Internal State Variable Model 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.

A Rate- and Temperature-Dependent Thermomechanical Internal State Variable Model of the Directed Energy Deposition Process

A Rate- and Temperature-Dependent Thermomechanical Internal State Variable Model of the Directed Energy Deposition Process

Correction: numerical implementation and application of an internal state variable model to analyze the time-dependent behavior of mining excavations in rock salt

Résumé Le comportement géomécanique du sel gemme a été largement étudié au fil des ans. Les différentes études expérimentales ont permis d'identifier des caractéristiques distinctives associées à une réponse inélastique non-linéaire qui est fonction du temps et de l'historique de chargement. Ces caractéristiques doivent être reproduites par les équations constitutives utilisées pour analyser la réponse du sel gemme autour des ouvertures souterraines. Dans cet article, un modèle constitutif relativement simple est introduit dans le code numérique FLAC et appliqué pour analyser le comportement différé d'excavations dans des mines de sel. Le modèle unifié en fluage et plasticité comprend une variable d“état interne avec une loi d”évolution qui induit un écrouissage progressif jusqu'à ce qu“un état stationnaire soit atteint. Des analyses numériques sont effectuées avec le modèle ISV-SH pour évaluer les paramètres du matériau et simuler la réponse d”une ouverture circulaire et d'excavations minières rectangulaires créées séquentiellement. Les résultats sont présentés et comparés à ceux obtenus avec l“équation bien connue de la loi de puissance de Norton, couramment utilisée dans l”ingénierie des mines de sel. Les résultats des calculs, présentés en termes de l’évolution des contraintes, des déformations et de vitesses de déformation, illustrent certains des aspects clés du comportement du sel gemme et ils mettent en évidence l“influence majeure de la loi de comportement utilisée pour les analyses numériques. L”approche de simulation proposée ici constitue une alternative pratique et flexible aux outils de modélisation basés sur le fluage stationnaire, souvent utilisés pour l“analyse des ouvertures souterraines dans les opérations d”extraction de sel gemme.1 Ceci est une traduction fournie par l'auteur du résumé en anglais.

Using an Internal State Variable Model Framework to Investigate the Influence of Microstructure and Mechanical Properties on Ballistic Performance of Steel Alloys

An internal state variable (ISV)-based constitutive model has been used within a Lagrangian finite element analysis (FEA) framework to simulate ballistic impact of monolithic rolled homogenous armor (RHA) steel plates by RHA steel spheres and cylinders. The ISV model predictions demonstrate good agreement with experimental impact data for spherical projectiles. A simulation-based parametric sensitivity study was performed to determine the influence of a variety of microstructural and mechanical properties on ballistic performance. The sensitivity analysis shows that the lattice hydrogen concentration, material hardness, and initial void volume fraction are dominant factors influencing ballistic performance. Finite element simulations show that variation of microstructure properties could explain the reduced ballistic performance of high hardness materials previously documented in the literature. The FEA framework presented in this work can be used to determine material properties conducive to ballistic-impact resistance.

Numerical implementation and application of an internal state variable model to analyze the time-dependent behavior of mining excavations in rock salt

The geomechanical behavior of rock salt has been investigated extensively over the years. Experimental studies have identified distinctive features associated with nonlinear inelastic response that shows strong time and loading history dependencies. In this paper, a relatively simple constitutive model is presented and implemented into the numerical code FLAC, and then applied to analyze the time-dependent behavior of excavations in salt mines. The unified creep-plasticity model includes an internal state variable (ISV) with an evolution law that induces progressive strain hardening (SH) until a steady state is reached. Numerical analyses are performed with the proposed ISV–SH model to evaluate material parameters, based on creep test results on natural rock salt, and to simulate the response of a circular opening and of rectangular mining excavations created in sequence. The results are analysed and compared with those obtained with the well-known Norton power law equation, commonly used in salt mine engineering. The time-dependent calculation results illustrate key aspects of rock salt behavior and highlight the major influence of transient inelastic behavior and stress redistribution on the response of underground openings. The proposed modelling approach constitutes an advantageous alternative to analyses based on stationary creep laws, often used in rock salt mining operations.

Internal state variable creep constitutive model for the rock creep behavior

Internal state variable creep constitutive model for the rock creep behavior

Mesoscale Simulation-Based Parametric Study of Damage Potential in Brain Tissue Using Hyperelastic and Internal State Variable Models.

Two-dimensional mesoscale finite element analysis (FEA) of a multilayered brain tissue was performed to calculate the damage-related average stress triaxiality and local maximum von Mises strain in the brain. The FEA was integrated with rate-dependent hyperelastic and internal state variable (ISV) models, respectively, describing the behaviors of wet and dry brain tissues. Using the finite element results, a statistical method of design of experiments (DOE) was utilized to independently screen the relative influences of seven parameters related to brain morphology (sulcal width/depth, gray matter (GM) thickness, cerebrospinal fluid (CSF) thickness and brain lobe) and loading/environment conditions (strain rate and humidity) with respect to the potential damage growth/coalescence in the brain tissue. The results of the parametric study illustrated that the GM thickness and humidity were the two most crucial parameters affecting average stress triaxiality. For the local maximum von Mises strain at the depth of brain sulci, the brain lobe/region was the most influential factor. The conclusion of this investigation gives insight for the future development and refinement of a macroscale brain damage model incorporating information from lower length scale.

An Internal State Variable Elastoviscoplasticity-Damage Model for Irradiated Metals

Abstract This study presents an irradiation-dependent internal state variable (ISV) elastoviscoplasticity-damage constitutive model that accounts for nuclear irradiation hardening and embrittlement of the irradiated polycrystalline materials. The irradiation effects were added to the coupled plasticity-damage kinetics with consideration of the structure–property relationships. The present irradiation-dependent elastoviscoplasticity-damage model was compared with the lab deformation experimental data of irradiated oxygen-free high conductivity (OFHC) copper, modified 9Cr-1Mo steel, and Ti-5Al-2.5Sn. The results show excellent agreement over the entire stress–strain curves at various irradiation doses. Because the ISV model, before the irradiation plasticity-damage addition, had been used on over 80 different metal alloys, it is anticipated that this nuclear irradiation supplement will also allow for application to many more irradiated metal alloys.

Physics-based and phenomenological plasticity models for thermomechanical simulation in laser powder bed fusion additive manufacturing: A comprehensive numerical comparison

The present study investigated the sensitivity of material constitutive models on thermomechanical responses in laser powder bed fusion additive manufacturing of Ti-6Al-4V. Uniform scan strategies with scan lengths of 0.5, 1, and 2 mm were applied so that wide ranges of thermal histories could be generated. The Johnson-Cook (JC) and Mechanical Threshold Stress (MTS) material plasticity models were chosen to capture the influence of strain, strain rate, and temperature. The JC model is a phenomenological model which is known for its easy implementation and excellent agreement with material testing results. On the other hand, the MTS model is a more complex physics-based internal state variable plasticity model that is expected to provide more accurate estimation, particularly for cases involving changes in strain rate and temperature. Numerical results revealed that both JC and MTS models provided a similar stress evolution, however, the plastic strain evolution was more realistic using the MTS model. Moreover, it was found that the maximum strain and the strain rate in the LPBF process are high compared to typical quasi-static testing, i.e., ~ 2% and ~ 4 s−1, respectively. Accordingly, the material models should be calibrated with data obtained under similar deformation conditions. The choice of scan length also strongly affects in-plane stress anisotropy. Ultimately, we show both qualitatively and quantitatively the dependency of mechanical behavior prediction in LPBF on the choice of material models.

Finite element analysis of a ram brain during impact under wet and dry horn conditions

In this study, ram impacts at 5.5 m/s are simulated through finite element analysis in order to study the mechanical response of the brain. A calibrated internal state variable inelastic constitutive model was implemented into the finite element code to capture the brain behavior. Also, constitutive models for the horns were calibrated to experimental data from dry and wet horn keratin at low and high strain rates. By investigating responses in the different keratin material states that occur in nature, the bounds of the ram brain response are quantified. An acceleration as high as 607 g's was observed, which is an order of magnitude higher than predicted brain injury threshold values. In the most extreme case, the maximum tensile pressure and maximum shear strains in the ram brain were 245 kPa and 0.28, respectively. Because the rams do not appear to sustain injury, these impacts could give insight to the threshold limits of mechanical loading that can be applied to the brain. Following this motivation, the brain injury metric values found in this research could serve as true injury metrics for human head impacts.

Experimental characterization and constitutive modeling of an aluminum 7085-T711 alloy under large deformations at varying strain rates, stress states, and temperatures

The strain rate, stress state, and temperature dependent mechanical behavior of a hot rolled aluminum 7085-T711 alloy is experimentally observed and subsequently modeled using a continuum Internal State Variable (ISV) plasticity-damage constitutive model. Structure-property relationships for the alloy are quantified using a series of microstructural analyses and mechanical property experiments. The calibrated ISV model for the Al 7085-T711 alloy is implemented in an implicit finite element code (Abaqus) to simulate the deformation of notch Bridgman tension specimens at a variety of stress states and temperatures. The ISV model accurately captures the material's elastoplastic behavior by predicting the stress state difference between tension, compression, and torsion and temperature dependent microstructure evolution under large deformations.

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.

Transient strain age hardening of Al–Mg alloys

The static strain aging response of an Al–Mg alloy is investigated by means of tensile pre-deformation followed by annealing. The mechanical response following re-testing in tension reveals a lower flow stress and raised work hardening rate at yield. The enhanced work hardening rate is shown to be transient, the behaviour returning to that of the non-aged material at high flow stress. In order to decouple the various effects arising from solute segregation and dislocation loss via recovery, a simplified version of the two internal state variable model originally proposed by Kubin and Estrin is introduced. It is proposed that the response, yield strength and work hardening rate, following pre-deformation and annealing can be ascribed to the combined effects of the loss of mobile dislocations and a transient hardening that is interpreted to come from dislocations that have had solute segregated to them.

Modeling of flow behavior and microstructure evolution for Mg-6Gd-5Y-0.3Zr alloy during hot deformation using a unified internal state variable method

The hot deformation behavior of Mg-6Gd-5Y-0.3Zr alloy at the temperatures of 673–748K and the strain rates of 0.01–1.0s−1 were investigated by hot compression experiments. On this basis, a physically based internal state variable (ISV) model coupling the microstructural evolution was established, in which the ISVs of the model included dislocation density, grain size and volume fraction of dynamic recrystallization (DRX). Besides, the recrystallized and un-recrystallized grain sizes were considered to characterize the evolution process of microstructure. By comparing the calculated and experimental flow stresses, the model was proved to be reliable. Then the model was imported into the finite element analysis (FEA) software ABAQUS to predict the ISV evolution of Mg-6Gd-5Y-0.3Zr alloy in the hot extrusion process. The comparative analysis of the simulation and experimental results further verified the applicability of the constitutive model established in this study.

Uncertainty propagation in reduced order models based on crystal plasticity

In this work, an uncertainty propagation study is performed based on simulated ensembles of statistical volume elements (SVE) used to inform a reduced order internal state variable model, focusing on model form and uncertainties in numerical simulations. A rate-dependent polycrystal plasticity finite element model (CP-FEM) of cartridge brass is calibrated to room-temperature uniaxial tension testing data of an annealed sample. Model forms are considered with and without the inclusion of back stress in the CP model. Three sizes of SVE are explored for simulation uncertainty effects. Effects are explored with regard to the parameters of a Bammann–Chiesa–Johnson (BCJ) macroscopic viscoplasticity model, calibrated element-wise to SVEs from an ensemble modeled using CP-FEM simulations exploring quasi-static uniaxial tension. Variability in stress–strain response at multiple defined length-scales is measured. The selection of model form is discussed. The numerical simulation uncertainty of the model reduction process is quantified.

A combined viscoelasticity-viscoplasticity-anisotropic damage model with evolving internal state variables applied to fiber reinforced polymer composites

A thermodynamically-consistent large strain viscoelasticity-viscoplasticity-damage Internal State Variable (ISV) constitutive material model integrated with a mixture theory combining each individual constituent is formulated to describe the highly nonlinear, rate-dependent thermomechanical behavior of Fiber Reinforced Polymer (FRP) composites. The model is formulated in an intermediate configuration where unloading the specimen is viscoelastic. In our model, a rate-dependent yield surface is employed to identify initial yielding of the material. At small deformations, which occurs before yielding, the viscoelastic behavior of FRPs is described using the generalized Maxwell model, while at large deformation, three physically based evolving ISVs are used to capture the hardening and softening behaviors caused by their polymeric constituents. The complicated damage state of the FRPs is captured by a second rank tensor, which is further decomposed to model the subscale damage phenomena of microvoids/cracks nucleation, growth and coalescence. Finally, the mixture theory in combination with certain homogenization procedures are adopted to formulate the overall stress, strain, and damage of the composite in terms of the individual constituents. To demonstrate the model’s usage, two examples are presented. First, the ISV model is calibrated to a Polyamide 6,6 reinforced with glass fibers with the aim of showing its capacity to predict the viscoelastic and damage coupled behaviors of a composite. Second, a glass fiber reinforced epoxy with the purpose of describing the temperature dependence and the interphase effect is illustrated.

A unified internal state variable material model for Ti2AlNb-alloy and its applications in hot gas forming

The high-temperature deformation behavior of Ti2AlNb sheet was studied by the uniaxial tensile tests at the temperatures from 910 °C to 1000 °C with the strain rates 0.0004–0.1 s−1. The microstructure evolutions of the as-received specimen and the hot deformation specimens were investigated by the electron back-scattered electron image and electron back-scattered diffraction. The flow stress and microstructure evolution are modeled via a set of internal-state variable unified constitutive equations based on the changes of dislocation density, dynamic recrystallization, grain size, phase volume fraction, globularization softening, damage evolution and deformation heating. The model constants were determined by the GA-based global optimization in MATLAB soft. The flow stress of the internal-state variable unified model was more precise than the flow stress of the Fields-Backofen model under the multistep strain rate conditions. The Ti2AlNb cup-shape parts were produced by the hot gas forming and the microstructures were predicted by the software ABAQUS using the VUMAT subroutine. The simulation predictions of microstructure evolution, i.e. dislocation density and materials damage, for the Ti2AlNb based alloy during hot gas forming process are in consistence with experimental results.

Predicting the reliability of an additively-manufactured metal part for the third Sandia fracture challenge by accounting for random material defects

We describe an approach to predict failure in a complex, additively-manufactured stainless steel part as defined by the third Sandia Fracture Challenge. A viscoplastic internal state variable constitutive model was calibrated to fit experimental tension curves in order to capture plasticity, necking, and damage evolution leading to failure. Defects such as gas porosity and lack of fusion voids were represented by overlaying a synthetic porosity distribution onto the finite element mesh and computing the elementwise ratio between pore volume and element volume to initialize the damage internal state variables. These void volume fraction values were then used in a damage formulation accounting for growth of these existing voids, while new voids were allowed to nucleate based on a nucleation rule. Blind predictions of failure are compared to experimental results. The comparisons indicate that crack initiation and propagation were correctly predicted, and that an initial porosity field superimposed as higher initial damage may provide a path forward for capturing material strength uncertainty. The latter conclusion was supported by predicted crack face tortuosity beyond the usual mesh sensitivity and variability in predicted strain to failure; however, it bears further inquiry and a more conclusive result is pending compressive testing of challenge-built coupons to de-convolute materials behavior from the geometric influence of significant porosity.

On the formulation of the kinematics and thermodynamics for polycrystalline materials undergoing phase transformation

Commonly implemented material processing routines not limited to quenching, welding or heat treatment requires exposure of a part to complex thermal and mechanical loading histories that manifest as residual stress and distortion. Of interest to material designers and fabricators is modeling and simulating the evolutionary process a part undergoes for the sake of capturing this observable residual stress states and geometric distortion accumulated after processing. To move toward an overall consistent modeling approach, we premise this investigation with a consistent thermodynamic framework for a generalized multiphase material. Following this, we extend the single phase Evolving Microstructural Model of Inelasticity (EMMI) internal state variable model to multiphase affirming that the interaction between phases is through an interfacial stress. We then use a self-consistent polycrystalline model to partition each individual phase's strain field ensuring a hybrid between compatibility and equilibrium. With a synthesis of the aforementioned ideas, the additional transformation plasticity (TRIP) is accounted for by changing each phase's flow rule to accommodate an interfacial stress. Following this, we couple the mechanical multiphase model equations with a previously developed non-diffusional phase transformation kinetics model. Contrary to the classical modeling approach, our material model is based on mixture theory wherein we track the behavior of each individual phase. A numerical evaluation of the coupled model is performed and applied to a simplified quenching boundary value problem.

Plasticity-Damage Modeling of Strain Rate and Temperature Dependence of Aluminum Alloy 7075-T651

This is the first study to correlate the observed damage evolution of ambient (298 K) and elevated (480 K) temperatures of quasi-static and dynamically-loaded wrought aluminum alloy 7075 (AA7075) to capture the strain rate and temperature influence on flow behavior with one set of material constants. In this research, an internal state variable (ISV) framework implementing a physically-based plasticity and damage constitutive model was used to capture the strain rate and temperature dependence of the wrought Al–Zn–Mg–Cu aluminum alloy. The model includes microstructural content processing history and is consistent with continuum level kinematics, kinetics, and thermodynamics. The ISV model captures deformations due to kinematic and isotropic stress-state dependent hardening and damage from the microscale that arise from microstructural features and defects in the wrought aluminum alloy. In addition, the ISV damage theory is based on void nucleation, void growth, and void coalescence. This research provides a foundation for capturing the structure–property relations from the microscale to the structural scale. Finite element methods coupled with internal state variables used to model the plasticity and damage state at the structural scale for high fidelity dynamic loading scenarios, such as blast and impact loading conditions.