FINAL EXAM REVIEW: Material covered—Mohr’s Circle until end. The exam covers recitations 8-12. (only 5 recitations!) TRANSFORMAION OF STRESS & STRAIN Stress transformation formulas for plane stress at a given location MOHR’s CIRCLE , R is equal to the maximum shear stress ELASTIC MODULI Linear elasticity--- modeling bonds between atoms- Lennard Jones Potential Review all equations from this, (aka. Write them on your sheets) remember: re is the distance where F(r)=dU(r)/dr=0Rubber Elasticity—Freely Jointed Chain Model Îa random walking polymer at finite T is a Hookean spring Why is this useful? Because these equations define the stretching of a single polymer chain. Loss of entropy upon stretching, means that there is a retractive force for recovery when external stress removed. This is why a rubber band returns to its original shape. A simple reminder of polymer statistics. Suppose the walk has N links: End to end distance R(N) From Rubber elasticity : r = instantaneous chain end-to-end separation distance (Draw on board--- squiggly lines with beginning and end separated by r) <r2> = na2 root mean square end to end distance a = statistical segment length—local chain stiffness n = # of a’sLc = contour length—length of fully extended chain. Probability of finding a free chain end a radial distance, r, away from a fixed chain end (origin) ~ omega = P(R) = (4b3r2)/sqrt(pi)*exp(-b2r2) where b = sqrt (3/(2na2)) This is Gaussian form Macrostate is defined by the length r. Microstates are the different random walks. So.. () () R)length withµstates of (# ~e~2223RRPNbRΩ− (N=n and b=a) Configurational Entropy (measure of disorder) = S = kb*ln[P(r)] Helmholtz Free Energy = A or H = -Tkb*ln[P(r)] Entropic elastic force, linear elasticity (hookean spring) f or F = -dA(r)/dr^2 Entropic chain stiffness = k = dF(r)/dr or second derivative of A. VISCOELASTICITY Review Spring-Dashpot models. Behavior between elastic solid and viscous fluid (hence the name viscoelastic) Creep test- Const. stress If viscoelastic, strain proportional to stress change Relaxation Test – Const. Strain Now, temperature effect on amorphous polymers 10^9GPa10^5GPaRelaxation modulus vs. Temp.Temp.TgGlassy regimeViscoelastic regimeRubbery regime10^9GPa10^5GPaRelaxation modulus vs. Temp.Temp.TgGlassy regimeViscoelastic regimeRubbery regimeSpring: E=stress/strain Dashpot: stress = viscosity(dε/dt) Åstrain rate drain Maxwell model—in series σs=σd=σtotal εs+εd=εtotal Voigt or Kelvin Model (in parallel) σs+σd=σtotal εs=εd=εtotal Neither give entire model of a polymer—need to sum both models together. Next, Stress-strain diagrams—now going into plasticity!AB'CC'The upperstress levelat which thematerialbehaveselasticallyNeckingMaterial canresist more loadincreaseYieldingElasticterial will deformpermanently and will eturn to itsorginal shape upon unloading. Thedeformation that occurs is calledσfσ′uσfσyσPLσ: The upper stresslimit that strainvaries linearly withstress. Materialfollows :Stress at which aslight increase instress will result inappreciably increasin strain withoutincrease in stressNeckingStress-strain using originalarea to calculateTrue Stress-strain usingactual area to calculateaterialwill return to its orginalshape if material is loadedand unloaded within thisrangeε :10 - 40 timeselastic strainyε Elastic Limit:Strain Hardening:Plastic Behaviour: Ma NOT rplastic deformationProportional LimitHooke's LawYield Stress Ultimate stressFailure stressElastic Behaviour: MYield strain If structure made of ductile materials is overloaded, it will present large deformation before failing Some ductile materials do not exhibit a well-defined yield point, we will use offset method to define a yield strength Some ductile materials do not have linear relationship between stress and strain, we call them nonlinear materialsσyσε0.002 or 0.2% offsetσε()fσε= The work done, which equals to the strain energy stored in the element, is ()(12dU dxdy dzσε=) or 12dU dVσε= VThe total strain energy stored in a material will be VUdσε=∫ The strain energy per unit volume or the strain energy density is 12dUudVσε== If the material is linear elastic ( Eσε=, Hooke’s Law holds), the strain energy density will be 21122uEEσσσ== Plasticity—3-D 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 σ1>σ2>σ3 You could take stress state, rotate and rewrite it. Mohr’s circle does this. For general stress state: τmax = (σ1-σ3)/2 In uniaxial test: σ1 (σ2=σ3 =0)τmax = (σ1)/2 so, you’d get yield when σ1=σY In general, you get yield when σ1 – σ3 = σY 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 σeq =sqrt(0.5*[(σ11 –σ22 )^2 + (σ22 –σ33 )^2 + (σ33 –σ11 )^2] +3σ23^2 + 3σ13^2 +3σ12^2 ) These are looking at effects of Shear on the Material. Good things to remember-- At R.T. many materials (esp. metals) have a well defined yield stress, σy σ < σy elastic recoverable deformation σ > σy 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 Æ even ceramics will yield. Really make sure to understand differences between elastic and plastic states of a system. Understand necking, fracture, and other deformations- understand why these occur. Good