cratepy.clustering.clusteringdata.def_gradient_from_log_strain

def_gradient_from_log_strain(log_strain)

Get deformation gradient from material logarithmic strain tensor.

Among the multitude of deformation gradients that may correspond to a given material logarithmic strain tensor, a particular choice stems from assuming that both tensors are coaxial, i.e., that the deformation gradient shares the eigenvectors with the material logarithmic strain tensor. In this case, the deformation gradient is symmetric and admits spectral decomposition as shown below.

Given the spectral decomposition of the elastic material logarithmic strain tensor

\[\boldsymbol{E}^{e} = \sum_{i=1}^{3} \lambda_{i}^{\boldsymbol{E}} \, \boldsymbol{l}_{i}^{\boldsymbol{E}} \otimes \boldsymbol{l}_{i}^{\boldsymbol{E}} \, ,\]

where \(\boldsymbol{E}^{e}\) is the elastic material logarithmic strain tensor, and \(\lambda_{i}^{\boldsymbol{E}}\) and \(\boldsymbol{l}_{i}^{\boldsymbol{E}}\), \(i=1,2,3\), are the eigenvalues and eigenvectors of \(\boldsymbol{E}^{e}\), the coaxial unique symmetric elastic deformation gradient comes

\[\boldsymbol{F}^{e} = \sum_{i=1}^{3} \lambda_{i}^{\boldsymbol{F}} \, \boldsymbol{l}_{i}^{\boldsymbol{E}} \otimes \boldsymbol{l}_{i}^{\boldsymbol{E}} \, ,\]

where \(\boldsymbol{F}^{e}\) is the elastic deformation gradient and \(\lambda_{i}^{\boldsymbol{F}} = \exp \left[\lambda_{i}^{\boldsymbol{E}}\right], \, i=1,2,3\), are the corresponding eigenvalues.

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Parameters:

log_strain (numpy.ndarray (2d)) – Material logarithmic strain tensor.

Returns:

def_gradient – Deformation gradient.

Return type:

numpy.ndarray (2d)