Constitutive Model Research Articles

Fractional Order Kelvin-Voigt Constitutive Model and Dynamic Damping Characteristics of Viscoelastic Materials

Viscoelastic damping materials are often under complex service conditions. To precisely characterize its dynamic damping properties, based on fractional calculus and viscoelastic theory, considering the periodic characteristics of viscoelastic materials, the distributed fractional order Kelvin-Voigt constitutive model (DFKV) was constructed. The model accuracy was validated by quasi-static experiments. The dynamic modulus expressions were derived, and the model parametric feature analysis was carried out. Dynamic Mechanical Analysis (DMA) and Separate Hopkinson Pressure Bar (SHPB) experiments were performed with silicone rubber as the subject. The results show that the silicone rubber has an obvious temperature and frequency dependence, and it has a strong strain rate correlation considering that the strain rate effect indexes that are under high strain is 29.331. At the same time, clearly the viscoelastic dynamic constitutive behavior has phased characteristics that are in line with the scientific assumption of the distribution order. The fitting accuracy of DFKV model is higher than the existing contrast models, which reflects DFKV model and favorably represents the dynamic mechanical properties of viscoelastic materials under wide temperature, frequency and strain rate. The model is highly accurate, has fewer parameters, and the physical meaning of its parameters is clearer. It can provide a theoretical reference for the research and design of viscoelastic damping materials.

Correction method of Mohr-Coulomb strength criterion for rock based on freeze-thaw and residual effects

Under confining pressure, rocks transition from brittle failure to plastic failure, and residual strength exists after complete failure. However, in the process of establishing rock damage constitutive models, the strength criteria used usually do not consider residual stress. In cold region engineering, the freeze-thaw effect caused by temperature changes should be considered in the constitutive model, and strength criteria should also be introduced. Considering the failure characteristics of micro element when rock is subjected to freeze-thaw and load, and based on the impact of reducing the effective bearing area on each damage, the total damage variable and constitutive model of rock under freeze-thaw and load are established. Starting from the characteristics of the entire process of rock deformation, the revised Mohr-Coulomb (M-C) strength criterion of rock is established by considering freeze-thaw cycles and residual effects. The results show that: The theoretical curve and test curve of the constitutive model are not completely consistent, but the variance R2 between the two does not exceed 0.6, indicating that the theoretical curve is in good agreement with the test curve and can reflect the entire process of rock deformation and failure, verifying the rationality of the constitutive model and damage variable description. With the increasing of freeze-thaw cycles, the strength decreases and the deformation increases of rock. As the number of freeze-thaw cycles increases, the strength decreases and deformation increases. When the freeze-thaw cycle reaches 40 cycles, the strength decreases by more than 60%. With the increasing of confining pressure, the strength and deformation also increase. When the confining pressure increases from 0 to 6 MPa, the strength of red sandstone increases by more than 80% under the same freeze-thaw cycles, which is consistent with the actual situation. The revised M-C strength criterion data does not exceed the test data and is relatively close to the test data. The difference between the two is less than 25%, indicating that the established strength criteria can be safely used as a basis for rock elemental failure, verifying the rationality of the strength standard considering freeze-thaw and residual effects. This Revised M-C strength criterion introduces the influence of freeze-thaw cycles and residual stress factors, which not only characterizes the relationship between internal stress parameters in rock limit states under different freeze-thaw cycles, but also addresses the disadvantage of Drucker-Prager (D-P) criterion being more conservative. This method is based on the measured results of rock triaxial test, which makes it more flexible and the calculation results closer to reality.

Neural networks with iterative parameter generation for determining parameters of constitutive models

Determining the parameters of constitutive models is typically a material-specific process that requires recalibration when the material changes. This procedure becomes notably time-consuming with an increased number of parameters and integral terms in the constitutive equations. Recently, a novel approach utilizing neural networks has emerged as an alternative. By training neural networks to replace constitutive equations, constitutive substitution method maps the extensive parameter space to computational results equivalent to those derived from original equations. Comparing these results with experimental data allows for efficient determination of optimal parameter vectors. The powerful fitting capabilities and rapid computational speed of neural networks significantly expedite the parameter determination process. Consequently, the task of solving for parameters transitions from a complex optimization problem to a straightforward computational search. In this work, we introduce an iterative parameter generation (IPG) algorithm to enhance this substitution method, thereby improving its ability to fit a diverse range of materials. Additionally, we explore the benefits of constructing a general parameter space applicable to various material classes.

Dynamics characteristics of variable stiffness curvilinear fiber-reinforced layered composite beam under moving load by sinusoidal shear flexible beam model including nonlinear and thermal effects

In this study, the dynamic response behavior of curvilinear fibers based layered-composite beam under transversely moving load is investigated considering linear and nonlinear analyses. The formulation is developed applying sinusoidal shear flexible beam theory, geometrical nonlinearity based on von Kármán’s assumption, and the constitutive equation fulfilling the plane stress situation through the width of arbitrarily lay-up composite beam. The load on the beam is treated as moving with uniform velocity, acceleration and also oscillatory motion with time. The governing nonlinear dynamic equations are framed employing Hamilton’s principle and by adopting finite element approach. The response of beam in time domain is obtained solving the governing equations through direct time integration method by Newmark integration scheme. and the results are viewed through the frequency-amplitude relationship. A detailed parametric study involving curvilinear fiber path angles, lay-up ply angles, thickness ratio, moving load velocity, and acceleration, and forcing frequency is made on the dynamic behavior of variable stiffness beam with curvilinear fiber beam. The effect of thermal environment and boundary conditions on the vibrational response behavior of beam is also predicted.

Finite Element Modeling of RC Beams Produced with Low-Strength Concrete and Strengthened for Bending and Shear with CFRP and GFRP

In this study, the analysis of reinforced concrete (RC) beams strengthened with Fiber Reinforced Polymer (FRP) composites against bending and shear loads was carried out with the finite element technique, using ABAQUS software, which is widely used in simulating experimental circumstances in numerical studies. It has been reported that buildings in areas damaged by earthquakes are generally constructed using low-strength concrete and inadequate reinforcement. Additionally, construction errors also contribute to reducing the load-bearing capacity of structural elements. For this purpose, nine rectangular cross-section RC beams were experimentally constructed using low-strength concrete and inadequate bending and shear reinforcement. These beams were strengthened by wrapping them in different configurations with Carbon and Glass FRP (CFRP and GFRP) composites to resist shear and bending forces in both transverse and longitudinal directions, and their load-displacement curves were obtained. Subsequently, a three-dimensional Finite Element Model (FEM) was created to validate the experimental results. The FEM validation demonstrated high accuracy in replicating experimental outcomes, emphasizing the influence of mesh size, dilation angle, and concrete constitutive models on simulation fidelity. Parametric studies revealed that increasing longitudinal reinforcement diameters had minimal effect on load capacity but highlighted the critical role of transverse reinforcement, as reducing stirrup spacing significantly improved load-bearing capacity. GFRP-reinforced beams exhibited superior ductility and a 15% higher strength compared to CFRP, suggesting their suitability for applications demanding enhanced displacement capacity. Furthermore, the findings underline the need for refined FEM models to better capture inclined fiber orientations and optimize structural reinforcement strategies.

Hyperelastic models for the human zona pellucida and their implications on shear modulus estimation in the clinical practice

It has long been speculated that the mechanical properties of the human oocyte can be an indicator for oocyte viability. Recent studies have demonstrated that embryo implantation rates, following Intra-Cytoplasmic Sperm Injection (ICSI) procedures, may be increased if the shear modulus value of the oocyte Zona Pellucida (ZP) is taken into consideration during embryo transfer. The shear modulus was determined by an iterative oocyte specific finite element (FE) analysis based on the clinical ICSI data. Nevertheless, the results obtained from the computational analysis may depend on the choice of the constitutive model used for the ZP. In the current study, three different hyper-elastic strain energy density functions (SEDF’s) are considered. The Neo-Hookean (NH), Mooney-Rivlin (MR) and Ogden models (OG) were used to determine the ZP shear modulus from ICSI clinical data. The sensitivity of each SEDF in extracting the oocyte specific shear modulus is examined, for the first time. It is demonstrated that the NH and MR models are more sensitive than the OG model for determining the ZP shear modulus from clinical ICSI data. It is also demonstrated that the accuracy of ZP shear modulus identification process greatly depends on the value of suction pressure applied during the ICSI procedure.

Biomechanical Constitutive Models, From The Atomistic Approach, For Blood Vessels

Objective: To present an analysis of the most relevant biomechanical atomistic constitutive models of blood vessels, which will serve as a basis for future proposals for modeling deformation phenomena in veins and arteries. Theoretical Framework: Myocardial ischemia is a disease that alters the mechanical response of blood vessel walls. Predicting the mechanical behavior of blood vessels based on the physiological state using biomechanical models is vital for diagnosis. Therefore, an analysis of biomechanical atomistic constitutive models is performed. Method: The bibliographic review method was used to identify, evaluate and synthesize the existing scientific evidence on the topic of atomistic modeling of blood vessels. Results and Discussion: A bibliographic study is presented, which highlights the benefits, limitations and potential of the following models: atomistic-continuous model of collagen mechanical properties, concurrent atomistic continuum for crystalline materials and multiscale hierarchical model based on atomistic and molecular simulation. Implications of the research: The considerations presented serve as a basis for future research and mathematical modeling of the phenomenon of degradation and deformation of blood vessels, setting the guidelines for the future, suggesting alternative treatment strategies to combat clinical conditions derived from the transformations of the mechanical properties of the same. Originality/Value: This study contributes by giving order to the multiple bibliographic information related to the subject. The relevance of this research is evidenced in the conclusions issued by analyzing the atomistic-continuous, concurrent-continuous and multiscale hierarchical models.

CONSTITUTIONAL MODEL FOR DETERMINING THE RIGHTS OF SUBJECTS IN THE FIELD OF SUBSOIL USE

The article analyzes the Constitutional model for determining subjects' rights in the field of subsoil use in Kazakhstan. The role of the Republic of Kazakhstan's Constitution in forming the legal basis for the use and protection of subsoil, including the rights of the state and individuals, is studied. The main provisions regulating access to natural resources and mechanisms for their protection are considered. It examines the pivotal role of the Constitution of the Republic of Kazakhstan in establishing a legal framework for the use and protection of subsoil resources, including the delineation of rights for both the state and individuals. Attention is focused on balancing interests between public and private entities and the need to comply with environmental standards. It analyzes comparative approaches from other countries to identify best practices and potential pitfalls that Kazakhstan could avoid. Furthermore, it emphasizes the need for greater stakeholder engagement in decision-making processes to ensure that the rights and interests of local communities are adequately represented and protected. As a result of the analysis, current problems of legal regulation are identified, and recommendations are proposed for improving legislation in subsoil use, aimed at ensuring sustainable and rational use of natural resources in Kazakhstan.

Influence of Fused Deposition Modeling Process Parameters on Constitutive Model of Hyperelastic Thermoplastic Polyurethane.

Thermoplastic polyurethanes (TPUs) are suited for fused deposition modeling (FDM) of parts that require high levels of flexibility and strength. Predicting the deformation of TPU parts produced using FDM may be difficult, especially under large deformations, as their constitutive models depend on the printing process parameters. The lack of understanding led to the absence of constitutive models for TPU parts produced using FDM. This work aims to identify accurate hyperelastic constitutive models. Six groups of uniaxial tensile specimens were produced using FDM. These groups were made with variations in two process parameters, which were infill geometry and extrusion nozzle temperature. Infill geometries either corresponded to a zero-deposition angle (wall-only) or an infill deposition of ±45° raster angle (infill-only). It was determined that a third-order Mooney-Rivlin constitutive model can accurately describe these six groups. A finite element analysis (FEA) of the experiments using the proposed constitutive models resulted in limited errors for all groups. The proposed approach was verified through a combination of experiments and FEA of FDM TPU components undergoing large deformation.

Isochoric unsteady multipolar spherical oscillations in compressible radial and cylindrical background flows

The multipolar spherical vortex solutions to the Euler and Navier–Stokes equations in background cylindrical flow with swirl admit an additional background divergent radial flow with arbitrary time-dependent amplitude. In this case the radial wavenumber $k$ , fundamental frequency $\omega$ and overall amplitude $U$ of the multipolar mode superposition become time-dependent and related functions. Assumption of an additional constraint, as a constitutive equation defining the time evolution of the spatially homogeneous divergence of the background flow, is required for the time evolution of the total flow to be completely evaluated from the initial conditions. It is found that flow compression implies an increase of the absolute values of the fundamental frequency $\omega$ and overall velocity amplitude $U$ of the oscillations.

A new shear deformation micro-beam model based on a nonlocal elasticity theory

In this article, a nonlocal refined shear deformation beam theory is proposed to analyze the free vibration behavior of FG nanobeams. The formulation is based on the nonlocal differential constitutive equations of Eringen that accounts for small scale effects. Material properties vary continuously through thickness according to a power law distribution. The study is performed within the framework of a new parametric shear deformation theory which provides parabolic transverse shear strains across the thickness direction and hence, does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the nanobeam are satisfied rigorously. Equations of motion of FG nanobeam are derived from both the nonlocal differential constitutive relations of Eringen and Hamilton’s principle. The system of differential equations is solved using the Navier solution technique. The parametric study is conducted to examine the effects of volume fraction index, length scale parameter and length ratio on the free frequencies of vibration of FG thin and thick nanobeams.

Constitutive model of lime-stabilized laterite considering curing time and softening characteristics

Constitutive model of lime-stabilized laterite considering curing time and softening characteristics

Vortical waves in a quantum fluid with vector, axial, and helical charges. I. Non-dissipative transport

Due to the spin-orbit coupling, Dirac fermions, submerged in a thermal bath with finite macroscopic vorticity, exhibit a spin polarisation along the direction parallel to the vorticity vector Ω. Due to the symmetries of the Lagrangian for free massless Dirac particles, there are three independent and classically conserved currents corresponding to the vector, axial, and helical charges. The constitutive relations for the charge currents and the stress-energy tensor at thermal equilibrium, derived in the framework of quantum field theory at finite temperature, reveal vorticity-induced contributions that deviate from the perfect fluid form. In this paper, we consider the mode structure of the corresponding hydrodynamical theory and derive collective excitations associated with coherent fluctuations of all three charges. We show that the chirally imbalanced rotating fluid should possess non-reciprocal gapless waves that propagate with different velocities along and opposite to the vorticity vector. We also uncover a strictly unidirectional mode, which we call the axial vortical wave, propagating in the background of the axial charge density. The emergence of this wave can be traced back to earlier studies of vortical chiral fluids in a hydrodynamic approach. We also point out an unexpected instability in the limit of degenerate matter and discuss possible solutions when helicity and axial charge non-conservation are taken into account.

Constitucionalismo do Brasil Imperial em Perspectiva Comparada

The 1824 Constitution was the result of the application of comparative constitutionalism, a technique that analyzes different constitutional models to identify the best forms of political organization. In Brazil, the debate between the “republican monarchy” model (national sovereignty, unicameral assembly, centralized state) and the “balanced monarchy” model (shared sovereignty, bicameralism, checks and balances) shaped the constituent process. The “republican monarchy” model was advocated by nativists and drew from the French Constitution of 1791 and the Spanish Constitution of 1812. Meanwhile, the “balanced monarchy” model, inspired by the British Constitution and the French Charter of 1814, was supported by Luso-Brazilians. The constitutional project of the French monarchist party, which proposed the monarch as the representative of the nation, was adopted by the Crown’s advisors. The acclamation of D. Pedro I was reinterpreted as a plebiscitary act, granting him legitimacy as the representative of the nation. The result was a hybrid model: national sovereignty, a centralized state, and a mixed system of government with a Moderating Power. The Constitution was sworn by D. Pedro I in 1824, an act justified by a doctrine that granted him the prerogative to enact it.

Model representations of the theory of thermal shock of viscoelastic bodies

Model representations of the theory of thermal shock of viscoelastic bodies based on two different approaches are considered. In the first approach, based on the introduction of stress and strain deviators using linear rheological models of Maxwell and Kelvin, new integral and differential relations are proposed, including simultaneously dynamic and quasi-static models for viscoelastic and elastic media, generalizing the results of previous studies. The proposed constitutive relations of the new form are applicable to describe the thermal response of bodies of canonical shape, limited by the boundaries of a rectilinear shape in Cartesian coordinates and are extended to the case of curvilinear boundaries in cylindrical and spherical coordinates. The second approach describes an elastic-viscoelastic analogy, which consists in the fact that the original problem of temperature stresses of a viscoelastic body can be reduced to the equivalent problem of thermoelasticity by replacing the shear modulus and Poisson’s ratio in the operational (according to Laplace) solution of the thermoelastic problem with their images as in the model Maxwell and in the Kelvin model. It is shown that after performing the inverse transformation, an analytical solution to the problem for a thermoviscoelastic medium is found. An illustrative example is given and the differences in the thermal response to sudden heating of an elastic and viscoelastic medium are analyzed.

Investigation of the Impact Response of Bi‐Continuous Nanoporous Solids via the Material Point Method: Verification Against Molecular Dynamics Predictions

ABSTRACTMolecular dynamics (MD) and the material point method (MPM) are both particle methods in spatial discretization. Molecular dynamics is a discrete particle method that is widely applied to predict fundamental physical properties and dynamic materials behaviors at nanoscale. The MPM is a continuum‐based particle method that was proposed about three decades ago to simulate large‐deformation problems involving multiphase interaction and failure evolution beyond the nanoscale. However, it is still a challenging task to validate MD responses against the experimental data due to the spatial limitation in impact and/or shock tests. The objective of this investigation is therefore to compare the MPM and MD solutions for the impact responses of porous solids at nanoscale. Since the governing equations for MD and explicit MPM are similar in temporal domain with different spatial discretization schemes, the MPM solutions could be verified against the MD ones, and the MD solutions might then be indirectly validated against the MPM ones as validated beyond the nanoscale. Since both MD forcing functions and MPM constitutive modeling are well‐formulated for metallic solids, we report a comprehensive comparative study of porous and non‐porous gold cubic targets impacted by full density non‐porous gold cubic flyers using the MPM and MD, respectively. The overall deformation patterns and particle‐velocity histories are demonstrated and analyzed, as obtained with the two particle methods. It appears that the MD and MPM solutions are consistent in capturing the physical responses, which shows the potential of using the MPM for multiscale simulations of extreme events involving porous solids, such as underground penetration and space exploration. In addition, MD solutions might be indirectly validated against the MPM ones for evaluating geological responses to extreme loadings, which provides an alternative route for multiscale verification and validation.

Sensitivity Analysis of the Johnson-Cook Model for Ti-6Al-4V in Aeroengine Applications

Titanium alloys, such as Ti-6Al-4V, are crucial for aeroengine structural integrity, especially during high-energy events like turbine blade-out scenarios. However, accurately predicting their behavior under such conditions requires the precise calibration of constitutive models. This study presents a comprehensive sensitivity analysis of the Johnson-Cook plasticity and progressive damage model parameters for Ti-6Al-4V in blade containment simulations. Using finite element models, key plasticity parameters (yield strength (A), strain-hardening constant (B), strain-rate sensitivity (C), thermal softening coefficient (m), and strain-hardening exponent (n)) and damage-related parameters (d1, d2, d3, d4, and d5) were systematically varied by ±5% to assess their influence on stress distribution, plastic deformation, and damage indices. The results indicate that the thermal softening coefficient (m) and the strain rate hardening coefficient (C) exhibit the most significant influence on the predicted casing damage, highlighting the importance of accurately characterizing these parameters. Variations in yield strength (A) and strain hardening exponent (n) also notably affect stress distribution and plastic deformation. While the damage evolution parameters (d1–d5) influence the overall damage progression, their individual sensitivities vary, with d1 and d4 showing more pronounced effects compared to others. These findings provide crucial guidance for calibrating the Johnson-Cook model to enhance aeroengine structural integrity assessments.

The Influence of Fresh Latex Coagulation on the Parameter Characteristics of the Yeoh Hyperelastic Constitutive Model for Natural Rubber.

The coagulation of fresh latex is one of the critical processes that impacts rubber quality during natural rubber processing. Constitutive relationships are the basis for the study of the mechanical properties of rubber materials and serve as a prerequisite for material simulation studies. However, studies on the effect of different coagulation methods on natural rubber constitutive relationships have yet to be carried out, and the current models used for natural rubber constitutive relationships need to be improved. In order to investigate the effects of different coagulation methods on the hyperelastic properties of natural rubber, the impact of natural coagulation, enzyme coagulation, acid coagulation, microbial coagulation, and enzyme-assisted microbial coagulation on the hyperelastic constitutive relationship of natural rubber were analyzed in detail based on tensile experiments and the Yeoh model. The results show that after introducing a strain rate-related factor, the Yeoh model can describe well the mechanical behavior of natural rubber carbon black composites in different deformation regions, and the rubber, studied with varying coagulation methods, exhibits different mechanical properties in different deformation regions. This study provides new evidence for the study of high-performance natural rubber and serves as a reference for process selection in the primary processing of natural rubber.

Analysis and Simulation Research on Static Softening Mechanism Following Multistage Hot Deformation of 1.3 GPa Grade Bulb Flat Steel

In this study, the flow stress behavior of 1.3 GPa grade bulb flat steel (BFS) is studied under different deformation temperature and strain rate using the Gleeble‐3800 thermomechanical simulator, and the Arrhenius‐type constitutive equation is established. A static recrystallization kinetics model of 1.3 GPa grade BFS is established based on double‐pass hot compression tests. The established constitutive equation and static recrystallization kinetics model are embedded into the finite‐element model of the hot‐rolling process, and the static recrystallization and residual stress distribution between each pass are analyzed. In the results, it is indicated that static softening happens rapidly after the end of the pass, and then the speed gradually slows down. Reasonably controlling the interval time between each pass helps to fully utilize the effect of static softening relaxation residual stress. After the K12–K8 and K7–K2 passes, the rolled pieces can undergo fully static recrystallization after being heated for 5 and 8–11 s, respectively, eliminating residual stresses during hot rolling and reducing the impact on plate shape. The results give data support for the optimization of the hot‐rolling process of 1.3 GPa grade BFS. It provides a new way to control the shape accuracy of hot‐rolling‐section steel.

Characterizing nonlinear piezoelectric dynamics through deep neural operator learning

Nonlinear hysteresis modeling is essential for estimating, controlling, and characterizing the behavior of piezoelectric material-based devices. However, current deep-learning approaches face challenges in generalizing effectively to previously unseen voltage profiles. This Letter tackles the limitation of generalization by introducing the notion of neural operators for modeling the nonlinear constitutive laws governing inverse piezoelectric hysteresis, specifically focusing on the relationship between voltage inputs and displacement responses. The study utilizes two neural operators—Fourier neural operator and the deep operator network—to predict material responses to unseen voltage profiles that are not part of the training data. Numerical experiments, including butterfly-shaped hysteresis curves, show that in accuracy and generalization to unseen voltage profiles, neural operators outperform traditional recurrent neural network-based models, including conventional gated networks. The findings highlight the potential of neural operators for modeling hysteresis in piezoelectric materials, offering advantages over existing methods in varying voltage scenarios.