Conversion of material parameters and other utilities.
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]
Transformations of constitutive law coefficients of 3D problems to 2D.
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
d3 : array
b3 : array
|
---|
Compute bulk modulus from Lamé parameters.
\gamma = \lambda + {2 \over 3} \mu
Compute bulk modulus corresponding to Young’s modulus and Poisson’s ratio.
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.
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}
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