A phenomenological and extended continuum approach for modelling non-equilibrium flows

This paper presents a new technique that combines Grad’s 13-moment equations (G13) with a phenomenological approach to rarefied gas flows. This combination and the proposed solution technique capture some important non-equilibrium phenomena that appear in the early continuum-transition flow regime. In contrast to the fully coupled 13-moment equation set, a significant advantage of the present solution technique is that it does not require extra boundary conditions explicitly; Grad’s equations for viscous stress and heat flux are used as constitutive relations for the conservation equations instead of being solved as equations of transport. The relative computational cost of this novel technique is low in comparison to other methods, such as fully coupled solutions involving many moments or discrete methods. In this study, the proposed numerical procedure is tested on a planar Couette flow case, and the results are compared to predictions obtained from the direct simulation Monte Carlo method. This test case highlights the presence of normal viscous stresses and tangential heat fluxes that arise from non-equilibrium phenomena, which cannot be captured by the Navier–Stokes–Fourier constitutive equations or phenomenological modifications.      

Analysis of matching conditions at the boundary surface of a fluid-saturated porous solid and a bulk fluid: the use of Lagrange multipliers

The compatibility conditions matching macroscopic mechanical fields at the contact surface between a fluid-saturated porous solid and an adjacent bulk fluid are considered. The general form of balance equations at that discontinuity surface are analyzed to obtain the compatibility conditions for the tangent and normal components of the velocity and the stress vector fields. Considerations are based on the procedure similar to that used in the phenomenological thermodynamics for derivation of constitutive relations, where the entropy inequality and the concept of Lagrange multipliers are applied. This procedure made possible to derive the compatibility conditions for the viscous fluid flowing tangentially and perpendicularly to the boundary surface of the porous solid and to formulate the generalized form of the so called slip condition for the fluid velocity field, postulated earlier by Beavers and Joseph, J. Fluid. Mech. 30, 197–207 (1967). PACS 47.55.Mh Communicated by Y.D. Shikhmurzaev     

A Phenomenological Constitutive Model for Describing Thermo-Viscoplastic Behavior of Al-Zn-Mg-Cu Alloy Under Hot Working Condition

In order to predict the high-temperature deformation behavior of Al-Zn-Mg-Cu alloy, the hot compression tests were conducted in the strain rate range of (0.001–0.1)s⁻¹ and the forming temperature range of (573–723) K. Based on the experimental results, Johnson-Cook model was found inadequate to describe the high-temperature deformation behavior of Al-Zn-Mg-Cu alloy. Therefore, a new phenomenological constitutive model is proposed, considering the coupled effects of strain, strain rate and forming temperature on the material flow behavior of Al-Zn-Mg-Cu alloy. In the proposed model, the material constants are presented as functions of strain rate. The proposed constitutive model correlates well with the experimental results confirming that the proposed model can give an accurate and precise estimate of flow stress for the Al-Zn-Mg-Cu alloy investigated in this study.     

Analysis of the rate-dependent coupled thermo-mechanical response of shape memory alloy bars and wires in tension

In this paper, the coupled thermo-mechanical response of shape memory alloy (SMA) bars and wires in tension is studied. By using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is derived. Taking into account the effect of generated (absorbed) latent heat during the forward (inverse) martensitic phase transformation, the local form of the first law of thermodynamics is used to obtain the energy balance relation. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a one-dimensional problem. An explicit finite difference scheme is used to discretize the governing initial-boundary-value problem of bars and wires with circular cross-sections in tension. Considering several case studies for SMA wires and bars with different diameters, the effect of loading–unloading rate and different boundary conditions imposed by free and forced convections at the surface are studied. It is shown that the accuracy of assuming adiabatic or isothermal conditions in the tensile response of SMA bars strongly depends on the size and the ambient condition in addition to the rate dependency that has been known in the literature. The data of three experimental tests are used for validating the numerical results of the present formulation in predicting the stress–strain and temperature distribution for SMA bars and wires subjected to axial loading–unloading.     

Evolutionary model of finite-strain thermoelasticity

The equation of state of finite-strain thermoelasticity is obtained using a formalized approach to constructing constitutive relations for complex media under the assumption of closeness of intermediate and current configurations. A variational formulation of the coupled thermoelastic problem is proposed. The constitutive equation, the heat-conduction equation, the relations for internal energy, free energy, and entropy, and the variational formulation of the coupled problem of finite-strain thermoelasticity are tested on the problem of uniaxial extension of a bar. The model adequately describes experimental data for elastomers, such as entropic elasticity, temperature inversion, and temperature variation during an adiabatic process. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 184–196, May–June, 2008.     

A Two-Scale Model for Coupled Electro-Chemo-Mechanical Phenomena and Onsager’s Reciprocity Relations in Expansive Clays: I Homogenization Analysis

The macroscopic model governing coupled electro-chemo-mechanical phenomena in expansive clays is revisited within a rigorous homogenization procedure applied to the microscopic governing equations which describe the local interaction between charged clay particles and a binary monovalent aqueous electrolyte solution. The up-scaling of the microscopic electro-hydro-dynamics leads to a two-scale approach wherein the macroscopic model appears governed by a fully coupled form of Onsager’s reciprocity relations, mass conservation equations and a modified Terzaghi’s effective stress principle. In addition, the two-scale approach provides microscopic representations for the effective coefficients which are exploited herein to obtain further insight in the constitutive behavior of the electrochemical parameters and the swelling pressure. Among other effects, we show that these microscopic closure relations are mainly dictated by the spatial variability of a microscale electric potential which satisfies a local version of the Poisson–Boltzmann problem in a periodic unit cell, The proposed framework allows to address various relevant still open issues regarding the constitutive behavior of swelling systems, Among them we give particular emphasis on the analysis of the influence of the fluctuation and distortion of the electrical double layer upon the magnitude of the electrochemical coefficients and the precise local conditions for the validity of the symmetry of Onsager’s relations.     

Flow in a Porous Matrix with Anisotropic Structure

The flow of a viscous fluid through a porous matrix undergoing only infinitesimal deformation is described in terms of intrinsic variables, namely, the density, velocity and stress occurring in coherent elements of each material. This formulation arises naturally when macroscopic interfaces are conceptually partitioned into area fractions of fluid–fluid, fluid–solid, and solid–solid contact. Such theory has been shown to yield consistent jump conditions of mass, momentum and energy across discontinuities, either internal or an external boundary, unlike the standard mixture theory jump conditions. In the previous formulation, the matrix structure has been considered isotropic; that is, the area fractions are independent of the interface orientation. Here, that is not assumed, so in particular, the cross-section area of a continuous fluid tube depends on its orientation, which influences the directional fluxes, and in turn the directional permeability, anisotropy of the structure. The simplifications for slow viscous flow are examined, and particularly for an isotropic linearly elastic matrix in which area partitioning induces anisotropic elastic response of the mixture. A final specialization to an incompressible fluid and stationary matrix leads to potential flow, and a simple plane flow solution is presented to illustrate the effects of anisotropic permeability.     

Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

The objective of this article is to derive a macroscopic Darcy’s law for a fluid-saturated moving porous medium whose matrix is composed of two solid phases which are not in direct contact with each other (weakly coupled solid phases). An example of this composite medium is the case of a solid matrix, unfrozen water, and an ice matrix within the pore space. The macroscopic equations for this type of saturated porous material are obtained using two-space homogenization techniques from microscopic periodic structures. The pore size is assumed to be small compared to the macroscopic scale under consideration. At the microscopic scale the two weakly coupled solids are described by the linear elastic equations, and the fluid by the linearized Navier–Stokes equations with appropriate boundary conditions at the solid–fluid interfaces. The derived Darcy’s law contains three permeability tensors whose properties are analyzed. Also, a formal relation with a previous macroscopic fluid flow equation obtained using a phenomenological approach is given. Moreover, a constructive proof of the existence of the three permeability tensors allows for their explicit computation employing finite elements or analogous numerical procedures.     

Limiting Chain Extensibility Constitutive Models of Valanis–Landel Type

A new general constitutive model in terms of the principal stretches is proposed to reflect limiting chain extensibility resulting in severe strain-stiffening for incompressible, isotropic, homogeneous elastic materials. The strain-energy density involves the logarithm function and has the general Valanis–Landel form. For specific functions in the Valanis–Landel representation, we obtain particular strain-energies, some of which have been proposed in the recent literature. The stress–stretch response in some basic homogeneous deformations is described for these particular strain-energy densities. It is shown that the stress response in these deformations is similar to that predicted by the Gent model involving the first invariant of the Cauchy–Green tensor. The models discussed here depend on both the first and second invariants.      

Wave propagation in a transversely isotropic solid cylinder of arbitrary cross-sections immersed in fluid

The wave propagation in an infinite, transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid is studied using the Fourier expansion collocation method, within the framework of the linearized, three-dimensional theory of elasticity. The equations of motion of solid and fluid are respectively formulated using the constitutive equations of a transversely isotropic cylinder and the constitutive equation of an inviscid fluid. Three displacement potential functions are introduced to uncouple the equations of motion along the radial, circumferential and axial directions. The frequency equations of longitudinal and flexural (symmetric and antisymmetric) modes are analyzed numerically for an elliptic and cardioidal cross-sectional transversely isotropic solid cylinder of arbitrary cross-section immersed in fluid. The computed non-dimensional wavenumbers are presented in the form of dispersion curves for the material zinc. The general theory can be used to study any kind of cylinder with proper geometric relations.     

A linear dynamic model for a saturated porous medium

A linear isothermal dynamic model for a porous medium saturated by a Newtonian fluid is developed in the paper. In contrast to the mixture theory, the assumption of phase separation is avoided by introducing a single constitutive energy function for the porous medium. An important advantage of the proposed model is it can account for the couplings between the solid skeleton and the pore fluid. The mass and momentum balance equations are obtained according to the generalized mixture theory. Constitutive relations for the stress, the pore pressure are derived from the total free energy accounting for inter-phase interaction. In order to describe the momentum interaction between the fluid and the solid, a frequency independent Biot-type drag force model is introduced. A temporal variable porosity model with relaxation accounting for additional attenuation is introduced for the first time. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be estimated from directly measurable phenomenological parameters. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for the two P waves and one S wave are calculated. The influences of the porosity relaxation coefficient on the velocities and attenuation coefficients of the three waves of the porous medium are discussed in a numerical example.     

Coupled nonlinear constitutive models for rarefied and microscale gas flows: subtle interplay of kinematics and dissipation effects

The constitutive relations of gases in a thermal nonequilibrium (rarefied and microscale) can be derived by applying the moment method to the Boltzmann equation. In this work, a model constitutive relation determined on the basis of the moment method is developed and applied to some challenging problems in which classical hydrodynamic theories including the Navier–Stokes–Fourier theory are shown to predict qualitatively wrong results. Analysis of coupled nonlinear constitutive models enables the fundamentals of gas flows in thermal nonequilibrium to be identified: namely, nonlinear, asymmetric, and coupled relations between stresses and the shear rate; and effect of the bulk viscosity. In addition, the new theory explains the central minimum of the temperature profile in a force-driven Poiseuille gas flow, which is a well-known problem that renders the classical hydrodynamic theory a global failure.     

Derivation of a universal relation between tangential stress and shear strain intensities in describing reversible martensitic deformation within the framework of a synthetic model

Conclusion The concept of slipping can be used in designing modern phenomenological models for the nonlinear deformation of polycrystals of various nature. Among the approaches based on the concept of slipping, the synthetic approach is one of the most effective and mathematically justified. The proposed synthetic model of phase deformation was used to describe a reversible isothermal martensitic reaction. The process of accumulation and recovery of strain under loading and unloading was described. Allowance for the microstructural peculiarities of martensitic transformations leads to understanding of macroscopic regularities in the deformation behavior of polycrystals. Use of the above averaging method enables one to describe analytically reversible changes in material properties for various types of stressed states. A universal relationship between the tangential stress and shear strain intensities is derived. The constitutive relations of the model are brought to a form analogous to the relations of the deformation theory of plasticity. Good qualitative agreement with the experimental data was obtained. In addition to the transition considered, phase reactions of the first kind under different strength and thermal conditions can be described within the framework of this model. L'vov Polytechnic University, L'vov 290013, Ukraine. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 37, No. 3, pp. 178–185, May–June, 1996.     

Unified analysis of liquefaction and the ground flow phenomenon

Loose saturated sand behaves as a solid before liquefaction but as a fluid when the excess pore water pressure equals the initial confining stress, after which it recovers its strength. A simple constitutive equation for loose saturated sand was developed to express the phase transformation between a solid and fluid during liquefaction and the ground flow phenomenon. This constitutive equation was used for a shaking table test, and its applicability was investigated by comparing numerical and experimental results Published in Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 129–144, August 2007. An erratum to this article is available at .     

An empirical constitutive equation of integral type for viscoelastic liquids

A 5-constant constitutive equation is proposed. The analytical form for the relaxation modulus as a function of flow conditions was chosen based on experimental data for stress-relaxation in solid polymers. The resulting formulae for the material functions in simple and oscillatory shear flow fulfil the empirical Cox-Merz rule as well as other phenomenological relations formulated by Coleman and Markowitz. The theoretical results are compared with experimental data obtained by Han for various polymer melts. Good agreement between theory and experiment is found.     

A Model for Flow and Deformation in Unsaturated Swelling Porous Media

A thermomechanical theory for multiphase transport in unsaturated swelling porous media is developed on the basis of Hybrid Mixture Theory (saturated systems can also be modeled as a special case of this general theory). The aim is to comprehensively and non-empirically describe the effect of viscoelastic deformation on fluid transport (and vice versa) for swelling porous materials. Three phases are considered in the system: the swelling solid matrix s, liquid l, and air a. The Coleman–Noll procedure is used to obtain the restrictions on the form of the constitutive equations. The form of Darcy’s law for the fluid phase, which takes into account both Fickian and non-Fickian transport, is slightly different from the forms obtained by other researchers though all the terms have been included. When the fluid phases interact with the swelling solid porous matrix, deformation occurs. Viscoelastic large deformation of the solid matrix is investigated. A simple form of differential-integral equation is obtained for the fluid transport under isothermal conditions, which can be coupled with the deformation of the solid matrix to solve for transport in an unsaturated system. The modeling theory thus developed, which involves two-way coupling of the viscoelastic solid deformation and fluid transport, can be applied to study the processing of biopolymers, for example, soaking of foodstuffs and stress-crack predictions. Moreover, extension and modification of this modeling theory can be applied to study a vast variety of problems, such as drying of gels, consolidation of clays, drug delivery, and absorption of liquids in diapers.     

Constitutive relations for isotropic or kinematic hardening at finite elastic–plastic deformations

The rate-type constitutive relations of rate-independent metals with isotropic or kinematic hardening at finite elastic–plastic deformations were presented through a phenomenological approach. This approach includes the decomposition of finite deformation into elastic and plastic parts, which is different from both the elastic–plastic additive decomposition of deformation rate and Lee’s elastic–plastic multiplicative decomposition of deformation gradient. The objectivity of the constitutive relations was dealt with in integrating the constitutive equations. A new objective derivative of back stress was proposed for kinematic hardening. In addition, the loading criteria were discussed. Finally, the stress for simple shear elastic–plastic deformation was worked out.     

The Influence of Moduli Slope of a Linearly Graded Matrix on the Bulk Moduli of Some Particle- and Fiber-Reinforced Composites

The stored energy functional of a homogeneous isotropic elastic body is invariant with respect to translation and rotation of a reference configuration. One can use Noether's Theorem to derive the conservation laws corresponding to these invariant transformations. These conservation laws provide an alternative way of formulating the system of equations governing equilibrium of a homogeneous isotropic body. The resulting system is mathematically identical to the system of equilibrium equations and constitutive relations, generally, of another material. This implies that each solution of the system of equilibrium equations gives rise to another solution, which describes the reciprocal deformation and solves the system of equilibrium equations of another material. In this paper we derive conservation laws and prove the theorem on conjugate solutions for two models of elastic homogeneous isotropic bodies – the model of a simple material and the model of a material with couple stress (Cosserat continuum). This revised version was published online in August 2006 with corrections to the Cover Date.     

Determination of the strength characteristics of a physically nonlinear inclusion in a linearly elastic medium

An isotropic elastic plane with a physically nonlinear inclusion with unknown properties is considered. The general relations between the stress-strain state of the inclusion and the loads applied at infinity are obtained. These relations are used to develop a method of determining the viscoelastoplastic properties of an inclusion that is based on measurement of the displacement vectors of two points that lie on the boundary of the inclusion and are nonsymmetrical with respect to its center. This makes it possible to find numerical values of the constants that enter the constitutive equations of an inclusion. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 178–184, July–August, 2000.