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(1) |
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(2) |
The Eulerian strain satisfies ([8]):
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(3) |
Finally, the deviatoric elastic left Cauchy-Green tensor is defined by ([22]):
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(4) |
where is the elastic Jacobian and
is the elastic deformation gradient. The above formulas apply for Carthesian coordinate systems.
The stress measure consistent with the Lagrangian strain is the second Piola-Kirchhoff stress S. This stress, which is internally used in CalculiX for all applications (the so-called total Lagrangian approach, see [6]), can be transformed into the first Piola-Kirchhoff stress P (the so-called engineering stress, a nonsymmetric tensor) and into the Cauchy stress t (true stress). All CalculiX input (e.g. distributed loading) and output is in terms of true stress. In a tensile test on a specimen with length L the three stress measures are related by:
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(5) |
where is the engineering strain defined by
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(6) |