Modeling Approaches and Some Physical Considerations Concerning Thermodynamics and the Theory of Mixtures Applied to Time-Dependent Behaviors in Heterogeneous Materials

“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 l^α 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 \( \mathrm{tr}\left(\frac{\partial \psi }{\partial {\mathbf{F}}^{\alpha }}{\dot{\mathbf{F}}}^{\alpha}\right)=\mathrm{tr}\left[{\mathbf{F}}^{\alpha }{\left(\frac{\partial \psi }{\partial {\mathbf{F}}^{\alpha }}\right)}^T{\mathbf{L}}^{\alpha}\right] \)

Γ_i may be any tensorial order but are represented here arbitrarily as second-order.

In the case of Rajagopal and Srinivasa [75], the implicit constitutive equations govern a class of nonlinear viscous fluids. Other applications to implicit constitutive equations can be found in Rajagopal [76] and later publications involving co-authors.

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.

Compare Hackl, et al. [78, 79] who independently address coupled deformational and temperature gradient-driven heating.

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.

A similar coupling, though obtained through somewhat different means, arises in the shape memory polymer crystallization treatment of Barot, et al. [99], and revisited for 2-way shape memory with thermal expansion as modeled by Hall, et al. [105].

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, (θ^a, η^a) “are not uniquely defined and have only a relative significance.” (Germain, et al. [110].)

E.g. de Groot and Mazur [74], Coleman and Gurtin [112].

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.

See Hall [3] at Equations (15a-c) for additional discussion.

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.

The term “interphase” is a commonly applied term in the materials science community, referring to a mixture of constituents involving compositional gradients resulting from either or all of: intentional constituent insertions, such as coatings and sizings; environmental influences such as oxidation and moisture; and chemical migrations and reactions between the nominal constituents, e.g. fibers and matrices and their respective chemical constituents operating under the influences of surface energy and chemical potential differences, etc. The term is used in the present section in the described context, and should not be confused with the proper definition of the word “phase,” which distinguishes between states of a substance of fixed chemical composition.

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.