Effective Constitutive Relations Research Articles

An improved continuum model for hypersonic thermal nonequilibrium flow in the near-continuum regime

In this work, the rarefied Couette flow of diatomic gases with thermal nonequilibrium effects is investigated by the direct simulation Monte Carlo (DSMC) method, and a macroscopic computational model is developed to consider the local rarefaction effects for diatomic gases in the near-continuum regime. The nonlinear transport properties of the diatomic gases are studied, indicating that effective viscosity and effective translational thermal conductivity in the shear nonequilibrium state are affected by translational nonequilibrium effects, which obey the same laws for both monatomic and diatomic gases. The transport coefficients of internal energy modes are affected by both translational nonequilibrium and internal energy relaxation, therefore, the effective rotational and vibrational thermal conductivities are related to the effective viscosity through a modified Eucken relation that accounts for internal energy relaxation. Conclusively, effective constitutive relations are newly established as a function of the shear nonequilibrium parameter and the modified Eucken factors for thermal nonequilibrium flows, and these are integrated into the macroscopic two-temperature model. Subsequently, it is assessed in the simulation of hypersonic flows over flat plates and cylinders at various Knudsen numbers. The results show that the surface shear stress and heat flux obtained by the proposed model agree well with the DSMC results, indicating significantly improved performance compared to the conventional Navier–Stokes two-temperature model for hypersonic flows in the near-continuum regime.

An active learning SPH method for generalized Newtonian free surface flows

This paper presents an active learning smoothed particle hydrodynamics (ALSPH) method to simulate generalized Newtonian free surface flows. First, an improved smoothed particle hydrodynamics (ISPH) method is established to obtain more reliable results for free surface flows by coupling the modified kernel gradient, the artificial viscosity, the density diffusive term, and the optimized particle shifting technique. Second, based on data and Gaussian process regression (GPR), an active learning strategy is developed to provide an effective constitutive relation. It is the first time that the ISPH method is combined with GPR to simulate generalized Newtonian free surface flows. Not only can the constitutive relation of any generalized Newtonian fluid in nature be accurately predicted, but a small amount of sampling data is also able to ensure accuracy over a wide range of the shear deformation rate. The challenging droplet impact and dam break are first modeled to validate the ISPH method. Due to the lack of an analytical constitutive relation for an arbitrary generalized Newtonian fluid in nature, the Cross model is then adopted and offers the required data to validate the ALSPH method. The results indicate that the learned constitutive relation is quite consistent with the analytical one and the simulation results match well. In addition, predictive accuracy and time consumption are proven. Furthermore, to verify the applicability of the learned constitutive relation, the jet buckling case and the jet entering the static fluid case are modeled. The good performance demonstrates the ALSPH method has a promising prospect of applications in simulating complex flows in nature.

Dynamic homogenization of heterogeneous piezoelectric media: A polarization approach using infinite-body Green’s function

Dynamic homogenization theories are powerful tools for describing and understanding the behavior of heterogeneous media such as composites and metamaterials. However, a major challenge in the homogenization theory is determining Green’s function of these media, which makes it difficult to predict their effective constitutive relations, particularly for the finite-size and/or non-periodic media in real-world applications. In this paper, we present a formulation for finding the elastodynamic effective constitutive relations for general heterogeneous media, including finite-size and non-periodic ones, via a polarization approach based on the Hashin–Shtrikman principle along with Green’s identities. Our proposed formulation relies on the infinite-body Green’s function of a homogeneous reference medium, making it free from the difficulty of determining Green’s function even for the homogenization of finite-size and/or non-periodic media. Additionally, we demonstrate the universal applicability of this formulation for both random and deterministic heterogeneous media. This work contributes to a better understanding of the homogenization theory and the design of next-generation metamaterials that require the accurate prediction of effective material characteristics for dynamic wave manipulation under desired operating environments.

Maximizing electro-momentum coupling in generalized 2D Willis Metamaterials

The coupling of momentum to strain in elastic metamaterials, known as the Willis coupling, has been widely studied in recent years for its potential in enabling novel phenomena in wave propagation. More recent work has shown that in piezoelectric composites, the momentum can also be coupled to the electrical stimulus, resulting in a new form of electro-momentum coupling, which offers a new approach to controlling elastic wave phenomena through a non-mechanical stimulus. In this study, we present a topology optimization approach to maximize the electro-momentum coupling in piezoelectric composites, where dynamic homogenization is utilized to obtain the effective mechanical, electrical, and electro-mechanical constitutive relations. We first validate the approach in one-dimension, then demonstrate that the electro-momentum coupling can enable asymmetric wave propagation in two-dimensions, both through mechanical and electrical loadings. This approach can enable the design of piezoelectric composites that support novel wave phenomena that can be excited through non-mechanical means.

Modeling of effective elastic-plastic properties of layered composites with a periodic structure in the framework of the anisotropic flow theory

The article is devoted to the development of a method for constructing theoretical strain diagrams. The method is based on the use of a model of effective constitutive relations for approximating the deformation diagrams of layered composites obtained using the asymptotic averaging method. To find the elastic constants of the model of a transversally isotropic composite, the method of minimizing the deviation of the approximation deformation diagrams from the diagrams obtained by the asymptotic homogenization (AH) method is used for a series of standard problems of deformation at small deformations. Minimization problems were solved using the Hooke-Jeeves method. The results of numerical simulation by the proposed method for layered composites are presented, which showed good approximation accuracy, which is achieved due to the proposed method for separating the coupled problems of micro- and macroscopic deformation.

Effective Constitutive Relations for Sintered Nano Copper Joints

Abstract Sintered copper nanoparticles are being considered as alternatives to solder and/or sintered silver in different applications. Like for the alternatives, the interpretation of accelerated fatigue test results does however require modeling, typically involving prediction of stresses and strains versus time and temperature based on constitutive relations. This poses a challenge as the inelastic deformation properties depend strongly on both the initial particles and details of the processing, i.e., unlike for solder general constitutive relations are not possible. This work provides a mechanistic description of the early transient creep of relevance in cycling, including effects of sintering parameters and subsequent oxidation. Inelastic deformation is dominated by diffusion, rather than dislocation motion. Generalized constitutive relations are provided to the extent that quantitative modeling of a specific structure only requires the measurement of a single creep curve for that.

Constitutive Relations of Anisotropic Polycrystals: Self-Consistent Estimates.

In this paper, the elastic constitutive relation of polycrystals contains the effect of the mesostucture coefficients. We consider a general case and derive the average elastic constitutive relation pertaining to polycrystals of cubic crystals with any symmetry of crystalline orientation in their statistical distribution. Following Budiansky and Wu, we used self-consistent estimates of eigenstrain to obtain the effective elastic constitutive relation of polycrystals in an explicit form. For the Voigt assumption and the Reuss assumption, the effective elastic constitutive relation of polycrystals on cubic crystals contains the the mesostructure coefficients up to linear terms. In general, the linear term expression works well for materials such as aluminum, the single crystal of which has weak anisotropy. However the same expression (which allows the anisotropic part of the effective elastic constitutive relation to depend only linearly on the mesostructure coefficients) does not suffice for materials such as copper, in which the single crystal is strongly anisotropic. Per the Taylor theorem, we expand the expression based on the self-consistent estimates with respect to the mesostructure coefficients up to quadratic terms for anisotropic polycrystals of cubic crystals. While our numerical data are very close to those of Morris, our expression is much simpler.

An alternative constitutive model for elastic particle-reinforced hyperelastic matrix composites with explicitly expressed Eshelby tensor

Herein, an alternative theoretical model is developed based on a two-phase Mori–Tanaka method within the framework of finite deformation continuum theory, in order to characterize the nonlinear mechanical behavior of particle-reinforced hyperelastic matrix composites (PRHMCs). The equivalent inclusion problems of a hyperelastic medium (HEM) and a reference elastic medium (REM) are analyzed, both of which have the same configuration and elastic property. Based on the equivalence between incremental stress fields in the HEM and REM, the Eshelby tensor for large deformation cases can be explicitly expressed as a closed-form function depending on the elastic modulus and tangent modulus of the hyperelastic matrix and the classical Eshelby tensor for linearly elastic composites. The Eshelby tensor for hyperelastic composites can be easily determined, without requiring time-consuming numerical ellipsoidal integrals in existing meso-mechanical models for PRHMCs. Incremental constitutive relations depending on the material tangent modulus are adopted to describe the mechanical behaviors of hyperelastic matrix and elastic particles. An incrementally effective constitutive relation of PRHMCs is finally achieved with a homogenization procedure, which is used to well reproduce the stress-strain responses of several types of particle-reinforced hyperelastic matrix composites in uniaxially and biaxially tensile experiments. These results demonstrate the feasibility and convenience of the present model not only in predicting the mechanical behavior of particle-reinforced elastomers, but also in guiding the design of advanced flexible composites with desirable mechanical performances.

Some personal reflections on acoustic metamaterials

This article is concerned with waves in n-component composites with random microgeometry. Such composites have been contemplated for decades but more recently have acquired prominence through the development of the so-called metamaterials. The account is a personal one and traces the development of variational formulations and the recognition of unconventional couplings in effective constitutive relations. It then addresses the inadequacy of treatments which consider only mean waves, beyond the “homogenization” or long-wavelength limit. The variational formulation can be employed to find approximations to the solution in all realizations, as well as for the mean waves. This has been illustrated by solving explicitly a problem of transmission and reflection at the interface between a composite and a homogeneous material. The solution has been published already, for the mean waves. A new development is the deduction of mean energy fluxes, which conform to conservation of energy. A striking phenomenon, beyond the reach of homogenization theory, is that a substantial fraction of the incident energy is reflected incoherently as frequency tends to zero, even when the homogeneous half-space and composite are acoustically matched.

Interface energy effect on effective elastoplastic behavior of spheroidal particle reinforced metal matrix nanocomposites

A nanomechanical framework is proposed to predict the effective elastoplastic behavior of the metal matrix nanocomposites (MMNCs) containing randomly distributed and randomly oriented spheroidal particles. The interface energy effect is investigated by considering both of the interface elastic stress and the interface residual stress on the idealized zero-thickness membrane interphase between the matrix and the inhomogeneities. The orientational average is performed to obtain the effective constitutive relations of the MMNCs with randomly oriented spheroidal particles. By employing the micromechanical homogenization approaches, an effective yield criterion is formulated, and the effective secant moduli of the MMNCs are derived. Comparisons are made between the present framework and the classical micromechanics theory. The interface energy effect on the effective secant moduli are discussed, and the particle size dependence of the effective secant moduli is revealed. Furthermore, the effects of the interface elastic stress and the interface residual stress on the plastic yielding of the nanocomposites are discussed. Lastly, the theoretical predictions obtained by the present framework are compared with the relevant experiments and numerical simulations.

Cluster based nonuniform transformation field analysis: An efficient homogenization for inelastic heterogeneous materials

AbstractIn this article, the authors propose an efficient numerical homogenization method, the so‐called cluster based nonuniform transformation field analysis (NTFA), for inelastic heterogeneous material. The method enables to predict the effective properties directly from the local constitutive relations for microstructural phases of heterogeneous material, thus no requiring preliminary assumptions and any derivations for the effective constitutive relation as in the conventional NTFA. Evolution of plastic strain field is expressed by combination of nonuniform plastic strain modes evaluated from the nonlinear finite element analysis for training directions previously given and snapshot proper orthogonal decomposition. Eventually, microscopic stress and strain field are represented by scalar reduced variables and plastic strain modes. A simple maximization of minimum distance is proposed in order to choose uniformly distributed training directions on a unit sphere in the load space. Also, scalar reduced variables are directly determined from the integration of local constitutive relation, where material points are compressed into a number of clusters by clustering scheme. Thus, evolution of reduced variables is evaluated by the integration of local constitutive relation only in clusters. Numerical examples that consider material models followed J2 plasticity and pressure‐dependent non‐associated flow rule show that the current approach may predict the effective properties of inelastic heterogeneous material accurately and efficiently. It should be noted that the current approach can be available for material followed not only associated flow rule but also non‐associated flow one.

All order effective action for charge diffusion from Schwinger-Keldysh holography

An effective action for diffusion of a conserved U(1) charge is derived to all orders in the derivative expansion within a holographic model dual to the Schwinger-Keldysh closed time path. A systematic approach to solution of the 5D Maxwell equations in a doubled Schwarzschild-AdS5 black brane geometry is developed. Constitutive relation for the stochastic charge current is shown to have a term induced by thermal fluctuations (coloured noise). All transport coefficient functions parameterising the effective action and constitutive relations are computed analytically in the hydrodynamic expansion, and then numerically for finite momenta.

Modeling of the effective universal constitutive relations for elastic laminated composites with finite strains

The article considers the modeling results of layered composites with finite strains deformation according to the individual layers characteristics. The article proposes an asymptotic averaging method version for layered composites with finite deformations and periodic structure. The method allows calculating the effective deformation diagrams connecting the averaged Piola-Kirchhoff stress tensors components and the strain gradient.

Universal models for effective constitutive relations of laminated composites with finite strains

The results of the development of the theory of constructing effective constitutive relations for laminated composites with finite deformations are presented. The theory is based on the application of the method of asymptotic averaging for periodic structures and the use of universal models of constitutive relations for composite layers, proposed by Yu.I. Dimitrienko. Universal models allow us to formulate the constitutive relations simultaneously for all the main conjugated stress – strain tensors pairs. The method of asymptotic averaging makes it possible to obtain a relationship between the effective constitutive relations of the composite and its constituent layers. Cases of incompressible layers and layers with compressibility are considered. A numerical algorithm for constructing effective deformation diagrams of layered composites and calculating the material constants in the effective constitutive relations of an anisotropic composite is proposed. Numerical examples of method implementation and numerical algorithms are given.

Active learning of constitutive relation from mesoscopic dynamics for macroscopic modeling of non-Newtonian flows

We simulate complex fluids by means of an on-the-fly coupling of the bulk rheology to the underlying microstructure dynamics. In particular, a continuum model of polymeric fluids is constructed without a pre-specified constitutive relation, but instead it is actively learned from mesoscopic simulations where the dynamics of polymer chains is explicitly computed. To couple the bulk rheology of polymeric fluids and the microscale dynamics of polymer chains, the continuum approach (based on the finite volume method) provides the transient flow field as inputs for the (mesoscopic) dissipative particle dynamics (DPD), and in turn DPD returns an effective constitutive relation to close the continuum equations. In this multiscale modeling procedure, we employ an active learning strategy based on Gaussian process regression (GPR) to minimize the number of expensive DPD simulations, where adaptively selected DPD simulations are performed only as necessary. Numerical experiments are carried out for flow past a circular cylinder of a non-Newtonian fluid, modeled at the mesoscopic level by bead-spring chains. The results show that only five DPD simulations are required to achieve an effective closure of the continuum equations at Reynolds number Re=10. Furthermore, when Re is increased to 100, only one additional DPD simulation is required for constructing an extended GPR-informed model closure. Compared to traditional message-passing multiscale approaches, applying an active learning scheme to multiscale modeling of non-Newtonian fluids can significantly increase the computational efficiency. Although the method demonstrated here obtains only a local viscosity from the polymer dynamics, it can be extended to other multiscale models of complex fluids whose macro-rheology is unknown.

Multi-band metamaterial absorber with arbitrary polarization and wide-incident angle

We propose a multi-band metamaterial perfect absorber with a deep-subwavelength thickness and near-unity absorption coefficient at microwave frequencies. This planar metamaterial is composed of periodically arranged concentric metallic rings and a continuous metallic ground plate separated by one or two lossy dielectric layers. The origin of the induced multi-band perfect absorption is attributed to the multiple impedance-matching to free space and can be achieved through tuning the effective constitutive relations by interactions of several electromagnetic resonances. Numerical simulations reveal the electric field and surface current distributions at each absorption peak, and further analysis indicates that the absorption mainly arises from the dielectric loss. Moreover, the absorption could be enhanced and broaden by overlapping the modes of different rings. The absorption is identical for arbitrary polarizations due to the rotational symmetry of the subwavelength unit. It also remains over 90% energy absorption for a wide angle of incidence up to 45 $$^\circ$$ , which is confirmed by the experiment. The designed multi-band absorber is also easy to manufacture by the mature printed circuit board technology, which is favorable for a wide variety of applications.

Effective constitutive relations for simulating CO2 capillary trapping in heterogeneous reservoirs with fluvial sedimentary architecture

Carbon dioxide (CO2) storage reservoirs commonly exhibit sedimentary architecture that reflects fluvial deposition. The heterogeneity in petrophysical properties arising from this architecture influences the dynamics of injected CO2. We previously used a geocellular modeling approach to represent this heterogeneity, including heterogeneity in constitutive saturation relationships. The dynamics of CO2 plumes in fluvial reservoirs were investigated during and after injection. It was shown that small-scale (centimeter–meter) features play a critical role in capillary trapping processes and have a primary effect on physical- and dissolution-trapping of CO2, and on the ultimate distribution of CO2 in the reservoir. Heterogeneity in saturation functions at that small scale enhances capillary trapping (snap-off), creates capillary pinning, and increases the surface area of the plume. The understanding of these small-scale trapping processes from previous work is used to develop effective saturation relationships that represent, at a larger scale, the integral effect of these processes. Though it is generally not computationally feasible to represent small-scale heterogeneity directly in a typical reservoir simulation, the effective saturation relationships for capillary pressure and relative permeability presented here, along with an effective intrinsic permeability, allow better representation of the total physical trapping at the scale of larger model grid cells, as typically used in reservoir simulations. Thus, the approach diminishes limits on cell size and decreases simulation time in reservoir simulations.

Numerical homogenization of the Richards equation for unsaturated water flow through heterogeneous soils

AbstractHomogenized equations and the corresponding effective constitutive relations are generally necessary for numerically modeling large‐scale unsaturated flow processes in soils. Recently, based on the Kirchhoff transformation and the two‐scale convergence theory, a homogenization method for the Richards equation with the Mualem‐van Genuchten model has been proposed, with a constant model parameter α relating to the inverse of the air‐entry pressure and the soil pore size distribution. The homogenized model is computationally efficient and convenient to use because of its explicit expression. In this study, we generalize this method, allowing α to be a spatially distributed random field and proposing a homogenized Richards equation in the mixed form (θ/h) under the condition that the effective hydraulic conductivity tensor is diagonal. This generalization eliminates the limitation of a constant α in practical applications; the proposed homogenized model is meaningful in most situations because the flow problems are influenced mainly by the diagonal terms of conductivity and the off‐diagonal terms are often neglected. Two‐dimensional numerical tests are conducted in soil profiles with different degrees of spatial heterogeneity structure to illustrate that the homogenized model can capture the fine‐scale flow behaviors on coarse grids effectively. Homogenization for the Richards equation with other two commonly used constitutive relations—the Brooks‐Corey model and the Gardner‐Russo model—is also illustrated in this study.

Micromechanics methods for evaluating the effective moduli of soft neo-Hookean composites

Most biological soft tissues are multiphase composite materials, and the determination of their effective constitutive relations is a major concern in medical engineering. In this paper, we consider a class of soft two-phase composites, in which both phases are isotropic and hyperelastic neo-Hookean materials. For such an isotropic composite consisting of inclusions uniformly distributed but randomly oriented in a matrix, two constitutive parameters are required to characterize its hyperelastic constitutive relation. Micromechanics methods, including dilute concentration method, Mori–Tanaka method, self-consistent method, and differential method are extended to predict the effective properties of this kind of composites. Analytical solutions are given for the hyperelastic neo-Hookean composites reinforced by spherical particles, long fibers, and penny-shaped platelets, respectively. Finite element simulations are performed to evaluate the accuracy of these theoretical methods.

Willis elastodynamic homogenization theory revisited for periodic media

The theory of elastodynamic homogenization initiated by J.R. Willis is revisited for periodically inhomogeneous media through a careful scrutiny of the main aspects of that theory in the 3D continuum context and by applying it to the thorough treatment of a simple 1D discrete periodic system. The Bloch theorem appears to be central to appropriately defining and interpreting effective fields. Based on some physical arguments, three necessary conditions are derived for the transition from the microscopic description to the macroscopic description of periodic media. The parameters involved in the Willis effective constitutive relation are expressed in terms of two localization tensors and specified with the help of the corresponding Green function in the spirit of micromechanics. These results are illustrated and discussed for the 1D discrete periodic system considered. In particular, inspired by Brillouin's study, the dependency of the effective constitutive parameters on the frequency is physically interpreted in terms of oscillation modes of the underlying microstructure.