Draft:Repeated superposition of finite strains
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 Comment: As mentioned by a previous review, this article draws heavily on the work of a single person, viz. V.A. Levin. This may place WP:UNDUE weight on his/her work. In addition, the title is very long and may be better renamed as "Finite strain superposition".
{{uwaddie96}} {talk}
11:08, 7 September 2019 (UTC)
 Comment: This seems to heavily draw on the work of V. A. Levin. This might not be problematic, but this might place WP:UNDUE weight on the work of Levin. I also feel there's a better/more snappy title, but I don't know what it would be. This one is very long. Possibly something like "finite strain superposition" Headbomb {t · c · p · b} 09:42, 5 August 2019 (UTC)
In solid mechanics, repeated superposition of finite strains is intended to simulate the deformation of bodies in some stages, when at each stage large (finite) strains occur in the body.^{[1]} The theory of repeated superposition of finite strains was developed in the 70s80s of the last century.^{[2]}^{[3]}
The causes for body deformation at each stage can be, for example, the following:
 application of external loads to the body;^{[4]}^{[5]}
 removal of part (s) of the body;^{[6]}^{[7]}
 addition of a new part (s) to the body;^{[8]}^{[9]}^{[10]}
 change in the mechanical properties of the material of the body or its part (s),^{[11]} including as a result of solidstate phase transitions;^{[12]}^{[13]}^{[14]}
 viscoelastic processes occurring in the body material;^{[15]}^{[16]}
 thermal or electromagnetic effects;^{[17]}
 chemical reactions.^{[18]}^{[19]}
The theory of repeated superposition of finite strains can be useful, for example, when modeling phenomena and processes such as crack initiation and growth,^{[20]}^{[21]}^{[22]} the biological tissue growth,^{[23]}^{[24]} or rocks evolution.^{[25]}^{[26]} The theory can also be applied to the additive manufacturing products strength analysis,^{[27]}^{[28]} in rubber and tire industry,^{[29]}^{[30]} in mining industry.^{[31]} This theory may also be useful in modeling residual stresses at finite strains.^{[32]}^{[33]}^{[34]}
The theory of repeated superposition of finite strains is a generalization of the theory of small strains superposed on finite strains.^{[35]}^{[36]}^{[37]} In the framework of this theory, additional strains at the second and subsequent stages of deformation are considered small.
To become familiar with the theory of repeated superposition of finite strains, it is desirable to know the basics of the theory of finite strains.^{[38]}^{[39]}
See also[edit]
References[edit]
 ^ Levin, V.A. (1998). "Theory of repeated superposition of large deformations: elastic and viscoelastic bodies". International Journal of Solids and Structures. 35: 2585–2600.
 ^ Levin, V. A.; Taras'ev, G. S. (1983). "One variant of the model of a viscoelastic body at large deformations" (PDF). Soviet Applied Mechanics. 19 (7): 615–618.
 ^ Levin, V. A. (1987). "Using the method of successive approximations in problems of superposition of finite deformations" (PDF). Soviet Applied Mechanics. 23 (5): 472–476.
 ^ Levin, V. A.; Bulatov, L. A. (1983). "Stress concentration around circular hole in body made of viscoelastic material" (PDF). Mechanics of Composite Materials. 19 (3): 307–310.
 ^ Levin, V. A. (1988). "Stress concentration near a hole, which is circular at the time of formation, in a body made of a viscoelastic material". Soviet Physics Doklady. 33: 296–298.
 ^ Levin, V.A.; Zingerman, K.M. (1998). "Interaction and microfracturing pattern for successive origination (introduction) of pores in elastic bodies: finite deformation". Trans. ASME. Journal of Applied Mechanics. 65 (2): 431–435.
 ^ Levin, V.A.; Morozov, E.M. (2002). "Nonlocal fracture criterion: finite strains". Doklady Physics. 47 (9): 680–681.
 ^ Lychev, S.A. (2011). "Universal deformations of growing solids". Mechanics of Solids. 46 (6): 863–876.
 ^ Ganghoffer, J.F.; Sokolowski, J. (2014). "A micromechanical approach to volumetric and surface growth in the framework of shape optimization". International Journal of Engineering Science. 74: 207–226.
 ^ Levin, V.A.; Zubov, L.M.; Zingerman, K.M. (2016). "An exact solution for the problem of flexure of a composite beam with preliminarily strained layers under large strains. Part 2. Solution for different types of incompressible materials". International Journal of Solids and Structures. 100—101: 558–565.
 ^ Zingerman, K.M.; Levin, V.A. (2009). "Redistribution of finite elastic strains after the formation of inclusions. Approximate analytical solution". Journal of Applied Mathematics and Mechanics. 73 (6): 710–721.
 ^ Levitas, V.I.; Levin, V.A.; Zingerman, K.M.; Freiman, E.I. (2009). "Displacive Phase Transitions at Large Strains: PhaseField Theory and Simulations". Physical Review Letters. 103 (025702).
 ^ Levin, V. A.; Levitas, V. I.; Lokhin, V. V.; Zingerman, K. M.; Sayakhova, L. F.; Freiman, E. I. (2010). "Solidstate stressinduced phase transitions in a material with nanodimensional inhomogeneities: Model and computational experiment". Doklady Physics. 55 (10): 507–511.
 ^ Levin, V. A.; Levitas, V. I.; Zingerman, K. M.; Freiman, E. I. (2013). "Phasefield simulation of stressinduced martensitic phase transformations at large strains". International Journal of Solids and Structures. 50 (19): 2914–2928.
 ^ Levin, V. A.; Zingerman, K. M. (2008). "A class of methods and algorithms for the analysis of successive origination of holes in a prestressed viscoelastic body. Finite strains". Communications in Numerical Methods in Engineering. 24 (12): 2240–2251.
 ^ Levin, V.A.; Zingerman, K.M.; Vershinin, A.V.; Freiman, E.I.; Yangirova, A.V. (2013). "Numerical analysis of the stress concentration near holes originating in previously loaded viscoelastic bodies at finite strains". International Journal of Solids and Structures. 50 (20–21): 3119–3135.
 ^ Levin, V.A.; Zubov, L.M.; Zingerman, K.M. (2018). "Multiple joined prestressed orthotropic layers under large strains". International Journal of Engineering Science. 133: 47–59.
 ^ Levitas, V.I.; Attariani, H. (2013). "Anisotropic Compositional Expansion and Chemical Potential for Amorphous Lithiated Silicon under Stress Tensor". Scientific Reports. 3 (1615).
 ^ Freidin, A.B. (2015). "On the Chemical Affinity Tensor for Chemical Reactions in Deformable Materials". Mechanics of Solids. 50 (3): 260–285.
 ^ Levin, V.A.; Lokhin, V.V.; Zingerman, K.M. (1995). "Growth of a narrow crack formed in a preloaded nonlinearelastic body: Analysis using the theory of repeated superposition of severe strains". PhysicsDoklady. 40 (8): 431–433.
 ^ Levin, V.A.; Morozov, E.M. (2007). "Nonlocal criteria for determining a prefracture zone in the process of defect growth for finite strains". Doklady Physics. 52 (7): 391–393.
 ^ Levin, V. A. (2008). "The construction of a model of the growth of a defect for finite strains: Nonlocal criteria". Journal of Applied Mathematics and Mechanics. 72 (3): 306–311.
 ^ Jin, L.; Liu, Y.; Cai, Z. (2018). "Asymptotic solutions on the circumferential wrinkling of growing tubular tissues". International Journal of Engineering Science. 128: 31–43.
 ^ Ahamed, T.; Dorfmann, L.; Ogden, R.W. (2016). "Modelling of residually stressed materials with application to AAA". J. Mech. Behav. Biomed. Mater. 61: 221–234.
 ^ Aller, J.; BobilloAres, N.C.; Bastida, F.; Lisle, R.J.; Menéndez, C.O. (2010). "Kinematic analysis of asymmetric folds in competent layers using mathematical modeling". Journal of Structural Geology. 32 (8): 1170–1184.
 ^ Abdelsalam, Mohamed G. (2010). "Quantifying 3D postaccretionary tectonic strain in the Arabian–Nubian Shield: Superimposition of the Oko Shear Zone on the Nakasib Suture, Red Sea Hills, Sudan". Journal of African Earth Sciences. 56 (4–5): 167–178.
 ^ Levin, V. A.; Zubov, L. M.; Zingerman, K. M. (2015). "An exact solution of the nonlinear bending problem for a composite beam containing a prestressed layer at large strains". Doklady Physics. 60 (1): 24–27.
 ^ Levin, V. A.; Zubov, L. M.; Zingerman, K. M. (2015). "An exact solution for the problem of flexure of a composite beam with preliminarily strained layers under large strains". International Journal of Solids and Structures. 67–68: 244–249.
 ^ Zingerman, K.M.; Levin, V.A. (2013). "Extension of the LaméGadolin problem for large deformations and its analytical solution". Journal of Applied Mathematics and Mechanics. 77 (2): 235–244.
 ^ Levin, V. A.; Zubov, L. M.; Zingerman, K. M. (2013). "Torsion of a composite nonlinear elastic cylinder with a prestressed inclusion". Doklady Physics. 58 (12): 540–543.
 ^ Levin, V.A.; Vershinin, A.V. (2008). "Nonstationary plane problem of the successive origination of stress concentrators in a loaded body. Finite deformations and their superposition". Communications in Numerical Methods in Engineering. 24 (12): 2229–2239.
 ^ Hoger, A. (1986). "On the determination of residual stress in an elastic body". Journal of Elasticity. 16 (3): 303–324.
 ^ Johnson, B.E.; Hoger, A. (1995). "The use of a virtual configuration in formulating constitutive equations for residually stressed elastic materials". Journal of Elasticity. 41 (3): 177–215.
 ^
Ogden, R.W. (2003). "Nonlinear Elasticity, Anisotropy, Material Stability and Residual Stresses in Soft Tissue". 441. Vienna: Biomechanics of Soft Tissue in Cardiovascular Systems. International Centre for Mechanical Sciences (Courses and Lectures). Cite journal requires
journal=
(help)  ^ Green, A.E.; Rivlin, R.S.; Shield, R.T. (1951). "General theory of small elastic deformations superposed on finite elastic deformations". Proc. Royal Soc.A. 211: 128–154.
 ^ Guz', A.N.; Han, L.M. (1976). "Wave propagation in composite layered materials with large initial deformations". Soviet Applied Mechanics. 12 (1): 1–7.
 ^ Zubov, L.M. (1971). "Variational principles of the nonlinear theory of elasticity. Case of superposition of a small deformation on a finite deformation". Journal of Applied Mathematics and Mechanics. 35 (5): 802–806.
 ^ Ogden, R. W. (1984). Nonlinear elastic deformations. Mineola, New York: Dover Publications.
 ^ Fu, Y.; Ogden, R. (2001). Nonlinear Elasticity: Theory and Applications. Cambridge: Cambridge University Press. ISBN 9780511526466.
Further reading[edit]
 Drozdov, A.D. (1998). Viscoelastic Structures. Mechanics of Growth and Aging (PDF). San Diego, London, Boston: Academic Press.
 Ganghoffer, J.F. (2010). "Mechanical modeling of growth considering domain variation. Part II: Volumetric and surface growth involving Eshelby tensors". Journal of the Mechanics and Physics of Solids. 58 (9): 1434–1459.
 Ganghoffer, J.F.; Rahouadj, R. (2017). "Thermodynamic formulations of continuum growth of solid bodies". Mathematics and Mechanics of Solids. 22 (5): 1027–1046.
 Lanir, Y. (2017). "Fibrous tissues growth and remodeling: Evolutionary micromechanical theory". Journal of the Mechanics and Physics of Solids. 107: 115–144.
 Levin, V.A.; Zubov, L.M.; Zingerman, K.M. (2014). "The torsion of a composite, nonlinearelastic cylinder with an inclusion having initial large strains". International Journal of Solids and Structures. 51 (6): 1403–1409.
 Levin, V. A.; Zubov, L. M.; Zingerman, K. M. (2015). "Influence of the prestressed layer on the nonlinear flexure of a rectangular beam made of compressible material". Doklady Physics. 60 (4): 167–170.
 Levin, V.A.; Zubov, L.M.; Zingerman, K.M. (2016). "Large bending strains in an orthotropic beam with a preliminarily stretched or compressed layer: Exact solution". Doklady Physics. 61 (8): 407–411.
 Levin, V.A. (2017). "Equilibrium of micropolar bodies with predeformed regions. The superposition of large deformations". Journal of Applied Mathematics and Mechanics. 81 (3): 223–227.
 Lin, W.J.; Iafrati, M.D.; Peattie, R.A.; Dorfmann, L. (2018). "Growth and remodeling with application to abdominal aortic aneurysms". Journal of Engineering Mathematics. 109 (1): 113–137.
 LlanaFúnez, Sergio; Rutter, Ernest H. (2008). "Strain localization in direct shear experiments on Solnhofen limestone at high temperature – Effects of transpression". Journal of Structural Geology. 30 (11): 1372–1382.
 Rodriguez, E.K.; Hoger, A.; McCulloch', A.D. (1994). "Stressdependent finite growth in soft elastic tissues". Journal of Biomechanics. 27 (4): 455–467.
 Rausch, M.K.; Kuhl, E. (2013). "On the effect of prestrain and residual stress in thin biological membranes". J. Mech. Phys. Solids. 61: 1955–1969.
 Witzenburg, C.M.; Holmes, J.W. (2017). "A Comparison of Phenomenologic Growth Laws for Myocardial Hypertrophy". Journal of Elasticity. 129 (1–2): 257–281.
Category:Continuum mechanics Category:Elasticity (physics) Category:Solid mechanics