Page 278 - Intro to Tensor Calculus

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EXERCISE 2.4

I 1. Verify the generalized Hooke’s law constitutive equations for hexagonal materials.

In the following problems the Young’s modulus E, Poisson’s ratio ν, the shear modulus or modulus
of rigidity µ (sometimes denoted by G in Engineering texts), Lame’s constant λ and the bulk modulus of
elasticity k are assumed to satisfy the equations (2.4.19), (2.4.24) and (2.4.25). Show that these relations
imply the additional relations given in the problems 2 through 6.

I 2.
µ(3λ +2µ) 9k(k − λ) 9kµ
E = E = E =
µ + λ 3k − λ µ +3k
λ(1 + ν)(1 − 2ν)
E = E =2µ(1 + ν) E =3(1 − 2ν)k
ν
I 3.
2
2
p
3k − E (E + λ) +8λ − (E + λ) E − 2µ
ν = ν = ν =
6k 4λ 2µ
λ 3k − 2µ λ
ν = ν = ν =
2(µ + λ) 2(µ +3k) 3k − λ
I 4.
p 2 2 E
(E + λ) +8λ +(E +3λ) k = 2µ(1 + ν)
k = 3(1 − 2ν) k =
6 3(1 − 2ν)
2µ +3λ µE λ(1 + ν)
k = k = k =
3 3(3µ − E) 3ν
I 5. p
2
2
3(k − λ) 3k(1 − 2ν) (E + λ) +8λ +(E − 3λ)
µ = µ = µ =
2 2(1 + ν) 4
λ(1 − 2ν) 3Ek E
µ = µ = µ =
2ν 9k − E 2(1 + ν)
I 6.
3kν 3k − 2µ νE
λ = λ = λ =
1+ ν 3 (1 + ν)(1 − 2ν)
µ(2µ − E) 3k(3k − E) 2µν
λ = λ = λ =
E − 3µ 9k − E 1 − 2ν
I 7. The previous exercises 2 through 6 imply that the generalized Hooke’s law

σ ij =2µe ij + λδ ij e kk

is expressible in a variety of forms. From the set of constants (µ,λ,ν,E,k) we can select any two constants
and then express Hooke’s law in terms of these constants.
(a) Express the above Hooke’s law in terms of the constants E and ν.
(b) Express the above Hooke’s law in terms of the constants k and E.
(c) Express the above Hooke’s law in terms of physical components. Hint: The quantity e kk is an invariant
hence all you need to know is how second order tensors are represented in terms of physical components.
See also problems 10,11,12.