cratepy.material.materialoperations.conjugate_material_log_strain

conjugate_material_log_strain(def_gradient, first_piola_stress)

Stress conjugate of material logarithmic strain tensor.

Material logarithmic strain tensor:

\[\boldsymbol{E} = \frac{1}{2} \ln (\boldsymbol{F}^{T} \boldsymbol{F})\]

where \(\boldsymbol{E}\) is the material logarithmic strain tensor and \(\boldsymbol{F}\) is the deformation gradient.

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Stress conjugate of material logarithmic strain tensor:

\[\boldsymbol{T} = \boldsymbol{R}^{T} \boldsymbol{P} \boldsymbol{F}^{T} \boldsymbol{R}\]

where \(\boldsymbol{T}\) is the stress conjugate of the material logarithmic strain tensor, \(\boldsymbol{R}\) is the rotation tensor, \(\boldsymbol{P}\) is the first Piola-Kirchhoff stress tensor, and \(\boldsymbol{F}\) is the deformation gradient.

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Parameters:

• def_gradient (numpy.ndarray (2d)) – Deformation gradient.

• first_piola_stress (numpy.ndarray (2d)) – First Piola-Kirchhoff stress tensor.

Returns:

• material_log_strain (numpy.ndarray (2d)) – Material logarithmic strain.

• stress_conjugate (numpy.ndarray (2d)) – Stress conjugate to material logarithmic strain.