Growth mechanics problems require the solution of mass balance equations that include supply terms and account for mass exchanges among constituents of a mixture. Though growth may often be accompanied by a variety of concomitant phenomena that increase modeling complexity, such as solid matrix deformation, evolving traction-free configurations, cell division, and active cell contraction, it is important to distinguish these accompanying phenomena from the fundamental growth process that consists of deposition or removal of mass from the solid matrix. Therefore, the objective of this study is to present a canonical problem of growth, namely, dissolution of a rigid solid matrix in a solvent. This problem illustrates a case of negative growth (loss of mass) of the solid in a mixture framework that includes three species, a solid, a solvent, and a solute, where the solute is the product of the solid dissolution. By analyzing both volumetric and surface dissolutions, the two fundamental modes of growth are investigated within the unified framework of mixture theory.
Issue Section:
Research Papers
1.
Skalak
, R.
, Dasgupta
, G.
, Moss
, M.
, Otten
, E.
, Dullumeijer
, P.
, and Vilmann
, H.
, 1982, “Analytical Description of Growth
,” J. Theor. Biol.
0022-5193, 94
(3
), pp. 555
–577
.2.
Rodriguez
, E. K.
, Hoger
, A.
, and McCulloch
, A. D.
, 1994, “Stress-Dependent Finite Growth in Soft Elastic Tissues
,” J. Biomech.
0021-9290, 27
(4
), pp. 455
–467
.3.
Cowin
, S. C.
, and Hegedus
, D. H.
, 1976, “Bone Remodeling—1. Theory of Adaptive Elasticity
,” J. Elast.
0374-3535, 6
(3
), pp. 313
–326
.4.
Skalak
, R.
, Farrow
, D. A.
, and Hoger
, A.
, 1997, “Kinematics of Surface Growth
,” J. Math. Biol.
0303-6812, 35
(8
), pp. 869
–907
.5.
Epstein
, M.
, and Maugin
, G. A.
, 2000, “Thermomechanics of Volumetric Growth in Uniform Bodies
,” Int. J. Plast.
0749-6419, 16
(7–8
), pp. 951
–978
.6.
Ateshian
, G. A.
, 2007, “On the Theory of Reactive Mixtures for Modeling Biological Growth
,” Biomech. Model. Mechanobiol.
1617-7959, 6
(6
), pp. 423
–445
.7.
Klisch
, S. M.
, Sah
, R. L.
, and Hoger
, A.
, 2000, “A Growth Mixture Theory for Cartilage
,” AMD (Am. Soc. Mech. Eng.)
0160-8835, 242
, pp. 229
–242
.8.
Humphrey
, J. D.
, and Rajagopal
, K. R.
, 2002, “A Constrained Mixture Model for Growth and Remodeling of Soft Tissues
,” Math. Models Meth. Appl. Sci.
0218-2025, 12
(3
), pp. 407
–430
.9.
Byrne
, H.
, and Preziosi
, L.
, 2003, “Modelling Solid Tumour Growth Using the Theory of Mixtures
,” IMA J. Math. Appl. Med. Biol.
0265-0746, 20
(4
), pp. 341
–366
.10.
Garikipati
, K.
, Arruda
, E.
, Grosh
, K.
, Narayanan
, H.
, and Calve
, S.
, 2004, “A Continuum Treatment of Growth in Biological Tissue: The Coupling of Mass Transport and Mechanics
,” J. Mech. Phys. Solids
0022-5096, 52
(7
), pp. 1595
–1625
.11.
Araujo
, R. P.
, and McElwain
, D. S.
, 2005, “A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues I: A General Formulation
,” SIAM J. Appl. Math.
0036-1399, 65
(4
), pp. 1261
–1284
.12.
Truesdell
, C.
, and Toupin
, R.
, 1960, “The Classical Field Theories
,” Handbuch der Physik
, S.
Flugge
, ed., Springer-Verlag
, Berlin
, Vol. III
/1.13.
Bowen
, R. M.
, 1976, “Theory of Mixtures
,” Continuum Physics
, A. E.
Eringen
, ed., Academic
, New York
, Vol. 3
, pp. 1
–127
.14.
Bowen
, R. M.
, 1980, “Incompressible Porous Media Models by Use of the Theory of Mixtures
,” Int. J. Eng. Sci.
0020-7225, 18
(9
), pp. 1129
–1148
.15.
Bowen
, R.
, and Wiese
, J.
, 1969, “Diffusion in Mixtures of Elastic Materials
,” Int. J. Eng. Sci.
0020-7225, 7
(7
), pp. 689
–722
.16.
Eringen
, A.
, and Ingram
, J.
, 1965, “Continuum Theory of Chemically Reacting Media—1
,” Int. J. Eng. Sci.
0020-7225, 3
, pp. 197
–212
.17.
Ateshian
, G. A.
, Costa
, K. D.
, Azeloglu
, E. U.
, Morrison
, B. I.
, and Hung
, C. T.
, 2009, “Continuum Modeling of Biological Tissue Growth by Cell Division, and Alteration of Intracellular Osmolytes and Extracellular Fixed Charge Density
,” J. Biomech. Eng.
0148-0731, 131
(10
), p. 101001
.18.
Wan
, W.
, Hansen
, L.
, and Gleason
, R. L.
, Jr., 2010, “A 3-D Constrained Mixture Model for Mechanically Mediated Vascular Growth and Remodeling
,” Biomech. Model. Mechanobiol.
1617-7959, 9
(4
), pp. 403
–419
.19.
Ateshian
, G. A.
, and Ricken
, T.
, 2010, “Multigenerational Interstitial Growth of Biological Tissues
,” Biomech. Model. Mechanobiol.
1617-7959, 9
(6
), pp. 689
–702
.20.
Wilson
, C. G.
, Bonassar
, L. J.
, and Kohles
, S. S.
, 2002, “Modeling the Dynamic Composition of Engineered Cartilage
,” Arch. Biochem. Biophys.
0003-9861, 408
(2
), pp. 246
–254
.21.
Sanz-Herrera
, J. A.
, García-Aznar
, J. M.
, and Doblaré
, M.
, 2009, “On Scaffold Designing for Bone Regeneration: A Computational Multiscale Approach
,” Acta Biomater.
1742-7061, 5
(1
), pp. 219
–229
.22.
Frenning
, G.
, 2004, “Theoretical Analysis of the Release of Slowly Dissolving Drugs From Spherical Matrix Systems
,” J. Controlled Release
0168-3659, 95
(1
), pp. 109
–117
.23.
Dokoumetzidis
, A.
, and Macheras
, P.
, 2006, “A Century of Dissolution Research: From Noyes and Whitney to the Biopharmaceutics Classification System
,” Int. J. Pharm.
0378-5173, 321
(1–2
), pp. 1
–11
.24.
Noyes
, A.
, and Whitney
, W.
, 1897, “The Rate of Solution of Solid Substances in Their Own Solutions
,” J. Am. Chem. Soc.
0002-7863, 19
, pp. 930
–934
.25.
Bowen
, R. M.
, 1982, “Compressible Porous Media Models by Use of the Theory of Mixtures
,” Int. J. Eng. Sci.
0020-7225, 20
(6
), pp. 697
–735
.26.
Gu
, W. Y.
, Lai
, W. M.
, and Mow
, V. C.
, 1993, “Transport of Fluid and Ions Through a Porous-Permeable Charged-Hydrated Tissue, and Streaming Potential Data on Normal Bovine Articular Cartilage
,” J. Biomech.
0021-9290, 26
(6
), pp. 709
–723
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