CONTINUUM MECHANICS for ENGINEERS

upon X, in which case the deformation is termed inhomogeneous. If F is
independent of X, the deformation is called homogeneous. In symbolic notation
Eq 4.6-7 appears in either of the forms
dx = F ⋅ dX or dx = FdX (4.6-9)
where, as indicated by the second equation, the dot is often omitted for
convenience. In view of the smoothness conditions we have imposed on the
mapping function χ we know that F is invertible so that the inverse F –1 exists
such that
dX A = X A,i dx i or dX = F –1 ⋅ dx (4.6-10)
In describing motions and deformations, several measures of deformation
are commonly used. First, let us consider that one based upon the change
during the deformation in the magnitude squared of the distance between
the particles originally at P and Q, namely,
(dx) 2 – (dX) 2 = dx i dx i – dX A dX A
which from Eq 4.6-7 and the substitution property of the Kronecker delta
δ AB may be developed as follows,
where the symmetric tensor
(dx) 2 – (dX) 2 = (x i,AdX A)(x i,BdX B) – δ ABdX AdX B
= (x i,A x i,B – δ AB )dX A dX B
= (C AB – δ AB)dX AdX B
(4.6-11)
C AB = x i,A x i,B or C = F T ⋅ F (4.6-12)
is called the Green’s deformation tensor. From this we immediately define the
Lagrangian finite strain tensor E AB as
2E AB = C AB – δ AB or 2E = C – I (4.6-13)
where the factor of two is introduced for convenience in later calculations.
Finally, we can write,
(dx) 2 – (dX) 2 = 2E ABdX AdX B = dX ⋅ 2E ⋅ dX (4.6-14)