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FEATool Multiphysics
v1.16.5
Finite Element Analysis Toolbox
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FEATool Multiphysics
v1.16.5
Finite Element Analysis Toolbox
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EX_AXISTRESSSTRAIN4 Axisymmetric heat induced stress for a brake disk.
[ FEA, OUT ] = EX_AXISTRESSSTRAIN4( VARARGIN ) Quasi-static axisymmetric simulation of a brake disk under braking. The braking process induces heat through friction with the brake pad, which in turn results in stresses in the brake disk.
Ref. A. Adamowicz, Axisymmetric FE Model to Analysis of Thermal Stresses in a Brake Disk, Journal of Theoretical and Applied Mechanics, 53, 2, pp. 357-370, Warsaw 2015.
Accepts the following property/value pairs.
Input Value/{Default} Description
-----------------------------------------------------------------------------------
deltad scalar {5.5e-3} 1/2 Disk thickness
rd scalar {66e-3} Disk inner radius
rp scalar {75.5e-3} Pad inner radius
Rd scalar {113.5e-3} Disk outer radius
sfun string {sflag2} Shape function for displacements
nlev scalar {2} Uniform grid refinement level
iplot scalar 0/{1} Plot solution (=1)
.
Output Value/(Size) Description
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fea struct Problem definition struct
out struct Output struct
cOptDef = { 'deltad', 5.5e-3;
'rd', 66e-3;
'rp', 75.5e-3;
'Rd', 113.5e-3;
'sfun', 'sflag2';
'nlev', 2;
'iplot', 1;
'fid', 1 };
[got,opt] = parseopt(cOptDef,varargin{:});
fid = opt.fid;
% Brake disk material parameters.
kd = 1.44e-5; % Thermal diffsivity [m2/s]
Kd = 51; % Thermal conductivity [W/m/K]
rhod = 7100; % Density [kg/m3]
cpd = Kd/kd/rhod; % Specific heat [J/Kg/K]
E = 99.97e9; % Modulus of elasticity [Pa]
nu = 0.29; % Poisson's ratio
alpha = 1.08e-5; % Coefficient of thermal expansion [1/K]
T0 = 20 + 273.15; % Initial temperature [K]
% Brake pad material parameters.
kp = 1.46e-5; % Thermal diffsivity [m2/s]
Kp = 34.3; % Thermal conductivity [W/m/K]
rhop = 4700; % Density [kg/m3]
cpp = Kp/kp/rhop; % Specific heat [J/Kg/K]
% Simulation parameters.
tmax = 3.96; % Maximum time.
nsteps = 200; % Number of time steps.
% Create computational geometry and grid.
fea.sdim = { 'r' 'z' };
fea.grid = rectgrid( 2^(opt.nlev-1)*45, 2^(opt.nlev-1)*5, [opt.rd opt.Rd;0 opt.deltad] );
[~,ix] = findbdr( fea, ['(r>=',num2str(opt.rp),') & (z>=',num2str(opt.deltad-sqrt(eps)),')'], 0 );
fea.grid.b(3,ix) = max(fea.grid.b(3,:)) + 1; % Add boundary for brake pad.
n_bdr = max(fea.grid.b(3,:)); % Number of boundaries.
% Equations and problem definition.
fea = addphys( fea, @axistressstrain );
fea.phys.css.eqn.coef{1,end} = { nu };
fea.phys.css.eqn.coef{2,end} = { E };
fea.phys.css.eqn.coef{6,end} = { alpha };
fea.phys.css.eqn.coef{7,end} = { ['T-',num2str(T0)] };
fea.phys.css.sfun = { opt.sfun opt.sfun };
fea = addphys( fea, {@heattransfer, 1} );
fea.phys.ht.eqn.coef{1,end} = { rhod };
fea.phys.ht.eqn.coef{2,end} = { cpd };
fea.phys.ht.eqn.coef{3,end} = { Kd };
fea.phys.ht.sfun = { opt.sfun };
% Equation coefficients, constants, and expressions.
fea.expr = { 'eta' [] [] {} ;
'f' [] [] {0.5} ;
'omega' [] [] {[]} ;
'p0' [] [] {1.47e6} };
% Boundary conditions.
bctype = mat2cell( zeros(2,n_bdr), [1 1], ones(1,n_bdr) );
bctype{2,1} = 1;
fea.phys.css.bdr.coef{1,5} = bctype;
eta = sqrt(Kd*rhod*cpd) / (sqrt(Kd*rhod*cpd) + sqrt(Kp*rhop*cpp));
f = 0.5;
p0 = 1.47e6;
qd = [num2str(eta*f*88.464*p0/(2*pi-0.8)),'*r*(1-t/',num2str(tmax),')'];
fea.phys.ht.bdr.sel(5) = 4;
fea.phys.ht.bdr.coef{4,end}{5}{1} = qd;
% Solve problem.
fea = parsephys( fea );
fea = parseprob( fea );
[fea.sol.u,tlist] = solvetime( fea, 'ischeme', 2, 'icub', 3, 'init', { 0 0 T0 }, 'tmax', tmax, 'tstep', tmax/nsteps, 'fid', fid );
% Postprocessing.
sr = fea.phys.css.eqn.vars{5,end};
st = fea.phys.css.eqn.vars{6,end};
svm = fea.phys.css.eqn.vars{1,end};
if( opt.iplot>0 )
figure
solnum = numel(tlist);
niso = 25;
subplot(2,2,1)
postplot( fea, 'surfexpr', 'T-273.15', 'isoexpr', 'T-273.15', 'isolev', niso, 'solnum', solnum )
title(['Temperature [deg C], t = ',num2str(tlist(solnum)),' s'])
subplot(2,2,2)
postplot( fea, 'surfexpr', ['(',sr,')*1e-6'], 'isoexpr', ['(',sr,')*1e-6'], 'isolev', niso, 'solnum', solnum )
title('radial stress [MPa]')
subplot(2,2,3)
postplot( fea, 'surfexpr', ['(',st,')*1e-6'], 'isoexpr', ['(',st,')*1e-6'], 'isolev', niso, 'solnum', solnum )
title('tangential stress [MPa]')
subplot(2,2,4)
postplot( fea, 'surfexpr', ['(',svm,')*1e-6'], 'isoexpr', ['(',svm,')*1e-6'], 'isolev', niso, 'solnum', solnum )
title('von Mieses stress [MPa]')
figure
u_stored = fea.sol.u;
np = 100;
times = [0.1 0.2 1 2 3 tmax];
rz = [ linspace(opt.rd,opt.Rd,np); opt.deltad*ones(1,np) ];
for i=1:numel(times)
t_i = times(i);
ix = max( find( tlist<t_i ) );
s = ( tlist(ix+1) - t_i )/( tlist(ix+1) - tlist(ix) );
fea.sol.u = u_stored(:,ix) + s*( u_stored(:,ix+1) - u_stored(:,ix) );
subplot(2,2,1)
T_i = evalexpr( 'T-273.15', rz, fea );
plot( rz(1,:), T_i ), hold on, grid on
text( rz(1,end), T_i(end), [num2str(t_i),' s'] )
title('Temperature [deg C]')
axis tight
subplot(2,2,2)
sr_i = evalexpr( ['(',sr,')*1e-6'], rz, fea );
plot( rz(1,:), sr_i ), hold on, grid on
text( rz(1,end-floor(np/4)), sr_i(end-floor(np/4)), [num2str(t_i),' s'] )
title('radial stress [MPa]')
axis tight
subplot(2,2,3)
st_i = evalexpr( ['(',st,')*1e-6'], rz, fea );
plot( rz(1,:), st_i ), hold on, grid on
text( rz(1,end-floor(np/4)), st_i(end-floor(np/4)), [num2str(t_i),' s'] )
title('tangential stress [MPa]')
axis tight
subplot(2,2,4)
svm_i = evalexpr( ['(',svm,')*1e-6'], rz, fea );
plot( rz(1,:), svm_i ), hold on, grid on
text( rz(1,end-floor(np/4)), svm_i(end-floor(np/4)), [num2str(t_i),' s'] )
title('von Mieses stress [MPa]')
axis tight
end
fea.sol.u = u_stored;
figure
hold on
npth = 72;
th = linspace( 0, 2*pi, npth );
x = opt.rd*cos(th);
z = opt.rd*sin(th);
y = opt.deltad*ones(size(x));
plot3(x, y,z,'k-')
plot3(x,-y,z,'k-')
x = opt.Rd*cos(th);
z = opt.Rd*sin(th);
plot3(x, y,z,'k-')
plot3(x,-y,z,'k-')
npr = 25;
g = ringgrid( npr-1, npth-1, opt.rd, opt.Rd );
r = sqrt( g.p(1,:).^2 + g.p(2,:).^2 );
T = evalexpr( 'T-273.15', [r;zeros(size(r))], fea );
h = patch( 'faces', g.c', 'vertices', [g.p(1,:)' zeros(size(g.p,2),1) g.p(2,:)'], 'facevertexcdata', T, 'facecolor', 'interp', 'linestyle', 'none' );
postplot( fea, 'surfexpr', 'T-273.15', 'colorbar', 'off' )
fea.grid.p(2,:) = -fea.grid.p(2,:);
postplot( fea, 'surfexpr', 'T-273.15', 'colorbar', 'off' )
fea.grid.p(2,:) = -fea.grid.p(2,:);
view(3)
axis off
axis tight
end
out = [];
[Tmin,Tmax] = minmaxsubd( 'T-273.15', fea );
Terr = [ abs((34.179-Tmin)/34.179) abs((88.224-Tmax)/88.224) ];
[stmin,stmax] = minmaxsubd( ['(',st,')*1e-6'], fea );
sterr = [ abs((-21.75-stmin)/-21.75) abs((45.1-stmax)/45.1) ];
[svmmin,svmmax] = minmaxsubd( ['(',svm,')*1e-6'], fea );
svmerr = [ abs((4.478-svmmin)/4.478) abs((46.4-svmmax)/46.4) ];
out.T = [Tmin Tmax];
out.Terr = Terr;
out.st = [stmin stmax];
out.sterr = sterr;
out.sv = [svmmin svmmax];
out.svmerr = svmerr;
out.pass = ~any( [ Terr>0.12 sterr>0.03 svmerr>[0.3 0.01] ] );
if( nargout==0 )
clear fea out
end