by introducing two first-order stress function vectors expanded into power series. For the important case of thin shells, the infinite number of two-dimensional equilibrium equations is truncated to obtain a ‘first-order’ model, where the equilibrated stress-space requires three vectorial stress function coefficients only. The formulation presented for thin shells is compared to the nonlinear equilibrium equations of the classical shell theories, written in terms of the first Piola-Kirchhoff stress resultants and stress couples and satisfied by the introduction of three first-order stress function vectors.

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On divergence-free stress fields and zero-energy stress functions in elasticity

Bertóti

2016

Mechanics Research Communications

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On the stress function approach in three-dimensional elasticity

Cited by 3 publications

References 6 publications

Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells

Derivation of a dual-mixed hp-finite element model for axisymmetrically loaded cylindrical shells

Equilibrated stress space for nonlinear dimensionally reduced shell models in terms of first-order stress functions

On divergence-free stress fields and zero-energy stress functions in elasticity

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