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Review of Basic Field Equations eij 1 (ui , j u j ,i ) 2 Strain-Displacement Relations eij ,kl ekl ,ij eik , jl e jl ,ik 0 Compatibility Relations ij , j Fi 0 ij ekk ij 2eij 1 eij ij kk ij E E Equilibrium Equations Hooke’s Law 15 Equations for 15 Unknowns ij , eij , ui Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Boundary Conditions T(n) S S St Su R R R S = S t + Su u Traction Conditions Displacement Conditions Rigid-Smooth Boundary Condition Allows Specification of Both Traction and Displacement But Only in Orthogonal Directions Symmetry Line y u0 x Elasticity Mixed Conditions Theory, Applications and Numerics M.H. Sadd , University of Rhode Island T y( n ) 0 Boundary Conditions on Coordinate Surfaces On Coordinate Surfaces the Traction Vector Reduces to Simply Particular Stress Components r y xy r x r y r xy x y x (Cartesian Coordinate Boundaries) Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island r (Polar Coordinate Boundaries) Boundary Conditions on General Surfaces On General Non-Coordinate Surfaces, Traction Vector Will Not Reduce to Individual Stress Components and General Traction Vector Form Must Be Used y n Tx( n ) x nx xy n y S cos Ty( n ) xy nx y n y S sin S x Two-Dimensional Example Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Example Boundary Conditions Traction Free Condition y Fixed Condition u=v=0 Traction Condition Tx( n ) x S, Ty( n ) xy 0 Traction Condition Tx( n) xy 0, Ty( n) y S S l x b S Tx( n ) 0 Ty( n ) 0 a y x Traction Free Condition Tx( n) xy 0, Ty(n) y 0 Coordinate Surface Boundaries Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Fixed Condition u=v=0 Traction Free Condition Non-Coordinate Surface Boundary Interface Boundary Conditions Material (1): ij(1) , ui(1) n s Material (2): Embedded Fiber or Rod Elasticity Interface Conditions: Perfectly Bonded, Slip Interface, Etc. ij( 2) , u i( 2) Layered Composite Plate Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Composite Cylinder or Disk Fundamental Problem Classifications Problem 1 (Traction Problem) Determine the distribution of displacements, strains and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the tractions are prescribed over the surface of the body, (n) (s) (s) Ti T(n) S R ( xi ) f i ( xi ) Problem 2 (Displacement Problem) Determine the distribution of displacements, strains and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the displacements are prescribed over the surface of the body. S R ui ( xi( s ) ) g i ( xi( s ) ) Problem 3 (Mixed Problem) Determine the distribution of displacements, strains and stresses in the interior of an elastic body in equilibrium when body forces are given and the distribution of the tractions are prescribed as per (5.2.1) over the surface St and the distribution of the displacements are prescribed as per (5.2.2) over the surface Su of the body (see Figure 5.1). Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island u St Su R Stress Formulation Eliminate Displacements and Strains from Fundamental Field Equation Set (Zero Body Force Case) Equilibrium Equations x yx zx 0 x y z xy y zy 0 x y z xz yz z 0 x y z Compatibility in Terms of Stress: Beltrami-Michell Compatibility Equations 2 (1 ) x 2 ( x y z ) 0 x 2 2 (1 ) y 2 ( x y z ) 0 y 2 2 (1 ) z 2 ( x y z ) 0 z 2 2 (1 ) xy ( x y z ) 0 xy 2 2 (1 ) yz ( x y z ) 0 yz 2 2 (1 ) zx ( x y z ) 0 zx 2 6 Equations for 6 Unknown Stresses Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Displacement Formulation Eliminate Stresses and Strains from Fundamental Field Equation Set (Zero Body Force Case) Equilibrium Equations in Terms of Displacements: Navier’s/Lame’s Equations u v w 0 x x y z u v w 0 2 v ( ) y x y z u v w 0 2 w ( ) z x y z 2 u ( ) 3 Equations for 3 Unknown Displacements Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Summary of Reduction of Fundamental Elasticity Field Equation Set General Field Equation System (15 Equations, 15 Unknowns:) {ui , eij , ij ; , , Fi } 0 1 (ui , j u j ,i ) 2 ij, j Fi 0 ij ( ) ekk ij 2eij eij,kl ekl ,ij eik , jl e jl ,ik 0 eij Stress Formulation Displacement Formulation (6 Equations, 6 Unknowns:) (3 Equations, 3 Unknowns: ui) ( t ) { ij ; , , Fi } ij,kk Elasticity ij, j Fi 0 1 kk ,ij ij Fk ,k Fi , j F j ,i 1 1 Theory, Applications and Numerics M.H. Sadd , University of Rhode Island ( u ) {ui ; , , Fi } ui ,kk ( )u k ,ki Fi 0 Principle of Superposition For a given problem domain, if the state {ij(1) , eij(1) , ui(1) } is a solution to the fundamental elasticity equations with prescribed body forces Fi (1) and surface tractions Ti (1) , and the state {ij( 2) , eij( 2) , ui( 2) } is a solution to the fundamental equations with prescribed body forces Fi ( 2) and surface tractions Ti ( 2) , then the state {ij(1) ij( 2) , eij(1) eij( 2) , ui(1) ui( 2) } will be a solution to the problem with body forces Fi (1) Fi ( 2) and surface tractions Ti (1) Ti ( 2) . (1)+(2) = (1) + (2) {ij(1) , eij(1) , ui(1) } {ij(1) ij( 2) , eij(1) eij( 2) , ui(1) ui( 2) } Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island {ij( 2) , eij( 2) , ui( 2) } Saint-Venant’s Principle The stress, strain and displacement fields due to two different statically equivalent force distributions on parts of the body far away from the loading points are approximately the same FR P (1) P P 2 2 P P P 3 3 3 (2) (3) Stresses approximately the same Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island T(n) S* Boundary loading T(n) would produce detailed and characteristic effects only in vicinity of S*. Away from S* stresses would generally depend more on resultant FR of tractions rather than on exact distribution General Solution Strategies Used to Solve Elasticity Field Equations Direct Method - Solution of field equations by direct integration. Boundary conditions are satisfied exactly. Method normally encounters significant mathematical difficulties thus limiting its application to problems with simple geometry. Inverse Method - Displacements or stresses are selected that satisfy field equations. A search is then conducted to identify a specific problem that would be solved by this solution field. This amounts to determine appropriate problem geometry, boundary conditions and body forces that would enable the solution to satisfy all conditions on the problem. It is sometimes difficult to construct solutions to a specific problem of practical interest. Semi-Inverse Method - Part of displacement and/or stress field is specified, while the other remaining portion is determined by the fundamental field equations (normally using direct integration) and boundary conditions. Constructing appropriate displacement and/or stress solution fields can often be guided by approximate strength of materials theory. Usefulness of this approach is greatly enhanced by employing SaintVenant’s principle, whereby a complicated boundary condition can be replaced by a simpler statically equivalent distribution. Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Mathematical Techniques Used to Solve Elasticity Field Equations Analytical Solution Procedures - Power Series Method - Fourier Method - Integral Transform Method - Complex Variable Method Approximate Solution Procedures - Ritz Method Numerical Solution Procedures - Finite Difference Method (FDM) - Finite Element Method (FEM) - Boundary Element Method (BEM) Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island