ISSN:
1432-0924
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
Notes:
Abstract 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 satisfied. 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 finite strain) states or bending-shearing states, whichever dominate.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s004660050287
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