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3 Continuum Mechanics and Nonlinear Elasticity
Fig. 3.7 Deformation
gradient tensor F transforms
length element dR in
undeformed body B into
length element dr in
deformed body b
F
dR
dr
B
b
R
r
F at that point. Consequently, in 3D we stipulate that F transforms a general line
element dR in the undeformed body into the element dr in the deformed body
(Fig. 3.7). This transformation is defined through the relation
dr = F · dR = dR · F
T ,
(3.42)
where the second form for dr follows from the fourth identity in Table 2.3. Here,
it is important to emphasize that dR changes its length and rotates while being
transformed into dr. Thus, the deformation gradient tensor contains information on
both deformation and rigid-body rotation.
To determine F, we consider a general deformation field described by the
mapping
r = r(R),
(3.43)
where the position vectors R and r define the locations of the elements dR and dr,
respectively. Replacing φ by r and a by R in Eq. (2.56) yields
dr = dR ·
∂r
∂R
= dR · ∇r,
(3.44)
where
∇ =
∂
∂R
(3.45)
is the gradient operator in the undeformed body. Comparison with Eq. (3.42) shows
that
F T = ∇r
→
F = (∇r) T .
(3.46)
Appendix A provides the components of F for general 3D deformation in Cartesian,
cylindrical, and spherical coordinates.