Phenomenological Constitutive Model Research Articles

Phenomenological Constitutive Model and Physical Constitutive Model of Hot Deformed Dual-Phase Ti-5Al-4Sn-2Zr-5Mo Alloy

Phenomenological Constitutive Model and Physical Constitutive Model of Hot Deformed Dual-Phase Ti-5Al-4Sn-2Zr-5Mo Alloy

Grain disintegration and dynamic recrystallization during impact tests of additively manufactured nickel-based alloy 718

The high-temperature dynamic mechanical response of Alloy 718 produced via laser-powder bed fusion (LPBF) was investigated through compressive Split-Hopkinson Pressure Bar (SHPB) tests. Simulating the typical service conditions of Alloy 718, the tests were conducted at temperatures ranging from 298 K to 773 K and at strain rates of 1000 s−1 to 1500 s−1. Phenomenological material constitutive models, such as the modified versions of Johnson-Cook and Hensel-Spittel models, were developed based on the SHPB test results. Analysis of microstructural evolution under impact conditions highlighted that columnar grains with high Schmid factors tend to undergo preferential activation and dislocation pile-up. This process leads to the formation of adiabatic shear bands, grain disintegration, and intense lattice rotation, particularly at higher strain rates. Furthermore, increasing the dynamic deformation temperature facilitates the activation of discontinuous dynamic recrystallization (DRX), with strain accumulation promoting localized grain nucleation along heavily dislocated dendritic boundaries. Recognizing the limitations of phenomenological material constitutive models in accurately representing the underlying microstructural evolution, an artificial neural network (ANN)-based constitutive model employing a three-layer backpropagation learning algorithm was implemented, reducing the Average Absolute Relative Error (AARE) to 0.17%.

Nonlinear geometrically exact dynamics of hyperelastic pipes conveying fluid: Comparative study of different hyperelastic models

This work presents a comparative analysis of the nonlinear dynamic responses of fluid-conveying pipes modelled by four different phenomenological hyperelastic constitutive models, namely the Mooney-Rivlin model, the Neo-Hookean model, the Yeoh model, and the Ogden model. Based on the strain energy density functions and the exact geometric relation, the geometrically exact dynamic models corresponding to different hyperelastic contitutive models are established. Then the Galerkin's truncation is adopted to transfer the obtained nonlinear integral-partial differential equations into the nonlinear ordinary differential equations. Subsequently, linear dynamic study and finite difference method are taken to explore the natural frequencies and stabilities and nonlinear dynamic responses of the pipe, respectively. And the fitting coefficients for various models obtained from Treloar's experiment data are utilized in the numerical calculation. It is found the Hopf bifurcation and the limit cycle oscillations in the post-flutter status are influenced by the choice of the hyperelastic models, particularly under large flow velocity. However, hyperelastic constitutive models do not exert a significantly extreme influence on the results, which can be attributed to the relatively small strains even though large deformations occur in the pipe. Mathematically, it is proven that the symmetry observed in the pipe's displacement response stems from the symmetry of its cross-section and the inextensibility assumption of the centerline, with hyperelasticity having no effect on this symmetry.

A comparative study of hot tensile deformation behavior of 6016 aluminum alloy under LSTM neural network and Arrhenius model

Abstract The isothermal tensile test of 6016-T6 aluminum alloy was carried out on Gleeble-3500 at the temperature range of 400 °C–550 °C and the strain rate range of 0.01–10 s−1. The results show that the thermal deformation mechanism of 6016-T6 is dynamic recovery and dynamic recrystallization. In this paper, the phenomenological Arrhenius constitutive model and the data-driven WOA-LSTM constitutive model for predicting the hot tensile deformation behavior of 6016-T6 aluminum alloy were studied in contrast. The whale optimization algorithm was used to optimize the hyperparameters of LSTM neural network to improve the prediction accuracy of flow stress. The optimization results show that the optimal hidden layer node, maximum training period, initial learning rate and mini batch size of WOA-LSTM are 46, 260, 0.0248 and 16, respectively. In addition, the influence of the number of hidden layers on the results of the network was discussed. The appropriate hidden layer of the network was determined to be 2. The result show that the prediction accuracy of WOA-LSTM constitutive model is better than the Arrhenius constitutive model. The mean absolute error and correlation coefficient are 0.9348% and 0.99952, respectively. Among them, in this study, the Arrhenius constitutive model has low precision and only has high precision within a single temperature range.

Phenomenological Muscle Constitutive Model With Actin-Titin Binding for Simulating Active Stretching.

The force produced by a muscle depends on its contractile history, yet human movement simulations typically employ muscle models that define the force-length relationship from measurements of fiber force during isometric contractions. In these muscle models, the total force-length curve can have a negative slope at fiber lengths greater than the fiber length at which peak isometric force is produced. This region of negative stiffness can cause numerical instability in simulations. Experiments have found that the steady-state force in a muscle fiber following active stretching is greater than the force produced during a purely isometric contraction. This behavior is called residual force enhancement. We present a constitutive model that exhibits force enhancement, implemented as a hyperelastic material in the febio finite element software. There is no consensus on the mechanisms responsible for force enhancement; we adopt the assumption that the passive fiber force depends on the sarcomere length at the instant that the muscle is activated above a threshold. We demonstrate the numerical stability of our model using an eigenvalue analysis and by simulating a muscle whose fibers are of different lengths. We then use a three-dimensional muscle geometry to verify the effect of force enhancement on the development of stress and the distribution of fiber lengths. Our proposed muscle material model is one of the few models available that exhibits force enhancement and is suitable for simulations of active lengthening. We provide our implementation in febio so that others can reproduce and extend our results.

Hot deformation behavior of 30MnB5V steel: Phenomenological constitutive model, ensemble learning algorithm, hot processing map and microstructure evolution

The hot deformation behavior of 30MnB5V steel was experimentally investigated and theoretically analyzed using phenomenological constitutive models and an ensemble learning algorithm. Hot compression experiments were conducted with an MMS-200 thermo-mechanical simulation machine on the 30MnB5V steel at strain rates of 0.01–10 s−1 and temperatures of 950–1150 °C. The characteristics of the true stress-strain curves under different conditions and the corresponding microstructural evolutions were examined in detail. The results indicate that raising the temperature or lowering the strain rate promotes dynamic recrystallization. As temperature decreases or strain rate increases, the degree of recrystallization softening declines, and stress-strain curves transition from recrystallization to sustained work hardening or dynamic recovery types. Subsequently, the strain-compensated Arrhenius model, modified Johnson-Cook (J-C) model, and Bayesian-optimized LightGBM (BO-LightGBM) model were constructed sequentially to predict the flow stress of the tested steel. The accuracy of the three models was evaluated using the coefficient of determination (R2), mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE). The results reveal that the accuracy of the BO-LightGBM model is significantly higher than that of the Arrhenius model and the modified J-C model. Finally, three-dimensional power dissipation maps and hot processing maps were established under various strain conditions. The results suggest that the tested steel demonstrates good workability at deformation temperatures of 1075–1125 °C and strain rates of 0.03–0.22 s−1.

A one‐dimensional phenomenological constitutive model of shape memory alloys considering the cyclic degradation of two‐way memory effect

AbstractThis study introduces a one‐dimensional phenomenological constitutive model designed to describe two‐way shape memory effect (TWSME) and its associated cyclic degradation. The model utilizes the logistic function to formulate the phase transformation equation, incorporates the expansion of key parameters into their respective fatigue functions to characterize the fatigue phenomenon, and integrates a phase transformation rate regulation function into the differential form of the phase transformation equation. This integration facilitates the control over the entire phase transformation process and the simulation of incomplete transformations. The model is distinguished by its comprehensive functionality, simple form, ease of calculation, and the clear and direct influence of key parameters. Furthermore, it offers a degree of flexibility because each function within the framework is replaceable. The simulation of the TWSME strain and recovery stress hysteresis loop deformation has been successfully conducted, enabling the description of internal hysteresis loops caused by incomplete transformation. The validity of the model is corroborated by comparing it with existing experimental results.

Micromechanism insights and constitutive modeling for deformation in a novel high manganese austenitic steel

We explore the mechanical behavior and microstructural evolution of a novel high manganese austenitic steel (HMAS), extensively applied in underground engineering and structural construction. Our investigation focuses on two variants of HMAS, namely HMAS-1.9 and HMAS-10, which exhibit distinct yield strengths and elongation behaviors. Electron backscatter diffraction (EBSD), transmission electron microscopy (TEM) and X-ray diffraction (XRD) were employed to elucidate the evolution of microstructures during plastic deformation. Based on these experimental results, we developed a new phenomenological multiscale constitutive model that captures the mechanical behavior of HMAS, integrating micromechanisms and microstructural evolutions. Optimized with the MATLAB genetic algorithm toolbox, this model was successfully implemented in a uniaxial tensile finite element model (FEM) within ABAQUS, proving its efficacy in predicting both macroscopic flow stress and microstructural evolutions. Our findings highlight the important roles of initial grain size, dislocation density, and twin fraction in determining the mechanical properties of HMAS. Throughout the deformation process, both HMAS variants demonstrate similar plastic deformation behaviors, characterized by comparable rates of increase in dislocation density and twinning fraction, accompanied by a consistent trend of grain refinement. These findings not only deepen our understanding of the complex relationship between microstructure and mechanical properties of the HMAS steel, but also provide an effective multiscale constitutive modeling approach for predicting its deformation behavior.

A simple hydrodynamic model for clay

Laboratory description of clay normally distinguishes the scale of atoms from the scale of clay particles and aggregates. Contemporary constitutive models for clay tend to ignore this scale separation, and rather focus on phenomenology. By considering scale separation, this paper introduces a robust physics-based phenomenological constitutive model for clay that qualitatively captures their broad spectrum of rate-dependent mechanical features. The model is derived using the thoroughly rigorous hydrodynamic procedure. While some imagine that by considering rigour and physics, their models would get complicated, the resulting set of equations reveal a surprising degree of simplicity. The derivation strongly benefits from the principle of two-stage irreversibility, which describes energy flow within the material from the continuum scale down to the atomistic micro-scale, through the meso-scale of clay aggregates. While thermal and meso-related temperatures capture atomistic and clay aggregate fluctuating motions, a sink term from the latter to the former underpins the direction of the energy flow. The model’s standout feature is in pinpointing new transport coefficients that drive both volumetric and shear plastic flows in a thermodynamically coupled manner. A novel scheme is then proposed to calibrate these coefficients from conventional steady-state observations. Thanks to the formulation the model shows a remarkable level of predictiveness, despite being relatively simple mathematically. In particular, the model readily explains the broad spectrum of rate-dependent phenomena during transient loading, along with creep and relaxation processes. Given the generality of hydrodynamics, it is anticipated that the new model could be expanded to capture fluid-solid transitions between liquid-like soft mud and plastic-like stiff clay responses, contingent on water content variations.

Experimental Investigation of Strain Rate Influence on Anisotropy of Uniaxial Tensile Mechanical Properties of CuFe2P Alloy Sheet.

Wire crimping, a process commonly used in the automotive industry, is a solderless method for establishing electrical and mechanical connections between wire strands and terminals. The complexity of predicting the final shape of a crimped terminal and the imperative to minimize production costs indicate the use of advanced numerical methods. Such an approach requires a reliable phenomenological elasto-plastic constitutive model in which material behavior during the forming process is described. Copper alloy sheets, known for their ductility and strength, are commonly selected as terminal materials. Generally, sheet metals exhibit significant anisotropy in mechanical properties, and this phenomenon has not been sufficiently investigated experimentally for copper alloy sheets. Furthermore, the wire crimping process is conducted at higher velocities; therefore, the influence of the strain rate on the terminal material behavior has to be known. In this paper, the influence of the strain rate on the anisotropic elasto-plastic behavior of the copper alloy sheet CuFe2P is experimentally investigated. Tensile tests with strain rates of 0.0002 s-1, 0.2 s-1, 1 s-1, and 5.65 s-1 were conducted on sheet specimens with orientations of 0°, 45°, and 90° to the rolling direction. The influence of the strain rate on the orientation dependences of the stress-strain curve, elastic modulus, tensile strength, elongation, and Lankford coefficient was determined. Furthermore, the breaking angle at fracture and the inelastic heat fraction were determined for each considered specimen orientation. The considered experimental data were obtained by capturing the loading process using infrared thermography and digital image correlation techniques.

Contact stress decomposition in large amplitude oscillatory shear of concentrated noncolloidal suspensions

The concentrated noncolloidal suspensions show complex rheological behavior, which is related to the existence of contact stress. However, determining the contact stress in time-varying flow like oscillatory shear is challenging. Herein, we propose a contact stress decomposition method to decompose the total stress directly into contact stress and hydrodynamic stress in large amplitude oscillatory shear (LAOS). The results of hydrodynamic stress and contact stress are consistent with those determined by the shear reversal experiment. The contact stress decomposition also explains the failure of the Cox–Merz rule in noncolloidal suspensions because the particle contacts exist in steady shear but are absent in small amplitude oscillatory shear. The intracycle and intercycle of contact stress are further analyzed through the general geometric average method. The intracycle behaviors exhibit strain hardening, strain softening, and shear thickening. The intercycle behaviors show bifurcations in stress-strain and stress-strain rate relations, where the transition strains at different concentrations define the state boundaries between the discrete particle contacts, the growing of particle contacts, and the saturated contacts. We also established a phenomenological constitutive model using a structural parameter to describe the shear effect on the buildup and breakdown of particle contacts. The contact stress of noncolloidal suspensions with wide ranges of particle concentrations and strain amplitudes under LAOS can be well described by the model.

Numerical simulation of microstructural failure of natural fibre–hydrogel composites

In this article, we conduct a numerical simulation to investigate the failure mechanism of natural fibre–hydrogel composites under tensile deformation and swelling. Firstly, a physical-based constitutive model is chosen to describe the deformation of the hydrogel matrix. Concurrently, a macroscopic phenomenological hyperelastic constitutive model, incorporated into the material database of Abaqus, is employed to describe the mechanical responses of natural fibres. Then, we implement the model for a numerical analysis. The material parameters of the model for the hydrogel matrix and fibres are determined by a trial-and-error method from the experimental curves of pure hydrogel and fibres; the numerically simulated stress–strain curve of the composites is compared with the experimental one to verify the effectiveness of the simulation. Subsequently, several numerical analyses are performed to discuss the failure mechanism of hydrogel composites caused by the microstructural features of fibres. The findings revealed that fibre protrusion, fibre orientation deviation, and interface strength significantly affect the local failure of the composites during tensile deformation and swelling; the composites’ strength and toughness mainly depend on the cooperation of various microstructural features. These results provide valuable insights into the design of fibre–hydrogel composites.

A new effective phenomenological constitutive model for semi‐crystalline and amorphous polymers

AbstractA new phenomenological constitutive model is proposed for the prediction of the tensile and compressive behavior of semi‐crystalline and amorphous polymers. The new model modifies the phenomenological model previously proposed by Zhou and Mallick (ZM model) to correctly predict the complex behavior of thermoplastic materials including the linear viscoelastic deformation, the non‐linear viscoelastic deformation, the yielding, the post‐yield strain softening and the post‐yield strain hardening. A validation activity based on literature data has been carried out for polyether‐ether‐ketone (PEEK) and polycarbonate (PC) materials. The new model proved effective in fitting with high accuracy all the phases of the flow stress behavior for the considered materials, across a wide range of strain rates and temperature conditions. Finally, the comparison with Zhu et al., Duan et al., Nasraoui et al., Mulliken‐Boyce and Zhou‐Mallick models showed the better fitting performance of the proposed phenomenological model.Highlights Formulation of a new phenomenological model. Validation on mechanical test of semi‐crystalline and amorphous thermoplastics. Comparison with different phenomenological models present in the literature. Superior prediction results achieved compared to previously developed models.

Stress localization investigation of additively manufactured GRCop-42 thin-wall structure

A full-field crystal plasticity (CP) framework is presented for the GRCop-42 alloy to study microscopic mechanical behavior and local stress heterogeneities. The microstructures of additively manufactured (AM) materials are often unique relative to conventionally processed materials, and the local thermal histories drive these differences during the build process. These thermal histories depend on the process parameters (laser power, scan speed, and scan strategy) and the part geometry. Prior research has shown that the mechanical properties of thin-walled structures can vary significantly with wall thickness due to changes in the thermal boundary conditions during manufacturing. It is, therefore, desirable to perform CP simulations based on the phenomenological constitutive model to predict the local mechanical responses induced by microstructural heterogeneities. This work generates representative microstructures based on experimentally collected grain information (i.e., texture) for grain scale stress analysis, and the material constitutive parameters are calibrated using the experimental mechanical testing data. We specifically investigated the effect of crystallographic texture and grain morphologies on the size-dependent mechanical properties of AM GRCop-42. The selection of appropriate material properties for implementing an effective free surface boundary condition and the influence of adjacent buffer layers are also discussed. Analysis of local field results reveals a strong correlation between stress localization and the initial grain orientation. However, no significant relationship between the misorientation of the individual adjacent grains and the average misorientation is observed.

A Modular Nitinol Model with Lode Angle-Based Asymmetry and Improved Representation of the Superelastic Loops for Medical Device Applications

Abstract We describe a phenomenological constitutive model for representing Nitinol in medical device applications. The intent is to capture some of the response features important to medical device applications that are neglected in prevalent state of the art models. These features include better representation of the response under compression which is needed to assess bending, and asymmetric plastic yield. We will refer to this implementation as the Thornton Tomasetti (TT) model to distinguish it from other existing implementations that have evolved from the baseline underlying theory. We implement the model in a modular form suitable for use in commercial finite element software and describe initialization from laboratory test data.

Microstructure generation and full-field multi-scale analyses for short fiber reinforced thermoplastics: Application to PA66GF composites

Short Fiber Reinforced Thermoplastics (SFRTs) like PA66GF composites are highly heterogeneous materials with a rather complex microstructure such that the characterization and the prediction of their mechanical behavior remain quite challenging. So far, most of the research efforts have employed phenomenological and mean-field multi-scale models to deal with SFRTs. The present contribution rather focuses on a full-field multi-scale approach that incorporates advanced techniques in terms of microstructural representation and material modeling, allowing a deep insight of the dominating deformation mechanisms occurring in PA66GF composites. The proposed approach is based on an automatic periodic mesh generation algorithm for matrix-inclusion Representative Volume Elements (RVE) with randomly positioned fibers that follow a given Orientation Distribution Function (ODF). At the microscopic scale, while the fibers are assumed to be elastic, the behavior of the thermoplastic matrix is described by a phenomenological multi-mechanism constitutive model accounting for viscoelasticity, viscoplasticity and ductile damage. It results an advanced multi-scale model that enables to visualize the local deformation and degradation mechanisms occurring at the microscopic scale while simultaneously analyzing their influence on the macroscopic response of the composite upon monotonic, persistent and cyclic loading. The potential of the proposed approach is further evaluated by comparing the predicted responses against experimental data.

Temperature and Strain Rate Related Deformation Behavior of UHMWPE Fiber-Reinforced Composites.

As they possess the qualities of high specific strength, high specific modulus, high specific energy absorption, and excellent designability, ultra-high molecular weight polyethylene (UHMWPE) fiber-reinforced composites have gradually replaced traditional materials such as ceramics and steel plates as the main ballistic protection materials. Using an improved test method, the uniaxial tensile tests of UHMWPE fiber-reinforced composites at two strain rates of 10-4 s-1 and 10-2 s-1 and a temperature range from -20 °C to 80 °C are carried out to study the effects of strain rate and temperature on the tensile behavior of UHMWPE fiber-reinforced composites. The experimental results indicate that the tensile responses exhibit nonlinear characteristics and the sensitivity of strain rate and temperature. The yield strength and modulus decrease with increasing temperature and increase with the increase in strain rate. A phenomenological viscoelastic constitutive model composed of a nonlinear spring and a nonlinear Maxwell element is proposed to characterize the temperature and strain rate dependent deformation behavior of UHMWPE fiber-reinforced composites before yielding. The results show that the model can accurately predict the tensile nonlinear viscoelastic responses of UHMWPE fiber-reinforced composites before yielding over a wide temperature range under quasi-static loading.

Asymmetric and anisotropic quench sensitivity of ZA81M magnesium alloy

The quench sensitivity of the ZA81M magnesium alloy is studied by combining experiment and crystal plasticity finite element simulation. The quench sensitivity is high in tension and low in compression making the single-valued quench sensitivity calculated from hardness insufficient. So, there is different loss rate of strength, along different test directions of the alloy, due to slow quenching, i.e., the quench sensitivity is asymmetric and anisotropic. The coarsening of the precipitate and widening of the precipitate-free zone are the origin of the quench sensitivity. Crystal plasticity finite element simulation is proposed to calculate the quench sensitivity. The phenomenological constitutive model is used to explicitly consider the strengthening effect of precipitates on different slip and twinning systems. So, the precipitate characteristics of the water-quenched and air-cooled alloy are easily captured. The anisotropy and asymmetry of the quench sensitivity in two perpendicular sample planes are determined by simulation. Such a method can be useful for determining the quench sensitivity of precipitation-hardening alloys with low crystal symmetry and with precipitates having different strengthening efficiencies on different slip or twinning systems.

Topology optimization structure design of shape memory alloy with multiple constraints

As an emerging functional material, shape memory alloy (SMA) exhibits remarkable mechanical properties and finds diverse applications across industries. This paper presents a topology optimization framework based on the bi-directional evolutionary structural optimization (BESO) method for designing SMA structures, which maximizes structural stiffness under multiple constraints of specified volume fraction, displacement, and fundamental frequency. A phenomenological constitutive model is utilized to simulate the mechanical behavior of SMA accurately. The unit virtual load method is employed to determine sensitivities. Several optimized SMA beam structures and simply-supported cube structures are designed under different thermal-mechanical loads, and their displacement, mean compliance, and fundamental frequency are evaluated throughout the optimization process. The results demonstrate that the proposed framework successfully customizes the SMA topology structure with adjustable displacement and fundamental frequency, and the optimized schemes exhibit more considerable deformation and more uniform mechanical properties than their initial counterparts. The proposed framework has higher computational efficiency than the traditional SIMP-based SMA topology optimization design method.

Virtual laboratory enabled constitutive modelling of dual phase steels

Accurate and efficient characterization and modelling of deformation responses are critically essential in the development of advanced metal forming processes. This work presents a virtual laboratory enabled constitutive framework for modelling complex deformation responses of dual phase (DP) steels under complex stress states, based on multi-phase full-field crystal plasticity (CP) and advanced phenomenological modelling. In the CP based virtual testing method, kinematic hardening and degradation of elastic modulus are modelled in particular to improve the capability for describing evolving microstructure induced mechanical responses of DP steels. The workflow of this framework is completely built and numerically implemented, including (i) representative volume element (RVE) generation based on microstructure characterization, (ii) identification of multiphase CP model parameters, (iii) prediction of elastic modulus degradation, (iv) prediction of yield stresses and plastic potentials in uniaxial tension and biaxial tension with various stress ratios, and (v) prediction of stress–strain curves in reverse tension compression. Using DP780 and DP980 as case materials, the corresponding physical experiments are conducted to verify the accuracy of the virtual tests, showing a good agreement between the virtual and experimental approaches. Both experimental and virtual tests are used to calibrate advanced phenomenological constitutive models that include non-linear elasticity, anisotropic yielding, and kinematic hardening. The calibrated models are implemented into the finite element (FE) codes to predict complex deformation and springback behaviours of DP780 and DP980 sheets in U-bending processes. In comparison with forming experiments of U-channel parts, the virtually calibrated models are validated and show good performance in predicting deformation and springback behaviours, providing a high capability for process analysis. The findings support that the virtual laboratory enabled modelling approach could be a substitute for extensive, expensive, and hard-to-access physically mechanical experiments required in the model calibration for a more effective and efficient analysis of metal forming processes.