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MS_effective_medium.m
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MS_effective_medium.m
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% MS_effective_medium - generate elastic constants from various effective
% medium theories.
%
% // Part of MSAT - The Matlab Seismic Anisotropy Toolkit //
%
% Generate elastic constants for the effective medium parameters based on
% various theories, identified by the string 'theory'. Subsequent required
% arguments depend on the theory invoked.
%
% Currently available theories:
%
% ** Tandon and Weng, 84 ('tandon' or 't&w')
% (Isotropic host matrix with unidirectionally aligned isotropic spheroid
% inclusions)
%
% [Ceff,rh]=MS_effective_medium('t&w',vpm,vsm,rhm,vpi,vsi,rhi,del,f) or
% Input parameters:
% vpm,vsm,rhm : isotropic parameters of the matrix (km/s, kg/m3)
% del : aspect ratio of spheroids:
% del < 1 = smarties
% del = 1 = spheres
% del > 1 = cigars
% f : volume fractions of inclusions
% vpi, vsi, rhi : isotropic parameters of the inclusions
%
% [Ceff,rh]=MS_effective_medium('t&w',Cm,rhm,Ci,rhi,del,f) or
% Input parameters:
% Cm,rh : elasticity and density of the matrix (GPa, kg/m3)
% del : aspect ratio of spheroids:
% del < 1 = smarties
% del = 1 = spheres
% del > 1 = cigars
% f : volume fractions of inclusions
% Ci, rhi : isotropic parameters of the inclusions
%
% Output parameters:
% Ceff : Elastic constants (GPa) (symmetry in X1 direction)
% rh : aggregate density (kg/m3)
%
%
% References:
% - Tandon, GP and Weng, GJ. The Effect of Aspect Ratio of Inclusions on the
% Elastic Properties of Unidirectionally Aligned Composites. Polymer
% Composites, 5, pp 327-333, 1984.
%
%
% See also: MS_elasticDB
% Copyright (c) 2011, James Wookey and Andrew Walker
% All rights reserved.
%
% Redistribution and use in source and binary forms,
% with or without modification, are permitted provided
% that the following conditions are met:
%
% * Redistributions of source code must retain the
% above copyright notice, this list of conditions
% and the following disclaimer.
% * Redistributions in binary form must reproduce
% the above copyright notice, this list of conditions
% and the following disclaimer in the documentation
% and/or other materials provided with the distribution.
% * Neither the name of the University of Bristol nor the names
% of its contributors may be used to endorse or promote
% products derived from this software without specific
% prior written permission.
%
% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS
% AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED
% WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
% WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
% PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL
% THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY
% DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
% CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
% PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
% USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
% CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
% CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE
% OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
% SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
function [Ceff,rh]=MS_effective_medium(theory, varargin) ;
if ~ischar(theory)
error('MS:EFFECTIVE_MEDIUM:BadTString', ...
'A string specifying the theory to use is required.') ;
end
switch lower(theory)
%-------------------------------------------------------------------------------
case {'tandon', 't&w'}
if length(varargin)~=8 & length(varargin)~=6
error('MS:EFFECTIVE_MEDIUM:TWWrongArgs', ...
'Tandon and Weng (1984) requires 6 or 8 input parameters.') ;
end
if length(varargin)==6 % elasticity matrix form
Cm = varargin{1} ; rhm = varargin{2} ;
Ci = varargin{3} ; rhi = varargin{4} ;
del = varargin{5} ; f = varargin{6} ;
% ** check the matrices
MS_checkC(Cm) ;
if MS_anisotropy( Cm) > 10*sqrt(eps) % not isotropic
error('MS:EFFECTIVE_MEDIUM:TWBadC', ...
'Tandon and Weng (1984) requires isotropic inputs matrices.') ;
end
MS_checkC(Ci) ;
if MS_anisotropy( Ci ) > 10*sqrt(eps) % not isotropic
error('MS:EFFECTIVE_MEDIUM:TWBadC', ...
'Tandon and Weng (1984) requires isotropic input matrices.') ;
end
% ** unload the
vpm = sqrt(Cm(3,3)*1e3./rhm) ; vsm = sqrt(Cm(6,6)*1e3./rhm) ;
vpi = sqrt(Ci(3,3)*1e3./rhi) ; vsi = sqrt(Ci(6,6)*1e3./rhi) ;
else % velocity form
vpm = varargin{1} ; vsm = varargin{2} ; rhm = varargin{3} ;
vpi = varargin{4} ; vsi = varargin{5} ; rhi = varargin{6} ;
del = varargin{7} ; f = varargin{8} ;
end
[Ceff,rh]=MS_tandon_and_weng(vpm,vsm,rhm,vpi,vsi,rhi,del,f) ;
%-------------------------------------------------------------------------------
otherwise
error('MS:EFFECTIVE_MEDIUM:UnknownTheory', ...
'Specified theory is not supported.') ;
%-------------------------------------------------------------------------------
end % of switch
end
function [CC,rh]=MS_tandon_and_weng(vp,vs,rho,vpi,vsi,rhoi,del,c)
% weighted average density
rh = (1.0-c)*rho + c*rhoi ;
vp = vp * 1e3 ; % convert to m/s
vs = vs * 1e3 ; % convert to m/s
amu = vs*vs*rho ;
amui = vsi*vsi*rhoi ;
alam = vp*vp*rho - 2.0*amu ;
alami = vpi*vpi*rhoi - 2.0*amui ;
bmi = alami + amui*2.0/3.0 ;
bmps = alam + amu ;
% Young's modulus for matrix
E0 = amu*(3.0*alam + 2.0*amu)/(alam + amu) ;
% Poisson's ratio of the matrix.
anu = alam/(2.0*(alam + amu)) ;
% Some time saving terms
t1 = del*del - 1.0 ;
t2 = 1.0 - anu ;
t3 = 1.0 - 2.0*anu ;
t4 = 3.0*del*del ;
t5 = 1.0 - del*del ;
%
% D1, D2 and D3 from Tandon and Weng (1984) (just before equation (18)).
D1 = 1.0 + 2.0*(amui - amu)/(alami - alam) ;
D2 = (alam + 2.0*amu)/(alami - alam) ;
D3 = alam/(alami-alam) ;
%
% g and g' terms (appendix of Tandon and Weng 1984). g is for spheroidal
% inclusions (del>1), whilst g' is for disc-like inclusions (del<1).
%
if (del >= 1)
acshdel = log(del + sqrt(t1)) ;
g =(del*sqrt(t1) - acshdel)*del/(sqrt(t1)^3) ;
else
% g' below
g =(acos(del) - del*sqrt(t5))*del/(sqrt(t5)^3) ;
end
%
% Eshelby's Sijkl tensor (appendix of Tandon and Weng 1984).
%
s11 = (t3 + (t4-1.0)/t1 - (t3 + t4/t1)*g)/(2.0*t2) ;
s22 = (t4/(t1*2.0) + (t3 - 9.0/(4.0*t1))*g)/(4.0*t2) ;
s33 = s22 ;
s23 = (del*del/(2.0*t1) - (t3 + 3.0/(4.0*t1))*g)/(4.0*t2) ;
s32 = s23 ;
s21 = (-2.0*del*del/t1 + (t4/t1 - t3)*g)/(4.0*t2) ;
s31 = s21 ;
s12 = (-1.0*(t3 + 1.0/t1) + (t3 + 3.0/(2.0*t1))*g)/(2.0*t2) ;
s13 = s12 ;
s44 = (del*del/(2.0*t1) + (t3 - 3.0/(4.0*t1))*g)/(4.0*t2) ;
s66 = (t3 - (t1+2.0)/t1 - (t3 - 3.0*(t1+2.0)/t1)*g/2.0)/(4.0*t2) ;
s55 = s66 ;
%
% Tandon and Weng's B terms (after equation 17).
B1 = c*D1 + D2 + (1.0-c)*(D1*s11 + 2.0*s21) ;
B2 = c + D3 + (1.0-c)*(D1*s12 + s22 + s23) ;
B3 = c + D3 + (1.0-c)*(s11 + (1.0+D1)*s21) ;
B4 = c*D1 + D2 + (1.0-c)*(s12 + D1*s22 + s23) ;
B5 = c + D3 + (1.0-c)*(s12 + s22 + D1*s23) ;
%
% Tandon and Weng's A terms (after equation 20).
A1 = D1*(B4 + B5) - 2.0*B2 ;
A2 = (1.0 + D1)*B2 - (B4 + B5) ;
A3 = B1 - D1*B3 ;
A4 = (1.0 + D1)*B1 - 2.0*B3 ;
A5 = (1.0 - D1)/(B4 - B5) ;
A = 2.0*B2*B3 - B1*(B4+B5) ;
%
% Tandon and Weng (1984) equations (25) (28) (31) (32)
E11 = E0 /(1.0+c*(A1+2.0*anu*A2)/A) ;
E22 = E0 ...
/(1.0+c*(-2.0*anu*A3 + (1.0-anu)*A4 + (1.0+anu)*A5*A)/(2.0*A)) ;
amu12 = amu*(1.0 + c/(amu/(amui-amu) + 2.0*(1.0-c)*s66)) ;
amu23 = amu*(1.0 + c/(amu/(amui-amu) + 2.0*(1.0-c)*s44)) ;
%
% Sayers equation (36)
anu31 = anu - c*(anu*(A1+2.0*anu*A2)+(A3-anu*A4)) ...
/(A + c*(A1+2.0*anu*A2)) ;
%
% T&W equation (36)
% aK12 term; bmps=plane strain bulk modulus
anum = (1.0+anu)*(1.0-2.0*anu) ;
denom = 1.0 - anu*(1.0+2.0*anu31) ...
+ c*(2.0*(anu31-anu)*A3 + (1.0-anu*(1.0+2.0*anu31))*A4)/A ;
aK23 = bmps*anum/denom ;
anu12tst = E11/E22 - (1.0/amu23 + 1.0/aK23)*E11/4.0 ;
%
% Cij - Sayers' (1992) equations (24)-(29).
% Conversion
%
CC(2,2) = amu23 + aK23 ;
CC(3,3) = CC(2,2) ;
CC(1,1) = E11 + 4.0*anu12tst*aK23 ;
CC(2,3) = -amu23 + aK23 ;
CC(1,2) = 2.0*anu31*aK23 ;
CC(1,3) = CC(1,2) ;
CC(5,5) = amu12 ;
CC(6,6) = CC(5,5) ;
CC(4,4) = (CC(2,2)-CC(2,3))/2.0 ;
% Fill out matrix by symmetry
% make symmetrical
for i=1:6
for j=i:6
CC(j,i) = CC(i,j) ;
end
end
% convert to GPA
CC = CC./1e9 ;
end