The strain state at a point on a surface is given by three strain components. Us

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The strain state at a point on a surface is given by three strain components. Usually these are denoted epsilon_x, epsilon_y and lambda_xy. The numerical values depend on the orientation of the coordinate system. Change the orientation and you get other values even though the values represent the same state of strain. This is analysed and visualized with Mohr's circle. Using Mohr's strain circle it is possible to calculate the strain state in an arbitrarily oriented coordinate system at a point where epsilon_x, epsilon_y, and lambda_xy are known in one coordinate system. Techniques to measure the state of strain are based on these facts. It is possible measure the normal strain with a strain gauge, cf. http;//en.wikipedia.Qrg/wiki/Strain gauge If we use a strain gauge with a specific orientation we measure the value of the normal strain in that direction. It is obviously not possible to determine three strain components epsilon_x, epsilon_y and lambda_xy from one strain component, cf. the course in Linear Algebra. We only get one equation from Mohr's strain circle to determine the three strain components. The trick is to use three strain gauges at the point For each measured value we get one equation from Mohr's strain circle. With three measured values, we get three linear equations for the three unknowns epsilon_x, epsilon_y and lambda_xy. Write a MATLAB function that can be used to calculate the state of strain from measurements of three normal strain components measured in three directions at a point. Input to the program is three normal strain components (epsilon_1, epsilon_2, epsilon_3) and the corresponding directions (alpha_1, alpha_2, alpha_3) in which the strains are measured. The angles are measured in degrees counter clockwise relative the x axis. Output of the program is the state of strain epsilon_x, epsilon_y, and lambda_xy. Write a MATLAB function that, for a linearly elastic material and a state of plane stress, calculates the matrix representation of the stress tensor if the strains in the xy plane and material parameters are known. Input to the program is the three strain components (epsilon_x, epsilon_y, lambda_xy), Young's modulus E and Poisson's ratio v. Output from the program is the matrix representation of the stress tensor S. Write a MATLAB function that for a given state of stress calculates the effective stress due to von Mises. Input to the program is the matrix representation of the stress tensor S. Output from the program is the von Mises effective stress sigma^M_E.

Explanation / Answer

Matlab Code

function [ ] = plane( exx,eyy,txy )
E = 200*10^9; %youngs modulus
v = 0.3;      %poissons ratio
sigmax = E/(1-v^2) * (exx + v*eyy);   %plane stress in x direction
sigmay = E/(1-v^2) * (v*exx + eyy);   %plane stress in y direction
sigmaxy = E/(2*(1+v))*txy;   %plane stress in xy direction
disp(sigmax);
disp(sigmay);
disp(sigmaxy);
end

Sample Output

>> plane(10^-6,10^-6,10^-6)
   2.8571e+05

   2.8571e+05

   7.6923e+04

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