To cast this in a formal way we represent the positionsof neighboring material points relative to an arbitrary originin the reference configuration by the vector dx. As a result of deformation, this vector is mapped into its image in the current configuration, dy = dx + du, where du is the differential total displacement vector. These vectors are related by the total, or shape, deformation gradient, F:
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where I is the second rank identity tensor. The second rank tensor formed from the partial derivatives of u with respect to x is known as the shape (also total) distortion tensor, b and is a perfect differential if the deformations that produce it do not introduce any discontinuities, i.e. gaps or cleavages, in the global body. That is, there exists a one-toone mapping of material points from the current state tothe reference state. The Lagrangian and Eulerian (Almansi) finite strain tensors, E and E, associated with a deformation (defined by the deformation gradient F), respectively, are symmetric tensors defined by: |