Nonlinear Strain Research Articles

Non-linear elasticity, earthquake triggering and seasonal hydrological forcing along the Irpinia fault, Southern Italy.

Pump-probe experiments investigate the strain sensitivity of crustal elastic properties, showing nonlinear variations during the strain cycle. In the laboratory, pre-seismic reductions in seismic velocity indicate that asperity contacts within the fault zone begin to fail before the macroscopic frictional sliding. The recognition of such effects in natural seismic-cycles has been challenging. Here we exploit seasonal hydrological strains, performing a natural analogue to a quasi-static laboratory pump-probe experiment to investigate the nonlinear strain sensitivity of crustal rocks and its role in seismic failure along the tectonically-active Irpinia Fault System (Southern Italy). By comparing 14-years-long series of spring discharge, strain, seismic velocity variations and earthquakes rate, we find that seismicity peaks during maximum hydrological forcing and minimum seismic velocity. Seasonal strains of ~10-6 are required for both earthquake triggering and significant nonlinearity effects arising from modulus reduction. We suggest that, for faults in a critical state, cyclical softening may lead to failure and seasonal seismicity.

Nonlinear large strain mechanics and failure of 45° woven CFRP laminates characterized by Full-Field measurements

Nonlinear large strain mechanics and failure of 45° woven CFRP laminates characterized by Full-Field measurements

Free vibration of flexible two links manipulator with variable cross-sectional areas

This paper demonstrated the free vibration of a planar two-link flexible manipulator with harmonic drivers and non-uniformity in the cross-section of links. A dynamic model has been developed that incorporates flexibilities in both link and joint experiencing harmonic vibration, with joint flexibility modeled as torsional spring-inertia components combination. Two links were modeled based on Von Karman’s nonlinear strain relations and theory of the Euler-Bernoulli beam. The nonlinear governing equations were derived using the extended Hamilton’s principle for planar two-link flexible manipulators with variable cross-sections for each link. The coefficients of the obtained partial differential equations were as a function of spatial coordinates and were reduced to the ordinary differential equations for a family of cross-section geometries with exponentially varying widths of rectangular sections of each link. The frequency equation was obtained and analytical solutions of the vibration were achieved for different exponential shapes of each link (widening and narrowing beam for each link). Natural frequencies and mode shapes were presented for varying cross-section areas of the links. The effects of the end masses, payload, beam mass density, flexural rigidity ratio, and joint frequency were investigated on the mode shapes and natural frequencies when the power of the exponential function of the cross-sectional areas of two links was varied. The numerical results were verified with experimental results.

Nonlinear buckling and free vibration analysis of auxetic graphene origami composite beams under nonuniform thermal environment

This study examines the thermo-mechanical behavior of auxetic metamaterial beams enhanced by graphene origami (GOri) under spatially varying nonuniform temperature distributions (SVTD). Utilizing Timoshenko beam theory considering von-Kármánn type nonlinear strain–displacement relationship, GOri beams are modeled as layered structures. The Ritz method is employed to solve equilibrium equations, analyzing the impact of GOri distribution patterns, content, and folding degree on post-buckling and vibration paths. The effects of five SVTDs, three end conditions, and three GOri distribution patterns on buckling, post-buckling behavior, and nonlinear free vibration characteristics are explored. Findings reveal that the parabolic temperature distribution with peak temperatures at beam ends (P-MAE) results in higher critical temperatures and nonlinear free vibration frequencies. This research provides crucial insights into the design and optimization of GOri-enabled metamaterial structures in complex thermal environments, highlighting the significant influence of nonuniform temperature distributions along the beam’s length.

Galerkin’s Method and Multivariate Newton’s Method for the Nonlinear Deformation of the Magneto-Electro-Elastic Bi-Layered Laminates

In this paper, mathematical modelling for the large deformation of a magneto-electro-elastic rectangular bi-layered laminate with general boundary conditions is presented. Constitutive equations involving the magneto-electro-elastic (MEE) material properties are introduced, Maxwell equations accounts for the electric and magnetic effects are also utilized. First-order shear deformation theory (FSDT) considering the von Karman nonlinear strain is adopted, and the plain strain/stress assumption applicable for thin plate analysis is used. A rather compact set of governing equations related to kinematical variables, electric/magnetic potentials and the Airy stress function is obtained as a consequence of the preliminary condensation for the electro-magnetic state to the plate kinematics. Semi-analytic solution for a bi-layered BaTiO3-CoFe2O4 laminate with specified boundary conditions subjected to various external applied loads is performed. By employing the Bubnov-Galerkin method, the set of nonlinear partial differential equations is transformed to a set of third-order nonlinear algebraic equations for the static deformation due to applied load. Numerical results are carried out by using the multivariate Newton's method with respect to various volume fractions indicating the volume ratio between piezoelectric BaTiO3 layer and piezomagnetic CoFe2O4. From the result, the nonlinearity of the von Karman strain appears to enhance system rigidity as smaller deformations will be detected when external load is applied. Also, some other interesting results are obtained which could be useful to future analysis and design of multiphase composite plates.

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

A Strain-Controlled Finite Strain Model for CRD Consolidation of Saturated Clays Considering Non-Linear Compression and Permeability Relationships

Consolidation is the combined phenomenon of the compression and groundwater seepage of clay. Accurate evaluation of the consolidation characteristic is essential for the design, construction, and long-term stability of geotechnical structures. In this study, a strain-controlled non-linear finite strain model for constant rate-of-deformation (CRD) consolidation was developed for quickly and reliably predicting the consolidation behavior of clay soils. The model can account for any form of non-linear compression and permeability relationships, thus considering variations in the coefficient of consolidation. Being strain-controlled, it overcomes the limitations of stress-controlled models which require complex numerical iteration. The validity and accuracy of this model were verified through rigorous comparisons with both numerical simulations and experimental data. For normally consolidated soils, a non-linear e-lgσ′compression model was used instead of a linear compression model. For overconsolidated soils, the Harris function compression model was determined to be recommended to overcome the discontinuities in total stress and pore pressure caused by the traditional piecewise e-lgσ′ model. It was also found that determining the steady state of consolidation for normally consolidated soils should use the non-linear method, while the linear method is suggested to be adopted for overconsolidated soils.

Anisotropic formability and deformation mechanism of near-α titanium alloy sheet under continuous nonlinear strain paths at high temperature

In the present study, the anisotropic formability and underlying deformation mechanism of near-α TA32 titanium alloy sheet under continuous nonlinear strain paths (CNSPs) at high temperature were investigated in depth. To achieve this goal, a new experimental method combining hot gas bulging with step-combined dies was proposed, and the hot CNSPs of metal sheets can be flexibly and conveniently realized by changing the number and shape of step-combined dies. Based on this method, the anisotropic deformation behavior and forming limits of TA32 sheet under fifteen CNSPs were tested at 800 ℃ with a strain rate of 0.001 s−1. Then, two advanced constitutive models with different scales were embedded into the classical Marciniak-Kuczyński (M-K) theory to predict the forming limits of TA32 sheet under different strain paths: the macro-scale viscoplastic model coupled with Hill48 yield criterion and non-associated flow rule (NAFR) as well as the meso-scale three-dimensional crystal plasticity finite element (CPFE) model coupled with cellular automata (CA). The results demonstrate that the CPFE-CA-MK coupled model exhibits higher accuracy in predicting the forming limits of TA32 sheet under linear and continuous nonlinear strain paths. Especially for the tension-tension strain paths, the CPFE-CA-MK coupled model improves accuracy by at least 3.1 % compared to macro-scale models. Due to the material anisotropy, the initial inclination angle of the groove in the CPFE-CA-MK model is closely related to the strain path and significantly affects the prediction accuracy. Based on the CPFE simulation, the effects of anisotropy and strain path change on the dislocation slip mode of different texture components was analyzed in depth, which provides a theoretical guidance for the optimization of hot forming process of TA32 titanium alloy complex components.

Thermal postbuckling and thermally induced postbuckled flutter of tri-directional functionally graded plates in yawed supersonic flow

The current work examines the thermal postbuckling, aero-elastic flutter and thermally induced postbuckled flutter about a static equilibrium state of tri-directional functionally graded (TFGM) rectangular and tapered plates in yawed supersonic flow. Based on the von Kármán nonlinear strains, the general higher-order shear deformation theory (GHSDT) and the first-order piston theory, the governing equations of motion for TFGM plates in yawed supersonic flow are established via the Hamilton's principle. The isogemetric analysis (IGA), the load continue strategy associated with the Newton Raphson iterative technique are exploited synthetically to capture the postbuckling paths, and then the postbuckled flutter behaviors are acquired with the static equilibrium state being obtained. Comparative and convergence studies are provided to improve reliabilities of the present material model, formulae and code implementations. Subsequently, detailed parametric investigations are carried out to evaluate the influences of material volume fractions, boundary conditions and airflow angles on the thermal postbuckling, aero-elastic flutter and postbuckled flutter behaviors of TFGM plates. The results show that it is the in-plane volume fractions, rather than the thickness volume fraction, that exert a greater influence on the thermal postbuckling and flutter behaviors of TFGM plates. Furthermore, TFGM plates exhibit more diverse postbuckling paths and more complex deformations due to the inhomogeneity of the in-plane material properties compared with z-directional FGM plates.

Dairy versus non-dairy cheese texture: Sensory and instrumental contrasts.

With growing consumer demand for plant-based products that mimic the eating experience of animal-based products, there is a need for improvement in instrumental measurements of sensory texture. This study aimed to characterize textural differences between dairy and non-dairy cheeses, and to investigate whether Large Amplitude Oscillatory Shear (LAOS) rheometry could discriminate sensory texture better than Texture profile analysis. Commercial dairy and non-dairy cheddar, mozzarella, and cream cheese were selected to provide a wide range of textures. Sensory evaluation used the check-all-that-apply methodology with 73 consumers. Texture profile analysis force-distance data were analyzed empirically, and also converted to stress and strain (see https://shiny.csiro.au/texture_dash). The major textural differences between dairy and non-dairy cheddar were related to structural cohesion, according to both instrumental measures (dairy cheddar had 1.5-fold higher failure stress and 2.2-fold higher failure strain) and sensory measurements (dairy cheddar was more chewy and less crumbly). In contrast, cream cheeses showed similar textural properties using sensory testing but significant instrumental differences (non-dairy cream cheese had 5.7-fold higher modulus of deformability, 4.7-fold higher failure stress). For mozzarella, there were large differences in both sensory attributes (chewy, crumbly, jelly-like, stretchy) and instrumental parameters (13.6-fold difference in modulus, 2.7-fold difference in failure stress). LAOS rheometry gave insights into the mechanisms by which samples absorbed or dissipated mechanical energy at nonlinear strains. The LAOS parameter correlated well with sensory attributes creamy, fatty/oily, and moist, indicating the potential of this technique to measure structural phenomena linked to sensory attributes that resonate with consumers.

Nonlinear dynamic modeling and vibration analysis for flexible tethered satellite systems under large deformations

The escalating threat posed by space debris necessitates innovative approaches for its removal. This article presents a comprehensive mathematical modeling and dynamic analysis of a flexible tethered satellite system (FTSS) in the post-capture phase, taking into account nonlinear strain effects for the flexible appendages. The FTSS consists of a rigid central body satellite, two identical flexible panels, and a towing mass that manipulates the satellite through a tether. We develop a detailed dynamic model that accurately represents the system’s three-dimensional motion, incorporating the rigid body’s 6 degrees of freedom (DOF), the tether’s 3 DOF motion, and two modal generalized coordinates for each panel. The extended Hamilton method is employed to derive this model, enabling us to investigate the conditions under which nonlinear strain considerations become critical. The findings of this article reveal that nonlinear strain effects can profoundly impact the system’s dynamics under certain geometric and forcing conditions, often overlooked in previous studies. In these scenarios, the panel strains exceed the linear regime, necessitating the use of nonlinear models for accurate representation. This work emphasizes the importance of incorporating nonlinear strain terms in the dynamic analysis of FTSSs, particularly when dealing with large deformations and high-amplitude vibrations encountered in debris removal applications.

Hamiltonian formalism for bistable-multilayered plates under non-mechanical stimuli

Bistable multilayered plates have attracted significant attention as morphing and adaptive structures, renowned for their unprecedented exceptional performance. However, accurately predicting both their shape and internal stresses poses formidable challenges due to their geometrically nonlinear nature. For the first time, this paper introduces a Hamiltonian formalism aimed at achieving high-resolution analysis of nonlinear multilayered plates subjected to non-mechanical stimuli, including hygro-thermo-electro-magneto-elastic responses. A canonical system is strategically developed to compute membrane behaviors, yielding symplectic dual differential equations for in-plane field variables. This method elegantly decouples out-of-plane variables from the full-state vector, leading to an exact analytical solution in the membrane problem. To predict the bending behaviors, the first variation of the Hamiltonian energy density function, expressed as a power series of the transverse deflection, ensures flexural equilibrium. The power series effectively captures all admissible out-of-plane deformations, enabling a smooth transition between linear and nonlinear plate responses, including pitchfork bifurcation and limit points. The validity and accuracy of the proposed method are rigorously assessed through convergence tests and satisfaction of boundary conditions across various equilibria, ranging from monostability to bistability. It is cross-verified with high-fidelity finite-element methods (FEM), showing excellent agreements in both deformations and stress resultants. This research presents a physics-based methodology that unveils a parametric interplay between in-plane and out-of-plane field variables, serving as an efficient approach for analyzing adaptive and morphing structures.

Constitutive description of distortional hardening in a TWIP steel: Addressing differential hardening under nonlinear strain paths

The present study aims to describe the in-plane differential hardening behaviour of the twinning induced plasticity sheet metal TWIP980 under various stress states, including uniaxial tension, plane strain tension, and pure shear, particularly focusing on non-proportional loading conditions. The true stress–strain curves for each stress states were inversely obtained from their corresponding load–displacement curves and modeled using a differential hardening model that can accommodate all three stress states simultaneously on plastic work (density) contours. For non-proportional loading tests, oversize specimens were initially stretched under uniaxial tension up to a 10% pre-strain along the rolling, diagonal, and transverse directions, respectively. Subsequently, the three stress states were applied to subsize specimens cut from the deformed oversize specimens along the rolling direction. To describe the hardening behaviours during non-proportional loading, a homogeneous anisotropic hardening model was adopted and calibrated using two-step uniaxial tension tests. Subsequently, the differential hardening model was successfully incorporated into the homogeneous anisotropic hardening model to describe both the differential hardening and the strain path change-induced hardening behaviours under the two-step loadings, i.e., uniaxial tension to pure shear and uniaxial tension to plane strain tension. Both experimental and simulation results underscore the necessity to consider differential hardening under non-proportional loading conditions.

Finite element analysis of a rubber bearing base isolator under vertical and horizontal loads using a nonlinear hyper-viscoelastic material model

This research is devoted to developing a finite element model using Abaqus code for the computer simulation of an in-house developed and manufactured rubber bearing subjected to static vertical and cyclic horizontal loads. A high-damping rubber compound was designed. The material behavior of the rubber was assumed to be described by the hyper-viscoelastic model. Both linear (Prony series) and nonlinear (strain hardening power law) viscoelastic relationships were used in conjunction with the Ogden-Roxburgh equation to take the addition of the stress softening phenomenon or Mullins effect into consideration. The parameters of the material model were determined using MCalibration code in which an optimization technique was used, and data obtained in experiments carried out on test specimens were fitted into the selected model. The results of the simulations were compared with their corresponding experimental data carried out on the rubber bearing. The force-displacement behavior, stress and strain fields, and computed energy were presented and discussed. It is shown that the nonlinear viscoelastic model accompanied by the Mullins effect gives the best results. Moreover, the model could accurately predict the energy variations during the earthquake loading.

Dynamic deformation characteristics of lightweight soil based on Davidenkov model

The dynamic deformation characteristics of lightweight soil were studied using consolidated-undrained dynamic triaxial tests to predict its non-linear stress and strain behaviour under a traffic load. The results showed that the influence of confining pressure on the attenuation of the dynamic shear modulus ratio became significant with increasing expanded polystyrene (EPS) particle content or cement content. The attenuation curve of the dynamic shear modulus ratio was evaluated using Davidenkov model parameters, A, B and γ 0. It was proved that the parameter A was mostly affected by EPS particle content and confining pressure, with its value ranging between 1.1 and 1.85. The parameter γ 0 was affected by cement content and confining pressure, and its value ranged between 0.0007 and 0.0016. The parameter B was found to range between 0.38 and 0.55, in general. The relationship curves between damping ratio and dynamic shear strain revealed two forms, S shaped and bell shaped. The verification tests showed that the Davidenkov model could predict accurately the dynamic deformation characteristics of lightweight soil both under a changing stress state and under constant gradient loading conditions. These results provide a theoretical basis for determining the deformation of a lightweight soil subgrade under a traffic load.

Micro-milling machinability prediction for crystalline materials via numerical-analytical hybrid modelling and strain rate-dependent grain-scale simulation

Micro-milling is a promising technology to produce miniature-sized components, while the machinability prediction of crystalline material micro-milling is difficult due to the inhomogeneous microstructure, anisotropic properties, nonlinear strain rate sensitivity and complex process characteristics. To achieve accurate and effective micro-milling machinability prediction of crystalline materials, this paper proposes a numerical-analytical hybrid method that comprehensively and simultaneously considers core material and process characteristics of the crystalline material micro-milling technique, which evaluates micro-milling forces and surface roughness via a series of adaptive strain rate-dependent micro-orthogonal cutting emulations in grain-scale and a set of analytical transformations. Taking Inconel-718 as the sample material, grain-scale mechanical parameters under 10−2–104 s−1 strain rate range were calibrated. Strain rate effect-considered grain-scale constitutive theory is proposed and strain rate-dependent micro-orthogonal cutting emulations were executed, which were experimentally verified to be accurate and robust on the forces and surface roughness prediction of micro-cutting. Furthermore, numerical-analytical hybrid modelling and machinability prediction of Incoenl-718 micro-milling process under 12 mm/min and 20 mm/min feed rates were carried out, the relative prediction errors of micro-milling forces and surface roughness are <15 % and 14 %, which confirm that the proposed hybrid method is accurate and robust on the micro-milling machinability prediction of crystalline materials.

Visco-Hyperelastic Constitutive Modeling for High-Damping Rubber Materials During Combined Quasi-Static Compression–Cyclic Shear Deformation Process

High-damping rubber materials utilized in high-damping rubber isolation bearings are frequently subjected to multiple deformations during the occurrence of earthquakes. Typically, large combined compression–shear deformations of the material could potentially cause compressive shear damage to rubber bearings. During this process, visco-hyperelastic properties of rubber materials will greatly change, which would significantly impact the seismic performance of rubber bearings. Thus, to give out a deep insight into their variations, it is necessary and urgent to develop a high-performance numerical method to investigate this process. This paper proposed a visco-hyperelastic constitutive modeling approach for high-damping rubber materials based on the experimental assessment of combined quasi-static compression–cyclic shear deformation process. Within the thermodynamic framework, the Clausius–Duhem inequality associated with the intrinsic dissipation of the material was firstly derived in accordance with the Lagrangian formulism. Then, stress–strain relations were obtained upon considering the occurrence of entropy production due to viscous dissipation. In the model, Stumpf–Marczak strain energy density function, which satisfies the Baker–Ericksen (B–E) inequality, was harnessed to describe the hyperelasticity of the material. By introducing higher orders of strain and strain rates and taking their couplings into account, a generalized viscous dissipation potential was proposed to capture nonlinear strain and strain rate-sensitivity effects of the material. To identify constitutive parameters, the deformation gradient was particularized for the combined quasi-static compression–cyclic shear deformation process. And, an inverse identification procedure was carried out at different levels of compression stress. The prediction results revealed that the proposed model exhibits remarkable prediction ability and adaptivity for different rubber materials during this process. Several new insights were highlighted on the variations of visco-hyperelastic characteristics of high-damping rubber materials with respect to the compression stress. The accuracy of the model was further validated by design parameters including initial shear modulus, secant shear modulus and equivalent viscous damping factor. This work could provide a fundamental guideline for the optimization and reliability analysis of high-damping rubber isolation bearings used in the field of seismic engineering.

Axisymmetric defect originated Al–CFRP interface thermo-elastic fracture study of type-III composite vessel and inverse identification of cohesive zone parameters

Debonding existence at the interface of Al liner and overwrapped CFRP laminates may endanger the structural integrity of type-III composite vessel. In this study, CFRP and GFRP wound composite pressure vessels with Teflon sheet inserted artificial axisymmetric defect in Al–CFRP interface and without any defect were manufactured and nitrogen gas inside cooling experiments of the vessels were conducted. During the cooling axial strain and temperature responses at distinct locations of outer GFRP hoop surface were monitored to study the thermoelastic crack growth along Al–CFRP interface. The nonlinear strain response obtained under temperature change in the defect bearing specimen was considered as a result of Al–CFRP interface crack growth. Furthermore, to expose the Al–CFRP interfacial cross section of post-cooling specimen, destructive cutting and optical microscopy were performed. The Al–CFRP cross sectional microscopic observations at cutting planes showed Al–CFRP interface crack growth due to cooling in the defective specimen. Afterwards, the GFRP hoop surface axial strain experimental data fitted inversely to the surface based cohesive zone modeling (CZM) simulation results to estimate the Al–CFRP interface cohesive strength, and fracture energy.

Nonlinear wave localization in an acoustic metamaterial with attached masses through one element of main chain

The nonlinear strain solitary waves in a metamaterial mass-in-mass lattice model with a di-atomic symmetry is studied when attached masses tare located through one element of main chain are studied in the continuum limit. The reasonable variables are introduced to account for strains. An asymptotic procedure is developed to reduce coupled continuum equations to a single equation for longitudinal strains. A correspondence with the classic model with a di-atomic symmetry is established. An improvement of the model is suggested to account for a switch-on/off of the parts of the metamaterial model arising due to the studied kind of symmetry. It allows us to change the behavior of the localized nonlinear strain waves. This improvement is important for description of a durability of the metamaterial. Also it helps to understand new heat processes, like thermal diode, in complex symmetric lattices.

Uniaxial tensile behavior of engineering cementitious composite reinforced with fiber textile grid and steel wire mesh

The Textile Reinforced Engineering Cementitious Composite (TRECC) is a cementitious composite material reinforced with short polyvinyl alcohol (PVA) or polyethylene (PE) fibers, possessing exceptional ductility. This unique attribute enhances the utilization of the textile grid’s strength and significantly improves the load-bearing and deformation capabilities of ECC members reinforced with textile grids. In this study, the effects of fiber textile grids and steel wire mesh on the tensile properties of ECC were investigated. The test results showed that the carbon textile grid significantly improved the peak strength of specimens. On the other hand, the glass textile grid significantly improved the ultimate strain of specimens. The carbon combined with glass textile grids can integrate effectively the advantages of both carbon and glass textile grids. When compared with single textile grids, the combination uses of steel wire mesh and fiber textile grids resulted in a significant improvement in ultimate strain and post-peak energy absorption values. Finally, based on the test data, this study proposed the peak strength calculation of TRECC and provided the stress–strain expressions of the initial non-linear strain hardening phase and the tensile softening phase and a related analytical model to describe the stress–strain behavior of TRECC.