Internal energy variational principles and governing equations in electroelastic analysis

The first thermodynamic law contains a universal thermodynamic variational principle. The complete internal energy variational principle in the electroelastic analysis is not discussed in previous papers. In this paper this principle will be discussed. From this principle the simple complete governing equations can be deduced, and the Maxwell stress can be naturally derived from this variational principle. It is shown that the Maxwell stress has slightly different forms determined by using internal energy or electric Gibbs free energy variational principle, but substantially they are the same. In the second-order precision the Maxwell stress is uniquely determined, and its expression has the same form for all deformable and rigid dielectrics. The electroelastic analyses in the dielectric should be studied together with its environment, because the electric field exists in all materials except the ideal conductor. The complete governing equations under finite deformation in the initial configuration are also discussed.     

Gradient elasticity and nonstandard boundary conditions

Gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality. The pertinent thermodynamic restrictions upon the gradient constitutive equations are derived, which are shown to include, besides the field (differential) stress–strain laws, a set of nonstandard boundary conditions. Consistently with the latter thermodynamic requirements, a surface layer with membrane stresses is envisioned in the strained body, which together with the above nonstandard boundary conditions make the body constitutively insulated (i.e. no long distance energy flows out of the boundary surface due to nonlocality). The total strain energy is shown to include a bulk and surface strain energy. A minimum total potential energy principle is provided for the related structural boundary-value problem. The Toupin–Mindlin polar-type strain gradient material model is also addressed and compared with the above one, their substantial differences are pointed out, particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations. A gradient one-dimensional bar sample in tension is considered for a few applications of the proposed theory.     

Fundamentals of viscoelastoplasticity and hardening theory revisited

Thermodynamic and statistical methods for setting up the constitutive equations describing the viscoelastoplastic deformation and hardening of materials are proposed. The thermodynamic method is based on the law of conservation of energy, the equations of entropy balance and entropy production in the presence of self-balanced internal microstresses characterized by conjugate hardening parameters. The general constitutive equations include the relationships between the thermodynamic flows and forces, which follow from nonnegative entropy production and satisfy the generalized Onsager’s principle, and the thermoelastic relations and the expression for entropy, which follow from the law of conservation of energy. Specific constitutive equations are derived by representing the dissipation rate as a sum of two terms responsible for kinematic and isotropic hardening and approximated by power and hyperbolic-sinus functions. The constitutive equations describing viscoelastoplastic deformation and hardening are derived based on stochastic microstructural concepts and on the linear thermoelasticity model and nonlinear Maxwell model for the spherical and deviatoric components of microstresses and microstrains, respectively. The problem of determining the effective properties and stress-strain state of a three-component material found using the Voigt-Reuss scheme leads to constitutive equations similar in form to those produced by the thermodynamic method __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 3–18, February 2008.     

耗散系统的虚功变分不等式及其应用

对于保守系统,能量变分原理为推导力学系统控制方程提供了简洁的途径.对于耗散系统,控制方程的建立往往需要引入经验的或理性的假定,增大了建模的难度.针对耗散系统,引入系统局部稳定的概念,并在此基础上,提出一类虚功变分不等式.这一不等式事实上揭示了耗散系统的一类虚功不等原理.该原理的物理含义为:使系统状态稳定的必要条件是,在该状态附近所有可能的虚拟路径上系统释放的势能不大于系统耗散的能量.研究表明:仅需结合虚功不等原理和能量守恒原理,即可导出准静态系统力学状态量的全部控制方程.作为应用,文章重新讨论了塑性力学,结合虚功不等原理与能量守恒原理,导出经典塑性力学的全部控制方程,并证明了经典的最大塑性耗散原理可以作为虚功不等原理的推论导出;同时,以Mohr-Coulomb强度准则为例,讨论了虚功不等原理在强度理论中的应用,说明基于应力的强度准则可以是基于能量的稳定性准则的推论.上述例子说明了虚功不等原理的广泛适用性和在建立耗散系统控制方程中的有效性.     

Mechanics and thermodynamics of fluid-saturated highly elastic materials

We consider a mixture that consists of a highly elastic material and a liquid dissolved in this material. Using the laws of classical thermodynamics, we state a variational principle describing the mixture equilibrium under static loading conditions. From this principle, we derive equilibrium equations and a system of constitutive relations characterizing the mixture elastic and thermodynamic properties. We state problems describing the stress-strain state of a swollen material and a statically loaded material in thermodynamic equilibrium with the liquid. We consider the case of incompressible mixture. The general theory is illustrated by examples concerned with the deformation behavior of inhomogeneously swollen cross-linked polymers and with their thermodynamics of strains and swelling in solvent media.     

Extension of Gibbs–Duhem equation including influences of external fields

Gibbs–Duhem equation is one of the fundamental equations in thermodynamics, which describes the relation among changes in temperature, pressure and chemical potential. Thermodynamic system can be affected by external field, and this effect should be revealed by thermodynamic equations. Based on energy postulate and the first law of thermodynamics, the differential equation of internal energy is extended to include the properties of external fields. Then, with homogeneous function theorem and a redefinition of Gibbs energy, a generalized Gibbs–Duhem equation with influences of external fields is derived. As a demonstration of the application of this generalized equation, the influences of temperature and external electric field on surface tension, surface adsorption controlled by external electric field, and the derivation of a generalized chemical potential expression are discussed, which show that the extended Gibbs–Duhem equation developed in this paper is capable to capture the influences of external fields on a thermodynamic system.     

A unified residual-based thermodynamic framework for strain gradient theories of plasticity

A unified thermodynamic framework for gradient plasticity theories in small deformations is provided, which is able to accommodate (almost) all existing strain gradient plasticity theories. The concept of energy residual (the long range power density transferred to the generic particle from the surrounding material and locally spent to sustain some extra plastic power) plays a crucial role. An energy balance principle for the extra plastic power leads to a representation formula of the energy residual in terms of a long range stress, typically of the third order, a macroscopic counterpart of the micro-forces acting on the GNDs (Geometrically Necessary Dislocations). The insulation condition (implying that no long range energy interactions are allowed between the body and the exterior environment) is used to derive the higher order boundary conditions, as well as to ascertain a principle of the plastic power redistribution in which the energy residual plays the role of redistributor and guarantees that the actual plastic dissipation satisfies the second thermodynamics principle. The (nonlocal) Clausius-Duhem inequality, into which the long range stress enters aside the Cauchy stress, is used to derive the thermodynamic restrictions on the constitutive equations, which include the state equations and the dissipation inequality. Consistent with the latter inequality, the evolution laws are formulated for rate-independent models. These are shown to exhibit multiple size effects, namely (energetic) size effects on the hardening rate, as well as combined (dissipative) size effects on both the yield strength (intrinsic resistance to the onset of plastic strain) and the flow strength (resistance exhibited during plastic flow). A friction analogy is proposed as an aid for a better understanding of these two kinds of strengthening effects. The relevant boundary-value rate problem is addressed, for which a solution uniqueness theorem and a minimum variational principle are provided. Comparisons with other existing gradient theories are presented. The dissipation redistribution mechanism is illustrated by means of a simple shear model.     

Higher Order Terms in Extended Thermodynamics

Extended Thermodynamics is an elegant and powerful material theory which yields, when applied, a symmetric hyperbolic system of evolution equations for the independent field variables. This system is obtained by imposing general physical principles, such as that of material objectivity and the entropy principle. With respect to thermodynamic equilibrium, the latter is carried through to second order. However, the following question arises: If this principle is imposed to even higher order terms, could it then yield restrictions also on the lower order terms, beyond those previously obtained in the literature? In this paper the entropy principle is applied to fourth order so that constitutive functions up to this order are obtained. In the process of computations many complicated equations involving only the lower order terms are deduced, which must be satisfied as identities. These equations, after long and tedious calculations, turn indeed out to be identically satisfied. This fact cannot be casual, thus confirming that this theory is trustworthy. Received October 28, 1997     

固体炸药爆轰与惰性介质相互作用的一种扩散界面模型

提出一种保持热力学一致性的扩散界面模型,用来数值模拟固体炸药爆轰与惰性介质的相互作用问题。基于混合网格内各组分物质间可以达到力学平衡状态而不能达到热学平衡状态的假设,由混合网格能量守恒以及压力相等条件,推导出每种组分物质的体积分数演化方程。由此获得的扩散界面模型包括组分物质的质量守恒方程、混合物质的动量及总能量守恒方程,同时包括组分物质的体积分数演化方程和混合物质的压力演化方程。该扩散界面模型的主要特点是考虑了化学反应以及热学非平衡的影响。提出的扩散界面模型在物质界面附近不会出现物理量的非物理振荡现象、适用于任意表达形式的物质状态方程以及任意数目的惰性介质。     

热动力学理论在粘弹性断裂力学中的应用

本文基于热动力学定理推导了线性粘弹性材料的本构方程，并根据已有文献的思想列出了线性粘弹性材料的热力学平衡方程，由此推得二种全局能量释放率的表达开式，同时根据虚功原理进一步推得裂尖局部能量释放率，也就说明了全局能量释放率和被广泛应用的局部能量释放率之间的等价性，本文最后给出了在正交各向异性和向同性粘弹性介质中裂尖能量释放率的具体表达式。     

Potentials for the modified Cam-Clay model

Energy and dissipation pseudo-potentials are employed to derive constitutive relationships, in the context of thermodynamic concepts, for the widely used Modified Cam-Clay (MCC) model for soil mechanics. A variational formulation of the MCC evolution equations is proposed in this paper. Since plastic collapse of MCC soils cannot be embedded in the classical limit analysis theory, finding the critical amplification of the load that produces plastic collapse is formulated in the form of a system of equations and inequalities. Then, a mixed minimization principle is proposed for the plastic collapse analysis of MCC soils. This principle is obtained by the application of the variational formulation for the flow law introduced in the first part of the article.     

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     

Some Considerations on Modeling Heat and Mass Transfer in Porous Media

In this paper some considerations are presented about the equations needed to set up a model of the process of heat and mass transfer in porous media. A clear classification is made of the various types of equations used and of their physical meaning. Special attention is paid to the thermodynamic equilibrium equations and to their derivation since they are too often taken for granted. The importance of the various transport mechanisms (of mass and energy) is analyzed and the consequences that can arise when some term is neglected are indicated.     

A nonhomogeneous nonlocal elasticity model

Nonlocal elasticity with nonhomogeneous elastic moduli and internal length is addressed within a thermodynamic framework suitable to cope with continuum nonlocality. The Clausius–Duhem inequality, enriched by the energy residual, is used to derive the state equations and all other thermodynamic restrictions upon the constitutive equations. A phenomenological nonhomogeneous nonlocal (strain difference-dependent) elasticity model is proposed, in which the stress is the sum of two contributions, local and nonlocal, respectively governed by the standard elastic moduli tensor and the (symmetric positive-definite) nonlocal stiffness tensor. The inhomogeneities of the elastic moduli and of the internal length are conjectured to be each the cause of additional attenuation effects upon the long distance particle interactions. The increased attenuation effects are accounted for by means of the standard attenuation function, but with the standard spatial distance replaced by a suitably larger equivalent distance, and with the spatially variable internal length replaced by the largest value within the domain. Formulae for the computation of the equivalent distance are heuristically suggested and illustrated with numerical examples. The solution uniqueness of the continuum boundary-value problem is proven and the related total potential energy principle given and employed for possible nonlocal-FEM discretizations. A bar in tension is considered for a few numerical applications, showing perfect numerical stability, provided the free energy potential is positive definite.     

A thermodynamic approach with internal variables using Lagrange formalism. Part II. Continuous symmetries in the case of the time–temperature equivalence

In the first part of this contribution, the Lie-symmetries of the principle of least action associated to the constitutive equations of the DNLR formalism of relaxation have been presented. We examine in this second part the continuous symmetries corresponding to the simple case of stress relaxation under isothermal conditions. The well-known principle of time/temperature equivalence is discussed in terms of variational symmetry for the Jacobi’s action functional, and connected to the Onsager’s relation near the thermodynamic equilibrium.     

A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi's effective stress principle

Summary The principle of virtual power is used to derive the equilibrium field equations of a porous solid saturated with a fluid, including second density-gradient effects; the intention is the elucidation and extension of the effective stress principle of Terzaghi and Fillunger. In the context of a first density-gradient theory for a saturated solid we interpret the porewater pressure as a Lagrange multiplier in the expression for the deformation energy, assuring that the saturation constraint is verified. We prove that this saturation pressure is distributed among the constituents according to their respective volume fraction (Delesse law) only if they are both true density-preserving. We generalize the Delesse law to the case of compressible constituents. If a material-dependent effective stress contribution is to arise, it is, in general, nonvanishing simultaneously in both the solid and fluid constituents. Moreover, saturation pressure, effective stresses and compressibility constitutive equations determine the exchange volume forces. In a theoretical formulation without non-isotropic strain measures, second density-gradient effects must be incorporated, not only to accommodate for the equilibrium-solid-shear stress and the interaction among neighboring solid-matrix pores, but also to describe internal capillarity effects. The earlier are accounted for by a dependence of the thermodynamic energy upon the density-gradient of the solid, while the latter derives from a mixed density-gradient dependence. Examples illustrate the necessity of these higher gradient effects for properly posed boundary value problems describing the mechanical behaviour of the disturbed rock zone surrounding salt caverns. In particular, we show that solid second-gradient strains allow for the definition of the concept of static permeability, which is distinct from the dynamic Darcy permeability. Received 1 February 1999; accepted for publication 9 March 1999     

A finite-deformation constitutive model for bulk metallic glass composites

The focus of this study is the development of an elastic-viscoplastic, three-dimensional, finite-deformation constitutive model to describe the large deformation behavior of bulk metallic glass (BMG) composite. A macroscopic theoretical formulation is proposed based on thermodynamic considerations to describe the response at ambient temperature and pressure, as well as at different strain rates. A constitutive equation is derived using the principle of thermodynamics and the augmenting of free energy. This is done by assuming that deformation within the constituent phases of the composite is affine; kinetic equations defining plastic shear and evolution of free volume concentration are then derived. The constitutive model is subsequently implemented in a finite-element program (Abaqus/Explicit) via a user-defined material subroutine. Numerical predictions are compared with experimental results from tests on La-based in situ BMG composite (La–Al–Cu–Ni) specimens cast in-house; this demonstrates that the model is able to describe the material behavior observed.     

On the equilibrium response of different modeling approaches for the porosity variation in dry granular matter

A selection of models for the variation in porosity in dry granular flows is investigated and compared on the basis of thermodynamic consistency to illustrate their performance and limitations in equilibrium situations. To this end, the thermodynamic analysis, based on the Müller–Liu entropy principle, is employed to deduce the ultimate constitutive equations at equilibrium. Results show that while all the models deliver appropriate equilibrium expressions of the Cauchy stress tensor for compressible grains, the model in which the variation in porosity is treated kinematically yields a spherical stress tensor for incompressible grains. Only the model in which the variation in porosity is modeled by a dynamic equation can give rise to a non-spherical stress tensor at equilibrium. The present study illuminates the validity and thermodynamic justification of the two modeling approaches for the porosity variation in dry granular matter.