MIT 1 033 - Elasticity and Elasticity Bounds (15 pages)

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Elasticity and Elasticity Bounds



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Elasticity and Elasticity Bounds

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Pages:
15
School:
Massachusetts Institute of Technology
Course:
1 033 - Mechanics of Material Systems: An Energy Approach
Mechanics of Material Systems: An Energy Approach Documents
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1 033 1 57 Mechanics of Material Systems Mechanics and Durability of Solids I Franz Josef Ulm Lecture MWF1 Recitation F 3 00 4 30 1 033 1 57 Mechanics of Material Systems Part III Elasticity and Elasticity Bounds 5 Thermoelasticity 1 033 1 57 Mechanics of Material Systems Content 1 033 1 57 Part I Deformation and Strain 1 Description of Finite Deformation 2 Infinitesimal Deformation Part II Momentum Balance and Stresses 3 Momentum Balance 4 Stress States Failure Criterion Part III Elasticity and Elasticity Bounds 5 Thermoelasticity 6 Variational Methods Part IV Plasticity and Yield Design 7 1D Plasticity An Energy Approac 8 Plasticity Models 9 Limit Analysis and Yield Design 1 033 1 57 Mechanics of Material Systems The Necessity of Material Laws UNKNOWNS EQUATIONS 6 strains ij ji 6 stresses ij ji 3 displacements i 3 Momentum Balance ij j fi 0 6 Strain Displacement Relations 2 ij i j j i 9 15 15 9 6 6 Missing Relations 1 033 1 57 Mechanics of Material Systems 1D Think Model of Elasticity Nonlinear Elastic Behavior L E Tangent Modulus E 1 L 1 E Linear Elastic Behavior Unique relation 1 033 1 57 Mechanics of Material Systems Elasticity Potential 1st 2nd Law dt d d 0 L Stored Energy Helmholtz Energy Work 1D Model 1 L E 1 2 3D Model ij 1 033 1 57 Mechanics of Material Systems 2 ij ij Isotropic Elastic Material Properties Triaxial Test Hydrostatic Test p I p p II II III m p K V V I III 2G I III Bulk Modulus Shear Modulus 1 033 1 57 Mechanics of Material Systems Shear Modulus Triaxial Test Mohr Representation 2G I III 2 I III 2 I II III a I II III b 1 033 1 57 Mechanics of Material Systems Boussinesq Problem P x O r y z w 1 033 1 57 Mechanics of Material Systems Direct Solving Methods in Elasticity Displacement Method Start Strain Displacement Linear Isotropic Elasticity 2 ij i j j i ij Oij 2 ij kk ij Momentum Balance ij j fi 0 Integration Displacement Stress Boundary Condition 1 033 1 57 Mechanics of Material Systems Solution Exercise Soil layer under uniform surface pressure Application of Displacement Method p Td 0 pex x 0 ex u X ex x H ud H 0 1 033 1 57 Mechanics of Material Systems Training Set Cylinder Tube Deep Tunnel z p0 R1 Illustration of cylinder strain components R0 ez l l 1 r f r e l r p1 er 1 033 1 57 Mechanics of Material Systems d Vessel Formula Revisited Elasticity 2max 0 4 e t 2 2 er R0 V R1 rr p0 2 1 2 R0 V Maximum Shear in Thick Cylinder Tube Mohr Representation 1 033 1 57 Mechanics of Material Systems Theorem of Superposition Applied to Deep Tunneling in Elastic Soil Rock Deep er p0 er R0 R0 0 0p 0 1 p01 Natural Isotropic Prestress r r R0 Td p0er Excavation Removal of Support Stress Thick tube solution for outer radius R1 1 033 1 57 Mechanics of Material Systems Linear Elastic Solution Deep er p0 er r R0 R0 0 p01 r R0 Td p0er p0 p0 p0 1 p0 p0 r p0 1 1 2 r R0 R0 r R0 1 033 1 57 Mechanics of Material Systems 1 R0 r


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