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1.033/1.57 Mechanics of Material SystemsMechanics of Material Systems(Mechanics and Durability of Solids I)Franz-Josef UlmLecture: MWF1 // Recitation: F 3:00-4:301.033/1.571.033/1.57 Mechanics of Material SystemsPart III: Elasticity and Elasticity Bounds5. Thermoelasticity1.033/1.57 Mechanics of Material SystemsPart 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 Approac8 Plasticity Models9 Limit Analysis and Yield DesignContent 1.033/1.571.033/1.57 Mechanics of Material Systems• UNKNOWNS– 6 strains εij= εji– 6 stresses σij= σji– 3 displacements ξ i• EQUATIONS– 3 Momentum Balance σij,j+ ρ fi= 0– 6 Strain-Displacement Relations: 2εij= ξ i,j + ξ j,iΣ = 15Σ = 9∆ = 15 − 9 = 66 Missing RelationsThe Necessity of Material Laws1.033/1.57 Mechanics of Material SystemsE1 [L]ε [L]1EσεE(ε)σLinear ElasticBehaviorNonlinear ElasticBehaviorTangent Modulusσ↔ε; σ= σ (ε ); ε= ε (σ )Unique relation1D Think Model of Elasticity1.033/1.57 Mechanics of Material SystemsE1 [L]ε [L]σϕdt = σ dε − dψ = 0 • 1st+ 2ndLaw:WorkStored Energy= Helmholtz Energy• 1D-Model:ψ = 1/2Εε 2σ =∂ψ∂ε• 3D-Model:ψ = ψ (εij)σij=∂ψ∂εijElasticity Potential1.033/1.57 Mechanics of Material SystemspppσI−σII=−σIII−σIIσm= −p=K ∆V/VBulk ModulusσI−σIII= 2G (εI−εIII)Shear ModulusHydrostatic TestTriaxial TestIsotropic Elastic Material Properties1.033/1.57 Mechanics of Material SystemsσIσII = σIIIστ(σI−σIII)/2εIεII = εIIIλγ(εI−εIII)/22G(a)(b)Shear Modulus – Triaxial TestMohr Representation1.033/1.57 Mechanics of Material Systemsxyz,wOrPBoussinesq Problem1.033/1.57 Mechanics of Material SystemsDisplacement Methodξ2εij= ξ i,j + ξ j,iσij= σij+ 2µεij+ λεkkδijOσij,j+ ρ fi= 0IntegrationSolutionDisplacement + StressBoundary ConditionStartStrain-DisplacementLinear Isotropic ElasticityMomentumBalanceDirect Solving Methods in Elasticity1.033/1.57 Mechanics of Material Systemsunder uniform surface pressureApplication of Displacement Methodexx = 0x = Hpξ = u(X) exud(H)=0Td(0)=pexExercise: Soil layer1.033/1.57 Mechanics of Material Systemszezereθp1p0R0R1rr+f(r)dθll’=l(1+εθθ)Illustration of cylinder strain componentsTraining Set: Cylinder Tube … Deep Tunnel1.033/1.57 Mechanics of Material SystemsR0tp0ereθV Vσ/ϖτ/ϖ1 22νR1R0σθθσrr2maxτ/σ0−2ϑ=π/2ϑ=−π/4(Elasticity)Maximum Shear inThick Cylinder TubeMohr RepresentationVessel Formula Revisited1.033/1.57 Mechanics of Material Systems=σ0=−p01p0R0r=R0: Td=−p0ererR0erξr+“Deep”Applied to Deep Tunneling in Elastic Soil/RockNaturalIsotropic PrestressExcavation = Removal of ‘Support’-Stress+=Linear Elastic SolutionThick tube solutionfor outer radius R1→∞σ0=−p01Theorem of Superposition1.033/1.57 Mechanics of Material Systems==+σ0=−p01p0R0r=R0: Td=−p0ererR0erξr-1σ/ p0τ/ p0-1σ/ p0τ/ p0R0r>R01r→∞-2σ/ p0τ/


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MIT 1 033 - Elasticity and Elasticity Bounds

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