Type Constitutive Equation Research Articles

Consistency-enhanced modified SAV time-stepping method with relaxation for binary mixture of and viscous fluids

In this work, the binary mixture of nematic liquid crystals and viscous fluids (NLC-VF) system consists of the Cahn–Hilliard (CH) equations for the phase-field variable for the free interface, the Allen–Cahn (AC) type constitutive equation for the nematic director, and the incompressible Navier–Stokes (NS) equation for the two fluids. To address the computational challenges posed by this complex system, we propose a scheme that is fully decoupled, energy-stable and highly consistent, based on a modified scalar auxiliary variable (MSAV) with stabilization. Our approach employs two auxiliary variables: one derived from the nonlinear terms in the original energy, and the other leveraging the “zero-energy-contribution (ZEC) ”property satisfied by some nonlinear terms. To ensure consistency between the continuous and discrete auxiliary variables, we employ relaxation techniques. Besides, the proposed method enables sequential solving of each variable, and the computational overhead of the relaxation technique is minimal, resulting in highly efficient computation. We demonstrate the accuracy, stability, consistency, and practicality of the method through comprehensive numerical experiments, and provide rigorous proof of its unconditional energy stability.

Prediction of single cell mechanical properties in microchannels based on deep learning

Traditional methods for measuring single-cell mechanical characteristics face several challenges, including lengthy measurement times, low throughput, and a requirement for advanced technical skills. To overcome these challenges, a novel machine learning (ML) approach is implemented based on the convolutional neural networks (CNNs), aiming at predicting cells’ elastic modulus and constitutive equations from their deformations while passing through micro-constriction channels. In the present study, the computational fluid dynamics technology is used to generate a dataset within the range of the cell elastic modulus, incorporating three widely-used constitutive models that characterize the cellular mechanical behavior, i.e., the Mooney-Rivlin (M-R), Neo-Hookean (N-H), and Kelvin-Voigt (K-V) models. Utilizing this dataset, a multi-input convolutional neural network (MI-CNN) algorithm is developed by incorporating cellular deformation data as well as the time and positional information. This approach accurately predicts the cell elastic modulus, with a coefficient of determination R2 of 0.999, a root mean square error of 0.218, and a mean absolute percentage error of 1.089%. The model consistently achieves high-precision predictions of the cellular elastic modulus with a maximum R2 of 0.99, even when the stochastic noise is added to the simulated data. One significant feature of the present model is that it has the ability to effectively classify the three types of constitutive equations we applied. The model accurately and reliably predicts single-cell mechanical properties, showcasing a robust ability to generalize. We demonstrate that incorporating deformation features at multiple time points can enhance the algorithm’s accuracy and generalization. This algorithm presents a possibility for high-throughput, highly automated, real-time, and precise characterization of single-cell mechanical properties.

Hot deformation and dynamic recrystallization of Mg-Gd-Y-Zn-Zr alloy containing high volume fraction 18R-LPSO phase

The deformation behavior including flow behavior, hot workability and dynamic recrystallization (DRX) mechanism of as-cast Mg-9Gd-3Y-3.75Zn-0.6Zr alloy containing 26.5 % volume fraction 18R long-period stacking ordered (LPSO) phase distributed in continuous network morphology were studied by hot compression experiments at deformation condition of 350 ℃-500 ℃, 0.01–1 s−1 strain rate and 0.75 true strain. The influence of deformation conditions on the microstructural evolution was analyzed by using optical microscopy (OM), scanning electron microscope (SEM), electron back scatter diffraction (EBSD) and transmission electron microscope (TEM). Results display that stress exponent and deformation activation energy are 5.58 and 241.39 kJ·mol−1 correspondingly, which suggested that dislocation climb is the principal deformation mechanism. The Arrhenius type constitutive equation is established as ε̇=4.09×1016sinh(9.922×10−3σ)5.58exp(−241394RT). Microstructure evolution can be characterized by Zener-Holomon parameter (Z) which reflects the combining effect of temperature and strain rate on microstructure. At low lnZ (lnZ<36), the 18R-LPSO phase dissolves from continuous network to blocks and high DRX volume (>50 %) is realized by particle-stimulated nucleation and discontinuous dynamic recrystallization. At moderate lnZ (36≤lnZ<40), a fine and coarse grains bimodal structure is formed at the original grain boundary by continuous dynamic recrystallization resulting in a DRX volume between 10 % and 50 %. At high lnZ (lnZ≥40), layered 14H‐LPSO phase precipitates and inhibits DRX (DRX volume＜10 %) by absorbing dislocations. With combination of dissipation coefficient, instability factor and microstructure analysis, 400 ℃/0.01–0.05 s−1 and 450 ℃/0.01 s−1 are confirmed as optimal hot working domains for the as-cast alloy, under which conditions bimodal microstructure and equiaxed grains microstructure (average grain size 10.3 μm) can be formed correspondingly.

Electro-mechanical vibro-acoustic characteristics of submerged functionally graded piezoelectric plates with general boundary conditions

The present paper focuses on studying the underwater electro-mechanical vibro-acoustic characteristics of functionally graded piezoelectric material (FGPM) plates with general boundary conditions. For this purpose, the collocation points method is firstly developed to handle vibro-acoustic coupling, which can directly calculate the virtual work of secondary sound pressure via an easy double integral, without pre-simplifying the complex quadruple integral. The proposed FGPM plate model is established in the framework of first-order shear deformation theory (FSDT) and second type constitutive equations of piezoelectric materials, and the macroscopically inhomogeneous electro-mechanical properties are considered by means of Voigt model. The multi-physical fields including electric, mechanical and acoustic fields are uniformly extended in Chebyshev polynomials. Meanwhile, penalty function method is introduced to satisfy the arbitrary mechanical and electric constraints. All energy formulations are processed by Rayleigh-Ritz procedure and a standard governing equation is finally derived. Based on the proposed unified model, the convergence and accuracy for solving vibration and acoustic radiation of FGPM plates are validated firstly. Additionally, the influence of some key parameters on underwater vibro-acoustic characteristics of FGPM plates is fully examined. The results of this paper may provide a reference for future application of FGPM-based smart structures.

Modeling Elastomer Compression: Exploring Ten Constitutive Equations.

This paper presents the results of research aimed at assessing the effectiveness of ten selected constitutive equations for hyperelastic bodies in numerical modeling of the first compression load cycle of a polyurethane elastomer with a hardness of 90 Sh A depending on the methodology for determining the material constants in the constitutive equations. An analysis was carried out for four variants for determining the constants in the constitutive equations. In three variants, the material constants were determined on the basis of a single material test, i.e., the most popular and available in engineering practice, the uniaxial tensile test (variant I), the biaxial tensile test (variant II) and the tensile test in a plane strain (variant III). In variant IV, the constants in the constitutive equations were determined on the basis of all three above material tests. The accuracy of the obtained results was verified experimentally. It has been shown that, in the case of variant I, the modeling results depend to the greatest extent on the type of constitutive equation used. Therefore, in this case it is very important to choose the right equation. Taking into account all the investigated constitutive equations, the second variant for determining the material constants turned out to be the most advantageous.

Implementation of the Lemaitre damage model with kinematic hardening in the ANSYS finite element complex

Currently, one of the most popular in fracture mechanics is taking into account damage and its effect on the stress-strain state of structural elements. In this work, the Lemaitre damage model was integrated taking, into account the combined hardening law, which combines the Armstrong - Frederick kinematic hardening law and the Voce isotropic hardening law, into the ANSYS finite element software. The model is implemented in the form of a dynamically linked library of user material for three-dimensional objects, which is tested on a cylindrical specimen with an external annular notch, both in an axisymmetric setting and in a three-dimensional one. The article presents model representations of the above-listed standard systems. This work demonstrates only one of the two stages of verification of the created program - comparison of damage fields under monotonic loading with data known in the literature - and doesn’t take into account the verification of cycle-by-cycle kinetics of plastic deformation accumulation with experimental data for low-cycle fatigue. The result of verification, consistent with similar experiments known in the literature, is confirmed in accordance with similar experiments. In addition, an analogy was found using the TSL law of the cohesive fracture mechanics approach. Despite the fact that two different types of constitutive equations are used in the cohesive model and Lemaitre, the physical meaning of these equations consists in one thing - visualization and identification of mechanisms and coordinates of damage.

Hot deformation behavior of a novel alpha + beta titanium alloy TIMETAL®407

The hot deformation behavior of a newly developed α + β titanium alloy TIMETAL®407 (Ti407) was characterized in starting transformed-β microstructure at different temperatures (T = 650–950 °C) and strain rates (ε̇ = 10−3 - 1 s−1) using hot-compression testing. The Arrhenius type constitutive equations are developed for Ti407 and the activation energy for deformation is determined as 333.1 and 179.8 kJ/mol in the α and β dominated phases, respectively. The processing and strain-rate sensitivity maps of Ti407 were generated at true strain, ε = 0.5 on the basis of modified dynamic materials model (DMM) proposed by Murty and Rao. Results obtained from the maps and microstructural analysis of the deformed specimens indicates three distinct stable domains (I-A, II-A and I-B) within the examined conditions. Out of the three stable domains, Domain I-A occurring in the α + β phase field (T = 800–860 °C and ε̇= 6.3 ×10−2 - 1 s−1) and Domain I-B in the β phase field (T = 900–950 °C and ε̇= 10−3 - 1 s−1) were optimum domains for possible hot working of Ti407. The peak in the efficiency of power dissipation of about 58% occurred in Domain I-A at 850 °C/1 s−1, where dynamic spherodization of α lamellae occurred. In Domain I-B, formation of fine dynamically recrystallized β grains was observed with peak in power dissipation efficiency of about 60% at 950 °C/10−3 s−1. Ti407 showed evidence of flow instabilities like flow localization and kinking of α lamellae in the predicted unstable zones identified from the processing map. The hot workability of Ti407 on the basis of expanding or shrinking stable regimes for processing is also compared with CP-Ti and Ti-6Al-4V alloys at identical strain rates and temperatures relative to the beta-transus. Finite element method (FEM) software DEFORM®− 3D was used for forging simulation at the peak in deformation efficiency condition in various stable domains to evaluate the flow response and distributions of stress/strain in the forged billets.

Flow Stress Modeling of Tube and Slab Route Sheets of Zircaloy-4 Using Machine Learning Techniques and Arrhenius Type Constitutive Equations

Flow Stress Modeling of Tube and Slab Route Sheets of Zircaloy-4 Using Machine Learning Techniques and Arrhenius Type Constitutive Equations

Hot deformation behavior of Sc/Nb modified AA2195 Al–Li–Cu alloys

The hot workability of third generation AA2195 (Al–Li–Cu–Ag–Mg–Zr) alloy with different Sc (0.15–0.25 wt%) and Nb (0.15–0.25 wt%) additions in the homogenized condition was investigated using processing maps approach. The additions of Sc and Nb were found to be beneficial in refining the as-cast grain size of AA2195 and changing the morphology from less dendritic grain structure to a globular or equiaxed form. Hot compression tests carried out in the temperature, T, range of 300–450 °C and strain rate, ε˙, of 10−3 – 10 s−1 were utilized to generate processing maps at a true strain, ε = 0.5. The flow curves of the alloys exhibit a typical dynamic recovery (DRV) characteristics and Arrhenius type constitutive equations successfully predicted flow stress of the alloys. The results obtained from the processing maps and complementary microstructural analysis of the deformed specimens revealed that 0.15 wt% additions of Sc and Nb (separately or in combination) are optimum to improve its workability. However, increasing Sc and Nb contents to 0.25 wt% relatively expanded flow instability regions. With the aid of the processing maps, deformed specimens microstructural analysis, activation energy and Zener-Hollomon parameter, Domain D-3 occurring at higher temperatures and strain rates were identified as the “safe” regimes for hot working of AA2195 with optimum Sc and Nb contents (0.15 wt% added separately or in combination). Limited pancake type grain morphology with higher fractions of dynamically recrystallized (DRX) grains was observed in Domain D-3. The microstructural manifestation in the flow instability regions was flow localization (at higher strain rates), cavities formation at the grain boundaries & precipitates, particles breakage and wedge cracking at the triple junctions at lower strain rates.

Hot compression behaviors and microstructural evolution of casting Be-38wt.%Al

The uniaxial compressive (UAC) performance of casting Be-38wt.%Al was investigated in the temperature range from 250 to 550 °C and in the strain rate from 0.001 to 1 s−1. Serrated flow and work hardening effect dominate the true strain-true stress curves. Arrhenius type constitutive equation based on strain is established. The microstructure is analyzed by OM, ECCI and EBSD. The optimum deformation parameter has wide range in 300 ∼ 500 ℃ and 0.001 ∼ 1 s−1. The EBSD test shows {101¯2}<101¯1> type twinning of Be phase and particle simulated recrystallization in Al phase contribute to the good coordination during deformation.

Modeling Hot Deformation of 5005 Aluminum Alloy through Locally Constrained Regression Models with Logarithmic Transformations

This study presents the adoption of locally constrained regression models (LCRMs) with logarithmic transformations in order to model the flow stress behavior of the high-temperature deformation of 5005 aluminum alloy. Hot tensile tests for 5005 aluminum alloy were conducted under the temperatures of 290 °C, 360 °C, 430 °C, and 500 °C, and the strain rates of 0.0003/s, 0.003/s, and 0.03/s. The flow stress behavior was analyzed based on variations in temperature and strain rate. The flow stress during the hot deformation was modeled using the traditional Arrhenius type constitutive equation and the neural network approach. Then, for improved prediction accuracy, the flow stress was modeled using LCRMs. The prediction accuracies of the models were compared by calculating the MAE (Maximum Absolute Error) and RMSE (Root-Mean-Squared Errors) values. The MAE and RMSE of the LCRMs were lower than the errors of the Arrhenius equation and the neural network model. The results show that LCRMs can be useful in modeling the flow stress of 5005 aluminum alloy, and that the developed model can accurately predict the flow stress.

Modeling of rubber-cord layers under quasi-static loading

Modeling a pneumatic tire with a strong change in shape causes the problem of choosing an adequate model for rubber-cord plies. Generally, the classical method of asymptotic homogenization is not suitable due to physical and geometrical nonlinearity for the strain up to 15 %. The known models used for simulation the entire tire as well as rubber-cord plies are analyzed. A choice is made in favor of modeling the plies using the stress anisotropic potential, which is an anisotropic function of the strain tensor invariants. The relationship of such a constitutive law with a quasi-linear constitutive equation in terms of stress and strain differentials is indicated. A convenience of these two types of constitutive equations in terms of numerical implementation is also given. A modification of the effective properties definition for an inhomogeneous layer is explained. The difference from the standard effective moduli definition is clarified. The arrangement of the cords is supposed to be approximately periodic and all cords to be equivalent to the effective fiber. Two models of a rubber-cord plies are described under such assumption. These are a model of an orthotropic material and a model of a transversely isotropic material. Computational experiments, which make it possible to determine the material parameters of anisotropic potentials, are pointed out. Real tests with a sample of rubber ply under slow quasi-static loading were conducted. Significant hysteresis was detected. It is shown that an additive model combining a hyper elastic material with Maxwell viscoelastic model provides good accuracy in stress dependence on the strain rate. The numerical procedure developed to calculate solution to the quasi-static problem of tire deformation is described. It is implemented in home-made computer code. A numerical example on the tire simulation is given. That is so-called breaking test, in which strong deformation is achieved and the developed model is applied to.

Hot compression behaviors and microstructural evolution of beryllium by vacuum hot pressing

The uniaxial compressive (UAC) performance of beryllium by vacuum hot pressing (VHP) was investigated in the temperature range from 250 to 550°C and strain rate from 0.01 to 10 s−1. Serrated flow and yield point showed in the stress–strain curve are caused by Portevin-Le Chatelier effect (PLCE). Arrhenius type constitutive equation based on true strain is established and the volume diffusion of Fe and C in solution is deduced by the calculated activation energy. The typical microstructure instability characters are interfacial cracking and intragranular cracking. The optimum deformation parameter is 350°C and 0.01 s−1, with uniform deformed microstructure and almost little defect increasing.

Hot compression behaviors and microstructural evolution of Be-38wt.%6061Al

The compressive deformation performance of Be-38wt.%6061Al coordination was investigated in the temperature range from 250 to 550 °C and strain rate from 0.01 to 10 s−1. The soften mechanism showed in the stress–strain curve is dynamic recovery (DRV). Arrhenius type constitutive equation is established based. Dispersion strengthening mechanism, caused by oxidation and powder metallurgy processing, is derived from calculated material constants. The typical microstructure instability mechanisms under the deforming craft above are interfacial debonding and 45° shear cracking. The optimum deformation parameter is 450℃ and 0.01 s−1, which contributes to the well-distribution of oxides and the alloy agglomeration, and coordinates the microstructure of each part to more consistency.

J-integral estimation by reference plastic slope method for poly-linear stress-strain curves

It is difficult to obtain J-integral values (J-values) for materials using various types of constitutive equations without performing numerical analyses. In this study, the reference plastic slope (RPS) method was applied to obtain J-values for engineering materials, for which the deformation property was expressed by a poly-linear stress-strain curve. The RPS method derives the J-value using bi-linear stress-strain curves obtained from the original stress-strain curve. First, the basic concept and characteristics of the RPS method is explained. It is shown that well-approximated J-values were obtained by assuming bi-linear stress-strain curves whose plastic slope were determined using the reference stress for representative materials (three carbon steels and two stainless steels). Second, it is demonstrated that the J-values for bi-linear stress-strain curves could be estimated by relatively simple equations for a center cracked plate subjected to a tensile load. Third, to apply the RPS method for practical problems, the equations are developed to estimate the J-values for bi-linear stress-strain curves for a straight pipe with a circumferential crack subjected to a tensile or bending load. Finally, it is confirmed that the developed equations could predict the failure load of which error was within 10% at the most unconservative case.

Regime-Switched Neural Networks: Flow Stress Modeling Strategy of 310s Stainless Steel During Hot Deformation

This study examines the high temperature deformation and constitutive modeling of flow stress for 310s stainless steel. To this end, hot tensile experiments were conducted under the temperatures of 700° and 800° at the strain rates of 0.0002/s, 0.002/s, and 0.02/s. Flow stress was modeled using the Arrhenius type constitutive equation and neural network approach. Specifically, Regime-Switched Neural Networks (RSNN), a set of neural networks with switching, has been newly proposed as a better predictive model for flow stress. The modeling performance of the RSNN model was evaluated through a comparison with traditional Arrhenius-type constitutive equations and the single neural network model. The results showed that the accuracy of the proposed RSNN was substantially higher than those of the existing Arrhenius-type equations, and the prediction performance was therefore significantly improved. In addition, the accuracy of the proposed RSNN was improved by approximately more than 24% in comparison with the existing global model—a single neural network—thus confirming the superiority of the proposed switching model.

Investigation of cusp catastrophe model of rock slope instability with general constitutive equations

The unsteadiness of rock slopes is often triggered by water-softening. Based on catastrophe theory and the general constitutive equations of the strain-softening section in the weak interlayer, the present study investigated the catastrophic failure mechanism of rock slopes. Using theoretical analysis, the cusp catastrophe model for the failure of a rock slope with a strain-softening section in the soft rock strata was established, and the balance equation, bifurcation set, and formulas of displacement jumping and energy jumping were deduced. The changes of displacement jumping and energy jumping following the variation of the systematic rigidity ratio are discussed with regard to the occurrence of water-softening. Regardless of which type of constitutive equation is adopted, the control variables and sudden jump values of the rock slope’s displacement and energy can be determined by the systematic rigidity ratio when the systematic rigidity ratio is constant. Whether a rock slope is stable or not depends only on the mechanical and geometrical parameters of the system itself, and is unrelated to the constitutive equation of the strain-softening section. The sudden jump values of the displacement and energy are both inversely proportional to the systematic rigidity ratio. By comparing the formula of the systematic rigidity ratio deduced under different constitutive equations with the results obtained by other studies, the results of the present study were verified through the consideration of actual engineering projects. Based on catastrophe theory, the present paper discusses the principle of correctly selecting a constitutive equation to assess the rock slope stability in practical engineering.

Understanding Hot Workability and Flow Stress Prediction through Processing Map with Microstructural Correlation for HY85 Steel

Abstract In this study, the hot compression tests of HY85 steel have been experimented by using thermomechanical simulator Gleeble 3800 in hot compression temperature range of 750°C–1,100°C under strain rates between 0.001–10 s−1 up to true strain 0.7. It is worth noticing from the results that the flow stresses are dependent on the combination of temperature and strain rate. Adiabatic heating corrections of experimental flow stress data were carried out and were used to model stress-strain curves. The predictability of these stress-strain curves was confirmed from experimental results. True stress-true strain curves show the peak stresses followed by softening probability because of the existence of dynamic recrystallization (DRX). The apparent activation energy is calculated to be 360 KJ/mol using the sinh type constitutive equation, and it is observed that its value decreases with increasing strain. Hot compression of HY85 steel is controlled by dislocation climb and glide (n = 6.3). The processing maps are plotted based on different dynamic materials modeled with strain rate versus temperatures. The flow localization is one of the primary factors with resulting instability, which is revealed from the microstructure. Moreover, the partial DRX was noticed besides the grain boundaries and compression bands. Workability regions were identified using processing maps in the domain of strain rate and temperature and verified with microstructures.

High Temperature Deformation Behavior and Constitutive Modeling for Flow Behavior of Alloy 718

In the present work, hot deformation behavior of Alloy 718 was investigated over a temperature range of 1223–1373 K and strain rate range of 10−2–10 s−1. The flow curves were corrected for adiabatic temperature rise, particularly at high strain rates. Arrhenius type constitutive equations were derived for Alloy 718 to model the peak flow stress from apparent and physically based approaches. A stress exponent of 5 was obtained from the power-law equation, indicating that the deformation is governed by the dislocation climb mechanism within the aforementioned processing domain. Further, to model the flow behavior, a generalized constitutive equation was derived in which the effect of strain on the flow stress was incorporated. In addition, artificial neural networks (ANN) method was also employed to model the flow behavior. Statistical parameters such as regression coefficient (R) and average absolute relative error (AARE) indicated that the ANN method was more accurate in predicting the flow behavior with R = 0.99 and AARE = 0.79% compared to the apparent-based constitutive equation with R = 0.99 and AARE = 4.5%. Accuracy of the derived constitutive equation as a material model in finite element (FE) simulation studies was also evaluated. Flow curve predictions obtained from the FE simulation were comparable to the experimental results. The microstructure and hardness at different locations in the deformed samples were consistent with the strain distribution map generated by the FE simulation.

Some Universal Solutions for a Class of Incompressible Elastic Body that is Not Green Elastic: The Case of Large Elastic Deformations

Summary Some universal solutions are studied for a new class of elastic bodies, wherein the Hencky strain tensor is assumed to be a function of the Kirchhoff stress tensor, considering in particular the case of assuming the bodies to be isotropic and incompressible. It is shown that the families of universal solutions found in the classical theory of nonlinear elasticity, are also universal solutions for this new type of constitutive equation.