sfepy.mechanics.matcoefs module

Conversion of material parameters and other utilities.

class sfepy.mechanics.matcoefs.ElasticConstants(young=None, poisson=None, bulk=None, lam=None, mu=None, p_wave=None, _regenerate_relations=False)

Conversion formulas for various groups of elastic constants. The elastic constants supported are:

• System Message: WARNING/2 (E)

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  : Young’s modulus

• System Message: WARNING/2 (\nu)

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  : Poisson’s ratio

• System Message: WARNING/2 (K)

  latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.
  : bulk modulus

• System Message: WARNING/2 (\lambda)

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  : Lamé’s first parameter

• System Message: WARNING/2 (\mu, G)

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  : shear modulus, Lamé’s second parameter

• System Message: WARNING/2 (M)

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  : P-wave modulus, longitudinal wave modulus

The elastic constants are referred to by the following keyword arguments: young, poisson, bulk, lam, mu, p_wave.

Exactly two of them must be provided to the __init__() method.

Examples

• basic usage:

  >>> from sfepy.mechanics.matcoefs import ElasticConstants
  >>> ec = ElasticConstants(lam=1.0, mu=1.5)
  >>> ec.young
  3.6000000000000001
  >>> ec.poisson
  0.20000000000000001
  >>> ec.bulk
  2.0
  >>> ec.p_wave
  4.0
  >>> ec.get(['bulk', 'lam', 'mu', 'young', 'poisson', 'p_wave'])
  [2.0, 1.0, 1.5, 3.6000000000000001, 0.20000000000000001, 4.0]
  

• reinitialize existing instance:

  >>> ec.init(p_wave=4.0, bulk=2.0)
  >>> ec.get(['bulk', 'lam', 'mu', 'young', 'poisson', 'p_wave'])
  [2.0, 1.0, 1.5, 3.6000000000000001, 0.20000000000000001, 4.0]
  

get(names)

Get the named elastic constants.

init(young=None, poisson=None, bulk=None, lam=None, mu=None, p_wave=None)

Set exactly two of the elastic constants, and compute the remaining. (Re)-initializes the existing instance of ElasticConstants.

class sfepy.mechanics.matcoefs.TransformToPlane(iplane=None)

Transformations of constitutive law coefficients of 3D problems to 2D.

tensor_plane_stress(c3=None, d3=None, b3=None)

Transforms all coefficients of the piezoelectric constitutive law from 3D to plane stress problem in 2D: strain/stress ordering: 11 22 33 12 13 23. If d3 is None, uses only the stiffness tensor c3.

Parameters:

c3 : array

The stiffness tensor.

d3 : array

The dielectric tensor.

b3 : array

The piezoelectric coupling tensor.

sfepy.mechanics.matcoefs.bulk_from_lame(lam, mu)

Compute bulk modulus from Lamé parameters.

System Message: WARNING/2 (\gamma = \lambda + {2 \over 3} \mu )

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sfepy.mechanics.matcoefs.bulk_from_youngpoisson(young, poisson, plane='strain')

Compute bulk modulus corresponding to Young’s modulus and Poisson’s ratio.

sfepy.mechanics.matcoefs.lame_from_youngpoisson(young, poisson, plane='strain')

Compute Lamé parameters from Young’s modulus and Poisson’s ratio.

The relationship between Lamé parameters and Young’s modulus, Poisson’s ratio (see [1],[2]):

System Message: WARNING/2 (\lambda = {\nu E \over (1+\nu)(1-2\nu)},\qquad \mu = {E \over 2(1+\nu)} )

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The plain stress hypothesis:

System Message: WARNING/2 (\bar\lambda = {2\lambda\mu \over \lambda + 2\mu} )

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[1] I.S. Sokolnikoff: Mathematical Theory of Elasticity. New York, 1956.

[2] T.J.R. Hughes: The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. New Jersey, 1987.

sfepy.mechanics.matcoefs.stiffness_from_lame(dim, lam, mu)

Compute stiffness tensor corresponding to Lamé parameters.

System Message: WARNING/2 ({\bm D}_{(2D)} = \begin{bmatrix} \lambda + 2\mu & \lambda & 0\\ \lambda & \lambda + 2\mu & 0\\ 0 & 0 & \mu \end{bmatrix} )

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System Message: WARNING/2 ({\bm D}_{(3D)} = \begin{bmatrix} \lambda + 2\mu & \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda + 2\mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda + 2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu\\ \end{bmatrix} )

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

sfepy.mechanics.matcoefs.stiffness_from_lame_mixed(dim, lam, mu)

Compute stiffness tensor corresponding to Lamé parameters for mixed formulation.

System Message: WARNING/2 ({\bm D}_{(2D)} = \begin{bmatrix} \widetilde\lambda + 2\mu & \widetilde\lambda & 0\\ \widetilde\lambda & \widetilde\lambda + 2\mu & 0\\ 0 & 0 & \mu \end{bmatrix} )

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System Message: WARNING/2 ({\bm D}_{(3D)} = \begin{bmatrix} \widetilde\lambda + 2\mu & \widetilde\lambda & \widetilde\lambda & 0 & 0 & 0\\ \widetilde\lambda & \widetilde\lambda + 2\mu & \widetilde\lambda & 0 & 0 & 0 \\ \widetilde\lambda & \widetilde\lambda & \widetilde\lambda + 2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu\\ \end{bmatrix} )

latex exited with error [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013/TeX Live for SUSE Linux) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 78 languages loaded. (/usr/share/texmf/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texmf/tex/latex/base/size12.clo)) (/usr/share/texmf/tex/latex/base/inputenc.sty ! LaTeX Error: File `utf8x.def’ not found. Type X to quit or <RETURN> to proceed, or enter new name. (Default extension: def) Enter file name: ! Emergency stop. <read *> l.131 \endinput ^^M No pages of output. Transcript written on math.log.

where

System Message: WARNING/2 (\widetilde\lambda = -{2\over 3} \mu )

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sfepy.mechanics.matcoefs.stiffness_from_youngpoisson(dim, young, poisson, plane='strain')

Compute stiffness tensor corresponding to Young’s modulus and Poisson’s ratio.

sfepy.mechanics.matcoefs.stiffness_from_youngpoisson_mixed(dim, young, poisson, plane='strain')

Compute stiffness tensor corresponding to Young’s modulus and Poisson’s ratio for mixed formulation.