Mechanics of Material Systems(Mechanics and Durability of Solids I)Franz-Josef UlmLecture: MWF1 // Recitation: F3:00-4:301.033/1.57Part IV: Plasticity and Yield Design7. 1D-Plasticity – An Energy ApproachPart I. Deformation and Strain1 Description of Finite Deformation2 Infinitesimal DeformationPart II. Momentum Balance and Stresses3 Momentum Balance4 Stress States / Failure CriterionPart III. Elasticity and Elasticity Bounds5 Thermoelasticity, 6 Variational MethodsPart IV. Plasticity and Yield Design7 1D-Plasticity – An Energy Approach8 Plasticity Models9 Limit Analysis and Yield DesignContent 1.033/1.57σkε pFriction Element(a) (b)E1 [L]ε [L]σε pk1Eσε1Eε (σ =0)=εpkDissipatedEnergyMaximum Free Energy1D-Think Model of Ideal PlasticityE1 [L]ε [L]σ1Eσεε pk1Eε (σ =0)=εpkχHH>0H=0H<01ETζ(χ)1D-Think Model of Hardening Plasticityκσγ pεE2E1A model for the origin of the hardening energyσε pζ =κ −σε pdU >0−=kσε pkDissipated Energyσε pζ =κ −σε pdU <0−=kσεpkDissipated EnergySofteningHardeningEnergy Dissipation in Hardening Plasticityσζσεk-kk1E1ET(a) (b)2k1D-Kinematic Hardening Modelσζσεkk1E1ET(a) (b)1D-Isotropic Hardening Model(a) (b) (c)E1 [L]ε [L]σ 0σ (t)kEε vpηε vp.ηε vp.(t0)(t)(t)1D-ViscoplasticityPlastic Creep & Plastic Relaxationkσ0/Et0σ0tσPrescribedPlastic Creepεεvpkε 0t0σ0tPrescribedPlastic RelaxationσεEσhhKkζ /2ζ /2 kε pγγTraining Set: Stefani ModelσhhKkkζ/2σHHKkkζ/2ζ/2(a) (b)Stefani Model: Multisurface Plasticity−KK−kkσζσ=K+2kζ=−2(K+k)σ=K−2kζ=−2(K−k)σ=−K−2kζ=2(K−k)σ=−(K+2k)ζ=2(K+k)fk=0fK=0Stefani Model: Generalized Force PlanehhKkσσpεdγdγdζ/2ζ/2hhKkσσγdζ/2 ζ/2 pεd(a) (b)Stefani Model: Dissipative MechanismsO123456236kεσσζ1Oσmaxεp-2kk2kStefani Model: Response under Cyclic LoadingFL123xy,δδ /LNKK1EA(a)(b)Homework Set