Module 4 Boundary value problems in linear elasticity

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Module 4Boundary value problems in linearelasticityLearning Objectives formulate the general Boundary value problem of linear elasticity in three dimensions understand the stress and displacement formulations as alternative solution approachesto reduce the dimensionality of the general elasticity problem solve uniform states of strain and stress in three dimensions specialize the general problem to planar states of strain and stress understand the stress function formulation as a means to reduce the general problemto a single differential equation. solve aerospace-relevant problems in plane strain and plane stress in cartesian andcylindrical Summary of field equationsReadings: BC 3 Intro, Sadd Equations of equilibrium ( 3 equations, 6 unknowns ): ji,j+fi= 0( ) Compatibility ( 6 equations, 9 unknowns): ij=12( ui xj+ uj xi)( )7778MODULE 4. Boundary value problems IN linear ELASTICITYe1e2e3Bbfb Buut BtbuFigure : Schematic of generic problem in linear elasticityor alternatively the equations of strain compatibility (6 equations, 6 unknowns), seeModule ij,kl+ kl,ij= ik,jl+ jl,ik( ) Constitutive Law (6 equations, 0 unknowns) : ij=Cijkl kl( ) ij=Sijkl kl( )In the specific case of linear isotropic elasticity : ij= kk ij+ 2 ij( ) ij=1E[(1 + ) ij kk ij]( ) Boundary conditions of

78 MODULE 4. BOUNDARY VALUE PROBLEMS IN LINEAR ELASTICITY e 1 e 2 e 3 B b f @B u b u t @B t b u Figure 4.1: Schematic of generic problem in linear elasticity or alternatively the equations of strain compatibility (6 equations, 6 unknowns), see

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