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Applied Elasticity and Plasticity Project 
Use a computer program to draw the yield surfaces of the yield criteria given in the table below (a) on deviatoric planes at ρ = 0 and four other different values, where ρ = I_{1}/√3 = √3σ_{m} = √3σ_{oct}; (b) on meridain planes at θ = 0^{o}, 15^{o}, 30^{o}, 45^{o}, and 60^{o}. (c) Also draw the 3D perspective views of the yield surfaces in the principalstress space from three different angles.
Yield Criteria

Uniaxial Compressive Yield Stress

Uniaxial Tensile Yield Stress

von Mises

50,000 psi

50,000 psi

MohrCoulomb

2,000 psi

200 psi

Yield criterion defined below

2,000 psi

200 psi

The yield surface of the yield criterion is expressed in the form:
F(σ_{∼}) = √3τ_{0}  r(θ, σ_{0}/σ_{yc}, k_{s}) = 0 (1)
where σ_{∼} is the stress tensor whose Cartesian components are σ_{ij}, σ_{0} = I_{1}/3, I_{1} = σ_{kk} = first invariant of stress tensor, τ_{0} = √(2J_{2}/3), J_{2} = s_{ij}s_{ij}/2 = second invariant of the stress deviator s_{ij} = σ_{ij}  1/3 δ_{ij}σ_{kk}, δ_{ij} = Kronecker delta, θ = 1/3 cos^{1}((3√3)/2)J_{3}J_{2}^{3/2}) = similarity angle which represents the polar angle in the deviatoric section, J_{3} = 1/3 s_{ij}s_{jk}s_{ki} = third invariant of s_{ij}, σ_{yc} is the uniaxial compressive yield stress, k_{s} is a constant which determines the size of the yield surface, and function r(θ, σ_{0}/σ_{yc}, k_{s}) represents the deviatoric section of the yield surface which can be expressed by the formula
where r_{t} and r_{c} represent the tensile and compression meridians, respectively, which are defined as:
The equations above involve eight parameters α_{0}, · · ·, α_{3}, β_{0},· · · , β_{3}. These eight parameters and constant k_{s} can be determined based on the following conditions:
1. The two apices of each curve of Eq. 3 and Eq. 4 on the horizontal axis must be common to both curves.
2. The point representing the yield stress under uniaxial tension has to be on the yield surface.
3. The point representing the yield stress under uniaxial compression has to be on the yield surface too.
The above conditions are not enough to determine a unique set of the constant values. Use any set of constants fulfill the above conditions for your project. Include a section in your project report to show the values you use and explain and show calculations how you determine the constants.
Note  It has to be done in Matlab. As for the report of 1000 words need Abstract, intro, procedure, discussion of results, conclusion.
Attachment: Assignment File.rar