Free Energy Density Function Research Articles

Thermodynamically consistent modeling of viscoelastic solids under finite strain

The present article is concerned with modeling the viscoelastic behavior of Polydimethylsiloxane (PDMS) in large-strain regime. Starting from the basic principles of thermodynamics, an incremental variational formulation is derived. Within this model, the free energy density and dissipation function determine elastic and viscous properties of the solid. The main contribution of this paper is the estimation of the parameters in the proposed phenomenological model from measurements conducted on Sylgard184 samples. This non-linear material model simplifies to a Prony-series representation in frequency domain in case of small deformations. The coefficients of this Prony-series are detected from dynamical temperature mechanical analysis measurements. Time-temperature superposition allows to combine measurements at different temperatures, such that a sufficiently large frequency range is available for subsequent fitting of Prony-parameters. A set of material parameters is thus provided. The incremental variational formulation directly lends itself to finite element discretization, where an efficient and stable choice of elements is proposed for radially symmetric problems. This formulation allows to verify the proposed model against experimental data gained in ball-drop experiments.

A Machine Learning Free Energy Functional for the 1D Reference Interaction Site Model: Towards Prediction of Solvation Free Energy for All Solvent Systems

Understanding the interactions between solutes and solvents is vital in many areas of the chemical sciences. Solvation free energy (SFE) is an important thermodynamic property in characterising molecular solvation and so accurate prediction of this property is sought after. The One-Dimensional Reference Interaction Site Model (RISM) is a well-established method for modelling solvation, but it is known to yield large errors in the calculation of SFE. In this work, we show that a single machine learning free energy functional for RISM can accurately model solvation thermodynamics in multiple solvents. A convolutional neural network is trained on solvation free energy density functions calculated by RISM for small organic molecules in approximately 100 different solvent systems. We achieve an average RMSE of 1.41 kcal/mol and an R2 of 0.89 across all solvent systems. We also compare the performance for the most and least commonly represented solvents and show that higher accuracy is generally seen with higher volumes of data, with RMSE values of 0.69–1.29 kcal/mol and R2 values of 0.78–0.97 for solvents with more than 50 data points. We have shown that machine learning can greatly improve solvation free energy predictions in RISM, while demonstrating that the methodology is generalisable across solvent systems. This represents a significant step towards a universal machine learning SFE functional for RISM.

A theoretical framework for multi-physics modeling of poro-visco-hyperelasticity-induced time-dependent fracture of blood clots

Fracture resistance of blood clots plays a crucial role in physiological hemostasis and pathological thromboembolism. Although recent experimental and computational studies uncovered the poro-viscoelastic property of blood clots and its connection to the time-dependent deformation behavior, the effect of these physical processes on clot fracture and the underlying fracture mechanisms are not well understood. This work aims to formulate a thermodynamically consistent, multi-physics theoretical framework for describing the time-dependent deformation and fracture of blood clots. This theory concurrently couples fluid transport through porous fibrin networks, non-linear visco-hyperelastic deformation of the solid skeleton, solid–fluid interactions, mechanical degradation of tissues, gradient enhancement of energy, and protein unfolding of fibrin molecules. The constitutive relations of tissue constituents and the governing equation of fluid transport are derived within the framework of porous media theory by extending non-linear continuum thermodynamics at large strains. A physics-based, compressible network model is developed for the fibrin network of blood clots to describe its mechanical response. The kinetic equations of the internal variables, introduced for describing the non-linear viscoelastic deformation, non-local damage driving force and protein unfolding, are formulated according to the thermodynamics principles by incorporating a non-equilibrium energy of fibrin networks, a gradient-enhanced energy, and a stretch-induced energy of fibrin molecules, respectively, into the total free energy density function. An energy-based damage model is developed to predict the damage and fracture of blood clots, and an evolving regularization parameter is proposed to limit the damage zone bandwidth. The proposed model is implemented into finite element code by writing subroutines and is experimentally validated using single-edge cracked clot specimens with different constituents. The fracture of blood clots subject to different loading conditions is simulated, and the mechanisms of clot fracture are systematically analyzed. Computational results show that this model can accurately capture the experimentally measured deformation and fracture. The viscoelasticity and fluid transport play essential roles in the fracture of blood clots under physiological loading.

A phase field framework for corrosion fatigue of carbon steel

Corrosion fatigue damage occurs when metallic materials are subjected to cyclic loading in a corrosive medium. In this study, a phase field framework is proposed to predict the corrosion fatigue of carbon steels. The coupling effect of fatigue and corrosion is explicitly implemented in the proposed phase field framework by coupling the displacement field, electrochemical field and phase field. A degradation function of the interface free energy density with the consideration of elastic and plastic strain energies is introduced to account for the fatigue damage accumulated during the corrosion fatigue process. The applicability of this framework is validated by accurately capturing the pure fatigue and corrosion fatigue behaviors of compact tension specimens, particularly the acceleration effect of corrosion on the fatigue crack growth. The propagation morphology and rate of the corrosion fatigue crack in single pit and multiple pit models are studied. The distribution of stress state and strain energy density induces the directionality of crack propagation. The influence of loading frequency on the corrosion fatigue process is discussed in detail. Due to the corrosion-fatigue coupling effect, the corrosion rate increases with increasing of the loading frequency, resulting in an accelerated corrosion fatigue process. Moreover, the significance of the plasticity in the prediction of corrosion fatigue is emphasized.

Framework for Laplacian-Level Noninteracting Free-Energy Density Functionals.

A framework for orbital-free Laplacian-level meta-generalized-gradient approximation (meta-GGA) for the noninteracting free-energy-density functionals based upon analysis of the fourth-order gradient expansion is developed. The framework presented here provides a new tool for developing advanced orbital-free thermal functionals at the meta-GGA level of theory. A nonempirical meta-GGA functional, which in the slowly varying density limit correctly reduces to the finite-temperature fourth-order gradient expansion for the noninteracting free energy, is constructed. Application to warm dense helium demonstrates that the developed meta-GGA functional drastically increases the accuracy of orbital-free-density functional theory simulations at temperatures below 40 eV as compared to the lower Thomas-Fermi and GGA rungs.

Tunable noninteracting free-energy density functionals for high-energy-density physics applications

In this work, we introduce the concept of a tunable noninteracting free-energy density functional and present two examples realized: (i) via a simple one-parameter convex combination of two existing functionals and (ii) via the construction of a generalized gradient approximation (GGA) enhancement factor that contains one free parameter and is designed to satisfy a set of incorporated constraints. Functional (i), constructed as a combination of the local Thomas–Fermi and a pseudopotential-adapted GGA for the noninteracting free-energy, has already demonstrated its practical usability for establishing the high temperature end of the equation of state of deuterium [Phys. Rev. B 104, 144104 (2021)] and CHON resin [Phys. Rev. E 106, 045207 (2022)] for inertial confinement fusion applications. Hugoniot calculations for liquid deuterium are given as another example of how the application of computationally efficient orbital-free density functional theory (OF-DFT) can be utilized with the employment of the developed functionals. Once the functionals have been tuned such that the OF-DFT Hugoniot calculation matches the Kohn–Sham solution at some low-temperature point, agreement with the reference Kohn–Sham results for the rest of the high temperature Hugoniot path is very good with relative errors for compression and pressure on the order of 2% or less.

Bridging scales with Machine Learning: From first principles statistical mechanics to continuum phase field computations to study order–disorder transitions in Li[formula omitted]CoO[formula omitted]

LixTMO2 (TM=Ni, Co, Mn) forms an important family of cathode materials for Li-ion batteries, whose performance is strongly governed by Li composition-dependent crystal structure and phase stability. Here, we use LixCoO2 (LCO) as a model system to benchmark a machine learning-enabled framework for bridging scales in materials physics. We focus on two scales: (a) assemblies of thousands of atoms described by density functional theory-informed statistical mechanics, and (b) continuum phase field models to study the dynamics of order–disorder transitions in LCO. Central to the scale bridging is the rigorous, quantitatively accurate, representation of the free energy density and chemical potentials of this material system by coarse-graining formation energies for specific atomic configurations. We develop active learning workflows to train recently developed integrable deep neural networks for such high-dimensional free energy density and chemical potential functions. The resulting, first principles-informed, machine learning-enabled, phase-field computations allow us to study LCO cathodes’ order–disorder transitions in terms of temperature, microstructure, and charge cycling. We highlight several insights gained to the dynamics of the phase transitions, and that have been made possible by the quantitatively rigorous scale bridging. To the best of our knowledge, such a scale bridging framework has not been previously demonstrated for LCO, or for materials systems of comparable technological interest. This approach can be expanded to other materials systems and can incorporate additional physics to that studied here.

Thermodynamics and Ferroelectric Properties of Pb1-xSrxTiO3 Solid Solutions

Pb1-xSrxTiO3 solid solutions have been utilized to compositionally tune the phase transitions and ferroelectric properties and thus are attractive for potential applications in tunable microwave devices and pyroelectric energy harvesters. However, a reliable thermodynamic description of the Pb1-xSrxTiO3 system is currently lacking. Here, we establish a thermodynamic free energy density function for the Pb1-xSrxTiO3 solid solution system taking into account both the polarization and octahedral tilt, and validated it by reproducing the experimentally measured polarization, lattice parameters, and dielectric constant values. Based on the thermodynamic model, we successfully explain the experimentally observed dielectric anomalies by analyzing the interactions between ferroelectric and octahedral tilt order parameters and constructed a comprehensive temperature-composition Pb1-xSrxTiO3 phase diagram which can be used to guide future experimental synthesis and thin film growth. The thermodynamic model developed in this work can be employed not only to predict the ferroelectric and structural states and their properties for compositions that have not yet been synthesized but also used as input to model the formation of mesoscale domain structures using the phase-field method.

Wave propagation in generalized thermo-microstretch elastic solid containing voids

A linear theory of generalized thermo-microstretch elastic solid containing voids is formulated. Lord and Shulman [2] theory of thermoelasticity is employed to incorporate thermal effects. Free energy density function is constructed to develop the constitutive relations and field equations for an isotropic homogeneous generalized thermo-microstretch elastic solid containing voids. The possibility of propagation of plane waves is investigated in the medium of infinite extent. It is found that there may exist four sets of coupled longitudinal waves, two sets of coupled transverse waves and an independent longitudinal microrotational wave traveling with distinct speeds. Each set of coupled longitudinal waves is found to be attenuating and dispersive in nature, while an independent longitudinal microrotational wave and the remaining two sets of coupled transverse waves are found to be dispersive but non-attenuating in nature. All the possible waves are influenced by the polar property of the medium; however, all the coupled longitudinal waves are influenced by the stretch, voids and thermal properties of the medium. It is also found that all coupled longitudinal waves exist for all non-negative frequencies, while the independent longitudinal microrotational wave and one of the sets of coupled transverse waves exist only after certain cutoff frequency.

The electro-mechanical coupling responses of functionally graded piezoelectric nanobeams with flexoelectric effect

The aim of this paper is to study the electro-mechanical coupling responses of functionally graded (FG) piezoelectric cantilever nanobeam under the concentrated load, and the material properties follow an exponential distribution in the thickness direction. The constitutive equations are derived from the electric Gibbs free energy density function. The governing equations and boundary conditions of the Euler–Bernoulli FG beam can be obtained by variational principle, and then, the analytical expressions for its deflection, polarization intensity, and induced potential are obtained. The results show that the flexoelectric effect has a significant impact on the deflection, polarization intensity, and induced potential of the FG cantilever nanobeam.

Cohesive zone phase field model for electromechanical fracture in multiphase piezoelectric composites

This paper introduces a coupled electromechanical finite deformation phase field model for crack propagation and interfacial decohesion in multiphase piezoelectric composites with interfaces. The crack phase field model is augmented with cohesive traction-separation laws at the material interfaces, derived from a cohesive potential function. A Gibbs free energy density function is proposed, allowing for the incorporation of the anisotropic elastic stiffness of the piezoelectric material. Numerical simulations exhibiting different failure mechanisms are carried out to demonstrate the efficacy of the model. Effects of external electric field on crack evolution and the competition between penetration and deflection of a crack impinging on an interface are investigated. Limited verification tests are conducted with theoretical results. Finally, the model is used to simulate fracture in nonuniform piezocomposite microstructures. The effect of crack propagation on the evolution of the electric field with different crack face conditions are analyzed. Differences in the electromechanical responses of piezocomposites due to different fiber distributions are also observed.

COUPLED POROMECHANICS AND ADSORPTION IN MULTIPLE-POROSITY SOLIDS

This paper presents a general constitutive model to describe the coupling among bulk solid stress, pore fluid pressure, and adsorption in a fluid-saturated multiple-porosity elastic solid. The general case of N coexisting porosities is considered. Each porosity system may contain fluid in the free or adsorbed phase, as well as in both phases. Thus, the utility of the developed constitutive relations extends to materials with complex pore structures, which may involve several scales of pore size. These relations are obtained from the thermodynamic free energy density function for deformation of the porous solid frame while incorporating the surface energy stored at the interface between the pore fluid and solid phase. A number of previously published models for coupled adsorption and poroelasticity in microporous and mesoporous materials are examined as special cases. Possible means of estimating the constitutive model parameters are discussed.

A computational framework for topology optimisation of flexoelectricity at finite strains considering a multi-field micromorphic approach

This paper presents a novel in-silico framework for the design of flexoelectric energy harvesters at finite strains using topology optimisation. The main ingredients of this work can be summarised as follows: (i) a micromorphic continuum approach is exploited to account for size dependent effects in the context of finite strains, thus permitting the modelling and simulation of flexoelectric effects in highly deformable materials such as dielectric elastomers. A key feature of the multi-field (mixed) formulation pursued is its flexibility as it permits, upon suitable selection of material parameters, to degenerate into other families of high order gradient theories such as flexoelectric gradient elasticity. (ii) A novel energy interpolation scheme is put forward, whereby different interpolation strategies are proposed for the various contributions that the free energy density function is decomposed into. This has enabled to circumvent numerical artifacts associated with fictitious high flexoelectric effects observed in the vicinity of low and intermediate density regions, where extremely high strain gradients tend to develop. (iii) A weighted combination of efficiency-based measures and aggregation functions of the stress is proposed to remedy the shortcomings of state-of-the-art efficiency-based functionals, which promotes the development of hinges with unpractical highly localised large strain gradients. Finally, a series of numerical examples are analysed, studying the development of direct flexoelectricity induced by bending, compression and torsional deformations.

A Landau–Devonshire analysis of strain effects on ferroelectric Al1−xScxN

We present a thermodynamic analysis of the recently discovered nitride ferroelectric materials using the classic Landau–Devonshire approach. Electrostrictive and dielectric stiffness coefficients of Al1−xScxN with a wurtzite structure (6 mm) are determined using a free energy density function assuming a hexagonal parent phase (6/mmm), with the first-order phase transition based on the dielectric stiffness relationships. The results of this analysis show that the strain sensitivity of the energy barrier is one order of magnitude larger than that of the spontaneous polarization in these wurtzite ferroelectrics, yet both are less sensitive to strain compared to classic perovskite ferroelectrics. These analysis results reported here explain experimentally reported sensitivity of the coercive field to elastic strain/stress in Al1−xScxN films and would enable further thermodynamic analysis via phase field simulation and related methods.

Phase Field vs Gradient Enhanced Damage Models: A Comparative Study

A comparison of the two different approaches for modelling damage in material in an infinitesimal strain setting is studied. The first approach is the nonlocal gradient enhanced damage model where the damage variable is taken as an independent variable which will be determined based on the local strain measure. Here, the nonlocal integral form is approximated to an implicit or explicit differential form using the Taylor’s series expansion for simpler numerical implementation. The second approach is the phase field damage model where a Helmholtz free energy density function is considered that includes a new energy degradation function along with a phase field non-conserved order parameter. The first variational principle on this energy density functional with respect to the corresponding order parameter variable will reach a stationarity value resulting in the non-conserved Allen-Cahn equation. The relationship of the order parameter with the damage variable gives the Allen-Cahn evolution equation for damage. A 1D bar example is considered for commenting on the similarities and differences of the two approaches to damage based on the mesh convergence studies based on the results obtained for various meshing profiles, changing length scale parameter values and the obtained damage profiles.

A parabolic approximation scheme for multi-phase-filed simulation of non-isothermal solidification

A parabolic approximation scheme is proposed to implement the multi-phase-filed simulation of non-isothermal solidification. Based on the second-order fitting of free energy density function, this scheme constructs an explicit expression for the calculation of phase compositions, which is more efficient than the traditional Newton’s iteration scheme used in the Kim-Kim-Suzuki model. As an application example of this numeral scheme, the peritectic solidification of Ti-Al alloy is simulated to present the coupling of multi-phase field and concentration field, which reveals the non-isothermal evolution principle of microstructure and micro-segregation. The present study not only improves the calculation efficiency through a new numeral scheme, but also contributes to the understanding of the thermodynamics and kinetics of Ti-Al peritectic solidification.

Crystal Plasticity Phase-Field Model with Crack Tip Enhancement Through a Concurrent Atomistic-Continuum Model

This paper develops a method for physics-based augmentation of the Helmholtz free energy density functionals, used in coupled crystal plasticity phase-field finite element (CP-PF FE) models of fracture in crystalline metallic materials. Specifically, the defect and crack surface energy components are enhanced with terms that mechanistically account for the presence of atomic-scale, crack-tip nucleated dislocations. The additional terms in the free energy representation are motivated and calibrated by energy equivalence between a concurrent atomistic–continuum scale model and the CP-PF FE model. The atomistic domain of the concurrent model incorporates a time-accelerated, molecular dynamics (MD) LAMMPS code, while the continuum domain is modeled by a dislocation-density crystal plasticity FE model. The concurrent model transfers and transforms discrete dislocations in the atomistic domain to dislocation densities in the crystal plasticity domain. Dislocation densities are transported in the continuum domain by solving the advection equation using a particle-based reproducing kernel particle method with collocation. A new form of the defect energy density is proposed by considering the effect of crack-tip nucleated dislocations. Parameters in the augmented defect and surface energies are evaluated by comparing with the concurrent model results. A comparison of crack propagation with and without contributions from the nucleated dislocations demonstrates a significant effect of nucleated dislocations on the evolution of the crack.

Hysteretic response of bulk magnetostrictive material employing a novel hyperbolic vector generalized magneto-thermoelastic constitutive model

Taylor series expansion of Gibbs free energy density function for elastic deformation, the physics of demagnetization and the Weiss molecular field coupled with the thermodynamic relations have been used to develop a generalized nonlinear hysteretic thermo-magneto-elastic vector model with potential application for the development of sensors and actuators. The nonlinear elastic strains in the magnetostrictive material are induced due to the magnetic domain rotation caused by tensile and compressive stresses in giant magnetostrictive materials. A vector function of the hyperbolic tangent is defined to take into account this nonlinear characteristic validating the elastic stress field boundary condition. The vector generalized Jiles-Atherton hysteresis model with modified Langevin equation is employed to incorporate the pinning of magnetic domain walls in the mathematical model. Thus, derived macroscopic model is put to the test for 1D, 2D and 3D nonlinear magnetostriction, magnetization and elastic constitutive relation with Terfenol-D (giant magnetostrictive materials) as a potential candid material. Finite element analyses for rods and films have been conducted using the general H(B) relation coded in the C programming language, as shown in the solution algorithm (Fig. 2). Employing the calibrated physical and experimental parameters (Tables 2 and 3), the confidence of the model is checked with existing analytical and experimental models. The magnetization and magnetostriction hysteresis responses have been evaluated considering the variation of a broad range of the prestress (up to 110 MPa), temperature (0 °C–80 °C) and applied magnetic field (0 kA/m–193.2 kA/m). The numerical simulation of this fully coupled phenomenological model demonstrates good agreement with the experimental data. The theoretical modelling error obtained from averaging the normalized root mean square error for each experimental data set is found to be maximum of 2.8 % and hence vindicates the efficacy of the present model.

Integrity bases for cubic nonlinear magnetostriction

A so-called smart material is a material that is the seat of one or more multiphysical coupling. One of the key points in the development of the constitutive laws of these materials, either at the local or at the global scale, is to formulate a free energy density (or enthalpy) from vectors, tensors, at a given order and for a class of given symmetry, depending on the symmetry classes of the crystal constituting the material or the symmetry of the representative volume element. This article takes as a support of study the stress and magnetization couple (σ,m) involved in the phenomena of magnetoelastic coupling in a cubic symmetry medium. Several studies indeed show a non-monotonic sensitivity of the magnetic susceptibility and magnetostriction of certain soft magnetic materials under stress. Modeling such a phenomenon requires the introduction of a second order stress term in the Gibbs free energy density. A polynomial formulation in the two variables stress and magnetization is preferred over a tensorial formulation. For a given material symmetry class, this allows to express more easily the free energy density at any bi-degree in σ and m (i.e. at any constitutive tensors order for the so-called tensorial formulation). A rigorous and systematic method is essential to obtain the high-degree magneto-mechanical coupling terms and to build a free energy density function at any order which is invariant by the action of the cubic (octahedral) group. For that aim, theoretical and computer tools in Invariant Theory, that allow for the mathematical description of cubic nonlinear magneto-elasticity, are introduced. Minimal integrity bases of the invariant algebra for the pair (m,σ), under the proper (orientation-preserving) and the full cubic groups, are then proposed. The minimal integrity basis for the proper cubic group is constituted of 60 invariants, while the minimal integrity basis for the full cubic group (the one of interest for magneto-elasticity) is made up of 30 invariants. These invariants are formulated in a (coordinate free) intrinsic manner, using a generalized cross product to write some of them. The counting of independent invariants of a given multi-degree in (m,σ) is performed. It is shown accordingly that it is possible to list without error all the material parameters useful for the description of the coupled magnetoelastic behavior from the integrity basis. The technique is applied to derive general expressions Ψ★(σ,m) of the free energy density at the magnetic domains scale exhibiting cubic symmetry. The classic results for an isotropic medium are recovered.

A thermomechanical‐phase‐field approach for modeling of residual stresses in fusion welding

AbstractThe welding process of metallic components, such as using gas tungsten arc welding (GTAW), is often accompanied by detrimental deformations and residual stresses, which affect the strength and functionality of these welded components. In this work, a phase‐field model, usually used to track the state of the phase‐change materials, is embedded in a thermo‐elasto‐plastic finite element model in order to simulate the welding process and the residual stresses. The phase‐field method (PFM) enables us to track the melting front of the metal, due to the welding heat source, and to split the analysis domain into soft and solid regions. In this way, temperature and phase‐field‐dependent material properties can properly be assigned to different regions in the domain. The phase‐field evolution is governed by Allen‐Cahn equation and relies on specification of the free energy density function as the main driving force for the phase change. Having a coupled thermomechanical process during the welding heating‐cooling cycle, the onset of residual stresses is associated with plastic deformations and can be observed after cooling to the ambient temperature. The plasticity model considers J2 plasticity with isotropic hardening. The coupled nonlinear system of equations is solved in the open‐source FE package, FEniCS, with time discretization based on the finite difference method.