Nonlinear Shell Theory With Finite Rotation and Stress-Function Vectors

Simmonds, J. G.; Danielson, D. A.

doi:10.1115/1.3422833

Cited by 116 publications

(29 citation statements)

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“…The challenge has spawned numerous publications, for example [1,4,5,34,42,43,62,71,72]. For use of finite rotations in mathematical models, particularly shells, see [60,70,77]. There has been an Euromech Colloquium devoted entirely to that topic [61].…”

Section: Continuum Mechanics Sourcesmentioning

confidence: 99%

A unified formulation of small-strain corotational finite elements: I. Theory

Felippa

Haugen²

2005

Computer Methods in Applied Mechanics and Engineering

354

266

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Section: Continuum Mechanics Sourcesmentioning

confidence: 99%

A unified formulation of small-strain corotational finite elements: I. Theory

Felippa

Haugen²

2005

Computer Methods in Applied Mechanics and Engineering

354

266

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“…Classical, minimum complementary energy-based, dimensionally reduced plate and shell models in terms of symmetric stresses can be derived using functional (8). In that case both the translational and rotational equilibrium equations, (13) and (16), should be satisÿed a priori. When dimensional reduction in terms of stresses is carried out, transverse variation of the stress components is approximated by polynomials of ÿnite degree in z. Equilibrium requirements (13) and (16) mean, however, that the polynomial dependence of the stress components on z cannot be arbitrary.…”

Section: Notation and Preliminariesmentioning

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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“…They were so far derived from Newton's law and Euler's theorem either by directly using the Cosserat model [36], [37], [38] or indirectly from 3D elasticity [12], [39]. When investigated by the direct approach, the dynamics of the directors t 3 are the most of the time, directly derived on S 2 through an alternative set of angular equations deduced from (80)-bottom (or (81)-bottom) by cross multiplying them on the right by t 3 .…”

Section: Reduction Of the Kinematic And Kinetic Modelsmentioning

confidence: 99%

Poincaré’s Equations for Cosserat Media: Application to Shells

Boyer

Renda

2016

J Nonlinear Sci

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In 1901 Henri Poincaré discovered a new set of equations for mechanics. These equations are a generalization of Lagrange's equations for a system whose conguration space is a Lie group which is not necessarily commutative. Since then, this result has been extensively rened through the Lagrangian reduction theory. In the present contribution, we apply an extended version of these equations to continuous Cosserat media, i.e. media in which the usual point particles are replaced by small rigid bodies, called micro-structures. In particular, we will see how the Shell balance equations used in nonlinear structural dynamics, can be easily deduced from this extension of the Poincaré's result. In future, these results will be used as foundations for the study of squid locomotion, which is an emerging topic relevant to soft robotics. AcknowledgmentThe nal publication is available at: Transformation at time t from material to geometric space

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Nonlinear Shell Theory With Finite Rotation and Stress-Function Vectors

Cited by 116 publications

References 0 publications

A unified formulation of small-strain corotational finite elements: I. Theory

A unified formulation of small-strain corotational finite elements: I. Theory

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

Poincaré’s Equations for Cosserat Media: Application to Shells

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