MATRIX EXPONENTIAL AND GEOMETRICAL MEANING OF LOGARITHMIC STRAIN

Abstract eng:
On the space of all symmetric positive definite matrices (the space of deformation tensor fields) one can introduce a Riemannian geometry, so that the matrix exponential represents a geodesic (i.e. a generalised straight line, the shortest connecting line of two points) emanating from an initial point - the identity matrix, in a direction given by a vector - the prescribed matrix. Based on this approach, we prove that the logarithmic strain can be interpreted as a vector, determined by a geodesic connecting an undeformed and a deformed states. This approach applies also to deformation tensor fields, but unlike previous papers the deformation process of the whole continuum will be described by mutually uninteracting tensors, developed in separate points. As a consequence of this interpretation, it follows that for hyperelastic materials there exists a one-to-one correspondence between solution of a problem in the framework of small, and finite deformations, which is given by this very matrix exponential. As a result, the solution in the framework of finite deformations is determined by the corresponding solution in the framework of small deformations.

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