338
6 Growth
for the axial Cauchy stress. The bar has an undeformed length L 0 , crosssectional area A 0 , and a modulus that varies according to the relation
c(X) = c 0 (1 + aX)
3 ,
where c 0 and a are constants, and X is the longitudinal material coordinate.
The ends of the bar are fixed to walls a distance L 0 apart. If the bar grows
uniformly with growth ratios G x = G and G y = G z = 1, determine the force
that the bar exerts on the walls. (Neglect end effects.) Hint: The total stretch
ratio varies along the bar.
6.11 Write a computer program to solve the problem studied in Sect. 6.11, which
deals with functional adaptation in arteries. Check the code by reproducing
the results in Fig. 6.27. Ambitious readers may want to try also computing
opening angles and reproducing the results in Fig. 6.28.
References
Ambrosi D, Pezzuto S, Riccobelli D, Stylianopoulos T, and Ciarletta P (2017) Solid tumors are
poroelastic solids with a chemo-mechanical feedback on growth. J. Elasticity 129:107–124
Ateshian GA (2007) On the theory of reactive mixtures for modeling biological growth. Biomech
Model Mechanobiol 6:423–445
Beloussov LV, Dorfman JG, Cherdantzev VG (1975) Mechanical stresses and morphological
patterns in amphibian embryos. J Embryol Exp Morph 34:559–574
Bergel DH (1960) The viscoelastic properties of the arterial wall. Vol. Ph.D. University of London,
London
Berne RM, Levy MN (1981) Cardiovascular physiology. C.V. Mosby Co., St. Louis
Burton RR, Goss RJ (1972) Heart growth and size in homeotherms. Regulation of organ and tissue
growth. Academic Press, New York, pp 101–125
Chung DT, Horgan CO, Abeyaratne R (1986) The finite deformation of internally pressurized
hollow cylinders and spheres for a class of compressible elastic materials. Int J Solids Struct 22
:1557–1570
Chuong CJ, Fung YC (1986) On residual stresses in arteries. J Biomech Eng 108:189–192
Chuong CJ, Fung YC, Schmid-Schonbein GW, Woo SLY, Zweifach BW (1986) Residual stress in
arteries. In: Frontiers in biomechanics. Springer, New York, pp 117–129
Cyron CJ, Humphrey JD (2017) Growth and remodeling of load-bearing biological soft tissues.
Meccanica 52:645–664
Eyckmans J, Boudou T, Yu X, Chen CS (2011) A hitchhiker’s guide to mechanobiology. Dev Cell
21:35–47
Fedorchak GR, Kaminski A, Lammerding J (2014) Cellular mechanosensing: getting to the
nucleus of it all. Prog Biophys Mol Biol 115:76–92
Freeman PL, Luff AR (1982) Contractile properties of hindlimb muscles in rat during surgical
overload. Am J Physiol 242:C259–264
Fung YC (1984) Biomechanics: circulation, 1st edn. Springer, New York
Fung YC (1990) Biomechanics: motion, flow, stress, and growth. Springer, New York
Fung YC (1991) What are the residual stresses doing in our blood vessels? Ann Biomed Eng
19:237–249
Fung YC (1997) Biomechanics: circulation, 2nd edn. Springer, New York
6 Growth
for the axial Cauchy stress. The bar has an undeformed length L 0 , crosssectional area A 0 , and a modulus that varies according to the relation
c(X) = c 0 (1 + aX)
3 ,
where c 0 and a are constants, and X is the longitudinal material coordinate.
The ends of the bar are fixed to walls a distance L 0 apart. If the bar grows
uniformly with growth ratios G x = G and G y = G z = 1, determine the force
that the bar exerts on the walls. (Neglect end effects.) Hint: The total stretch
ratio varies along the bar.
6.11 Write a computer program to solve the problem studied in Sect. 6.11, which
deals with functional adaptation in arteries. Check the code by reproducing
the results in Fig. 6.27. Ambitious readers may want to try also computing
opening angles and reproducing the results in Fig. 6.28.
References
Ambrosi D, Pezzuto S, Riccobelli D, Stylianopoulos T, and Ciarletta P (2017) Solid tumors are
poroelastic solids with a chemo-mechanical feedback on growth. J. Elasticity 129:107–124
Ateshian GA (2007) On the theory of reactive mixtures for modeling biological growth. Biomech
Model Mechanobiol 6:423–445
Beloussov LV, Dorfman JG, Cherdantzev VG (1975) Mechanical stresses and morphological
patterns in amphibian embryos. J Embryol Exp Morph 34:559–574
Bergel DH (1960) The viscoelastic properties of the arterial wall. Vol. Ph.D. University of London,
London
Berne RM, Levy MN (1981) Cardiovascular physiology. C.V. Mosby Co., St. Louis
Burton RR, Goss RJ (1972) Heart growth and size in homeotherms. Regulation of organ and tissue
growth. Academic Press, New York, pp 101–125
Chung DT, Horgan CO, Abeyaratne R (1986) The finite deformation of internally pressurized
hollow cylinders and spheres for a class of compressible elastic materials. Int J Solids Struct 22
:1557–1570
Chuong CJ, Fung YC (1986) On residual stresses in arteries. J Biomech Eng 108:189–192
Chuong CJ, Fung YC, Schmid-Schonbein GW, Woo SLY, Zweifach BW (1986) Residual stress in
arteries. In: Frontiers in biomechanics. Springer, New York, pp 117–129
Cyron CJ, Humphrey JD (2017) Growth and remodeling of load-bearing biological soft tissues.
Meccanica 52:645–664
Eyckmans J, Boudou T, Yu X, Chen CS (2011) A hitchhiker’s guide to mechanobiology. Dev Cell
21:35–47
Fedorchak GR, Kaminski A, Lammerding J (2014) Cellular mechanosensing: getting to the
nucleus of it all. Prog Biophys Mol Biol 115:76–92
Freeman PL, Luff AR (1982) Contractile properties of hindlimb muscles in rat during surgical
overload. Am J Physiol 242:C259–264
Fung YC (1984) Biomechanics: circulation, 1st edn. Springer, New York
Fung YC (1990) Biomechanics: motion, flow, stress, and growth. Springer, New York
Fung YC (1991) What are the residual stresses doing in our blood vessels? Ann Biomed Eng
19:237–249
Fung YC (1997) Biomechanics: circulation, 2nd edn. Springer, New York