TODO
Import an Elasticity Tensor
Let us start by importing the elastic stiffness tensor of an Olivine crystal in reference orientation from a file.
fname = fullfile(mtexDataPath,'tensor','Olivine1997PC.GPa');
cs = crystalSymmetry('mmm',[4.7646 10.2296 5.9942],'mineral','Olivin');
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
Christoffel Tensor
The Christoffel Tensor is symmetric because of the symmetry of the elastic constants. The eigenvalues of the 3x3 Christoffel tensor are three positive values of the wave moduli which corresponds to \rho Vp^2 , \rho Vs1^2 and \rho Vs2^2 of the plane waves propagating in the direction n. The three eigenvectors of this tensor are then the polarization directions of the three waves. Because the Christoffel tensor is symmetric, the polarization vectors are perpendicular to each other.
% It is computed for a specific direction x by the
% command <tensor.ChristoffelTensor.html ChristoffelTensor>.
T = ChristoffelTensor(C,vector3d.X)
T = ChristoffelTensor (Olivin)
rank: 2 (3 x 3)
320.5 0 0
0 78.7 0
0 0 77
Elastic Wave Velocity
The Christoffel tensor is the basis for computing the direction dependent wave velocities of the p, s1, and s2 wave, as well as of the polarization directions. Therefore, we need also the density of the material, e.g.,
rho = 3.355
rho =
3.3550
which we can write directly into the elastic stiffness tensor
C = addOption(C,'density',rho)
C = stiffnessTensor (Olivin)
unit : GPa
density: 3.355
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
the single crystal wave velocities are now computed by the command stiffnessTensor.velocity.html velocity
>
[vp,vs1,vs2,pp,ps1,ps2] = velocity(C)
vp = S2FunTri
vertices: 1 x 18338
values: 1 x 18338
vs1 = S2FunTri
vertices: 1 x 18338
values: 1 x 18338
vs2 = S2FunTri
vertices: 1 x 18338
values: 1 x 18338
pp =
S2AxisFieldTri with properties:
tri: [1×1 S2Triangulation]
values: [1×18338 vector3d]
vertices: [1×18338 S2Grid]
antipodal: 0
ps1 =
S2AxisFieldTri with properties:
tri: [1×1 S2Triangulation]
values: [1×18338 vector3d]
vertices: [1×18338 S2Grid]
antipodal: 0
ps2 =
S2AxisFieldTri with properties:
tri: [1×1 S2Triangulation]
values: [1×18338 vector3d]
vertices: [1×18338 S2Grid]
antipodal: 0
As output we obtain three spherical functions vp
, vs1
and vs2
representing the velocities of P, and fast and slow S-waves respectively in dependency of the propagation direction. The remaining three output variables pp
, ps1
, ps2
are spherical vector fields representing the polarization directions of these wave as functions of the propagation direction.
In order to visualize these quantities, there are several possibilities. Let us first plot the direction dependent wave speed of the p-wave
plot(vp,'complete','upper')
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Next, we plot on the top of this plot the p-wave polarization direction.
hold on
plot(pp)
hold off
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We may even compute with these spherical functions as width ordinary values. E.g. to visualize the speed difference between the s1 and s2 waves we do.
plot(vs1-vs2,'complete','upper')
hold on
plot(ps1)
hold off
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When projected to a plane the different wave speeds
planeNormal = vector3d.X;
% options for sections
optSec = {'color','interp','linewidth',6,'doNotDraw'};
% options for quiver
optQuiver = {'linewidth',2,'autoScaleFactor',0.35,'doNotDraw'};
optQuiverProp = {'color','k','linewidth',2,'autoScaleFactor',0.25,'doNotDraw'};
prop = S2VectorFieldHarmonic.normal; % the propagation direction
% wave velocities
%close all
plotSection(vp,planeNormal,optSec{:},'DisplayName','Vp')
hold on
plotSection(vs1,planeNormal,optSec{:},'DisplayName','Vs1')
plotSection(vs2,planeNormal,optSec{:},'DisplayName','Vs2')
% polarisation directions
quiverSection(vp,pp,planeNormal,'color','c',optQuiver{:},'DisplayName','pp')
quiverSection(vs1,ps1,planeNormal,'color','g',optQuiver{:},'DisplayName','ps1')
quiverSection(vs2,ps2,planeNormal,'color','m',optQuiver{:},'DisplayName','ps2')
% plot propagation directions as reference
quiverSection(vp,prop,planeNormal,optQuiverProp{:},'DisplayName','x')
quiverSection(vs1,prop,planeNormal,optQuiverProp{:})
quiverSection(vs2,prop,planeNormal,optQuiverProp{:})
hold off
axis off tight
legend('Vp','Vs1','Vs2','pp','ps1','ps2','x','Location','eastOutSide')
mtexTitle('Phase velocity surface (km/s)')
mtexColorMap blue2red
mtexColorbar('Title','(km/s)','location','southOutSide')
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Similarly, we can visualize the slowness surfaces (s/km)
% plot slowness surfaces
plotSection(1./vp,planeNormal,optSec{:},'DisplayName','Vp')
hold on
plotSection(1./vs1,planeNormal,optSec{:},'DisplayName','Vs1')
plotSection(1./vs2,planeNormal,optSec{:},'DisplayName','Vs2')
% polarisation directions
quiverSection(1./vp,pp,planeNormal,'color','c',optQuiver{:},'DisplayName','pp')
quiverSection(1./vs1,ps1,planeNormal,'color','g',optQuiver{:},'DisplayName','ps1')
quiverSection(1./vs2,ps2,planeNormal,'color','m',optQuiver{:},'DisplayName','ps2')
% plot propagation directions as reference
quiverSection(1./vp,prop,planeNormal,optQuiverProp{:},'DisplayName','x')
quiverSection(1./vs1,prop,planeNormal,optQuiverProp{:})
quiverSection(1./vs2,prop,planeNormal,optQuiverProp{:})
hold off
axis off tight
legend('Vp','Vs1','Vs2','pp','ps1','ps2','x','Location','eastOutSide')
mtexTitle('Slowness surface (km/s)')
mtexColorMap blue2red
mtexColorbar('Title','(s/km)','location','southOutSide')
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set back default colormap
setMTEXpref('defaultColorMap',WhiteJetColorMap)