166
4 Problems in Soft Tissue Biomechanics
(D) Determine the Cartesian components of the first Piola-Kirchhoff stress tensor
using (1) the equation P = J F −1 · σ as provided by (3.237); and (2) the
constitutive relation
P =
∂W
∂F T − Jp F
−1 .
(4.26)
4.3.2 Solution
Since this problem involves shear relative to the specified coordinates, we cannot
use the simplified equations of Sect. 3.7.3. Rather than jumping immediately to the
scalar equations for Cartesian coordinates, we use the general tensor relations of
Sect. 3.7.1. This allows us to more easily employ matrix algebra, for which symbolic
manipulation software is convenient.
Part A: Strain With Eqs. (4.25) defining the deformed position vector r = x e x +
y e y + z e z , the transpose of the deformation gradient tensor is given by
F
T
= ∇r =
e x
∂
∂X
+ e y
∂
∂Y
+ e z
∂
∂Z
x(X, Y )e x + y(Y )e y + z(Z)e z
= e x e x
∂x
∂X
+ e y e x
∂x
∂Y
+ e y e y
∂y
∂Y
+ e z e z
∂z
∂Z
= e x e x + ke y e x + e y e y + e z e z .
Taking the transpose of both sides yields the component matrices
F =
⎡
⎣
1 k 0
0 1 0
0 0 1
⎤
⎦ ,
F
−1
=
⎡
⎣
1 −k 0
0 1 0
0 0 1
⎤
⎦ ,
(4.27)
where the inverse is computed using Eq. (2.25). Consistent with the observation that
the prescribed deformation does not change the volume of the block (see Fig. 4.6),
the incompressibility condition J = det F = 1 is satisfied identically for any value
of k.
The expression
E =
1
2
F
T
· F − I
for the Lagrangian strain tensor provides the Cartesian strain components
⎡
⎣
E xx E xy E xz
E yx E yy E yz
E zx E zy E zz
⎤
⎦ =
⎡
⎣
0
1
2 k 0
1
2 k
1
2 k 2 0
0 0 0
⎤
⎦ .
(4.28)