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Abstract

A multiscale approach to growth and recession is adopted that is compatible with the theory of mixtures. The growth may result from chemical reactions or infiltrations, or combinations thereof. The emphasis of the work is on the construction of approximate constituent elastic deformation gradients, associated with an arbitrary overall deformation, that reflect an initially circularly cylindrical geometry relative to which growth occurs. A decomposition of the deformation into cylindrical and non-cylindrical parts is employed. A cylindrically symmetric reference configuration for a given constituent is adopted in which chemical growth, chemothermal (and/or equivalently moisture) expansion, and appearance or disappearance of the constituent may occur; the reference configuration of a constituent thus possesses evolving spatial dimensions but fixed geometric and material symmetries. The traditional composite micromechanics assumption of common radial stress is adopted, in combination with the assumptions of either common axial deformation rate or common axial deformation, for the cylindrical component of deformation. Elastic deformations of constituents are obtained through comparison, of the actual deformation, to the assumed stress-free deposition configurations and subsequent chemothermal stress-free expansions. The requirement that the local and homogenized cylindrical deformations (or rates) match in the axial and outermost radial directions is used to define a homogenized non-cylindrical component that is applied across the local constituents to couple the cylindrical deformation to the overall deformation. The general objective of the study may be stated as incorporating ordered microstructure-based stresses incorporating growth, chemothermal expansion, inhomogeneous natural states, and applied deformations into a mixture theory-compatible framework to provide an important refinement to the assessments of constituent states. The model is intended to enable a new efficiency of multiscale representation in the modeling of composite materials with evolving microstructures, considering both manufacturing processes and environmental influences.

Keywords

Mixture theory composite finite deformation microstructure cylindrical chemical thermal deposition growth recession

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References

Truesdell

. Sulle basi della thermomeccanica. Rend Lincei 1957; 22: 33–38.

Truesdell

. Sulle basi della thermomeccanica. Rend Lincei 1957; 22: 158–166.

Truesdell

. Mechanical basis of diffusion. J Chem Phys 1962; 37: 2336–2344.

Green

Naghdi

. On basic equations for mixtures. Q J Mech Appl Math 1969; 22: 427–438.

Bowen

. Theory of mixtures. In: Eringen

(ed.) Continuum physics, Vol. III. New York: Academic Press, 1976.

Atkin

Craine

. Continuum theories of mixtures: basic theory and historical developments. Q J Appl Math 1976; XXIX: 209–244.

Bedford

Drumheller

. Theories of immiscible and structured mixtures. Int J Eng Sci 1983; 21: 863–960.

Samohyl

. Thermodynamics of irreversible processes in fluid mixtures. Leipzig: GDR, 1987.

Rajagopal

Tao

. Mechanics of mixtures. Singapore: World Scientific Publishers, 1995.

10.

Hall

Rajagopal

. Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory. Math Mech Solid 2012; 17: 131–164.

11.

Christensen

. Mechanics of composite materials. New York: John Wiley, 1979.

12.

Hashin

Rosen

. The elastic moduli of fiber-reinforced materials. J Appl Mech 1964; 31: 223.

13.

Christensen

. Solutions for effective shear properties in three phase sphere and cylinder models. J Mech Phys Solid 1979; 27: 315–330.

14.

Pagano

Tandon

. Elastic response of multi-directional coated-fiber composites. Comp Sci Tech 1988; 31: 273–292.

15.

You

. An efficient method for thermal analysis of composites containing anisotropic continuous fibers and an arbitrary number of interfacial layers. Comp Mater Sci 2004; 29: 49–66.

16.

Tsui

Clyne

. An analytical model for predicting residual stresses in progressively deposited coatings, Part 2: Cylindrical geometry. Thin Solid Film 1997; 306: 34–51.

17.

Hay

. Growth stress in SiO2 during oxidation of SiC fibers. J Appl Phys 2012; 111: 063527.

18.

Garikipati

Rao

. Recent advances in models for thermal oxidation of silicon. J Comput Phys 2001; 174: 138–170.

19.

Ben Amar

Goriely

. Growth and instability in elastic tissues. J Mech Phys Solid 2005; 53: 2284–2319.

20.

Humphrey

Rajagopal

. A constrained mixture model for growth and remodeling of soft tissues. Math Model Meth Appl Sci 2002; 12: 407–430.

21.

Soares

Rajagopal

Moore

. Deformation-induced hydrolysis of a degradable polymeric cylindrical annulus. Biomech Model Mechanobiol 2010; 9: 177–186.

22.

Humphrey

Rajagopal

. A constrained mixture model for arterial adaptations to a sustained step change in blood flow. Biomech Model Mechanobiol 2003; 2: 109–126.

23.

Baek

Rajagopal

Humphrey

. Competition between radial expansion and thickening in the enlargement of an intracranial saccular aneurysm. J Elast 2005; 80: 13–31.

24.

Baek

Rajagopal

Humphrey

. A theoretical model of enlarging intracranial fusiform aneurysms. J Biomech Eng 2006; 128: 142–149.

25.

Jones

. Mechanics of composite materials. New York: McGraw-Hill, 1975.

26.

Hall

Rao

. Thermodynamics and thermal expansion of crystallizable shape memory polymers using logarithmic strain. Mech Time Depend Mater 2014; 18: 437–452.

27.

Asaro

Lubarda

. Mechanics of solids and materials. New York: Cambridge University Press, 2006.

28.

Fung

Y C

. Foundations of solid mechanics. Englewood Cliffs, NJ: Prentice-Hall, 1965.

29.

Hall

. Effective moduli of cellular materials. J Reinf Plastic Composite 1993; 12: 186–197.

30.

Hall

Rajagopal

. A mixture theory of n-constituent finitely-deforming anisotropic materials subject to thermo chemical changes with diffusion, chemothermal expansion and entropy production, to be submitted.

31.

Schapery

. Thermal expansion coefficients of composite materials based on energy principles. J Comput Mater 1968; 2: 380–404.

32.

Rosen

Hashin

. Effective thermal expansion coefficients and specific heats of composite materials. Int J Eng Sci 1970; 8: 157–173.

33.

Levin

. On the coefficients of thermal expansion of heterogeneous materials (in Russian). Mekhanika Tverdogo Tela 1967; 2: 88–94.

34.

Dvorak

Chen

. Thermal expansion of three-phase composite materials. J Appl Mech 1989; 56: 418–422.

35.

Malvern

. Introduction to the mechanics of a continuous medium. Englewood Cliffs, NJ: Prentice-Hall, 1969.

36.

Kannan

Rajagopal

. A thermodynamical framework for chemically reacting systems. Z Angew Math Phys 2011; 62: 331–363.

37.

Dvorak

Benveniste

. On transformation strain and uniform fields in multiphase elastic media. Proc R Soc Lond A 1992; 437: 291–310.

38.

Dvorak

. Transformation field analysis of inelastic composite materials. Proc R Soc Lond A 1992; 437: 311–327.

39.

Eshelby

. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc Lond A 1957; 241: 376–396.

40.

Hill

. A self-consistent mechanics of composite materials. J Mech Phys Solid 1965; 13: 213–222.

41.

Mori

. and Tanaka, K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metal 1973; 21: 571–574.

A mixture-compatible theory of chemothermal deposition and expansion in n -constituent finitely deforming composite materials with initially circularly cylindrical microstructures

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