82
3 Continuum Mechanics and Nonlinear Elasticity
Since I = e R e R + e e + e Z e Z , the above equations give the components of F
as
F RR = 1 +
∂u R
∂R
F = 1 +
1
R
u R +
∂u
∂∂
F RR =
1
R
∂u R
∂∂
− u
F R =
∂u
∂R
.
The components of E are
E RR =
∂u R
∂R
+
1
2
∂u R
∂R
2
+
∂u
∂R
2
E =
1
R
u R +
∂u
∂∂
+
1
2R 2
u R +
∂u
∂∂
2
+
∂u R
∂∂
− u
2
E RR = E R =
1
2
∂u
∂R
+
1
R
∂u R
∂∂
− u
+
1
2R
∂u
∂R
u R +
∂u
∂∂
+
∂u R
∂R
∂u R
∂∂
− u
.
These strain components also could be determined using E =
1
2 (F T · F − I) and the
F I J given above.
Perhaps it is becoming clear why problems in nonlinear elasticity are often so
complicated. (And the equations in the last example are only for two dimensions!)
Largely because of geometric nonlinearities, analytic solutions exist for only
relatively simple problems. (Actually, to a lesser extent, the same could be said
for linear elasticity.) To solve problems involving realistic geometries, numerical
methods such as finite elements are often used. Nevertheless, for many problems,
much insight into fundamental mechanics can be gained by first considering a model
based on simplified geometry, such as using a cylinder to approximate the trachea,
an artery, or even the elliptical left ventricle. From this starting point, additional
complexities can be included one step at a time, with the previous model used as a
special case to check the next model.