208
4 Problems in Soft Tissue Biomechanics
3R
2 r
3 d 2 r
dR 2 − 2Rr
3 dr
dR
+ 2R
4
dr
dR
4
= 0.
4.13 In the undeformed configuration, an incompressible circular cylinder (with
no hole) has mass density ρ 0 and radius b 0 . The cylinder is free of surface
tractions and rotates at constant angular velocity ω about its axis (the z-axis).
In this case, a differential element a distance r from the z-axis is subjected to
the centripetal (inward) acceleration rω 2 , and the deformation is axisymmetric
with stretch ratios in cylindrical coordinates given by Eq. (4.52), i.e.,
λ r =
∂r
∂R
,
λ θ =
r
R
,
λ z = λ.
(a) For a general strain-energy function W (λ r , λ θ , λ z ), determine the wall
stresses as functions of r and the unknown stretch ratio λ. Hint: First, find
σ r from the equation of motion (3.238) 1 ; then p = ¯
σ r − σ r .
(b) Find an explicit algebraic equation to solve for λ.
References
Davies JA (2005) Mechanisms of morphogenesis: the creation of biological form. Elsevier, San
Diego
Fung YC (1997) Biomechanics: circulation, 2nd edn. Springer, New York
Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues, and organs. Springer, New
York
Libai A, Simmonds JG (2005) The nonlinear theory of elastic shells. Cambridge University Press,
Cambridge
Mirsky I (1973) Ventricular and arterial wall stresses based on large deformation analyses. Biophys
J 13:1141–1159
Poynting JH (1909) On pressure perpendicular to the shear planes in finite pure shears, and on the
lengthening of loaded wires when twisted. Proc R Soc Lond A 82:546–559.
Savin T, Kurpios NA, Shyer AE, Florescu P, Liang H, Mahadevan L, Tabin CJ (2011) On the
growth and form of the gut. Nature 476:57–62
Szilard R (1974) Theory and analysis of plates. Prentice-Hall, Englewood Cliffs
Taber LA (2014) Morphomechanics: transforming tubes into organs. Curr Opin Genet Dev 27:7–
13
Wang V, Nielsen P, Nash M (2015) Image-based predictive modeling of heart mechanics. Ann Rev
Biomed Eng 17:351–383
Woods RH (1892). A few applications of a physical theorem to membranes in the human body in
a state of tension. J Anat Physiol 26:362–370
Yin FC (1981) Ventricular wall stress. Circ Res 49:829–842
4 Problems in Soft Tissue Biomechanics
3R
2 r
3 d 2 r
dR 2 − 2Rr
3 dr
dR
+ 2R
4
dr
dR
4
= 0.
4.13 In the undeformed configuration, an incompressible circular cylinder (with
no hole) has mass density ρ 0 and radius b 0 . The cylinder is free of surface
tractions and rotates at constant angular velocity ω about its axis (the z-axis).
In this case, a differential element a distance r from the z-axis is subjected to
the centripetal (inward) acceleration rω 2 , and the deformation is axisymmetric
with stretch ratios in cylindrical coordinates given by Eq. (4.52), i.e.,
λ r =
∂r
∂R
,
λ θ =
r
R
,
λ z = λ.
(a) For a general strain-energy function W (λ r , λ θ , λ z ), determine the wall
stresses as functions of r and the unknown stretch ratio λ. Hint: First, find
σ r from the equation of motion (3.238) 1 ; then p = ¯
σ r − σ r .
(b) Find an explicit algebraic equation to solve for λ.
References
Davies JA (2005) Mechanisms of morphogenesis: the creation of biological form. Elsevier, San
Diego
Fung YC (1997) Biomechanics: circulation, 2nd edn. Springer, New York
Humphrey JD (2002) Cardiovascular solid mechanics: cells, tissues, and organs. Springer, New
York
Libai A, Simmonds JG (2005) The nonlinear theory of elastic shells. Cambridge University Press,
Cambridge
Mirsky I (1973) Ventricular and arterial wall stresses based on large deformation analyses. Biophys
J 13:1141–1159
Poynting JH (1909) On pressure perpendicular to the shear planes in finite pure shears, and on the
lengthening of loaded wires when twisted. Proc R Soc Lond A 82:546–559.
Savin T, Kurpios NA, Shyer AE, Florescu P, Liang H, Mahadevan L, Tabin CJ (2011) On the
growth and form of the gut. Nature 476:57–62
Szilard R (1974) Theory and analysis of plates. Prentice-Hall, Englewood Cliffs
Taber LA (2014) Morphomechanics: transforming tubes into organs. Curr Opin Genet Dev 27:7–
13
Wang V, Nielsen P, Nash M (2015) Image-based predictive modeling of heart mechanics. Ann Rev
Biomed Eng 17:351–383
Woods RH (1892). A few applications of a physical theorem to membranes in the human body in
a state of tension. J Anat Physiol 26:362–370
Yin FC (1981) Ventricular wall stress. Circ Res 49:829–842