Nonlinear Elastic Shell Theory

Libai, A.; Simmonds, J. G.

doi:10.1016/s0065-2156(08)70245-x

Cited by 87 publications

(56 citation statements)

References 82 publications

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“…Similar conditions for the non-material shell base surface were first proposed by Libai and Simmonds [13,22]. In lines of principal curvatures of the shell base surface M these conditions become equivalent to those proposed by Reissner [12].…”

Section: Introductionsupporting

confidence: 64%

“…To have some insight into the problem, let us analyse the 3D-to-2D reduction of the exact 3D stress power density (22) for the special case of the resultant linear six-field theory of isotropic elastic shells. In this case, time derivatives of various fields become increments of the fields from the undeformed shell state, and the 2D stress power reduces to the 2D stress working.…”

Section: Gradients Of 2d Shell Measures In the Resultant Stress Workingmentioning

confidence: 99%

“…In this case, time derivatives of various fields become increments of the fields from the undeformed shell state, and the 2D stress power reduces to the 2D stress working. In equilibrium problems, the first two terms in (22), but without overdots, represent the effective 2D stress working expressed entirely in terms of 2D shell stress and strain measures. Any possible non-classical terms may only be hidden in the last three integrals of (22) containing the intrinsic deviation vector e(θ α ,ξ ).…”

Section: Gradients Of 2d Shell Measures In the Resultant Stress Workingmentioning

confidence: 99%

“…In the statically admissible stress distributions needed in (22) it is sufficient for our purpose to take into account only the main terms (see [34], Eq. (30c), and [20], Eqs.…”

Section: Gradients Of 2d Shell Measures In the Resultant Stress Workingmentioning

confidence: 99%

“…The first two terms of (22) represent the 2D effective stress power density expressible entirely in terms of the 2D resultant stress measures and corresponding work-conjugate 2D strain measures. The last three integrals in (22) represent the additional part of the 2D stress power density which is not expressible through the resultant 2D shell stress and strain measures alone.…”

Section: Introductionmentioning

confidence: 99%

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The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

Pietraszkiewicz

2015

Z Angew Math Mech

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Key words Theory of shells, linear six-field model, static-geometric analogy, complex shell relations, Cesáro type formulas, gradients of strain measures.Basic relations of the resultant linear six-field theory of shells are established by consistent linearization of the resultant 2D non-linear theory of shells. As compared with the classical linear shell models of Kirchhoff-Love and TimoshenkoReissner type, the six-field linear shell model contains the drilling rotation as an independent kinematic variable as well as two surface drilling couples with two work-conjugate surface drilling bending measures are present in description of the shell stress-strain state. Among new results obtained here within the six-field linear theory of elastic shells there are: 1) formulation of the extended static-geometric analogy; 2) derivation of complex BVP for complex independent variables; 3) description of deformation of the shell boundary element; 4) the Cesáro type formulas and expressions for the vectors of stress functions along the shell boundary contour; 5) discussion on explicit appearance of gradients of 2D stress and strain measures in the resultant stress working.

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Section: Introductionsupporting

confidence: 64%

Section: Gradients Of 2d Shell Measures In the Resultant Stress Workingmentioning

confidence: 99%

Section: Gradients Of 2d Shell Measures In the Resultant Stress Workingmentioning

confidence: 99%

“…In the statically admissible stress distributions needed in (22) it is sufficient for our purpose to take into account only the main terms (see [34], Eq. (30c), and [20], Eqs.…”

Section: Gradients Of 2d Shell Measures In the Resultant Stress Workingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

Pietraszkiewicz

2015

Z Angew Math Mech

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

Chróścielewski

Makowski

Stumpf

1992

Numerical Meth Engineering

130

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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Thin shells with finite rotations formulated in biot stresses: Theory and finite element formulation

Wriggers

Gruttmann

1993

Numerical Meth Engineering

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SUMMARYA bending theory for thin shells undergoing finite rotations is presented, and its associated finite element model is described. The kinematic assumption is based on a shear elastic Reissner-Mindlin theory. The starting point for the derivation of the strain measures are the resultant equilibrium equations and the associated principle of virtual work. Within this formulation the polar decomposition of the shell material deformation gradient leads to symmetric strain measures. The associated work-conjugate stress resultants and stress couples are integrals of the Biot stress tensor. This tensor is invariant with respect to rigid body motions and, therefore, appropriate for the formulation of constitutive equations. Finite rotations are introduced via Eulerian angles.The finite element discretization of arbitrary shells is based on the isoparametric concept formulated with respect to a plane reference configuration. The numerical model is applied to different non-linear plate and shell problems and compared with existing formulations. Due to a consistent linearization, the step size of a load increment is only limited by the local convergence behaviour of Newton's method.

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Nonlinear Elastic Shell Theory

Cited by 87 publications

References 82 publications

The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

The resultant linear six‐field theory of elastic shells: What it brings to the classical linear shell models?

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

Thin shells with finite rotations formulated in biot stresses: Theory and finite element formulation

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