Constitutive Model Research Articles

A spatiotemporal correlation and attention-based model for pipeline deformation prediction in foundation pit engineering

In foundation pit engineering, the deformation prediction of adjacent pipelines is crucial for construction safety. Existing approaches depend on constitutive models, grey correlation prediction, or traditional feedforward neural networks. Due to the complex hydrological and geological conditions, as well as the nonstationary and nonlinear characteristics of monitoring data, this problem remains a challenge. By formulating the deformation of monitoring points as multivariate time series, a deep learning-based prediction model is proposed, which utilizes the convolutional neural network to extract the spatial dependencies among various monitoring points, and leverages the bi-directional long-short memory unit network to extract temporal features. Notably, an attention mechanism is introduced to adjust the trainable weights of spatial-temporal features extracted in the prediction. The evaluation of a real-world subway project demonstrates that the proposed model has advantages compared with current models, particularly in long-term prediction. It improves the Adjusted R2 index averagely by from 19.4 to 61.6%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} compared with existing models. The proposed model also exhibits a decrease in mean absolute error ranging from 51.5 to 70.3%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} compared to others. Experiments and analyses verify that the spatial-temporal dependencies in time series and the attention learning for spatial-temporal features can improve the prediction of such engineering problems.

A Cam Clay constitutive relation for semi-analytical elasto-plastic modeling of wheel-soil interaction for fast applications

In this paper, a semi-analytical elasto-plastic (SAEP) model is proposed to simulate wheel motion on soft soil using a Modified Cam Clay (MCC) constitutive model. The focus is on enhancing computational efficiency by integrating the MCC model into the framework. The MCC-based model outperforms the Drucker-Prager with Cap hardening (DPC)-based model in terms of reduced computation time, stability, and convergence. Simulations with a 30% slip ratio show a significant reduction in solution time compared to the DPC-based model. For slip ratios of 50% and 90%, the computation time decreases even further. Overall, the MCC-based model demonstrates over 100 times reduction in the solution time compared to the DPC-based model, particularly with high slip ratios. Indeed, the DPC-based model was found to be highly sensitive to high values of slip ratios, whereas the MCC-based model is not. In comparison with existing methods in the literature, the proposed formulation is superior in terms of significantly reduced solution time and improved numerical stability, making it ideal for advanced applications where fast solutions are crucial.

Tensile properties and constitutive modeling of Kevlar29 fibers: from filaments to bundles

This paper experimentally investigates the tensile properties of Kevlar29 filaments and fiber bundles, revealing the size and strain rate effects on the mechanical properties of fiber bundles. The fracture modes and mechanisms of the fibers were analyzed, and viscoelastic constitutive models at the fiber bundle scale and constitutive models for filaments-bundles based on the Weibull distribution were established. The results show that Kevlar29 fiber bundles exhibit significant strain rate effects: as the strain rate increases, tensile strength and modulus increase, while fracture strain and toughness decrease. Under quasi-static loading, the fracture modes of the fibers are mainly fibrillation or shear flow, but as the strain rate increases, the failure modes shift towards brittle fracture. Both constitutive models can accurately predict the tensile properties of fiber bundles at different strain rates. The accuracy and applicability of the filament-to-bundle constitutive model were verified through numerical simulations, demonstrating that the model better describes the size effect of fibers and quantifies the damage to the fiber bundles.

Comprehensive analysis of mode-I cracking in ice: Exploring full-range rate dependency

Ice, as a crystalline material, demonstrates a unique response in its mode-I fracture toughness to variations in loading rates, characterized by an initial decrease and subsequent increase in toughness. This phenomenon has been explored in limited studies. In this work, an extensive numerical analysis of mode-I ice cracking is conducted by using the distinct lattice spring model (DLSM). Norton-Bailey Drucker-Prager DLSM (NB-DP-DLSM) is initially employed to investigate the classical explanations of creep and stress relaxation for the anomaly observed in ice’s fracture toughness at lower loading rates, but this approach does not successfully replicate the experimentally observed strain rate dependency. Then, two rate-dependent constitutive models are introduced to further examine the mode-I fracture mechanics of ice. Our numerical simulations of three-point bending tests show that the relationship between fracture toughness and strain rate at lower levels is more accurately captured by rate-dependent models. For higher strain rates, our numerical modeling of notched semi-circular bending tests indicates that a rate-independent constitutive model can replicate the loading rate dependency of ice’s mode-I fracture toughness. In conclusion, these observations suggest that a reverse-stage-like dynamic constitutive model for DLSM can potentially capture the full range of loading rate dependencies observed in ice.

Constitutive modeling of creep behavior considering microstructure evolution for directionally solidified nickel-based superalloys

During creep at elevated temperatures, the performance of directionally solidified nickel-based superalloys experiences progressive degradation, accompanied by significant microstructure evolution. In this study, creep tests of varying durations were conducted on smooth specimens, revealing typical microstructure evolution, including dissolution, coarsening, and rafting of the γ′ phase. The process of microstructure evolution during creep was precisely quantified utilizing an advanced image processing technique. Subsequently, a phenomenological model was formulated to predict the evolution of the γ/γ′ microstructure. Furthermore, with the introduction of the microstructure evolution model, a multiscale creep constitutive model was established within the framework of crystal plasticity. This model encompasses various dislocation strengthening mechanisms, including dislocation bypassing, dislocation pairs shearing, and dislocation hardening. The constitutive model can accurately describe both the microstructure evolution and creep deformation of the DZ406 superalloy at various temperatures, with maximum errors of 18.13% and 24.31%, respectively. Finally, the model under multiaxial stress conditions was validated through creep tests on specimens with a film-cooling hole. The maximum prediction errors for microstructure evolution and creep life were 30.46% and 28.00%, respectively.

Comparison of the mechanical properties and constitutive models of carbon fiber-reinforced coral concrete cubes and prisms under uniaxial compression

Carbon fiber reinforced coral concrete (CFRCC) plays a crucial role in island construction. In engineering applications, CFRCC is often subjected to complex stress states. Multiaxial stress studies usually use cubes, and in order to maintain research consistency, uniaxial studies also often use cubes. Traditional uniaxial stress studies mainly use prisms. However, the differences in uniaxial compression mechanical properties between these two types of specimens have not been thoroughly studied. To fill this gap, this study conducted uniaxial compression tests on cubes and compared the results with previous tests on prisms.The results showed that the failure patterns of the two types of specimens were different. The main cracks of the cubes were vertically distributed, while the main cracks of the prisms mostly unfolded diagonally. The addition of carbon fibers (CFs) reduced the number of large cracks and increased the number of microcracks. The stress-strain curves, peak stress, peak strain, and initial elastic modulus of the two types of specimens show similar variations with CFs, and CFs significantly improve the mechanical properties of coral concrete. The constitutive models based on prisms cannot accurately describe the mechanical behavior of cubes, therefore, we proposed revised constitutive models. The revised constitutive models can accurately describe the mechanical behavior of cubes. This study laid the foundation for subsequent research on multiaxial stress.

A multi-linear constitutive relation considering the temperature effect on quasi-static mode I delamination in UD/MD laminates

In this study, a multi-linear constitutive relation taking into account temperature and fiber bridging is proposed for characterizing delamination behavior in composite laminates under various temperature conditions. An approach combining analytical solution and J-integral is also established for determining the cohesive parameters in the multi-linear constitutive relation. To validate the proposed constitutive relation, mode I quasi-static delamination experiments of unidirectional (UD) and multidirectional (MD) carbon/bismaleimide laminates are carried out at 25 ℃ (room temperature), 80 °C and 130 ℃. The experimental results show that the increasing temperature resulted in a monotonic increase in the fracture toughness of the UD laminates while affect the fracture toughness of MD laminates slightly. A FE model is established with the implementation of the proposed multi-linear constitutive relation using UMAT subroutine. Good agreements between the experimental and simulated results demonstrate the validity of the proposed constitutive relation, with the relative difference of peak load between predicted and experimental values less than 8.2 % and the relative difference of initial and steady-state fracture toughness between predicted and tested results less than 15 %. This study provides the possibility to numerically study the temperature effect on the delamination behavior of laminates and has promising applications in the damage tolerance design of composite structures.

OXFORD-UMAT: An efficient and versatile crystal plasticity framework

The crystal plasticity-based finite element method is widely used, as it allows complex microstructures to be simulated and allows direct comparison with experiments. This paper presents the OXFORD-UMAT for Abaqus®, a novel crystal plasticity code that is publicly available online for researchers interested in using crystal plasticity. The model is able to simulate a wide range of materials and incorporates two different solvers based on the solution of slip increments and Cauchy stress with variants of state update procedures including explicit, semi-implicit, and fully-implicit for computational efficiency that can be set by the user based on the application. Constitutive laws are available for a range of materials with single or multiple phases for slip, creep, strain hardening, and back stress. The model includes geometrically necessary dislocations that can be computed using finite element interpolation functions by four alternative methods, including the total form with and without a correction for the dislocation flux, a widely used rate form, and a slip-gradient formulation. In addition, the initial strengthening and subsequent softening seen in irradiated materials can also be simulated with the model. The analysis is available in 2D (plane stress and plane strain) and 3D, including linear and quadratic elements. Here we include full derivations of the key equations used in the code and then demonstrate the capability of the code by modeling single-crystal and large-scale polycrystal cases. Comparison of OXFORD-UMAT with other available crystal plasticity codes for Abaqus® reveals the efficiency of the proposed approach, with the backup solver offering greater versatility for handling convergence issues commonly found in practical applications.

Experimental investigation on the post-fire mechanical properties of YSt-355FR (0.1 % Mo) low-alloyed fire-resistant steel

Fire-resistant steel (YSt-355FR (0.1 % Mo)) has become the most utilized structural steel due to its superior performance at elevated temperatures compared to traditional structural steel at both the material and structural levels. However, its performance after exposure to fire remains questionable. This study investigates the post-fire performance of cold-formed fire-resistant steel at the material level. A total of 132 steel coupons and 84 Charpy test specimens were exposed to temperatures ranging from 100°C to 1000°C. These temperature-exposed steel coupons were subjected to four different cooling conditions: furnace cooling (FC), air cooling (AC), water-jet cooling (WJC), and water cooling (WC). Tensile tests were performed on the cooled steel coupons to obtain post-fire stress-strain curves, from which the residual mechanical properties were determined. Charpy impact tests were also conducted on the cooled steel specimens to obtain impact energies and identify the fracture patterns. Key observations regarding the post-fire behavior of fire-resistant steel in terms of strength, deformability, stiffness, and toughness parameters were made. Both the exposure temperature and the cooling condition significantly influence the post-fire performance of fire-resistant steel. Fire-resistant steel demonstrates superior post-fire behavior compared to traditional structural steel. Constitutive equations were proposed to predict the mechanical properties of fire-resistant steel under various post-fire conditions, and a sound reliability analysis was performed to examine the reliability of the predicted design parameters compared to the test results. The findings presented in this paper are significant as no prior investigation has assessed the post-fire performance of fire-resistant steel; thus, they may encourage the steel construction industry to prefer fire-resistant steel over traditional structural steels and promote the reuse of temperature-exposed fire-resistant steel for sustainable construction.

A rate-dependent crack bridging model for dynamic fracture of CNT-reinforced polymers

Carbon nanotubes (CNTs) improve the fracture toughness of polymer-based matrix composites by bridging the crack growth path. This research presents a finite element (FE) rate-dependent crack bridging model of Mode-I dynamic crack growth in CNT-reinforced polymers accounting for rate of loading and rapid crack growth. Two distinct CNT bridging and matrix cracking damage mechanisms are taken into account in the fracture process zone (FPZ) accounting for the dissipation of fracture energy. A viscoelastic-viscoplastic material model is adapted for the matrix phase to capture the strain rate effects. The crack in the matrix phase is modeled using cohesive zone elements characterized by experimental data. A rate-dependent traction-separation law obtained from CNT pull-out simulation at relevant crack opening speeds is used to simulate the CNT bridging in the FPZ. Considering the given traction-separation law as a constitutive equation, the CNTs are replaced with non-linear spring elements to facilitate the FE simulation of crack bridging with numerous CNTs in the FPZ. The proposed rate-dependent FE model for crack bridging enables the study of the effect of key CNT factors such as length, orientation, waviness, volume fraction, and agglomeration on fracture energy dissipation at various crack speeds. The developed model can simultaneously consider the interactive effects of all CNT parameters which is useful for considering all processing-induced uncertainties in analyzing the effects of CNTs on the dynamic fracture toughness of nanocomposites.

Data-driven homogenisation of viscoelastic porous elastomers: feedforward versus knowledge-based neural networks

A computational framework is established to implement time-dependent data-driven surrogate constitutive models for the homogenised mechanical response of porous elastomers at large strains. The aim is to enhance the computational efficiency of multiscale analyses through the use of these surrogate models. To achieve this, explicit finite element (FE) simulations are conducted to predict the homogenised response of a cubic unit cell of a porous elastomer, using two different viscoelastic descriptions of the parent material, subject to pseudo-random, multiaxial, non-proportional histories of macroscopic strains. The histories of homogenised variables extracted from each set of FE predictions form a training dataset, which is used to train two different surrogate models, both relying on artificial neural networks (NNs). The first model predicts the increment in macroscopic stress over a simulation step, as a function of the macroscopic stress and strain at the beginning of the step, of the prescribed macroscopic strain increment, and of the corresponding time increment. The second model uses the same inputs and outputs but tests a knowledge-based modelling approach: it relies on the aid of an additional nonlinear elastic constitutive model, which is time- and path-independent and known a priori. This model represents an existing base of knowledge which is augmented and corrected by an NN after training on viscoelastic data. The data-driven surrogate model, therefore, learns the viscoelastic behaviour of the unit cell starting from knowledge of its elastic response. The two surrogate models are found to have comparable and very high accuracies at capturing the homogenised unit cell response to highly complex loading cases across a range of viscoelastic properties. Hyperparameter optimisation shows that the second, knowledge-based model requires a simpler NN and therefore incurs a smaller computational cost.

Constitutive model and deterioration mechanisms of uniaxial compressive behavior for calcium aluminate cement-based UHPC under simulated fire temperatures

Constitutive model and deterioration mechanisms of uniaxial compressive behavior for calcium aluminate cement-based UHPC under simulated fire temperatures

Studying the Constitutive Model of Damage for a Stainless Steel Argon–Oxygen Decarburization Slag Mixture

The purpose of this study was to fully explore the mechanical properties of five different doses of an Argon–Oxygen Decarburization slag mixture in an unconfined compressive strength test. The peak stress, elastic modulus, and stress–strain curve of the mixture were studied for 90 days. Based on the experimental data and according to the theory of damage mechanics, the concept of damage threshold (t) was introduced to construct a damage constitutive model. Referring to the damage threshold of concrete, that of the mixture was determined to be 0.7 times higher than the peak strain, and the correlation coefficient between the established model and the test curve was above 0.85. These results indicate that the addition of AOD slag and fly ash can cause hydration reactions, increase the quantity of hydration products, and enhance the peak stress and elastic modulus of the mixture. The maximum increases were 94.9% and 43.1%, respectively. Parameters a and b reflect the peak stress and brittleness of the mixture, respectively. The incorporation of Argon–Oxygen Decarburization slag can make the mixture less brittle and improve its properties. The incorporation of Argon–Oxygen Decarburization slag can protect the mixture from damage. The maximum decrease is 40.2%.

Effect of the flow behavior and dynamic recrystallization of EH420 marine steel on the morphology evolution of martensite substructures

The hot deformation behavior of EH420 marine steel was investigated by isothermal compression tests in the temperature range of 850~1050°C and the strain rate range of 0.01~10s−1. A high-temperature constitutive model and a hot processing map were developed. The results indicated that the microstructure of EH420 marine steel after hot deformation was mainly bainite and lath martensite. The martensite blocks gradually coarsened with the increase in temperature within the range of 950~1050°C/0.01s-1, and the average width increased from about 0.7 μm to 2.3 μm. At high strain rates, martensite blocks became coarse and disordered due to flow instability. The flow stress curve showed a single peak when the strain rate was low. There were more dynamic recrystallization (DRX) structures, resulting in fine and ordered martensite blocks. Under the deformation conditions of 0.1~10s-1/1000°C, the proportion of high-angle grain boundaries (HAGBs) decreased from 53.5% to 40.7% as the strain rate increased. Meanwhile, the hot deformation mechanism shifted from discontinuous dynamic recrystallization (DDRX) to the combined effect of DDRX and dynamic recovery (DRV) and finally transformed into DRV. The flow stress derived from the constitutive model was consistent with the experimental results. The activation energy of EH420 marine steel was 3.16×105J/mol. The optimal hot processing parameters were 950~1050°C and 0.01~0.1s-1.

Anisotropy cyclic plasticity constitutive modelling for Ni-based single-crystal superalloys based on Kelvin decomposition

Nickel-based single-crystal (SC) superalloys exhibited excellent exceptional mechanical properties at high temperatures due to the elimination of internal grain boundaries, contributed a strong orientation-dependent material response. The anisotropy of SC superalloys was modeled viscoplastically from a macroscopic viewpoint based on the Kelvin decomposition theory [1] which was a decomposition of the stress space according to the elastic matrix eigen-directions to control the viscoplastic flow by Kelvin stress decoupled from each other. Compared to the classical phenomenological macro model, the proposed model effectively captures the slip deformation mechanism of SC superalloys with the inherent ability to simulate anisotropic because of the two criterions framework controlled by Kelvin stress. Compared with others, the proposed model was able to simulate time-dependent inelastic deformation and cyclic deformation behavior under complex loading. The kinematic hardening and isotropic hardening models incorporated microscopic quantities, such as dislocation density and channel phase width, connecting the macroscopic mechanical response with the microscopic state to achieve multiscale constitutive modelling. The parameter identification and finite element implementation were conducted on a SC superalloy [2]. Simulation results demonstrated the accuracy of the proposed model in predicting deformation behavior under various orientations, rate-dependent effects, isothermal and non-isothermal cyclic deformation. Comparison with the classical anisotropic matrix macroscopic phenomenological approaches highlights the superior capability of the proposed model to simulate the orientation-dependent mechanical properties of single-crystal alloys.

Low-velocity impact of castor oil-based rigid polyurethane foams: Experiments, microstructure effects and constitutive modelling

Low-velocity impact of castor oil-based rigid polyurethane foams: Experiments, microstructure effects and constitutive modelling

Multiaxial low cycle fatigue behavior and constitutive model of 316L under various loading paths at high-temperature

The work is devoted into investigating the multiaxial low cycle fatigue behavior and constitutive model of 316L under various strain amplitudes, strain ratios, and phase angles at 550 °C. Experimental results show that both axial and shear stress amplitudes present three stages of cyclic hardening, softening and fracture. Internal stress analysis reveals that initial cyclic hardening is influenced by both friction and back stresses, while cyclic softening is primarily controlled by friction stress. Moreover, the Mises equivalent stress–strain relationship effectively accommodates different strain amplitudes and strain ratios, but cannot account for the non-proportional hardening arising from back stress. Pearson correlation analysis highlights a correlation between fatigue life and the equivalent stress amplitude and plastic strain energy density, and that elastic modulus is influenced by strain ratio and phase angle, not the strain amplitude. Based on the Chaboche unified viscoplastic constitutive theory, an improved constitutive model incorporating new hardening rules and Hooke’s law is proposed. In the proposed model, three classical loading path-dependent coefficients’ ability for description of non-proportional hardening and stiffness weakening behaviors are evaluated. Simulation results reveal that the proposed model can effectively capture the non-proportional hardening of back stress, stiffness weakening, non-masing effect, and varied softening rate.

A modified Holmquist‒Johnson‒Cook (HJC) constitutive model and its application to numerical simulations of explosions and impacts in rock materials

A modified Holmquist‒Johnson‒Cook (HJC) constitutive model and its application to numerical simulations of explosions and impacts in rock materials

Mechanical properties and constitutive model of deep marble under alternating creep-fatigue action

Given the time dependence and susceptibility to disturbance of deep rocks, this paper analyzes mechanical properties, acoustic emission (AE) signals, and pore radius distribution characteristics of deep marble under alternating creep-fatigue (ACF) action at different confining pressures by experiment and theoretical analysis. The results show that: (1) the alternation of creep and fatigue accelerates the rock damage, and the number of alternations increases with the increasing confining pressure. (2) Strain increments exhibit a pattern of first decreasing and then increasing, and they undergo a transformation from “V”-type to “U”-type with the increase of confining pressure. (3) The AE ringing count is mainly generated in the fatigue segment. The radius of large pores inside the rock is constrained by the confining pressure. (4) The ACF constitutive model is established, where model parameters show a nonlinear decreasing trend of the power exponential function with the strain increment.

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.