Mechanics of Growing Deformable Solids: A Review
Publication: Journal of Engineering Mechanics
Volume 120, Issue 2
Abstract
Growing deformable solids increase their mass during the deformation process by way of the gradual addition of material onto the external surface of the solids. Only the processes of continuous growth (accretion) are under consideration in this paper. The mathematical formulation of the boundary‐value problem for continuously growing solids has some general distinctive features, which differentiate it from the formulation for solids of constant mass. First, the full strains within the growing body are incompatible due to arbitrary straining of each material element before its attachment to the body. Second, the boundary conditions on the surface of growth must prescribe the total stress state of the surface element, i.e., the full stress tensor on this part of the external boundary of the body. The writer undertakes an attempt to display such features and to present a brief historical sketch of the topic.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Arutyunyan, N. Kh. (1977). “Boundary value problem of the theory of creep for a body with accretion.” J. Appl. Math. and Mech., 41(5), 804–810.
2.
Arutyunyan, N. Kh., and Drozdov, A. D. (1984). “Mechanics of growing viscoelastic bodies subject to aging at finite strains.” Doklady AN SSSR, 276(4), 821–825 (in Russian).
3.
Arutyunyan, N. Kh., and Drozdov, A. D. (1985). “Build‐up of viscoelastic aging solids under conditions of a phase transition.” Mech. of Solids, 20(6), 138–146.
4.
Arutyunyan, N. Kh., Drozdov, A. D., and Naumov, V. E. (1987). “Mechanics of growing visco‐elasto‐plastic bodies.” Moscow, Nauka (in Russian).
5.
Arutyunyan, N. Kh., Grigoryan, S. S., and Naumov, V. E. (1988). “The problem of a growing icicle.” J. Appl. Math. and Mech., 52(2), 198–206.
6.
Arutyunyan, N. Kh., Manzhirov, A. V., and Naumov, V. E. (1991). “Contact problems in mechanics of growing solids.” Moscow, Nauka (in Russian).
7.
Arutyunyan, N. Kh., and Metlov, V. V. (1982). “Some problems in the theory of creep of inhomogeneous‐aging bodies with variable boundaries.” Mech. of Solids, 17(5), 92–100.
8.
Arutyunyan, N. Kh., and Metlov, V. V. (1983). “Nonlinear problems of creep of accreting bodies subject to aging.” Mech. of Solids, 18(4), 140–151.
9.
Arutyunyan, N. Kh., and Naumov, V. E. (1984a). “The boundary value problem of the theory of viscoelastic plasticity of a growing body subject to aging.” J. Appl. Math. and Mech., 48(1), 1–10.
10.
Arutyunyan, N. Kh., and Naumov, V. E. (1984b). “One mechanism of formation of growing viscoelastic bodies.” Mech. of Solids, 19(1), 52–59.
11.
Arutyunyan, N. Kh., Naumov, V. E., and Radaev, Yu. N. (1990). “A mathematical model of a dynamically accreted deformable body. Part 1. Kinematics and measure of deformation of the growing body.” Mech. of Solids, 25(6), 86–99.
12.
Arutyunyan, N. Kh., Naumov, V. E., and Radaev, Yu. N. (1991). “Mathematical model of a dynamically accreted deformable body. Part 2. Evolutionary boundary‐value problem of the theory of growing bodies.” Mech. of Solids, 26(1), 67–80.
13.
Arutyunyan, N. Kh., Naumov, V. E., and Radaev, Yu. N. (1992a). “Dynamic accretion of an elastic layer. 1. Motion of deposited particles at variable velocity.” Mech. of Solids, 27(5), 6–24.
14.
Arutyunyan, N. Kh., Naumov, V. E., and Radaev, Yu. N. (1992b). “Dynamic accretion of an elastic layer. 2. Case of incidence of accreted particles at constant velocity.” Mech. of Solids, 27(6), 90–102.
15.
Arutyunyan, N. Kh., and Zevin, A. A. (1981). “Optimal shape of an accreting column.” Mech. of Solids, 16(5), 111–115.
16.
Bažant, Z. P. (1977). “Viscoelasticity of solidifying porous material—concrete.” J. Engrg. Mech. Div., ASCE, 103(6), 1049–1067.
17.
Bažant, Z. P. (1979). “Thermodynamics of solidifying or melting viscoelastic materials.” J. Engrg. Mech. Div., ASCE, 105(6), 933–952.
18.
Brown, C. B. (1967). “Forces on rigid culverts under high fills.” J. Struct. Div., ASCE, 93(5), 195–215.
19.
Brown, C. B., Evans, R. J., and LaChapelle, E. R. (1972). “Slab avalanching and state of stress in fallen snow.” J. Geophysical Res., 77(24), 4570–4580.
20.
Brown, C. B., and Goodman, L. E. (1963). “Gravitational stresses in accreted bodies.” Proc., Royal Soc. of London, Ser. A, 276(1367), 571–576.
21.
Brown, C. B., Green, D. R., and Pawsey, S. (1968). “Flexible culverts under high fills.” J. Struct. Div., ASCE, 94(4), 905–917.
22.
Bykovtsev, G. I., and Lukanov, A. S. (1985). “Some problems of the theory of solidifying and incrementing media.” Mech. of Solids, 20(5), 112–115.
23.
Bykovtsev, G. I., and Lukanov, A. S. (1990). “Deformation of a wedge in the process of accretion.” Mech. of Solids, 25(1), 190–193.
24.
Christiano, P. P., and Chuntranuluk, S. (1974). “Retaining wall under action of accreted backfill.” J. Geotech. Engrg. Div., ASCE, 100(4), 471–476.
25.
Goodman, L. E., and Brown, C. B. (1963). “Dead load stresses and the instability of slopes.” J. Soil Mech. and Found. Div., ASCE, 89(3), 103–134.
26.
Hector, L. G. Jr. (1991). “A theory of growth instability during the early stages of metal casting.” Mech. Res. Communications, 18(1), 51–60.
27.
Hector, L. G. Jr., Aksel, B., and Fridy, J. (1991). “Growth instability during non‐uniform directional solidification of a pure metal plate.” J. Appl. Mech., 58(2), 326–333.
28.
Kharlab, V. D. (1966). “Linear theory of creep for a growing body.” Proc. Leningrad Inst. Civ. Engrg., 49, 93–119 (in Russian).
29.
Kolmanovskii, V. B., and Metlov, V. V. (1983). “Optimal shape of a reinforced column built up at a random rate.” Mech. Solids, 18(1), 83–93.
30.
Meshcherskii, I. V. (1897). “Dynamics of a particle of variable mass,” PhD thesis, St. Petersburg (in Russian).
31.
Metlov, V. V. (1985). “On the accretion of inhomogeneous viscoelastic bodies under finite deformations.” J. Appl. Math. and Mech., 49(4), 490–498.
32.
Metlov, V. V., and Turusov, R. A. (1985). “Formation of stressed state of viscoelastic solids that grow under conditions of frontal hardening.” Mech. of Solids, 20(6), 147–161.
33.
Naumov, V. E. (1988). “Indentation of a growing die into a viscoelastic rod‐type foundation.” Mech. of Solids, 23(3), 80–89.
34.
Potapov, V. D. (1986). “Stability of a growing viscoelastic rod subjected to ageing.” J. Appl. Math. and Mech., 50(6), 782–788.
35.
Rashba, E. I. (1953). “Stress determination in bulks due to own weight taking into account the construction sequence.” Proc. Inst. Struct. Mech. Acad. Sci. Ukrainian SSR, 18, 23–27 (in Russian).
36.
Richmond, O., Hector, L. G. Jr., and Fridy, J. M. (1990). “Growth instability during nonuniform directional solidification of pure metals.” J. Appl. Mech., 57(3), 529–536.
37.
Southwell, R. V. (1941). An introduction to the theory of elasticity for engineers and physicists. 2nd ed., Oxford Univ. Press, Oxford, England.
38.
Trincher, V. K. (1984). “Formulation of the problem of determining the stress‐strain state of a growing body.” Mech. of Solids, 19(2), 119–125.
39.
Zabaras, N., and Richmond, O. (1990). “Analysis and finite element approximations of deformations and thermal stresses in solidifying bodies.” Computer modeling and simulation of manufacturing processes, B. Singh, Y. T. Im, I. Haque and C. Altan, eds. 89–105.
40.
Zabaras, N., Ruan, Y., and Richmond, O. (1990). “Front tracking thermomechanical model for hypoelastic‐viscoplastic behavior of a solidifying body.” Comp. Methods in Appl. Mech. and Engrg., 81(3), 333–364.
41.
Zabaras, N., Ruan, Y., and Richmond, O. (1991). “On the calculation of deformations and thermal stresses during axially symmetric solidification.” J. Appl. Mech., 58(4), 865–871.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 120 • Issue 2 • February 1994
Pages: 207 - 220
Copyright
Copyright © 1994 American Society of Civil Engineers.
History
Received: Oct 23, 1992
Published online: Feb 1, 1994
Published in print: Feb 1994
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.