Three-dimensional elasticity problems are very difficult to solve.Thus we will first develop governing equations for two-dimensional problems, and will explore four different theories: - Plane Strain - Plane Stress - Generalized Plane Stress - Anti-Plane Strain The basic theories of plane strain and plane stress represent the fundamental plane problem in elasticity. While these two theories apply to significantly different types of two-dimensional bodies, their formulations yield very similar field equations. Since all real elastic structures are three-dimensional, theories set forth here will be approximate models. The nature and accuracy of the approximation will depend on problem and loading geometry Two vs Three Dimensional Problems x y z x y z Three-Dimensional Two-Dimensional x y z Spherical Cavity Plane Strain Consider an infinitely long cylindrical (prismatic) body as shown. If the body forces and tractions on lateral boundaries are independent of the z-coordinate and have no z-component, then the deformation field can be taken in the reduced form x y z R 0 , ) , ( , ) , ( = = = w y x v v y x u u Plane Strain Field Equations 0 , 2 1 , , = = = | | . | \ | c c + c c = c c = c c = yz xz z xy y x e e e x v y u e y v e x u e 0 , 2 ) ( ) ( 2 ) ( , 2 ) ( = t = t µ = t o + o v = + ì = o µ + + ì = o µ + + ì = o yz xz xy xy y x y x z y y x y x y x x e e e e e e e e e Strains Stresses Equilibrium Equations 0 0 = + c o c + c t c = + c t c + c o c y y xy x xy x F y x F y x Navier Equations 0 ) ( 0 ) ( 2 2 = + | | . | \ | c c + c c c c µ + ì + V µ = + | | . | \ | c c + c c c c µ + ì + V µ y x F y v x u y v F y v x u x u Strain Compatibility y x e x e y e xy y x c c c = c c + c c 2 2 2 2 2 2 Beltrami-Michell Equation | | . | \ | c c + c c v ÷ ÷ = o + o V y F x F y x y x 1 1 ) ( 2 Examples of Plane Strain Problems x y z x y z P Long Cylinders Under Uniform Loading Semi-Infinite Regions Under Uniform Loadings Plane Stress Consider the domain bounded two stress free planes z = ±h, where h is small in comparison to other dimensions in the problem. Since the region is thin in the z-direction, there can be little variation in the stress components through the thickness, and thus they will be approximately zero throughout the entire domain. Finally since the region is thin in the z- direction it can be argued that the other non-zero stresses will have little variation with z. Under these assumptions, the stress field can be taken as 0 ) , ( ) , ( ) , ( = t = t = o t = t o = o o = o yz xz z xy xy y y x x y x y x y x x y z R 2h yz xz z t t o , , Plane Stress Field Equations Strains Strain Displacement Relations Equilibrium Equations 0 0 = + c o c + c t c = + c t c + c o c y y xy x xy x F y x F y x Navier Equations Strain Compatibility y x e x e y e xy y x c c c = c c + c c 2 2 2 2 2 2 Beltrami-Michell Equation 0 , 1 ) ( 1 ) ( ) ( 1 , ) ( 1 = = t v + = + v ÷ v ÷ = o + o v ÷ = vo ÷ o = vo ÷ o = yz xz xy xy y x y x z x y y y x x e e E e e e E e E e E e 0 2 1 , 0 2 1 2 1 , , , = | . | \ | c c + c c = = | | . | \ | c c + c c = | | . | \ | c c + c c = c c = c c = c c = x w z u e y w z v e x v y u e z w e y v e x u e xz yz xy z y x 0 ) 1 ( 2 0 ) 1 ( 2 2 2 = + | | . | \ | c c + c c c c v ÷ + V µ = + | | . | \ | c c + c c c c v ÷ + V µ y x F y v x u y E v F y v x u x E u | | . | \ | c c + c c v + ÷ = o + o V y F x F y x y x ) 1 ( ) ( 2 Examples of Plane Stress Problems Thin Plate With Central Hole Circular Plate Under Edge Loadings Plane Elasticity Boundary Value Problem R S o S i S = S i + S o x y Displacement Boundary Conditions Stress/Traction Boundary Conditions S on y x v v y x u u b b ) , ( , ) , ( = = S on n n y x T T n n y x T T y b y x b xy b y n y y b xy x b x b x n x ) ( ) ( ) ( ) ( ) ( ) ( ) , ( ) , ( o + t = = t + o = = Plane Strain Problem - Determine in- plane displacements, strains and stresses {u, v, e x , e y , e xy , o x , o y , t xy } in R. Out-of- plane stress o z can be determined from in-plane stresses via relation (7.1.3) 3 . Plane Stress Problem - Determine in- plane displacements, strains and stresses {u, v, e x , e y , e xy , o x , o y , t xy } in R. Out-of- plane strain e z can be determined from in-plane strains via relation (7.2.2) 3 . Correspondence Between Plane Formulations Plane Strain Plane Stress 0 ) ( 0 ) ( 2 2 = + | | . | \ | c c + c c c c µ + ì + V µ = + | | . | \ | c c + c c c c µ + ì + V µ y x F y v x u y v F y v x u x u 0 0 = + c o c + c t c = + c t c + c o c y y xy x xy x F y x F y x | | . | \ | c c + c c v ÷ ÷ = o + o V y F x F y x y x 1 1 ) ( 2 0 ) 1 ( 2 0 ) 1 ( 2 2 2 = + | | . | \ | c c + c c c c v ÷ + V µ = + | | . | \ | c c + c c c c v ÷ + V µ y x F y v x u y E v F y v x u x E u 0 0 = + c o c + c t c = + c t c + c o c y y xy x xy x F y x F y x | | . | \ | c c + c c v + ÷ = o + o V y F x F y x y x ) 1 ( ) ( 2 Transformation Between Plane Strain and Plane Stress Plane strain and plane stress field equations had identical equilibrium equations and boundary conditions. Navier’s equations and compatibility relations were similar but not identical with differences occurring only in particular coefficients involving just elastic constants. So perhaps a simple change in elastic moduli would bring one set of relations into an exact match with the corresponding result from the other plane theory. This in fact can be done using results in the following table. E v Plane Stress to Plane Strain 2 1 v ÷ E v ÷ v 1 Plane Strain to Plane Stress 2 ) 1 ( ) 2 1 ( v + v + E v + v 1 Therefore the solution to one plane problem also yields the solution to the other plane problem through this simple transformation scheme. Generalized Plane Stress The plane stress formulation produced some inconsistencies in particular out- of-plane behavior and resulted in some three-dimensional effects where in- plane displacements were functions of z. We avoided these issues by simply neglecting some of the troublesome equations thereby producing an approximate elasticity formulation. In order to avoid this unpleasant situation, an alternate approach called Generalized Plane Stress can be constructed based on averaging the field quantities through the thickness of the domain. Using the averaging operator defined by } ÷ ¢ = ¢ h h dz z y x h y x ) , , ( 2 1 ) , ( all plane stress equations are satisfied exactly by the averaged stress, strain and displacements variables; thereby eliminating the inconsistencies found in the original plane stress formulation. However, this gain in rigor does not generally contribute much to applications . Anti-Plane Strain An additional plane theory of elasticity called Anti-Plane Strain involves a formulation based on the existence of only out-of-plane deformation starting with an assumed displacement field ) , ( , 0 y x w w v u = = = y w e x w e e e e e yz xz xy z y x c c = c c = = = = = 2 1 , 2 1 0 yz yz xz xz xy z y x e e µ = t µ = t = t = o = o = o 2 , 2 0 0 0 = = = + c t c + c t c y x z yz xz F F F y x 0 2 = + V µ z F w Strains Stresses Equilibrium Equations Navier’s Equation Airy Stress Function Method Numerous solutions to plane strain and plane stress problems can be determined using an Airy Stress Function technique. The method will reduce the general formulation to a single governing equation in terms of a single unknown. The resulting equation is then solvable by several methods of applied mathematics, and thus many analytical solutions to problems of interest can be found. This scheme is based on the general idea of developing a representation for the stress field that will automatically satisfy equilibrium by using the relations y x x y xy y x c c | c ÷ = t c | c = o c | c = o 2 2 2 2 2 , , where | = |(x,y) is an arbitrary form called Airy’s stress function. It is easily shown that this form satisfies equilibrium (zero body force case) and substituting it into the compatibility equations gives 0 2 4 4 4 2 2 4 4 4 = | V = c | c + c c | c + c | c y y x x This relation is called the biharmonic equation and its solutions are known as biharmonic functions. Airy Stress Function Formulation The plane problem of elasticity can be reduced to a single equation in terms of the Airy stress function. This function is to be determined in the two- dimensional region R bounded by the boundary S as shown in the figure. Appropriate boundary conditions over S are necessary to complete a solution. Traction boundary conditions would involve the specification of second derivatives of the stress function; however, this condition can be reduced to specification of first order derivatives. R S o S i S = S i + S o x y 0 2 4 4 4 2 2 4 4 4 = | V = c | c + c c | c + c | c y y x x y x y y x xy n y y x y xy x x n x n x n y x n n T n y x n y n n T 2 2 2 ) ( 2 2 2 ) ( c | c + c c | c ÷ = o + t = c c | c ÷ c | c = t + o = o x t xy o y Polar Coordinate Formulation Plane Elasticity Problem | . | \ | ÷ c c + u c c = | . | \ | u c c + = c c = u u u u u r u r u u r e u u r e r u e r r r r r 1 2 1 1 0 , 2 ) ( ) ( 2 ) ( 2 ) ( Strain Plane = t = t µ = t o + o v = + ì = o µ + + ì = o µ + + ì = o u u u u u u u u u rz z r r r r z r r r r e e e e e e e e e 0 , 1 ) ( 1 ) ( ) ( 1 ) ( 1 Stress ane Pl = = t v + = + v ÷ v ÷ = o + o v ÷ = vo ÷ o = vo ÷ o = u u u u u u u u rz z r r r r z r r r e e E e e e E e E e E e Strain-Displacement Hooke’s Law Polar Coordinate Formulation 0 2 1 0 ) ( 1 r r = + t + u c o c + c t c = + o ÷ o + u c t c + c o c u u u u u u F r r r F r r r r r r r 0 1 1 ) 1 ( 2 0 1 ) 1 ( 2 Stress Plane 0 1 1 ) ( 0 1 ) ( Strain Plane 2 2 2 2 = + | . | \ | u c c + + c c u c c v ÷ + V µ = + | . | \ | u c c + + c c c c v ÷ + V µ = + | . | \ | u c c + + c c u c c µ + ì + V µ = + | . | \ | u c c + + c c c c µ + ì + V µ u u u u u u u u F u r r u r u r E u F u r r u r u r E u F u r r u r u r u F u r r u r u r u r r r r r r r r r r r r Equilibrium Equations Compatibility Equations Navier’s Equations 2 2 2 2 2 2 1 1 u c c + c c + c c = V r r r r | . | \ | u c c + + c c v + ÷ = o + o V | . | \ | u c c + + c c v ÷ ÷ = o + o V u u u u F r r F r F F r r F r F r r r r r r 1 ) 1 ( ) ( Stress Plane 1 1 1 ) ( Strain Plane 2 2 Polar Coordinate Formulation Airy Stress Function Approach | = |(r,θ) | . | \ | u c | c c c ÷ = t c | c = o u c | c + c | c = o u u r r r r r r r r 1 1 1 2 2 2 2 2 0 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 = | | | . | \ | u c c + c c + c c | | . | \ | u c c + c c + c c = | V r r r r r r r r R S x y u r - Airy Representation Biharmonic Governing Equation ) , ( , ) , ( u = u = u u r f T r f T r r Traction Boundary Conditions t ru o r o u
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