Deformation Constitutive Model Research Articles

Investigation on the thermal deformation mechanisms and constitutive model of Ti-55511 titanium alloy

This study presents a comprehensive analysis of the hot deformation behavior of the Ti-55511 alloy over a temperature range of 700 °C to 950 °C and strain rates from 0.001 s⁻1 to 1 s⁻1, utilizing isothermal compression testing alongside electron backscatter diffraction (EBSD) and transmission electron microscopy (TEM). The investigation focuses on the influence of various hot deformation parameters on the microstructural evolution of distinct phase regions within the alloy. At higher strain rates (0.1 s⁻1 to 1 s⁻1), the primary softening mechanisms for β grains in the α+β two-phase region are identified as continuous dynamic recrystallization (CDRX) and dynamic recovery (DRV), while in the β single-phase region, discontinuous dynamic recrystallization (DDRX) and DRV predominate. Conversely, at lower strain rates (0.001 s⁻1 to 0.01 s⁻1), the softening of β grains in the two-phase region is primarily governed by DRV, whereas in the single-phase region, DDRX and CDRX are the dominant mechanisms. Additionally, based on stress-strain data corrected for friction and temperature effects, an elastoplastic constitutive model is developed, incorporating strain rate and temperature to predict peak stress in the two-phase regions of the Ti-55511 alloy. This examination of the thermal deformation behavior and microstructural evolution of the Ti-55511 alloy provides significant theoretical and practical insights for optimizing the manufacturing processes of aviation die forging, tailoring microstructures, and enhancing hot working techniques.

Study on Thermal Deformation Constitutive Model and Microstructure of Forged N06625 Nickel-Based Alloy

Study on Thermal Deformation Constitutive Model and Microstructure of Forged N06625 Nickel-Based Alloy

Hot Deformation Behavior and Constitutive Modeling of High Strength Low-Carbon Alloyed Steel Manufactured by Wire and Arc Additive Manufacturing

Hot Deformation Behavior and Constitutive Modeling of High Strength Low-Carbon Alloyed Steel Manufactured by Wire and Arc Additive Manufacturing

A hot deformation constitutive model applicable for complete austenite and dynamic ferrite transformation interval

Dynamic ferrite transformation is a vital microstructure evolution mechanism for carbon alloy steels and an essential technology for manufacturing high-performance steel materials. In practice, carbon steel hot working processes are usually carried out at various temperatures, involving the evolutionary behavior of the single-phase austenite and the ferrite dynamic transformation process. Few models can simultaneously predict the microstructural state and deformation resistance in the single-phase austenitic interval and the dynamic ferrite transformation interval. The deformation mechanisms involve work-hardening, dynamic recovery, and dynamic recrystallization of austenite and ferrite phases, as well as the dynamic transformation of austenite to ferrite and coordinated deformation between the two phases. Therefore, this paper extends the application range of the model based on the viscoplastic unified constitutive model by considering the above deformation mechanisms. The constitutive model proposed in this paper applies to both the single-phase austenite interval and the dynamic ferrite transformation interval. Moreover, it can accurately predict carbon alloy steel's microstructure state and deformation resistance in different intervals. This paper can help to construct a hot deformation constitutive model in a wide range of deformation temperature intervals of steel materials.

A 3D finite deformation constitutive model for anisotropic shape memory polymer composites integrating viscoelasticity and phase transition concept

The phase transition theory of shape memory polymers (SMPs) often involves a phenomenological assumption that the reference configuration of the newly transformed phase deviates from that of the initial phase. This distinction serves as a crucial mechanism in the manifestation of the shape memory effect. However, elucidating the precise definition of the reference configuration of the transformed phase poses a significant challenge in the formulation of the constitutive model. To tackle this challenge, a three-dimensional (3D) finite deformation constitutive model incorporating effective phase evolution for SMPs has been developed. This model merges insights from the classical viscoelastic framework with the phase transition theory. The anisotropic thermo-viscoelastic constitutive model is further developed by introducing hyperelastic fibers, which integrate the anisotropy of the fibers into a continuous thermodynamic framework through structure tensors. Implemented within the ABAQUS software via a user material (UMAT) subroutine, the proposed model has been meticulously validated against experimental data, showcasing its prowess in simulating stress-strain responses and shape memory characteristics of SMPs and their composites (SMPCs). This innovative model stands as an invaluable instrument for the design and of sophisticated SMP and SMPC structures.

Hot deformation and constitutive modeling of a Ti-Al-Sn-Zr-Mo-Cr-Nb alloy

Titanium alloys are the ideal structural materials for automotive and aerospace industries, and hot deformation is widely used to modify the microstructure of the alloy. In this work, a Ti-Al-Sn-Zr-Mo-Cr-Nb alloy was hot deformed at various strain rates and temperatures. The deformation behavior, microstructure, processing maps, and constitutive modeling are investigated. The results show that the hot deformation behavior of the studied alloy is sensitive to the processing parameters, the revised Johnson-Cook model with an average relative error of 1.463 %, and a correlation coefficient of 0.991 can accurately describe the hot deformation behavior of the alloy. Based on the established processing maps, the safe processing areas are 1133–1213 K, 0.004–0.1 s−1 and 1093–1213 K, 0.001–0.004 s−1. With the increased deformation temperature, both the fraction and size of the equiaxed α and lamellar α phases are gradually decreased, and the spheroidization degree of the lamellar α phase gradually decreases too. The strain rates have different influences on the evolution of α and β phases. With the increased strain rates, the fraction of recrystallized and deformed β phase is remarkably decreased, the fraction of sub-structured microstructures of β phase is obviously increased, while the fraction of deformed structures of the α phase is less affected.

Tailoring Stress-Strain Curves of Flexible Snapping Mechanical Metamaterial for On-Demand Mechanical Responses via Data-Driven Inverse Design.

By incorporating soft materials into the architecture, flexible mechanical metamaterials enable promising applications, e.g., energy modulation, and shape morphing, with a well-controllable mechanical response, but suffer from spatial and temporal programmability towards higher-level mechanical intelligence. One feasible solution is to introduce snapping structures and then tune their responses by accurately tailoring the stress-strain curves. However, owing to the strongly coupled nonlinearity of structural deformation and material constitutive model, it is difficult to deduce their stress-strain curves using conventional ways. Here, a machine learning pipeline is trained with the finite element analysis data that considers those strongly coupled nonlinearities to accurately tailor the stress-strain curves of snapping metamaterialfor on-demand mechanical response with an accuracy of 97.41%, conforming well to experiment. Utilizing the established approach, the energy absorption efficiency of the snapping-metamaterial-based device can be tuned within the accessible range to realize different rebound heights of a falling ball, and soft actuators can be spatially and temporally programmed to achieve synchronous and sequential actuation with a single energy input. Purely relying on structure designs, the accurately tailored metamaterialsincrease the devices' tunability/programmability. Such an approach can potentially extend to similar nonlinear scenarios towardspredictable or intelligent mechanical responses.

Cryogenic deformation behavior, constitutive modeling and microstructure evolution of solution-treated 2195 Al–Li alloy at high strain rates

In this paper, the flow behavior, constitutive model and microstructure evolution of solution-treated 2195 Al–Li alloy at temperatures from 123 K to 298 K and strain rates from 2000 s−1 to 5000 s−1 were studied. Experimental results show that the flow stress of the solution-treated 2195 Al–Li alloy is more sensitive to deformation temperature than strain rate. As the temperature declines from 298 K to 123 K, the flow stress at 2000 s−1 and 5000 s−1 increases by nearly 32.8% and 34.5%, respectively; whereas as the strain rate increases from 2000 s−1 to 5000 s−1, the flow stress at 298 K and 123 K only increases by 5.1% and 6.5%, respectively. The average strain hardening rate tends to decline with the increase in strain rate and temperature, and its value at 123 K and 2000 s−1 is about 1.5 times greater than that at 298 K and 5000 s−1. To overcome the very limited applicability of the conventional Arrhenius model with strain compensation, an Arrhenius-based model (c-Arrhenius model) is developed by coupling an independent strain term and then derived by using normalized flow stress and linear regression. To quantify the strain rate effect and temperature effect, a Johnson-Cook-based model (c-Johnson-Cook model) with exponential terms is developed. These two new constitutive models are validated via a comparative analysis, showing good predictability and accuracy. A further microstructure observation shows that the dislocation density is the highest at a strain rate of 5000 s−1 and cryogenic temperature of 123 K, which is beneficial for subsequent aging strengthening.

Deformation behavior, constitutive modeling and microstructure evolution of 2195 Al-Li alloy deformed at high strain rates and cryogenic temperatures

High-speed and cryogenic-temperature forming technologies have exhibited great potential for fabricating advanced Al-Li alloy components for aircraft due to their ability to enhance materials’ formability. However, there remains an urgent need to study the high-accuracy constitutive model and complex deformation mechanism of Al-Li alloys deformed at high strain rates and cryogenic temperatures. In this study, the flow stress behavior and microstructure evolution of 2195 Al-Li alloy were investigated over a wide stain rate range (from 2000 s−1 to 5000 s−1) and temperature range (from 298 K to 123 K). The flow stress increases by ∼ 32% at 2000 s−1 and ∼ 37% at 5000 s−1 as the deformation temperature drops from 298 K to 123 K, but only increases by ∼ 4% at 298 K and ∼ 8% at 123 K as the strain rate increases from 2000 s−1 to 5000 s−1. At the high strain rate of 5000 s−1, the T1 phases exhibit bending (at 298 K) and fracturing (at 123 K), with increased lattice distortion and substructure generation around the intersections of T1 phases. A new constitutive model (g-Johnson-Cook model) is developed, which shows a higher accuracy than both the Arrhenius model and Johnson-Cook model in describing the deformation behavior.

A thermo-mechanical constitutive model for triple-shape and two-way shape memory polymers

Semicrystalline polymers often have diverse molecular configurations in response to the variation of temperature, making it easier to realize multiple or two-way shape memory effects (SMEs). In order to provide guidance for the design of relevant smart structures, it is necessary to develop corresponding constitutive models to better characterize the thermal-mechanical behavior of such shape memory polymers (SMPs). For the first time, a thermo-mechanical finite deformation constitutive model is presented to reveal the deformation mechanism of the triple-shape two-way SME of semicrystalline SMPs in our work, based on the theory of thermodynamics with internal state variables. To verify the validity of the model, the model results are compared with the test data for a typical shape memory cycle. It is found that the model can be employed to describe the non-equilibrium response of semicrystalline SMPs with two types of crystallites in the vicinity of crystallization and melting. Since the model results fit well with the test data, the effectiveness of the model in predicting the triple two-way shape memory behavior is demonstrated.

Investigation on dynamic deformation behavior of 7A75 aluminum alloy over wide strain rates and constitutive model establishment

The aluminum alloy components on automobile often experience high strain rate deformation during accidents. The dynamic deformation behavior and constitutive model are crucial for analysis of vehicle body collisions and failures. In this work, the 7A75 aluminum alloy plate was heat treated according to standard body manufacturing procedures. Quasi-static and dynamic tensile experiments were conducted on the aged 7A75 alloy in a wide strain rate range from 0.001 s−1 to 500 s−1. The dynamic deformation behavior of the alloy was revealed, and the constitutive models were established. It was found that the 7A75 alloy has strong strain rate sensitivity, and its mechanical properties can be significantly improved with the increase of strain rate. As the strain rate increases, the necking instability strain shifts towards higher values, which leads to a larger uniform deformation ability and fracture elongation. This behavior results in a better energy absorption capacity at high strain rates, thus providing a good collision protection property. The constitutive models, including Johnson-Cook (J-C) and Zerilli-Armstrong (Z-A) types, were established and compared. Note that the conventional J-C and Z-A models exhibit severe prediction distortion. Therefore, a modified J-C model incorporating strain rate compensation was introduced, which decrease the errors by 149% and achieves accurate predictions of both dynamic and static deformation behaviors over a wide range of strain rates.

A damage constitutive model of layered slate under the action of triaxial compression and water environment erosion

Based on the macroscopic structure control theory, The slate with a significant bedding plane is a composite rock mass composed of rock blocks containing microscopic defects, joint surface closure elements, and shear deformation elements. Considering the coupling damage effect of water erosion and triaxial compressive load on bedding structure plane, the transversely isotropic damage constitutive model of slate under triaxial compressive load is derived with the dip angle of bedding and confining pressure as the variable. Firstly, based on the statistical theory of continuous damage mechanics and the maximum tensile strain criterion, the transversely isotropic deformation constitutive model of rock block with micro-defects is given; Secondly, based on the phenomenological theory of closed deformation and shear-slip deformation mechanism of layered structural plane under the coupling action of water erosion and triaxial compression load, the calculation formula of axial deformation of layered structural plane under the coupling action is given; Finally, to verify the accuracy of the established constitutive model, triaxial compression tests are carried out to study the influence of dip angle and confining pressure on the macroscopic mechanical properties and mechanism of slate. The results show that: the established triaxial compression damage constitutive model of bedding slate can accurately describe the stress–strain relationship of bedding slate after water environment erosion. With the increase of bedding dip angle, the strength and deformation capacity of the bedding slate first decreases and then increases, showing a U-shaped distribution as a whole. There are three main types of failure: tension shear composite failure, shear slip failure, and splitting tension failure.

Dynamic mechanical behaviors of pearlitic U71MnG rail steel: Deformation mechanisms and constitutive model

Pearlitic rail steels with excellent mechanical and wear properties are widely employed for high-speed railways. During the service period, the presence of track irregularities, switches, as well as crossings can cause intense dynamic loadings, which accumulate the severe deformation of rails, and sometimes even facilitate crack initiation and failure. Knowledge of the dynamic mechanical response and damage behaviors of the rail steels is therefore important for the service performance estimation of railways. In this study, the pearlite U71MnG rail steel was deformed using a split Hopkinson pressure bar (SHPB) apparatus to reach a wide range of strain rates and temperatures. Microstructures including the grain morphology, deformation characteristic of cementite lamellae, dislocation substructures in ferrite lamellae, and phase transformation at pearlite colony boundaries were characterized. Based on the microstructural observations, the rate-controlling strengthening mechanism and temperature-dependent softening mechanism of the steel were revealed when the process for the dynamic loading-induced cementite decomposition at boundaries was discussed. In consideration of the athermal deformation mechanism, thermally activated mechanism, and viscous-drag effect, a physically based constitutive model was proposed to describe the dynamic mechanical response of the pearlite rail steel. The predicted flow stresses were found to be consistent with the experimental one for all strain rates and temperatures.

Plastic deformation mechanisms and constitutive modeling of WE43 magnesium alloy at various strain rates and temperatures

In this study, we investigated the mechanical behavior and deformation mechanisms of as-cast and as-rolled WE43 at various strain rates and temperatures. Compression tests were carried out at ambient temperature under 900/s, 1400/s, and 2000/s, and at temperatures of 100 °C, 200 °C, 250 °C, and 300 °C under 1400/s, respectively. The results showed that, with the rise of strain rate, the deformation mechanism of as-cast and as-rolled WE43 both changed from twinning dominant combined with slipping to slipping dominant. As the temperature increased, the deformation mechanism of WE43 changed from twinning to slipping, controlled by non-basal slips at various strain rates and temperatures. As for as-rolled WE43, it was observed that intense dynamic recrystallization and adiabatic shear bands (ASBs) occurred under high strain rate and high temperature conditions. Moreover, this study employed the Johnson-Cook (JC) model to formulate the plastic deformation model of the WE43 alloy under conditions of elevated temperatures and high strain rates. And a dynamic mechanical analyzer (DMA) was successfully employed to refine the accuracy of elastic modulus.

Deformation Characteristics and Constitutive Model of Construction and Demolition Waste Stabilized with Alkali-Activated Fly Ash

This study employed triaxial compression tests to investigate the deformation characteristics and damage evolution of construction and demolition waste (C&D) stabilized with alkali-activated fly ash (FA). Furthermore, the study explored the mechanisms of microcracks initiation, propagation, and the stress-strain behavior of stabilized C&D under varying stress conditions. A strain-softening damage model was employed to investigate the mechanism of damage evolution in the specimens. The test results revealed that the internal damage process of the stabilized recycled concrete aggregate (RCA) exhibited five stages: elastic recovery, damage initiation, damage acceleration, damage deceleration, and damage completion. The confining pressure had a significant influence on the deformation characteristics of the specimens. Additionally, the deformation characteristics of the stabilized C&D and rock materials exhibited remarkable similarities. By employing a model of rock strain softening and intrinsic structural damage the damage evolution trends and the equation of the intrinsic structure of the main component RCA-FA. The results demonstrated a good agreement between the test data and the predictions of the proposed constitutive model.

A Novel Deformation Analytical Solution and Constitutive Model for Fractured Rock Masses

In order to study the deformation and stability of a fractured rock mass, existing research suggests that fracture deformation, which is usually obtained by evaluating the equivalent deformation modulus, dominates the deformation of a fractured rock mass. However, this parameter is difficult to obtain in theory and practice, which limits the application of rock mass deformation analysis methods. In order to calculate the deformation of fractured rock masses, the mass of the rock is regarded as a sponge-like material, and it is assumed that the deformation of a water-saturated fractured rock mass under external force load is approximately equal to the net flow of fracture water. Based on this assumption, firstly, the relationship between rock mass deformation and fracture flow is studied through a single-fracture rock mass model. The hydraulic properties of the fracture are characterized by the permeability coefficient, and the fracture deformation in the rock mass is equivalent to the fracture flow. The fracture deformation calculation formula is derived from the fracture hydraulics calculation formula, and this is compared with the measured data. The rationality of the calculation formula was verified. On this basis, the calculation formula for rock mass deformation, including multiple groups of fracture surfaces, is proposed, and the stress-strain constitutive relationship of a complex rock mass is established. The correctness of the calculation method was verified by comparing it with other theoretical calculation results.

Study on thermal deformation constitutive model of NiCoFeAlCrMo high-entropy alloy with FCC / L12 coherent structure

Due to its multi-principal component characteristics, high-entropy alloys exhibit excellent comprehensive mechanical properties and become a new generation of engineering structure alternative materials. At present, a single constitutive model is used to describe the change of flow stress in the study of hot deformation behavior of high-entropy alloys, and there are some problems such as inaccurate prediction accuracy or inapplicability of the model in some hot processing intervals. In this paper, three different constitutive models are combined to fully consider the applicability of the model in different temperature ranges, and the change of flow stress behavior is accurately described, which is of great significance for subsequent deformation mechanism research, hot processing process optimization, numerical simulation and so on. Thermal compression experiments on cast Ni28.5Co18.6Fe18.2Al16.3Cr10.5Mo4 Ti2.3Nb0.6W0.9C0.2 high-entropy alloy were carried out using a Gleeble-3800 thermal simulation tester with deformation temperatures in the range from 900 ℃ to 1150 ℃ and strain rates from 0.001 s−1 to 1 s−1. The constitutive relationships between true stress and strain, temperature and strain rate were established, while the heat deformation process of this alloy under different deformation conditions by Arrhennius, Modified Johnson-Cook and Modified Fields-Backofen models were characterized. The results indicate that Arrenius model describe the rheological behavior of the alloy better in a wider temperature range and at a wider strain rate range. Comparatively, the model shows the highest prediction accuracy in the high-temperature region and slightly lower accuracy in the mesothermal and low-temperature regions. The modified J-C and F-B models have better prediction accuracy in the low-temperature and medium-temperature regions, respectively. Due to the different deformation conditions, dislocation configurations such as dislocation tangles, dislocation networks, and subcrystalline junctions can be observed in the plastic deformation region. The dislocation density of this alloy increases with increasing strain rate and decreasing deformation temperature after deformation.

Dynamic deformation behavior and constitutive model of a Zr–W alloy

In this paper, a Zr–5W alloy was fabricated via casting. In order to obtain the mechanical properties of the material, quasi-static compression tests at room temperature and split Hopkinson pressure bar tests at various temperatures were carried out. The x-ray diffraction result showed that the main components of the alloy were αZr and W2Zr, where αZr is the matrix and W2Zr is the reinforcement. The metallographic characterization results showed that there were two main forms of W2Zr in the material, namely, large particle boundary and small diffuse submicrometer precipitates. The reinforcements of both distributions have the effect of increasing the strength of the material, but the small submicrometer W2Zr precipitates would cause microcrack nucleation during the late plastic deformation stage, resulting in damage softening. In order to make theoretical calculations of the mechanical properties of materials, the Johnson–Cook (JC) constitutive model and Zerilli–Armstrong (ZAM) constitutive model of the material were obtained. It was found that the JC constitutive model had poor consistency in describing material properties. Although the consistency of the ZAM constitutive model was higher than that of the JC constitutive model, it still had obvious shortcomings. Combined with the deformation mechanism of the alloy, a modified constitutive relation was established by adding damage softening terms based on the hexagonal close-packed metal constitutive model inferred by the kinetics of heat-activated dislocations. The relative error results of all working conditions show that the correlation consistency of the improved constitutive model in this paper is significantly better than that of JC constitutive and ZAM constitutive.

Deformation failure mechanism and constitutive model of gradient aluminum foam under impact loading

The dynamic mechanical response and energy absorption characteristics of gradient aluminum foam under impact loading with strain rates of 150 s−1, 340 s−1, and 550 s−1 have been studied based on the Split Hopkinson Pressure Bar (SHPB) test in this paper. X-CT and numerical simulations were carried out to analyze the deformation of the micro-porous structure of gradient aluminum foam during impact compression. A constitutive model for gradient aluminum foam under impact load was further established. The results show that the mechanical properties of aluminum foam are positively correlated with the strain rate. The better energy absorption effect of gradient aluminum foams than the homogeneous aluminum foam under impact loading was confirmed. Negative gradient aluminum foam exhibited a 46.9 % higher specific energy absorption rate than homogeneous aluminum foam. The failure mode of homogeneous and positive gradient aluminum foam was observed to be 'from the proximal to the distal', whereas negative gradient aluminum foam exhibited a 'from the distal to the proximal' failure mode. The Sherwood-Frost model, considering the shape, density, and strain rate function correction, could accurately predict the impact stress–strain curve of gradient aluminum foam, which can be served as a reference for the optimal design of gradient aluminum foams.

Is the Jaumann Stress Rate Applicable? — Discussions on Choosing Proper Rate Expressions of a Finite Deformation Constitutive Model

The rate form constitutive expression plays an important role in finite deformation theory and is widely used in the finite element method (FEM) software. However, the choice of objective stress and strain rates in constitutive expressions has caused many controversies. This paper aims to clarify these controversies by distinguishing the related concepts: constitutive behavior, constitutive model and constitutive expression, and discuss the influence of constitutive model choices and constitutive expression choices separately. As a certain constitutive model may correspond to many different rate form constitutive expressions, the conversion relationships among them are derived, which can help clarify many controversies. First, it is proven that three different stress–strain curves in the controversial simple shear example actually correspond to three constitutive models. For each constitutive model, the corresponding different rate form constitutive expressions yield consistent calculation results. We also demonstrate that the applicability of a constitutive model is completely independent of the choice of objective stress rates in the constitutive expressions. Second, we find that for FEM simulation, the element tangent stiffness matrix can serve as an indicator to evaluate the validity of different constitutive expressions and their corresponding FEM formulations. Moreover, even though non-work-conjugate stress-strain rates are adopted, such as the Jaumann stress rate/deformation rate, different rate form constitutive expressions can correctly reflect the constitutive model with the proper tangent modulus tensors. All these conclusions have been verified by examples. For convenience, we recommend using the constitutive expressions in terms of the second Piola–Kirchhoff stress versus Green’s strain tensor and Truesdell stress rate versus deformation rate for experimental measurements and FEM simulations. Of course, other objective stress rates are all applicable and have no effect on the calculation results as long as the correct conversion relationships are used.