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Plot the singular stress components, sxx, syy and txy for a cracked body under Mode I, Mode II and Mode III loading with a stress intensity factor Ki = 10 MPa·m1/2 (i = I, II and III). Plot the stresses near the crack tip (r 2 mm)


Question 1
1) Plot the singular stress components, sxx, syy and txy for a cracked body under Mode I, Mode II and Mode III loading with a stress intensity factor Ki = 10 MPa·m1/2 (i = I, II and III). Plot the stresses near the crack tip (r 2 mm). In order to obtain a meaningful figure, take into account that the stresses for very short r tend to infinity (singularity). It is recommended to use Matlab (file guide_plot2D.m may provide some guidelines).
2) Consider the Mode I case. Assume that the material exhibits a yield stress of 150 MPa. Make use of the Von Misses yield criteria to plot the contour of the zone near the crack tip were the material is expected to yield. Answer this question for i) Plane stress ii) Plane strain
Compare both results in the same plot. Again, it is recommended to use Matlab (file guide_contour2D.m may provide some guidelines). The Von Misses stress criteria in local coordinates reads:
Assume that the Poisson ratio, ?, is 0.3
Question 2
A metallic plate under an uniaxial remote load, has an inclined crack, which is small compared with the dimensions of the plate.
Plot the critical load vs. the inclination angle (ß).
Geometry of the plate:
W = 400 mm
B = 13 mm
2a = 30 mm
Material properties: sys = 320 MPa KIC = 22 MPa·m½
Plot inclination angle (ß) from 0 to 90 every 5 degree.
Matlab help of plot 2D
% Sample program to generate 2D contour plots clear;
% Mesh generation with polar coordinates (r, theta)
% rmin = 1e-5 rmax= 5e-3 increm=100
interval = (rmax-rmin)/increm
[theta,r] = meshgrid (-pi:2*pi/increm:pi+0.1, rmin:interval:rmax);
%
% Sample function
%
sample = sin(theta*2).*(1./(1+100*r));
%
% cartesian coordinates
%
x = r.* cos(theta); y = r.* sin(theta);
%
%
% ———————————————————————— %% Create figure for ‘sample’
figure1 = figure(‘PaperPosition’,[0.6345 6.345 20.3
15.23],’PaperSize’,[20.98 29.68]); %% Create axes
axes1 = axes(‘Layer’,’top’,’Parent’,figure1);
% selection of the plot domain axis(axes1,[-0.005 0.005 -0.005 0.005]); title(axes1,’sample function’); xlabel(axes1,’X (a.u.)’); ylabel(axes1,’Y (a.u.)’); hold(axes1,’all’);
%% Create contour
% with ‘LevelList’ it is decided the number of
contours and at which level
% appears each one. In this case, 25 levels: from -1 to -.96, from -.96 to
% -.92, etc.
contour1 = contour(x,…
y, sample,…
‘DisplayName’,’sample’,…
‘Fill’,’on’,…
‘LevelList’,[-1:0.04:1],…
‘LineColor’,[0 0 0],…
‘LineStyle’,’none’,…
‘XDataSource’,’x’,…
‘YDataSource’,’y’,…
‘ZDataSource’,’sample’); %% Create colorbar colorbar1 = colorbar(‘peer’,…
axes1,’EastOutside’,… ‘Box’,’on’); hold off
% ————————————————————————
Matlab example plot contours
% Sample program to generate 2D contour plots clear;
% Mesh generation with polar coordinates (r, theta)
% rmin = 1e-5 rmax= 5e-3 increm=100
interval = (rmax-rmin)/increm
[theta,r] = meshgrid (-pi:2*pi/increm:pi+0.1,
rmin:interval:rmax);
%
% Sample function
%
sample = sin(theta*2).*(1./(1+100*r));
%
% cartesian coordinates
%
x = r.* cos(theta); y = r.* sin(theta);
% %
% ———————————————————————— %% Create figure for ‘sample’
figure1 = figure(‘PaperPosition’,[0.6345 6.345 20.3
15.23],’PaperSize’,[20.98 29.68]); %% Create axes
axes1 = axes(‘Layer’,’top’,’Parent’,figure1);
% selection of the plot domain axis(axes1,[-0.005 0.005 -0.005 0.005]); title(axes1,’sample function’); xlabel(axes1,’X (a.u.)’); ylabel(axes1,’Y (a.u.)’); hold(axes1,’all’);
%% Create contour
% with ‘LevelList’ it is decided the number of contours and at which level
% appears each one. In this case, 25 levels: from -1 to -.96, from -.96 to
% -.92, etc.
contour1 = contour(x,…
y, sample,…
‘DisplayName’,’sample’,…
‘Fill’,’on’,…
‘LevelList’,[-1:0.04:1],…
‘LineColor’,[0 0 0],…
‘LineStyle’,’none’,…
‘XDataSource’,’x’,…
‘YDataSource’,’y’,…
‘ZDataSource’,’sample’); %% Create colorbar colorbar1 = colorbar(‘peer’,…
axes1,’EastOutside’,… ‘Box’,’on’); hold off
% ————————————————————————
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