Helmholtz Free Energy Function Research Articles

On the development of a constitutive model for steel subjected to fire and explosion

The behavior of steel under thermal and mechanical loading including high transient temperatures and strain rates (e.g., fire, creep or explosion) is extremely complex. To study this behavior, we propose a constitutive model accordance with the laws of thermodynamic, the principle of virtual power and experimental results. The thermodynamic framework considers the Clausius-Duhem inequality and maximum dissipation rate principle, as well as the theory of finite deformations, linear isotropic viscoelasticity, nonlinear isotropic and kinematic hardening, local and non-local viscoplasticity, local and non-local anisotropic viscodamage and parameters associated to material behavior dependent on loadings. The components of the thermodynamic conjugate forces are defined from the Helmholtz free energy function and the rate of energy dissipation, which are postulated considering the creep phenomenon, thermal-mechanical transient phenomenon, cyclic phenomenon and plastic wave propagation phenomenon. The balance of microforces is defined based on the principle of virtual power. A novel rule of non-associated thermo-viscoplastic flow of the Perzyna type and a novel rate dependent local and non-local damage evolution law are introduced from constitutive model. The verification of the model was carried out in two benchmark studies composed of fire tests. The predictions of the model have shown in according with the results of the experiments. A sensitivity study was also performed with a view to analysis the influence of the parameters in achieved results. From this study a simplified constitutive model is presented. Benchmark studies composed of explosion tests are part of future work.

Phase-field modeling of fracture for ferromagnetic materials through Maxwell’s equation

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. Piezoelectric material, as a special class of active materials, can produce an internal electric field when subjected to mechanical stress or strain. In return, there is the converse piezoelectric response, which expresses the induction of the mechanical deformation in the material when it is subjected to the application of the electric field. This work presents a variational-based computational modeling approach for failure prediction of ferromagnetic materials. In order to solve this problem, a coupling between magnetostriction and mechanics is modeled, then the fracture mechanism in ferromagnetic materials is investigated. Furthermore, the failure mechanics of ferromagnetic materials under the magnetostrictive effects is studied based on a variational phase-field model of fracture. Phase-field fracture is numerically challenging since the energy functional may admit several local minima, imposing the global irreversibility of the fracture field and dependency of regularization parameters related discretization size. Here, the failure behavior of a magnetoelastic solid body is formulated based on the Helmholtz free energy function, in which the strain tensor, the magnetic induction vector, and the crack phase-field are introduced as state variables. This coupled formulation leads to a continuity equation for the magnetic vector potential through well-known Maxwell’s equations. Hence, the energetic crack driving force is governed by the coupled magneto-mechanical effects under the magneto-static state. Several numerical results substantiate our developments.

Modelling of pressure-induced magnetic and magnetocaloric effects in dissipative magnetic shape memory alloy systems

This article presents a coupled magneto-thermo-mechanical model of pressure-dependent magneto-caloric effect (MCE) and magnetization responses for polycrystalline magnetic shape memory alloys (MSMA). Coupled constitutive equations are derived from a Helmholtz free energy function in a consistent thermodynamic way. The hysteretic and dissipative characteristics of phase transformations in MSMAs are captured by the internal state variables approach with their evolution equations. A relation of adiabatic temperature change is established for such a dissipative material system. The model is calibrated and validated with the existing experimental data. The validated constitutive model is then exploited to predict MCEs at different pressure and magnetic field level. Some predicted results are compared with the available experimental data.

Bitangent planes of surfaces and applications to thermodynamics

The classical van der Waals equation, applied to one or two mixing fluids, and the Helmholtz (free) energy function A yield, for fixed temperature T, a curve in the plane R2 (one fluid) or a surface in 3-space R3 (binary fluid). A line tangent to this curve in two places (bitangent line), or a set of planes tangent to this surface in two places (bitangent planes) have a thermodynamic significance which is well documented in the classical literature. Points of contact of bitangent planes trace ‘binodal curves’ on the surface in R3. The study of these bitangents is also classical, starting with D.J. Korteweg and J.D. van der Waals at the end of the 19th century, but continuing into modern times. In this paper we give a summary of the thermodynamic background and of other mathematical investigations and then present a new mathematical approach which classifies a wide range of situations in R3 where bitangents occur. In particular, we are able to justify many of the details in diagrams of binodal curves observed by Korteweg and others, using techniques from singularity theory.

A thermodynamic-based model for modeling thermo-elastoplastic behaviors of saturated clayey soils considering bound water dehydration

The non-isothermal deformation of soft mudrocks or clay soils is one of the most critical issues in energy and environmental related geotechnics. Clay-related geomaterials hold complex microstructure and mineral composition, which brings difficulty in investigating their thermo-mechanical behaviors. Previous studies pay little attention to the difference between a thermal plastic strain and the strain from clay dehydration. In this study, a new constitutive model is proposed for describing the thermo-elastoplastic behaviors of clayey soils under water-saturated condition. The effect of temperature variation and mechanical loading on elastoplastic strains and dehydration are investigated. The thermodynamics laws and the unconventional plasticity are applied to quantify the thermo-mechanical behavior. The irreversible strain is captured by using Cam-Clay plasticity and subloading yield surface concept. The dehydration strain is described by utilizing a novel method based on generalized thermodynamics approach and Helmholtz free energy function. The internal variables, and the first and second laws of thermodynamics are applied in the model. The hardening rule is established by implementing the laws of physical conservation, energy dissipation, and plastic flow. The proposed model is validated using specially designed thermal consolidation tests on laboratory prepared heavily consolidated clayey soils and some published data of clayey soils with different geological origins.

Impact of hyper-elasticity on cyclic sand modelling: A numerical study based on SANISAND-MS

Accurate simulation of foundation response subjected to repeated loading require reliable sand constitutive models to reproduce realistic sand cyclic behaviour. Many elastoplastic constitutive models adopted the hypoelasticity law, which results in physically inaccurate strain accumulation prediction of soil when subjected to elastic cyclic loading. To address this issue, the existing elastoplastic bounding surface model SANISAND-MS has been upgraded with a hyperelastic formulation that is derived from the Helmholtz free energy function. The complete form of the derived nonlinear elasticity functions enhanced by stress induced anisotropy is presented in this work. The calibration of the upgraded SANISAND-MS is briefly discussed. It is demonstrated that the hyperelastic formulation with volumetric-deviatoric coupling: (1) guarantees no accumulation of elastic strain upon elastic cyclic loading, and (2) captures the stress-induced anisotropy of sand under reverse loading. This upgraded SANISAND-MS model with a physical adaptation and thermodynamically consistent elasticity law can predict strain accumulation accurately and is physically appropriate. Moreover, the upgraded SANISAND-MS is suitable for simulating problems with high-cyclic loads and is able to be used in boundary value problem simulations in areas including offshore engineering and railway engineering.

A thermodynamically consistent anisotropic damage model for metallic glasses

AbstractThe recent success in manufacturing large‐size, also called bulk metallic glasses (BMGs) using 3D‐printing based on laser powder bed fusion (LPBF) opens an avenue for the broad application of this material class. To explore the great potential of both as‐cast and 3D‐printed BMGs, a comprehensive understanding and an accurate prediction of the plastic deformation and damage behaviour of this material class are indispensable. In this study, we develop a thermodynamically consistent anisotropic damage model incorporating tension‐compression asymmetry (TCA) to describe the unique inelastic deformation and damage behaviours of metallic glasses. The widely observed normal stress sensitivity and plastic dilatancy in metallic glasses are considered by using an extended Mohr‐Coulomb criterion in the constitutive description. Furthermore, a second‐order damage tensor is adopted for describing the anisotropic damage behaviour. By augmenting the Helmholtz free energy function to be dependent on the gradient of the free volume concentration and the gradient of the nonlocal damage parameter, the governing equations for the corresponding internal variables are derived within the framework of finite deformation. The damage localisation and mesh dependency are correspondingly alleviated. The simulation result shows that the shear band patterns are in good agreement with the experimental observations from literature.

Modeling of photo-thermo-sensitive hydrogels by applying the temperature expansion analogy

Hydrogels are an outstanding material for sensor and actuator applications, for example, chemosensors and microfluidics, and have been extensively studied in both, experiments as well as in modeling within the last years. The theoretical investigations of hydrogels are key factors for the development of new hydrogel-based concepts in research and engineering. The recent approaches in modeling of the light-sensitive behavior of hydrogels are often complicated and very detailed. Commonly, they are based on the Helmholtz free energy function within a continuum-mechanical framework. In contrast, the Stimulus Expansion Model (SEM) is a simple and very effective approach to embed the swelling properties of a hydrogel into a continuum mechanical framework. Originally, the SEM was applied to chemical stimulation processes based on available experimental swelling curves. The current work provides an extension of the SEM for photo-thermo-sensitive hydrogels. The present approach considers (i) the attenuation of light by applying Lambert-Beer’s law as well as (ii) the energy transfer of light into heat. In this study, PNIPAm hydrogels with incorporated light-absorbing particles of copper-chlorophyllin are investigated. To demonstrate the capabilities of the presented approach, the effect of the variation of (i) light power, (ii) particle volume fraction, and (iii) ambient temperatures on the swelling behavior is analyzed. The obtained results show an excellent correlation with experimental results from literature. Concluding, the extended Stimulus Expansion Model provides further opportunities to design and simulate photo-thermo-sensitive hydrogels for engineering applications.

Data-driven anisotropic finite viscoelasticity using neural ordinary differential equations

We develop a fully data-driven model of anisotropic finite viscoelasticity using neural ordinary differential equations as building blocks. We replace the Helmholtz free energy function and the dissipation potential with data-driven functions that a priori satisfy physics-based constraints such as objectivity and the second law of thermodynamics. Our approach enables modeling viscoelastic behavior of materials under arbitrary loads in three-dimensions even with large deformations and large deviations from the thermodynamic equilibrium. The data-driven nature of the governing potentials endows the model with much needed flexibility in modeling the viscoelastic behavior of a wide class of materials. We train the model using stress–strain data from biological and synthetic materials including human brain tissue, blood clots, natural rubber and human myocardium and show that the data-driven method outperforms traditional, closed-form models of viscoelasticity.

Thermomechanical response of nonlinear viscoelastic materials

In this article, the thermomechanical properties of the classical and nonclassical viscoelastic materials with particular emphasis on the entropy are studied in the framework of continuum thermodynamics. Specifically, considering the Clausius integral as “Part I” of the second law of thermodynamics in the classical formulation of continuum thermodynamics, the entropy function for various classes of nonclassical thermo-viscoelastic materials is shown to exist. This is achieved by employing the first law of thermodynamics and a scalar potential known as the Helmholtz free energy function. In addition, another function known as the internal rate of production of entropy defined for the classical and nonclassical thermo-viscoelastic materials is adopted to obtain a balance law for the entropy. Finally, upon linearizing the theory, a coupled system of differential equations is obtained for the nonclassical thermo-viscoelastic materials, which under some special circumstances, permits propagation of damped harmonic waves in the viscoelastic medium.

Constitutive modeling for the training process of two-way shape memory effect under thermal cyclic loading

Shape memory alloys (SMAs)—if trained properly—can induce two-way shape memory effect which is applicable in actuator design using SMAs. In this study, a three-dimensional phenomenological model is proposed for SMAs within the framework of irreversible thermodynamics in order to predict the training process under cyclic loading and the induced two-way shape memory effect. In this regard, transformation and permanent strains are considered as internal variables and Helmholtz free energy function is introduced using several material parameters accounting for elastic, transformation and permanent deformation mechanisms. In order to demonstrate the capabilities of the proposed model, we first compare the model results with several experimental results and prove that the current model fits well with various experiments under cyclic loading as well as two-way shape memory effect. In addition, the capabilities of the present model in predicting the two-way shape memory effect, degradation effects, evolution rate, stabilization of permanent strain as well as multiaxial loading have been carefully examined through parametric studies. It is concluded that the present model is able to predict the training process from the initial cycles to stabilization and two-way effect for different materials and conditions.

Modeling of load responses and aging of high strength fibers considering UV-radiation

Long-term illumination can damage the mechanical properties of high strength fiber composites, resulting in the material aging and loss of mechanical properties. To reveal the mechanism of ultraviolet- (UV-) induced damage, in this paper, a nonlinear constitutive model is developed to describe the elastic and inelastic deformation behavior of high strength fibers under UV-radiation and mechanical loadings. Two internal variables are employed to measure the effects of damage caused by UV-radiation and by mechanical loadings, respectively. The Helmholtz free energy function and the evolution equations of these two internal variables are investigated elaborately and given explicitly for high strength fibers. The proposed theoretical model is then applied to analyze the mechanical degradation of high-performance high strength fibers under uniaxial tension. The modulus and strength of the material predicted by the theoretical model match the aging experimental results rather well, which verifies the rationality and accuracy of the present model. This model is expected to predict the evolution of mechanical properties of high-strength fibers under UV-radiation, which can also provide a theoretical basis for the development of anti-UV technology of fiber reinforced composites.

Gradient-enhanced modelling of deformation-induced anisotropic damage in metallic glasses

Recent experimental and computational studies at different scales reveal an apparent flow-induced anisotropy of the inelastic deformation behaviour in metallic glasses (MGs). However, the anisotropic damage behaviour accompanied by the formation of shear bands is not adequately described in the previous constitutive modelling work. In this study, we develop a thermodynamically-consistent, anisotropic damage model incorporating tension–compression asymmetry (TCA) to describe the unique deformation and damage behaviours. The elastic–plastic response, including the normal stress sensitivity and the plastic dilatancy, is captured by the extended Mohr–Coulomb criterion. A second-order damage tensor is adopted for describing the anisotropic damage behaviour of MGs. By augmenting the Helmholtz free energy function to be dependent on the gradient of the free volume concentration and the gradient of the nonlocal damage parameter, the kinetic equations for the corresponding internal variables are derived within the framework of finite deformation continuum thermodynamics. Different material degradations under tension and compression are considered by decomposing the elastic strain tensor into a positive mode and a negative mode in the free energy function, where the tension-strain-induced damage is fully switched on while the compression-strain-induced damage is only partially activated. The influence of the TCA parameter on the damage initiation, crack propagation and shear bands density, as well as the influence of the gradient regularisation parameter on the width of the damage zone, are systematically studied. The proposed damage model is validated by experiments carried on unnotched and notched microcantilever beams. The simulation results show that both the displacement–load curves and the shear band patterns are in good agreement with the experimental observations.

A state-dependent hyperelastic-plastic constitutive model considering shear-induced particle breakage in granular soils

An elastic–plastic constitutive model considering particle breakage for simulation of crushable granular soils behavior is proposed. In the model, elastic strain rates are derived from a modified Helmholtz free energy function, and the influence of plastic shear work-induced particle breakage on the elastic properties of sand is taken into account as an elastic–plastic coupling mechanism. A stress ratio-driven mechanism is employed for calculation of the plastic strain rates. The proposed model is capable of tracking the evolution of the grain size distribution (GSD) due to shear-induced particle breakage. The evolving breakage index of Einav (2007) (J Mech Phys Solids 55(6):1274–1297, 2007) is interrelated to the plastic shear work to avoid overestimation of shear-induced particle breakage in loose sands. A direct comparison between the model simulations and laboratory data has been carried out for five series of drained/undrained monotonic and cyclic triaxial tests covering a wide range of initial states. For the sake of comparison, predicted behaviors from a hypoplastic constitutive model specially developed for crushable granular soils are also included. It is shown that the proposed constitutive model can provide reasonable predictions using a single set of parameters for each series of the laboratory data.

Control-Oriented Modeling and Analysis of Tubular Dielectric Elastomer Actuators Dedicated to Cardiac Assist Devices

This letter deals with the control-oriented modeling of a multilayered dielectric elastomer actuator based tube. The actuator is clamped at both sides and performs a radial displacement. The hyperelastic deformation and the viscoelastic performance, together with the electro-mechanical coupling are considered in the model. With the chosen Yeoh’s model for the Helmholtz free energy function, we aim to model the snap-through effect of this tubular actuator at high applied voltage. Simulation results of the nonlinear dynamic model are later compared with experimental ones in order to identify unknown parameters of the dielectric elastomer, especially to fit the snap-through effect and the high frequency performance. With proper identification process, our model has a good accuracy to fit the physical system (more than 90.27% for a ramp input and more than 82.7% for a step input) and is ready for the future controller design.

Study of short range and long range molecular interactions in binary liquid mixtures of N,N-dimethylformamide (DMF) and 1-propanol

In the present study precisely measured properties; static dielectric constant (ε0), refractive index (n) and density (ρ) of binary liquid mixtures of N,N-dimethylformamide (DMF) and 1-propanol (1-PN) have been used to determine linear correlation parameter (grr), mutual linear correlation parameter (gAB), excess Helmholtz free energy functions (ΔF) and excess molar polarization (ΔP). The dipolar contributions to these functions arising from long range electrostatic interaction, short range interaction between identical molecules and short range interaction between dissimilar molecules have been assessed separately. The excess thermodynamic functions calculated from the experimental data supports the formation of clusters with antiparallel or destructive angular correlation (β-clusters). The variation of excess static dielectric constant, excess refractive index and excess molar volume have been used to discuss the type, the strength and the nature of intermolecular interactions betweenconstituent species. Comparison of the predicted values of static dielectric constant and refractive index of the liquid mixtures using various mixing rules with the experimentally determined values have been assessed in terms of Average Percentage Deviation (APD).

Hyper-dual number-based numerical differentiation of eigensystems

Considering application to nonlinear material models, this study proposes a novel numerical method for computing arbitrary-order derivatives of eigenvalues and eigenvectors (eigensystems) for second-order tensors (matrices) based on hyper-dual numbers (HDNs). This study introduces two numerical differentiation approaches for eigensystems: one obtains derivatives of both eigenvalues and eigenvectors by inductively solving linear equations derived from an HDN formulation of eigensystem, and the other specializes in solving derivatives of a function represented by eigenvalues, such as the Helmholtz free energy function in the Ogden model, a representative example of the eigenvalue-based hyperelastic material models. Because these approaches differ considerably in applicability and computational cost, this study proposes a numerical differentiation method that uses both complementarily. The proposed method can accurately compute higher-order derivatives even if the second-order tensors have multiple solutions for eigenvalues. Moreover, applying the proposed method to an incremental variational formulation (IVF) leads to robust and accurate computations for the Ogden material model. These results imply that IVF facilitates the implementation of several constitutive models in a unified manner, and the proposed method broadens the scope of these models, including eigensystem expressions.

Pauli oscillator in noncommutative space

In this study, we investigate the Pauli oscillator in a noncommutative space. In other words, we derive wave function and energy spectrum of a spin half non-relativistic charged particle that is moving under a constant magnetic field with an oscillator potential in noncommutative space. We obtain critical values of the deformation parameter and the magnetic field, which they counteract the normal and anomalous Zeeman effects. Moreover, we find that the deformation parameter has to be smaller than [Formula: see text]. Then, we derive the Helmholtz free energy, internal energy, specific heat and entropy functions of the Pauli oscillator in the non commutative space. With graphical methods, at first, we compare these functions with the ordinary ones, and then, we demonstrate the effects of magnetic field on these thermodynamic functions in the commutative and noncommutative space, respectively.

A new anisotropic elasto-plastic-damage model for quasi-brittle materials using strain energy equivalence

An anisotropic elasto-plastic-damage model is developed for quasi-brittle materials within the thermodynamics framework. The anisotropic damage is characterized by second-order damage tensors that are different under tensile and compressive loadings. The hypothesis of strain energy equivalence is adopted with two components, including the hypothesis of elastic strain energy equivalence and the hypothesis of plastic strain energy equivalence. The hypothesis of plastic strain energy equivalence is extended from metals to quasi-brittle materials so as to derive the relations of hardening stresses, which are previously assumed between the undamaged and damaged configurations for quasi-brittle materials. A plastic yield criterion is adopted simultaneously with a damage criterion to account for the coupled elasto-plastic-damage behavior under different loadings. The damage criterion can capture larger axial damage in uniaxial tension and smaller axial damage in uniaxial compression as compared with the lateral damage. The Helmholtz free energy functions are derived for three components: elastic, plastic, and damage. The constitutive laws are derived for three scenarios: the elastoplastic behavior without further damage evolution, the damage behavior without further plasticity evolution, and the coupled elasto-plastic-damage behavior. The anisotropic elasto-plastic-damage model is implemented in ABAQUS as a user material (UMAT) subroutine. The coupled constitutive laws in the continuum domain are implemented in an uncoupled way in the discrete domain with three steps, elastic predictor, plastic corrector, and damage corrector. The UMAT is validated in different conditions, including uniaxial, biaxial, and three-point bending tests. The advantage of considering the plastic free energy in anisotropic damage is illustrated through the strength envelope in biaxial loadings. The numerical results generally agree with experimental results, including the stress-strain curves in uniaxial tests, strength envelope in biaxial tests, and load-deflection curves in three-point bending tests.

High-temperature effect on the material constants and elastic moduli for solid rocks

Abstract Thermally coupled constitutive relations are generally used to determine material constants and elastic moduli (Young's modulus and shear modulus) of solid media. Conventional studies on this issue are mainly based on the linear temperature dependence of elastic moduli, whereas analytical difficulties are often encountered in theoretical studies on nonlinear temperature dependence, particularly at high temperatures. This study investigates the thermally coupled constitutive relations for elastic moduli and material constants using the assumption of axisymmetric fields, with applications to geologic materials (marble, limestone and granite). The Taylor power series of the Helmholtz free energy function within dimensionless temperatures could be used to develop the thermally coupled constitutive relations. The thermoelastic equivalent constitutive equations were formulated under the generalized Hooke's law. The material constants of solid rocks were determined by fitting experimental data using axisymmetric stress and strain fields at different temperatures, based on their thermomechanical properties. For these geologic materials, the resultant equivalent elastic moduli and deformations were in good agreement with those from the experimental measurements. Thermal stresses, internal moisture evaporation and internal rock compositions significantly affected the experimental results. This study provides a profound understanding of the thermally coupled constitutive relations that are associated with the thermomechanical properties of solid rocks exposed to high temperatures.