CONTINUUM MECHANICS for ENGINEERS

is obtained by differentiating Eq 4.6-20a. Therefore, to the first order of
approximation for the case of small displacement gradients, it is unimportant
whether we differentiate the displacement components with respect to the
material or spatial coordinates. In view of this, we may display the equivalent
relative displacement gradients for small deformation theory as either u i,A
or u i,j. Similarly, it can be shown that in the linear theory u A,B and u A,j are
equivalent. It follows that to the same order of approximation, from Eqs 4.7-1
and 4.7-2,
(4.7-4)
and it is customary to define a single infinitesimal strain tensor for which we
introduce the symbol ε ij as
∂u
∂u
∂u
∂u
i
j
i j
2εij = δ δ
(4.7-5)
Aj + Bi = + = ui,j + uj,i
∂X
∂X
∂x
∂x
Because the strain tensors E AB, e ij, and ε ij are all symmetric, second-order
tensors, the entire development for principal strains, strain invariants, and
principal strain directions may be carried out exactly as was done for the
stress tensor in Chapter Three. Thus, taking ε ij as the typical tensor of the
group, we summarize these results by displaying its matrix relative to principal
axes in the alternative forms,
together with the strain invariants
A
E ≈ e δ δ
AB ij iA jB
I ε = ε ii = tr ε = ε I + ε II + ε III
(4.7-6)
(4.7-7a)
II ε = (ε iiε jj – ε ijε ji) = ε Iε II + ε IIε III + ε IIIε I (4.7-7b)
III ε = ε ijk ε 1iε 2jε 3k = ε Iε IIε III
B
ε(
)
ε
* [ εij]
ε(
)
ε
ε(
)
ε
=
⎡
⎤
1 0 0 ⎡ I 0 0 ⎤
⎢
⎥ ⎢
⎥
⎢ 0 2 0 ⎥ =
⎢
0 II 0
⎥
⎢
⎥
⎣ 0 0 ⎦ ⎣
⎢
⎦
⎥
3 0 0 III
1
2
(4.7-7c)
The components of ε have specific physical interpretations which we now
consider. Within the context of small deformation theory we express
Eq 4.6-14 in the modified form
j
(dx) 2 – (dX) 2 = 2ε ijdX idX j = dX ⋅ 2ε ⋅ dX (4.7-8)
i