×

A theory of volumetric growth for compressible elastic biological materials. (English) Zbl 1022.74027

Summary: A general theory of volumetric growth for compressible elastic materials is presented. We derive a complete set of governing equations for an elastic material undergoing a continuous growth process. In particular, we obtain two constitutive restrictions from a work-energy energy principle. First, we show that a growing elastic material behaves as a Green elastic material. Second, we obtain an expression that relates the stress power due to growth to the rate of energy change due to growth. Then, the governing equations for a small increment of growth are derived from the more general theory. The equations for the incremental growth boundary value problem provide an intuitive description of quantities that describe growth, and are used to implement the theory. The main features of the theory are illustrated with specific examples employing two strain energy functions that have been used to model biological materials.

MSC:

74L15 Biomechanical solid mechanics
74B20 Nonlinear elasticity
92C10 Biomechanics
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Pauwels, F., Biomechanics of the Normal and Diseased Hip (1976)
[2] Ogden, J. A., Behavior of the Growth Plate (1988)
[3] Brickley-Parson, D., Spine 9 pp 148– (1984)
[4] Sandy, J. D., Arthritis and Rheumatism 27 pp 388– (1984)
[5] Kiviranta, I., Journal of Orthopaedic Research 6 pp 188– (1988)
[6] Kiviranta, I., Journal of Orthopaedic Research 12 pp 161– (1994)
[7] Palmoski, M., Arthritis and Rheumatism 22 pp 508– (1979)
[8] Hall, A. C., Journal of Orthopaedic Research 9 pp 1– (1991)
[9] Sah, R. L., Journal of Orthopaedic Research 7 pp 619– (1989)
[10] Ateshian, G. A., Journal of Biomechanics 30 pp 1157– (1997)
[11] Klisch, S. M., Journal of Biomechanical Engineering 122 pp 180– (2000)
[12] Hsu, F., Journal of Biomechanics 1 pp 303– (1968)
[13] Cowin, S. C., Journal of Elasticity 6 pp 313– (1976) · Zbl 0335.73028
[14] Skalak, R., Journal of Theoretical Biology 94 pp 555– (1982)
[15] Skalak, R., Journal of Mathematical Biology 34 pp 889– (1996) · Zbl 0858.92005
[16] Skalak, R., Journal of Mathematical Biology 35 pp 869– (1997) · Zbl 0883.92005
[17] Rodriguez, E. K., Journal of Biomechanics 27 pp 455– (1994)
[18] Lin, I. E., Journal of Biomechanical Engineering 117 pp 343– (1995)
[19] Taber, L. A., Journal of Theoretical Biology 180 pp 343– (1996)
[20] Taber, L., Journal of Biomechanical Engineering 120 pp 348– (1998)
[21] Chen, Y., Journal of Elasticity 59 pp 175– (2000) · Zbl 0987.74009
[22] Hoger, A., Journal of Elasticity. · Zbl 0793.73002
[23] Van Dyke, T J., Journal of Theoretical Biology.
[24] Atkin, R. J., Quarterly Journal of Mechanics and Applied Mathematics 29 pp 209– (1976) · Zbl 0339.76003
[25] Ogden, R. W., Non-Linear Elastic Deformations (1984) · Zbl 0541.73044
[26] Beatty, M. F., Nonlinear Effects in Fluids and Solids (1996)
[27] Almeida, E. S., Computer Methods and Applications in Mechanical Engineering 151 pp 513– (1998) · Zbl 0920.73350
[28] Wagner, D. R., Proceedings of the 1999 Summer Bioengineering Conference, ASME pp 363–
[29] Klisch, S. M., Journal of Biomechanics 32 pp 1027– (1999)
[30] Truesdell, C., Handbuch der Physik 11/3 (1965)
[31] Cowin, S. C., Journal of Biomechanics 29 pp 647– (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.