Helmholtz Free Energy Potential Research Articles

A relaxation model for the non-isothermal Navier-Stokes-Korteweg equations in confined domains

The Navier-Stokes-Korteweg (NSK) system is a classical diffuse interface model which is based on van der Waals' theory of capillarity. Diffuse interface methods have gained much interest to model two-phase flow in porous media. However, for the numerical solution of the NSK equations two major challenges have to be faced. First, an extended numerical stencil is required due to a third-order term in the linear momentum and the total energy equations. In addition, the dispersive contribution in the linear momentum equations prevents the straightforward use of contact angle boundary conditions. Secondly, any real gas equation of state is based on a non-convex Helmholtz free energy potential which may cause the eigenvalues of the Jacobian of the first-order fluxes to become imaginary numbers inside the spinodal region.In this work, a thermodynamically consistent relaxation model is presented which is used to approximate the NSK equations. The model is complimented by thermodynamically consistent non-equilibrium boundary conditions which take contact angle effects into account. Due to the relaxation approach, the contribution of the Korteweg tensor in the linear momentum and total energy equations can be reduced to first-order terms which enables a straightforward implementation of contact angle boundary conditions in a numerical scheme. Moreover, the definition of a modified pressure function enables to formulate first-order fluxes which remain strictly hyperbolic in the entire spinodal region. The present work is a generalization of a previously presented parabolic relaxation model for the isothermal NSK equations.A high-order discontinuous Galerkin spectral element method which supports curved elements and hanging nodes is employed to discretize the system. The relaxation model and its corresponding boundary conditions are validated using solutions of the original NSK model and analytical results for one-, two- and three-dimensional test cases. The simulation of a spinodal decomposition in a three-dimensional porous structure underlines the capability of the presented approach.

Comparative study of galaxy clustering using halo approximation and mean-field theory in the light of modified theories of gravity

In this manuscript, we have explored the large-scale structure (LSS) of the Universe using the halo approximation of galaxies and the mean-field (mean-field approximation) theory of galaxy clusters. Employing the modified gravitational potential, which relies on Boltzmann’s statistics, we have analyzed the effect of galaxy clusters on the large-scale structure of the Universe. With both of these approximations, we have observed the significance of the modified potential. Also, the gravitational partition function obtained from the applied approximations has been used to determine the thermodynamic properties of these galaxy clusters. Besides this, the impact of these approximations on the derived quantities, such as Helmholtz free energy, entropy, internal energy, pressure, and chemical potential, was observed and is also evident from their respective plots along with their distribution functions. Importantly, the divergence of configurational integrals has been removed through the said approach.

A gradient micromorphic modeling for plasticity softening

Localization of internal quantities, which induce material softening, potentially leads to an ill-posed global partial differential equation. Along with further developments of singularity issues in the simulations, an unwanted divergent numerical solution may occur. The work at hand attempts to formulate a ductile softening model by considering a size reduction of the plastic yield surface. An internal variable to reduce the size of the plastic yield surface is constituted by a gradient micromorphic approach. In detail, a degradation function induced by an internal softening quantity is employed to multiply the thermodynamic force of plastic hardening in the yield function. In this regard, a gradually shrinking plastic yield surface is constituted to model the overall softening response. The evolution of the internal softening variable is governed by a Kuhn–Tucker condition together with its non-local extension. The present constitutive model of the coupled softening problem is derived based on a thermodynamically consistent algorithm from a well-defined Helmholtz free energy potential, which is implemented into the context of a conventional Finite Element Method. A representative and meaningful numerical example is studied to demonstrate the capability of the present model.

Bottom-up coarse-grain modeling of plasticity and nanoscale shear bands in α-RDX.

Computationally inexpensive particle-based coarse-grained (CG) models are essential for use in molecular dynamics (MD) simulations of mesoscopically slow cooperative phenomena, such as plastic deformations in solids. Molecular crystals possessing complex symmetry present enormous practical challenges for particle-based coarse-graining at molecularly resolved scales, when each molecule is in a single-site representation, and beyond. Presently, there is no published pairwise non-bonded single-site CG potential that is able to predict the space group and structure of a molecular crystal. In this paper, we present a successful coarse-graining at a molecular level from first principles of an energetic crystal, hexahydro-1,3,5-trinitro-s-triazine (RDX) in the alpha phase, using the force-matching-based multiscale coarse-graining (MSCG/FM) approach. The new MSCG/FM model, which implements an optimal pair decomposition of the crystal Helmholtz free energy potential in molecular center-of-mass coordinates, was obtained by force-matching atomistic MD simulations of liquid, amorphous, and crystalline states and in a wide range of pressures (up to 20GPa). The MSCG/FM potentials for different pressures underwent top-down optimization to fine-tune the mechanical and thermodynamic properties, followed by consolidation into a transferable density-dependent model referred to as RDX-TC-DD (RDX True-Crystal Density-Dependent). The RDX-TC-DD model predicts accurately the crystal structure of α-RDX at room conditions and reproduces the atomistic reference system under isothermal (300K) hydrostatic compression up to 20GPa, in particular, the Pbca symmetry of α-RDX in the elastic regime. The RDX-TC-DD model was then used to simulate the plastic response of uniaxially ([100]) compressed α-RDX resulting in nanoscale shear banding, a key mechanism for plastic deformation and defect-free detonation initiation proposed for many molecular crystalline explosives. Additionally, a comparative analysis of the effect of core-softening of the RDX-TC-DD potential and the degree of molecular rigidity in the all-atom treatment suggests a stress-induced short-range softening of the effective intermolecular interaction as a fundamental cause of plastic instability in α-RDX. The reported RDX-TC-DD model and overall workflow to develop it open up possibilities to perform high quality simulation studies of molecular energetic materials under thermal and mechanical stimuli, including extreme conditions.

A thermodynamically consistent fractional visco-elasto-plastic model with memory-dependent damage for anomalous materials

We develop a thermodynamically consistent, fractional visco-elasto-plastic model coupled with damage for anomalous materials. The model utilizes Scott-Blair rheological elements for both visco- elastic/plastic parts. The constitutive equations are obtained through Helmholtz free-energy potentials for Scott-Blair elements, together with a memory-dependent fractional yield function and dissipation inequalities. A memory-dependent Lemaitre-type damage is introduced through fractional damage energy release rates. For time-fractional integration of the resulting nonlinear system of equations, we develop a first-order semi-implicit fractional return-mapping algorithm. We also develop a finite-difference discretization for the fractional damage energy release rate, which results into Hankel-type matrix–vector operations for each time-step, allowing us to reduce the computational complexity from O(N3) to O(N2) through the use of Fast Fourier Transforms. Our numerical results demonstrate that the fractional orders for visco-elasto-plasticity play a crucial role in damage evolution, due to the competition between the anomalous plastic slip and bulk damage energy release rates.

Modified theory of gravity and clustering of multi-component system of galaxies

In this paper, we analyze the clustering of galaxies using a modified theory of gravity, in which the field content of general relativity has been be increased. This increasing in the field content of general relativity changes the large distance behavior of the theory, and in weak field approximation, it will also modify the large distance behavior of Newtonian potential. So, we will analyzing the clustering of multi-component system of galaxies interacting through this modified Newtonian potential. We will obtain the partition function for this multi-component system, and study the thermodynamics of this system. So, we will analyze the effects of the large distance modification to the Newtonian potential on Helmholtz free energy, internal energy, entropy, pressure and chemical potential of this system. We obtain also the modified distribution function and the modified clustering parameter for this system, and hence observe the effect of large distance modification of Newtonian potential on clustering of galaxies.

Closure for the Ornstein-Zernike equation with pressure and free energy consistency.

The Ornstein-Zernike (OZ) integral equation theory is a powerful approach to simple liquids due to its low computational cost and the fact that, when combined with an appropriate closure equation, the theory is thermodynamically complete. However, approximate closures proposed to date exhibit pressure or free energy inconsistencies that produce inaccurate or ambiguous results, limiting the usefulness of the Ornstein-Zernike approach. To address this problem, we combine methods to enforce both pressure and free energy consistency to create a new closure approximation and test it for a single-component Lennard-Jones fluid. The closure is a simple power series in the direct and total correlation functions for which we have derived analytical formulas for the excess Helmholtz free energy and chemical potential. These expressions contain a partial molar volumelike term, similar to excess chemical potential correction terms recently developed. Using our bridge approximation, we have calculated the pressure, Helmholtz free energy, and chemical potential for the Lennard-Jones fluid using the Kirkwood charging, thermodynamic integration techniques, and analytic expressions. These results are compared with those from the hypernetted chain equation and the Verlet-modified closure against Monte Carlo and equations-of-state data for reduced densities of ρ^{*}<1 and temperatures of T^{*}=1.5, 2.74, and 5. Our closure shows consistency among all thermodynamic paths, except for one expression of the Gibbs-Duhem relation, whereas the hypernetted chain equation and the Verlet-modified closure exhibit consistency between only a few relations. Accuracy of the closure is comparable to the Verlet-modified closure and a significant improvement to results obtained from the hypernetted chain equation.

Thermomechanical framework for modelling elastoplastic damaged materials

In the paper a thermomechanical framework for modelling elasto-plastic damaged materials is presented. Basic assumptions and concepts are given, leading to formulation of constitutive equations using two potentials only: Helmholtz free energy and dissipation potential. Consecutive steps of the procedure are shown for simplified one-dimensional case, followed by three-dimensional example concerning Beltrami-Michell failure condition.

A comprehensive comparison between thermodynamic perturbation theory and first-order mean spherical approximation: Based on discrete potentials with hard core

Using the TL (Tang and Lu, 1993) method, Ornstein-Zernike integral equation is solved perturbatively under the mean spherical approximation (MSA) for fluid with potential consisting of a hard sphere plus square-well plus square-shoulder (HS+SW+SS) to obtain first-order analytic expressions of radial distribution function (RDF), second-order direct correlation function, and semi-analytic expressions for common thermodynamic properties. A comprehensive comparison between the first-order MSA and high temperature series expansion (HTSE) to third-, fifth- and seventh-order is performed over a wide parameter range for both a HS+SW and the HS+SW+SS model fluids by using corresponding “exact” Monte Carlo results as a reference; although the HTSE is carried out up to seventh-order, and not to the first order as the first-order MSA the comparison is considered fair from a calculation complexity perspective. It is found that the performance of the first-order MSA is dramatically model-dependent: as target potentials go from the HS+SW to the HS+SW+SS, (i) there is a dramatic dropping of performance of the first-order MSA expressions in calculating the thermodynamic properties, especially both the excess internal energy and constant volume excess heat capacity of the HS+SW+SS model cannot be predicted even qualitatively correctly. (ii) One tendency is noticed that the first-order MSA gets more reliable with increasing temperatures in dealing with the pressure, excess Helmholtz free energy, excess enthalpy and excess chemical potential. (iii) Concerning the RDF, the first-order MSA is not as disappointing as it displays in the cases of thermodynamics. (iv) In the case of the HS+SW model, the first-order MSA solution is shown to be quantitatively correct in calculating the pressure and excess chemical potential even if the reduced temperatures are as low as 0.8. On the other hand, the seventh-order HTSE is less model-dependent; in most cases of the HS+SW and the HS+SW+SS models, the seventh-order HTSE improves the fifth- and third-order HTSE in both thermodynamic properties and RDF, and the improvements are very demonstrable in both the excess internal energy and constant volume excess heat capacity; for very limited cases, the seventh-order HTSE improves the fifth-order HTSE only within lower density domain and even shows a bit of inadaptation over higher density domain.

A meshless approach to non-local damage modelling of concrete

A non-linear continuum damage mechanics model for concrete constructions is analysed using a radial point interpolation meshless method (RPIM). The fundamental mathematical relations and the material model are fully characterized. The 2D plane stress RPIM formulation is extended to a rate-independent standard (local) damage model considering both tension and compression static states. Additionally, in this work, the local damage formulation is modified considering a non-local constitutive damage criterion with regard to a Helmholtz free energy potential. Here, the internal variational fields, such as local and non-local damage variables, are determined by a return-mapping damage algorithm. Due to the non-linear nature of the phenomenon, a displacement controlled Newton-Raphson iterative approach is adopted to attain the non-linear damage solution. In the end, the performance of the proposed non-local damage model is evaluated using an experimental test of a notched-three-point bending beam available in the literature. The obtained solution shows that the meshless methods are capable to effectively analyse concrete structures assuming a non-linear non-local continuum damage model.

A consistent and efficient localized damage model for concrete

Damage induced localized failure is of vital importance to evaluate residual safety and to prevent catastrophic collapse of concrete structures. Aiming for the efficient and robust modeling of localized failure in quasi-brittle solids and structures, this paper addresses a thermodynamically consistent and numerically efficient localized damage model for concrete. The thermodynamic framework is presented in such a way that not only the continuum model for the bulk material, but also the localized model for the discontinuity, can be established in a consistent approach. For the latter, a localized Helmholtz free energy potential is postulated, mimicking the classical continuum damage model. Specifically, a localized damage variable is introduced to characterize degradation of the initially-rigid discontinuity with a well-defined reference stiffness. Consistent evolution law for the localized damage variable is derived from an appropriate traction-based failure criterion and its equivalent separation-based counterpart, e.g. the novel hyperbolic damage criterion introduced in this work. The proposed model can be regularized and in particular, upon the assumption of continuous stress field, an orthotropic damage model in the context of smeared crack methods is recovered. This coincidence not only justifies the proposed model, but also sheds new lights on other classical methods. Numerically, the proposed localized damage model is incorporated into the improved stable eXtended FEM. Compared to those plasticity-based localized models, an explicit numerical algorithm can be used to update the cohesive tractions transferred across the discontinuity with no iteration on the constitutive level. This feature, together with the well-conditioned system matrix ensured by the improved stable eXtended FEM, guarantees noticeable efficiency and robustness of the overall numerical performances. The proposed model is verified against several classical benchmark tests of concrete in both mode-I and mixed-mode failure. The numerical predictions agree well with experimental test data and those reported in the literature. Remarkably, all the numerical results are mesh-size and mesh-alignment independent, and no spurious stress locking is observed, showing validity of the proposed model in the modeling of localized failure in concrete.

Plasticity coupled with thermo-electric fields: Thermodynamics framework and finite element method computations

A consistent thermodynamic-based theoretical framework and three-dimensional finite element formulation is presented, capable of coupling elastic, thermal and electric fields. The complete set of governing equations is obtained from conservation principles for electric charge, energy and momentum. The second principle of thermodynamics is taken into account to introduce the irreversible phenomena, such as plastic dissipation or Joule heating. The constitutive relations are derived consistently from the Helmholtz free-energy potential for each corresponding dual variable in terms of the defined set of state variables. We consider the case of linear isotropic hardening model for plasticity, and provide the consistent form of the tangent thermo–electro–elastoplastic modulus through dual variable computations. The latter plays the crucial role in ensuring fast convergence properties of the finite element computations with the proposed coupled plasticity model. The implementation is carried out in a research version of the well-known computer code FEAP. Several numerical simulations are presented in order to illustrate the proposed model and formulation capabilities for providing an enhanced formulation of an important practical application in terms of Peltier cells.

Effects of cosmological constant on clustering of Galaxies

In this paper, we analyse the effect of the expansion of the universe on the clustering of galaxies. We evaluate the configurational integral for interacting system of galaxies in an expanding universe by including effects produced by the cosmological constant. The gravitational partition function is obtained using this configuration integral. Thermodynamic quantities, specifically, Helmholtz free energy, entropy, internal energy, pressure and chemical potential are also derived for this system. It is observed that they depend on the modified clustering parameter for this system of galaxies. It is also demonstrated that these thermodynamical quantities get corrected because of the cosmological constant.

Initial strain energy-based thermo-elastoviscoplastic two-parameter damage–self-healing models for bituminous composites—Part I: Formulations

The initial strain energy-based thermo-elastoviscoplastic two-parameter damage–self-healing formulations for asphalt concrete materials are proposed and implemented for numerical simulations of experimental data. A class of elastoviscoplastic two-parameter constitutive damage–self-healing models, based on a continuum thermodynamic framework, is developed within an initial-elastic strain energy-based formulation. The Helmholtz free energy potential is employed and uncoupled with an Arrhenius-type temperature term in order to account for the effect of temperature. In particular, the governing incremental damage and healing evolutions are coupled in volumetric and deviatoric parts and characterized through the net stress concept in conjunction with the hypothesis of strain equivalence. The conceptual illustration is shown for the proposed governing incremental damage and healing evolutions. The (undamaged) energy norms of the volumetric and deviatoric strain tensors as the equivalent volumetric and deviatoric strains are redefined and employed for the two-parameter damage and healing criteria. A rate-dependent (viscous) volumetric and deviatoric damage model with a structure analogous to viscoplasticity of the Perzyna type is used for rate sensitivity of bituminous composites. Completely new computational algorithms are systematically proposed in Part II of this sequel, based on the two-step operator splitting methodology. Comparisons with experimental measurements and model predictions are demonstrated in Part II of this sequel as well.

New strain energy-based thermo-elastoviscoplastic isotropic damage–self-healing model for bituminous composites—Part I: Formulations

New initial strain energy-based thermo-elastoviscoplastic isotropic damage–self-healing formulations for bituminous materials are developed and implemented for numerical predictions of experimental measurements. A class of elastoviscoplastic constitutive damage–self-healing model, based on a continuum thermodynamic framework, is proposed within an initial elastic strain energy-based formulation. An Arrhenius-type temperature term is uncoupled with Helmholtz free energy potential to account for the effect of temperature. In particular, the governing incremental damage and healing evolutions are coupled and characterized through the net stress concept in conjunction with the hypothesis of strain equivalence. The viscoplastic flow is introduced by means of an additive split of the stress tensor. The (undamaged) energy norm of strain tensor is redefined and employed as equivalent strain. A rate-dependent (viscous) damage model with a structure analogous to viscoplasticity of the Perzyna type is used for rate sensitivity of bituminous materials. In net space, Drucker–Prager yield function and Perzyna viscoplastic model are employed. Completely new computational algorithms are systematically developed in Part II of this sequel, based on the two-step operator splitting methodology. Comparisons with experimental results and numerical simulations are undertaken in Part II of this work.

A competitive mechanism driven damage-plasticity model for fatigue behavior of concrete

In this paper, an innovative three-dimensional plastic-damage model is proposed to describe the behavior of concrete material under fatigue loading. In the proposed model, two damage variables are defined to describe the fatigue degradation of the mechanical properties of concrete under tension and shear, respectively. Meanwhile, by adopting the elastoplastic damage energy release rates, which are derived from the elastoplastic Helmholtz free energy potentials, a novel fatigue damage evolution law is developed. Specifically, the damage evolution process is governed by the competition between the damage healing effect and the damage driving effect induced by the propagation and closure of micro cracks under fatigue loading. The proposed model also allows the accumulation of fatigue damage within the classical damage surface by adopting the loading/unloading irreversibility concept. Moreover, the evolution of plastic strains is considered within the framework of the effective stress space plasticity. In the end, a set of numerical tests is presented, and the validity and effectiveness of the proposed model are illustrated according to the simulated results and the comparisons with the experimental data in literatures.

Unconditionally stable methods for simulating multi-component two-phase interface models with Peng–Robinson equation of state and various boundary conditions

In this paper, we consider multi-component dynamic two-phase interface models, which are formulated by the Cahn–Hilliard system with Peng–Robinson equation of state and various boundary conditions. These models can be derived from the minimum problems of Helmholtz free energy or grand potential in the realistic thermodynamic systems. The resulted Cahn–Hilliard systems with various boundary conditions are fully coupled and strongly nonlinear. A linear transformation is introduced to decouple the relations between different components, and as a result, the models are simplified. From this, we further propose a semi-implicit unconditionally stable time discretization scheme, which allows us to solve the Cahn–Hilliard system by a decoupled way, and thus, our method can significantly reduce the computational cost and memory requirements. The mixed finite element methods are employed for the spatial discretization, and the approximate errors are also analyzed for both space and time. Numerical examples are tested to demonstrate the efficiency of our proposed methods.

Strain gradient plasticity modeling of the cyclic behavior of laminate microstructures

Two recently proposed Helmholtz free energy potentials including the full dislocation density tensor as an argument within the framework of strain gradient plasticity are used to predict the cyclic elastoplastic response of periodic laminate microstructures. First, a rank-one defect energy is considered, allowing for a size-effect on the overall yield strength of micro-heterogeneous materials. As a second candidate, a logarithmic defect energy is investigated, which is motivated by the work of Groma et al. (2003). The properties of the back-stress arising from both energies are investigated in the case of a laminate microstructure for which analytical as well as numerical solutions are derived. In this context, a new regularization technique for the numerical treatment of the rank-one potential is presented based on an incremental potential involving Lagrange multipliers. The results illustrate the effect of the two energies on the macroscopic size-dependent stress–strain response in monotonic and cyclic shear loading, as well as the arising pile-up distributions. Under cyclic loading, stress–strain hysteresis loops with inflections are predicted by both models. The logarithmic potential is shown to provide a continuum formulation of Asaro's type III kinematic hardening model. Experimental evidence in the literature of such loops with inflections in two-phased FFC alloys is provided, showing that the proposed strain gradient models reflect the occurrence of reversible plasticity phenomena under reverse loading.

Adsorption preference reversal phenomenon from multisite-occupancy theory for two-dimensional lattices

The statistical thermodynamics of polyatomic species mixtures adsorbed on two-dimensional substrates was developed on a generalization in the spirit of the lattice-gas model and the classical Guggenheim-DiMarzio approximation. In this scheme, the coverage and temperature dependence of the Helmholtz free energy and chemical potential are given. The formalism leads to the exact statistical thermodynamics of binary mixtures adsorbed in one dimension, provides a close approximation for two-dimensional systems accounting multisite occupancy and allows to discuss the dimensionality and lattice structure effects on the known phenomenon of adsorption preference reversal.

Conduction-only transport phenomena in compressible bivelocity fluids: diffuse interfaces and Korteweg stresses.

"Diffuse interface" theories for single-component fluids—dating back to van der Waals, Korteweg, Cahn-Hilliard, and many others—are currently based upon an ad hoc combination of thermodynamic principles (built largely upon Helmholtz's free-energy potential) and so-called “nonclassical” continuum-thermomechanical principles (built largely upon Newtonian mechanics), with the latter originating with the pioneering work of Dunn and Serrin [Arch. Ration. Mech. Anal. 88, 95 (1985)]. By introducing into the equation governing the transport of energy the notion of an interstitial work-flux contribution, above and beyond the usual Fourier heat-flux contribution, namely, jq = -k∇T, to the energy flux, Dunn and Serrin provided a rational continuum-thermomechanical basis for the presence of Korteweg stresses in the equation governing the transport of linear momentum in compressible fluids. Nevertheless, by their failing to recognize the existence and fundamental need for an independent volume transport equation [Brenner, Physica A 349, 11 (2005)]—especially for the roles played therein by the diffuse volume flux j v and the rate of production of volume πν at a point of the fluid continuum—we argue that diffuse interface theories for fluids stand today as being both ad hoc and incomplete owing to their failure to recognize the need for an independent volume transport equation for the case of compressible fluids. In contrast, we point out that bivelocity hydrodynamics, as it already exists [Brenner, Phys. Rev. E 86, 016307 (2012)], provides a rational, non-ad hoc, and comprehensive theory of diffuse interfaces, not only for single-component fluids, but also for certain classes of crystalline solids [Danielewski and Wierzba, J. Phase Equilib. Diffus. 26, 573 (2005)]. Furthermore, we provide not only what we believe to be the correct constitutive equation for the Korteweg stress in the class of fluids that are constitutively Newtonian in their rheological response to imposed stresses but, equally importantly, we establish the explicit functional forms of Korteweg's phenomenological thermocapillary coefficients appearing therein.