5.5 Case Study: Cardiac Mechanics
249
where P 1 and P 2 are first Piola-Kirchhoff stresses, λ 1 and λ 2 are total stretch
ratios of the base and apex in the x-direction, and λ ∗
2 = λ 2 /K.
When the apex contracts by a given K, the cell deforms as shown in
Fig. 5.21b. Neglecting changes in membrane thickness, write four equations
to be solved for a 1 , a 2 , d, and the fluid pressure p f . Hint: Draw a free-body
diagram of half the deformed cell with a vertical cut through its center.
5.5 An artery with undeformed radius a 0 and wall thickness h 0 is filled with an
incompressible fluid, and its ends are fixed between two rigid walls. Assume
the following:
• There is no leakage, and gravity can be neglected.
• To a first approximation, the artery can be treated as a thin-walled cylindrical
membrane with circumferential contractile fibers.
• The passive and active strain-energy density functions are
W p = c p
(I 1 − 3)
2
+ γ e
β
λ 2
f −1
2
W a = c a
λ
∗
f − 1
4
,
where λ f is the fiber stretch ratio, and all material coefficients are known.
Here, the volume fractions are incorporated into c p and c a , so Eq. (5.52) can
be written as σ = ¯
σ p + σ a − p I.
• When the artery is mounted and filled, it is passive and all stresses are zero.
If the smooth muscle undergoes uniform contraction, compute the following
in terms of the contraction ratio K:
(a) The passive and active forces that the artery exerts on the supports.
(b) The total force that the fluid exerts on the supporting walls.
In addition, sketch the expected shape of the artery if contraction is stronger
near the center than near the ends of the vessel.
5.6 To a first approximation, the left ventricle is modeled as an isotropic, incompressible, spherical membrane. When diastolic filling begins, the radius is a 0
and the wall thickness is h 0 . During each beat, the ventricle traverses a pressurevolume loop like that shown in Fig. 5.14, where p A , p B , p C, and p D are the
cavity pressures at begin filling, end diastole, begin ejection, and end systole,
respectively. If p A = 0, determine the pressures at the other corners of the loop,
given the following information and assumptions:
• The wall is thin (h 0 << a 0 ).
• Muscle velocity and inertia effects can be neglected.
• The ventricle remains spherical (λ φ = λ θ = λ) while undergoing isotropic
systolic contraction K(t) parallel to the surface. Passive and active material
properties are defined by
249
where P 1 and P 2 are first Piola-Kirchhoff stresses, λ 1 and λ 2 are total stretch
ratios of the base and apex in the x-direction, and λ ∗
2 = λ 2 /K.
When the apex contracts by a given K, the cell deforms as shown in
Fig. 5.21b. Neglecting changes in membrane thickness, write four equations
to be solved for a 1 , a 2 , d, and the fluid pressure p f . Hint: Draw a free-body
diagram of half the deformed cell with a vertical cut through its center.
5.5 An artery with undeformed radius a 0 and wall thickness h 0 is filled with an
incompressible fluid, and its ends are fixed between two rigid walls. Assume
the following:
• There is no leakage, and gravity can be neglected.
• To a first approximation, the artery can be treated as a thin-walled cylindrical
membrane with circumferential contractile fibers.
• The passive and active strain-energy density functions are
W p = c p
(I 1 − 3)
2
+ γ e
β
λ 2
f −1
2
W a = c a
λ
∗
f − 1
4
,
where λ f is the fiber stretch ratio, and all material coefficients are known.
Here, the volume fractions are incorporated into c p and c a , so Eq. (5.52) can
be written as σ = ¯
σ p + σ a − p I.
• When the artery is mounted and filled, it is passive and all stresses are zero.
If the smooth muscle undergoes uniform contraction, compute the following
in terms of the contraction ratio K:
(a) The passive and active forces that the artery exerts on the supports.
(b) The total force that the fluid exerts on the supporting walls.
In addition, sketch the expected shape of the artery if contraction is stronger
near the center than near the ends of the vessel.
5.6 To a first approximation, the left ventricle is modeled as an isotropic, incompressible, spherical membrane. When diastolic filling begins, the radius is a 0
and the wall thickness is h 0 . During each beat, the ventricle traverses a pressurevolume loop like that shown in Fig. 5.14, where p A , p B , p C, and p D are the
cavity pressures at begin filling, end diastole, begin ejection, and end systole,
respectively. If p A = 0, determine the pressures at the other corners of the loop,
given the following information and assumptions:
• The wall is thin (h 0 << a 0 ).
• Muscle velocity and inertia effects can be neglected.
• The ventricle remains spherical (λ φ = λ θ = λ) while undergoing isotropic
systolic contraction K(t) parallel to the surface. Passive and active material
properties are defined by