Page 351 - Introduction to Continuum Mechanics

Problems 335

PROBLEMS
5.1. Show that the null vector is the only isotropic vector.
(Hint: Assume that a is an isotropic vector, and use a simple change of basis to equate the
primed and the unprimed components)
5.2. Show that the most general isotropic second-order tensor is of the form a I, where a is a
scalar and I is the identity tensor.
5.3. Show that for an anisotropic linear elastic material, the principal directions of stress and
strain are usually not coincident.
5.4. If the Lame constants for a material are

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A = 119.2 GPa(17.3xl0 psi), fi = 79.2 GPa (11.5 xl0 psi),
find Young's modulus, Poisson's ratio, and the bulk modulus.
5.5. Given Young's modulus Ey - 103 GPa and Poisson's ratio v = 0.34, find the Lame
constants A and^w. Also find the bulk modulus.
5.6. Given Young's modulus Ey =193 GPa and shear modulus /* = 76 GPa, find Poisson's
ratio v, Lame's constant A and the bulk modulus k

5.7. If the components of strain at a point of structural steel are

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find the stress components, A = 119.2 GPa(17.3 x 10 psi), n = 79.2 GPa (ll.SxlO psi).
5J. Do Problem 5.7 if the strain components are

5.9. (a) If the state of stress at a point of structural steel is

what are the strain components? Ey = 207 GPa, ft = 79.2 GPa, v = 0.30

(b) Suppose that a five centimeter cube of structural steel has a constant state of stress giver
in part (a). Determine the total change in volume induced by this stress field.
5.10. (a) For the constant stress field below, find the strain components