Internal Variable Model Research Articles

On the thermo-mechanical coupling of the Bammann plasticity-damage internal state variable model

In this study, thermodynamic incompatibility issues of the thermo-mechanical coupling of the Bammann-temperature-dependent plasticity-damage internal state variable (ISV) model are investigated. The exclusion of the thermal expansion phenomena from the Helmholtz free energy, as assumed in the model, is proven to contradict the First and Second Law of Thermodynamics, as well as the omnipresence principle. Four different approaches are discussed to address those issues, and the inclusion of the thermal expansion as a dependent variable in the Helmholtz free energy is considered the most appropriate and efficient. Based on these findings, a multiphysics ISV theory that couples the elasto-visco-plasticity-damage model of Bammann with thermal expansion is presented in which the kinematics, thermodynamics, and kinetics are internally consistent. Other material models may benefit from the findings of this study and apply similar modifications with their thermo-mechanical couplings.

A multiphase internal state variable model with rate equations for predicting elastothermoviscoplasticity and damage of fiber-reinforced polymer composites

This paper explores the integration of an internal state variable (ISV) model for polymers (Bouvard et al. in Acta Mech 213(1):77–96, 2010; Int J Plast 42:168–193, 2013) with damage evolution (Horstemeyer and Gokhale in Int J Solids Struct 36:5029–5055, 1999; Horstemeyer et al. in Theor Appl Fract Mech 33(1):31–47, 2000; Francis et al. in Int J Solids Struct 51:2765–2776, 2014) into a multiphase ISV framework (Rajagopal and Tao in Advances in mathematics for the applied sciences, World Scientific, Singapore, 1995; Bammann in Proceedings of 2nd international conference on quenching and the control of distortion, vols 4–7, 1996) that features a finite strain theoretical framework for fiber-reinforced polymer (FRP) composites under various stress states, temperatures, strain rates, and history dependencies. In addition to the inelastic ISVs for the polymer matrix and interphase, new ISVs associated with the interaction between phases are introduced. A scalar damage variable is employed to capture the damage history of the FRP, which comprises three damage modes: matrix cracking, fiber breakage, and deterioration of the fiber–matrix interface. The constitutive model developed herein employs standard postulates of continuum mechanics with the kinematics, thermodynamics, and kinetics being internally consistent, whose ISVs can be either calculated from molecular dynamics simulations or calibrated through microstructural characterizations for specific FRPs. The developed elastothermoviscoplasticity and damage modeling framework is then employed to model the internal damage evolution of a glass fiber-reinforced polyamide 66 (Rolland et al. in Compos Part B Eng 90:65–377, 2016) in terms of above three damage mechanisms. A detailed description of the model parameter identification process is given by using the example of a unidirectional glass fiber-reinforced epoxy, and the mechanical behaviors and properties of the composites at varying temperature and fiber volume fraction are predicted by the model, which are in good agreement with the experimental result.

Thermomechanical constitutive modeling of fiber reinforced shape memory polymer composites based on thermodynamics with internal state variables

Shape memory polymer composites (SMPCs) can be used to improve the mechanical properties of the shape memory polymers (SMPs). They also have the potential to enhance or enable new stimulus approaches and novel shape memory effects (SMEs). In this paper, a thermoviscoelastic finite deformation constitutive model is developed for thermally activated unidirectional continuous elastic fiber reinforced SMPCs. Since the structural relaxation and viscous flow mainly exist in the SMP matrix, an internal state variable modeling approach is used to describe the thermomechanical behavior of the SMPCs. Recent works mainly focus on the thermomechanical behavior of carbon fiber reinforced SMPCs in the small strain range, for the tensile tolerant strain of carbon fiber is small. The model that is developed here shows that the unidirectional continuous carbon fiber reinforced SMPCs with proper fiber inclination angle and volume fraction can also be used in finite deformation range for the first time. The finite deformation thermomechanical response of unidirectional continuous carbon fiber reinforced SMPCs with different inclination angles and volume fractions is addressed here. Therefore, this study provides useful guidance for reasonable design of unidirectional continuous elastic fiber reinforced SMPCs.

Predicting microstructure and strength for AISI 304L stainless steel forgings

Specially designed billets of AISI 304L stainless steel were forged at two temperatures, 843 °C and 941 °C, and with varying transfer times. The machined 304L billets were thick near the center (51 mm) and thin near the periphery (10 mm) such that forging with a flat platen resulted in regions of high strain (center) and low strain (periphery). Multiple metallographic samples were extracted from one experimental forging for each temperature (water quenched forging with nominal transfer times). Microstructural analysis revealed that recrystallization initiated at about 1.4 plastic strain for forgings made at 843 °C and 1.1 plastic strain for 941 °C. Mechanical tests—performed using tensile bars from multiple locations from each forging—revealed that the resulting yield strength increased with strain up ~10–20% recrystallization, then decreased. Coupled thermo-mechanical simulations estimated a temperature increase of up to 162 °C in the center of the forging due to rapid deformation in the High Energy Rate Forging (HERF) press. Simulation using updated BCJ model parameters for 304L stainless steel predicted the strain distributions using the scalar value, equivalent plastic strain. The predicted trends for percent recrystallization and yield strength from this simulation (using the internal state variable BCJ model) agreed qualitatively with experimental tensile test results, but the quantitative comparison leaves room for improvement. Both the simulation and the experimental results clarify that the microstructure and strength of 304L forgings are difficult to control in a forging with excessive strain variation. It is recommended that strain variation be minimized and overall strain be kept under ~0.5 to best control the strength and grain structure of 304L stainless steel. Excessive transfer time appeared to have little effect on the forging yield strength, but air cooling and die cooling after forging indicated softening for the forgings made at 941 °C.

Thermomechanical model for monotonic and cyclic loading of PEEK

An internal state variable (ISV) model for polyether ether ketone (PEEK) is developed to capture experimental measurements conducted below its glass transition temperature. The model is based on the response of two relaxing components put in parallel, one having a back stress response that softens with plastic flow. This softening allows reproducing both monotonic and cyclic loading, and is possibly associated with detangling, melting of network junctions, and/or change in crystallinity. The resulting model, which is thermodynamic and multidimensional, was used to capture the response of PEEK 450G using thermal expansion (23 °C to 120 °C), heat capacity (−40 °C to 140 °C), monotonic extension and compression (−85 °C to 150 °C; 0.0001 s−1 to 3000 s−1; strains up to 40–80%), equilibrium stress measurements in compression (23 °C to 120 °C; strains up to 60%), and ultrasonic longitudinal and shear wave speed measurements along and transverse to the directions of compression (23 °C to 120 °C; up to 50% plastic compression). The model is compared to the response under monotonic and cyclic shear and internal dissipation is assessed using the equivalent adiabatic temperature rise.

Evolution law of continuous dynamic recrystallization in forming of AA7075 tee valve by multi-direction loading

In forming of AA7075 tee valve by multi-direction loading, continuous dynamic recrystallization (CDRX) will occur and significantly affect the final microstructure and performance of formed parts. In this paper, the developed physically-based internal state variable model for CDRX and finite element (FE) simulation were used to investigate the influence law of forming parameters (deformation temperature, punch fillet radius, friction factor, and the loading speed of punch) on CDRX in the intersecting region (complex deformation and possible weak zone) of the formed tee valve. By considering the grain size and its distributing uniformity in the intersecting region, the reasonable forming parameters were obtained, under which, the evolution behavior and law of the dislocation density, subgrain size, average misorientation of subgrain boundary and average grain size were revealed. The results can provide a guide for the quick prediction of CDRX and selection of the reasonable forming parameters for multi-direction loading forming of tee valves.

Continuous dynamic recrystallization prediction in multi-direction loading forming of 7075 aluminum alloy tee valve

In multi-direction loading forming of 7075 aluminum tee valve, continuous dynamic recrystallization (CDRX) will occur and affect the final microstructure and performance. In this paper based on the developed physically-based internal state variable model and finite element (FE) simulation, under the reasonable forming parameters, the evolution behavior and law of the dislocation density, subgrain size, average misorientation of subgrain boundary and average grain size were revealed, especially in the intersecting region at different multi-direction loading forming stages. It is also shown that when the horizontal and vertical punches are loaded simultaneously, the CDRX degree decreases while grain size increases gradually from the inner to outer wall of branch tube. In the intersecting region, the average grain size increases from the inside to outer corner. The results can provide a guide for the quick prediction of CDRX and selection of the reasonable forming parameters for multi-direction loading forming of tee valve.

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.

Stress-level-dependency and bimodal precipitation behaviors during creep ageing of Al-Cu alloy: Experiments and modeling

Understanding the deformation and strengthening mechanism at different stress levels is crucial for accurate prediction of creep ageing behaviors, especially for alloys subjected to a complex stress-state. A comprehensive investigation has been performed on the creep deformation and age hardening during creep ageing of an aluminum-copper alloy AA2219-T4 under various stress levels that lead to elastic or initial plastic deformation. Microstructural variables including dislocations and precipitates have been characterized in detail by X-ray diffractometer, scanning/transmission electron microscopy and differential scanning calorimetry. The creep mechanism is found to be dominated by diffusion creep under small stress level and then by dislocation climb with increasing stress. The high-stress-induced dislocations raise the creep strain considerably and cause a notable extension of the primary creep stage. The existence of preferential alignment of θ" precipitates (stress-orienting effect) in creep-aged samples results in an evident decline in the age hardening potential. A typical bimodal distribution of precipitates comprised of θ" homogeneously dispersed in the matrix and relatively large stable θ′ heterogeneously nucleated at dislocations forms during high-stress ageing. The θ′ precipitates exhibit no stress-orienting effect and lead to a considerable enhancement of hardening response. Based on detailed microscopic observations, a unified constitutive model is developed and calibrated by experimental data to predict the observed stress-level dependency of creep ageing behaviors. The dislocation evolution during both the loading stage and the creep stage as well as the bimodal precipitation of shearable and nonshearable phases are quantified and incorporated in a dislocation-based internal variable creep model. The proposed model is able to well describe the concurrent evolution of dislocations and precipitation, whose interplay gives rise to a characteristic upturn behavior of stress effects on creep strain and age hardening response. The model can be applied to simulate the creep age forming of complex aluminum alloy component structures.

A novel unified model predicting flow stress and grain size evolutions during hot working of non-uniform as-cast 42CrMo billets

The cast preformed forming process (CPFP) is increasingly considered and applied in the metal forming industries due to its short process, low cost, and environmental friendliness, especially in the aerospace field. However, how to establish a unified model of a non-uniform as-cast billet depicting the flow stress and microstructure evolution behaviors during hot working is the key to microstructure prediction and parameter optimization of the CPFP. In this work, hot compression tests are performed using a non-uniform as-cast 42CrMo billet at 1123–1423 K and 0.01–1 s−1. The effect laws of the non-uniform state of the as-cast billet with different initial grain sizes on the flow stress and microstructure are revealed deeply. Based on experimental results, a unified model of flow stress and grain size evolutions is developed by the internal variable modeling method. Verified results show that the model can well describe the responses of the flow stress and microstructure to deformation conditions and initial grain sizes. To further evaluate its reliability, the unified model is applied to FE simulation of the cast preformed ring rolling process. The predictions of the rolling force and grain size indicate that it could well describe the flow stress and microstructure evolutions during the process.

Molecular dynamics simulations showing 1-palmitoyl-2-oleoyl-phosphatidylcholine (POPC) membrane mechanoporation damage under different strain paths

Continuum finite element material models used for traumatic brain injury lack local injury parameters necessitating nanoscale mechanical injury mechanisms be incorporated. One such mechanism is membrane mechanoporation, which can occur during physical insults and can be devastating to cells, depending on the level of disruption. The current study investigates the strain state dependence of phospholipid bilayer mechanoporation and failure. Using molecular dynamics, a simplified membrane, consisting of 72 1-palmitoyl-2-oleoyl-phosphatidylcholine (POPC) phospholipids, was subjected to equibiaxial, 2:1 non-equibiaxial, 4:1 non-equibiaxial, strip biaxial, and uniaxial tensile deformations at a von Mises strain rate of 5.45 × 108 s−1, resulting in velocities in the range of 1 to 4.6 m·s−1. A water bridge forming through both phospholipid bilayer leaflets was used to determine structural failure. The stress magnitude, failure strain, headgroup clustering, and damage responses were found to be strain state-dependent. The strain state order of detrimentality in descending order was equibiaxial, 2:1 non-equibiaxial, 4:1 non-equibiaxial, strip biaxial, and uniaxial. The phospholipid bilayer failed at von Mises strains of .46, .47, .53, .77, and 1.67 during these respective strain path simulations. Additionally, a Membrane Failure Limit Diagram (MFLD) was created using the pore nucleation, growth, and failure strains to demonstrate safe and unsafe membrane deformation regions. This MFLD allowed representative equations to be derived to predict membrane failure from in-plane strains. These results provide the basis to implement a more accurate mechano-physiological internal state variable continuum model that captures lower length scale damage and will aid in developing higher fidelity injury models.

Experimental procedure and simplified modeling for the high strain-rate and transient hardness evolution of aluminum AA1050

This work investigates the strain-rate influence on Vickers hardness evolution of a commercially pure aluminum (AA1050). The material hardness properties are assessed by means of laboratory tests employing quasi-static indentation tests performed on samples previously submitted to dynamic plastic compression considering a wide range of strain and strain-rate levels. Total strains of approximately 1.5 are imposed at different strain-rates, ranging from quasi-static to high strain-rate (1.1×104s−1) conditions. The experiments consist of nearly constant strain-rate incremental compressions, decremental strain-rate and strain-rate jump tests. High strain-rate tests are conducted employing a specifically designed impact testing machine. It is observed that both the material hardness and the corresponding hardening-rate increase with loading-rate. However, strain-rate effects are more pronounced at earlier deformations stages. Specific experiments with abrupt (decremental or jump) changes in strain-rate reveal the loading path influence on corresponding material hardness evolution. Specifically, short inverted transients are evidenced from decremental tests. In order to macroscopically describe the experimental material hardness behavior in terms of strain and strain-rate histories, a single scalar internal variable model is formulated. The associated model parameters are calibrated using constant strain-rate experimental data, and the model validation is performed against sequential strain-rate experiments. Overall, the model performs reasonable well within considered strain and strain-rate ranges, while remaining tractable for material identification procedure. However, the inverted transient response observed in decremental tests is not fully well captured, which is clearly a counterpart of operating with a single variable approach. Additionally, the employed methodology can be viewed as an easy-to-perform and low-cost procedure to macroscopically assess the corresponding yield stress and hardening behavior of a metallic material at high strain-rate, since expensive instrumentation and high speed data acquisition systems are not necessary. Its effectiveness will rely on the ability to formulate and calibrate constitutive or empirical correlations between the current yield stress and corresponding material hardness.

A two-stage constitutive model of X12CrMoWVNbN10-1-1 steel during elevated temperature

In order to clarify the competition between work hardening (WH) caused by dislocation movements and the dynamic softening result from dynamic recovery (DRV) and dynamic recrystallization (DRX), a new two-stage flow stress model of X12CrMoWVNbN10-1-1 (X12) ferrite heat-resistant steel was established to describe the whole hot deformation behavior. And the parameters were determined by the experimental data operated on a Gleeble-3800 thermo- mechanical simulation. In this constitutive model, a single internal variable dislocation density evolution model is used to describe the influence of WH and DRV to flow stress. The DRX kinetic dynamic model can express accurately the contribution of DRX to the decline of flow stress, which was established on the Avrami equation. Furthermore, The established new model was compared with Fields-Bachofen (F-B) model and experimental data. The results indicate the new two-stage flow stress model can more accurately represent the hot deformation behavior of X12 ferrite heat-resistant steel, and the average error is only 0.0995.

Finite element analysis of large body deformation induced by a catastrophic near impact event

Finite element simulations of near impacts of terrestrial bodies are presented to investigate possible deformation behavior induced by a massive body during the creation week and/or Genesis Flood. Using the universal law of gravitation, a gravitationally loaded objected is subjected to the ‘pull’ of a near passing fly-by object, and the resulting surface deformations are studied. An Internal State Variable (ISV) pressure dependent plasticity model for silicate rocks (Cho et al., 2018) is used to model the deformation behavior and to capture the history effects involved during the complex surface loading/unloading found in a near impact event. The model is used to simulate the earth and a “fly-by” object interaction and is able to accurately reproduce the internal pressure profiles of the earth and fly-by object. In this context, the fly-by object can be the original Moon, a meteor, or another type of large object that has moved through space to interact with the Earth. Due to the wide range of features that can drive surface deformations during a near impact event, a Design Of Experiments (DOE) methodology was used to independently investigate the influences of five parameters (stationary body size, core material, core/mantle thickness ratio, passing object mass, and passing object distance) concerning surface deformation. The results indicate that the passing body distance, stationary body size, and core/mantle ratio are the most dominant influence parameters on surface deformation. Examination of the ISV parameters of the mantle during deformation shows a complex relationship between the hardening and recovery terms of the model and the resulting plastic strain and surface deformation induced from the near pass event. Surface rise from the near passage of a Moon sized object could be as high as 800 m, in turn causing large tsunamis and possibly causing the Earth’s crust to crack. For this first of its kind study, the conclusions provide understanding of the possible ranges of deformations observed from a near pass event and provides insights into possible catastrophic deformation mechanisms relevant to the young Earth paradigm.

Nonlinear waves in solids with slow dynamics: an internal-variable model.

In heterogeneous solids such as rocks and concrete, the speed of sound diminishes with the strain amplitude of a dynamic loading (softening). This decrease, known as 'slow dynamics', occurs at time scales larger than the period of the forcing. Also, hysteresis is observed in the steady-state response. The phenomenological model by Vakhnenko et al. (2004 Phys. Rev. E 70, 015602. (doi:10.1103/PhysRevE.70.015602)) is based on a variable that describes the softening of the material. However, this model is one dimensional and it is not thermodynamically admissible. In the present article, a three-dimensional model is derived in the framework of the finite-strain theory. An internal variable that describes the softening of the material is introduced, as well as an expression of the specific internal energy. A mechanical constitutive law is deduced from the Clausius-Duhem inequality. Moreover, a family of evolution equations for the internal variable is proposed. Here, an evolution equation with one relaxation time is chosen. By construction, this new model of the continuum is thermodynamically admissible and dissipative (inelastic). In the case of small uniaxial deformations, it is shown analytically that the model reproduces qualitatively the main features of real experiments.

A constitutive model for amorphous shape memory polymers based on thermodynamics with internal state variables

Shape memory polymers (SMPs) are a class of smart materials which can recover a large pre-deformed shape in response to external stimulus. A thermoviscoelastic finite deformation constitutive model incorporated with structural and stress relaxation to capture the material behavior in the vicinity of the glass transition is developed for thermally activated amorphous SMPs in the paper. The incorporation of the Adam–Gibbs model for structure relaxation and the modified Eying model for viscous flow into a thermoviscoelastic finite deformation modeling framework is the main feature of the model. Differing from the traditional phenomenological fictive temperature modeling approach, an internal state variable modeling approach developed recently based on thermodynamics is used in the model. Besides, the temperature dependence of the model parameters is studied in the research. Comparisons between the simulation results and the experimental data show good agreement. Furthermore, the predictability of the model is examined by a parametric study and the results also demonstrate the validity of the model.

Study on the microstructure evolution during radial-axial ring rolling of IN718 using a unified internal state variable material model

Microstructure control during radial-axial ring rolling (RARR) of IN718 is important for increasing the performance in service of IN718 rings. However, RARR is an extremely complex dynamic rolling process with non-uniform local deformation and non-uniform temperature distribution, making the microstructure control difficult. This paper presented an internal state variable (ISV) material model which enables the unified prediction of flow behavior and microstructure evolution during dynamic and post dynamic regime. Based on user defined subroutine, a multiscale finite element (FE) model with adaptive motion control of rolls was established to study the evolution of dislocation density, recrystallized fraction and grain size during RARR of IN718. RARR experiment was conducted to verify the multiscale FE model. The predicted outer diameter of the ring and the radial rolling force as well as the microstructure distribution on the ring cross section were in good agreement with the measured results. The evolution and distribution of ISVs were discussed, and the effect of mandrel diameter, main roll diameter, initial temperature, and rolling ratio on the microstructure evolution of the ring was analyzed. Sensitivity analysis of rolling parameters was conducted. The initial temperature is the most sensitive parameter and the initial temperature should be as high as possible with the pinning of δ phase particle to promote the recrystallization process. It should be careful to increase the rolling ratio when the rolling ratio is higher than 1.667. Because the refine effect of grain structure decreases and the rolling force may increase dramatically due to the low temperature.

An internal state variable thermodynamic model for determining the Taylor-Quinney coefficient of glassy polymers

In this study, a theoretical model was established for predicting the evolution of the Taylor–Quinney coefficient (β) for glassy polymers, according to the internal state variables thermodynamic framework. The proposed model considers both the strain-softening and the strain-hardening processes simultaneously, making it different from and more complex than previous models. It was found that β depends on both the strain and the strain rate. The Taylor–Quinney coefficient first increases to the maximum value and then decreases with increasing strain, which accords with the experimental results for different strain rates. The evolution of the Taylor–Quinney coefficient was caused by the competition mechanism between the strain-softening part and the strain-hardening part. A simple one-parameter method for predicting the temperature increase at different strain rates was proposed, which fits well with the experimental data and can be applied in practice.

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.

Strain Rate and Stress-State Dependence of Gray Cast Iron

An investigation of the mechanical strain rate, inelastic behavior, and microstructural evolution under deformation for an as-cast pearlitic gray cast iron (GCI) is presented. A complex network of graphite, pearlite, steadite, and particle inclusions was stereologically quantified using standard techniques to identify the potential constituents that define the structure–property relationships, with the primary focus being strain rate sensitivity (SRS) of the stress–strain behavior. Volume fractions for pearlite, graphite, steadite, and particles were determined as 74%, 16%, 9%, and 1%, respectively. Secondary dendrite arm spacing (SDAS) was quantified as 22.50 μm ± 6.07 μm. Graphite flake lengths and widths were averaged as 199 μm ± 175 μm and 4.9 μm ± 2.3 μm, respectively. Particle inclusions comprised of manganese and sulfur with an average size of 13.5 μm ± 9.9 μm. The experimental data showed that as the strain rate increased from 10−3 to 103 s−1, the averaged strength increased 15–20%. As the stress state changed from torsion to tension to compression at a strain of 0.003 mm/mm, the stress asymmetry increased ∼470% and ∼670% for strain rates of 10−3 and 103 s−1, respectively. As the strain increased, the stress asymmetry differences increased further. Coalescence of cracks emanating from the graphite flake tips exacerbated the stress asymmetry differences. An internal state variable (ISV) plasticity-damage model that separately accounts for damage nucleation, growth, and coalescence was calibrated and used to give insight into the damage and work hardening relationship.