204
4 Problems in Soft Tissue Biomechanics
4.4 Consider combined extension and shear of a block described by
x = λ x X + kY
y = Y
z = λ z Z
in Cartesian coordinates, where λ x , λ y , and k are constants. The block is
composed of compressible isotropic material with
W = c
I 1 − 3 + β
−1
I
−β
3 − 1
,
where β = ν/(1 − 2ν), and Eqs. (3.69) give the strain invariants in terms of
Lagrangian strains. Both λ x and k are specified, and the block is stress free in
the z-direction.
(a) Determine the Lagrange strain tensor in terms of λ x , λ z , and k.
(b) Write the components of second Piola-Kirchhoff stress tensor (S ij ) as
functions of the strain components.
(c) Determine λ z using the plane-stress condition S zz = 0.
(d) Show that incompressibility is satisfied when ν = 0.5.
4.5 Consider combined extension and torsion of a cylindrical bar with undeformed
radius b 0 (see Sect. 4.5).
(a) Write a computer program to solve the problem and use it to reproduce
the results in Fig. 4.12a, c.
(b) Suppose the bar is composed of neo-Hookean material with W = c(I 1 −
3). Show that Eqs. (4.87) give the axial force and twisting moment as
N = 2πb
2
0 c
λ −
1
λ 2 −
ψ 2 b 2
0
4λ 2
M = πb
4
0 ψc/λ,
where ψ is the angle of twist per unit undeformed length, and λ is the
axial stretch ratio.
4.6 In the undeformed state, a tube has an inner radius a 0 and outer radius b 0 .
With (R, ,, Z) and (r, θ, z) being cylindrical coordinates to a point before
and after deformation, respectively, the tube undergoes combined extension,
inflation, and torsion described by
r = r(R)
θ = + ψZ
z = λZ,
4 Problems in Soft Tissue Biomechanics
4.4 Consider combined extension and shear of a block described by
x = λ x X + kY
y = Y
z = λ z Z
in Cartesian coordinates, where λ x , λ y , and k are constants. The block is
composed of compressible isotropic material with
W = c
I 1 − 3 + β
−1
I
−β
3 − 1
,
where β = ν/(1 − 2ν), and Eqs. (3.69) give the strain invariants in terms of
Lagrangian strains. Both λ x and k are specified, and the block is stress free in
the z-direction.
(a) Determine the Lagrange strain tensor in terms of λ x , λ z , and k.
(b) Write the components of second Piola-Kirchhoff stress tensor (S ij ) as
functions of the strain components.
(c) Determine λ z using the plane-stress condition S zz = 0.
(d) Show that incompressibility is satisfied when ν = 0.5.
4.5 Consider combined extension and torsion of a cylindrical bar with undeformed
radius b 0 (see Sect. 4.5).
(a) Write a computer program to solve the problem and use it to reproduce
the results in Fig. 4.12a, c.
(b) Suppose the bar is composed of neo-Hookean material with W = c(I 1 −
3). Show that Eqs. (4.87) give the axial force and twisting moment as
N = 2πb
2
0 c
λ −
1
λ 2 −
ψ 2 b 2
0
4λ 2
M = πb
4
0 ψc/λ,
where ψ is the angle of twist per unit undeformed length, and λ is the
axial stretch ratio.
4.6 In the undeformed state, a tube has an inner radius a 0 and outer radius b 0 .
With (R, ,, Z) and (r, θ, z) being cylindrical coordinates to a point before
and after deformation, respectively, the tube undergoes combined extension,
inflation, and torsion described by
r = r(R)
θ = + ψZ
z = λZ,