2.2.28. addStrainTensors


Purpose

Append to an ASCII table strain columns derived from given stretches of requested deformation gradient column(s).

Operates on given file(s) or STDIN right STDOUT.


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Usage

> addStrainTensors options [file(s)]


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Options

--right / -u
calculate strains based on the spectral decomposition of the »material stretch tensor« $ \tnsr U $ following from the »right Cauchy–Green deformation tensor«: $ \tnsr C = \tnsr F^\text T\tnsr F = \tnsr U^\text T \tnsr R^\text T \tnsr R\,\tnsr U = \tnsr U^2 = \lambda_i^2\,\vctr u_i \otimes \vctr u_i $
--left / -v
calculate strains based on the spectral decomposition of the »spatial stretch tensor« $ \tnsr V $ following from the »left Cauchy–Green deformation tensor«: $ \tnsr B = \tnsr F\,\tnsr F^\text T = \tnsr V \tnsr R \tnsr R^\text T \tnsr V^\text T = \tnsr V^2 = \lambda_i^2\,\vctr v_i \otimes \vctr v_i $
--logarithmic / -l / -0
$ \ln(\lambda_i)\,\vctr n_i \otimes \vctr n_i $ (»material« or »spatial Hencky«)
--biot / -b / -1
$ (\lambda_i-1)\,\vctr u_i \otimes \vctr u_i $ (»material Biot«)
$ (1-{\lambda_i}^{-1})\,\vctr v_i \otimes \vctr v_i $ (»spatial Biot«)
--green / -g / -2
$ \frac{1}{2}({\lambda_i}^2-1)\,\vctr u_i \otimes \vctr u_i $ (»material Green«)
$\frac{1}{2}(1-{\lambda_i}^{-2})\,\vctr v_i \otimes \vctr v_i $ (»spatial Almansi«)
--deformation / -f string [ f ]
(list of) heading of columns containing deformation gradient. Separate multiple items by comma: first,second,third,...


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Note

the »material stretch tensor« $ \tnsr U $ following from the »right Cauchy–Green deformation tensor«: $ \tnsr C = \tnsr F^\text T\tnsr F = \tnsr U^\text T \tnsr R^\text T \tnsr R\,\tnsr U = \tnsr U^2 = \lambda_i^2\,\vctr u_i \otimes \vctr u_i $ the »spatial stretch tensor« $ \tnsr V $ following from the »left Cauchy–Green deformation tensor«: $ \tnsr B = \tnsr F\,\tnsr F^\text T = \tnsr V \tnsr R \tnsr R^\text T \tnsr V^\text T = \tnsr V^2 = \lambda_i^2\,\vctr v_i \otimes \vctr v_i $
  1. : $ \ln(\lambda_i)\,\vctr n_i \otimes \vctr n_i $ (»material« or »spatial Hencky«)
  2. : $ (\lambda_i-1)\,\vctr u_i \otimes \vctr u_i $ (»material Biot«)
    $ (1-{\lambda_i}^{-1})\,\vctr v_i \otimes \vctr v_i $ (»spatial Biot«)
  3. : $ \frac{1}{2}({\lambda_i}^2-1)\,\vctr u_i \otimes \vctr u_i $ (»material Green«)
    $\frac{1}{2}(1-{\lambda_i}^{-2})\,\vctr v_i \otimes \vctr v_i $ (»spatial Almansi«)

Strain formulas are taken from chapter 2.3 in

A. Bertram
Elasticity and Plasticity of Large Deformations: An Introduction
3rd edition, Springer, 2012
ISBN:9783642246142
Topic revision: r10 - 20 Feb 2016, MartinDiehl


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