Formulation and Solution Strategies
Formulation and Solution Strategies
Outline. Review of Field Equations· Types of BCs: BCs on Coordinate Surfaces. BCs on Oblique Surface? Line of Symmetry BCs· Interface BCs: Problem ClassificationStress Formulation Displacement FormulationPrinciple of Superposition Uniqueness of Elastic Solution: Saint-Venant's Principle: Solution Strategies: Mathematical Techniques2
Outline • Review of Field Equations • Types of BCs • BCs on Coordinate Surfaces • BCs on Oblique Surface • Line of Symmetry BCs • Interface BCs • Problem Classification • Stress Formulation • Displacement Formulation • Principle of Superposition • Uniqueness of Elastic Solution • Saint-Venant’s Principle • Solution Strategies • Mathematical Techniques 2
Governing Equations in linear elasticity(6 eqns)Strain-displacement relations: u+12Bik,j - jl,ik = 0 (6 eqns)Strain compatibility:6u,ki + kl.u -(3 eqns) Equilibrium:αu., + F, = 0Isotropic Hooke's Law:1 +(6 eqns)0,=a8ko,+2G6,EE. 15 equations for 15 unknowns (3 displacements, 6 strains6 stresses).May define the entire system as元.G.FaS3
, , 1 2 i j i j j i u u , , , , 0 ij k l k l ij ik jl jl ik , 0 ij j i F 1 2 ; . i j k k i j i j i j i j k k i j G E E Governing Equations in linear elasticity • Strain-displacement relations: • Strain compatibility: • Equilibrium: • Isotropic Hooke’s Law: • 15 equations for 15 unknowns (3 displacements, 6 strains, 6 stresses). • May define the entire system as , { , ; , , } 0 i i j i j i f u G F 3 (6 eqns) (6 eqns) (3 eqns) (6 eqns)
Traction and Displacement Boundary ConditionsrnSSSRRRMixed ConditionsDisplacementConditionsTraction Conditions(c)(b)(a)A
Traction and Displacement Boundary Conditions 4 (a) (b) (c)
Boundary Conditions on Coordinate Surfaces. The traction specification can be reduced to a stressspecificationis irrelaventisirrelavent.atxytre0TxyTre6xC(CartesianCoordinateBoundaries)(PolarCoordinateBoundaries)5
Boundary Conditions on Coordinate Surfaces • The traction specification can be reduced to a stress specification. , . i s i r r e la v e n t . , . i s i r r e la v e n t . x x y x y y y y x x y x T T T T x x y y , , r r r r r T T T T r r θ θ 5
Boundary Conditions on Obligue Surfaces: On general non-coordinate surfaces traction vector willnot reduce to individual stress components and generalform of traction vector must then be usedItxa(a.Xcosαssinα6
• On general non-coordinate surfaces traction vector will not reduce to individual stress components and general form of traction vector must then be used. c o s s i n x x x y y x x y x y y y n n T S n n T S n n x S y n Boundary Conditions on Oblique Surfaces x x x y y x z z x x y x y y y z z y x z x y z y z z z n n n T n n n T n n n T n n n x T n y T n z T n 6
Example of Boundary ConditionsExample (1)TractionConditionFixedConditionT)=0x= S,T,= txy= 0u=V=0I*.aTractionFreeConditionT(= txy =0, T()= 0y= 0(CoordinateSurfaceBoundaries)1
Example of Boundary Conditions Example (1) 7
Example of Boundary ConditionsExample (2)Traction ConditionT(n) = -txy =0, T,(n)XT(m)二=xnx+txyny=0T(n = trynx +yny=0yFixedConditionu=V=oTractionFreeCondition(Non-Coordinate SurfaceBoundary)8
Example of Boundary Conditions Example (2) 8
Line of Symmetry Boundary ConditionsRigid-SmoothSymmetry LineBoundaryConditionu=0T = 0VLineofsymmetryboundarycondition9
Line of Symmetry Boundary Conditions 9
Interface Boundary ConditionsCompositeCylinderorDiskLayeredCompositePlateEmbeddedFiberorRodMaterial (1):InterfaceConditions:anPerfectlyBonded,SlipInterface,Etc9)Material (2):10
Embedded Fiber or Rod Layered Composite Plate Composite Cylinder or Disk Material (1): ( 1 ) ( 1 ) , ij i u ( 2 ) ( 2 ) , ij i Material (2): u Interface Conditions: Perfectly Bonded, Slip Interface, Etc. n s Interface Boundary Conditions 10