3.3 Analysis of Deformation
81
Example 3.13 In two dimensions, the displacement vector in polar coordinates
(R, ,) can be written as
u(R, ,) = u R e R + u e ,
and the gradient operator has the form
∇ = e R
∂
∂R
+
e
R
∂
∂∂
relative to the undeformed configuration. Write the 2D components of the deformation gradient tensor and the Lagrangian strain tensor as functions of the
displacements u R and u .
Solution
In dyadic form, the deformation gradient and strain tensors are
F = F RR e R e R + F RR e R e + F R e e R + F e e
E = E RR e R e R + E RR e R e + E R e e R + E e e .
In terms of displacements, Eqs. (3.47) and (3.57) give
F = I + (∇u)
T
E =
1
2 [∇u + (∇u)
T
+ (∇u) · (∇u)
T
].
These relations require the displacement gradient, which is provided by
∇u =
e R
∂
∂R
+ e
1
R
∂
∂∂
(u R e R + u e )
= e R
∂u R
∂R
e R + u R
∂e R
∂R
+
∂u
∂R
e + u
∂e
∂R
+
e
R
∂u R
∂∂
e R + u R
∂e R
∂∂
+
∂u
∂∂
e + u
∂e
∂∂
=
∂u R
∂R
e R e R +
∂u
∂R
e R e
+
1
R
∂u R
∂∂
− u
e e R +
1
R
u R +
∂u
∂∂
e e ,
in which Eqs. (2.3) have been used for the derivatives of the unit vectors.