On large strain deformations of shells

Stumpf, H.; Makowski, Jerzy

doi:10.1007/bf01176879

Cited by 28 publications

(26 citation statements)

References 6 publications

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“…The dependence of e 33 on n certainly cannot be neglected for the shell analyzed in Stumpf and Makowski (1986), which is thick and/or strongly curved with haR 0X2 and undergoes 20% stretches of the middle surface. Note that bodies of such a geometry are traditionally considered as remaining beyond the area of application of shell equations, and are analysed using three-dimensional equations and ®nite elements.…”

Section: Examplementioning

confidence: 97%

“…In Stumpf and Makowski (1986) the example of an inversion of a spherical cap is considered in detail. The deformed con®guration of this shell also has a spherical shape, and an explicit form of kf can be obtained, cf.…”

Section: Examplementioning

confidence: 99%

“…To capture this effect the kinematical hypothesis is generalized, e.g. in Stumpf and Makowski (1986) and Schieck et al (1992) the generalized Kirchhoff hypothesis is used, while in Makowski and Stumpf (1986) the generalized Reissner hypothesis is exploited. In these papers a scalar extension function over the thickness is introduced and determined from the incompressibility condition.…”

Section: Incompressible Materialsmentioning

confidence: 99%

“…For instance, in Stumpf and Makowski (1986) the current position vector is assumed as xf x 0 kf" n, where "…”

Section: Incompressible Materialsmentioning

confidence: 99%

“…4) For the incompressible materials, the generalized kinematical hypothesis with the scalar extension function are often used, e.g. Stumpf and Makowski (1986), Makowski and Stumpf (1986) and Schieck et al (1992). We show, for thin shells, that the generalized Reissner hypothesis with the scalar extension parameter or the standard Reissner hypothesis also can be used if the normal strain is properly recovered from the incompressibility condition, and a suitable form of the constitutive equations is exploited.…”

Section: Introductionmentioning

confidence: 99%

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A shell theory with independent rotations for relaxed Biot stress and right stretch strain

Wisniewski

1998

Computational Mechanics

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A shell theory incorporating rotations as the independent variable, among them the drilling rotation, is derived systematically from three-dimensional equations of a classical (non-polar) continuum.The three-dimensional balance equations are expressed in terms of the Biot stress, the deformation gradient and rotations, which are introduced via a polar decomposition equation for the deformation gradient. This equation is relaxed, by replacing the rotation tensor by an arbitrary orthogonal one, and appended as a rotation constraint. Two methods of imposing the rotation constraint are discussed; a penalty method, and a penalty method combined with a staggered scheme. A relaxed right stretch strain, deduced from the symmetric part of the weak form of the balance equations, yields the right stretch strain when the rotation constraint is satis®ed.The weak form shell equations are obtained from the three-dimensional ones by Taylor approximations in the thickness direction, (linear for the deformation gradient and constant for the rotations) and an integration over the shell thickness. The relaxed right stretch strain for Reissner kinematics naturally includes drilling rotations. To facilitate a description in co-rotational bases the rotated-forward shell measures and a co-rotational variation of the Green-McInnis-Naghdi type are introduced.The constitutive equations for the shell stress and couple resultants are found for a linear material and a Mooney-Rivlin incompressible material. They are derived without any simplifying assumptions for the strain energy given in terms of the right stretching tensor, linearly approximated across the thickness. The normal strain is recovered for the Reissner kinematics on use of the plane stress condition for the linear material, and the incompressibility condition for the Mooney-Rivlin material.Considering prospective numerical implementations we try to establish whether the present approach exploiting the relaxed right stretch strain is advantageous over the one based on the Green strain. Limitations resulting from the Taylor approximation of the rotation constraint are also addressed. A scalar parameter is introduced to secure correctness of either membrane (even ®nite strain) states or bending-shearing states, whichever dominate.

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Section: Examplementioning

confidence: 97%

Section: Examplementioning

confidence: 99%

Section: Incompressible Materialsmentioning

confidence: 99%

“…For instance, in Stumpf and Makowski (1986) the current position vector is assumed as xf x 0 kf" n, where "…”

Section: Incompressible Materialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A shell theory with independent rotations for relaxed Biot stress and right stretch strain

Wisniewski

1998

Computational Mechanics

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Mechanics of Curvilinear Electronics

Wang

Xiao

Song

et al. 2012

Nano and Cell Mechanics

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Genuinely resultant shell finite elements accounting for geometric and material non‐linearity

Chróścielewski

Makowski

Stumpf

1992

Numerical Meth Engineering

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132

114

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SUMMARYA general theoretical framework is presented for the fully non-linear analysis of shells by the finite element method. The governing equations are derived exclusively in terms of resulting quantities through a logical and straightforward descent from three-dimensional continuum mechanics without appealing to simplifying assumptions (hence the name genuinely resultant). As a result, the underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. The underlying mathematical structure and the variational formulation of the theory are examined. This appears to be crucial for the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method. The treatment of finite rotations through an arbitrary parametrization of the rotation group and the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis are other issues studied in this paper. A numerical analysis is presented in order to assess the effectiveness of the proposed formulation. Small strain problems as well as finite strain deformation of rubber-like shells undergoing finite rotations are considered. Special attention is devoted to the assessment of the relevance of the drilling degree-of-freedom and highly non-uniform through-the-thickness deformation in the case of shells made of incompressible material.

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On large strain deformations of shells

Cited by 28 publications

References 6 publications

A shell theory with independent rotations for relaxed Biot stress and right stretch strain

A shell theory with independent rotations for relaxed Biot stress and right stretch strain

Mechanics of Curvilinear Electronics

Genuinely resultant shell finite elements accounting for geometric and material non‐linearity

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