RECITATION 12 DECEMBER 2 2003 I think craze shear deformation zones and rubber toughening of glassy polymers was covered well in lecture so I will focus on the more complicated equations dealing with the 3 D deformation Please ask me if you d like more clarification and I can cover it in the final recitation Dec 9 PLASTICITY Plastic behavior 3 D Plastic Deformation Review of stress states for 3 D Von Mises Tresca Plasticity At R T many materials esp metals have a well defined yield stress y y elastic recoverable deformation y plastic irrecoverable Plastic behavior is important for Material design for strengthening alloying Work hardening Hardness friction and wear Some materials fracture before yielding ceramics brittle BUT fracture can be suppressed by a hydrostatic stress even ceramics will yield This next stuff low temperature plasticity not high temp creep behavior As a review We ve talked about a couple types of tests 1 Uniaxial Tensile Test a Destructive to sample b Measures ductility c Shows where yield stress is located at 0 2 strain 2 Hardness Test a Indenter pushed into polished surface w a known force b Measure the size of indent you should have done this will do this in 3 081 c Gives hardness which is related to yield stress d Different versions of indenters sphere pyramid cone i Advantages 1 non destructive easy to do 2 can use this on ceramics note for hardness test it needs to be a polished material maybe 500um diameter Equations of plasticity Assumptions 1 Plastic flow occurs at constant volume a V Vo 0 11 22 33 0 note strains are plastic strains in x y and z direction 2 Modest hydrostatic pressures to not affect yield stress strength a modest E 100 3 Neglect work hardening a Assume elastic region got to perfect plastic region 4 Assume Material is Isotropic Criteria for Yielding Under Multi Axial Stresses Yield unaffected by hydrostatic component of stress mean stress 11 12 33 m m m 11 m 12 33 m where m 11 12 33 3 sigma m is the mean stress component everything but mean stress is related to the amount of shearing in the material called the deviatoric component of Yield is governed by the deviatoric part of shearing is what s really controlling yield Yield Criteria Tresca Criterion Sometimes called the Maximum shear stress criterion 1 get yield when max shear stress tau in component under general stress state equals the ma tau in a uniaxial tensile test at yield 2 principle stresses 1 2 3 You could take stress state rotate and rewrite it Mohr s circle does this For general stress state max 1 3 2 In uniaxial test 1 2 3 0 max 1 2 so you d get yield when 1 Y In general you get yield when 1 3 Y Tresca says you take the difference between the biggest and smallest stress to get yield stress Yield Criteria Von Mises Criterion You get yield when equivalent sigma yield stress eq sqrt 0 5 11 22 2 22 33 2 33 11 2 3 23 2 3 13 2 3 12 2 How is this related to shearing We re applying stress The stress energy for mean stress energy for shear multiaxial stress state Show example of comparison of Tresca and Von Mises for Biaxial stress sigma22 0 Tresca if stress state inside yield criteria elastic If On yield criteria plastic For Von Mises you have ellipsoid criteria There is a gap between them that correlates to pure shear it s not too far apart for either criteria the maximum difference is 13 Tresca criterion is more conservative than Von Mises criterion most used Example Problems with Von Mises Tresca analysis 2 When coins are minted they are stamped out in a die
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