The Cauchy-Green Strain Measure

The deformation gradient F contains geometrical information that is not related to strain (i.e. dimensional or shape change) namely, it includes rigid-body rotations. The matrix associated with rotation is familiar from its use in changing the representation of a vector when the axis set is rotated. It also defines the rigid-body rotation of an element of material. 

1-2-3 axes, the 2 -axis direction cosines (In, In, 23) with respect to the 1-2-3 axes and the new 3-axis direction cosines ( 31, 32, 33) with respect to the 1-2-3 axes. Then the matrix of rotation R is given by 

This gives the new co-ordinate vector (xi,X2,X3) when operating on the old co-ordinates (Xi,X2,Xs). Note that in two dimensions, it reduces to 

Suppose that the deformation gradient F consists of a pure deformation V, ( pure in the sense that it does not include any rigid-body rotation), and a rotation R, then according 

unlike F, is always symmetric (f,j =fy, i = 1,2, 3). When a deformation gradient F is itself symmetric, it already corresponds to a deformation with no rigid-body motions, a pure deformation V. It is possible to derive such a quantity from the Cauchy-Green measure C. This requires knowledge of how to obtain principal values of C or V, and of how to transform C and V between different axis sets. 

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Any second-order tensor has a number of invariants associated with it. One such is the trace of the tensor, equal to the sum of its diagonal terms, applicable to any strain tensor. We define the first invariant h as the trace of the Cauchy-Green strain measure tr(C) ... 

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