Abstract
Following a brief review of the theory of continuum mixtures, recent developments and applications emphasizing time-dependent behaviors of heterogeneous materials are described. Common approximations of mixture theory and related continuum homogenization schemes such as assigning material properties, boundary conditions and body forces are considered. Approaches to imposing restrictions due to the second law are discussed; traditional employment of the Clausius-Duhem inequality enforced for arbitrary processes is contrasted with maximization of the rate of entropy production applied to a functional space suitably restricted to conform with constitutive postulates. Remarks are made concerning related homogenization and thermodynamic developments, including poroelasticity, volume-averaging, local accompanying state, and linear irreversible thermodynamics. Applications to fibrous composite materials, and to additional classes of heterogeneous materials, are briefly discussed.
Similar content being viewed by others
Notes
“Heterogeneous” implies a length scale. Modeling of the collection of celestial bodies can be considered as a continuum at large enough length scales. “Heterogeneity” implies a window of suitable dimensions such that when different parts of the body are viewed through it, they appear to have different properties. All bodies are of course heterogeneous at sufficiently small scales.
In most cases, the first author’s journal publications were associated with prior SEM conference publications.
Additional elaboration is provided in Section 3 concerning the balance postulates for mixtures.
The terminology “volume-averaging” is employed both by advocates of mixture-type approaches, based in general on the full set of thermodynamic balance equations as described herein, as well as by the composite mechanics community based on “concentration factor” transfer matrices; these matrices relate composite stresses and strains to those of the constituents, which generally have explicitly-described geometrical representations (see e.g. Christensen [2]). An approach employing the geometrical features and associated stress representations of a concentric cylinder representation at the constituent level in combination with a mixture framework is given by Hall [3], in the context of a theory including chemothermal deposition and expansion.
In mixture theories, interactive body forces, and possibly body couples, are used to balance the equations of linear and angular momenta for the constituents in a mass-density, or volumetric, sense, as opposed to the actual forces transmitted between the surfaces of constituents. For example, the forces acting on the roughly cylindrical surfaces of a local group of fibers, due to a surrounding matrix, are replaced in mixture theory by a single body force; it represents the average of the forces experienced by the fiber constituent in a representative region, i.e. an RVE, which is then projected to the scale of a point. The fiber stress, on the other hand, represents the force per unit area obtained by solving the fiber continuum boundary value problem obtained by partitioning the overall applied force to obtain that applied to the fibers at the overall body boundaries, considering the body geometry, constraints, body forces and in general, body couples, acting on the fiber constituent continuum; the analogue can be said about the matrix stresses. The interactions occurring between these continua, i.e. at the fiber-matrix interfaces, are represented by the aforementioned interactive body forces, and possibly body couples; these are applied locally to each of the constituent continua, by the other, and are accounted for in the balance equations leading to the coupled constituent continuum boundary value problems. Much more detailed models of the constituents and their interfaces, e.g. finite element discretizations, usually associated with greater computational cost, could be used to illustrate the association of locally-averaged forces and couples to a constituent mass or volume. The manner in which the externally applied tractions and couples are transmitted to the individual constituents can be a concern if near-surface details are needed, but at positions removed from their application a Saint-Venant-type effect renders the different approximations equivalent; see Rajagopal [52] and Rajagopal and Tao [20]. The combination of effects at a point arising from the inputs of the constituent behaviors leads, in general, to more complex partial differential equations than those in a discrete constituent (i.e. micromechanics) treatment, which may pose their own computational challenges.
Note that lα is considered independent of the macro-angular momentum (see Eringen [1]).
For discussions of these higher-order theories, see e.g. Malvern [53], Tiersten and Bleustein [54], Eringen [55]. If one views the higher-order displacement gradient terms as associated with Taylor-series representations, over space coordinates, of motions relative to a point, such theories involve departures from the ideas of “local action” which prevail in classical theories of “simple” materials (see e.g. Eringen [55], Malvern [53]). See also footnote 16.
Exceptions are discussed in the sequel.
The idea of maximization of the rate of dissipation was first used widely by Ziegler and co-workers (see Ziegler [68], Ziegler [69], Ziegler and Wehrli [70]) though within the context of the result of assumed normality and orthogonality behaviors, and primarily with regard to small strain theories. Also there are several drawbacks in those works with regard to how the maximization is enforced; it cannot be used to yield several classes of constitutive relations that are necessary to describe the observed behavior of materials (see Rajagopal and Srinivasa [66] for a detailed discussion of the relevant issues). The genesis of the idea can be traced back to Kelvin, Rayleigh and Maxwell. The postulate causes no conflicts with notions that the rate of entropy production is minimized once a response function is chosen for the rate of dissipation (Onsager [71], Prigogine [72], Glansdorff & Prigogine [73]) culminating in equilibrium.
Noting \( \mathrm{tr}\left(\frac{\partial \psi }{\partial {\mathbf{F}}^{\alpha }}{\dot{\mathbf{F}}}^{\alpha}\right)=\mathrm{tr}\left[{\mathbf{F}}^{\alpha }{\left(\frac{\partial \psi }{\partial {\mathbf{F}}^{\alpha }}\right)}^T{\mathbf{L}}^{\alpha}\right] \)
Γi may be any tensorial order but are represented here arbitrarily as second-order.
The cited constraint proposed by Mills [77] is often applied in mixture theory, for the purpose of ensuring solution quality when the total volume is nearly constant, though the individual constituent volumes may vary. As stated by Hall and Rajagopal [51], if it is assumed that the local subvolumes, occupied by each separate constituent in its respective reference configuration, sum to the local total mixture volume occupied at the corresponding reference time, and that the current total volume and the reference time total volume are equal, it follows that \( \left({\rho}^{solid}/{\rho}_{\operatorname{Re}\mathrm{ference}}^{solid}+{\rho}^{fluid}/{\rho}_{\operatorname{Re}\mathrm{ference}}^{fluid}=1\right) \), where ρ indicates densities.
As noted by Rajagopal and Srinivasa [67], “It is possible to have zero entropy production even though the affinities do not vanish. However, … if some of the fluxes … are non-zero, it would be impossible for the process to be non-dissipative. Thus dissipative processes always correspond to non-zero fluxes while non-dissipative processes correspond to the vanishing of the fluxes. The same is not true of the affinities. This is particularly true of yielding behaviour.”
The “principle of local action” (Truesdell and Noll [85]) states that in determining the stress at a given particle X [of a continuum body], the motion outside an arbitrary neighborhood of X may be disregarded.” This is an oft-applied feature of the so-called “Rational Mechanics” school of thought (ref. Malvern [53], Truesdell [18]), in particular reference to so-called “simple” materials models, which obey the principle by definition.
In the previous section, as opposed to the Coleman-Noll [65] formalism, the requirement is implied that the considered space of processes must be limited such as to maintain the postulated functional relationships.
Or “local-equilibrium hypothesis.” Some associated requirements, according to proponents, are, 1. “The system under study can be…split into a series of cells sufficiently large to allow them to be treated as macroscopic thermodynamic subsystems, but sufficiently small that equilibrium is very close to being realized in each cell.” 2. “The relationships in equilibrium… between state variables remain valid outside of equilibrium provided that they are stated locally at each instant of time.” These requirements are fundamental to the so-called “Classical [or Linear] Irreversible Thermodynamics (CIT or LIT)” school of thought, “as developed by Onsager, Prigogine and many other authors.” Other features defining CIT/LIT are, “the existence of a non-negative rate of entropy production, [and] the existence of linear constitutive laws [i.e.] the Onsager-Casimir reciprocal relations.” (Jou, et al. [107]) Requirement (2) of this school of thought is in agreement with the need stated in the previous section to limit the process space under consideration, although the previous section considers a much wider class of behaviors including non-linear force-vs-flux relationships.
Rate of conversion of mechanical working to entropy.
Compare Truesdell [18], Eq. (2.26). The first law is stated locally as \( {\rho}_0\dot{\varepsilon}=\mathbf{S}.\dot{\mathbf{E}}+{\dot{Q}}_0 \) using the definitions of Equation (26). It follows from Equation (14), considering the case of only one material. \( \mathbf{S}.\dot{\mathbf{E}}=\mathrm{tr}\left[{\mathbf{S}}^T\dot{\mathbf{E}}\right] \). Equation (26) is expressed here in reference coordinates to facilitate direct comparison with the kinematic form Equation (29) chosen by Germain, et al. [110].
“Surroundings,” in this case, refers to everything external to the point of reference; the analogous integral statement over space relates to the surroundings of the entire body.
Germain et al. [110].
Proponents argue that absolute temperature and entropy “have no physical content outside equilibrium and its neighborhood,” and that, depending on the relation of modeled time-scale to physical relaxation mechanisms and internal variables selected to represent a process, (θa, ηa) “are not uniquely defined and have only a relative significance.” (Germain, et al. [110].)
Via a “principle of least irreversible force” in velocity space and its corresponding “principle of least velocity” in irreversible force space, which impose orthogonality/normality. While the work gave impetus to many years of foundational developments in continuum thermodynamics, we contrast this approach to the maximization of the rate of entropy production as described by Rajagopal and Srinivasa [67], where the normality of forces to the entropy production rate surface and solution uniqueness properties follows rigorously.
Sometimes called “reaction-limited.”
Sometimes called “diffusion-limited.”
As a simplification, common deformations of the constituents (constrained mixture) may be assumed, dependent on the chosen method of fabrication of the composite material.
The radial reference coordinate direction, for purposes of mapping to the overall body deformation, is taken to lie within a reference ply and orthogonal to the unidirectional fibers of that ply.
Assumed as a scalar in Equations (7), (9)–(10) and (12)–(14) and annotated here to reflect dependence on the reference configuration, which may possess evolving densities.
Supplemented here by additional details regarding the kernels and limits.
References
Eringen AC (1968) Theory of micropolar elasticity. In: Liebowitz H (ed) Fracture – an advanced treatise. Academic Press, New York, Vol. II, p 623
Christensen RM (1979) Mechanics of composite materials. John Wiley, New York
Hall RB (2015) A mixture-compatible theory of chemothermal deposition and expansion in n-constituent finitely-deforming composite materials with initially circularly cylindrical microstructures. Mathematics and Mechanics of Solids 20:228–248. https://doi.org/10.1177/1081286514544853
Dvorak GJ, Bahei-El-Din YA (1987) A bimodal plasticity theory of fibrous composite materials. Acta Mech 69:219–241
Hall RB (1997) Matrix-dominated viscoplasticity theory for fibrous metal matrix composites. Composites Part A 28A:769–780
Dvorak GJ, Bahei-El-Din YA, Macharet Y, Liu CH (1988) An experimental study of elastic-plastic behavior of a fibrous boron-aluminum composite. J Mech Phys Solids 36(6):655–687
Nigam H, Dvorak GJ, Bahei-El-Din YA (1994) An experimental investigation of elastic-plastic behavior of a fibrous boron-aluminum composite: I. Matrix-dominated mode Int J Plasticity 10:23–48
Nigam H, Dvorak GJ, Bahei-El-Din YA (1994) An experimental investigation of elastic-plastic behavior of a fibrous boron-aluminum composite: II. Fiber-dominated mode. Int J Plasticity 10:49–62
Fick A (1855) Uber diffusion. Ann Phys 94:59–86
Darcy H (1856) Les Fontaines Publiques de La Ville de Dijon. Victor Dalmont, Paris
Truesdell C (1957) Sulle basi della thermomeccanica. Accademia Nazionale dei Lincei, Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali 22(8):33–38, 158–166
Truesdell C (1962) Mechanical basis of diffusion. J Chem Phys 37:2336–2344
Bowen RM (1976) Theory of mixtures. In: Eringen AC (ed) Continuum physics, Vol. III. Academic Press, New York
Atkin RJ, Craine RE (1976) Continuum theories of mixtures: basic theory and historical developments. Quart J Mech Appl Math 29:209–244
Atkin RJ, Craine RE (1976) Continuum theories of mixtures: applications. J Inst Math Appl 17:153–207
Bedford A, Drumheller DS (1983) Theories of immiscible and structured mixtures. Intl Journal of Engineering Science 21:863–960
Rajagopal KR (2007) On a hierarchy of approximate models for flows of incompressible fluids through porous solids. Math Models Methods Appl Sci 17(2):215–252
Truesdell C (1984) Rational thermodynamics. Springer-Verlag, New York
Pekař M, Samohýl I (2014) The thermodynamics of linear fluids and fluid mixtures. Springer, New York. isbn:978-3-319-02513-1
Rajagopal KR, Tao L (1995) Mechanics of mixtures. World Scientific Publishers, Singapore
Crochet MJ, Nagdhi PM (1967) Small motions superposed on large static deformations. Acta Mech 4:315–335
Adkins JE (1963) Nonlinear diffusion I. diffusion and flow of mixtures of fluids. Phil Trans R Soc A 255A:607–633
Adkins JE (1963) Non-linear diffusion, 2. Constitutive equations for mixtures of isotropic fluids. Phil Trans R Soc London 255A:635–648
Green AE, Adkins JE (1964) A contribution to the theory of non-linear diffusion. Arch Rational Mechanics Analysis 15:235–246
Gray WG (1983) General conservation equations for multiphase systems 4. Constitutive theory including phase change. Adv Water Resources 6:130–140
Hassanizadeh M, Gray WG (1979) General conservation equations for multiphase systems, 2. Mass, momentum, energy and entropy equations. Adv Water Resources 2:191–203
Hassanizadeh M, Gray WG (1980) General conservation equations for multiphase systems, 3. Constitutive theory for porous media flow. Adv Water Resources 3:25–40
Whitaker S (1986) Flow in porous media I: a theoretical derivation of Darcy’s law. Transp Porous Media 1:3–25
Whitaker S (1967) Diffusion and dispersion in porous media. AICHE J 13:420–427
Whitaker S (1977) Simultaneous heat mass and momentum transfer in porous media: a theory of drying. Advances in heat transfer 13, academic press, New York
Whitaker S (1999) The method of volume averaging. Kluwer, Boston
Slattery JC (1967) Flow of viscoelastic fluids through porous media. AICHE J 13:1066–1071
Coussy O (1995) Mechanics of porous continua. Wiley, Chichester
Biot MA (1956) Theory of elastic waves in a fluid-saturated porous solid. I Low frequency range J Acoustic Soc Am 28:168–178
Biot MA (1956) Theory of elastic waves in a fluid-saturated porous solid. II High frequency range J Acoustic Soc Am 28:179–191
Biot MA (1962) Mechanics of deformation and acoustic propagation in porous media. J Appl Phys 33:1482–1498
de Boer R (1996) Highlights in the historical development of the porous media theory: toward a consistent macroscopic theory. Appl Mech Rev 49(4):201–262
Rajagopal KR, Tao L (2005) On the propagation of waves through porous solids. International Journal of Non-Linear Mechanics 40:373–380
Coussy O, Dormieux L, Detournay E (1998) From mixture theory to Biot’s approach for porous media. Int J Solids Structures 35(34–35):4619–4635
Müller I (1968) A thermodynamic theory of mixtures of fluids. Arch Rat Mech Anal 28:1–39
Anderson TB, Jackson R (1967) A fluid mechanical description of fluidized beds. Ind Eng Chem (Fund) 6:527–538
Bowen RM, Weise JC (1969) Diffusion in mixtures of elastic materials. Int J Eng Sci 7:689–722
Ingram JD, Eringen AC (1967) A continuum theory of chemical reacting media- II- constitutive equations of reacting fluid mixtures. Intl J of Engr Sci 5:289–332
Dunwoody N, Müller I (1968) A thermodynamic theory of two chemically reacting ideal gases with different temperatures. Arch Rat Mech Anal 29:344–369
Steel TR (1968) Determination of the constitutive coefficients for a mixture of two solids. Int J Solids Struct 4:1149–1160
Bowen RM, Garcia DJ (1970) On thermodynamics of mixtures with several temperatures. Int J Eng Sci 8:63–83
Bowen RM, Garcia DJ (1971) On nonlinear heat conduction in mixtures. Arch Mech 23:289–301
Craine RE, Green AE, Naghdi PM (1970) A mixture of viscous elastic materials with different constituent temperatures. Q J Mechanics and Applied Mathematics 23:171–184
Bedford A, Stern M (1972) Multi continuum theory for composite elastic-materials. Acta Mech 14:85–102
Bowen RM, Reinicke KM (1978) Plane progressive waves in a binary mixture of linear elastic-materials. J Applied Mechanics 45:493–499
Hall RB, Rajagopal KR (2012) Diffusion of a fluid through an Anisotropically chemically reacting Thermoelastic body within the context of mixture theory. Mathematics and Mechanics of Solids 17(2):131–164. https://doi.org/10.1177/1081286511407754
Rajagopal KR (2003) Diffusion through polymeric solids undergoing large deformations. Mater Sci Technol 19:1175–1180. https://doi.org/10.1179/026708303225004729
Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Cliffs
Tiersten H., Bleustein, JL (1974) Generalized elastic continua. In: G Herrmann (ed) R. D. Mindlin and applied mechanics. Pergamon Press, New York. eBook ISBN: 9781483155548
Eringen AC (1999) Microcontinuum field theories: foundations and solids. Springer, New York
Hall RB (2017) A Theory of Coupled Anisothermal Chemomechanical Degradation for Finitely-Deforming Composite Materials with Higher-Gradient Interactive Forces. In: Antoun B. et al. (eds) Challenges in Mechanics of Time Dependent Materials, Volume 2. Conference Proceedings of the Society for Experimental Mechanics (2016). Springer. https://doi.org/10.1007/978-3-319-41543-7
Gajendran H, Hall R, Masud A (2017) Edge stabilization and consistent tying of constituents at Neumann boundaries in multi-constituent mixture models. Int J Num Methods Eng 110:1142–1172. https://doi.org/10.1002/nme.5446
Gajendran H, Hall RB, Masud A, Rajagopal KR (2018) Chemo-mechanical coupling in curing and material-interphase evolution in multi-constituent materials. Acta Mech 229:3393–3414. https://doi.org/10.1007/s00707-018-2170-y
Prasad SC, Rajagopal KR, Rao IJ (2006) A continuum model for the anisotropic creep of single crystal nickel-based superalloys. Acta Materiala 54:1487–1500
Gurtin ME, de La Penha GM (1970) On the thermodynamics of mixtures. I Mixtures of rigid heat conductors Arch Rat Mech Anal 36(5):390–410
Bowen RM, Rankin RL (1973) Acceleration waves in ideal fluid mixtures with several temperatures. Arch Rat Mech Anal 51:261–277
Pecker C, Deresiewicz H (1973) Thermal effects on wave propagation in liquid-filled porous media. Acta Mech 16(1–2):45–64
Bowen RM, Chen PJ (1974) Shock waves in ideal fluid mixtures with several temperatures. Arch Rat Mech Anal 53:277–294
Bowen RM, Chen PJ (1979) Some properties of curved shock waves in ideal fluid mixtures with multiple temperatures. Acta Mech 33(4):265–280
Coleman BD, Noll W (1963) The thermodynamics of elastic materials with heat conduction and viscosity. Arch Ration Mech Anal 13(1):167–178
Rajagopal KR, Srinivasa AR (1998) Mechanics of the inelastic behavior of materials: part II - inelastic response. Int J Plast 14:969–995
Rajagopal KR, Srinivasa AS (2004) On thermomechanical restrictions of continua. Proc R Soc Lond A 460:631–651
Ziegler H (1963) Some extremum principles in irreversible thermodynamics. In: Sneddon N, Hill R (eds) Progress in solid mechanics, Vol 4, I. North-Holland, Amsterdam, New York
Ziegler H (1983) An introduction to thermodynamics, 2d ed. North-Holland Series in Applied Mathematics and Mechanics, North-Holland
Ziegler H, Wehrli C (1987) The derivation of constitutive equations from the free energy and the dissipation function. In: Wu TY, Hutchinson JW (eds) Advances in applied mechanics, vol 25. Academic Press, New York, pp 183–238
Onsager L (1931) Reciprocal relations in irreversible thermodynamics. I Phys Rev 37:405–426
Prigogine I (1967) Introduction to thermodynamics of irreversible processes, 3rd edn. Wiley Interscience, Hoboken
Glansdorff P, Prigogine I (1971) Thermodynamic theory of structure, stability and fluctuations. Wiley Interscience, Hoboken
De Groot, SR, Mazur P (1984) Non-equilibrium thermodynamics, 2d ed. Dover Publications, New York. (Unabridged, corrected republication of 1st ed. published by North-Holland Publishing Company, Amsterdam, 1962)
Rajagopal KR, Srinivasa AR (2008) On the thermodynamics of fluids defined by implicit constitutive relations. Mech Res Commun 35:483–489
Rajagopal KR (2003) On implicit constitutive equations. Appl Math 48:279–319
Mills N (1966) Incompressible mixtures of Newtonian fluids. Int J Eng Sci 4:97–112
Hackl K, Fischer FD, Svoboda J (2011) A study on the principle of maximum dissipation for coupled and non-coupled non-isothermal processes in materials. Proc Roy Soc A 467:1186–1196
Idem A (2011) Proc. Roy Soc A 467:2422–2426
Hron J, Kratochvil J, Malek J, Rajagopal KR, Tuma K (2012) A thermodynamically compatible rate type fluid to describe the response of asphalt. Math Comput Simul 82:1853–1873
Rajagopal KR, Srinivasa AS (2013) An implicit thermomechanical theory based on a Gibbs potential formulation for describing the response of thermoviscoelastic solids. International Journal of Eng Science 70:15–28
Rajagopal KR, Srinivasa AR (1998) Mechanics of the inelastic behavior of materials: part I - theoretical underpinnings. Int J Plast 14:945–967
Rajagopal KR, Srinivasa AS (2004) On the thermomechanics of materials that have multiple natural configurations. Part I: Viscoelasticity and classical plasticity Z angew Math Phys 55:861–893
Rajagopal KR, Srinivasa AS (2004) On the thermomechanics of materials that have multiple natural configurations. Part II: Twinning and solid to solid phase transformation Z angew Math Phys 55:1074–1093
Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flugge S (ed) Encyclopedia of physics, Vol 3/3. Springer-Verlag, Berlin
Rajagopal KR, Srinivasa AR (2000) A thermodynamic framework for rate type fluid models. J Non-Newtonian Fluid Mech 88:207–227
Rao IJ, Rajagopal KR (2001) A study of strain-induced crystallization of polymers. Int J Solids Struct 38:1149–1167
Málek J, Rajagopal KR (2007) Incompressible rate type fluids with pressure and shear-rate dependent material moduli. Nonlinear Analysis: Real World Applications 8:156–164
Rajagopal KR, Srinivasa AR (2001) Modeling anisotropic fluids within the framework of bodies with multiple natural configurations. J Non-Newtonian Fluid Mech 99:1–16
Murali Krishnan J, Rajagopal KR (2001) Review of the uses and modeling of bitumen from ancient to modern times. Appl Mech Rev 56:149–214
Murali Krishnan J, Rajagopal KR (2004) A thermomechanical framework for the constitutive modeling of asphalt concrete: theory and applications. ASCE Journal of Materials in Civil Engineering 16:155–166
Rao IJ, Rajagopal KR (2002) A thermodynamic framework for the study of crystallization in polymers. Z Angew Math Phys 53:365–406
Kannan K, Rajagopal KR (2004) A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid. Mathematics and Mechanics of Solids 9:37–59
Kannan K, Rao IJ, Rajagopal KR (2006) A thermomechanical framework for describing solidification of polymer melts. J Eng Mater Technol 128:55–63
Rao IJ, Humphrey JD, Rajagopal KR (2003) Biological growth and remodeling: a uniaxial example with possible application to tendons and ligaments. Computer Modeling in Engineering & Sciences 4:439–455
Rajagopal KR, Srinivasa AR (1995) On the inelastic behavior of solids - part 1: twinning. Int J Plast 11:653–678
Rajagopal KR, Srinivasa AR (1997) Inelastic behavior of materials - part 2: energetics associated with discontinuous deformation twinning. Int J Plast 13:1–35
Rajagopal KR, Srinivasa AR (1999) On the thermodynamics of shape memory wires. Z Angew Math Phys 50:459–496
Barot G, Rao IJ, Rajagopal KR (2008) A thermodynamic framework for the modeling of crystallizable shape memory polymers. Int J Eng Sci 46:325–351
Prasad SC, Rajagopal KR (2006) On the diffusion of fluids through solids undergoing large deformations. Mathematics and Mechanics of Solids 11:291–305
Málek J, Rajagopal KR (2006) On the modeling of inhomogeneous incompressible fluid-like bodies. Mech Mater 38:233–242
Málek J, Rajagopal KR (2008) A thermodynamic framework for a mixture of two liquids. Nonlinear Analysis: Real World Applications 9(4):1649–1660
Hall RB (2008) Combined thermodynamics approach for anisotropic, finite deformation overstress models of Viscoplasticity. Int J Eng Sci 46:119–130
Rajagopal KR, Srinivasa AR, Wineman AS (2007) On the shear and bending of a degrading polymer beam. Int J Plast 23:1618–1636
Hall RB, Rao IJ, Qi HJ (2014) Thermodynamics and thermal expansion of Crystallizable shape memory polymers using logarithmic strain. Mechanics of Time-Dependent Materials 18(2):437–452. https://doi.org/10.1007/s11043-014-9236-6
Hall RB, Gajendran H, Masud A (2015) A mixture theory based model for diffusion of a chemically reacting fluid through a nonlinear Hyperelastic solid. Mathematics and Mechanics of Solids 20:204–227. https://doi.org/10.1177/1081286514544852
Jou D, Casas-Vasquez J, Lebon G (2010) Extended irreversible thermodynamics, 4th edn. Springer, New York
Halphen B, Nguyen QS (1975) Sur les Matériaux Standards Généralisés. J de Mécanique 14:39–63
Hall RB (2016) A theory of multi-constituent finitely-deforming composite materials subject to thermochemical changes with damage. In: Ralph C, Silberstein M, Thakre PR, Singh R (Eds) Mechanics of Composite and Multi-functional Materials, Vol. 7. Springer, pp. 269–275, ISBN 978–3–319-21761-1, Proceedings of the 2015 Annual Conference on Experimental & Applied Mechanics (SEM, Costa Mesa, CA, June 8–11)
Germain P, Nguyen QS, Suquet P (1983) Continuum thermodynamics. J Appl Mech 50:1010–1020
Moreau JJ (1970) Sur les Lois de Frottement, de Viscosité et de Plasticité. C R Acad Sc Paris 271:608–611
Coleman BD, Gurtin ME (1967) Thermodynamics with internal state variables. J Chem Phys 47(2):597–613
Hellan K (1984) Introduction to fracture mechanics. McGraw-Hill, New York
Hall RB (2020) Anisotropic viscous damage governed by maximum rate of entropy production, to be submitted
Nelson JB (1983) Thermal aging of graphite polyimide composites. In: O’Brien TK (ed) Long-Term Behavior of Composites, ASTM STP 813: American Society for Testing and Materials, Philadelphia, 206–221
Schoeppner GA, Tandon GP, Ripberger ER (2007) Anisotropic oxidation and weight loss in PMR-15 composites. Compos Part A 38:890–904
Schoeppner GA, Tandon GP, Pochiraju KV (2008) Predicting thermo-oxidative degradation and performance of high-temperature polymer matrix composites. In: Allen DH, Talreja R (eds) Kwon YW. Multiscale Modeling and Simulation of Composite Materials and Structures, Springer US
Tandon GP, Pochiraju KV, Schoeppner GA (2006) Modeling of oxidative development in PMR-15 resin. Polym Degrad Stab 91:1861–1869
Anguiano M, Gajendran H, Hall RB, Masud A (2016) Coupled chemomechanical degradation solutions in one dimension. XIII SEM International Congress and Exposition on Experimental & Applied Mechanics, Jun 6–9, 2016, Orlando, FL
Anguiano M, Gajendran H, Hall, RB, Masud A (2017) Coupled anisothermal chemomechanical degradation solutions in one dimension. Proceedings of the XIV SEM International Congress and Exposition on Experimental & Applied Mechanics, Jun 12–15, 2017, Indianapolis, IN, Springer
Anguiano M, Gajendran H, Hall RB, Masud A (2017) Numerical solution of coupled anisothermal advection-diffusion-reaction PDEs: Process modeling of silicon carbide (SiC) oxidation in one dimension. EMI-2017, 2017 Conference of the engineering mechanics institute, June 4–7 2017, San Diego
Anguiano M, Gajendran H, Hall RB, Rajagopal K, Masud A (2020) Chemo-mechanical coupling and material evolution in finitely deforming solids with advancing fronts of reactive fluids, submitted
Ruiz E, Trochu F (2005) Thermomechanical properties during cure of glass-polyester RTM composites: elastic and viscoelastic modeling. J Compos Mater 39(10):881–916
Ruiz E, Trochu F (2005) Numerical analysis of cure temperature and internal stresses in thin and thick RTM parts. Compos A: Appl Sci Manuf 36(6):806–826
Vanlandingham MR, Eduljee RF, Gillespie JW Jr (1999) Relationships between stoichiometry, microstructure, and properties for amine-cured epoxies. J Appl Polym Sci 71(5):699–712
Dix EH (1940) Acceleration of the rate of corrosion by high constant stresses. Trans AIME 137:11
Mears R, Brown R, Dix E (1945) a generalized theory of stress corrosion of alloys. In: Warwick C, parsons a (eds) symposium on stress-corrosion cracking of metals, STP64, ASTM international, west Conshohocken, PA
Clarke DR, Faber KT (1987) Fracture of ceramics and glasses. J Phy Chem Solids 48(11):1115–1157
Sieradzki K, Newman RC (1987) Stress-corrosion cracking. J Phys Chem Solids 48(11):1101–1113
Akid R (1997) The role of stress-assisted localized corrosion in the development of short fatigue cracks. STP1298, ASTM International, West Conshohocken, PA
Najjar D, Magnin T, Warner TJ (1997) Influence of critical surface defects and localized competition between anodic dissolution and hydrogen effects during stress corrosion cracking of a 7050 aluminum alloy. Materials science and Eng A238: 293:302
Gilbert JL, Buckley CA, Jacobs JJ (1993) In vivo corrosion of modular hip prosthesis components in mixed and similar metal combinations. The effect of crevice, stress, motion, and alloy coupling. J Biomedical Mat Res Pt A 27(12):1533–1544
Mao SX, Li M (1998) Mechanics and thermodynamics on the stress and hydrogen interaction in crack tip stress corrosion: experiment and theory. J Mech Phys Solids 46(6):1125–1137
Desch P, Dillon JJ (2004) Case histories of stress-assisted corrosion in boilers. Proceedings of Corrosion 2004, 28 Mar – 1 Apr 2004, New Orleans LA, NACE-04516, NACE International
Yang D, Singh PM, Neu RW (2007) Initiation and propagation of stress-assisted corrosion (SAC) cracks in carbon steel boiler tubes. J Eng Mat Technol 129(4):559–566
Weitsman YJ (1995) Effects of fluids on polymeric composites – a review. Office of Naval Research contract technical report. ADA297030, dtic.mil
Bedford A, Stern M (1971) Toward a diffusing continuum theory of composite materials. J Appl Mech 38(1):8–14
Stern M, Bedford A (1972) Wave propagation in elastic laminates using a multi-continuum theory. Acta Mech 15(1–2):21–38
Tiersten HF, Jahanmir M (1977) A theory of composites modeled as interpenetrating solid continua. Arch Rat Mech Anal 65(2):153–192
Pop JJ, Bowen RM (1978) A theory of mixtures with a long range spatial interaction. Acta Mech 29:21–34
Hall RB (2018) A mixture theory with interactive body forces for composite interphases. In: Arzoumanidis A, Silberstein M, Amirkhizi A (eds) Challenges in mechanics of time dependent materials, volume 2. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi-org.wrs.idm.oclc.org/10.1007/978-3-319-63393-0_7. isbn:978-3-319-63392-3
Mindlin RD, Tiersten HF (1962) Effects of couple-stresses in linear elasticity. Arch Rat Mech Anal 11(1):425–448
Mindlin RD (1963) Influence of couple-stresses on stress concentrations. Exp Mech 3:1–7
Westbrook K et al (2010) Constitutive modeling of shape memory effects in semicrystalline polymers with stretch induced crystallization. J Eng Mater Technol 312:041010
Chung T, Romo-Uribe A, Mather PT (2008) Two-way reversible shape memory in a semicrystalline network. Macromolecules 41:184–192
Rao IJ, Rajagopal KR (2000) Phenomenological modelling of polymer crystallization using the notion of multiple natural configurations. Interfaces Free Bound 2:73–94
Barot G, Rao IJ (2006) Constitutive modeling of the mechanics associated with crystallizable shape memory polymers. Z Angew Math Phys 57:652–681
Wineman AS, Rajagopal KR (1990) On a constitutive theory for materials undergoing microstructural changes. Archives of Mechanics 42:53–75
Rajagopal KR, Wineman AS (1992) A constitutive equation for nonlinear solids which undergo deformation induced microstructural changes. Int J Plast 8:385–395
Qi HJ, Nguyen TD, Castro F, Yakacki CM, Shandas R (2008) Finite deformation thermo-mechanical behavior of thermally induced shape memory polymers. Journal of the Mechanics and Physics of Solids 56:1730–1751
Arruda EM, Boyce MC (1993) Three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J Mech Phys Solids 41(2):389–412
Silberstein MN, Cremar LD, Beiermann BA, Kramer SB, Martinez TJ, White SR, Sottos NR (2014) Modeling mechanophore activation within a viscous rubbery network. Journal of the Mechanics and Physics of Solids 63:141–153
Ong MT, Leiding J, Hongli T, Virshup AM, Martinez TJ (2009) First principles dynamics and minimum energy pathways for mechanochemical ring opening of cyclobutene. J Am Chem Soc Commun 131:6377–6379
Mow VC, Gu WY, Chen FH (2005) Structure and function of articular cartilage and meniscus. In: Mow VC, Huiskes R (eds) Basic Orthopaedic Biomechanics & Mechano-Biology, 3rd edn. Lippincott Williams & Wilkins, Philadelphia
Mow VC (2011) Rigorous mechanics and elegant mathematics on the formulation of constitutive laws for complex materials: an example from biomechanics. A tribute to a colleague and friend: Prof. Y. H. Pao. In: Hutter K, Wu T-T, Shu Y-C (eds) From waves in complex systems to dynamics of generalized continua: tributes to professor Yih-Hsing Pao on his 80th birthday. World Scientific, New Jersey
Ambrosi D, Ateshian GA, Arruda EM, Cowin SC, Dumais J, Goriely A, Holzapfel GA, Humphrey JD, Kemkemer R, Kuhl E, Olberding JE, Taber LA, Garikipati K (2011) Perspectives on biological growth and remodeling. Journal of the Mechanics and Physics of Solids 59:863–883
Humphrey JD, Rajagopal KR (2002) A constrained mixture model for growth and remodeling of soft tissues. Mathematical Models and Methods in Applied Sciences 12(3):407–430
Anand M, Rajagopal K, Rajagopal KR (2003) A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood. Journal of Theoretical Medicine 5(3–4):183–218
Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media, 2nd edn. John Wiley and Sons, New York
Tiersten HF (1964) Coupled magnetomechanical equation for magnetically saturated insulators. J Math Phys 5:1298–1318
Tiersten HF (1971) On the nonlinear equations of thermoelectroelasticity. Int J Engng Sci 9:587–604
Tiersten HF, Tsai CF (1972) On the interaction of the electromagnetic field with heat conducting deformable insulators. J Math Phys 13:361–378
de Lorenzi HG, Tiersten HF (1975) On the interaction of the electromagnetic field with heat conducting deformable semiconductors. J Math Phys 16:938–957
Ancona MG, Tiersten HF (1987) Macroscopic physics of the silicon inversion layer. Phys Rev B 35:7959–7965
Ancona MG, Tiersten HF (1980) Fully macroscopic description of bounded semiconductors with an application to the Si-SiO2 interface. Phys Rev B 22:6104–6119
Ancona MG, Tiersten HF (1983) Fully macroscopic description of electrical conduction in metal-insulator-semiconductor structures. Phys Rev B 27:7018–7045
Pepper M (1977) An introduction to silicon inversion layers. Contemp Phys 18(5):423–454
Ancona MG, Iafrate GJ (1989) Quantum correction to the equation of state of an electron gas in a semiconductor Phys Rev B 39:9536
Yang JS (2006) In memoriam: Harry F. Tiersten, professor and Ph.D. IEEE Trans Ultrason Ferroelectr Freq Control 53:1399–1403
Acknowledgements
The first author gratefully acknowledges the support of Air Force Research Laboratory, as well as discussions with A. Masud, H. Gajendran and M. Anguiano.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hall, R.B., Rajagopal, K.R. Modeling Approaches and Some Physical Considerations Concerning Thermodynamics and the Theory of Mixtures Applied to Time-Dependent Behaviors in Heterogeneous Materials. Exp Mech 60, 591–609 (2020). https://doi.org/10.1007/s11340-020-00582-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11340-020-00582-9