DEPARTMENT OF MECHANICAL ENGINEERING INDIAN INSTITUTE OF TECHNOLOGY - GUWAHATI ME 664 : Theory of Elasticity 2013-2014, IInd semester ASSIGNMENT - 2 on Strain-Displacement Relations 1. Consider the relation ai = −1/2 ϵijk ajk where aij is a anti-symmetric tensor. Show that this expression can be inverted leading to the expression ajk = −ϵijk ai . The vector ai is called the dual vector of aij . (Hint: Use the e − δ identity and argue the resulting expression using the property of anti-symmetric tensor.) 2. Derive the strain-displacement relation in cylindrical coordinate system. (Hint: Use the relation for gradient of a vector in cylindrical coordinate system to calculate the gradient of displacement vector in cylindrical coordinate system.) 3. Derive the strain-displacement relation in spherical coordinate system. (Hint: Use the relation for gradient of a vector in spherical coordinate system to calculate the gradient of displacement vector in spherical coordinate system.) 4. Using the strain-displacement relation in cylindrical coordinates (in two-dimensional case), determine the strains er , eθ , and erθ for the following displacement fields: (i) ur = A , r uθ = B cos θ, (ii) ur = A r2 , uθ = B r sin θ, (iii) ur = A sin θ + B cos θ, uθ = A cos θ − B sin θ + Cr, where A, B, and C are arbitrary constants. 5. The displacement field in a solid body is given by u = u1 (x1 , x2 , x3 ) e1 + u2 (x1 , x2 , x3 ) e2 + u3 (x1 , x2 , x3 ) e3 (1) where u1 (x1 , x2 , x3 ) = 3x21 x3 + 60x1 , u2 (x1 , x2 , x3 ) = 5x23 + 20x1 x2 , and u3 (x1 , x2 , x3 ) = 6x23 + 2x1 x2 x3 . Find the components of the strain tensor at a point P whose coordinates are (3, 4, 0.5) units. Also, determine the principal strains and the principal axes. Then, diagonalize the strain tensor. 1 6. The strain tensor (with the tensorial shearing strains) at a point in a body is given by   12 3 4  3 8 −4  4 −4 18 Determine the principal strains and the principal axes. 7. The strain field e (with engineering shear) in a body is given by   0.005 0.003 0.002x2 x3  0.003 0.004 0.003 + 0.002x1 x3  0.002x2 x3 0.003 + 0.002x1 x3 0.002x1 x2 Check if it is a compatible strain field. 8. The strain field e (with engineering shear) in a body is given by   0.005x3 0.003x1 x2 0.001x2  0.003x1 x2 0.001x1 −0.001x1 x3  0.001x2 −0.001x1 x3 −0.002x1 x2 Check whether it is a compatible strain field. 9. The strain field e (with engineering shear) in a body is given by   a1 (x21 + x22 ) a4 x1 x2 a6 x3 x1  a4 x1 x2 a2 (x22 + x23 ) a5 x2 x3  2 a6 x3 x1 a5 x2 x3 a3 (x3 + x21 ) Determine relation between the constants a1 − a6 so that the strain field is a possible strain field. 10. Determine the strain tensor e and rotation tensor ω for the displacement field given by u1 (x1 , x2 , x3 ) = Ax21 , u2 (x1 , x2 , x3 ) = Bx1 x2 , and u3 (x1 , x2 , x3 ) = Cx1 x2 x3 where A, B, and C are arbitrary constants. 11. A two dimensional problem of a rectangular bar stretched by uniform end loadings results in the following constant strain field e:   A 0 0  0 −B 0  0 0 0 where A and B are arbitrary constants. Assuming the field depends only on x1 and x2 , integrate the strain-displacement relations to determine the displacement components and identify any rigid-body motion terms. 12. A three-dimensional elasticity problem of a uniform bar stretched under its own 2 weight gives the following strain field e:   Ax3 0 0  0 Ax3 0  0 0 Bx3 where A and B are constants. Integrate the strain-displacement relations to determine the displacement components and identify any rigid-body motion terms. 13. A rectangular parallelepiped with original volume V0 is oriented such that its edges are parallel to the principal directions of strain. For small strains, show that the dilation is given by change in volume ∆V ekk = . (2) = original volume V0 14. Consider the compatibility relations given by Eqs. (13 a - 13 f) in Lecture 6. Show that the L.H.S. equals R.H.S. using the strain-displacement relation given by Eq. (16) in Lecture 5. 15. (i) Use the e − δ identity to show that the compatibility relation given by Eq. (12) in Lecture 6 can be written in short as ηij = ϵikl ϵjmp elp,km . (ii) Show that ηij = ∇ × e × ∇. (iii) Show that ηij,j = 0. 16. In cylindrical polar coordinates, the strain-displacement relations for the ”in-plane” strains are err = erθ = eθθ = ∂ur , ∂r( ) 1 1 ∂ur ∂uθ uθ + − , 2 r ∂θ ∂r r ur 1 ∂uθ + . r r ∂θ Use these relations to obtain a compatibility equation that must be satisfied by the three strains. 17. If no stresses occur in a body, an increase in temperature T causes unrestrained thermal expansion defined by the strains e11 = e22 = e33 = αT , e12 = e23 = e31 = 0 . Show that this is possible only if T is a linear function of x1 , x2 , x3 and that otherwise stresses must be induced in the body, regardless of the boundary conditions. (Hint: Use compatibility relations to show that the gradient of temperaturez is constant.) 3