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Three-dimensional elasticity problems are difficult to solve. Thus we first develop governing equations for two-dimensional problems, and explore four different theories: - Plane Strain - Plane Stress - Generalized Plane Stress - Anti-Plane Strain 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. 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. Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Two vs Three Dimensional Problems Three-Dimensional Two-Dimensional x y y z z z Spherical Cavity y x Elasticity x Theory, Applications and Numerics M.H. Sadd , University of Rhode Island 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 u u ( x, y ) , v v ( x, y ) , w 0 y x R z Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Plane Strain Field Equations Strains ex u v 1 u v , ey , exy , ez exz e yz 0 x y 2 y x x (ex e y ) 2ex , y (ex e y ) 2e y Stresses z (e x e y ) ( x y ) xy 2exy , xz yz 0 Equilibrium Equations x xy Fx 0 x y xy y Fy 0 x y Navier Equations Elasticity 2 u ( ) u v Fx 0 x x y 2 v ( ) u v Fy 0 y x y Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Strain Compatibility 2 2 exy 2 ex e y 2 2 y 2 x xy Beltrami-Michell Equation 2 ( x y ) 1 Fx Fy 1 x y Examples of Plane Strain Problems y P z x x y z Long Cylinders Under Uniform Loading Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island 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 z , xz , yz through the thickness, and thus they will be approximately zero throughout the entire domain. Finally since the region is thin in the zdirection 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 y x x ( x, y ) 2h y y ( x, y ) xy xy ( x, y ) z xz yz 0 R z x Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Plane Stress Field Equations Strains 1 1 ( x y ) , e y ( y x ) E E e z ( x y ) (e x e y ) E 1 1 exy xy , exz e yz 0 E ex Equilibrium Equations x xy Fx 0 x y xy y Fy 0 x y Navier Equations Elasticity 2 u E u v Fx 0 2(1 ) x x y 2 v E u v Fy 0 2(1 ) y x y Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Strain Displacement Relations u v w 1 u v , ey , ez , exy x y z 2 y x 1 v w 1 u w 0 , exz e yz 0 2 z y 2 z x ex Strain Compatibility 2 2 exy 2 ex e y 2 2 y 2 x xy Beltrami-Michell Equation F Fy 2 ( x y ) (1 ) x x y Examples of Plane Stress Problems Thin Plate With Central Hole Circular Plate Under Edge Loadings Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Plane Elasticity Boundary Value Problem Displacement Boundary Conditions u ub ( x, y ) , v vb ( x, y) on S Si So Stress/Traction Boundary Conditions Txn Tx( b ) ( x, y ) (xb ) n x (xyb ) n y T T n y (b) y ( x, y ) n x n y (b) xy (b) y on S y R S = S i + So x Plane Strain Problem - Determine inplane displacements, strains and stresses {u, v, ex , ey , exy , x , y , xy} in R. Out-ofplane stress z can be determined from in-plane stresses via relation (7.1.3)3. Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Plane Stress Problem - Determine inplane displacements, strains and stresses {u, v, ex , ey , exy , x , y , xy} in R. Out-ofplane strain ez can be determined from in-plane strains via relation (7.2.2)3. Correspondence Between Plane Formulations Plane Strain Plane Stress 2 u ( ) u v Fx 0 x x y 2 u E u v Fx 0 2(1 ) x x y 2 v ( ) u v Fy 0 y x y 2 v E u v Fy 0 2(1 ) y x y x xy Fx 0 x y xy y Fy 0 x y 2 ( x y ) Elasticity 1 Fx Fy 1 x y Theory, Applications and Numerics M.H. Sadd , University of Rhode Island x xy Fx 0 x y xy y Fy 0 x y F Fy 2 ( x y ) (1 ) x y x 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. Plane Stress to Plane Strain Plane Strain to Plane Stress E E 1 2 E (1 2) (1 ) 2 1 1 Therefore the solution to one plane problem also yields the solution to the other plane problem through this simple transformation scheme. Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Generalized Plane Stress The plane stress formulation produced some inconsistencies in particular outof-plane behavior and resulted in some three-dimensional effects where inplane 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 ( x, y ) 1 h ( x, y , z )dz h 2h 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 . Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island 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 u v 0 , w w( x, y ) Strains ex e y ez exy 0 exz Elasticity 1 w 1 w , e yz 2 x 2 y Stresses x y z xy 0 xz 2exz , yz 2e yz Equilibrium Equations Navier’s Equation xz yz Fz 0 x y Fx Fy 0 2 w Fz 0 Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Airy Stress Function Method Numerous solutions to plane strain and plane stress problems can be determined using an Airy Stress Function technique. The method reduces 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 2 2 2 x 2 , y 2 , xy y x xy 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 4 4 4 2 2 2 4 4 0 4 x x y y This relation is called the biharmonic equation and its solutions are known as biharmonic functions. Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island 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 twodimensional region R bounded by the boundary S as shown. Appropriate boundary conditions over S are necessary to complete the 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. 4 4 4 2 2 2 4 4 0 4 x x y y (n) x T 2 2 x nx xy n y 2 nx ny y xy T (n) y y 2 2 xy nx y n y nx 2 n y xy x Si R S = S i + So Theory, Applications and Numerics M.H. Sadd , University of Rhode Island y xy x x Elasticity So Polar Coordinate Formulation Plane Elasticity Problem Strain-Displacement ur r u 1 e u r r 1 1 ur u u er 2 r r r er Hooke’s Law P lane Strain r ( er e ) 2er ( er e ) 2e z ( e r e ) ( r ) r 2er , z rz 0 Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Pane l Stress 1 er ( r ) E 1 e ( r ) E ez ( r ) ( e r e ) E 1 1 er r , ez erz 0 E Polar Coordinate Formulation Navier’s Equations P lane Strain 2 u r ( ) u r u r 1 u Fr 0 r r r r 2 u ( ) 1 u r u r 1 u F 0 r r r r Equilibrium Equations r 1 r ( r ) Fr 0 r r r r 1 2 r F 0 r r r Compatibility Equations P lane Stress E u r u r 1 u Fr 0 2(1 ) r r r r E 1 u r u r 1 u 2 u F 0 2(1 ) r r r r 2 u r 2 1 1 2 2 r r r r 2 2 2 Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island Plane Strain 1 Fr Fr 1 F 1 r r r Plane Stress 2 ( r ) F F 1 F 2 ( r ) (1 ) r r r r r Polar Coordinate Formulation Airy Stress Function Approach = (r,θ) Airy Representation 1 1 2 r r r r 2 2 2 2 r 1 r r r Biharmonic Governing Equation 2 1 1 2 2 1 1 2 2 2 2 2 2 2 0 r r r r r r r r 4 S r r R y Traction Boundary Conditions Tr f r (r, ) , T f (r, ) r x Elasticity Theory, Applications and Numerics M.H. Sadd , University of Rhode Island