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(2) |
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(3) |
The Eulerian strain satisfies ([21]):
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(4) |
Finally, the deviatoric elastic left Cauchy-Green tensor is defined by ([78]):
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(5) |
where is the elastic Jacobian and
is the elastic deformation gradient. The above formulas apply for Cartesian 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 [9]), can be transformed into the first Piola-Kirchhoff stress P (the so-called engineering stress, a non-symmetric 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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(6) |
where is the engineering strain defined by
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(7) |
The treatment of the thermal strain depends on whether the analysis is
geometrically linear or nonlinear. For isotropic material the thermal strain
tensor amounts to
, where
is the
expansion coefficient,
is the temperature change since the initial
state and
is the second order identity tensor. For
geometrically linear calculations the thermal strain is subtracted from the
total strain to obtain the mechanical strain:
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(8) |
In a nonlinear analysis the thermal strain is subtracted from the deformation gradient in order to obtain the mechanical deformation gradient. Indeed, assuming a multiplicative decomposition of the deformation gradient one can write:
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(9) |
where the total deformation gradient
is written as the
product of the mechanical deformation gradient and the thermal deformation
gradient. For isotropic materials the thermal deformation gradient can be
written as
and
consequently:
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(10) |
Therefore one obtains:
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|
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(11) |
Based on the mechanical deformation gradient the mechanical Lagrange strain is calculated and subsequently used in the material laws:
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(12) |