Softening Effect Research Articles

Systematic softening in universal machine learning interatomic potentials

Machine learning interatomic potentials (MLIPs) have introduced a new paradigm for atomic simulations. Recent advancements have led to universal MLIPs (uMLIPs) that are pre-trained on diverse datasets, providing opportunities for universal force fields and foundational machine learning models. However, their performance in extrapolating to out-of-distribution complex atomic environments remains unclear. In this study, we highlight a consistent potential energy surface (PES) softening effect in three uMLIPs: M3GNet, CHGNet, and MACE-MP-0, which is characterized by energy and force underprediction in atomic-modeling benchmarks including surfaces, defects, solid-solution energetics, ion migration barriers, phonon vibration modes, and general high-energy states. The PES softening behavior originates primarily from the systematically underpredicted PES curvature, which derives from the biased sampling of near-equilibrium atomic arrangements in uMLIP pre-training datasets. Our findings suggest that a considerable fraction of uMLIP errors are highly systematic, and can therefore be efficiently corrected. We argue for the importance of a comprehensive materials dataset with improved PES sampling for next-generation foundational MLIPs.

Strength-ductility synergy of Mg-7.5 wt.% Sn binary alloys promoted by semisolid isothermal treatment coupled with hot extrusion

In this paper, we propose to adopt a new process for as-extruded Mg-7.5wt.%Sn binary alloy, i.e., semisolid isothermal heat treatment coupled with hot extrusion. Semi-solid isothermal treatment achieved nanosizing of micron-layer lamellar eutectic in the Mg-7.5wt.% Sn binary alloy, while subsequent hot extrusion crushed this nanoeutectic to nanoparticles, resulting in high strength (303 ± 2.8MPa for UTS and 246 ± 0.7MPa for YS,) and high elongation (11.5 ± 0.1% for EL) of the Mg-7.5wt.% Sn binary alloy. Grain refinement, multi-scale particles (broken nanoscale and submicron particles, and dynamically precipitated nanoscale phases), and high-density dislocations are the main contributors to the high strength. Meanwhile, the good ductility of the alloy is promoted by the softening effect due to dynamic recrystallisation as well as the weakening and random distribution of the texture.

Differences in Rejuvenation Mechanisms and Physical Properties of Aged Styrene-Butadiene-Styrene (SBS)-Modified Bitumen by Mono-Epoxy and Di-Epoxy Compounds.

Studying the mechanisms and effects of rejuvenators on SBS-modified bitumen is crucial for repairing degraded SBS and recycling aged SBS-modified bitumen (ASMB), thereby contributing to the sustainable development of bitumen pavements. This research examines the roles of mono-epoxy Alkyl (C12-C14) glycidyl ether (AGE) and di-epoxy 1,6-Hexanediol diglycidyl ether (HDE) under the catalysis of N,N-dimethyl benzyl amine (BDMA) in repairing degraded SBS chains. Aromatic oil (ORSMB)-, AGE-aromatic oil (ARSMB)-, and HDE-aromatic oil (HRSMB)-rejuvenated bitumen are analyzed for their chemical structures, physical properties, and rheological properties. Fluorescence microscopy (FM) and Fourier transform infrared spectroscopy (FTIR) reveal that HDE chemically reconnects degraded SBS chains, enhancing ASMB properties, while AGE improves ASMB properties through physical softening. HDE balances high-temperature properties and improves mid-temperature fatigue resistance through a rigid repair effect and flexible chain structure. AGE enhances mid-temperature fatigue resistance but significantly reduces high-temperature rutting resistance due to a softening effect. The findings demonstrate that HDE restores ASMB ductility chemically, while AGE improves crack resistance through physical softening. These differences in rejuvenation mechanisms provide a theoretical basis for optimizing rejuvenator design and advancing bitumen pavement recycling.

LDED-based additive-subtractive hybrid manufacturing of Inconel 718 superalloy: evolution of microstructure and residual stress

ABSTRACT The development of additive-subtractive hybrid manufacturing (ASHM) offers a near-perfect solution for the integrated preparation of complex structural components. In this study, we prepared additive manufacturing (AM), ASHM-fabricated Inconel 718 (IN718) superalloy based on laser-directed energy deposition (LDED), including two subtractive manufacturing (SM) situations, i.e. milling after cooling (room temperature SM) and immediately milling after AM (high-temperature SM). The evolution of microstructure, residual stress and milling forces for IN718 during a complete ASHM process (including AM and subsequent SM) were explored by experiments, finite element and molecular dynamics simulation. The SM process induces gradient plastic deformation on the sample surface, which led to the formation of nano-grains, low-angle grain boundaries and residual compressive stress. Owing to the higher internal temperatures present at SM, the thermal softening effect results in lower milling forces being subjected to high-temperature SM sample. Also, the dynamic recovery results in lesser nano-grains, low-angle grain boundaries and residual compressive stress in high-temperature SM sample. Thus, SM immediately after AM has a lower degree of plastic deformation on the sample surface compared to room temperature SM, and shows advantages in the preparation of difficult-to-machine materials. Highlights Inconel 718 (IN718) superalloy was prepared by LDED-based additive-subtractive hybrid manufacturing (ASHM). The evolution of microstructure, residual stress and nano-hardness for IN718 during ASHM were investigated. Subtractive manufacturing process of ASHM introduces nano-grains and residual compressive stress. Thermal softening effects and dynamic recovery affect the plastic deformation degree of ASHM-fabricated IN718.

Study on Damage Characteristics and Failure Patterns of Sandstone Under Temperature–Water Interactions

In modern tunnel construction, complex environments with high geothermal gradients and abundant groundwater are frequently encountered. To investigate the damage and failure mechanisms of sandstone under the combined effects of temperature and water, uniaxial compression tests were conducted on sandstone at different temperatures (25 °C, 55 °C, 85 °C, and 95 °C) and soaking durations (0.5 h, 1 h, and 3 h). The acoustic emission (AE) signals and energy evolution during the damage and failure processes were analyzed, revealing the damage characteristics and failure mechanisms of sandstone. The results indicate the following: (1) As the temperature increases, under the 3 h condition, the water content of sandstone is highest at 55 °C, reaching 3.01%, and the thermal expansion effect of sandstone is not obvious. Under the conditions of 85 °C and 95 °C, the thermal expansion effect leads to a decrease in the water content, enhances the water absorption softening effect, increases the plastic deformation capacity of sandstone, and weakens its brittle failure capacity. (2) When soaked for 0.5 h and 1 h, the maximum acoustic emission ring count and maximum acoustic emission energy of sandstone increase initially, then decrease, and subsequently increase again as the temperature rises, while the cumulative acoustic emission ring count gradually increases with temperature. Under the 3 h soaking condition, the maximum ring count, maximum energy, and cumulative ring count of sandstone at all temperatures show a consistent increasing trend with temperature. (3) The increase in soaking time reduced the damage variable of sandstone, with the largest reduction of 54.17% under the 3 h condition. At different temperatures, the damage variable of sandstone was smallest at 55 °C, only 0.33. (4) Sandstone primarily experiences tensile failure under different temperatures and soaking times. The extension of soaking time promotes the development of shear cracks, while the increase in temperature can effectively promote the expansion of tensile cracks. The research results provide certain theoretical references for the damage and failure of surrounding rock in modern tunnel construction.

Role of lattice distortion on spallation of CoCrCuFeNi high-entropy alloy

High-entropy alloys (HEAs), known for their high strength and enhanced ductility, have promising applications across various fields. Lattice distortion is a key factor in their strengthening, yet its role in dynamic fracture strength or spall strength remains unclear. This study employs large-scale nonequilibrium molecular dynamics simulations to investigate the dynamic responses of equiatomic CoCrCuFeNi HEA under shock velocities ranging from 0.6 to 1.45 km/s. By comparing the systems described using an average-atom interatomic potential, we uncover the role of lattice distortion. Our results reveal that spall strength exhibits complex behavior depending on the HEA's shock response. As shock velocity increases, the deformation mechanism transitions from elastic to dislocation and stacking fault (SF) dominated, eventually leading to a face-centered cubic to body-centered cubic phase transition. Lattice distortion significantly alters the active slip planes of dislocations and SFs, resulting in more SF intersections, while its effect on compression-induced phase transition is minor. During shock-induced spallation, residual defects after decompression significantly affect spall strength. Lattice distortion introduces additional stress and strain concentration sites, facilitating void formation and reducing spall strength. The temperature at the spall region is identified as a major factor governing spall strength variation under different shock velocities. Although lattice distortion can mitigate the softening effect of elevated temperature, it ultimately reduces spall strength, challenging the traditional views on its strengthening role. Moreover, the effects of lattice distortions on spall strength are quantified in terms of lattice misfit under varying loading strain rates and initial temperatures.

Effect of Surfactant on Properties of Bio-Based Polyurethane Foams

This study explores the impact of epoxidized palm oil (EPO) content and surfactant type on the mechanical properties of polyurethane foams. The resilience, hardness, and compressive strength of the foams were systematically analyzed at different NCO/OH molar ratios. The findings revealed that increasing EPO content generally decreased both hardness and resilience values, indicating enhanced viscoelastic properties due to the plasticizing effect of EPO's hydrocarbon chains. However, specific surfactants significantly influenced these mechanical properties. Concentrol STB PU-2254 and Tegostab® B82001 VE surfactants enhanced compressive strength by promoting a compact cellular structure with smaller, more numerous cells, effectively distributing loads and counteracting the softening effect of high EPO content. Conversely, the use of Tegostab® B8462 resulted in reduced hardness due to increased porosity from larger cell formation. At an NCO/OH ratio of 1.0, higher pMDI content improved compressive strength by increasing hard segment formation. These results underscore the importance of surfactant selection and NCO/OH ratio optimization in tailoring the mechanical properties of polyurethane foams, offering valuable insights for their application-specific design and optimization.

Analysis of Stress Softening Effects and Crack Properties for Polymer Network-Based Soft Materials under Biaxial Stretching

Analysis of Stress Softening Effects and Crack Properties for Polymer Network-Based Soft Materials under Biaxial Stretching

Numerical modelling of pile jacking in highly sensitive clays

This paper presents large deformation finite element (FE) modelling of penetration of solid cylindrical piles into highly sensitive soft clay. The simulations are performed using a Coupled Eulerian–Lagrangian (CEL) FE modeling technique. The pile is penetrated at a constant rate, and the analyses are performed for undrained conditions to simulate the response during penetration. The soil model considers the effects of strain softening and strain rate on the undrained shear strength. The FE-calculated results are compared with available analytical and numerical solutions for idealized soil profiles. Simulations are also performed for two instrumented piles previously installed into highly sensitive clay at Saint-Alban in Québec, Canada. The installation-induced changes in stresses, degradation of undrained shear strength, and tip resistance obtained from FE analyses are consistent with the field test results. Large plastic shear strains develop near the pile, which can significantly remould the soil near the pile shaft. A parametric study shows that a quicker post-peak degradation of undrained shear strength of highly sensitive clay creates a smaller zone of high plastic shear strain near the pile, while the plastic zone is wider for low- to non-sensitive clays.

Study on high temperature tensile constitutive behavior and deformation mechanism of Ti-47.5Al-2.5 V-1.0Cr-0.2Zr alloy

The high temperature tensile test of Ti-47.5Al-2.5 V-1.0Cr-0.2Zr alloy was carried out by electronic universal testing machine under the condition of 750–900 °C/10−5–10−3 s−1. The hot tensile stress-strain curves were analyzed, and the constitutive model under hot tensile conditions was established. The microstructure transformation rule and deformation mechanism during tensile process were determined. The results show that the hot tensile curve is longer in the steady-state flow stage corresponding to lower strain rate and higher tensile temperature, the true stress declines and the elongation at break increases. The constitutive equation was established based on the tensile curve data, and the corresponding thermal activation energy was 310.3 kJ/mol. As the rise of tensile temperature and the decline of strain rate, the proportion about cross-layer fracture in the tensile fracture morphology decreases, the number of dimples increases, more lamellar structures change into recrystallized structures, and the softening effect of the alloy is more obvious. The dislocation deformation mechanism mainly includes the existence of dislocation slip and climb, dislocation intersection and dislocation ring or dislocation network formation. Twinning deformation is also another important mechanism of high temperature tensile deformation of TiAl alloy. Twins are formed in parallel with each other, so that the deformation can be further carried out. The dislocations and twins in the deformed microstructure will provide nucleation conditions for recrystallized grains. The dynamic recrystallization(DRX) grains are preferentially formed around the grain boundary and the vicinity of dislocation and twin is also preferred nucleation site for DRX. The DRX behavior is the main softening mechanism, and DRX size and volume fraction corresponding to lower tensile rate and higher tensile temperature are larger.

Analysis of mechanical properties of copper tantalum alloy material

Abstract Alloy material is one of the important factors affecting the penetration ability of shaped charge warhead. It is of great significance to study the mechanical properties of copper-tantalum alloy material for its application in shaped charge warhead. The mechanical tests of alloy materials at different temperatures and different strain rates were carried out by material universal testing machine and split Hopkinson pressure bar. The experimental results show that under quasi-static compression and dynamic compression, the plastic flow stress of Cu-40Ta material increases with the increase of strain rate, and has significant strain hardening effect. The plastic flow stress decreases with the increase of temperature, and there is thermal softening effect. The yield strength increases with the increase of strain rate, and the material has obvious strain rate strengthening effect.

Influence of ultrasonic surface rolling on fatigue performance of high carbon low alloy quenching-partitioning-tempering steel

In order to expand the application scope of the quenching-partitioning-tempering (Q-P-T) steel in the industrial field, ultrasonic rolling treatment (USRP) is carried out, and the influence of USRP on the fatigue properties of the Q-P-T steel is elucidated. Compared with the Q-P-T specimen (580 MPa), the fatigue limit of the USRP3 specimen increases to 620 MPa, and the crack initiation location is transferred from the surface to the core. The primary reasons for this fatigue strength incensement are as follows: a higher surface hardness effectively inhibits surface fatigue crack initiation; residual compressive stress reduces the driving force at crack tips and impedes crack propagation. Moreover, there is a continuous increase in hardness for the USRP3 specimen during cyclic loading due to dominant phase transformation strengthening effect caused by transformation from austenite to martensite. On the other hand, the USRP6 specimen possesses a gradient grain size structure with higher hardness and deeper range, which can decelerate crack propagation rate. However, surface damage caused by excessive ultrasonic rolling as well as the cyclic softening effect of the surface during fatigue ultimately counterbalance the positive influence of surface strengthening on fatigue properties.

Investigation on the impact of thermal softening effect induced by nanosecond pulsed laser-assisted cutting on the deformation mechanism of SiCp/Al composites

The machining of SiCp/Al composites containing high hardness SiC particles poses a significant challenge. The nanosecond pulsed laser-assisted cutting method has been selected to improve the machinability of SiCp/Al composites with a volume fraction of 45%. User-written subroutine is used to establish two dimensional and three dimensional cutting simulation models. The deformation mechanism of the Al matrix and SiC particles in SiCp/Al composites is investigated using cutting simulation, digital image correlation (DIC) experiments, and nanosecond pulsed laser-assisted cutting experiments. The effect of the various pulsed laser parameters on the surface quality of the workpiece is also investigated. The simulation and experimental results demonstrate that the damage of SiC particles and their arrangement hinder the plastic deformation of the Al matrix, and the nanosecond pulsed laser induces thermal softening effect and improve the plastic deformation of the Al matrix. Compared to the conventional cutting, the SiC particles exhibit a denser arrangement within the chip, and the plastic deformation of the Al matrix governs the chip deformation. The aim of this present study is to enhance the plastic deformation of the Al matrix, mitigate damage to SiC particles, improve the arrangement of SiC particles, and reduce the surface roughness of the workpiece in order to enhance the machinability of SiCp/Al composites.

A new semi–analytical method for elastic–strain softening circular tunnel with hydraulic–mechanical coupling

This study presents a novel coupled hydraulic–mechanical algorithm for analysing nonlinear seepage in circular tunnels with elastic strain softening characteristics. The model integrates porosity changes with permeability coefficients to formulate a set of nonlinear seepage equations. The Mohr–Coulomb criterion is used to determine the of rock stress yielding state, and the plastic strain εθp is used as a softening parameter to assess the degradation of rock strength. The proposed model is validated through a comparison with established numerical and analytical solutions. The pore water pressure distribution exhibits a distinctive three-stage curve under coupling conditions, with the seepage behaviour transitioning to a classical Laplace-type equation as the coupling coefficient weights diminish. Parametric analysis reveals that the brittleness index β is a pivotal factor governing the extent of the softening region. A decrease in residual strength σc∗ intensifies strain softening, leading to a more extensive softening zone. Conversely, a reduction in the angle of internal friction φ affects only the rock's strength without altering the softening intensity. The study also demonstrated that the pore water pressure diminishes the effective stress within the rock mass, resulting in a larger plastic zone than that under dry conditions. Finally, internal reinforcement and external support can effectively mitigate the effects of strain softening and seepage on the stability of the surrounding rock.

Thermal Regulation of the Acoustic Bandgap in Pentamode Metamaterials

This study used the finite element method to investigate the acoustic bandgap (ABG) characteristics of three-dimensional pentamode metamaterial (PM) structures under the thermal environment, and a method for controlling the PM ABG based on external temperature variation is also proposed. The results indicate that the complete acoustic bandgap can be obtained for a PM in the thermal environment, which makes the PM combine the bandgap characteristics of phononic crystals. More than that, the bandwidth and locations of ABGs can be effectively manipulated by controlling the temperature. Considering the softening effect of thermal stresses, the ABG gradually moves to lower frequencies as the temperature increases. Based on this, different degrees of ABG tunability can be achieved by changing the thermal environment to propagate or suppress acoustic waves of different frequencies. This work provides the possibility for PMs to realize intelligent regulation of the bandgap.

Effects of ultrasonic shot peening process on the microstructure and mechanical properties of nickel-based superalloys formed by selective laser melting

Selective laser melting (SLM) is an attractive additive manufacturing technique for fabricating high-performance superalloys engineering components. Unfortunately, these as-built parts generally exhibit unsatisfied mechanical properties because of their natural thermal residual stress and non-equilibrium microstructures. Ultrasonic shot peening (USP) can effectively inhibit defect expansion and even close pores, thereby improving the mechanical properties of alloys. In this study, the internal defects, residual stress and microhardness redistribution, tensile properties, as well as microstructural response of the GH3230 alloy treated by different ultrasonic peening time are systematically studied. USP treatment induces a significant carbides fragmentation and synthesizes surface gradient heterogeneous structure, which displays a microstructure gradient in grain size, carbides size and dislocation density from surface to core. Long-time USP treatment induces work softening of GH3230 alloys. The work softening mechanism of the GH3230 alloys after long-time USP treatment is ascribed to the increase of surface temperature and high stored strain energy. Both the microhardness and residual compressive stress in the softened region of GH3230 alloys are reduced. Work softening effect increases the surface plasticity of the GH3230 sample, which induces deeper gradient structure. The various strengthening mechanisms of gradient heterogeneous structures, as well as the multiple effects of hetero deformation induced (HDI) hardening, and densification strengthening are responsible for achieving strength-ductility synergy. The research findings provide new insights based on USP for improving the mechanical properties of heterogeneous superalloys engineering components.

Investigation of Combined Aging and Mullins Stress Softening of Rubber Nanocomposites.

The present study investigated the effects of thermal aging, ultraviolet radiation (UV), and stress softening on the performance properties of rubber modified with Cloisite Na+ or Cloisite 20A. Tensile strength (TS), strain at break (SB), modulus, and the retention coefficient were measured before and after aging. Results showed that TS and SB decreased by about 50% after 7 days of aging for all tested samples due to the breakage of the chemical bonds between rubber and nanoparticles. The modulus at 300% elongation increased by 20%, 15%, and 7% after thermal aging for the unmodified sample, nanocomposites with Cloisite Na+, and Cloisite 20A, respectively. The shape retention coefficient of all samples was not affected by heat, except for the virgin rubber sample, which exhibited a decrease of about 15% under thermal aging. The virgin matrix and nanocomposites showed different values of aging coefficient during thermal aging and UV radiation. The dissipated energy of samples that were aged after stretching was slightly higher than that of samples that were aged after stretching due to the breakdown of the bonds within the nanocomposites. Loading-reloading energy results showed that the level of stress softening was lower when Mullins was applied after the aging of the samples. Differential scanning calorimetry results indicated a slight decrease in Tg1 in the aged and stretched samples and an increase in the temperature of the first endothermic peak due to the addition of nanofillers in the stretched and aged samples. Thermogravimetric analysis revealed that all tested samples exhibited similar thermograms, regardless of their state of stretching or aging. Scanning electron microscopy analysis showed that the fracture surface of the virgin unaged sample was rough with some holes, while it was flatter and less rough after aging.

Effect of moisture on mechanical and physical properties of coals: a uniaxial compression study

The estimation of mechanical and physical properties of coal reservoirs is important for the successful exploration and development of coalbed methane (CBM). Unlike conventional sandstone reservoirs, coal reservoirs exhibit greater sensitivity to stress, resulting in distinct mechanical and physical behaviors. In this study, uniaxial compression tests were performed on both low-rank and high-rank coal samples under different moisture conditions to reveal the mechanical and physical property changes with stress. The results indicate that during axial stress loading (up to 2.8 MPa), axial strain initially increases rapidly and subsequently at a slower rate, with an axial strain of 0.13–0.25% observed at the maximum axial stress. The instantaneous Young’s modulus increases linearly before stabilizing, ranging from 618.01 to 4861.10 MPa, while the Poisson’s ratio remains relatively constant or increases linearly, ranging from 0.002 to 0.165. This results in a negative exponential decrease in both porosity and permeability, with maximum reductions of 1.77–4.21% and 5.38–12.25%, respectively. The mechanical properties of coal are influenced by both the cementation effect of water at low water saturation and the softening effect at high water saturation, which results in axial strain decreases and then increases as the water saturation increases. Concurrently, the elastic modulus initially increases and then decreases, while the Poisson’s ratio exhibits a less pronounced change or tends to increase. Consequently, there is a trend in which porosity and permeability first increase and then decrease. In addition, during stress unloading, the influence of water in coal induces a notable strain hysteresis phenomenon in water-containing coal samples, and this phenomenon is more obvious in the low-rank coals.

Dynamic deformation behaviors and structure evolution of TiZrHf hexagonal closed-packed medium-entropy alloy

In this study, the deformation behavior of a hexagonal closed-packed (HCP) TiZrHf medium-entropy alloy (MEA) was investigated across a wide range of strain rates from 10−4 s−1 to 4990 s−1. The alloy exhibits an exceptional combination of strength and plasticity during dynamic loading, as well as a noticeable strain rate hardening effect. The strain rate hardening effect is associated with the strong dislocation drag resulting from the fast dislocation velocity at high strain rates. Microstructure evolution analyses demonstrate that various deformation mechanisms occur within shear bands under dynamic loading, including the formation of deformation twins, dislocation cells, microbands, amorphous bands, and dynamic recrystallization. The dynamic deformation is influenced by the competition between hardening mechanisms and thermal softening effects. Dislocations, deformation twins, and amorphous bands dominate the strain hardening effect, while temperature rise induced by adiabatic shear contributes to thermal softening effects. Additionally, dynamic recrystallization and amorphization also lead to a decrease in dislocation density during dynamic loading.

Steel stresses and shear forces in reinforcing bars due to dowel action

AbstractReinforcing bars in structural concrete are typically designed to carry axial forces. Nevertheless, due to their bending stiffness, the bars can also carry transverse forces, that are associated with the localized bending mechanism (dowel action) resulting from the relative displacements (or slip) wherever a crack interface or a discontinuity interface (between two concrete parts cast at different times) intercept the bar. Such localized bending induces stress concentrations in both the bars and the concrete. The relative displacement can occur at interfaces either perpendicular to the bar or inclined with respect to its axis. Thanks to steel ductility, the bending stresses in the bars due to dowel action do not impair the sectional capacity at the ultimate limit state. Fatigue verifications, however, require an accurate evaluation of these stresses under imposed transverse displacements or shear forces. As well known, dowel action can be described by means of the traditional unidimensional Winkler's model (beam on an elastic foundation), where the bearing stiffness of the concrete embedment is typically introduced through a couple of parameters, namely the bar diameter and the concrete strength in compression. The actual behavior of a dowel, however, is definitely more complex and for such a reason, improvements are needed for the Winkler's model to introduce other parameters typical of actual structures. Hence, a new formulation is introduced in this study for the bearing stiffness, that is calibrated based on mechanical considerations and measurements with optical fibers. The proposed formulation also accounts for the following parameters: angle between the crack and the bar, concrete‐cover thickness, number of load cycles and the softening effect caused by the local secondary cracks radiating from bar ribs during the pull‐out process. The predictions of the model—implemented with the proposed bearing stiffness—fit fairly well the test results under both monotonic and cyclic loads, in terms of shear force–transverse displacement response and peak stress in the reinforcing bars.