Internal State Variable Constitutive Model Research Articles

An internal state variables constitutive model for Ni-based superalloy considering the influence of second phase and its application in flexible skew rolling

Microstructure prediction and control of Ni-based alloy during plastic forming is crucial for obtaining high-performance products. Taking GH4169 alloy as an example for study, firstly, hot compression experiments were conducted using both solution treated (ST) and aging treated (AT) alloys at different true strains (0.223–0.916), temperatures (900℃-1100℃), and strain rates (0.1 s−1-10 s−1). Then, an internal state variable constitutive model was established based on experimental data. In the developed model, the evolution of dislocation density, dynamic recovery, and dynamic recrystallization (DRX) behavior are reasonably described. Specifically, by introducing the volume fraction and size of the initial δ-phase, the interaction between δ-phase and dislocations, the influence of δ-phase on DRX behavior and critical strain, and its pinning effect on grain boundaries are considered. The calculation results indicate that accurate predictions can be achieved within a large parameter range. Subsequently, the model was applied to flexible skew rolling (FSR) process, a novel plastic forming method for forming shaft parts and bars. The finite element (FE) simulations and rolling experiments were carried out. The results indicate that the FE model embedded with the constitutive model can effectively predict the forming dimensions and microstructure distribution of GH4169 alloy. The established constitutive model can provide reference for the high-temperature plastic forming process of alloys containing the second phase.

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.

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.

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.

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.

A unified static and dynamic recrystallization Internal State Variable (ISV) constitutive model coupled with grain size evolution for metals and mineral aggregates

A history dependent and physically-motivated Internal State Variable (ISV) constitutive model is presented that simultaneously accounts for the effects of static recrystallization, dynamic recrystallization, and grain size with respect to the mechanical behavior under different strain rates, temperatures, and pressures. A unique aspect of our ISV constitutive model is that grain size and recrystallized volume fraction can be directly included along with its associated rate of change under deformation and time in a coupled manner. The present ISV constitutive model was calibrated to several metals (oxygen-free high conductivity copper, AZ31 magnesium alloy, pure nickel, and 1010 low carbon steel) and geological materials (olivine and clinopyroxene). The model calibration shows good agreement with the experimental stress-strain behavior and average grain size data. Validation of the ISV constitutive model was accomplished by applying complex and history sensitive thermomechanical problems once the model was calibrated: i) sequential transitions of different loading conditions and ii) a multistage tubing process. The history dependence naturally provided by ISVs enabled the present model to effectively capture the complex boundary value problems with changing boundary conditions.

A microstructure sensitive grain boundary sliding and slip based constitutive model for machining of Ti-6Al-4V

A composite dual phase internal state variable constitutive model was developed for Ti-6Al-4V. The proposed model includes diffusion assisted grain boundary sliding based physics in addition to a traditional slip-based plasticity. Influence of microstructure on the flow stress is introduced via dislocation density and mean grain size internal state variables. The dislocation density evolves according to a physics based law that considers dislocation nucleation and annihilation processes. Grain refinement is driven by dynamic recrystallization, which is modeled phenomenologically. The model is calibrated with uniaxial stress–strain data that ranges between quasi-static and dynamic rates across a wide range of temperatures. Validation against machining data shows that the model predicts chip segmentation frequency, machining forces, and tool temperatures reasonably well. The newly introduced grain boundary sliding physics was found to dominate deformation following sufficient grain refinement. This deformation mode provides softening at the constitutive level without the need for invoking damage based softening mechanisms. This physical interpretation is something that has not previously been explored in the machining literature.

Analysis of Deformation in a High Entropy Alloy Using an Internal State Variable Model

Deformation in the model high entropy alloy CoCrFeMnNi is assessed using an internal state variable constitutive model. A remarkable property of these alloys is the extraordinarily high strain hardening rates they experience in the plastic region of the stress strain curve. Published stress-strain measurements over a range of temperatures are analyzed. Dislocation obstacle interactions and the observed high rate of strain hardening are characterized in terms of state variables and their evolution. A model that combines a short-range obstacle and a long-range obstacle is shown to match experimental measurements over a wide range of temperatures and grain sizes. The long-range obstacle is thought to represent interactions of dislocations with regions of incomplete mixing or partial segregation. Dynamic strain aging also is observed at higher temperatures. Comparisons with measurements in austenitic stainless steel show some common trends.

On the Definition of State Variables for an Internal State Variable Constitutive Model Describing Metal Deformation

The quest for an internal state variable constitutive model describing metal deformation is reviewed. First, analogy is drawn between a deformation model and the Ideal Gas Law. The use of strain as a variable in deformation models is discussed, and whether strain serves as an internal state variable is considered. A simple experiment that demonstrated path dependence in copper is described. The importance of defining appropriate internal state variables for a constitutive law relates to the ability to accurately model temperature and strain-rate dependencies in deformation simulations.

An Internal State Variable Constitutive Model for Deformation of Austenitic Stainless Steels

Austenitic stainless steels—particularly the 304 and 316 families of alloys—exhibit similar trends in the dependence of yield stress on temperature. Analysis of temperature and strain-rate dependent yield stress literature data in alloys with varying nitrogen content and grain size has enabled the definition of two internal state variables characterizing defect populations. The analysis is based on an internal state variable constitutive law termed the mechanical threshold stress model. One of the state variables varies solely with nitrogen content and is characterized with a larger activation volume. The other state variable is characterized by a much smaller activation volume and may represent interaction of dislocations with solute and interstitial atoms. Analysis of the entire stress–strain curve requires addition of a third internal state variable characterizing the evolving stored dislocation density. Predictions of the model are compared to measurements in 304, 304L, 316, and 316L stainless steels deformed over a wide range of temperatures (up to one-half the melting temperature) and strain rates. Model predictions and experimental measurements deviate at temperatures above ∼600 K where dynamic strain aging has been observed. Application of the model is demonstrated in irradiated 316LN where the defect population induced by irradiation damage is analyzed. This defect population has similarities with the stored dislocation density. The proposed model offers a framework for modeling deformation in stable austenitic stainless steels (i.e., those not prone to a martensitic phase transformation).

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.

An Internal State Variable Constitutive Description for the Deformation of Metals at Elevated Temperatures

In the present investigation, an internal state variable constitutive model for the description of metals that are deformed under hot-working conditions in the range of thermal recovery, where positive work hardening is displayed, is proposed. The flow stress of the material is expressed in terms of the deformation temperature and strain rate, besides the contribution of the mechanical strength of the different obstacles that oppose dislocation motion. The constitutive description is validated employing the flow stress, temperature, strain rate, and strain increment data obtained from tests carried out both on Al-5.5 pct Mg and C-Mn steel. The model thus advanced assumes that the main obstacles to dislocation motion are solute atoms either in substitutional or interstitial solid solution and forest dislocations, although other contributions to the flow stress can also be readily incorporated. The formulation developed describes quite accurately the flow stress and work-hardening rate of the material under the temperature and strain rate conditions investigated. Also, it is able predict the occurrence of significant strain transients when deformation conditions are suddenly changed. Appropriate parameters involved in the model allow the comparison of the contribution of different strengthening mechanisms to the flow stress of different materials

Strain Rate Sensitivity of Epoxy Resin in Tensile and Shear Loading

The mechanical response of E-862 and PR-520 resins is investigated in tensile and shear loadings. At both types of loading the resins are tested at strain rates of about 5x10(exp 5), 2, and 450 to 700 /s. In addition, dynamic shear modulus tests are carried out at various frequencies and temperatures, and tensile stress relaxation tests are conducted at room temperature. The results show that the toughened PR-520 resin can carry higher stresses than the untoughened E-862 resin. Strain rate has a significant effect on the response of both resins. In shear both resins show a ductile response with maximum stress that is increasing with strain rate. In tension a ductile response is observed at low strain rate (approx. 5x10(exp 5) /s), and brittle response is observed at the medium and high strain rates (2, and 700 /s). The hydrostatic component of the stress in the tensile tests causes premature failure in the E-862 resin. Localized deformation develops in the PR-520 resin when loaded in shear. An internal state variable constitutive model is proposed for modeling the response of the resins. The model includes a state variable that accounts for the effect of the hydrostatic component of the stress on the deformation.

Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling

Shape memory polymers (SMPs) can retain a temporary shape after pre-deformation at an elevated temperature and subsequent cooling to a lower temperature. When reheated, the original shape can be recovered. Relatively little work in the literature has addressed the constitutive modeling of the unique thermomechanical coupling in SMPs. Constitutive models are critical for predicting the deformation and recovery of SMPs under a range of different constraints. In this study, the thermomechanics of shape storage and recovery of an epoxy resin is systematically investigated for small strains (within ±10%) in uniaxial tension and uniaxial compression. After initial pre-deformation at a high temperature, the strain is held constant for shape storage while the stress evolution is monitored. Three cases of heated recovery are selected: unconstrained free strain recovery, stress recovery under full constraint at the pre-deformation strain level (no low temperature unloading), and stress recovery under full constraint at a strain level fixed at a low temperature (low temperature unloading). The free strain recovery results indicate that the polymer can fully recover the original shape when reheated above its glass transition temperature ( T g). Due to the high stiffness in the glassy state ( T < T g), the evolution of the stress under strain constraint is strongly influenced by thermal expansion of the polymer. The relationship between the final recoverable stress and strain is governed by the stress–strain response of the polymer above T g. Based on the experimental results and the molecular mechanism of shape memory, a three-dimensional small-strain internal state variable constitutive model is developed. The model quantifies the storage and release of the entropic deformation during thermomechanical processes. The fraction of the material freezing a temporary entropy state is a function of temperature, which can be determined by fitting the free strain recovery response. A free energy function for the model is formulated and thermodynamic consistency is ensured. The model can predict the stress evolution of the uniaxial experimental results. The model captures differences in the tensile and compressive recovery responses caused by thermal expansion. The model is used to explore strain and stress recovery responses under various flexible external constraints that would be encountered in applications of SMPs.

Integration of Basic Materials Research Into the Design of Cast Components by a Multi-Scale Methodology

We present a formal methodology to integrate basic materials research results into a design procedure for engineering components. Although the methodology is generic, we demonstrate its development and application in designing a cast automotive control arm. Specifically, we propose a multi-scale analysis whereby the local mechanisms of deformation are accounted for by an internal state variable constitutive model embedded within a non-linear elastic-plastic finite element analysis. Multiple fundamental materials research approaches at different length scales are utilized to quantify the pertinent deformation mechanisms, such as atomistic simulations, microscopy, micromechanical finite element simulations, and mechanical testing. The current methodology is a step toward a larger goal of interactively integrating basic materials science research into engineering system design. [S0094-4289(00)01903-4]

Thermal stress and fatigue analysis of plated-through holes using an internal state variable constitutive model

Previous related research on plated-through hole (PTH) fatigue investigations has been based on the so-called effective stress/strain methods, which did not account for the fact that fatigue crack nucleation and growth is observed to occur on planes of specific orientation. Moreover, previous related thermal stress/strain analyses were at most based on bilinear constitutive relations for modeling copper plating along with a linear kinematic hardening assumption, and this cannot capture many aspects of cyclic stress/strain behavior during thermal excursions. In this paper, thermal stress analyses using internal state variable (ISV) models of metallic constituents of PTHs are conducted using the finite element code ABAQUS (1996). Two thermal history profiles having two repeated cycles were applied for the PTHs of a double layered printed wiring board (PWB) uniformly: (1) MIL-T-CYC (between −65°C and 125°C), and (2) IEC OIL-T-SHOCK (between 25°C and 260°C). A critical plane theory was used for purposes of multiaxial fatigue life prediction. The stress/strain results were reported and compared at the PTH corner and barrel. For both cases, the thermomechanical mismatch between the FR4 and copper constituents of the PWB generates nonproportional stress/strain responses. This complicates PTH thermal fatigue investigation.

Constitutive equations for modelling flow softening due to dynamic recovery and heat generation during plastic deformation

The flow softening of metals at elevated temperatures is modelled in this paper based on the theory of viscoplasticity. A single internal state variable constitutive model is presented to represent the deformation behavior of metals that exhibit flow softening caused by the competing hardening and recovery processes and heat generation during plastic deformation. The constitutive equations are validated by hot torsion testing that has advantage over tension and compression in generating flow stress to large strains. The material constants determination methods have been developed and successfully applied to aluminium alloy 5083 over a wide range of strain rates and temperatures. The prediction of the flow stress in aluminium alloy 5083 as a function of strain rate, temperature and strain has shown to be in good agreement with the test data.

Object-oriented analysis of network flows at pore and reservoir scales

When low porosity rock samples are loaded beyond a certain threshold, microcracking occurs. This threshold can be seen as a yielding condition in an elasto-plastic model, or as a damage initiation criterion in an internal state variable constitutive model. In this paper, the authors present a simple multiaxial formulation used to describe this threshold in the conventional stress space. The criterion is applied to a hard rock (granite) and to a soft rock (rocksalt), and its significance is discussed.

A damage initiation criterion for low porosity rocks

Abstract When low porosity rock samples are loaded beyond a certain threshold, microcracking occurs. This threshold can be seen as a yielding condition in an elasto-plastic model, or as a damage initiation criterion in an internal state variable constitutive model. In this paper, the authors present a simple multiaxial formulation used to describe this threshold in the conventional stress space. The criterion is applied to a hard rock (granite) and to a soft rock (rocksalt), and its significance is discussed.

A solution for the contact between two spherical particles undergoing large deformation

A unit model consisting of two particles in contact and under compression, leading to the development of an internal state variable (ISV) constitutive model incorporating the particle-to-particle contact during large deformations, is studied. Following Matthews' approximation that a general form of the Hertz elastic contact solution holds for nonlinear materials, an analytical solution based on the ISV constitutive model and valid for the small contact stage is first derived for the unit problem. To account for large contact areas, the analytical solution is modified based on numerical results obtained from a detailed finite element analysis of the compression of two identical spherical aluminum particles in contact. A set of unified relations is derived from this analysis that accounts for the strain rate effects and accurately represents the particle-to-particle contact behavior for both small and large contact areas. The performance of the proposed model is validated through comparisons with detailed numerical results obtained for a wide range of strain rates.