# A general deformation matrix for three-dimensions

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## 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 (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 steady-state progressive deformations, from the direction cosines matrix that defines the orientation of shear strains in the x_{i} 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 (*W*_{k} 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 shear coaxial progressive deformation velocity gradient tensor deformation tensor vorticity tensor strain ellipsoid passive linear marker### Notation

*a*_{ij}Components of the velocity gradient tensor L

*c*_{ij}Constant coefficients of Equation (19) [see Eq. (B.10)]

**D**Deformation tensor

*k*_{i}Principal strain of the coaxial progressive deformation

*l*_{ij}Direction cosines of a

_{ij}**L**Velocity gradient tensor (matrix a)

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

*s*_{ij}Components of the simple shearing velocity gradient tensor

*S*Stretching tensor

*S*_{i}Principal stretch of the strain ellipsoid

- v
Velocity field tensor

*w*Magnitude of the vorticity vector

**W**Vorticity or spin tensor

**W**_{k}Kinematic vorticity number (varies between 1 for simple shearing, and 0 for pure shearing)

*x*_{i}Coordinate system in the deformed state for the coaxial progressive deformation

*x*_{i}^{′}Coordinate system in the deformed state for the simple shearing progressive deformation

*X*_{i}Coordinate system in the undeformed state

*α*_{ij}Angle between

*x*_{i}^{′}and*X*_{j}*γ*_{ij}Simple shear strain along the

_{i}^{′}-x_{j}^{′}plane*gg*_{ij}Instantaneous simple shear strain rate along the

_{i}^{′}- x_{j}^{′}plane- δ
Dilation or anisotropic volume change

*δ*_{ij}Kronecker delta (unit diagonal matrix)

*ge*_{ij}Principal strain rate of the coaxial progressive deformation

*θ*Orientation of any line in the undeformed state with respect to

*x*_{1}*θ′*Orientation of any line in the deformed state with respect to x

_{1′}(e.g., principal stretch of the strain ellipse)*Φ*Angle between the flow direction of simple shearing and the principal strain

*k*_{1}of pure shearing (between x_{1′}and x_{1})- X
_{i} 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)

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