CONTINUUM MECHANICS for ENGINEERS

so that the relative differential displacement is seen to be the result of a rigid
body rotation about the axis of the rotation vector . On the other hand, if
ω ij ≡ 0, the relative displacement will be the result of pure strain.
Finally, if we consider the six independent strain-displacement relations,
Eq 4.7-5
as a system of partial differential equations for determining the three displacement
components u i (assuming the ε ij are known as functions of x i), the
system is over-determined, and we cannot in general find three single-valued
functions u i = u i (x j) satisfying the six partial differential equations. Therefore,
some restrictive conditions must be imposed upon the strain components
(actually upon derivatives of the strain components) if the equations above
are to be satisfied by a single-valued displacement field. Such conditions are
expressed by the strain compatibility equations
ε ij,km + ε km,ij – ε ik,jm – ε jm,ik = 0 (4.7-32)
There are 3 4 = 81 equations in all (four free indices) in Eq 4.7-32, but only six
of these are distinct. It may be shown that these six conditions are both
necessary and sufficient for a single-valued displacement field of a body
occupying a simply connected domain.
For plane strain in the x 1x 2 plane, the six unique equations in Eq 4.7-32
reduce to a single equation,
ε 11,22 + ε 22,11 = 2ε 12,12
(4.7-33)
which may be easily verified as a necessary condition by a simple differentiation
of Eq 4.7-5 for a range of two on the indices i and j.
4.8 Stretch Ratios
∂ u ∂ u
i j
+ =2εij
∂ x ∂ x
j
Referring again to Figure 4.2, we define the ratio of the magnitudes of dx
and dX to be the stretch ratio, Λ (or simply the stretch). In particular, for the
differential element in the direction of the unit vector at P, we write
ˆN
Λ ( ˆ )
N = dx
dX
i
(4.8-1)