AE350_Elasticity

AE350_Elasticity - Introduction to Elasticity Elasticity -...

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Introduction to Elasticity Elasticity - Study of strain - displacement relations, equilibrium equations, and constitutive equations for elastic (not rigid, not plastic) structures. Displacement is the change of position during deformation. p(x , y , z) p’(x’ , y’ , z’) x’ = x + u or u = x’ – x y’ = y + v or v = y’ – y z’ = z + w or w = z’ – z Displacement U = ui + vj + wk z y x r r’ p p’ AE/ME350 Jha Elasticity-1
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Normal and Shear Strains Normal Strain ( = for uniform strain in axial members ) Shear Strain x u x u x xx 0 lim O L L z w y v zz yy  ; y x ) , ( y y x R ) , ( ' y y u x R 1 2 ) , ( y x x Q u v x y ) ' , ' ( ' ), , ( y x P y x P '( , ) Qx xy v  AE/ME350 Jha Elasticity-2
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Shear Strains Points (P,Q,R) before deformation; points (P’,Q’,R’) after deformation Rotation of PQ due to deformation ( assumed to be small) Rotation of PR due to deformation Total change of angle between PQ and PR after deformation is defined as the shear strain (in x-y plane) x v x v x 0 1 lim 1 y u y u y 0 2 lim y u x v yx xy 2 1   AE/ME350 Jha Elasticity-3
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Shear Strains Shear strains in y-z plane and x-z plane are given by General deformation can be described by three normal strains and three shear strains See example 2.1 in Text (Sun) z u x w zx xz   z v y w yz zy   zz yy xx , , xz yz xy , , AE/ME350 Jha Elasticity-4
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Rigid Body Motion Translation if every point has the same displacement along X- axis (or Y-axis or Z-axis) Another rigid body motion is the rigid body rotation, and is represented in the X-Y plane by displacements, Rigid body motion (translation or rotation) does not generate any strain constant o uu ;; 0 uy v x w  AE/ME350 Jha Elasticity-5
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Example Problem Consider the displacement field in a body given below and find normal strain, shear strain, and the location of the point (5,0,0) after deformation.
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AE350_Elasticity - Introduction to Elasticity Elasticity -...

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