A finite strain beam formulation. The three-dimensional dynamic problem. I
type: Item
instance of: scholarly article
title: A finite strain beam formulation. The three-dimensional dynamic problem. I
zbMATH Open document ID: 0583.73037
DOI: 10.1016/0045-7825(85)90050-7
published in: Computer Methods in Applied Mechanics and Engineering
publication date: 1985-01-01T00:00:00Z
review text: In the context of a director type of approach, \textit{S. S. Antman} has developed a three-dimensional extension of the classical Kirchhoff-Love rod model including finite extension and finite shearing [Q. Appl. Math. 32, 221-240 (1974; Zbl 0302.73031)]. This paper should be regarded as a convenient parametrization of this extension, realized by constraining the 3-dimensional theory with the introduction of the kinematic assumption. In section 1, the formulation of the basic kinematics of the beam is given in terms of a 3-dimensional orthogonal moving frame defined so that one of its vectors, denoted by n, remains normal to a typical cross- section in any configuration. Corresponding expressions for the linear and angular momentum are obtained in section 2. The former is associated with the acceleration of the centroid and the latter with the time derivative of the vorticity vector of the moving frame. Next, the basic laws of motion are summarized in section 3, and the spatial and material descriptions of the resultant force acting on a cross-section are discussed. In section 4, starting with the general 3-dimensional expression for the internal power, the author develops spatial and material reduced expressions involvig the resultant force and torque and their conjugate strain rates. In the material description the reduced expression for the internal power enables one to identify the appropriate strain measures. In particular, the strain measure conjugate to the resultant torque has a simple geometric interpretation as the axial vector of a skew-symmetric tensor associated with the moving frame. In the spatial description, in addition to the appropriate strain measures, the reduced expression of the internal power determines the appropriate (objective) strain rate as the rate of change relative to an observer spinning with the moving frame. Finally, in section 5, i) it is shown that for the plane problem the present development reduces to that proposed by \textit{E…
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 74S05
zbMATH DE Number: 3934502
zbMATH Keywords: director type of approach
zbMATH Keywords: finite extension
zbMATH Keywords: finite shearing
zbMATH Keywords: constraining the 3-dimensional theory
zbMATH Keywords: basic kinematics
zbMATH Keywords: 3-dimensional orthogonal moving frame
zbMATH Keywords: linear and angular momentum
zbMATH Keywords: acceleration of the centroid
zbMATH Keywords: time derivative of the vorticity vector
zbMATH Keywords: basic laws of motion
zbMATH Keywords: spatial and material descriptions
zbMATH Keywords: resultant force
zbMATH Keywords: internal power
zbMATH Keywords: conjugate strain rates
zbMATH Keywords: reduced expression for the internal power
zbMATH Keywords: appropriate strain measures
zbMATH Keywords: plane problem
zbMATH Keywords: parametrization
zbMATH Keywords: resulting geometric stiffness
zbMATH Keywords: numerical implementation
author: J. C. Simo
reviewed by: Michel Bernadou
MaRDI profile type: Publication
full work available at URL: 0045-7825(85)90050-7
OpenAlex ID: W2043042987
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description: scientific article; zbMATH DE number 3934502