Internal Variable Theory Research Articles

A multiaxial multiscale compressive thermo-mechanical damage model for Fiber Reinforced Cementitious Composites (FRCC)

In the current study, a multiaxial multiscale compressive stress-strain model for Fiber Reinforced Cementitious Composites (FRCC) was innovatively developed. The proposed model was on the basis of the theory of thermodynamics, multiscale theory and internal variable theory. The free energy function and dissipation function were established to characterize the elastic and plastic deformation of FRCC, respectively. The multiscale behaviors of FRCC in meso-scale and macro-scale was taken into account. In meso-scale, the energy dissipation functions of fiber in tension and bond-slip deformation in the interface transition zone (ITZ) were established. The resistance to cracking of cementitious matrix through the fiber bond stress were taken into account. In macro-scale, the thermo-mechanical coupling contributions of plastic damage, thermal damage, yield surface and hardening evolution were creatively considered in the potential function within thermodynamics framework. The influence of types and volume fractions of fibers on the reinforcement mechanism was discussed. When the temperature is higher than the melting point of fiber, the fiber influence coefficients was employed. The sensitivity of yield surface was discussed. The numerical predictions of this developed model were carried out for validation. The nonlinear stress-strain responses of concrete were accurately captured at temperatures ranges from 20 °C to 800 °C compared with the experimental data, which indicates the accuracy and applicability of the proposed model.

Experimental and theoretical investigation on magnetorheological elastomers containing carbonyl iron particles coated with silane coupling agent

Magnetorheological (MR) elastomer composites, comprising soft silicone rubber, various additives, and different weight fractions of carbonyl iron particles (CIPs) coated with silane coupling agent, are produced via a novel manufacturing process in an anisotropic state. This study encompasses both experimental and modeling investigations into the dynamic viscoelastic properties of magnetorheological elastomer (MREs) in shear mode under varying magnetic fields, displacement amplitudes, and frequencies. Two MRE vibration mitigation devices are fabricated to experimentally assess the shear storage modulus and the loss factor of MREs. The experimental findings reveal a pronounced MR effect in the MRE devices, where both the shear storage modulus and the loss factor increase with rising magnetic fields, frequencies, and particle weight fractions, yet decrease with higher displacement amplitudes. A modified fractional-derivative equivalent parametric model, grounded in a magnetic field- and frequency-dependent shear modulus model along with internal variable theory, is proposed to describe the effects of these key influencing factors on the MREs’ dynamic viscoelastic properties. Comparative analysis of experimental and numerical data demonstrates that this refined mathematical model can accurately represent the dynamic viscoelastic properties of MREs.

Internal variable gradient model for active earth pressure of rigid retaining wall moving with translation

The instability of retaining wall is a key factor for many geo-hazards, such as landslides. To estimate the stability of retaining wall, the distribution of earth pressure is necessary. The results of in-situ observations and indoor experiments demonstrate that the distribution of earth pressure behind the retaining wall exhibits remarkable nonlinearity. When the results are analyzed in details, the oscillation and quasi-periodicity of the distribution of earth pressure are observed, which has not been given widely concerns and cannot be described by the existing analytical models. Based on the internal variable gradient theory and operator averaging method, a gradient-enhanced softening constitutive model is proposed in this paper to describe the oscillation and quasi-periodicity of the distribution of earth pressure acting on the retaining wall, by introducing the high-order gradient terms of the hydrostatic pressure into Mohr-Coulomb yield condition. In order to check the applicability of the proposed formulation, the predictions from the formulations are compared with the full-scale and laboratory-scale test results as well as the existing formulations. It is noted from the comparisons between predicted and measured values that the results of gradient-dependent softening constitutive model provides the comparable approximations for active earth pressure and describes the oscillation and quasi-periodicity very well. This model may enhance the comprehension of soil mechanics and provide a novel view for the design of the retaining wall.

A Multiphysics Thermoelastoviscoplastic Damage Internal State Variable Constitutive Model including Magnetism.

We present a macroscale constitutive model that couples magnetism with thermal, elastic, plastic, and damage effects in an Internal State Variable (ISV) theory. Previous constitutive models did not include an interdependence between the internal magnetic (magnetostriction and magnetic flux) and mechanical fields. Although constitutive models explaining the mechanisms behind mechanical deformations caused by magnetization changes have been presented in the literature, they mainly focus on nanoscale structure-property relations. A fully coupled multiphysics macroscale ISV model presented herein admits lower length scale information from the nanoscale and microscale descriptions of the multiphysics behavior, thus capturing the effects of magnetic field forces with isotropic and anisotropic magnetization terms and moments under thermomechanical deformations. For the first time, this ISV modeling framework internally coheres to the kinematic, thermodynamic, and kinetic relationships of deformation using the evolving ISV histories. For the kinematics, a multiplicative decomposition of deformation gradient is employed including a magnetization term; hence, the Jacobian represents the conservation of mass and conservation of momentum including magnetism. The first and second laws of thermodynamics are used to constrain the appropriate constitutive relations through the Clausius-Duhem inequality. The kinetic framework employs a stress-strain relationship with a flow rule that couples the thermal, mechanical, and magnetic terms. Experimental data from the literature for three different materials (iron, nickel, and cobalt) are used to compare with the model's results showing good correlations.

Compaction elastoplastic damage constitutive model of rock under hydro-mechanical coupling: A heuristic modeling method based on internal state variables theory

Based on the internal state variables (ISV) theory, this paper employs a heuristic modeling approach to investigate the compaction elastoplastic damage constitutive model of rocks under hydro-mechanical coupling. The heuristic modeling method demonstrated herein can seamlessly integrate with existing constitutive theories, facilitating a broader spectrum of research on hydro-mechanical coupling constitutive models. Furthermore, a numerical solution was conducted in the principal axes via stress spectrum decomposition, which can well reflect the distinct characteristics of compaction deformation and the influence of water pressure on rock stress distribution. Utilizing the classic Drucker-Prager (D-P) criterion, the modeling method and numerical algorithm were implemented to derive a hydro-mechanical coupling compaction elastic–plastic damage constitutive model. This contribution expands the application domain of the classic D-P criterion within the context of hydro-mechanical coupling environments. The rationality of the model was confirmed through compression test data under various water pressures and confining pressures. Water pressure increases nonlinear compaction deformation by changing the opening degree of microcracks under low confining pressure, which has not been paid enough attention in existing research. This study thoroughly considers the influence of water pressure on compaction deformation. The developed process of “heuristic modeling method-numerical method-model application” is helpful for enriching the constitutive theory of rock under hydro-mechanical coupling.

Study on the Damage Characteristics and Internal Variable Modeling of Single-Fracture Sandstone under the Coupling Effect of Freeze–Thaw and Fatigue Load

The fractured rock mass in the western cold region is affected by freezing and thawing disasters and is prone to local damage and fracture along the fissures’ ends. The fatigue damage induced by repeated frost heave and traffic loads seriously endangers the stability of cold region roadbed. This paper selects sandstone as the research object. Firstly, 20 freeze–thaw cycles were performed on fractured sandstone samples with different inclination angles of 30°, 45°, 60°, and 90°. Subsequently, triaxial compression and triaxial fatigue loading tests were conducted to explore the mechanical properties and fracture morphology evolution mechanism during the compression process of freeze–thaw fractured sandstone. Nuclear magnetic resonance technology (NMR) was used to measure the H-containing fluid inside rock pores. The microscopic damage characteristics inside the rock were analyzed from the NMR T2 relaxation spectrum signal and pore size distribution characteristics. Based on the internal variable theory of continuum mechanics, a fatigue model of freeze–thaw fractured sandstone with different inclination angles was established. The results show that sandstone strength was negatively correlated with the fracture dip angle, and the axial deformation and shear failure angle were positively correlated with the fracture dip angle. The mechanical properties of the sandstone were deteriorated by fatigue loading. When the crack angle was 90°, the fatigue failure strength of the rock sample was the lowest. The T2 spectrum distribution of the fractured sandstone mainly had three peaks and the pore size was mainly medium and small pores. There was a small leftward shift after freeze–thaw cycles and fatigue loading. The T2 spectrum area was significantly affected by fatigue loading, with the highest rate of change at a crack angle of 30°. Through the fine correspondence between the axial residual deformation and the deformation modulus, a fatigue model with different crack inclination angles was established using the axial residual deformation as the internal variable, and the rationality of the model was verified by fatigue loading tests.

An analysis of fluid–structure interaction coupling mechanisms in liquid-filled viscoelastic pipes subject to fast transients

An extension of a recently developed quasi-2D flow model for fluid transients in viscoelastic pipes to handle fluid–structure interaction mechanisms is presented. In a context in which the fluid flow is devised as a structured pseudo-mixture and the pipe’s viscoelasticity is rooted in an internal variable theory, the axial movement of the pipe wall is allowed to occur, giving rise to friction, Poisson, and junction coupling mechanisms. The resulting governing equations of the model form a quasi-linear hyperbolic system of partial differential equations, which approximated solution is achieved by means of the method of characteristics. The proposed approach is validated against pressure traces acquired from a reservoir–pipe–valve experimental setup found in the literature. In the course of the validating process, different pipe anchoring conditions are employed to study the system responses. Focus is given to pipe–fluid interface interactions, energy dissipation, and transfer of energy between both media.

Macro-micro damage model of the effect of freeze–thaw on jointed rocks considering compaction deformation

Jointed rock mass contains microscopic cracks, joints and other defects. Moisture migration during freeze–thaw cycles affects crack initiation and propagation of defects of different scales. The change of defects in turn affects the mechanical and deformation properties of jointed rock mass, especially in the compaction deformation stage. Considering the effects of freeze–thaw cycles on compaction deformation, plastic deformation, macroscopic damage and microscopic damage, a macro–micro damage constitutive model of jointed rock mass under freeze–thaw actions was proposed under the framework of thermodynamics. Based on the internal state variables (ISV) theory, the Helmholtz free energy of jointed rock mass under freeze–thaw actions was obtained by taking compaction deformation, macroscopic damage and microscopic damage as internal state variables. The compaction, elasticity, plasticity, and damage stages were analyzed respectively, and the macro–micro damage model of jointed rock considering compaction deformation were deduced by regarding the total strain as three parts (compaction deformation, elastic deformation and plastic deformation). The validity of the proposed model was verified by compression tests on rock samples with different induced crack inclination angles that have undergone different amounts of freeze–thaw cycles. The model considers the moisture migration mechanism of freeze–thaw damage and the friction and slip mechanism of microcrack surfaces. The effect of moisture migration on the compaction deformation during freeze–thaw cycles is significant, and the proposed model can accurately simulate the full stress–strain curves (compaction-elasticity-plasticity-damage) of jointed rock mass under freeze–thaw actions, which provides quantitative guidance for reducing freezing damage by attenuating moisture migration.

Modeling inelastic responses using constrained reactive mixtures

This study reviews the progression of our research, from modeling growth theories for cartilage tissue engineering, to the formulation of constrained reactive mixture theories to model inelastic responses in any solid material, such as theories for damage mechanics, viscoelasticity, plasticity, and elasto-plastic damage. In this framework, multiple solid generations α can co-exist at any given time in the mixture. The oldest generation is denoted by α=s and is called the master generation, whose reference configuration Xs is observable. The solid generations α are all constrained to share the same velocity vs, but may have distinct reference configurations Xα. An important element of this formulation is that the time-invariant mapping Fαs=∂Xα/∂Xs between these reference configurations is a function of state, whose mathematical formulation is postulated by constitutive assumption. Thus, reference configurations Xα are not observable (α≠s). This formulation employs only observable state variables, such as the deformation gradient Fs of the master generation and the referential mass concentrations ρrα of each generation, in contrast to classical formulations of inelastic responses which rely on internal state variable theory, requiring evolution equations for those hidden variables. In constrained reactive mixtures, the evolution of the mass concentrations is governed by the axiom of mass balance, using constitutive models for the mass supply densities ρˆrα. Classical and constrained reactive mixture approaches share considerable mathematical analogies, as they both introduce a multiplicative decomposition of the deformation gradient, also requiring evolution equations to track some of the state variables. However, they also differ at a fundamental level, since one adopts only observable state variables while the other introduces hidden state variables. In summary, this review presents an alternative foundational approach to the modeling of inelastic responses in solids, grounded in the classical framework of mixture theory.

Experimental study and mathematical modeling of viscoelastic dampers with wider temperature range based on blended rubber matrix

The glass transition temperature of existing viscoelastic (VE) materials is low, usually under 0 °C, thus the desired damping performance for VE dampers cannot be achieved at room temperature. On this issue, this study investigated the feasibility of utilizing blended rubber matrix to widen the working temperature range of VE materials, thereby improving the energy dissipation capacity at room temperature. A series of VE material samples based on blended rubber matrix were prepared, and the optimal formula was selected and applied in the fabrication of VE dampers. Then, the performance tests of VE dampers based on blended rubber matrix and single rubber matrix were conducted and compared. Results show that VE dampers based on blended rubber matrix have wider temperature range and better energy dissipation capacity than those based on single rubber matrix. A mathematical model for VE dampers based on blended rubber matrix was proposed by combining the internal variable theory and the temperature-frequency equivalence principle, and the accuracy of the proposed model was verified by comparisons between the calculated results and experimental data.

A thermodynamically consistent model for hydraulic transients in metallic pipes undergoing elasto-viscoplastic deformations

This work presents an extension of the fluid transient analysis to include the elasto-viscoplastic behavior of the pipe material. A previously developed quasi-2D flow model is employed as a basic framework, and the pipe’s elasto-viscoplastic behavior is added by considering constitutive equations formulated via an internal variables theory. The resulting governing equations form a non-linear hyperbolic system of partial differential equations whose approximated solutions are obtained by employing the method of characteristics followed by the Newton–Raphson method. The model is thermodynamically consistent, a feature that is consistently detailed in this work to uphold the application of energy-based theoretical analyses of the phenomenon. Taking as reference the transient responses obtained from an experimental installation that simulates an accident, the present approach is found to be fully consistent and valuable for simulating fast hydraulic transients in metallic materials experiencing plastic deformation. By comparing the pressure fluctuations of models with and without elasto-viscoplasticity, more attenuation and dispersion of the pressure responses are found when such behavior is considered. The elasto-viscoplasticity effect on the pressure responses is better understood when the connection between the viscoplastic mechanisms and energy dissipation is exposed. During the fluid transient, the pipe dissipates meaningful amounts of energy that cause a significant alteration in the whole fluid–pipe dynamics. Another numerical example, based on a hypothetical water hammer flow that causes plastic deformation in the pipeline, is also addressed. Such a case puts in evidence that not only the anelastic behavior of the pipe must be taken into account but also that an accurate description of the fluid friction may be significant to portray realistically fluid transient responses. At last, an assessment of the pipe integrity by means of a scalar damage variable theory is carried out to emphasize the application prospects of the present research.

Experimental and Theoretical Investigation of Viscoelastic Damper by Applying Fractional Derivative Method and Internal Variable Theory

Viscoelastic dampers are conventional passive vibration control devices with excellent energy dissipation performance. The fractional derivative has a simple form and high accuracy in the modelling of viscoelastic materials/dampers. The internal variables reflect the internal state evolution of materials, and are often used to analyze the deformation and thermal process of materials. In the present work, the mechanical properties of a plate-shear-type viscoelastic damper at room temperature are tested under sinusoidal displacement excitations. The impacts of frequency and displacement amplitude on the dynamic properties of the viscoelastic damper in a wide frequency domain (0.1–25 Hz) are investigated. The higher-order fractional derivative model and the temperature–frequency equivalent principle are employed to characterize the frequency and temperature influence, and the internal variable theory considering the internal/microscale structure evolutions is introduced to capture the displacement affection. The higher-order fractional derivative model modified with the internal variable theory and temperature–frequency equivalent principle (ITHF) is accurate enough in describing the dynamic behaviors of viscoelastic dampers with varying frequencies and displacement amplitudes.

Fluid transients in viscoelastic pipes via an internal variable constitutive theory

This work presents a fluid transient model capable of handling the viscoelastic behavior of the pipe. A previously developed quasi-2D flow model is employed as a base, and the viscoelastic behavior of the pipe is incorporated by considering constitutive equations formulated in a thermodynamically consistent framework of an internal variable theory. Such an approach straightforwardly provides expressions for computing the rates of energy dissipation in the fluid and pipe accurately and separately. This novel feature discerns the local and overall impacts of the energy dissipation on the pressure oscillations caused by each medium. The governing equations of the model form a hyperbolic system of partial differential equations whose approximated solutions are obtained by the method of characteristics. Taking as reference pressure signals obtained by a classic reservoir-pipe-valve experiment found in the literature, it is shown that the model predictions are fully consistent. A comparison between the pressure responses of viscoelastic and elastic pipes reveals that in addition to delaying the pressure oscillations, the viscoelastic behavior causes a faster attenuation of them. The rates of energy dissipation in the viscoelastic pipe attain significant magnitudes during the first moments of the fluid transient and alters the hydrodynamical behavior of the flow. Such interference is exposed by comparing the responses of the same experimental setup when two different viscoelastic pipe materials are considered. It is also shown that the knowledge of the parcels of energy dissipated in the fluid and pipe individually can improve the comprehension of the phenomenon and be utilized for theoretical and applied research in the field.

Internal Variable Theory in Viscoelasticity: Fractional Generalizations and Thermodynamical Restrictions

Here, we study the internal variable approach to viscoelasticity. First, we generalize the classical approach by introducing a fractional derivative into the equation for time evolution of the internal variables. Next, we derive restrictions on the coefficients that follow from the dissipation inequality (entropy inequality under isothermal conditions). In the example of wave propagation, we show that the restrictions that follow from entropy inequality are sufficient to guarantee the existence of the solution. We present a numerical solution to the wave equation for several values of the parameters.

Internal Variables as a Tool for Extending Navier-Stokes Equations

Abstract The formalism of the internal variable theory is applied to extend Navier-Stokes equations. The internal variable theory provides a thermodynamically consistent derivation of constitutive relations and equations of motion without a priori specifying the nature of internal variables. Both single and dual internal variables cases are thoroughly examined. The similarities and differences of the approaches are emphasized. In the single internal variable framework, the elimination of the internal variable results in Maxwell-type constitutive relations and hyperbolic equations of motion. The dual internal variable technique enables us to create even more sophisticated fluid flow models with coupled equations for fluid motion and internal variable evolution.

Experimental Investigation and Multiscale Modeling of VE Damper Considering Chain Network and Ambient Temperature Influence

Viscoelastic (VE) dampers are one of the most promising techniques for reducing vibration in engineering structures caused by earthquakes and wind. This work aims to develop a kind of high-dissipation VE damper for civil structures at low frequency and large amplitude in shear mode. First, nitrile rubber (NBR)/organic small-molecule composite VE materials are optimized and then made into VE damper. In order to test the mechanical performance and energy dissipation performance of the VE damper, the dynamic mechanical performance experiments at different temperatures, frequencies, and amplitudes were implemented. The experimental results show that the VE damper exhibits great stiffness and excellent energy dissipation capacity under different loading conditions. Second, a fractional derivative model based on Gauss microchain, Williams–Landel–Ferry (WLF) equation, and internal variable theory is proposed to accurately describe the effects of temperature, frequency, and amplitude on the dynamic mechanical properties of VE dampers. Finally, the accuracy of the mathematical model of VE damper is verified by comparing the calculated results with the experimental results. The study provides a theoretical basis for effective vibration reduction of civil structures with VE dampers at low frequency.

A NEURAL ORDINARY DIFFERENTIAL EQUATION FRAMEWORK FOR MODELING INELASTIC STRESS RESPONSE VIA INTERNAL STATE VARIABLES

We propose a neural network framework to preclude the need to define or observe incompletely or inaccurately defined states of a material in order to describe its response. The neural network design is based on the classical Coleman-Gurtin internal state variable theory. In the proposed framework the states of the material are inferred from observable deformation and stress. A neural network describes the flow of internal states and another represents the map from internal state and strain to stress. We investigate tensor basis, component, and potential-based formulations of the stress model. Violations of the second law of thermodynamics are prevented by a constraint on the weights of the neural network. We extend this framework to homogenization of materials with microstructure with a graph-based convolutional neural network that preprocesses the initial microstructure into salient features. The modeling framework is tested on large datasets spanning inelastic material classes to demonstrate its general applicability.

STABILITY CONSIDERATIONS FOR ELASTIC-PLASTIC PROCESSES WITH DAMAGE

Classical theory of plasticity is fairly complete with flow rules, convexity of yield surfaces, extremum principles, and the uniqueness theorem. For the strain-hardening plasticity, Drucker’s postulates are established and proven based on the plastic-work and energy principles. Plasticity models have been further applied to heterogeneous and cementitious materials with certain degrees of success. In this paper, the stability statements of strain-hardening and strain-softening processes in concrete are examined by utilizing thermodynamic potential functions in the stress space and by applying Euler’s theorem of homogenous functions. It is shown that by specifying a strain-hardening parameter to account for the plastic strains and a damage parameter to represent the effect of microcracking, the dissipation inequality can be used to establish the Drucker’s stability postulate for the plastic flow within the framework of the internal variable theory of thermodynamics. Using the same approach and assuming uncoupling between plastic flow and microcracking, the formation leads to a softening stability statement for damage processes in concrete.

Accelerated computational micromechanics and its application to polydomain liquid crystal elastomers

We present an approach to solving problems in computational micromechanics that is amenable to massively parallel calculations through the use of graphical processing units (GPUs) and other accelerators. We apply it to study microstructure evolution in polydomain liquid crystal elastomers (LCEs). LCEs are rubber-like solids where rod-like nematic molecules are incorporated into the main or a side polymer chain. They undergo isotropic to nematic phase transition accompanied by spontaneous deformation which can be exploited for actuation. Further, they display a soft behavior at low temperatures due to the reorientation of the nematic directors. The problem of understanding nematic reorientation in the presence of realistic defects (non-ideality) is computationally expensive, and we address this by efficiently exploiting GPUs. The approach is broadly applicable to various phenomena including crystal plasticity and phase transitions that are described by internal variable theories. We verify the approach against previous calculations and establish its performance by studying long wavelength instability of finite elasticity. Our numerical studies of LCEs provide insights into the director distribution and reorientation in polydomain specimens, and how these lead to soft behavior under multiaxial loading. The results show good agreement with experimental observations.

A historical review of the traditional methods and the internal state variable theory for modeling composite materials

A review of the development and the usages of traditional non-internal state variable (ISV) models and ISV theory for modeling behaviors of composite materials are presented in this paper. The history of earliest composites models for predictions of material stiffness, failure, viscoelasticity and plasticity is discussed first. Then following the ISV theory, a fundamental thermodynamic framework that governs the dissipative effects caused by the inelastic phenomena is introduced. Applying the thermodynamic restrictions obtained from that framework, different researchers have employed the ISV theory to develop constitutive models for composites to capture their viscoelasticity, rate-independent plasticity, viscoplasticity, and continuum damage. ISVs introduced into those models for describing each inelastic behavior of composites as well as their evolution rules are also discussed. The knowledge obtained from this review in terms of the usages of the ISV theory for composite material modeling forms a theoretical background for the development of a general modeling framework to predict all inelastic phenomena of a wide variety of composite materials.