An Approach to Constructing a Rheological Model of a Strain-Hardening Medium

The one-dimensional constitutive equations of strain-hardening materials subject to nonlinear creep are derived. The solution is found using the hypothesis of unified deformation curve based on the similarity of the tensile and isochronic creep curves. A generalized rheological model is constructed which accounts for the instantaneous strain rate, loading rate, and the mode of strain hardening. This model is used to derive one-dimensional constitutive equations for linear viscoelastic, nonlinear viscoelastic, and linear- and nonlinear-hardening viscoelastoplastic materials. It is shown that the creep of linear viscoelastic and linear-hardening viscoelastoplastic materials is transient. For nonlinear viscoelastic and nonlinear-hardening viscoelastoplastic materials, all the characteristic stages of creep are present     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

Fundamentals of two-phase flow by the method of irreversible thermodynamics

The paper explores thermodynamic aspect of modelling two-phase systems by the methods of irreversible thermodynamics in both classical (CIT) and extended (EIT) formulation. The conservation laws for two-phase model-continuum are derived. Then, the entropy production is analysed for two-fluid and homogeneous systems. Different equations of state are taken into consideration, namely that corresponding to the accompanying equilibrium state of physical element and more complex resulting from EIT. Obtained expressions for rate of entropy production per unit volume allow to identify the dissipative mechanisms in the two-phase system and suggest the forms of phenomenological relations to be adopted in the constitutive equations.     

Nonproportional loading effects on elastic-plastic behavior based on stress resultants for thin plates of strain hardening materials

A stress resultant constitutive law in rate form is constructed for power-law hardening materials. The change of plate thickness is considered in the constitutive law. The elastic-plastic behavior of a plate element based on the stress resultant constitutive law under uniaxial combined tension and bending is determined under a limited number of nonproportional and unloading paths. The results based on the stress resultant constitutive law and the through-the-thickness integration method are compared within the context of both the small-strain and finite deformation approaches. The results indicate that the selection of the normalized equivalent stress resultant and the corresponding work-conjugate normalized equivalent generalized strain is appropriate for describing the hardening behavior in the stress resultant space. However, the hardening rule in a power law form must be modified for low hardening materials at large plastic deformation when finite deformation effects are considered.     

On Thermodynamic Consistency of Turbulent Closures

The problem of the implementation of the second law of thermodynamics for the determination of the thermodynamic consistency of solutions determined by turbulent closures is considered for incompressible fluids. The possibility of the application of the methods of thermodynamics to constraining constitutive laws describing turbulent flow features, but not material behaviour, is discussed. It is shown that the ordinary realizability conditions requiring non-negative values of the averaged squared fluctuations are necessary and sufficient conditions determining the thermodynamic consistency of a process governed by a closure model. Because turbulent closures are not universal, using the second law of thermodynamics to constrain them can impose unnecessary restrictions on the models, when the turbulent entropy is considered as a constitutive quantity. The notion and validity of different forms of the turbulent entropy is discussed. It is found that the form of the turbulent entropy originating from the analogy between the turbulent kinetic energy and absolute temperature contradicts the principle of irreversibility. In a particular case of small temperature fluctuations, the second law yields correct constraints, if the turbulent entropy is assumed not to be a constitutive quantity, but a variable governed by an evolution equation of special form generated by the balance equation for internal energy. Received 14 October 2000 and accepted 30 May 2001     

On thermoelastic dielectrics

Constitutive equations for a linear thermoelastic dielectric are derived from the energy balance equation assuming dependence of the stored energy function on the strain tensor, the polarization vector, the polarization gradient tensor and entropy. A method is indicated for constructing a hierarchy of constitutive equations for materials with arbitrary symmetry by introducing various thermodynamic potentials. Maxwell's relations are constructed for the thermodynamic potential W^L. The entropy inequality is used to obtain stability conditions for an elastic dielectric in equilibrium under prescribed boundary constraints. Frequencies are explicitly determined for a plane wave propagating along the x₁-axis in an infinite centro-symmetric isotropic thermoelastic dielectric.     

Volume-weighted mixture theory for granular materials

In the present work we treat granular materials as mixtures composed of a solid and a surrounding void continuum, proposing then a continuum thermodynamic theory for it. In contrast to the common mass-weighted balance equations of mass, momentum, energy and entropy for mixtures, the volume-weighted balance equations and the associated jump conditions of the corresponding physical quantities are derived in terms of volume-weighted field quantities here. The evolution equations of volume fractions, volume-weighted velocity, energy, and entropy are presented and explained in detail. By virtue of the second law of thermodynamics, three dissipative mechanisms are considered which are specialized for a simple set of linear constitutive equations. The derived theory is applied to the analysis of reversible and irreversible compaction of cohesionless granular particles when a vertical oscillation is exerted on the system. In this analysis, a hypothesis for the existence of a characteristic depth within the granular material in its closely compacted state is proposed to model the reversible compaction.     

A thermodynamic model of turbulent motions in a granular material

This paper is devoted to a thermodynamic theory of granular materials subjected to slow frictional as well as rapid flows with strong collisional interactions. The microstructure of the material is taken into account by considering the solid volume fraction as a basic field. This variable is of a kinematic nature and enters the formulation via the balance law of the configurational momentum, including corresponding contributions to the energy balance, as originally proposed by Goodman and Cowin [1], but modified here. Complemented by constitutive equations, the emerging field equations are postulated to be adequate for motions, be they laminar or turbulent, if the resolved length scales are sufficiently small. On large length scales the sub-grid motion may be interpreted as fluctuations, which manifest themselves in correspondingly filtered equations as correlation products, like in the turbulence theory. We apply an ergodic (Reynolds) filter to these equations and thus deduce averaged equations for the mean motions. The averaged equations comprise balances of mass, linear and configurational momenta, energy, and turbulent kinetic energy as well as turbulent configurational kinetic energy. They are complemented by balance laws for two internal fields, the dissipation rates of the turbulent kinetic energy and of the turbulent configurational kinetic energy. We formulate closure relations for the averages of the laminar constitutive quantities and for the correlation terms by using the rules of material and turbulent objectivity, including equipresence. Many versions of the second law of thermodynamics are known in the literature. We follow the Müller-Liu theory and extend Müllers entropy principle to allow the satisfaction of the second law of thermodynamics for both laminar and turbulent motions. Its exploitation, performed in the spirit of the Müller-Liu theory, delivers restrictions on the dependent constitutive quantities (through the Liu equations) and a residual inequality, from which thermodynamic equilibrium properties are deduced. Finally, linear relationships are proposed for the nonequilibrium closure relations.Received: 21 March 2003, Accepted: 1 September 2003, Published online: 11 February 2004PACS: 05.70.Ln, 61.25.Hq, 61.30.-vCorrespondence to: I. Luca     

A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part II: Finite deformations

This paper generalizes to finite deformations our companion paper [Gurtin, M.E., Anand, L., 2004. A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part I: Small deformations. Journal of the Mechanics and Physics of Solids, submitted]. Specifically, we develop a gradient theory for finite-deformation isotropic viscoplasticity in the absence of plastic spin. The theory is based on the Kröner–Lee decomposition F = F^eF^p of the deformation gradient into elastic and plastic parts; a system of microstresses consistent with a microforce balance; a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; a constitutive theory that allows:

• the microstresses to depend on D^p, the gradient of the plastic stretching,

• the free energy ψ to depend on the Burgers tensor G = F^pCurlF^p.

The microforce balance when augmented by constitutive relations for the microstresses results in a nonlocal flow rule in the form of a tensorial second-order partial differential equation for F^p. The microstresses are strictly dissipative when ψ is independent of the Burgers tensor, but when ψ depends on G the microstresses are partially energetic, and this, in turn, leads to backstresses and (hence) Bauschinger-effects in the flow rule. The typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with viscoplastic flow, and, as an aid to numerical solution, a weak (virtual power) formulation of the nonlocal flow rule is derived. Finally, the dependences of the microstresses on D^p are shown, analytically, to result in strengthening and possibly weakening of the body induced by viscoplastic flow.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CON-TINUUM THEORIES (Ⅱ)—MICROMORPHIC CONTINUUM THEORY AND COUPLE STRESS THEORY

The purpose is to reestablish the balance laws of momentum, angular momentum and energy and to derive the corresponding local and nonlocal balance equations for micromorphic continuum mechanics and couple stress theory. The desired results for micromorphic continuum mechanics and couple stress theory are naturally obtained via direct transitions and reductions from the coupled conservation law of energy for micropolar continuum theory, respectively. The basic balance laws and equations for micromorphic continuum mechanics and couple stress theory are constituted by combining these results derived here and the traditional conservation laws and equations of mass and microinertia and the entropy inequality. The incomplete degrees of the former related continuum theories are clarified. Finally, some special cases are conveniently derived. Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931≈)     

准脆性材料的弹塑性各向异性损伤耦合本构模型研究

针对准脆性材料的非线性特征：强度软化和刚度退化、单边效应、侧限强化和拉压软化、不可恢复变形、剪胀及非弹性体胀,在热动力学框架内,建立了准脆性材料的弹塑性与各向异性损伤耦合的本构关系。对准脆性材料的变形机理和损伤诱发的各向异性进行了诠释,并给出了损伤构形和有效构形中各物理量之间的关系。在有效应力空间内,建立了塑性屈服准则、拉压不同的塑性随动强化法则和各向同性强化法则。在损伤构形中,采用应变能释放率,建立了拉压损伤准则、拉压不同的损伤随动强化法则和各向同性强化法则。基于塑性屈服准则和损伤准则,构建了塑性势泛函和损伤势泛函,并由正交性法则,给出了塑性和损伤强化效应内变量的演化规律,同时,联立塑性屈服面和损伤加载面,给出了塑性流动和损伤演化内变量的演化法则。将损伤力学和塑性力学结合起来,建立了应变驱动的应力-应变增量本构关系,给出了本构数值积分的要点。以单轴加载-卸载往复试验识别和校准了本构材料常数,并对单轴单调试验、单轴加载-卸载往复试验、二轴受压、二轴拉压试验和三轴受压试验进行了预测,并与试验结果作了比较,结果表明,所建本构模型对准脆性材料的非线性材料性能有良好的预测能力。     

THE EXTREMITY LAWS OF HYDRO-THERMODYNAMICS

This paper presents the law of maximum rate of energy dissipa-tion in hydrodynamics and also in general continuum dynamicsas an addition to the classical conservation laws expressed inthe equation of continuity and the equations of motion.Thecorollary of the law is B(?)langer-B(?)ss theorem of minimum reser-ved specific energy in applied hydraulics.The mechanical energy dissipated is transformed into heatreserved in the substance.The rate of energy dissipation ata time at a given temperature gives rise to the increase in en-tropy production.Hence the maximum rate of energy dissipationsuggests itself the idea of reformulation of the second law ofthermodynamics that the rate of entropy production in mech-anical motion is always the maximum possible.The proposed extremity law in continuum dynamics has beenderived from the variational principle and the reformulatedsecond law of thermodynamics analyzed microscopically in thepaper.The two laws together form the extremity laws of hydro-thermodynamics.     

A new thermodynamically consistent continuum model for hardening plasticity coupled with damage

A phenomenological model for hardening–softening elasto-plasticity coupled with damage is presented. Specific kinematic internal variables are used to describe the mechanical state of the system. These, in the hypothesis of infinitesimal changes of configuration, are partitioned in the sum of a reversible and an irreversible part. The constitutive equations, developed in the framework of the Generalised Standard Material Model, are derived for reversible processes from an internal energy functional, postulated as the sum of the deformation energy and of the hardening energy both coupled with damage, while for irreversible phenomena from a dissipation functional.Performing duality transformations, the conjugated potentials of the complementary elastic energy and of the complementary dissipation are obtained. From the latter a generalised elastic domain in the extended space of stresses and thermodynamic forces is derived. The model, which is completely formulated in the space of actual stresses, is compared with other formulations based on the concept of effective stresses in the case of isotropic damage. It is observed that such models are consistent only for particular choices of the damage coupling. Finally, the predictions of the proposed model for some simple processes are analysed.     

显式方法精确模拟形状记忆聚合物热力学行为

通过构建一个热耦合的多轴可压缩应变能函数,得到应力-应变、应力-温度和应变-温度之间的函数关系,建立形状记忆聚合物的本构方程.本文引入三个基于对数应变的不变量使得模型（i）可以模拟可压缩情况;（ii）适用于单轴拉伸和等双轴拉伸至少两个基准实验;（iii）多轴有效.通过显式方法（i）给出自由能和熵的具体表达,证明模型热力学定律;（ii）给出应变-应力,温度-应力以及,温度-应变的形函数具体表达.多轴模型在特定的情况下可以自动退化到各自的单轴情况. 通过调节形函数的参数,最终得到的模型结果和实验结果能够精确匹配.新方法建立的本构模型得到的结果能更加准确地指导形状记忆聚合物的工程设计。     

Galilean relativistic fluid mechanics

Single-component nonrelativistic dissipative fluids are treated independently of reference frames and flow-frames. First the basic fields and their balances are derived, then the related thermodynamic relations and the entropy production are calculated and the linear constitutive relations are given. The usual basic fields of mass, momentum, energy and their current densities, the heat flux, pressure tensor and diffusion flux are the time- and spacelike components of the third-order mass–momentum–energy density-flux four-tensor. The corresponding Galilean transformation rules of the physical quantities are derived. It is proved that the non-equilibrium thermodynamic frame theory, including the thermostatic Gibbs relation and extensivity condition and also the entropy production, is independent of the reference frame and also the flow-frame of the fluid. The continuity-Fourier–Navier–Stokes equations are obtained almost in the traditional form if the flow of the fluid is fixed to the temperature. This choice of the flow-frame is the thermo-flow. A simple consequence of the theory is that the relation between the total, kinetic and internal energies is a Galilean transformation rule.     

Formulation of thermo-hydro-mechanical coupling behavior of unsaturated soils based on hybrid mixture theory

Thermo-Hydro-Mechanical （THM） coupling pro- cesses in unsaturated soils are very important in both theoretical researches and engineering applications. A coupled formulation based on hybrid mixture theory is derived to model the THM coupling behavior of unsaturated soils. The free-energy and dissipative functions for different phases are derived from Taylor＇s series expansions. Constitutive relations for THM coupled behaviors of unsaturated soils, which include deformation, entropy change, fluid flow, heat conduction, and dynamic compatibility conditions on the interfaces, are then established. The number of field equations is shown to be equal to the number of unknown variables; thus, a closure of this coupling problem is established. In addition to modifications of the physical conservation equations with coupling effect terms, the constitutive equations, which consider the coupling between elastoplastic deformation of the soil skeleton, fluid flow, and heat transfer, are also derived.     

显式方法精确模拟形状记忆聚合物热力学行为

通过构建一个热耦合的多轴可压缩应变能函数,得到应力-应变、应力-温度和应变-温度之间的函数关系,建立形状记忆聚合物的本构方程.本文引入三个基于对数应变的不变量使得模型（i）可以模拟可压缩情况;（ii）适用于单轴拉伸和等双轴拉伸至少两个基准实验;（iii）多轴有效.通过显式方法（i）给出自由能和熵的具体表达,证明模型热力学定律;（ii）给出应变-应力,温度-应力以及,温度-应变的形函数具体表达.多轴模型在特定的情况下可以自动退化到各自的单轴情况. 通过调节形函数的参数,最终得到的模型结果和实验结果能够精确匹配.新方法建立的本构模型得到的结果能更加准确地指导形状记忆聚合物的工程设计。     

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY(Ⅰ)—NONLINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

IntroductionSoilisthemostcommonlyusedconstructionmaterialincivilengineeringandhydraulicengineering .Thecharacteristicsofsoilhavebeeninvestigatingfornearlyonehundredyears.Butbecauseofitscomplexstructure,changeableenvironmentandbeingsensitivetotheoutsideconditions,thesoiloftenshowsvariedproperties[1,2 ].Themaindifficultytothedevelopmentofgeotechnicalmechanicsishowtosetupconstitutiveequationswhichcouldsatisfactorilyaccountforengineeringpropertiesofsoil[3].Manyconstitutivemodelshavebeenformedinth…