Internal State Variable Theory Research Articles

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.

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.

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.

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.

An electroplastic internal state variable (ISV) model for nonferromagnetic ductile metals

This study presents a multiphysics Internal State Variable (ISV) theory that couples the thermo-elasto-viscoplastic-damage model of Bammann with electroplasticity and other electricity-related electromagnetic phenomena. The model framework features internally consistent kinematics, thermodynamics, and kinetics. The abstraction of the developed ISV model can capture the full spectrum of multiphysics electro-mechanical behavior of the nonferromagnetic ductile metals. However, in the absence of electromagnetic fields the model reverts to the existing large deformation Bammann plasticity ISV model with the Horstemeyer damage ISV formalism.

Multiscale ceramic matrix composite thermomechanical damage model with fracture mechanics and internal state variables

This work presents a multiscale damage model that accounts for ceramic matrix composite (CMC) length scale dependent behavior, including scale-specific brittle matrix damage initiation and propagation. Internal state variable theory is implemented to obtain the stress–strain constitutive relationship for damaged ceramic matrix material using damage variables which capture the effects of matrix cracking and nucleation and growth of matrix porosity. Matrix cracking is modeled using a damage variable determined using fracture mechanics and the self-consistent scheme. This methodology provides an effective way to model the effects of matrix cracks which initiate when stress intensity factors exceed the fracture toughness of the material, and whose growth is governed by crack growth kinetics. An additional source of material nonlinearity, matrix porosity, occurs as a result of material diffusion around grain boundaries and is related to the material entropy dissipation. These effects are captured using a porosity state variable that is explicitly related to the volumetric strain. The nonlinear predictive capabilities of the material model are demonstrated for monolithic silicon carbide, unidirectional (UD) carbon fiber/silicon carbide matrix (C/SiC) CMC, and five-harness satin (5HS) woven C/SiC CMC. The model predictions are in excellent agreement with experimental observations for 2D woven C/SiC CMCs from literature.

Peridynamic model for nonlinear viscoelastic creep and creep rupture of Polypropylene

This paper presents the peridynamic numerical method for nonlinear viscoelastic creep behaviour which consists of primary, secondary, tertiary creep stages and creep rupture. A nonlinear viscoelastic creep constitutive equation based on internal state variable (ISV) theory which covers four creep stages is examined. The viscoelastic equation is substituted into material parameter in the peridynamic equation to derive a new peridynamic method with two time parameters i.e. numerical time and real time. The parameters of the viscoelastic equation is analyzed and evaluated. In validating this peridynamic method, a comparison is made between numerical and experimental data. The peridynamic method for nonlinear viscoelastic creep behaviour (VE-PD) is approved by the good similarity between numerical and experimental creep strain curves with overall difference of 10.67%. The nonlinearity of experimental and numerical data is adequately similar as the error between experimental and numerical curves of secondary stage strain rate against load is 8.022%. The shapes of fractured numerical specimen show good resemblance with the experimental result as well.

Constitutive modeling of the cyclic dissipation in thin and thick rubber specimens

AbstractA thermo‐viscoelastic‐damage constitutive model, in accordance with the thermodynamic principles, is proposed to describe the cyclic dissipation in rubbers. The general form of the model can be represented by a thermal branch responsible for the stress‐free thermal expansion joined to parallel mechanical multi‐branches responsible for the cyclically damaged‐elastic response and the inelastic effects, i.e. fatigue‐induced stress‐softening and hysteresis. The viscoelastic and damage dissipations contributing to heat build‐up are considered in the model formulation using the internal state variable theory. The model is identified and verified using experimental observations on stress‐softening, hysteresis and dissipative heating obtained on rubber flat specimens containing different amounts of carbon‐black and cyclically loaded under different minimum stretch levels. The model is then used to predict damage and thermal patterns in thick specimens involving different triaxial stress states in the median cross‐section.

Development of an elastothermoviscoplasticity damage model for injection molded short fiber reinforced thermoplastics with anisotropic damage evolutions

ABSTRACTThis paper presents a finite deformation anisotropic and temperature-dependent constitutive model, which is developed for predicting elastoviscoplasticity and progressive damage behavior of injection molded short fiber reinforced thermoplastics (SFRPs) by the usage of the internal state variable (ISV) theory. In the present model, the SFRP is considered as an anisotropic single lamina and the viscoplasticity of the SFRP is captured with the help of three inelastic ISVs. A second-order damage tensor is introduced to model the evolution of the micro cracks/voids in the SFRP composites while also reflecting the anisotropic nature of the damage. The constitutive model developed herein arises employing standard postulates of continuum mechanics with the kinematics, thermodynamics, and kinetics being internally consistent. The developed model is then calibrated to the injection molded 35 wt% short glass fiber reinforced polyamide 6,6 (designation PA66GF-35) with all material constants obtained by fitting the model to the experimental data. A parametric study is conducted on the model to numerically investigate the effects of strain rate, temperature, and fiber orientation effects on the elastoviscoplasticity and damage performance of such material.

Pre-stretch dependency of the cyclic dissipation in carbon-filled SBR

This work explores for the first time the inelastic fatigue process in filled rubbers using the internal state variable theory. The theory is used to qualitatively and quantitatively analyze the complex history-dependent cyclic response of filled rubbers, but also to reveal the underlying physical mechanisms. Experimental observations are reported on a styrene-butadiene rubber (SBR) containing different amounts of carbon-black and cyclically loaded under a wide range of pre-stretch levels. The effects of pre-stretch and filler content on the carbon-filled SBR history-dependent cyclic response, characterized by stress-softening and hysteresis along with dissipative heating, are examined. The intrinsic dissipation, regarded as a consequence of two types of rearrangements in the rubber-filler material system, i.e. viscoelastic and damage mechanisms, is quantified and used to propose a plausible explanation of the underlying inelastic fatigue mechanisms consistent with our experimental observations.

Study on time-dependent behavior and stability assessment of deep-buried tunnels based on internal state variable theory

The creep model based on thermodynamics with internal state variables theory can simulate complex time-dependent deformation of rock mass, describe process of energy dissipation of material system, and can be used to evaluate the long-term stability of underground structures quantitatively. In this paper, the creep model proposed by author is improved further and recast to be central difference equation. The redevelopment interface of FLAC3D is used to develop a new calculation program, which is based on thermodynamics for vsico-plasticity (PTV-P). Program validation has been conducted by comparing the results from FLAC3D and Matlab software under uniaxial compression condition. Then the developed program has been applied to analyze the time-dependent behavior of deep-buried double tunnels. The integral values of energy dissipation rate and its time derivative in domain can be calculated and are used to evaluate the long-term stability of tunnels quantitatively, and the evaluation criterion is also proposed. Moreover, the contour map of energy dissipation rate is used to exhibits the local non-equilibrium region clearly.

A creep model with damage based on internal variable theory and its fundamental properties

The creep damage is discussed within Rice irreversible internal state variable (ISV) thermodynamic theory. An ISV small-strain unified creep model with damage is derived by giving the complementary energy density function and kinetic equations of ISVs. The proposed model can describe viscoelasticity and, preferably, three phases of creep deformation. Creep strain results from internal structural adjustment, and different creep stages accompany different thermodynamic properties in terms of flow potential function and energy dissipation rate. During the viscoelastic process, the thermodynamic state of the material system tends to equilibrate spontaneously. The thermodynamic state of the material system without damage tends to equilibrate or achieve steady state after loading. Kinetic equations of ISVs can be derived by one single flow potential function, and the energy dissipation rate decreases monotonically over time. In the entire creep damage process, multiple potentials are needed to characterise evolution of ISVs, rotational fluxes are presented in affinity space, and the thermodynamic state of material system tends to depart from the steady or equilibrium state. The energy dissipation rate can be a measure of the distance between the current thermodynamic state and the equilibrium state. The time derivative of the rate can characterise the development trend of the material, and the integral value in the domain may be regarded as indices to evaluate the long-term stability of the structure.

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.

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.

Application of the Bammann inelasticity internal state variable constitutive model to geological materials

SUMMARY We describe how the Bammann internal state variable (ISV) constitutive approach, which has proven highly successful in modelling deformation processes in metals, can be applied with great benefit to silicate rocks and other geological materials in modelling their deformation dynamics. In its essence, ISV theory provides a constitutive framework to account for changing history states that arise from inelastic dissipative microstructural evolution of a polycrystalline solid. In this paper, we restrict our attention to a Bammann ISV elastic-viscoplastic model with temperature and strain rate dependence and use isotropic hardening and anisotropic hardening as our two ISVs. We show the Bammann model captures the inelastic behaviour of olivine aggregates (with and without water), lherzolite (with and without water), Carrara marble and rock salt using some experimental data found in the literature. These examples illustrate that when more experimental stress–strain data are gathered on other rock materials, much more realistic numerical simulation of rock behaviour becomes feasible. Though not available in the literature, we outline a set of experiments to obtain unique Bammann ISV model constants.

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.

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.

A second gradient theoretical framework for hierarchical multiscale modeling of materials

A theoretical framework for the hierarchical multiscale modeling of inelastic response of heterogeneous materials is presented. Within this multiscale framework, the second gradient is used as a nonlocal kinematic link between the response of a material point at the coarse scale and the response of a neighborhood of material points at the fine scale. Kinematic consistency between these scales results in specific requirements for constraints on the fluctuation field. The wryness tensor serves as a second-order measure of strain. The nature of the second-order strain induces anti-symmetry in the first-order stress at the coarse scale. The multiscale internal state variable (ISV) constitutive theory is couched in the coarse scale intermediate configuration, from which an important new concept in scale transitions emerges, namely scale invariance of dissipation. Finally, a strategy for developing meaningful kinematic ISVs and the proper free energy functions and evolution kinetics is presented.

An extended multiscale principle of virtual velocities approach for evolving microstructure

A hierarchical multiscale approach is presented for modeling microstructure evolution in heterogeneous materials. Preservation of momentum across each scale transition is incorporated through the application of the principle of virtual velocities at the fine scale giving rise to the appropriate continuum momentum balance equations at the coarse scale. In addition to satisfying momentum balance and invariance of momentum among scales, invariance of elastic free energy, stored free energy, and dissipation between two scales of observation is regarded as crucial to the physics of each scale transition. The preservation of this energy partitioning scheme is obtained through construction of constitutive relations within the framework of internal state variable theory. Internal state variables that are directly computed from the fine scale response are introduced to augment the state equations and describe the inelastic energy storage and dissipation within the fine scale. By virtue of a second gradient kinematic decomposition, the framework naturally gives rise to couple stresses.

Non-local separation constitutive laws for interfaces and their relation to nanoscale simulations

A framework is proposed to characterize fracture through continuum interface separation constitutive laws that are motivated by molecular dynamics (MD) simulations using embedded-atom method potentials. These laws are distinguished from previous continuum models in that discrete atomistics are used to determine a set of nanoscale effects, accounting for the influence of atomic structure and imperfections on interface separation or fracture. This conceptual framework utilizes internal state variable (ISV) theory to embed a set of interface attributes into cohesive interface separation constitutive relations. Structural rearrangement of the interface region during separation can be explicitly tracked through a set of differential equations for the ISVs. To guide forms for the interface constitutive laws, MD simulations are presented for a planar, tilt nanograin boundary interface subjected to tension and shear deformations. The proposed list of ISVs accounts for geometry, composition, defect density, and damage within the interface region. Continuum calculations motivated by discrete nanoscale computations facilitate a more complete description of real material separation processes.