Mathematical Geology

, Volume 29, Issue 1, pp 93–130

# A general deformation matrix for three-dimensions

Article

DOI: 10.1007/BF02769621

Soto, J.I. Math Geol (1997) 29: 93. doi:10.1007/BF02769621

## Abstract

A deformation that is obtained by any simultaneous combination of two steady-state 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 non-orthogonal 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 (xi system), where the principal and intermediate strains are two horizontal coordinate axes, and another for the simple shear (xit’ system), with any orientation in space. For steady-state progressive deformations, from the direction cosines matrix that defines the orientation of shear strains in the xi coordinate system, an asymmetric finite-deformation matrix is derived. From this deformation matrix, the orientation and ellipticity of the strain ellipse, or the strain ellipsoid for three-dimensional deformations, can be determined. This deformation matrix also can be described as a combination of a rigid-body rotation and a stretching represented by a general coaxial progressive deformation. The kinematic vorticity number (Wk is derived for the general deformation matrix to characterize the non-coaxiality of the three-dimensional deformation. An application of the deformation matrix concept is given as an example, analyzing the changes in orientation and stretching that variously-oriented passive linear markers undergo after a general two-dimensional 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.

### Key Words

simple shearcoaxial progressive deformationvelocity gradient tensordeformation tensorvorticity tensorstrain ellipsoidpassive linear marker

### Notation

aij

Components of the velocity gradient tensor L

cij

Constant coefficients of Equation (19) [see Eq. (B.10)]

D

Deformation tensor

ki

Principal strain of the coaxial progressive deformation

lij

Direction cosines of aij

L

Velocity gradient tensor (matrix a)

R

Rigid-body rotation matrix (direction cosines matrix) and axial ratio or ellipticity of the strain ellipse

sij

Components of the simple shearing velocity gradient tensor

S

Stretching tensor

Si

Principal stretch of the strain ellipsoid

v

Velocity field tensor

w

Magnitude of the vorticity vector

W

Vorticity or spin tensor

Wk

Kinematic vorticity number (varies between 1 for simple shearing, and 0 for pure shearing)

xi

Coordinate system in the deformed state for the coaxial progressive deformation

xi

Coordinate system in the deformed state for the simple shearing progressive deformation

Xi

Coordinate system in the undeformed state

αij

Angle betweenxi andXj

γij

Simple shear strain along thei-xj plane

ggij

Instantaneous simple shear strain rate along thei- xj plane

δ

Dilation or anisotropic volume change

δij

Kronecker delta (unit diagonal matrix)

geij

Principal strain rate of the coaxial progressive deformation

θ

Orientation of any line in the undeformed state with respect tox1

θ′

Orientation of any line in the deformed state with respect to x1′ (e.g., principal stretch of the strain ellipse)

Φ

Angle between the flow direction of simple shearing and the principal straink1 of pure shearing (between x1′ and x1)

Xi

Eigenvalue of the velocity gradient tensor

II

Second moment of the tensor S

√λ

Length of a unit line after deformation (i.e., square of the quadratic elongation or extension, λ)