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)[source]¶

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

$E$ : Young’s modulus

$\nu$ : Poisson’s ratio

$K$ : bulk modulus

$\lambda$ : Lamé’s first parameter

$\mu, G$ : shear modulus, Lamé’s second parameter

$M$ : 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)[source]¶: Get the named elastic constants.

init(young=None, poisson=None, bulk=None, lam=None, mu=None, p_wave=None)[source]¶: 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)[source]¶

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

tensor_plane_stress(c3=None, d3=None, b3=None)[source]¶

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:

c3array: The stiffness tensor.
d3array: The dielectric tensor.
b3array: The piezoelectric coupling tensor.

sfepy.mechanics.matcoefs.bulk_from_lame(lam, mu)[source]¶: Compute bulk modulus from Lamé parameters.

$\gamma = \lambda + {2 \over 3} \mu$

sfepy.mechanics.matcoefs.bulk_from_youngpoisson(young, poisson, plane='strain')[source]¶: Compute bulk modulus corresponding to Young’s modulus and Poisson’s ratio.

sfepy.mechanics.matcoefs.lame_from_stiffness(stiffness, plane='strain')[source]¶: Compute Lamé parameters from an isotropic stiffness tensor.

sfepy.mechanics.matcoefs.lame_from_youngpoisson(young, poisson, plane='strain')[source]¶

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]):

$\lambda = {\nu E \over (1+\nu)(1-2\nu)},\qquad \mu = {E \over 2(1+\nu)}$

The plain stress hypothesis:

$\bar\lambda = {2\lambda\mu \over \lambda + 2\mu}$

[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)[source]¶: Compute stiffness tensor corresponding to Lamé parameters.

${\bm D}_{(2D)} = \begin{bmatrix} \lambda + 2\mu & \lambda & 0\\ \lambda & \lambda + 2\mu & 0\\ 0 & 0 & \mu \end{bmatrix}$

${\bm D}_{(3D)} = \begin{bmatrix} \lambda + 2\mu & \lambda & \lambda & 0 & 0 & 0\\ \lambda & \lambda + 2\mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda + 2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu\\ \end{bmatrix}$

sfepy.mechanics.matcoefs.stiffness_from_lame_mixed(dim, lam, mu)[source]¶

Compute stiffness tensor corresponding to Lamé parameters for mixed formulation.

${\bm D}_{(2D)} = \begin{bmatrix} \widetilde\lambda + 2\mu & \widetilde\lambda & 0\\ \widetilde\lambda & \widetilde\lambda + 2\mu & 0\\ 0 & 0 & \mu \end{bmatrix}$

${\bm D}_{(3D)} = \begin{bmatrix} \widetilde\lambda + 2\mu & \widetilde\lambda & \widetilde\lambda & 0 & 0 & 0\\ \widetilde\lambda & \widetilde\lambda + 2\mu & \widetilde\lambda & 0 & 0 & 0 \\ \widetilde\lambda & \widetilde\lambda & \widetilde\lambda + 2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu\\ \end{bmatrix}$

where

$\widetilde\lambda = -{2\over 3} \mu$

sfepy.mechanics.matcoefs.stiffness_from_youngpoisson(dim, young, poisson, plane='strain')[source]¶: Compute stiffness tensor corresponding to Young’s modulus and Poisson’s ratio.

sfepy.mechanics.matcoefs.stiffness_from_youngpoisson_mixed(dim, young, poisson, plane='strain')[source]¶: Compute stiffness tensor corresponding to Young’s modulus and Poisson’s ratio for mixed formulation.

sfepy.mechanics.matcoefs.stiffness_from_yps_ortho3(young, poisson, shear)[source]¶

Compute 3D stiffness tensor ${\bm D}$ of an orthotropic linear elastic material. Young’s modulus ( $[E_1, E_2, E_3]$ ), Poisson’s ratio ( $[\nu_{12}, \nu_{13}, \nu_{23}]$ ), and shear modulus ( $[G_{12}, G_{13}, G_{23}]$ ) are given.

${\bm C}_{(3D)} = \begin{bmatrix} 1/E_1 & -\nu_{21}/E_2 & -\nu_{31}/E_3 & 0 & 0 & 0\\ -\nu_{12}/E_1 & 1/E_2 & -\nu_{32}/E_3 & 0 & 0 & 0\\ -\nu_{13}/E_1 & -\nu_{23}/E_2 & 1/E_3 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/G_{12} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1/G_{13} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1/G_{23} \end{bmatrix}$

${\bm D}_{(3D)} = \mathrm{inv}({\bm C}_{(3D)})$

$\nu_{21} = \nu_{12}\frac{E_2}{E_1},\quad \nu_{31} = \nu_{13}\frac{E_3}{E_1},\quad \nu_{32} = \nu_{23}\frac{E_3}{E_2}$

[1] R.M. Jones: Mechanics of composite materials. 1999.

sfepy.mechanics.matcoefs.wave_speeds_from_youngpoisson(young, poisson, rho)[source]¶

Compute the P- and S-wave speeds from the Young’s modulus $E$ and Poisson’s ratio $\nu$ in a homogeneous isotropic material.

$v_p^2 = {E (1 - \nu) \over \rho (1 + \nu) (1 - 2 \nu)} = {(\lambda + 2 \mu) \over \rho}$

$v_s^2 = {E \over 2 \rho (1 + \nu)} = {\mu \over \rho}$

Parameters:

youngfloat or array: The Young’s modulus.
poissonfloat or array: The Poisson’s ratio.
rhofloat or array: The density.

Returns:

vpfloat or array: The P-wave speed.
vsfloat or array: The S-wave speed.

sfepy.mechanics.matcoefs.youngpoisson_from_stiffness(stiffness, plane='strain')[source]¶: Compute Young’s modulus and Poisson’s ratio from an isotropic stiffness tensor.

sfepy.mechanics.matcoefs.youngpoisson_from_wave_speeds(vp, vs, rho)[source]¶

Compute the Young’s modulus $E$ and Poisson’s ratio $\nu$ from the P- and S-wave speeds in a homogeneous isotropic material.

$E = {\rho v_s^2 (3 v_p^2 - 4 v_s^2) \over (v_p^2 - v_s^2)}$

$\nu = {(v_p^2/2 - v_s^2) \over (v_p^2 - v_s^2)}$

Parameters:

vpfloat or array: The P-wave speed.
vsfloat or array: The S-wave speed.
rhofloat or array: The density.

Returns:

youngfloat or array: The Young’s modulus.
poissonfloat or array: The Poisson’s ratio.