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Theory Manual Version 3.6
 Section 2.3: Deformation, Strain and Stress Up Section 2.3: Deformation, Strain and Stress Subsection 2.3.2: Strain 

2.3.1 The deformation gradient tensor

Consider the deformation of an object from an initial or reference configuration to a deformed or current configuration. The location of the material particles in the reference configuration are denoted by and are known as the material coordinates. Their location in the current configuration is denoted by and known as the spatial coordinates. The deformation map , which is a mapping from to , maps the coordinates of a material point to the spatial configuration:
figure ../Figures/FigDeformationMap.png

The deformation map
The displacement map is defined as the difference between the spatial and material coordinates: The deformation gradient is defined as The deformation gradient relates an infinitesimal vector in the reference configuration to the corresponding vector in the current configuration: The determinant of the deformation tensor is called the volume ratio; it gives the volume change, or equivalently the change in density: Here is the density in the reference configuration and is the current density.
When dealing with incompressible and nearly incompressible materials it will prove useful to separate the volumetric and the deviatoric (distortional) components of the deformation gradient. Such a separation must ensure that the deviatoric part of the deformation gradient, namely , does not produce any change in volume. Noting that the determinant of the deformation gradient gives the volume ratio, the determinant of must therefore satisfy, This condition can be achieved by choosing as, Using the polar decomposition of a second order tensor, the deformation gradient can be written as a product of a positive definite symmetric tensor (or ) and a proper orthogonal tensor : is called the left stretch tensor, is called the right stretch tensor and the orthogonal tensor is called the rotation.
 Section 2.3: Deformation, Strain and Stress Up Section 2.3: Deformation, Strain and Stress Subsection 2.3.2: Strain