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 0 Rubber Elasticity Freely Jointed Chain Model a 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 s Lc 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 3 R 2 P R e 2 Nb R of states with length R 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 2 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 Relaxation modulus vs Temp 10 9 GPa Glassy regime Viscoelastic regime 10 5 GPa Rubbery regime Tg Temp Spring E stress strain Dashpot stress viscosity d dt strain rate drain Maxwell model in series s d total s d total Voigt or Kelvin Model in parallel s d total s d total Neither give entire model of a polymer need to sum both models together Next Stress strain diagrams now going into plasticity Yield Stress y Stress at which a slight increase in stress will result in appreciably increas in strain without increase in stress f C True Stress strain using actual area to calculate B Ultimate stress u Failure stress f Elastic Limit The upper stress level at which the material behaves elastically C A Proportional Limit PL The upper stress limit that strain varies linearly with stress Material follows Hooke s Law Elastic Stress strain using original area to calculate Necking Yield strain y 10 40 times elastic strain Yielding Strain Hardening Material can resist more load increase Necking Plastic Behaviour Material will deform permanently and will NOT return to its orginal shape upon unloading The Elastic Behaviour Material deformation that occurs is called will return to its orginal plastic deformation shape if material is loaded and unloaded within this range 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 materials f y 0 002 or 0 2 offset The work done which equals to the strain energy stored in the element is 1 1 dU dxdy dz or dU dV 2 2 The total strain energy stored in a material will be U dV V The strain energy per unit volume or the strain energy density is u dU 1 dV 2 If the material is linear elastic E Hooke s Law holds the strain energy density will be 1 1 2 u 2 E 2 E 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 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 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 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 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 Luck