Micromechanics of spatially uniform heterogeneous media: A critical review and emerging approaches

Outside of the classical microstructural detail-free estimates of effective moduli, micromechanical analyses of macroscopically uniform heterogeneous media may be grouped into two categories based on different geometric representations of material microstructure. Analysis of periodic materials is based on the repeating unit cell (RUC) concept and the associated periodic boundary conditions. This contrasts with analysis of statistically homogeneous materials based on the representative volume element (RVE) concept and the associated homogeneous boundary conditions. In this paper, using the above classification framework we provide a critical review of the various micromechanical approaches that had evolved along different paths, and outline recent emerging trends. We begin with the basic framework for the solution of micromechanics problems independent of microstructural representation, and then clarify the often confused RVE and RUC concepts. Next, we describe classical models, including the available RVE-based models, and critically examine their limitations. This is followed by discussion of models based on the concept of microstructural periodicity. In the final part, two recent unit cell-based models, which continue to evolve, are outlined. First, a homogenization technique called finite-volume direct averaging micromechanics theory is presented as a viable and easily implemented alternative to the mainstream finite-element based asymptotic homogenization of unit cells. The recent incorporation of parametric mapping into this approach has made it competitive with the finite-element method. Then, the latest work based on locally-exact solutions of unit cell problems is described. In this approach, the interior unit cell problem is solved exactly using the elasticity approach. The exterior problem is tackled with a new variational principle that successfully overcomes the non-separable nature of the overall unit cell problem.     

A model for the ignition and extinction of liquid fuel droplets

The model of the ignition and extinction of liquid fuel droplets presented in this paper is an application of stability theory, and specifically of Liapunov's first (indirect) method, which is well developed for lumped parameter autonomous systems. Since the behavior of liquid droplets is usually represented by a distributed nonautonomous system, an important part of the paper deals with the method by which the equations governing droplet combustion are reduced to the form appropriate to this analysis.The model consists of two unsteady autonomous coupled ordinary differential equations in temperature and oxygen mass fraction. Far field assumptions have made it necessary to treat the fuel mass fraction as a parameter. The results include the familiar S curves of temperature versus Damköhler number, as well as a great deal of detailed information about the static and the dynamic stability of the transitions between steady state evaporation and steady state combustion.     

Thermo-mechanical fatigue modeling of advanced metal matrix composites in the presence of microstructural details

An analytical model developed for predicting the inelastic response of metal matrix composites subjected to axisymmetric loading is employed to investigate the behavior of SiC---Ti composites under thermo-mechanical fatigue loading. The model is based on the concentric cylinder assemblage consisting of arbitrary numbers of elastic or inelastic sublayers with isotropic, transversely isotropic, or orthotropic, temperature-dependent properties. In the present work, the inelastic response of the titanium matrix is modeled by the Bodner-Partom unified viscoplastic theory. These features of the model allow the investigation of microstructural effects (such as the layered morphology of the SCS-6 fiber, including the weak carbon coating, and matrix microstructure) and rate-dependent response of the matrix on the fatigue behavior.

In this paper, we employ the predictions of the multiple concentric cylinder model to study the effects of the layered morphology of the SCS-6 SiC fiber and two-phase microstructure of the Ti-15-3 matrix on the response of a SiC---Ti composite under thermo-mechanical fatigue loading. It is shown that ignoring the microstructure can lead to significant errors in the predictions of the internal stress and inelastic strain distributions.     

Higher-order theory for functionally graded materials

This paper presents the full generalization of the Cartesian coordinate-based higher-order theory for functionally graded materials developed by the authors during the past several years. This theory circumvents the problematic use of the standard micromechanical approach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites by explicitly coupling the local (microstructural) and global (macrostructural) responses. The theoretical framework is based on volumetric averaging of the various field quantities, together with imposition of boundary and interfacial conditions in an average sense between the subvolumes used to characterize the composite's functionally graded microstructure. The generalization outlined herein involves extension of the theoretical framework to enable the analysis of materials characterized by spatially variable microstructures in three directions. Specialization of the generalized theoretical framework to previously published versions of the higher-order theory for materials functionally graded in one and two directions is demonstrated. In the applications part of the paper we summarize the major findings obtained with the one-directional and two-directional versions of the higher-order theory. The results illustrate both the fundamental issues related to the influence of microstructure on microscopic and macroscopic quantities governing the response of composites and the technologically important applications. A major issue addressed herein is the applicability of the classical homogenization schemes in the analysis of functionally graded materials. The technologically important applications illustrate the utility of functionally graded microstructures in tailoring the response of structural components in a variety of applications involving uniform and gradient thermomechanical loading.     

Finite-volume micromechanics of periodic materials: Past,present and future

The finite-volume method is now a well-established tool in the numerical engineering community for simulation of a wide range of problems in fluid and solid mechanics. Its acceptance by the mechanics of heterogeneous media community, however, continues to be slow, often characterized by confusion with the finite-element method or so-called higher-order theories. Herein, we provide a brief historical perspective on the evolution of this important technique in the fluid mechanics community, its transition to the solution of solid mechanics boundary-value problems initiated in Europe in 1988, and the recent developments aimed at the solution of unit cell problems of periodic heterogeneous media. The differences and similarities with the finite-element method are highlighted, and the resulting tangible advantages of the finite-volume technique discussed and illustrated. Finally, our most recent results in this area are presented which demonstrate the method’s capability of solving unit cell problems with complex architectures in a variety of settings and applications, while revealing undocumented effects of interest in the development of new material microstructures with targeted response. Recent attempts to develop alternative versions of this technique are also discussed, together with our ongoing work to generalize the finite-volume micromechanics approach in order to further enhance its predictive capabilities and efficiency.     

Thermomechanical Analysis of Functionally Graded Thermal Barrier Coatings with Different Microstructural Scales

The recently developed two-dimensional version of the higher-order theory for functionally graded materials (denoted as HOTFGM-2D in previous communications) has been used to investigate the effects of microstructural architectures in graded thermal barrier coatings (TBCs) on stress distributions in the presence of a through-thickness temperature gradient. In particular, the response of TBCs with different levels of functionally graded microstructural refinement and different arrangements has been investigated, and the results for the through-thickness stress distributions are compared with those based on the standard micromechanical homogenization scheme. The examples presented illustrate the shortcomings of the standard micromechanics-based approach that is applied to the analysis of functionally graded TBCs, particularly if the presence of creep effects is included in the analysis.     

Role of the Material Constitutive Model in Simulating the Reusable Launch Vehicle Thrust Cell Liner Response

The reusable launch vehicle thrust cell liner, or thrust chamber, is a critical component of the space shuttle main engine. It is designed to operate in some of the most severe conditions seen in engineering practice. These conditions give rise to characteristic deformations of the cooling channel wall exposed to high thermal gradients and a coolant-induced pressure differential, characterized by the wall’s bulging and thinning, which ultimately lead to experimentally observed “dog-house” failure modes. In this paper, these deformations are modeled using the cylindrical version of the higher-order theory for functionally graded materials in conjunction with two inelastic constitutive models for the liner’s constituents, namely Robinson’s unified viscoplasticity theory and the power-law creep model. Comparison of the results based on these two constitutive models under cyclic thermomechanical loading demonstrates that, for the employed constitutive model parameters, the power-law creep model predicts more precisely the experimentally observed deformation leading to the “dog-house” failure mode for multiple short cycles, while also providing much improved computational efficiency. The differences in the two models’ predictions are rooted in the differences in the short-term creep and relaxation responses.     

Thermo-elastic moduli of periodic multilayers with wavy architectures

The recently developed parametric finite-volume direct averaging micromechanics theory for periodic materials is employed to investigate the effective moduli and thermal expansion coefficients of lamellar composites with wavy architectures. In the parametric version, a reference square subvolume is mapped onto a quadrilateral subvolume in the actual discretized microstructure to accurately capture the in situ microstructural details. The mapping is used to construct local stiffness matrices of quadrilateral subvolumes which are employed in the local/global stiffness matrix solution strategy for the unit cell problem within a homogenization framework. Complete set of homogenized moduli and thermal expansion coefficients of multilayers comprised of alternating soft and hard laminae with two types of waviness is generated for the first time as a function of the volume content of the hard phase for two amplitude-to-wavelength ratios. The observed changes in the homogenized mechanical and thermal properties relative to the reference flat-layer configuration depend on the wavy microstructure orientation and become greater with increasing amplitude-to-wavelength ratios. Examination of local stress fields explains the differences observed in the homogenized moduli of multilayers with sinusoidal and corrugated waveforms for the two amplitude-to-wavelength ratios.