Strains (Rigorous Version) PDF

Strains (Rigorous Version) Strain: a measure of deformation – Normal strain: change in length of a line segment – Shear strain: change in angle between two perpendicular line segments Displacement of P = (u1, u2, u3) Displacement of Q & R ∂u1 ∂u1 u1Q =u1 +...

Strains (Rigorous Version) Strain: a measure of deformation – Normal strain: change in length of a line segment – Shear strain: change in angle between two perpendicular line segments Displacement of P = (u1, u2, u3) Displacement of Q & R ∂u1 ∂u1 u1Q =u1 + ∆x1 u1R =u1 + ∆x2 ∂x1 ∂x2 R' ∂u2 ∂u2 u2Q =u2 + ∆x1 u2R =u2 + ∆x2 ∂x1 ∂x2 Q' ∂u3 ∂u3 R P'(x1+u1, x2+u2, x3+u3) u3Q =u3 + ∆x1 R u3 =u3 + ∆x2 ∆x2 ∂x1 ∂x2 x 2 ∆x1 P(x1,x2,x3) Q x1 x3 Displacement Field Coordinates of P, Q, and R before and after deformation P : (x1 , x2 , x3 ) Q : (x1 + ∆x1 , x2 , x3 ) R : (x1 , x1 + ∆x2 , x3 ) P′ : (x1 + u1P , x2 + u2P , x3 + u3P ) = (x1 + u1 , x2 + u2 , x3 + u3 ) Q′ : (x1 + ∆x1 + u1Q , x2 + u2Q , x3 + u3Q ) ∂u1 ∂u ∂u = (x1 + ∆x1 + u1 + ∆x1 , x2 + u2 + 2 ∆x1 , x3 + u3 + 3 ∆x1 ) ∂x1 ∂x1 ∂x1 R′ : (x1 + u1R , x2 + ∆x2 + u2R , x3 + u3R ) ∂u1 ∂u ∂u = (x1 + u1 + ∆x2 , x2 + ∆x2 + u2 + 2 ∆x2 , x3 + u3 + 3 ∆x2 ) ∂x2 ∂x2 ∂x2 Length of the line segment P'Q' ( ) +( ) +( ) 2 2 2 P′Q′ = x1P′ − x1Q′ x2P′ − x2Q′ x3P′ − x3Q′ Deformation Field Length of the line segment P'Q' 2 2 2  ∂u   ∂u   ∂u  ∆x1  1 + 1  +  2  +  3  P′Q′ =  ∂x1   ∂x1   ∂x1  2 1/2  2 2 ∂u1  ∂u1   ∂u2   ∂u3   = ∆x1  1 + 2 +  +  +    ∂x1  ∂x1   ∂x1   ∂x1      ∂u1 1  ∂u1  2 1  ∂u2  2 2 1  ∂u3    ∂u1  ≈ ∆x1  1 + +   +   +    ≈ ∆x  1 +   ∂x1 2  ∂x1  2  ∂x1  2  ∂x1   ∂x    1  Linear Nonlinear Ignore H.O.T. when displacement gradients are small Linear normal strain P′Q′ − PQ ∂u1 ε11 = PQ ∂x1 ∂u2 ∂u3 =ε22 , =ε33 ∂x2 ∂x3 Deformation Field Shear strain 𝛾𝛾𝑥𝑥𝑥𝑥 – change in angle between two lines originally parallel to x– and y–axes x2Q′ − x2QP’ ∂u2 x1R′ − x1RP’ ∂u1 =θ1 = =θ2 = ∆x1 ∂x1 ∆x2 ∂x2 ∂u1 ∂u2 γ12 = θ1 + θ2 = + ∂x2 ∂x1 ∂u2 ∂u3 γ23 = + Engineering shear strain ∂x3 ∂x2 ∂u3 ∂u1 γ13= + ∂x1 ∂x3 Different notations 1  ∂u1 ∂u2  1  ∂ui ∂uj  = ε12 + 2  ∂x2 ∂x1   = εij  +  2  ∂xj ∂xi  1  ∂u2 ∂u3  = ε23 + 2  ∂x3 ∂x2   ε= ij 1 (u + uj,i ) 2 i,j 1  ∂u3 ∂u1  = ε13 + 2  ∂x1 ∂x3   =ε sym(∇u)