Anisotropic Elasticity edit page

The linear theory of ellasticity in anisotropic materials is essentialy based on the fourth order stiffness tensor C. Such a tensor is represented in MTEX by a variable of type stiffnessTensor. Such a variable can either by set up using a symmetric 6x6 matrix or by importing it from an external file. The following examples does so for the stiffness tensor for Olivine

% file name
fname = fullfile(mtexDataPath,'tensor','Olivine1997PC.GPa');

% crytsal symmetry
cs = crystalSymmetry('mmm',[4.7646 10.2296 5.9942],'mineral','Olivin');

% define the tensor
C = stiffnessTensor.load(fname,cs)
C = stiffnessTensor (Olivin)
  unit: GPa              
  rank: 4 (3 x 3 x 3 x 3)
 
  tensor in Voigt matrix representation:
 320.5  68.2  71.6     0     0     0
  68.2 196.5  76.8     0     0     0
  71.6  76.8 233.5     0     0     0
     0     0     0    64     0     0
     0     0     0     0    77     0
     0     0     0     0     0  78.7

Hooke's Law

The stiffness tensor tensor of a material is defined as the stress the material expreances for a given strain

eps = strainTensor(diag([1,1.1,0.9]),cs)
eps = strainTensor (Olivin)
  type: Lagrange 
  rank: 2 (3 x 3)
 
   1   0   0
   0 1.1   0
   0   0 0.9

Now Hooke's law states that the resulting stress can be computed by

sigma = C : eps
sigma = stressTensor (Olivin)
  rank: 2 (3 x 3)
 
 459.9     0     0
     0 353.4     0
     0     0 366.2

The other way the compliance tensor S = inv(C) translates stress into strain

inv(C) : sigma
ans = strainTensor (Olivin)
  type: Lagrange 
  rank: 2 (3 x 3)
 
   1   0   0
   0 1.1   0
   0   0 0.9

The ellastic energy of the strain eps can be computed equivalently by the following equations

% the elastic energy
U = sigma : eps
U = EinsteinSum(C,[-1 -2 -3 -4],eps,[-1 -2],eps,[-3 -4]);

U = (C : eps) : eps;
U =
   1.1783e+03

Young's Modulus

Young's modulus is also known as the tensile modulus and measures the stiffness of elastic materials. It is computed for a specific direction d by the command YoungsModulus.

d = vector3d.X;
E = C.YoungsModulus(d)
E =
  286.9284

If the direction d is ommited Youngs modulus is returned as a spherical function.

% compute Young's modulus as a directional dependent function
E = C.YoungsModulus

% which can be evaluated at any direction
E.eval(d)

% or plot it
setMTEXpref('defaultColorMap',blue2redColorMap);
plot(C.YoungsModulus,'complete','upper')
E = S2FunHarmonicSym (Olivin)
 bandwidth: 16
 antipodal: true
ans =
  286.9259

Linear Compressibility

The linear compressibility is the deformation of an arbitrarily shaped specimen caused by an increase in hydrostatic pressure and can be described by a second rank tensor. Similar to the Young's modulus it can be computed by the command linearCompressibility for specific directions d or as a spherical function

% compute as a spherical function
beta = linearCompressibility(C)

% plot it
plot(beta,'complete','upper')

% evaluate the function at a specific direction
beta.eval(d)
beta = S2FunHarmonicSym (Olivin)
 bandwidth: 2
 antipodal: true
ans =
    0.0018

Poisson Ratio

The rate of compression / decompression in a direction n normal to the pulling direction p is called Poisson ratio.

% the pulling direction
p = vector3d.Z;

% two orthogonal directions
n = [vector3d.X,vector3d.Y];

% the Poisson ratio
nu = C.PoissonRatio(p,n)
nu =
    0.1515    0.3383

If we ommit in the call to PoissonRatio the last argument

nu = C.PoissonRatio(p)
nu = S2FunHarmonicSym (Olivin)
 bandwidth: 4
 antipodal: true

we again obtain a spherical function. However, this time it is only meaningful to evaluate this function at directions perpendicular to the pulling direction p. Hence, a good way to visualize this function is to plot it as a section in the x/y plane

plotSection(nu,p,'color','interp','linewidth',5)
axis off
mtexColorbar

Shear Modulus

The shear modulus is TODO

% shear plane
n = Miller(0,0,1,cs);

% shear direction
d = Miller(1,0,0,cs);

G = C.shearModulus(n,d)
G =
   6.2807e+04
newMtexFigure('layout',[1,3])
% shear plane
n = Miller(1,0,0,cs);
plotSection(C.shearModulus(n),n,'color','interp','linewidth',5)
mtexTitle(char(n))
axis off

nextAxis
n = Miller(1,1,0,cs);
plotSection(C.shearModulus(n),n,'color','interp','linewidth',5)
mtexTitle(char(n))

nextAxis
n = Miller(1,1,1,cs)
plotSection(C.shearModulus(n),n,'color','interp','linewidth',5)
mtexTitle(char(n))
hold off

setColorRange('equal')
mtexColorbar
drawNow(gcm,'figSize','large')
n = Miller (Olivin)
  h k l
  1 1 1

Wave Velocities

Since elastic compression and decompression is mechanics of waves traveling through a medium anisotropic compressibility causes also anisotropic waves speeds. The analysis of this anisotropy is explained in the section wave velocities.