Helmholtz Free Energy Function Research Articles

Research on the dissipation field of a nonlinear continuum based on the operational irreversible thermodynamics

The dissipation field of a nonlinear continuum is studied by a new operational theory of irreversible thermodynamics, the evolution equations of the dissipation field can be obtained under the condition of knowing the phenomenological nonlinear constitutive relation of the continuum. In this theory, the dissipation equation of the basic state, which is corresponding to the minimum dissipation principle, is utilized to solve the distribution of dissipation field regarding the equilibrium problem of quasi-statics, and the dissipation equations of higher order are directly related to the dynamics of dissipation fields. The paper also presents a method showing the dissipation force operator by using Helmholtz's free energy function. This paper will also show important phenomena of the localized deformation, such as the plastic instability of necking and shear band, which are predicted in the form of an analytical formula.

The effect of steepness of soft-core square-well potential model on some fluid properties

The effect of repulsive steepness of the soft-core square well (SCSW) potential model on the second virial coefficient, critical behaviour (two- phase region and the position of critical point), and coordination number are investigated. The soft-core thermodynamic perturbation theory (TPT) presented by Weeks-Chandler-Anderson (WCA) recently developed by Ben-Amotz and Stell (BAS) has been used for the reference system, and the Barker-Henderson TPT for the perturbed system. The Barker-Henderson macroscopic compressibility approximation has been used for all order perturbation terms in which the second-order one is improved by assuming that the molecules in every two neighbouring shells are correlated upon the original assumption. By using the hard-sphere isothermal compressibility consistency for the radial distribution function (RDF), an analytical closed expression has been derived for the Helmholtz free energy function contained effective hard-sphere diameter. The accuracy of the model has been examined for the hard-core system, and an appropriate range found for the attractive width of the potential well (R), then the effect of steepness parameter on the critical quantities, coordination number, and the inversion temperature of the second virial coefficient, has been investigated qualitatively. The predicted results are in good agreement with the computer simulation data for the critical constants, and coordination number at the limit of the hard-core square-well potential model at least qualitatively, and for the attractive range 1.55 ≤ R ≤ 1.7, quantitatively. It was found that the steepness of the potential model has a marginal effect on the critical behaviour, and also every thermodynamic quantity at low and medium temperatures for which the molecular penetration is negligible, but since the penetration at high temperatures is significant, the role of the steepness of potential on the inversion temperature of the second virial coefficient and coordination number is highlighted.

Prediction of stored energy in polycrystalline materials during cyclic loading

The effect of initial texture on the stored energy is investigated. Uniaxially loaded polycrystalline materials with initial textures based on the Goss component and the Brass component are analyzed. For reference purposes a single crystal and an initial isotropic crystal orientation distribution are also analyzed. Special attention is directed at the thermomechanical behavior of polycrystalline material during cyclic loading, the temperature evolution and change in stored energy are studied. Cyclic loading of Cook’s membrane is also considered. The simulations are done using a rate-dependent crystal plasticity model for large deformations formulated within a thermodynamic framework. It is shown that incorporation of the latent-hardening into the Helmholtz free energy function and use of evolution laws of appropriate form allows a thermodynamically consistent heat generation due to plastic work.

Coupled viscoplasticity damage constitutive model for concrete materials

A coupled viscoplasticity damage constitutive model for concrete materials is developed within the framework of irreversible thermodynamics. Simultaneously the Helmholtz free energy function and a non-associated flow potential function are given, which include the internal variables of kinematic hardening, isotropic hardening and damage. Results from the numerical simulation show that the model presented can describe the deformation properties of the concrete without the formal hypotheses of yield criterion and failure criteria, such as the volume dilatancy under the compression, strain-rate sensitivity, stiffness degradation and stress-softening behavior beyond the peak stress which are brought by damages and fractures. Moreover, we could benefit from the application of the finite element method based on this model under complex loading because of not having to choose different constitutive models based on the deformation level.

Two ways for predicting the hysteresis minimisation for shape memory alloys

In a recent communication, James and Zhang proposed four criteria for hysteresis width minimization for alloys exhibiting a first order solid–solid phase transformation. In the present work, four different phase transformations are investigated through four different alloys: cubic → trigonal R-phase for NiTi, cubic → monoclinic II for Cu–Zn–Al, cubic → monoclinic I for NiTi and cubic → tetragonal for Ni–Mn–Ga. The first aim of this paper is to validate the efficiency of these four criteria of reversibility. The second aim of the paper is to show how the crystallographic theory of martensite, built to predict microstructure under stress free state or dead loads, can be used for continuous loadings. Except for Ni–Mn–Ga, the criteria predictions are in good agreement with the experimental observations and with the choice of the Helmholtz free energy function expression.

Thermodynamic Property Model of Wide-Fluid Phase Propane

A new thermodynamic property model for propane is expressed in form of the Helmholtz free energy function. It consists of eight terms of the ideal-gas part and eighteen terms of the residual part. Accurate experimental data of fluid properties and theoretical approach from the intermolecular potential were simultaneously considered in the development to insure accuracy and to improve reliability of the equation of state over wide range of pressures and temperatures. Based on the state range of experimental data used in the model development, the validity range is judged from the triple-point of 85.48 K to temperature of 450 K and pressure up to 60 MPa. The uncertainties with respect to different properties are estimated to be within 0.03% in ideal-gas isobaric specific heat, 0.2% in liquid phase density, 0.3% in gaseous phase density 1% in specific heats, 0.1% in vapor-pressure except at very low temperatures, 0.05% in saturated-liquid density, 0.02% in speed of sound of the gaseous phase and 1% in speed of sound of the liquid phase.

On the Description of Growth in Saturated Living Tissues

AbstractA comprehensive model for biological tissues must include the anisotropic tissue structure, the interstitial liquid wich saturated the tissue and the growth mechanism of the tissue. In the present contribution this is done by use of a three phasic model with a solid, liquid and nutrient phase in the framework of the porous media theory (TPM). In order to characterize the transversal isotropic skeleton behavior, an invariant formulation of the Helmholtz free energy function and the permeability tensor is suggested. The growth mechanism is characterizes by a mass transfer between the nutrient and solid phase. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Spatial representation of evolving anisotropy at large strains

A phenomenological model for evolving anisotropy at large strains is presented. The model is formulated using spatial quantities and the anisotropic properties of the material is modeled by including structural variables. Evolution of anisotropy is accounted for by introducing substructural deformation gradients which are linear maps similar to the usual deformation gradient. The evolution of the substructural deformation gradients is governed by the substructural plastic velocity gradients in a manner similar to that for the continuum. Certain topics related to the numerical implementation are discussed and a simple integration scheme for the local constitutive equations is developed. To demonstrate the capabilities of the model it is implemented into a finite element code. Two numerical examples are considered: deformation of uniform plate with circular hole and the drawing of a cup. In the two examples it is assumed that initial cubic material symmetry applies to both the elastic and plastic behavior. To be specific, a polyconvex Helmholtz free energy function together with a yield function of quadratic type is adopted.

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.

On a Model of Heterogenous Deformation of Elastic Bodies by the Mechanism of Multiple Appearance of New Phase Layers

A model describing a possible scenario of martensite type phase transformations is examined. A new phase is supposed to nucleate in the form of plane parallel layers. As the boundary condition, average strains are imposed. Then, the governing parameters of the two-phase structure are the concentration of new phase layers, their orientation and also the orientation of anisotropy axes. The parameters depend on the average strains and are determined by the requirement to minimize the average Helmholtz free-energy function. Once a general procedure has been discussed, average strain–stress diagrams are constructed for two cases. In the first case, for the simplicity sake, both phases are assumed to be isotropic. In the second case anisotropy is produced by a non-spherical phase transformation strain tensor. For both cases phase transition zones (PTZs) are constructed. The PTZ is formed in the space of strains by those which can exist on equilibrium interfaces. Loading and unloading paths, corresponding to uniaxial stretching and plane stretching/compression, are examined and related with the PTZ. Effects of internal stresses induced by the nucleation of new phase areas and the anisotropy of new phase are discussed.

Prediction of phase behavior of nanoconfined Lennard-Jones fluids with density functional theory based on the first-order mean spherical approximation

The recently proposed first-order mean spherical approximation (FMSA) [Y. Tang, J. Chem. Phys. 121, 10605 (2004)] for inhomogeneous fluids is extended to study the phase behavior of nanoconfined Lennard-Jones fluids, which is consistent with the phase equilibria calculation of the corresponding bulk fluid. With a combination of fundamental measure theory, FMSA provides Helmholtz free energy and direct correlation function to formulate density functional theory, which implementation is as easy as the mean-field theory. Following previous success in predicting density profiles inside slit pores, this work is focused specially on the vapor-liquid equilibrium of the Lennard-Jones fluids inside these pores. It is found that outside the critical region FMSA predicts well the equilibrium diagram of slit pores with the sizes of 5.0, 7.5, and 10 molecular diameters by comparing with available computer simulation data. As a quantitative method, FMSA can be treated as an extension from its bulk calculation, while the mean-field theory is only qualitative, as its bulk version.

An equation of state for associating systems

The thermodynamic framework proposed by Hu et al. (1984) for generating the Helmholtz free energy functions for associating systems is rectified. A generic cubic equation of state is incorporated with the infinite linear association model and the monomer‐dimer association model as well as two different sets of mixing rules to result in four different forms of equations of state. The Peng‐Robinson equation of state is accordingly reformulated and the applicability of one of the reformulated equations is tested by using the selected equation to correlate the vapor pressures and liquid densities of water and selected alcohols. The values calculated by means of the reformulated Peng‐Robinson equation are in good agreement with the experimental data reported in the literature.

Thermodynamic Properties of the Binary Mixture of Methane and Hydrogen Sulfide

Abstract From the interest of the thermodynamic properties for natural gas system, we have developed the thermodynamic property model for the binary mixture of methane and hydrogen sulfide system by the Helmholtz free energy function. Our model represents divergence of the critical curve and vapor–liquid–liquid equilibrium of this system as well as thermodynamic properties with high accuracy. The behavior of critical curve and phase equilibrium is discussed in detail with comparison of that for the mixture of methane and ethane system. Thermodynamic properties of methane and hydrogen sulfide system have been calculated using our model and their behavior as a function of composition is also reported.

A constitutive model of the posterior cruciate ligament

Further to our previous work on the development of a general constitutive framework for transversely isotropic viscohyperelasticity (Limbert, G, Middleton, J. A transversely isotropic viscohyperelastic material. Application to the modelling of biological soft connective tissues. Int J Solids Struct 2004;41(15):4237–60.), we propose a phenomenological constitutive law to describe the anisotropic viscohyperelastic behaviour of the human posterior cruciate ligament (PCL) at high strain rates. The mechanical formulation is based on the definition of a Helmholtz free energy function containing a hyperelastic and a viscous potential. The equations are valid for arbitrary kinematics and satisfy elemental thermodynamic principles. Identification of the constitutive model with experimental data obtained from human PCL specimens was performed and showed the ability of the model to capture accurately the mechanical characteristics of the PCL at various strain rates. Influence of the isotropic and directional viscous stress responses on the global mechanical response are discussed in connection with the modelling hypotheses. This work was motivated by the need to provide an accurate constitutive model of the PCL to be used in finite element analyses of human occupants in car crash simulations. Besides uniaxial tests along the natural fibre orientation of the PCL, additional tests such as equibiaxial, strip biaxial compression–tension and shear tests were also performed in order to assess the physical response of the model in different loading situations. It was found that the model performed as well in these conditions.

Predictions of Some Internal Microstructural Models for Polymer Melts and Solutions in Shear and Elongational Flows

AbstractSummary: In this work, the behavior of some internal microstructural models with different mobility tensors has been studied for polymer melts and solutions under steady and transient simple shear and elongational flows. The time evolution equations for conformation and stress tensors in the models reviewed have their root in the Generalized Poisson bracket formalism. Two different families of conformational models have been selected for this study. The first family is based on the Modified Finitely Extensible Nonlinear Elastic (FENE‐P) energy while the second uses a Volume Preserving Conformational Rheological (VPCR) model based on the Hookean Helmholtz free energy function. Several expressions for the mobility tensor based on the previously mentioned energy functions are used to obtain the models. The sensitivity of both families of models to the choice of the mobility tensors on the prediction of material functions in the transient and steady flows is discussed. Also, effects of shear rate on the material functions in start‐up and relaxation shear flows for both models are studied. The predictions of both models are compared with experimental data taken from the literature for some polymer melts. These results show that the family of VPCR models is able to predict the steady shear and elongational flow material functions in an extended range of deformation rates whereas the family of FENE‐P models can predict the behavior of only some specified polymer melts.Experimental data and VPCR model predictions for steady and elongational viscosity for PS [data of H. Munstedt, 1980].imageExperimental data and VPCR model predictions for steady and elongational viscosity for PS [data of H. Munstedt, 1980].

A Helmholtz free-energy function for a Cu–Al–Ni shape memory alloy

In this paper we develop a free energy function for a Cu–Al–Ni alloy undergoing cubic–orthorhombic phase transitions. We use the irreducible Lagrangian strain polynomial invariants of the cubic austenite parent phase to construct a polynomial expansion for the Helmholtz free energy. Our expansion retains quartic terms of the strain components in order to describe the two-phase material—the cubic austenite phase and six variants of the orthorhombic martensite phase. The coefficients of the free energy polynomial function are temperature dependent and are fitted to appropriate experimental data for austenite and martensite phases from literature. The resulting Helmholtz free energy function is given by (3.20), (4.11). We examine the response predicted by the model for shear in the twinning directions.

A transversely isotropic viscohyperelastic material: Application to the modeling of biological soft connective tissues

A general transversely isotropic viscohyperelastic constitutive law including strain rate effects was proposed. It is based on a definition of a general Helmholtz free energy function which depends explicitly on the right Cauchy–Green deformation tensor, its material time derivative and a structural tensor characterizing the preferred direction from which anisotropy arises. The elastic and viscous potentials that defined the free energy function were assumed to be decoupled, thus facilitating the identification process. This law was valid for arbitrary kinematics and aimed at modeling the mechanical behavior of biological soft tissues at high strain rates and at the finite strain regime. This is of high relevance for dynamic analyses of human occupants in car crash simulations (finite element analyses) and for situations where dynamic loads are significant (sport injury, etc). Explicit expression of the stress, elasticity and viscosity tensors were established. As an application of the constitutive law, the general expressions of the stress tensors were particularized for a specific Helmholtz free energy function describing the mechanical characteristics of the human anterior cruciate ligament. The constitutive model was shown to capture the strain rate effects and other essential characteristics of ligaments such as finite strain, anisotropy and nearly incompressibility. The model was also tested for various multi-axial loading situations.

Simple extension of the Gouy–Chapman theory including hard sphere effects.: Diffuse layer contribution to the differential capacity curves for the electrode ∣ electrolyte interface

An extension of the Gouy–Chapman theory based on a field theoretical approach including the hard sphere effect is presented. This approach focuses on the influence of ions on the properties of the electrode ∣ electrolyte interface. The Hamiltonian consists of the electrostatic energy and the Helmholtz free energy function for the hard sphere mixtures. The calculated differential capacity curves depend on the ionic sizes and concentration. In dilute electrolytes, close to the potential of zero charge we observe the Gouy–Chapman minimum which becomes a maximum when the concentration of electrolyte increases. This minimum is surrounded by two maxima. The capacity at these maxima depends on the size of ions and their concentration in the solution. At high charge densities the ionic density becomes constant and the capacity depends only on the size of the counterion which counterbalances the charge on the electrode, being independent of its concentration. These features are observed on the capacity curves obtained for various electrolytes on various metals.

An anisotropic ductile damage model based on irreversible thermodynamics

The paper deals with fundamental constitutive issues in the elastic–plastic-damage rate theory and the numerical modelling of the large strain elastic–plastic deformation behavior of anisotropically damaged ductile metals. The proposed model is based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics taking into account kinematic description of damage. It employs the consideration of damaged as well as fictitious undamaged configurations related via metric transformations which lead to the definition of damage strain tensors. The modular structure of the continuum theory is accomplished by the kinematic decomposition of strain rates into elastic, plastic and damage parts. To be able to address both the plastic flow and the anisotropic damage process, respective Helmholtz free energy functions of the fictitious undamaged configuration and of the current damaged configuration are introduced separately. A generalized yield condition based on invariants of the effective stress tensor is used to adequately describe the plastic flow properties of ductile metals and the plastic strain rate tensor is determined by a non-associated flow rule. Considering the damaged configurations a damage criterion is formulated using stress components referred to the elastically unloaded damage configuration. The damage strain rate tensor takes into account isotropic as well as anisotropic effects providing a realistic physical representation of ductile material degradation. Identification of material parameters is discussed in some detail. The applicability of the proposed continuum damage theory is demonstrated by numerical simulation of the inelastic deformation process of tension specimens.

A theoretical framework of one-dimensional sharp phase fronts in shape memory alloys

In this paper, we propose a one-dimensional (1D) sharp phase front-based theoretical framework for shape memory alloys (SMAs). An assumption of equality of the chemical potential at the phase front leads to a generalized Clausius–Clapeyron equation, which then gives the condition for the evolution of the phase front during phase transformation. The theoretical framework is general enough to incorporate any Helmholtz free energy function, and for that reason, if the Helmholtz free energy function is completely characterized, then so is the entire system of equations (including the condition for the phase boundary evolution). The small strain, quasistatic approximation of the theory in conjunction with a trilinear Helmholtz free energy function is used to predict the following two experiments: (i) constant deformation-rate phase transformation in a SMA single crystal; and (ii) constant load, temperature-induced transformation in a SMA polycrystalline wire.