Indeterminacy of first order stress functions and the stress- and rotation-based formulation of linear elasticity

Bertóti, E.

doi:10.1007/bf00370076

Cited by 13 publications

(10 citation statements)

References 14 publications

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“…Depending on the structure of the Christoffel symbols, several combinations are available. It should also be mentioned that 27 possible groups of stress function triads can be selected independently from the coordinate system [6]. In this paper…”

Section: The Three-dimensional Fraeijs De Veubeke Variational Principlementioning

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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Section: The Three-dimensional Fraeijs De Veubeke Variational Principlementioning

confidence: 99%

“…In addition, to avoid redundant approximation of the stress function components, the indeterminacy property of the stress function tensor should be taken into account. This means that only six out of the nine components of the first-order stress function tensor are independent [6].…”

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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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“…Substituting (20)- (28) into the three-dimensional stress boundary conditions (14) and taking into account that n ± = ± e z , we obtain the following equations:…”

Section: (22)mentioning

confidence: 99%

“…Since the three-dimensional stress components with the above twodimensional stress coe cients satisfy the three-dimensional translational equilibrium equations (13) as well as the stress boundary conditions (14) on the top and bottom surfaces, they can be inserted in functional (7). Stress boundary conditions (15), prescribed on the lateral surface of the plate, can be satisÿed a priori in the course of the (ÿnite element) solution procedure.…”

Section: Satisfaction Of Translational Equilibriummentioning

confidence: 99%

Dual‐mixed p and hp finite elements for elastic membrane problems

Bertóti

2001

Numerical Meth Engineering

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This paper is dedicated to Professor Barna A. SzabÃ o on the occasion of his 65th birthday SUMMARY A complementary energy-based, dimensionally reduced plate model using a two-ÿeld dual-mixed variational principle of non-symmetric stresses and rotations is derived. Both the membrane and bending equilibrium equations, expressed in terms of non-symmetric mid-surface stress components, are satisÿed a priori introducing ÿrst-order stress functions. It is pointed out that (i) the membrane-, shear-and bending energies of the plate written in terms of ÿrst-order stress functions are decoupled, (ii) although unmodiÿed 3-D constitutive equations are applied, the energy parts do not contain the 1=(1 − 2 ) term for isotropic, linearly elastic materials. These facts mean that the ÿnite element formulation based on the present plate model should be free from shear locking when the thickness tends to zero and free from incompressibility locking when the Poisson ratio converges to 0.5, irrespective of low-order h-, or higher-order p elements are used.Curvilinear dual-mixed hp ÿnite elements with higher-order stress approximation and continuous surface tractions are developed and presented for the membrane (2-D elasticity) problem. In this case the formulation requires the approximation of three independent variables: two components of a ÿrst-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dualmixed elements is presented and compared to displacement-based hp ÿnite elements when the Poisson ratio converges to the incompressible limit of 0.5. The numerical results show that, as expected, the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, for both h-and p-extensions.

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“…After that, some researchers combined the SFE method with the stochastic perturbation approach [26,27] and the boundary element method [28]. In order to gain the advantages of the DFE method, the hybrid method was developed, which uses assumed displacement and stress functions [27,[29][30][31]. In addition, there are various books providing details about the SFE method [32,33].…”

Section: Introductionmentioning

confidence: 99%

Curvature‐ and displacement‐based finite element analyses of flexible slider crank mechanisms

Kuo

Cleghorn

2010

Numer Methods Biomed Eng

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SUMMARYThe paper presents the applications of the curvature-and displacement-based finite element methods to flexible slider crank mechanisms. The displacement-based method usually needs more elements or highdegree polynomials to obtain highly accurate solutions. The curvature-based method assumes a polynomial to approximate a curvature distribution, and the expressions are investigated to obtain the displacement and rotation distributions. During the process, the boundary conditions associated with displacement, rotation, and curvature are imposed, which leads the great reduction of the number of degrees of freedom that are required. The numerical results demonstrate that the errors obtained by applying the curvature-based method are much smaller than those by applying the displacement-based method, based on the comparison of the same number of degrees of freedom.

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Indeterminacy of first order stress functions and the stress- and rotation-based formulation of linear elasticity

Cited by 13 publications

References 14 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

Dual‐mixed p and hp finite elements for elastic membrane problems

Curvature‐ and displacement‐based finite element analyses of flexible slider crank mechanisms

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