FEATool  v1.9
Finite Element Analysis Toolbox
ex_axistressstrain4.m File Reference

Description

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 fricition 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
-----------------------------------------------------------------------------------
fea         struct                 Problem definition struct
out         struct                 Output struct

Code listing

 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{5,end} = { alpha };
 fea.phys.css.eqn.coef{6,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