CONTINUUM MECHANICS for ENGINEERS

where the factor of two on the shear strain components is introduced in
keeping with Eq 4.7-14. From these definitions, Hooke’s law is now written
σα = Cαβ εβ or = Cε
(6.1-8)
with Greek subscripts having a range of six. In matrix form Eq 6.1-8 appears
as
⎡σ
1⎤
⎡C
C C C C C
⎢
σ
⎥ ⎢
⎢ 2 C C C C C C
⎥ ⎢
⎢σ
⎥ ⎢ 3 C C C C C C
⎢ ⎥ = ⎢
⎢σ
4⎥
⎢C
C C C C C
⎢σ
⎥ ⎢
5 C C C C C C
⎢ ⎥ ⎢
⎣⎢
σ 6⎦⎥
⎣⎢
C C C C C C
11 12 13 14 15 16
21 22 23 24 25 26
31 32 33 34 35 36
41 42 43 44 45 46
51 52 53 54 55 56
61 62 63 64 65 66
⎤ ⎡ε1
⎤
⎥ ⎢
ε
⎥
⎥ ⎢ 2⎥
⎥ ⎢ε
⎥ 3
⎥ ⎢ ⎥
⎥ ⎢ε4
⎥
⎥ ⎢ε
⎥
5
⎥ ⎢ ⎥
⎦⎥
⎣⎢
ε6
⎦⎥
(6.1-9)
We point out that the array of the 36 constants C αβ does not constitute a tensor.
In view of our assumption to neglect thermal effects at this point, the
energy balance Eq 5.7-13 is reduced to the form
˙u= 1 ijDij ρ σ
which for small-deformation theory, by Eq 4.10-18, becomes
˙ ˙
u= 1 ij ij
ρ σε
(6.1-10a)
(6.1-10b)
The internal energy u in these equations is purely mechanical and is called
the strain energy (per unit mass). Recall now that, by the continuity equation
in Lagrangian form, ρ o = ρJ and also that to the first order of approximation
⎛ ∂ u ⎞ i ∂ ui
J = det F = det⎜δ iA +
(6.1-11)
⎝ ∂ X
⎟ ≈ +
⎠ ∂ X
1
Therefore, from our assumption of small displacement gradients, namely
∂u i/∂X A