×

zbMATH — the first resource for mathematics

Compatibility and the genesis of residual stress by volumetric growth. (English) Zbl 0858.92005
Summary: The equations of compatibility which are pertinent for growth strain fields are collected and examples are given in simply-connected and multiply-connected regions. Compatibility conditions for infinitesimal strains are well known and the possibilities of Volterra dislocations in multiply-connected regions are enumerated. For finite growth strains in a multiply-connected region, each case must be examined individually and no generalizations in terms of Volterra dislocations are available. Any incompatible growth strains give rise to residual stresses which are known to occur in many tissues such as the heart, arterial wall, and solid tumors.

MSC:
92C10 Biomechanics
92C15 Developmental biology, pattern formation
74L15 Biomechanical solid mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Blume, J. A. 1989, Compatibility conditions for a left Cauchy-Green strain field. J. Elasticity 21:271–308 · Zbl 0725.73041 · doi:10.1007/BF00045780
[2] Boley, B. A. and Weiner, J. H. 1960, Theory of Thermal Stresses, Wiley and Sons, New York · Zbl 0095.18407
[3] Boucher, Y., Baxter, L. T. and Jain, R. K. 1990, Interstitial pressure gradients in tissue-isolated and subcutaneous tumors: Implications for therapy. Cancer Res. 50: 4478–4484
[4] Boucher, Y. and Jain, R. K. 1992, Microvascular pressure in the principal driving force for interstitial hypertension in solid tumors: Implications for vascular collapse. Cancer Res. 52:5110–5114
[5] Cesaro, E. 1906, Sulle formole del Volterra, fundamentali nella teoria delle distorisioni elastische. Rend. Napoli (3a), vol. 12, 311–321 · JFM 37.0832.01
[6] Cowin, S. 1993, Bone stress adaptation models. J. Biomechanical Engineering 115: 528–533 · doi:10.1115/1.2895535
[7] Fredrickson, A. G. 1964, Principles and Applications of Rheology. Prentice-Hall, Englewood Cliffs, NJ
[8] Fung, Y. C. 1965, Foundations of Solid Mechanics. Prentice-Hall, Englewood Cliffs, NJ
[9] Fung, Y. C. 1990, Biomechanics: Motion, Flow, Stress, and Growth. Springer-Verlag, New York · Zbl 0743.92007
[10] Gairola, B. K. D. 1979, Nonlinear elastic problems. In: Dislocations in Solids, vol. 1, the Elastic Theory, F. R. N. Nabarro (ed.), North-Holland Pub. Co., Amsterdam, pp. 223–342
[11] Green, A. E. and Adkins, J. E. 1960, Large Elastic Deformations and Non-Linear Continuum Mechanics. Clarendon Press, Oxford · Zbl 0090.17501
[12] Gurtin, M. E. 1972, The Linear Theory of Elasticity, In: Mechanics of Solids, vol. 2, C. Truesdell (Ed.) Springer-Verlag, pp. 1–295
[13] Hirth, J. P. and Lothe, J. 1981, Theory of Dislocations, 2nd Edition, John Wiley and Sons, New York · Zbl 1365.82001
[14] Hoger, A. and Carlson, D. E. 1984, On the derivative of the square root of a tensor and Guo’s rate theorems, J. of Elasticity, 14: 329–336 · Zbl 0576.73004 · doi:10.1007/BF00041141
[15] Hoger, A. 1986, On the determination of residual stress in an elastic body, J. of Elasticity, 16: 303–324 · Zbl 0616.73033 · doi:10.1007/BF00040818
[16] Hoger, A. 1993, The constitutive equation for finite deformations of a transversely isotropic hyperelastic material with residual stress, J. of Elasticity, 33: 107–118 · Zbl 0799.73016 · doi:10.1007/BF00705801
[17] Jain, R. K. and Baxter, L. T. 1988, Mechanisms of heterogeneous distribution of monoclonal antibodies and other macromolecules in tumors: Significance of intersitial pressure. Cancer Res. 48: 7022–7032
[18] Jain, R. K. 1994, Barriers to drug delivery in solid tumors. Scientific American 271: 58–65 · doi:10.1038/scientificamerican0794-58
[19] Kondo, K. 1955, Geometry of Deformations and Stresses. In: Unifying Study of the Basic Problems in engineering Sciences by Means of Geometry. vol. I. K. Kondo (Ed.). Gakujustu Bunken Fukyu-Kai, Tokyo, pp. 1–17
[20] Kosevich, A. M. 1979, Crystal dislocations and the theory of elasticity. In: Dislocations in Solids, vol. 1, The Elastic Theory, F. R. N. Nabarro (Ed.), North-Holland Pub. Co. Amsterdam, pp. 33–141
[21] Love, A. E. H. 1927, A Treatise on the Mathematical Theory of Elasticity. Cambridge Univ. Press, Cambridge · JFM 53.0752.01
[22] Nabarro, F. R. N. 1967, Theory of Crystal Dislocations. Clarendon Press, Oxford
[23] Netti, P. A., Roberge, S., Boucher, Y., Baxter, L. T. and Jain, R. K. 1995, Intersitial hypertension coupled with high vascular permeability can impair tumor blood flow, in preparation
[24] Rodriguez, E. K., Hoger, A. and McCulloch, A. D. 1994, Stress-dependent finite growth in soft elastic tissue, Journal of Biomechanics, 27: 455–467 · doi:10.1016/0021-9290(94)90021-3
[25] Roesler, H. 1987, The history of some fundamental concepts in bone biomechanics. J. Biomechanics 20: 1025–1034 · doi:10.1016/0021-9290(87)90020-0
[26] Shield, R. T. 1973, The rotation associated with large strains. SIAM J. Appl. Math. 25: 483–491 · Zbl 0276.73007 · doi:10.1137/0125048
[27] Silk, W. K. and Erickson, R. O. 1979, Kinematics of plant growth. J. Theor. Biology 76: 481–501 · doi:10.1016/0022-5193(79)90014-6
[28] Skalak, R. 1981, Growth as a finite displacement field. In: Proceedings of the IUTAM Symposium on Finite Elasticity. D. E. Carlson and R. T. Shield (Eds.) Martinus Nijhoff, The Hague, pp. 347–355
[29] Synge, J. L. and Schild, A. 1949, Tensor Calculus. Univ. of Toronto Press,Toronto
[30] Thompson, D’Arcy W. 1969, On Growth and Form. 2nd Edn. Cambridge Univ. Press, Cambridge
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.