A general deformation matrix for threedimensions
 Juan Ignacio Soto
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A deformation that is obtained by any simultaneous combination of two steadystate progressive deformations: simple shearing and a coaxial progressive deformation, involving or not a volume change, can be expressed by a single transformation, or deformation matrix. In the general situation of simple shearing in a direction nonorthogonal with the principal strains of the coaxial progressive deformation, this deformation matrix is a function of the strain components and the orientation of shearing. In this example, two coordinate systems are defined: one for the coaxial progressive deformation (x_{i} system), where the principal and intermediate strains are two horizontal coordinate axes, and another for the simple shear (x _{i} ^{t’} system), with any orientation in space. For steadystate progressive deformations, from the direction cosines matrix that defines the orientation of shear strains in the x_{i} coordinate system, an asymmetric finitedeformation matrix is derived. From this deformation matrix, the orientation and ellipticity of the strain ellipse, or the strain ellipsoid for threedimensional deformations, can be determined. This deformation matrix also can be described as a combination of a rigidbody rotation and a stretching represented by a general coaxial progressive deformation. The kinematic vorticity number (W _{k} is derived for the general deformation matrix to characterize the noncoaxiality of the threedimensional deformation. An application of the deformation matrix concept is given as an example, analyzing the changes in orientation and stretching that variouslyoriented passive linear markers undergo after a general twodimensional deformation. The influence of the kinematic vorticity number, the simple and pure shear strains, and the obliquity between the two deformation components, on the linear marker distribution after deformation is discussed.
 Title
 A general deformation matrix for threedimensions
 Journal

Mathematical Geology
Volume 29, Issue 1 , pp 93130
 Cover Date
 199703
 DOI
 10.1007/BF02769621
 Print ISSN
 08828121
 Online ISSN
 15738868
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 simple shear
 coaxial progressive deformation
 velocity gradient tensor
 deformation tensor
 vorticity tensor
 strain ellipsoid
 passive linear marker
 Industry Sectors
 Authors

 Juan Ignacio Soto ^{(1)}
 Author Affiliations

 1. Instituto Andaluz de Ciencias de la Tierra and Departamento de Geodinamica, C.S.I.C.University of Granada, Faculty of Sciences, Campus Fuentenueva, 18008, Granada, Spain