function varargout=okada85(varargin)
%OKADA85 Surface deformation due to a finite rectangular source.
% [uE,uN,uZ,uZE,uZN,uNN,uNE,uEN,uEE] = OKADA85(E,N,...
% DEPTH,STRIKE,DIP,LENGTH,WIDTH,RAKE,SLIP,OPEN,NU)
% computes Okada's 1985 solution for displacements, tilts and strains in a
% geographic referential (East,North,Up). Coordinates (0,0) and depth
% correspond to the fault centroid.
%
% E,N : Coordinates of observation points (relative to fault centroid)
% DEPTH : Depth of the fault centroid (DEP > 0)
% STRIKE : Strike-angle from North (in degrees)
% DIP : Dip-angle (in degrees, must be scalar)
% LENGTH : Fault length in strike direction (LEN > 0)
% WIDTH : Fault width in dip direction (WIDTH > 0)
% RAKE : Slip-angle direction on fault plane (in degrees)
% SLIP : Dislocation in rake direction (length unit)
% OPEN : Dislocation in tensile component (same unit as SLIP)
% NU : Poisson's ratio (optional, default is 0.25)
%
% returns the following variables (same matrix size as E and N):
% uN,uE,uZ : Displacements (unit of SLIP and OPEN)
% uZE,uZN : Tilts (in radian)
% uNN,uNE,uEN,uEE : Strains (unit of SLIP and OPEN)/(Unit of N,E,..,WIDTH)
% POSITIVE = COMPRESSION
%
% Length unit consistency: E, N, DEPTH, LENGTH and WIDTH must have the same unit
% (e.g. km), which can be different from that of SLIP and OPEN (e.g. m).
%
% It is also possible to produce partial outputs, with following syntax:
% [uE,uN,uZ] = OKADA85(...) for displacements only;
% [uE,uN,uZ,uZE,uZN] = OKADA85(...) for displacements and tilts;
% [uE,uN,uZ,uNN,uNE,uEN,uEE] = OKADA85(...) for displacements and strains;
% [uZE,uZN] = OKADA85(...) for tilts only;
% [uZE,uZN,uNN,uNE,uEN,uEE] = OKADA85(...) for tilts and strains;
% [uNN,uNE,uEN,uEE] = OKADA85(...) for strains only.
%
% [...] = OKADA85(...,'plot') or OKADA85(...) without output argument produces
% a 3-D figure with fault geometry and dislocation at scale (if all of the fault
% parameters are scalar).
%
% Example:
% [E,N] = meshgrid(linspace(-10,10,50));
% [uE,uN,uZ] = okada85(E,N,2,30,70,5,3,-45,1,1,'plot');
% figure, surf(E,N,uN)
% considers a 5x3 fault at depth 2, N30-strike, 70-dip, and unit dislocation
% in all directions (reverse, senestral and open). Displacements are computed
% on a regular grid from -10 to 10, and North displacements are plotted.
%
% Equations are all vectorized excepted for argument DIP which has to be
% a scalar; all other arguments can be scalar or matrix of the same size.
%
% Formulas and notations similar to Okada [1985], excepted for strain convention
% (here positive strain means compression), and for fault parameters
% after Aki & Richards [1980]:
% STRIKE = fault trace direction (0 to 360 relative to North),
% defined so that the fault dips to the right side of the trace.
% DIP = angle of the fault (0 to 90, relative to horizontal).
% RAKE = direction the hanging wall moves during rupture, measured
% relative to the fault strike (-180 to 180).
% Examples:
% DIP=90, RAKE=0 : left lateral (senestral) strike slip
% DIP=90, RAKE=180 : right lateral (dextral) strike slip
% DIP=70, RAKE=90 : reverse fault
% DIP=70, RAKE=-90 : normal fault
%
%
% Author: Franois Beauducel <beauducel@ipgp.fr>
% Institut de Physique du Globe de Paris
% Created: 1997
% Updated: 2011-03-06
%
% References:
% Aki K., and P. G. Richards, Quantitative seismology, Freemann & Co,
% New York, 1980.
% Okada Y., Surface deformation due to shear and tensile faults in a
% half-space, Bull. Seismol. Soc. Am., 75:4, 1135-1154, 1985.
%
% Acknowledgments: Dmitry Nicolsky, University of Alaska
% Development history:
% [2011-03-06]: new optional argument to plot fault geometry with
% output arguments, and bug correction for the fault centroid position
% (in calculation and plot).
% [2010-11-29]: change coordinates and depth to fault centroid
% (instead of middle top edge).
% [2010-09-24]: bugs correction in the syntax of I1, K2 and uyy_tf
% functions, affecting some components. Detected by Dmitry Nicolsky.
%
% Copyright (c) 1997-2010, Franois Beauducel, covered by BSD License.
% 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
%
% 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 OWNER 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.
error(nargchk(10,12,nargin))
for i = 1:10
if ~isnumeric(varargin{i})
error('Input arguments E,N,DEPTH,STRIKE,DIP,LENGTH,WIDTH,RAKE,SLIP,OPEN must be numeric.')
end
end
if numel(varargin{5}) ~= 1
error('DIP argument must be scalar.')
end
% Default values for optional input arguments
plotflag = 0; % no plot
nu = 0.25; % isotropic Poisson's ratio
% Assigns input arguments
e = varargin{1};
n = varargin{2};
depth = varargin{3};
strike = varargin{4}*pi/180; % converting STRIKE in radian
dip = varargin{5}*pi/180; % converting DIP in radian ('delta' in Okada's equations)
L = varargin{6};
W = varargin{7};
rake = varargin{8}*pi/180; % converting RAKE in radian
slip = varargin{9};
U3 = varargin{10};
switch nargin
case 11
if isnumeric(varargin{11})
nu = varargin{11};
else
makeplot = varargin{11};
end
case 12
makeplot = varargin{12};
end
if exist('makeplot','var')
if strcmp(makeplot,'plot')
plotflag = 1;
else
error('Unknown last argument.')
end
end
if plotflag & any([numel(depth),numel(strike),numel(L),numel(W),numel(rake),numel(slip),numel(U3)]>1)
warning('Cannot make plot with fault geometry parameters other than scalars.')
plotflag = 0;
end
% Defines dislocation in the fault plane system
U1 = cos(rake).*slip;
U2 = sin(rake).*slip;
% Converts fault coordinates (E,N,DEPTH) relative to centroid
% into Okada's reference system (X,Y,D)
d = depth + sin(dip).*W/2; % d is fault's top edge
ec = e + cos(strike)*cos(dip).*W/2;
nc = n - sin(strike)*cos(dip).*W/2;
x = cos(strike).*nc + sin(strike).*ec + L/2;
y = sin(strike).*nc - cos(strike).*ec + cos(dip).*W;
% Variable substitution (independent from xi and eta)
p = y.*cos(dip) + d.*sin(dip);
q = y.*sin(dip) - d.*cos(dip);
% Displacements
if any(nargout==[3, 5, 7, 9])
ux = -U1/(2*pi) * chinnery(@ux_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@ux_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@ux_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uy = -U1/(2*pi) * chinnery(@uy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uz = -U1/(2*pi) * chinnery(@uz_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uz_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uz_tf,x,p,L,W,q,dip,nu); ... % tensile fault
% Rotation from Okada's axes to geographic
ue = sin(strike).*ux - cos(strike).*uy;
un = cos(strike).*ux + sin(strike).*uy;
end
% Tilt
if any(nargout==[2, 5, 6, 9])
uzx = -U1/(2*pi) * chinnery(@uzx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uzx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uzx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uzy = -U1/(2*pi) * chinnery(@uzy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uzy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uzy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
% Rotation from Okada's axes to geographic
uze = -sin(strike).*uzx + cos(strike).*uzy;
uzn = -cos(strike).*uzx - sin(strike).*uzy;
end
% Strain
if any(nargout==[4, 6, 7, 9])
uxx = -U1/(2*pi) * chinnery(@uxx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uxx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uxx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uxy = -U1/(2*pi) * chinnery(@uxy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uxy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uxy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uyx = -U1/(2*pi) * chinnery(@uyx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uyx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uyx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uyy = -U1/(2*pi) * chinnery(@uyy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) * chinnery(@uyy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) * chinnery(@uyy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
% Rotation from Okada's axes to geographic
unn = cos(strike)^2*uxx + sin(2*strike)*(uxy + uyx)/2 + sin(strike)^2*uyy;
une = sin(2*strike)*(uxx - uyy)/2 + sin(strike)^2*uyx - cos(strike)^2*uxy;
uen = sin(2*strike)*(uxx - uyy)/2 - cos(strike)^2*uyx + sin(strike)^2*uxy;
uee = sin(strike)^2*uxx - sin(2*strike)*(uyx + uxy)/2 + cos(strike)^2*uyy;
end
% Assigns output arguments
switch nargout
case 2
varargout = {uze, uzn};
case 3
varargout = {ue, un, uz};
case 4
varargout = {unn, une, uen, uee};
case 5
varargout = {ue, un, uz, uze, uzn};
case 6
varargout = {uze, ezn, unn, une, uen, uee};
case 7
varargout = {ue, un, uz, unn, une, uen, uee};
case 9
varargout = {ue, un, uz, uze, uzn, unn, une, uen, uee};
case 0
plotflag = 1;
otherwise
disp('Unvalid number of output arguments.')
end
% no output argument: plots geometry of the fault and dislocation
if plotflag
figure
plot(e,n,'.r','MarkerSize',.1)
alpha = pi/2 - strike;
x_fault = L/2*cos(alpha)*[-1,1,1,-1] + sin(alpha)*cos(dip)*W/2*[-1,-1,1,1];
y_fault = L/2*sin(alpha)*[-1,1,1,-1] + cos(alpha)*cos(dip)*W/2*[1,1,-1,-1];
z_fault = -d + sin(dip)*W*[1,1,0,0];
ddx = U1*cos(alpha) - U2*sin(alpha)*cos(dip) + U3*sin(alpha)*sin(dip);
ddy = U1*sin(alpha) + U2*cos(alpha)*cos(dip) - U3*cos(alpha)*sin(dip);
ddz = U2*sin(dip) + U3*cos(dip);
patch(x_fault,y_fault,z_fault,.3*[1,1,1],'EdgeColor','k','LineWidth',2)
patch(x_fault+ddx/2,y_fault+ddy/2,z_fault+ddz/2,.6*[1,1,1], ...
'EdgeColor','k','LineWidth',1,'FaceAlpha',.5)
patch(x_fault-ddx/2,y_fault-ddy/2,z_fault-ddz/2,.6*[1,1,1], ...
'EdgeColor','k','LineWidth',1,'FaceAlpha',.5)
xlabel('East'), ylabel('North'), zlabel('Vertical')
view(3), grid on, axis equal, rotate3d
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Notes for I... and K... subfunctions:
%
% 1. original formulas use Lame's constant as mu/(mu+lambda) which
% depends only on the Poisson's ratio = 1 - 2*nu
% 2. tests for cos(dip) == 0 are made with "cos(dip) > eps"
% because cos(90*pi/180) is not zero but = 6.1232e-17 (!)
% NOTE: don't use cosd and sind because of incompatibility
% with Matlab v6 and earlier...
% =================================================================
% Chinnery's notation [equation (24) p. 1143]
% -----------------------------------------------------------------
function u=chinnery(f,x,p,L,W,q,dip,nu)
u = feval(f,x,p,q,dip,nu) ...
- feval(f,x,p-W,q,dip,nu) ...
- feval(f,x-L,p,q,dip,nu) ...
+ feval(f,x-L,p-W,q,dip,nu);
% =================================================================
% Displacement subfunctions
% strike-slip displacement subfunctions [equation (25) p. 1144]
% -----------------------------------------------------------------
function u=ux_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./(R.*(R + eta)) ...
+ atan(xi.*eta./(q.*R)) ...
+ I1(xi,eta,q,dip,nu,R).*sin(dip);
% -----------------------------------------------------------------
function u=uy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = (eta*cos(dip) + q*sin(dip)).*q./(R.*(R + eta)) ...
+ q.*cos(dip)./(R + eta) ...
+ I2(eta,q,dip,nu,R).*sin(dip);
% -----------------------------------------------------------------
function u=uz_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = (eta*sin(dip) - q*cos(dip)).*q./(R.*(R + eta)) ...
+ q.*sin(dip)./(R + eta) ...
+ I4(db,eta,q,dip,nu,R).*sin(dip);
% dip-slip displacement subfunctions [equation (26) p. 1144]
% -----------------------------------------------------------------
function u=ux_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q./R ...
- I3(eta,q,dip,nu,R).*sin(dip).*cos(dip);
% -----------------------------------------------------------------
function u=uy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = (eta*cos(dip) + q*sin(dip)).*q./(R.*(R + xi)) ...
+ cos(dip).*atan(xi.*eta./(q.*R)) ...
- I1(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);
% -----------------------------------------------------------------
function u=uz_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = db.*q./(R.*(R + xi)) ...
+ sin(dip).*atan(xi.*eta./(q.*R)) ...
- I5(xi,eta,q,dip,nu,R,db).*sin(dip).*cos(dip);
% tensile fault displacement subfunctions [equation (27) p. 1144]
% -----------------------------------------------------------------
function u=ux_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2 ./(R.*(R + eta)) ...
- I3(eta,q,dip,nu,R).*sin(dip).^2;
% -----------------------------------------------------------------
function u=uy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = -(eta*sin(dip) - q*cos(dip)).*q./(R.*(R + xi)) ...
- sin(dip).*(xi.*q./(R.*(R + eta)) ...
- atan(xi.*eta./(q.*R))) ...
- I1(xi,eta,q,dip,nu,R).*sin(dip).^2;
% -----------------------------------------------------------------
function u=uz_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = (eta*cos(dip) + q*sin(dip)).*q./(R.*(R + xi)) ...
+ cos(dip).*(xi.*q./(R.*(R + eta)) ...
- atan(xi.*eta./(q.*R))) ...
- I5(xi,eta,q,dip,nu,R,db).*sin(dip).^2;
% I... displacement subfunctions [equations (28) (29) p. 1144-1145]
% -----------------------------------------------------------------
function I=I1(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
I = (1 - 2*nu) * (-xi./(cos(dip).*(R + db))) - sin(dip)./cos(dip) .*I5(xi,eta,q,dip,nu,R,db);
else
I = -(1 - 2*nu)/2 * xi.*q./(R + db).^2;
end
% -----------------------------------------------------------------
function I=I2(eta,q,dip,nu,R)
I = (1 - 2*nu) * (-log(R + eta)) - I3(eta,q,dip,nu,R);
% -----------------------------------------------------------------
function I=I3(eta,q,dip,nu,R)
yb = eta*cos(dip) + q*sin(dip);
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
I = (1 - 2*nu) * (yb./(cos(dip)*(R + db)) - log(R + eta)) ...
+ sin(dip)./cos(dip) * I4(db,eta,q,dip,nu,R);
else
I = (1 - 2*nu)/2 * (eta./(R + db) + yb.*q./(R + db).^2 - log(R + eta));
end
% -----------------------------------------------------------------
function I=I4(db,eta,q,dip,nu,R)
if cos(dip) > eps
I = (1 - 2*nu) * 1/cos(dip) * (log(R + db) - sin(dip)*log(R + eta));
else
I = -(1 - 2*nu) * q./(R + db);
end
% -----------------------------------------------------------------
function I=I5(xi,eta,q,dip,nu,R,db)
X = sqrt(xi.^2 + q.^2);
if cos(dip) > eps
I = (1 - 2*nu) * 2./cos(dip) ...
.* atan((eta.*(X + q.*cos(dip)) + X.*(R + X).*sin(dip))./(xi.*(R + X).*cos(dip)));
else
I = -(1 - 2*nu) * xi.*sin(dip)./(R + db);
end
% =================================================================
% Tilt subfunctions
% strike-slip tilt subfunctions [equation (37) p. 1147]
% -----------------------------------------------------------------
function u=uzx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = -xi.*q.^2.*A(eta,R).*cos(dip) ...
+ ((xi.*q)./R.^3 - K1(xi,eta,q,dip,nu,R))*sin(dip);
% -----------------------------------------------------------------
function u=uzy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = (db.*q./R.^3).*cos(dip) ...
+ (xi.^2.*q.*A(eta,R)*cos(dip) - sin(dip)./R + yb.*q./R.^3 - K2(xi,eta,q,dip,nu,R))*sin(dip);
% dip-slip tilt subfunctions [equation (38) p. 1147]
% -----------------------------------------------------------------
function u=uzx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = db.*q./R.^3 ...
+ q*sin(dip)./(R.*(R + eta)) ...
+ K3(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uzy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*db.*q.*A(xi,R) ...
- (2*db./(R.*(R + xi)) + xi*sin(dip)./(R.*(R + eta)))*sin(dip) ...
+ K1(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% tensile fault tilt subfunctions [equation (39) p. 1147]
% -----------------------------------------------------------------
function u=uzx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2./R.^3*sin(dip) ...
- q.^3.*A(eta,R)*cos(dip) ...
+ K3(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uzy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = (yb*sin(dip) + db*cos(dip)).*q.^2.*A(xi,R) ...
+ xi.*q.^2.*A(eta,R)*sin(dip)*cos(dip) ...
- (2*q./(R.*(R + xi)) - K1(xi,eta,q,dip,nu,R))*sin(dip)^2;
% -----------------------------------------------------------------
function a=A(x,R)
a = (2*R + x)./(R.^3.*(R + x).^2);
% K... tilt subfunctions [equations (40) (41) p. 1148]
% -----------------------------------------------------------------
function K=K1(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
K = (1 - 2*nu) * xi/cos(dip) .* (1./(R.*(R + db)) - sin(dip)./(R.*(R + eta)));
else
K = (1 - 2*nu) * xi.*q./(R + db).^2;
end
% -----------------------------------------------------------------
function K=K2(xi,eta,q,dip,nu,R)
K = (1 - 2*nu) * (-sin(dip)./R + q*cos(dip)./(R.*(R + eta))) - K3(xi,eta,q,dip,nu,R);
% -----------------------------------------------------------------
function K=K3(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
if cos(dip) > eps
K = (1 - 2*nu) * 1/cos(dip) .* (q./(R.*(R + eta)) - yb./(R.*(R + db)));
else
K = (1 - 2*nu) * sin(dip)./(R + db) .* (xi.^2./(R.*(R + db)) - 1);
end
% =================================================================
% Strain subfunctions
% strike-slip strain subfunctions [equation (31) p. 1145]
% -----------------------------------------------------------------
function u=uxx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.^2.*q.*A(eta,R) ...
- J1(xi,eta,q,dip,nu,R)*sin(dip);
% -----------------------------------------------------------------
function u=uxy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = xi.^3.*db./(R.^3.*(eta.^2 + q.^2)) ...
- (xi.^3.*A(eta,R) + J2(xi,eta,q,dip,nu,R))*sin(dip);
% -----------------------------------------------------------------
function u=uyx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./R.^3*cos(dip) ...
+ (xi.*q.^2.*A(eta,R) - J2(xi,eta,q,dip,nu,R))*sin(dip);
% -----------------------------------------------------------------
function u=uyy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*q./R.^3*cos(dip) ...
+ (q.^3.*A(eta,R)*sin(dip) - 2*q*sin(dip)./(R.*(R + eta)) ...
- (xi.^2 + eta.^2)./R.^3*cos(dip) - J4(xi,eta,q,dip,nu,R))*sin(dip);
% dip-slip strain subfunctions [equation (32) p. 1146]
% -----------------------------------------------------------------
function u=uxx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./R.^3 ...
+ J3(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uxy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*q./R.^3 ...
- sin(dip)./R ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uyx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.*q./R.^3 ...
+ q*cos(dip)./(R.*(R + eta)) ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% -----------------------------------------------------------------
function u=uyy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta*cos(dip) + q*sin(dip);
u = yb.^2.*q.*A(xi,R) ...
- (2*yb./(R.*(R + xi)) + xi*cos(dip)./(R.*(R + eta)))*sin(dip) ...
+ J2(xi,eta,q,dip,nu,R)*sin(dip)*cos(dip);
% tensile fault strain subfunctions [equation (33) p. 1146]
% -----------------------------------------------------------------
function u=uxx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q.^2.*A(eta,R) ...
+ J3(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uxy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
u = -db.*q./R.^3 ...
- xi.^2.*q.*A(eta,R)*sin(dip) ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uyx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2./R.^3*cos(dip) ...
+ q.^3.*A(eta,R)*sin(dip) ...
+ J1(xi,eta,q,dip,nu,R)*sin(dip)^2;
% -----------------------------------------------------------------
function u=uyy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
u = (yb*cos(dip) - db*sin(dip)).*q.^2.*A(xi,R) ...
- q*sin(2*dip)./(R.*(R + xi)) ...
- (xi.*q.^2.*A(eta,R) - J2(xi,eta,q,dip,nu,R))*sin(dip)^2;
% J... tensile fault subfunctions [equations (34) (35) p. 1146-1147]
% -----------------------------------------------------------------
function J=J1(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
if cos(dip) > eps
J = (1 - 2*nu) * 1/cos(dip) * (xi.^2./(R.*(R + db).^2) - 1./(R + db)) ...
- sin(dip)/cos(dip)*K3(xi,eta,q,dip,nu,R);
else
J = (1 - 2*nu)/2 * q./(R + db).^2 .* (2*xi.^2./(R.*(R + db)) - 1);
end
% -----------------------------------------------------------------
function J=J2(xi,eta,q,dip,nu,R)
db = eta*sin(dip) - q*cos(dip);
yb = eta*cos(dip) + q*sin(dip);
if cos(dip) > eps
J = (1 - 2*nu) * 1/cos(dip) * xi.*yb./(R.*(R + db).^2) ...
- sin(dip)/cos(dip)*K1(xi,eta,q,dip,nu,R);
else
J = (1 - 2*nu)/2 * xi*sin(dip)./(R + db).^2 .* (2*q.^2./(R.*(R + db)) - 1);
end
% -----------------------------------------------------------------
function J=J3(xi,eta,q,dip,nu,R)
J = (1 - 2*nu) * -xi./(R.*(R + eta)) ...
- J2(xi,eta,q,dip,nu,R);
% -----------------------------------------------------------------
function J=J4(xi,eta,q,dip,nu,R)
J = (1 - 2*nu) * (-cos(dip)./R - q*sin(dip)./(R.*(R + eta))) ...
- J1(xi,eta,q,dip,nu,R);
