CONTINUUM MECHANICS for ENGINEERS

FIGURE 4.7
Differential velocity field at point p.
Consider the velocity components at two neighboring points p and q. Let
the particle currently at p have a velocity v i, and the particle at q a velocity
v i + dv i as shown in Figure 4.7. Thus the particle at q has a velocity relative
to the particle at p of
Note that
or in symbolic notation
∂vi
dvi = dxj or dv = L ⋅ dx (4.10-5)
∂x
j
∂v
∂ v ∂ X d ⎛ ∂ x ⎞
i i A
i ∂ X
= =
∂ x ∂ X ∂ x dt
⎜
⎝ ∂ X
⎟
⎠ ∂ x
j
A
j
(4.10-6a)
L = ⋅ F –1 (4.10-6b)
˙F
where we have used the fact that material time derivatives and material
gradients commute. Therefore,
˙ F
= L ⋅ F (4.10-7)
Consider next the stretch ratio Λ = dx/dX where Λ is as defined in Eq 4.8-
4, that is, the stretch of the line element dX initially along and currently
along . By the definition of the deformation gradient, dxi = xi,A dXA, along
with the unit vectors ni = dxi /dx and NA = dXA /dX we may write
ˆ N
ˆn
dx n i = x i,A dX N A
A
A
j