土的本构方程与热力学

介绍一种基于热力学理论建立土力学本构方程的一般性理论框架. 这一方法利用两个势函数即自由能势函数和耗散势函数(或屈服函数)以及固定的过程和框架, 建立土的本构方程. 简要介绍了建立热力学本构方程中所用到的热力学内变量理论, 利用Legendre变换建立了热力学势函数之间以及各耗散函数与屈服函数之间的关系;利用自由能势函数和耗散势函数(或屈服函数)建立土的本构方程及其具体步骤. 最后讨论了土力学本构方程研究的意义以及它和应用之间的关系.      

Thermomechanical material modelling based on a hybrid free energy density depending on pressure,isochoric deformation and temperature

In order to represent temperature-dependent mechanical material properties in a thermomechanical consistent manner it is common practice to start with the definition of a model for the specific Helmholtz free energy. Its canonical independent variables are the Green strain tensor and the temperature. But to represent calorimetric material properties under isobaric conditions, for example the exothermal behaviour of a curing process or the dependence of the specific heat on the temperature history, the temperature and the pressure should be taken as independent variables. Thus, in the field of calorimetry the Gibbs free energy is usually used as thermodynamic potential whereas in continuum mechanics the Helmholtz free energy is normally applied. In order to simplify the representation of calorimetric phenomena in continuum mechanics a hybrid free energy density is introduced. Its canonical independent variables are the isochoric Green strain tensor, the pressure and the temperature. It is related to the Helmholtz free energy density by a Legendre transformation. In combination with the additive split of the stress power into the sum of isochoric and volumetric terms this approach leads to thermomechanical consistent constitutive models for large deformations. The article closes with applications of this approach to finite thermoelasticity, curing adhesives and the glass transition.     

On tensorial forms of thermodynamic potentials in mixtures theory

Four thermodynamic tensorial quantities, equivalents of the thermodynamic potentials, namely the chemical potential tensor, the tensor of enthalpy, the tensor of free energy and the tensor of internal energy are presented. The last three of them are proposed in this paper and connections between all of these tensors are derived. The tensorial forms of the thermodynamic potentials are expressed in terms of four possible pairs of independent state variables. A set of sixteen alternative expressions, four for each tensorial form of the thermodynamic potential are derived and their importance discussed.     

Shape memory behaviour: modelling within continuum thermomechanics

A phenomenological material model to represent the multiaxial material behaviour of shape memory alloys is proposed. The material model is able to represent the main effects of shape memory alloys: the one-way shape memory effect, the two-way shape memory effect due to external loads, the pseudoelastic and pseudoplastic behaviour as well as the transition range between pseudoelasticity and pseudoplasticity.The material model is based on a free energy function and evolution equations for internal variables. By means of the free energy function, the energy storage during thermomechanical processes is described. Evolution equations for internal variables, e.g. the inelastic strain tensor or the fraction of martensite are formulated to represent the dissipative material behaviour. In order to distinguish between different deformation mechanisms, case distinctions are introduced into the evolution equations. Thermomechanical consistency is ensured in the sense that the constitutive model satisfies the Clausius–Duhem inequality.Finally, some numerical solutions of the constitutive equations for isothermal and non-isothermal strain and stress processes demonstrate that the various phenomena of the material behaviour are well represented. This applies for uniaxial processes and for non-proportional loadings as well.     

An energy release rate-based plastic-damage model for concrete

A new plastic-damage constitutive model for concrete is proposed in this paper. A tensile and a shear damage variable are adopted to describe the degradation of the macromechanical properties of concrete. Within the framework of continuum damage mechanics, the elastic Helmholtz free energy is defined to establish the plastic-damage constitutive relation with the internal variables. Regarding the specific format for the effective stress space plasticity, the evolution law for the plastic strains and the explicit expression for the elastoplastic Helmholtz free energy are determined and the damage energy release rates that are conjugated to the damage variables are derived. Thus, damage energy release rate-based damage criteria can be established in conformity to thermodynamical principles. In accordance with the normality rule, evolution laws for the damage variables are obtained to complete the proposed plastic-damage model. Some computational aspects concerning the numerical algorithm implementation are discussed as well. Several numerical simulations are presented at the end of the paper, whose results allow for validating the capability of the proposed model for reproducing the typical nonlinear performances of concrete structures under different monotonic and cyclic load conditions.     

A thermomechanical framework for rate-independent dissipative materials with internal functions

This paper builds on previous work by Houlsby and Puzrin (Int. J. Plasticity 16 (2000) 1017) in which a framework was set out for the derivation of rate-independent plasticity theory from thermodynamic considerations. A key feature of the formalism is that the entire constitutive response is determined by knowledge of two scalar functions. The loading history is effectively captured through the use of internal variables. In this paper, we extend the concept of internal variables to that of internal functions, which represent infinite numbers of internal variables. In this case the thermodynamic functions are replaced by functionals. We set out the formalism necessary to derive constitutive behaviour within this approach. The principal advantages of this development is that it can provide realistic modelling of kinematic hardening effects and smooth transitions between elastic and elastic–plastic behaviour.     

A thermo-mechanical continuum theory with internal length for cohesionless granular materials

A thermodynamically consistent continuum theory for single-phase, single-constituent cohesionless granular materials is presented. The theory is motivated by dimensional inconsistencies of the original Goodman-Cowin theory [1–3]; it is constructed by removing these inconsistencies through the introduction of an internal length ℓ. Four constitutive models are proposed and discussed in which ℓ is (i) a material constant (Model I), (ii) an independent constitutive variable (Model II), (iii) an independent dynamic field quantity (Model III) and (iv) an independent kinematic field quantity (Model IV). Expressions of the constitutive variables emerging in the systems of the balance equations in these four models in thermodynamic equilibrium are deduced by use of a thermodynamic analysis based on the Müller-Liu entropy principle. Comments on the validity of these four models are given and discussed; the results presented in the current study show a more general formulation for the constitutive quantities and can be used as a basis for further continuum-based theoretical investigations on the behaviour of flowing granular materials. Numerical results regarding simple plane shear flows will be discussed and compared in Part II of this work.     

On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions

Expressions for thermodynamic potentials (internal energy, Helmholtz energy, Gibbs energy and enthalpy) of a thermoelastic material are developed under the assumption of small strains and finite changes in the thermal variable (temperature or entropy). The literature provides expressions for the Helmholtz energy in terms of strain and temperature, most often as expansions to the second order in strain and to a higher order in temperature changes, which ensures an affine stress–strain relation and a certain temperature dependence of the moduli of the material. Expressions are here developed for the four potentials in terms of all four possible pairs of independent variables. First, an expression is obtained for each potential as a quadratic function of its natural mechanical variable with coefficients depending on its natural thermal variable that are identified in terms of the moduli of the material. The form of the coefficients’ dependence on the thermal variable is not specified beforehand so as to obtain the most general expressions compatible with an affine stress–strain relation. Then, from each potential expressed in terms of its natural variables, expressions are derived for the other three potentials in terms of these same variables using the Gibbs–Helmholtz equations. The paper provides a thermodynamic framework for the constitutive modeling of thermoelastic materials undergoing small strains but finite changes in the thermal variables, the properties of which are liable to depend on the thermal variables.     

Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects

Recently it has been demonstrated that, on the basis of the separation D=D^e+D^p arising from the split of the stress power and two consistency criteria for objective Eulerian rate formulations, it is possible to establish a consistent Eulerian rate formulation of finite elastoplasticity in terms of the Kirchhoff stress and the stretching, without involving additional deformation-like variables labelled “elastic” or “plastic”. It has further been demonstrated that this consistent formulation leads to a simple essential structure implied by the work postulate, namely, both the normality rule for plastic flow D^p and the convexity of the yield surface in Kirchhoff stress space. Here, we attempt to place such an Eulerian formulation on the thermodynamic grounds by extending it to a general case with thermal effects, where the consistency requirements are treated in a twofold sense. First, we propose a general constitutive formulation based on the foregoing separation as well as the two consistency criteria. This is accomplished by employing the corotational logarithmic rate and by incorporating an exactly integrable Eulerian rate equation for D^e for thermo-elastic behaviour. Then, we study the consistency of the formulation with thermodynamic laws. Towards this goal, simple forms of restrictions are derived, and consequences are discussed. It is shown that the proposed Eulerian formulation is free in the sense of thermodynamic consistency. Namely, a Helmholtz free energy function in explicit form may be found such that the restrictions from the thermodynamic laws can be fulfilled with positive internal dissipation for arbitrary forms of constitutive functions included in the constitutive formulation. In particular, that is the case for the foregoing essential constitutive structure in the purely mechanical case. These results eventually lead to a complete, explicit constitutive theory for coupled fields of deformation, stress and temperature in thermo-elastoplastic solids at finite deformations.     

A physical model of the structure and attenuation of shock waves in metals

In this paper, a physical model of the structure and attenuation of shock waves in metals is presented. In order to establish the constitutive equations of materials under high velocity deformation and to study the structure of transition zone of shock wave, two independent approaches are involved. Firstly, the specific internal energy is decomposed into the elastic compression energy and elastic deformation energy, and the later is represented by an expansion to third-order terms in elastic strain and entropy, including the coupling effect of heat and mechanical energy. Secondly, a plastic relaxation function describing the behaviour of plastic flow under high temperature and high pressure is suggested from the viewpoint of dislocation dynamics. In addition, a group of ordinary differential equations has been built to determine the thermo-mechanical state variables in the transition zone of a steady shock wave and the thickness of the high pressure shock wave, and an analytical solution of the equations can be found provided that the entropy change across the shock is assumed to be negligible and Hugoniot compression modulus is used instead of the isentropic compression modulus. A quite approximate method for solving the attenuation of shock wave front has been proposed for the flat-plate symmetric impact problem.     

A continuum damage model for piezoelectric materials

In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helrnhotlz free energy of piezoelectric mate- rials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja＇s tensor valued internal state damage variables as well as the Helrnhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.     

Constraints, Constitutive Limits, and Instability in Finite Thermoelasticity

This paper examines all the possible types of thermomechanical constraints in finite-deformational elasticity. By a thermomechanical constraint we mean a functional relationship between a mechanical variable, either the deformation gradient or the stress, and a thermal variable, temperature, entropy or one of the energy potentials; internal energy, Helmholtz free energy, Gibbs free energy or enthalpy. It is shown that for the temperature-deformation, entropy-stress, enthalpy-deformation, and Helmholtz free energy-stress constraints equilibrium states are unstable, in the sense that certain perturbations of the equilibrium state grow exponentially. By considering the constrained materials as constitutive limits of unconstrained materials, it is shown that the instability is associated with the violation of the Legendre–Hadamard condition on the internal energy. The entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints do not exhibit this instability. It is proposed that stability of the rest state (or equivalently convexity of internal energy) is a necessary requirement for a model to be physically valid, and hence entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints are physical, whereas temperature-deformation constraints (including the customary notion of thermal expansion that density is a function of temperature only), entropy-stress constraints, enthalpy-deformation constraints, and Helmholtz free energy-stress constraints are not.     

Hyperelastic dielectrics with saturated polarization2

Variational and invariance principles of modern continuum mechanics are used to establish the field equations, boundary conditions and constitutive relations of a non-linear hyperelastic dielectric with constant magnitude ‘saturated’ polarization. Euclidean invariance places restrictions on the Lagrangian and implies the basic conservation laws. The principles of objectivity and material symmetry restrict the form of the constitutive equations. Four equivalent forms of the free energy functional are listed and for one of these forms the minimal isotropy integrity basis. consisting of eleven invariants, is constructed. The positive definiteness of the energy functional is used to derive various inequalities for the material constants of isotropic dielectrics.     

A principle of virtual work for combined electrostatic and mechanical loading of materials

The equations governing mechanics and electrostatics are formulated for a system in which the material deformations and electrostatic polarizations are arbitrary. A mechanical/electrostatic energy balance is formulated for this situation in terms of the electric enthalpy, in which the electric potential and the electric field are the independent variables, and charge and electric displacement, respectively, are the conjugate thermodynamic forces. This energy statement is presented in the form of a principle of virtual work (PVW), in which external virtual work is equated to internal virtual work. The resulting expression involves an internal material virtual work in which (1) material polarization is work-conjugate to increments of electric field, and (2) a combination of Cauchy stress, Maxwell stress and a product of polarization and electric field is work-conjugate to increments of strain. This PVW is valid for all material types, including those that are conservative and those that are dissipative. Such a virtual work expression is the basis for a rigorous formulation of a finite element method for problems involving the deformation and electrostatic charging of materials, including electroactive polymers and switchable ferroelectrics. The internal virtual work expression is used to develop the structure of conservative constitutive laws governing, for example, electroactive elastomers and piezoelectric materials, thereby determining the form of the Maxwell or electrostatic stress. It is shown that the Maxwell or electrostatic stress has a form fully constrained by the constitutive law and cannot be chosen independently of it. The structure of constitutive laws for dissipative materials, such as viscoelastic electroactive polymers and switchable ferroelectrics, is similarly determined, and it is shown that the Maxwell or electrostatic stress for these materials is identical to that for a material having the same conservative response when the dissipative processes in the material are shut off. The form of the internal virtual work is used further to develop the structure of dissipative constitutive laws controlled by rearrangement of material internal variables.     

On the representation of chemical ageing of rubber in continuum mechanics

In order to represent the chemical ageing behaviour of rubber under finite deformations a three-dimensional theory is proposed. The fundamentals of this approach are different decompositions of the deformation gradient in combination with an additive split of the Helmholtz free energy into three parts. Its first part belongs to the volumetric material behaviour. The second part is a temperature-dependent hyperelasticity model which depends on an additional internal variable to consider the long-term degradation of the primary rubber network. The third contribution is a functional of the deformation history and a further internal variable; it describes the creation of a new network which remains free of stress when the deformation is constant in time. The constitutive relations for the stress tensor and the internal variables are deduced using the Clausius–Duhem inequality. In order to sketch the main properties of the model, expressions in closed form are derived with respect to continuous and intermittent relaxation tests as well as for the compression set test. Under the assumption of near incompressible material behaviour, the theory can also represent ageing-induced changes in volume and their effect on the stress relaxation. The simulations are in accordance with experimental data from literature.     

Consequences of the second law for a turbulent fluid flow

Second Law statements in thermomechanics applicable to turbulent fluid flow, in which the internal energy in a macroscopic field theory includes contributions both from molecular vibrations and from turbulent fluctuations, are discussed. In the absence of turbulence, these statements naturally reduce to the known and accepted Second Law statements for a nonturbulent medium. The usual version of the Second Law statements — which deny the existence of perpetual motion and place restrictions on the constitutive equations —is extended here in the presence of turbulence; and an additional statement is introduced associated with the tendency of turbulent fluctuations to decay in the absence of external work or the addition of thermal heat. The mathematical representations of various Second Law statements are then used to derive several restrictions on the response variables of the macroscopic turbulence theory. Examples of such variables include the rates of production and dissipation of turbulent fluctuations, the rate of thermal entropy production, internal energy (involving constitutive coefficients which may be taken to be the thermal and turbulent specific heats), turbulent viscosity coefficients and other response functions which control the degree of flow anisotropy in the medium. These Second Law restrictions are then applied to a recent theory of macroscopic turbulent flow by the present authors in which fairly general constitutive equations are presented for the dependent variables of the theory. It is found that not only is the range of values of several constitutive coefficients limited by these Second Law restrictions, but the presence of a number of terms in the constitutive equations is entirely denied.     

On the thermodynamic framework of generalized coupled thermoelastic-viscoplastic-damage modeling

A complete potential based framework utilizing internal state variables is put forth for the derivation of reversible and irreversible constitutive equations. In this framework, the existence of the total (integrated) form of either the (Helmholtz) free energy or the (Gibbs) complementary free energy are assumed a priori. Two options for describing the flow and evolutionary equations are described, wherein option one (the fully coupled form) is shown to be over restrictive and the second option (the decoupled form) provides significant flexibility. As a consequence of the decoupled form, a new operator, that is, the compliance operator, is defined, which provides a link between the assumed Gibb's and complementary dissipation potential and ensures a number of desirable numerical features, for example, the symmetry of the resulting consistent tangent stiffness matrix. An important conclusion reached is that although many theories in the literature do not conform to the general potential framework outlined, it is still possible in some cases, by slight modifications of the employed forms, to restore the complete potential structure.     

Enhancing the Bolzon–Schrefler–Zienkiewicz Constitutive Model for Partially Saturated Soil

This paper presents an enhanced version of the elasto-plastic model for partially saturated soil first proposed by Bolzon, Schrefler and Zienkiewicz in 1996, “BSZ” model, which uses the effective stress tensor and suction as independent stress variables. It is recalled that the effective stress tensor proposed by Lewis and Schrefler in 1982 is thermodynamically consistent and, compared with other choices of stress tensors, results particularly suitable for partially saturated soil mechanics. A hydraulic constitutive relationship and a hydraulic hysteresis are introduced in the model, to take into account the irreversible deformation during cyclic drying and wetting until structural collapse. For this reason the plastic rate of strain is split into the sum of two components: one depending on the effective stress tensor and the other one on suction. This is the new feature of the BSZ model. This enhanced model is then cast into a thermodynamical framework at macroscopic level and it is shown that it is possible to derive the constitutive law from the Helmholtz free energy and a dissipation function, both for associative and non- associative plasticity. Finally the model predictions have been compared with experimental data for Sion slime, with particular emphasis on the deviatoric part, and model predictions of hysteretic behaviour have been investigated in case of a wetting and drying cycle on compacted betonite–kaolin.