Internal Variable Theory Research Articles

A constitutive model for the frequency dependence of magnetic hysteresis

The magnetic properties of materials are characterized by the variation of flux density with magnetic field. The hysteresis loop is generally dependent on the frequency of excitation. It is well known that the dependence is attributed to the effects of eddy current loss and anomalous (excess) loss. The present work deals with a new approaching method to model the frequency dependence of magnetic hysteresis within the framework of internal state variable theory, the fundamental structure of which is originally based on viscoplasticity theory in continuum mechanics. The hysteresis equations are formulated to be consistent with the general principles of irreversible thermodynamics with internal variables.

Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional

In this paper, time-independent plasticity is addressed within the thermodynamic framework with internal variables by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455). It is shown in this paper that the existence of a free energy function along with thermodynamic equilibrium conditions directly leads to associated flow rules. The time-independent inelastic behaviors can be fully determined by the Hessian matrix at the nondegenerate critical point of the free energy function. The normality rule of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25, pp. 448–461) or the Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech. 25, pp. 746–750) postulate is just a stability requirement of the thermodynamic equilibrium. The existence of a free energy functional which is not a direct function of the internal variables, along with thermodynamic equilibrium conditions also leads to associated flow rules. The time-independent inelastic behaviors with the free energy functional can be fully determined by the quasi Hessian matrix at the quasi critical point of the free energy functional. With the free energy functional, the thermodynamic forces conjugate to the internal variables are nonconservative and are constructed based on Darboux theorem. Based on the constructed nonconservative forces, it is shown that there may exist several possible thermodynamic equilibrium mechanisms for the thermodynamic system of the material sample. Therefore, the associated flow rules based on free energy functionals may degenerate into nonassociated flow rules. The symmetry of the conjugate forces plays a central role for the characteristics of time-independent plasticity.

Effect of deformation induced transformation of ɛ-martensite on ductility enhancement in a Fe-12 Mn steel at cryogenic temperatures

Metastable ɛ-martensite (ɛ-Ms) formed during a prior heat treatment of Fe-12Mn steel has been reported to transform into α-Ms during subsequent inelastic deformation. This deformation induced phase transformation (DIPT) from ɛ-Ms to α-Ms has been investigated in the present study within the framework of kinetics relation proposed based upon an internal variable theory of inelastic deformation. The ɛ-Ms phase was found to become more stable at lower temperatures to provide more prolonged DIPT from ɛ-Ms to α-Ms during an inelastic deformation at cryogenic temperatures, contrary to the case of austenite phase in various transformation induced plasticity steels being more stable at higher temperatures. This reversed stability-temperature relationship in Fe-12Mn steel appeared to provide a significant ductility enhancement at lower temperatures as well as significant strengthening effect.

Temperature effect on twin formation kinetics and deformation behavior of Fe-18Mn-0.6C TWIP steel

Temperature effect on deformation behavior has been investigated in relation to formation kinetics of twins in a Fe-18Mn-0.6C TWIP steel. Total elongation was found to reach a maximum value of 88% at 200 °C and then decreased continuously with the increase in test temperature from 300 °C up to 600 °C. This reversed temperature dependence on ductility could be attributed to the formation kinetics of deformation twins, as was prescribed by an internal variable theory of inelastic deformation. It was found that twins became more difficult to form at higher temperatures due to insufficient internal strain energy accumulated to reduce ductility progressively in this temperature range. Dislocation glide mechanism became, however, dominant at higher temperatures above 600 °C to increase total elongation following the usual temperature dependence. Finally the stacking fault energy was related with the stability parameter, β, used in the transformation kinetics relation.

Formulation of a macroscale corrosion damage internal state variable model

A new consistent formulation coupling kinematics, thermodynamics, and kinetics with damage using an extended multiplicative decomposition of the deformation gradient that accounts for corrosion effects is proposed. The corrosion model, based upon internal state variable (ISV) theory, captures the effects of general corrosion, pit nucleation, pit growth, pit coalescence, and intergranular corrosion. The different geometrically-affected rate equations are given for each mechanism after the ISV formalism and have a thermodynamic force pair that acts as an internal stress. Pit nucleation is defined as the number density that changes as a function of time driven by the local galvanic electrochemical potential between base matrix material and second phase material. Pit growth is defined as pit surface area growth. Pit coalescence is the interaction of the pits as they grow together and is often characterized by transgranular corrosion and is mathematically constructed from Coulomb’s Law and the Maxwell stress. General corrosion is signified by thickness loss of the material and is characterized by a modified Faraday’s Law. The intergranular corrosion rate is related to the grain boundary effects so that it is characterized by the misorientation between grains. The total damage (void volume or area fraction) is the addition of the general, pitting, and intergranular corrosion. The ability of the model to predict aspects of the corrosion mechanisms and aging history effects of an engineering material are then illustrated by comparison with experimental data of an extruded AZ31 magnesium alloy.

Influence of starch sodium octenyl succinate on rheological behaviour of wheat flour dough systems

Experimental and theoretical influence of addition of various amounts of three types starch sodium octenyl succinate (OSA) granules (0–20%): (a) non-physically modified, (b) pregelatinized and (c) hydrolyzed spray-dried on rheological behavior of wheat flour dough systems under oscillatory strain conditions was considered.Mathematical model was developed based on the internal variable theory by introducing the fractional derivatives to describe anomalous nature of energy dissipation. Two model parameters were used for quantitative description of the rheological behavior of the systems: the effective modulus and the dumping coefficient.The most rigid system with pronounced dumping effects was the dough supplemented with 20% of the non-physically modified OSA starch granules (maximum of the effective modulus and minimum of the dumping coefficient), while the softest system was the dough with 20% of the pregelatinized OSA starch. The obtained results revealed that the rheological behavior of OSA starch supplemented dough depended on the OSA starch granule rigidity, i.e. extent of OSA starch granule disintegration and polysaccharide degradation.

High temperature deformation mechanisms of nano-structured ferritic alloys in the context of internal variable theory of inelastic deformation

The stress relaxation behavior of 14YWT and ODS-Eurofer97 was examined within the framework of an internal-variable theory of inelastic deformation. Stress versus strain rate curves obtained by stress relaxation tests for 14YWT were described well by the equations for grain-matrix deformation while those of ODS-Eurofer97 were fitted with the combined curves of grain matrix deformation and grain boundary sliding. The sudden drop of total elongation of 14YWT was discussed in light of fracture surface observations. Grain boundary decohesion at 900°C was identified.

A new interpretation of internal-variable theory in finite thermo-viscoelasticity

Based on the non-equilibrium thermodynamics, an internal- variable theory in thermo-viscoelasticity at finite deformation was proposed by Huang in 1999. In this theory, a modified stretch of the molecular chain was introduced, and hence the molecular network model in rubber elasticity was extended to take into account the viscous and thermal effects of the material. The viscous dissipation of the material can then be described by means of these internal variables, which appear in the expression of the modified stretch. In order to give a clearer explanation on the physical implication of the internal variables, a connection between the internal-variable theory and theoretical formulation based on the multiplicative decomposition of the deformation gradient in existing literature is presented in this paper, which allows the above internal-variable theory to be more systematic.

Frequency and Amplitude Dependent Dynamic Model of Rotor Elastomeric Damper with Refined Nonlinear Stiffness

Elastomeric damper is a very important component for helicopter rotor system; its dynamic property has strong nonlinear behavior characterized by complex hysteresis loops, and dependence on excitation frequency, amplitude and temperature. Based on internal variables theory, combined with the nonlinear spring model, a time domain nonlinear dynamic model of elastomeric damper used for helicopter rotor load prediction is presented. The model was characterized by using the genetic algorithm, the hysteresis loop of elastomeric dampers made of different elastomeric materials under several excitation frequencies and strain amplitudes was calculated with this model and compared with experimental data. It is shown that the presented model can predict the hysteresis loop of the elastomeric dampers with little relative errors, and the model is able to catch the variation of dynamic stiffness. Therefore, the presented method can be used for helicopter rotor load prediction and aeroelastic analysis.

High Temperature Deformation Behavior of 8090 Al-Li Alloy

High temperature deformation behavior, especially the superplasticity of an 8090 Al-Li alloy, was studied within the recent framework of the internal variable theory of structural superplasticity. In this study, a series of load relaxation tests were conducted at various temperatures ranging from 200°C to 530°C to obtain the flow curves of log ε˙versus log ε. The effect of grain size was also examined by varying the grain sizes through a proper thermomechanical treatment. The flow curves were found to be composite curves consisting of contributions from grain boundary sliding (GBS) and grain matrix deformation (GMD) at superplastic temperatures. The activation energy obtained for GMD was 124.9 kJ/mole in the temperature range from 470°C to 530°C, very similar to that for self-diffusion in pure Al.

A serial two-stage viscoelastic–viscoplastic constitutive model with thermodynamical consistency for characterizing time-dependent deformation behavior of asphalt concrete mixtures

A serial two-stage viscoelastic–viscoplastic constitutive model is developed using internal variables theory and orthogonality principle of thermodynamics for modeling time-dependent behavior of asphalt concrete mixtures. In one stage when the actual stress is lower than the yield stress, the constitutive model only exhibit viscoelastic component, while in another stage when the actual stress exceeds the yield stress, the constitutive model exhibit both viscoelastic and viscoplastic components. Model parameters of viscoelastic component are estimated using data obtained from dynamic modulus test, whereas model parameters of viscoplastic component are consecutively estimated from creep test, assuming a known viscoelastic component.

Internal-variable analysis of high-temperature deformation behavior of Ti–6Al–4V: A comparative study of the strain-rate-jump and load-relaxation tests

The high-temperature deformation mechanisms of Ti–6Al–4V with either a fine or coarse alpha particle size were quantified using an internal-variable theory. For this purpose, strain rate jump tests (SRJT) and load relaxation tests (LRT) were conducted at 700, 800, and 900°C to determine the strain rate sensitivity and to establish constitutive behavior. Stress–strain rate plots obtained by both SRJT and LRT were in good agreement with the theoretical predictions based on the activation of grain-matrix deformation and particle/grain-boundary sliding (P/GBS). The relative contribution of the two mechanisms varied with the microstructure, temperature, and strain rate, which affected the flow stress and strain rate sensitivity of the alloys. A clear difference in the strain rate sensitivity was observed depending on the experimental method. In all cases, the SRJT values were higher than those from the LRT. The discrepancy in strain rate sensitivity could be attributed to a variation in prestrain between the two methods. This variation resulted in microstructural differences, such as the fraction of alpha/beta interfaces and the misorientation of alpha grain boundaries, and hence affected the contribution of P/GBS to the overall deformation.

A mesoscopic thermodynamical description of rheological models

We propose a simple and systematic description of the most used viscoelastic models by using a non-equilibrium thermodynamic approach, generally referred to as the internal variables theory. The main idea is to elevate the conformation tensor \(\textsf{C}\) describing the geometrical configuration of the macromolecules to the status of independent variable. It is shown that the sole knowledge of the Helmholtz free energy— whose explicit expression is provided by the kinetic theory—allows to derive the expressions of the polymeric stress tensor and the time evolution equation of the internal variable \(\textsf{C}\), which represents the main ingredients of the rheological models. As illustrations, the Hookean, FENE-P Giesekus, and Bird’s encapsulated dumbbell models are discussed. For completeness, the results are compared with these provided by extended irreversible thermodynamics.

An internal variable approach of orientation dependent deformation mechanism of rolled AZ31 magnesium alloy

Load relaxation behavior of an AZ31 Mg alloy has been studied in relation to temperature and orientation dependence. The rolled plate with 50 mm thickness was first homogenized at 400 °C for 4 h before preparing test specimens in the directions of 0° (RD), 45°, and 90° (ND) from the rolling direction. A series of tensile tests was consequently carried out under a strain rate of 10−2/s at room temperature (RT), 100 °C, and 200 °C. The RD specimens were found to deform mainly by dislocation slips without twinning. The 45° and 90° specimens were, on the other hand, found to deform in a combined mode of twinning and dislocation slips. Load relaxation tests were also performed to obtain flow curves in terms of stress and strain rate at the three different temperatures. The flow curves in terms of stress vs. strain rate were also found to consist of a plasticity curve due to dislocation glides in the lower strain rate region and a twin curve in the high strain rate range for the 45° and 90° specimens as prescribed by an internal variable theory. These results were consistent with the tensile test results and were further confirmed by microstructure observations.

Capillary Hysteresis Internal Variable Model of Unsaturated Soils

Soil-water characteristic curve describes the relationship between suction and volumetric water content of unsaturated soils. Air-enter have great influence on the drying scanning curves while the starting points of the scanning curve were at the highly saturated range. Based on capillary hysteresis internal variable model and conventional soil-water characteristics theory of unsaturated soils, a modified model which can simulate soil-water characteristic relationship that could consider the influence of air-enter is proposed. The influence of the air-enter on the drying scanning curves while the starting points were at the highly saturated range could be considered while this model is used. Numerical calculations are carried out on several kinds of soils. The simulating results show that the modified model is able to simulate the drying scanning curves while the starting points of the scanning curve were at the highly saturated range better than the capillary hysteresis internal variable model does.

Study on Non-Local Damage Constitutive Model of Concrete under Freeze-Thaw Action

Concrete is multi-phase composites. Due to the inhomogeneity of mechanical properties and complexity of physical properties, constitutive relations of concrete are more complicated. Starting from irreversible thermodynamics theory, internal state variable theory and nonlocal field theory, non-local damage constitutive model of concrete under freeze-thaw action is established in this paper. In the model, non-local influence functions are discussed which are used to describe interplay of damage between adjacent point.

Hamilton’s principle for Green-inelastic bodies

Abstract In this paper, Hamilton’s principle for Green-elastic bodies with conservative external forces (see Fung and Tong, 2001. Classical and Computational Solid Mechanics. World Scientific, Singapore, Chapter 11.1) is extended to Green-inelastic bodies. Inelastic constitutive laws are addressed within the thermodynamic framework with internal variables by Rice (1971. Inelastic constitutive relations for solids: an internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455). Generalized thermodynamic forces conjugate to internal variables are termed internal forces in this paper. The materials whose free energy functions are point functions of internal variables, are termed here Green-inelastic materials which embody Green-elastic materials as special cases and ensures that the internal forces are potential forces. It is shown that Hamilton’s principle for Green-elastic bodies can be extended to Green-inelastic bodies just by replacing the strain energy of a moving body with the corresponding free energy plus the potential energy of the internal forces.

A Micromechanical Model for Deformation Behavior of Nanocrystalline Copper

Molecular dynamics simulations have show that nanocrystalline (NC) materials can be treated as composite materials consisting of two phases of grain and grain boundary. In this paper, the incremental stress-strain relation is derived based on deformation mechanism of NC materials and internal variable theory from micromechanics point of view. The developed model is exemplified by the pure copper subjected to uniaxial tension. Implicated iteration algorithm is then employed to obtain the stress-strain relation. Moreover, the effects of grain shape and statistical distribution of grain sizes are also discussed, and predicted results are compared with experimental values to verify the model.

Work-hardening and ductility enhancement mechanism of cold rolled multiphase TRIP steels

The work hardening behavior of a high strength sheet steel has been studied in relation to the deformation induced martensitic transformation (DIMT) of retained austenite (γR). The work hardening rate was observed to decrease gradually in a continuously oscillating manner, leading to a significant ductility enhancement in these multiphase steels, in contrast to specimens without γR. Ductility enhancement has been found to depend on the austenite stability (β), with the largest tensile elongation of ∼35 % exhibited in the case with the minimum value of β. The DIMT has thus been interpreted in this study as the successive processes of accumulation and relaxation of internal strain energy produced during inelastic deformation, as has been prescribed by an internal variable theory. This process appears to be responsible for the locally fluctuating work hardening behavior. A new ductility enhancement mechanism is finally proposed in this connection to account for the observed work hardening as well as for the transformation behavior.

Historical review of internal state variable theory for inelasticity

A review of the development and the usages of internal state variable (ISV) theory are presented in this paper. The history of different developments leading up the formulation of the watershed paper by Coleman and Gurtin is discussed. Following the Coleman and Gurtin thermodynamics, different researchers have employed the ISV theory for dislocations, creep, continuum damage mechanics (CDM), unified-creep-plasticity (UCP), polymers, composites, biomaterials, particulate materials, multiphase and multiphysics materials, materials processing, multiscale modeling, integrating materials science (structure–property relations) into applied mechanics formulations, and design optimization under uncertainty for use in practical engineering applications.