78
3 Continuum Mechanics and Nonlinear Elasticity
is the right Cauchy-Green deformation tensor. 5 Because C T = (F T · F) T = F T ·
(F T ) T = F T · F = C, C is a symmetric tensor.
Analogous to the 1D definition of strain given by Eq. (3.30) 3 , the 3D Lagrangian
strain tensor E is defined by the relation
ds
2
− dS
2
= 2 dR · E · dR.
(3.54)
Substituting Eqs. (3.52) yields
ds
2
− dS
2
= dR · C · dR − dR · dR = dR · (C − I) · dR
(3.55)
and hence
E =
1
2 (C − I) =
1
2 (F T · F − I),
(3.56)
which also is a symmetric tensor. Finally, substituting (3.47) into this equation and
expanding the dot product yield the strain-displacement relation
E =
1
2 [∇u + (∇u) T + (∇u) · (∇u) T ].
(3.57)
The nonlinear term (∇u) · (∇u) T considerably complicates analysis. Ignoring this
term yields the linear strain tensor.
Next, we derive the component forms of these equations. In orthogonal curvilinear coordinates, the various tensors in this section can be written as
F = F ij e i e j ,
C = C ij e i e j
E = E ij e i e j ,
I = δ ij e i e j
(3.58)
in terms of the unit base vectors e i . Inserting the first of these relations into (3.53)
yields
C = F
T
· F
= (F ij e i e j )
T
· (F kl e k e l ) = (F ij e j e i ) · (F kl e k e l )
= F ij F kl δ ik e j e l = F ij F il e j e l
= F ki F kj e i e j ,
in which indices have been exchanged in the last line. Hence, C ij = F ki F kj and
Eq. (3.56) gives
5 The left Cauchy-Green deformation tensor, defined as B = F · F T , is not used in this book.