Atkin R. J. - An Introduction to the Theory of Elasticity

(1.11.3)

(1.15.1)

• 1. The motion represents a uniform rotation in which particles move in circles with angular speed
  
• 2. J = ,
  
• 4. Principal axes are inclined to the 1,2-axes at an angle .
• 5. The plane X 2 = is deformed into the plane x2 = tan ()x1. , where A is an arbitrary constant.
• 6. dl = {(1 + )2 + 2 + 2(1 + ) sin .
• 7. Each particle in the plane X 3 = constant is displaced a distance proportional to X 3 and to the distance from the X 3-axis, the direction of the displacement being at right angles to the radius vector ( X 1, X 2).
  

  The deformation is not isochoric (unless = 0). The cylinder deforms into the hyperboloid of revolution

  
• 9. t = 2(1, 1, 1), = 4, = 23. The principal stresses are -2, 1, 4.

  Direction cosines of the principal axes are

  

  respectively.

• 10.
  
• 11. The components of the surface traction on each face are:
  
• 12. Body force is -2( x 1, x 2, 0).
• 13.
• 14. T ( r ) = A/r 2, where A is an arbitrary constant.

• 1. < 1
• 2. Shearing of the planes X 2 = constant of amount together with stretches in the 1- and 3-directions and in the 2-direction.
• 5.
  
• 6. p = 2( C 1 C 2) + 4 C 2 A 2 8 k 2( C 1 + C 2)A2. The resultant force is 8( C 1+ C 2) A 2 L .
• 7. , where from the boundary conditions
  
• 8.
  

• 1.
• 3. where
  
• 4.
• 7.
  • (i) Support on rigid plinth with profile .
  • (ii) Apply a uniform pressure p gl over the cross-section.