Determine a unique set of the constant values
Course:- Mechanical Engineering
Reference No.:- EM132187179

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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 = I1/√3 = √3σm = √3σoct; (b) on meridain planes at θ = 0o, 15o, 30o, 45o, and 60o. (c) Also draw the 3-D perspective views of the yield surfaces in the principal-stress space from three different angles.

Yield Criteria

Uniaxial Compressive Yield Stress

Uniaxial Tensile Yield Stress

von Mises

50,000 psi

50,000 psi


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) = √3τ0 - r(θ, σ0yc, ks) = 0 (1)

where σ is the stress tensor whose Cartesian components are σij, σ0 = I1/3, I1 = σkk = first invariant of stress tensor, τ0 = √(2J2/3), J2 = sijsij/2 = second invariant of the stress deviator sij = σij - 1/3 δijσkk, δij = Kronecker delta, θ = 1/3 cos-1((3√3)/2)J3J2-3/2) = similarity angle which represents the polar angle in the deviatoric section, J3 = 1/3 sijsjkski = third invariant of sij, σyc is the uniaxial compressive yield stress, ks is a constant which determines the size of the yield surface, and function r(θ, σ0yc, ks) represents the deviatoric section of the yield surface which can be expressed by the formula


where rt and rc represent the tensile and compression meridians, respectively, which are defined as:


The equations above involve eight parameters α0, · · ·, α3, β0,· · · , β3. These eight parameters and constant ks 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

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    It has to be done in matlab, as for the report of 1000 words i need Abstract, intro, procedure, discussion of results, conclusion. The project is due on the day of the last lecture of this class. Hand in a hardcopy of typed project report by the due date. The report should at least include your thinking and explanation in working out the problem, the computer software you used, how to generate the input data, how to run the program, the computer-generated plots of the yield surfaces, your comments and discussions, and appendixes for the computer printouts of your input data and output results with essential information highlighted.

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    Organize your report according to the following general guidelines: Cover page: Giving title and dates, etc. Table of Contents: Listing the sections and figures. Abstract: Summarizing the report including important findings. Introduction: Describing the importance of the work done. Procedure and Results: Including figures and/or tables. Discussion of Results: Giving a thorough analysis and interpretation of results. Conclusions: Giving a summary of the findings. Appendix: Showing important calculations and formulas. Figures and tables should be numbered and must have captions explaining what the figures and tables are illustrating. You must always show units of axes for figures and the units for the numbers listed in the tables.

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