the elastic stiffness C, sigma = C : eps; a 6 x 6 Voigt matrix in GPa or the full array

Input

M 6-by-6 Voigt matrix, full (…, 3, 3, 3, 3) coefficients, or another tensor
args optional reference frame
kwargs tensor constructor options, including density, frame, unit and Kelvin; unit defaults to ‘GPa’

Example

Construct an illustrative cubic stiffness tensor in GPa, rotate it into specimen coordinates, and compare Young’s modulus along the same physical direction before and after rotation.

import numpy as np
import matplotlib.pyplot as plt
from mtex import (crystalFrame, stiffnessTensor, orientation, Miller, degree,
                  plot, mtexColorbar)

cs = crystalFrame('m-3m')
C = stiffnessTensor([[200, 120, 120, 0, 0, 0],
                     [120, 200, 120, 0, 0, 0],
                     [120, 120, 200, 0, 0, 0],
                     [0, 0, 0, 80, 0, 0],
                     [0, 0, 0, 0, 80, 0],
                     [0, 0, 0, 0, 0, 80]], cs, unit='GPa')
direction = Miller(1, 0, 0, cs, 'uvw')
ori = orientation.byEuler(30*degree, 40*degree, 20*degree, cs)
rotated = ori * C
print('Crystal direction modulus (GPa):', round(float(C.YoungsModulus(direction)), 3))
print('Rotated direction modulus (GPa):', round(float(rotated.YoungsModulus(ori*direction)), 3))
plot(C.YoungsModulus(), resolution=5*degree)
mtexColorbar(title='GPa')
plt.show()

The two numerical values agree because tensor and direction were rotated together. The plot shows the directional variation for these model coefficients, not measured material data.

Crystal direction modulus (GPa): 110.0
Rotated direction modulus (GPa): 110.0

Example output for stiffnessTensor.stiffnessTensor

Members

bulkModulus the bulk modulus of an isotropic polycrystal, the Voigt and Reuss bounds
ChristoffelTensor the elastic Christoffel tensor for a wave propagation direction
colon the double contraction A : B, the last two indices of A against the first two of B, MATLAB’s A:B
energyVector the energy velocity of a lossless elastic plane wave, the vector C_ijkl p_j n_k p_l / (rho v)
hashinShtrikmanBounds the Hashin-Shtrikman bounds of the isotropic moduli, the tightest pair for a quasihomogeneous polycrystal: (lower=(K, G), upper=(K, G)), each the extreme estimate over the references whose residual is semidefinite, found on the boundary of that convex set by a golden section over the shear and a bisection over the bulk modulus
HashinShtrikmanModulus the Hashin-Shtrikman estimate of the isotropic moduli of a polycrystal from a reference medium K0, G0, over arrays of references: the bulk and shear moduli, and the definiteness of C - C0 (1 where the estimate is a lower bound, -1 an upper one, 0 where it is neither and the moduli are NaN)
inv the inverse: rank two by the matrix, rank four through the Voigt matrix, which swaps the convention and with it the class, a stiffness inverting to a compliance
linearCompressibility the linear compressibility, the length change per pressure in the direction v, S_ijkk v_i v_j
mtimes scale a tensor, multiply rank-two tensors, or contract an elastic tensor with a strain
plotSeismicVelocities the overview of the seismic wave velocities of a stiffness tensor, six pole figures
PoissonRatio the Poisson ratio of the tension direction x and the transverse direction y, minus the transverse over the longitudinal strain, -S_ijkl x_i x_j y_k y_l / S_mnop x_m x_n x_o x_p
rand a random tensor with the symmetries its class and frame ask for
shearModulus the shear modulus of the plane h and the shear direction u, 1 / (4 S_ijkl h_i u_j h_k u_l)
stiffnessTensor the elastic stiffness C, sigma = C : eps
velocity the speeds of the three elastic plane waves in a direction and their polarisations, the eigenvalues and eigenvectors of the Christoffel tensor
volumeCompressibility the volume change per pressure, S_iikk
YoungsModulus Young’s modulus in the direction v, the reciprocal of S_ijkl v_i v_j v_k v_l

Used on

DefinitionAsCoordinateTransform, CPOSeismicProperties, IsotropicTheory, TensorAverage, TensorDefinition

MATLAB’s stiffnessTensor_index

source

Citing this page. This page is part of the documentation of MTEX, a free and open source MATLAB toolbox for analyzing and modeling crystallographic textures. It was written by The MTEX Developers and is published at https://mtex-toolbox.github.io/stiffnessTensor_index_py.html. If you use MTEX, or reuse text or figures from this page, in your research, please cite

F. Bachmann, R. Hielscher, H. Schaeben: Texture Analysis with MTEX - Free and Open Source Software Toolbox, Solid State Phenomena 160 (2010), 63-68. 10.4028/www.scientific.net/SSP.160.63

BibTeX 
@article{bachmann2010mtex,
  author  = {F. Bachmann and R. Hielscher and H. Schaeben},
  title   = {Texture Analysis with MTEX - Free and Open Source Software Toolbox},
  journal = {Solid State Phenomena},
  volume  = {160},
  pages   = {63-68},
  year    = {2010},
  doi     = {10.4028/www.scientific.net/SSP.160.63},
  url     = {https://doi.org/10.4028/www.scientific.net/SSP.160.63}
}

Other papers describing specific MTEX methods are listed under Publications — please cite the one that best fits your application. The MTEX source code is licensed under the GNU General Public License v2.0; the text and figures of this documentation are licensed under CC BY 4.0, which permits reuse — including by automated systems — provided The MTEX Developers and this page are credited.