Tutorial on Rotations in the Theories of Finite Deformation and Micropolar (Cosserat) Elasticity

Abstract

Although earthquake source studies have had a great impact on tectonics studies, there are at least two important problems for which seismology seems unable to provide answers. One of them refers to the rotation about vertical axes of crustal blocks in continental areas of diffuse deformation. The other problem is the stress rotations observed after large earthquakes. In both cases there are a number of competing explanations but none is supported by hard evidence. These problems are unlikely to be solved by conventional seismology, but the situation may be different if rotation data are acquired. In the near field of large earthquakes the linearized theory may not apply or a different theory may be needed. In this tutorial we consider rotations from two different points of view: the classical nonlinear theory and a nonclassical linear theory. In the nonlinear theory the deformation tensor can be expressed as the product of two tensors, one corresponding to a rotation and the other to strain, applied sequentially. In contrast, in the linearized theory the deformation tensor is the sum of a rotation and a strain tensor and the order of their application is immaterial. A linear theory that includes rotations not considered by classical elasticity (linear or not) is the micropolar theory, which deals with materials with microstructure. This theory assigns to each point in space six degrees of freedom, three corresponding to position and three corresponding to rotations. The specification of a linear micropolar isotropic body requires six elastic moduli, two of which are the classical Lame's parameters. Wave propagation in a micropolar medium is more complicated than in a linear elastic medium, with two coupled wave equations. The micropolar theory has been successful with media having periodic inner structures, but there is very little experimental work on solids with more complicated structure.

Measurement of Rotational Events in Regions Prone to Seismicity: A Review

“…In the micromorphic approach, solid material is taken consisting of numerous small (not infinitesimal) bodies, often called shells, which are capable of deformations and rotations. Various modes of passing them from one body to the other have been proposed [31], and this leads to the independent rotational waves' concept [32,33]. Individual small body in the undeformed state is usually seen as a cube; each of its sides may react not only to a vector of stress but also to their couple.…”

Introduction to the Special Issue on Rotational Seismology and Engineering Applications

“…The Cosserat theory is of particular interest to rotational seismology because it includes rotation in its formulation; see, for example, the tutorial by Pujol (2009) and the two articles by Kulesh (2009) and . Other general continua, such as asymmetric continuum, are discussed by Teisseyre and Górski (2009) …”

A Glossary for Rotational Seismology

“…Cosserat elasticity: An elasticity theory introduced by the Cosserat brothers (Eugène and François Cosserat, 1909), in which each material point has six degrees of freedom, three of which correspond to translation, as in the classical theory, and the other three to rotation. See Pujol (2009). Cosserat length: A characteristic length associated with the independent rotational field and defined for the given type of continuum.…”

“…For terms that cannot be found here, readers may consult, in particular, the tutorials by Peters (2009), Pujol (2009), Teisseyre (2009), and Zembaty (2009a) and the review by Trifunac (2009). Evans and International Working Group on Rotational Seismology (IWGoRS, 2009) describe notation conventions for rotational seismology, while Grekova and Lee (2009) provide suggested readings in continuum mechanics (including elasticity theories) and earthquake seismology.…”