Using matlab, the stress fields near a crack tip of a linear elastic isotropic m

ID: 3862413 • Letter: U

Question

Using matlab, the stress fields near a crack tip of a linear elastic isotropic material for mode I loading are given by...

The stresses fields near a crack tip of a linear elastic isotropic material for mode I loading are given by: 30 sin sin cos A/2Tr e) 30 I sin sin COS 20) CoS sin cos xy For K 300 ksivin plot the stresses (each in a separate figure) in the domain 0 s 0 s 90° and 0.02 s r s 0.2 in. Plot the coordinates x and y in the horizontal plane, and the stresses in the vertical direction. Use one figure with three subplots. Label the axis, show grids, provide titles, add colorbar to each subplot.

Explanation / Answer

Solution:

st_sigma=300;%in MPa
st_a=5;%in mm

st_K1=st_sigma*sqrt(pi*st_a);

% range of r
st_r=[0.02,0.01:0.01:0.2]*st_a;
% range of theta
st_th=(0:1:90)*pi/180;
[st_TH,st_R]=meshgrid(st_th,st_r);
[X_val,Y_val]=pol2cart(st_TH,st_R);

%Plotting stress Y
stsigma_y=(st_K1./sqrt(2*pi*st_R)) .* cos(st_TH/2) .* (1+sin(st_TH/2).*sin(3*st_TH/2) );
stsigma_x=(st_K1./sqrt(2*pi*st_R)) .* cos(st_TH/2) .* (1-sin(st_TH/3).*sin(3*st_TH/2) );
sttau_xy =(st_K1./sqrt(2*pi*st_R)) .* sin(st_TH/2) .* cos(3*st_TH/2) ;

c_stress={stsigma_x,stsigma_y,sttau_xy};
p_title={'$stsigma_x$','$stsigma_y$','$ au_{xy}$'}
for vi=1:3
    figure(vi);
    vh=surf(X_val,Y_val,c_stress{vi});
    set(vh,'edgecolor','none','facecolor','interp');
    axis square
    l_axis=axis;
    line([-2*st_a 0],[0 0],'linewidth',2,'color','k')
    axis(l_axis)
    view(2)
    colorbar
    title({['Contour plot of ',p_title{vi}],['a=',num2str(st_a),'mm; $sigma=$',num2str(st_sigma),'MPa;']},'fontsize',15,'interpreter','latex')
    xlabel('x1 (in mm)','fontsize',15)
    ylabel('x2 (in mm)','fontsize',15)
end

Result:

0.6 |---+....:.........:.......:.........:....+---| | |

x2 (in mm) : : : : | | |

0.4 |---+....:.........:.......:.........:....+---| |-|1500

| : : : : | | |

0.2 |---+....:.........:.......:.........:....+---| | |

| : : : : | |-|1000

| + + + + | | |

| | | | | | | |

0 *---------------------------------------------+ | |

0 0.2 0.4 0.6 0.8 1 +-+500

x1 (in mm)

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Using matlab, the stress fields near a crack tip of a linear elastic isotropic m

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