334
6 Growth
Fig. 6.33 Membrane with
attached springs
(Problem 6.3)
x
y
b
a
V 0
V 0
k
transversely isotropic, consisting of an isotropic matrix (volume fraction φ m )
with embedded fibers (volume fraction φ f ) aligned parallel to the x-axis. The
strain-energy density functions for the matrix and fibers, respectively, are
W m = c m (λ
2
x + λ
2
y + λ
2
z − 3)
W f = c f (λ x − 1)
2 ,
and the total Cauchy stress tensor is given by Eqs. (6.105) in the form
¯
σ = φ m ¯
σ m + φ f σ f − p I.
The fibers do not grow, but the matrix grows according to the growth tensor
G = G(e x e x + e y e y ) + e z e z ,
where G is a known constant.
(a) When stretched a distance , the springs exert a force kk per unit
undeformed area of the edge of the membrane. Express the Cauchy stress
σ s exerted by the springs on the membrane in terms of λ y and G.
(b) Given σ 0 and G, derive two equations to be solved for λ x and λ y . Hint:
The total stretch ratio λ x is the same for the matrix and the fibers.
6.4 Write a computer program to solve the problem studied in Sect. 6.7, which
involves residual stress in a tube caused by specified patterns of growth. First,
check the code by reproducing the results in Fig. 6.11. Then, for uniform
circumferential growth, explore how spatial variations in the modulus c in
Eq. (6.74) affect residual stress.
6.5 Consider a thin-walled membrane model for an artery. The circumferential
wall stress σ and fluid shear stress τ are given by the relations
σ =
pa
h
,
τ=
4μQ
πa 3 ,
6 Growth
Fig. 6.33 Membrane with
attached springs
(Problem 6.3)
x
y
b
a
V 0
V 0
k
transversely isotropic, consisting of an isotropic matrix (volume fraction φ m )
with embedded fibers (volume fraction φ f ) aligned parallel to the x-axis. The
strain-energy density functions for the matrix and fibers, respectively, are
W m = c m (λ
2
x + λ
2
y + λ
2
z − 3)
W f = c f (λ x − 1)
2 ,
and the total Cauchy stress tensor is given by Eqs. (6.105) in the form
¯
σ = φ m ¯
σ m + φ f σ f − p I.
The fibers do not grow, but the matrix grows according to the growth tensor
G = G(e x e x + e y e y ) + e z e z ,
where G is a known constant.
(a) When stretched a distance , the springs exert a force kk per unit
undeformed area of the edge of the membrane. Express the Cauchy stress
σ s exerted by the springs on the membrane in terms of λ y and G.
(b) Given σ 0 and G, derive two equations to be solved for λ x and λ y . Hint:
The total stretch ratio λ x is the same for the matrix and the fibers.
6.4 Write a computer program to solve the problem studied in Sect. 6.7, which
involves residual stress in a tube caused by specified patterns of growth. First,
check the code by reproducing the results in Fig. 6.11. Then, for uniform
circumferential growth, explore how spatial variations in the modulus c in
Eq. (6.74) affect residual stress.
6.5 Consider a thin-walled membrane model for an artery. The circumferential
wall stress σ and fluid shear stress τ are given by the relations
σ =
pa
h
,
τ=
4μQ
πa 3 ,