Fatigue Crack Propagation Life Research Articles

Evaluation of RPV closure head penetration integrity considering fitting and residual stresses

To ensure the structure integrity of the Reactor Pressure Vessel (RPV), the main challenge is the embrittlement of beltline material. However, the stress is more complex for the regions of Closure Head Penetration (CHP). Feedback of leakage at the CHP has been found during in-service inspection. Although the detailed mechanical evaluation of the RPV head and the CHP has been carried out by many international institutions, there are limited studies to concern structure integrity of CHP tube. The systematic analysis of the influence of material strength matching, assemble stress, the Welding Residual (WR) stress and work pressure load on the fracture is also limited. In this paper, the systematic influence of these factors is analyzed, and the distribution characteristics of Stress Intensity Factor (SIF) and the crack propagation characteristics are studied. The 2D model and accurate 3D model are carried out using the finite element method. The results show that the WR stress on the outer surface of the CHP and the area adjacent to the weld is larger than the yield strength of the material. There is a gap between the RPV and the upper part of the CHP. The attenuation rate of the WR stress in the outlet direction of RPV is lower than that in the inlet direction, so cracks are more likely to occur in the out of direction. The influence of WR stress on the circumferential stress distribution and fracture parameters in the wall thickness direction of CHP is bigger than the influence of assemble load and design condition load. However, the overall stress of CHP is low. The SIF values along the crack front have an asymmetric distribution, and the SIF value is very small. As the fatigue crack propagation life is much longer than the assumed limit in the design lifetime, the risk of fracture and crack fatigue propagation caused by mechanical stress is low.

Mechanical-thermal coupling fatigue failure of CoCrFeMnNi high entropy alloy

High entropy alloys exhibit excellent performance under different-temperature conditions. In order to investigate the effect of temperature on the mechanical-thermal coupling fatigue failure of classical CoCrFeMnNi high entropy alloy to accelerate the application of high entropy alloys, the uniaxial tensile and fatigue tests at temperatures ranging from RT to 500 °C were conducted, and statics simulation analysis of fatigue specimens under fatigue loading conditions was conducted. The results of the uniaxial tensile tests illustrated that the yield stress and tensile strength of the alloy decreased with increasing temperature. In contrast, the elongation after the fracture of the alloy increased with temperature. The fatigue test results revealed that the fatigue life decreased as the temperature increased. The fracture morphology of the fatigue specimens and the flank morphology and dislocation distribution near the crack nucleation zone were collected to support the investigation. Accordingly, the reasons for the high-temperature-induced decrease in fatigue crack nucleation and propagation lives were systematically investigated. The main reasons for the higher-temperature-induced decrease in crack nucleation lives were a decrease in the yield stress of the alloy, more easily triggered dislocation motion and stress homogenization. The reasons for the decrease in fatigue crack propagation lives induced by higher temperatures were the more prone occurrence of dislocation motion, a decrease in yield stress, and a decrease in the area of the crack propagation region.

Numerical Analysis of Crack Propagation in an Aluminum Alloy under Random Load Spectra

This study develops a rapid algorithm coupled with the finite element method to predict the fatigue crack propagation process and select the enhancement factor for the equivalent random load spectrum of accelerated fatigue tests. The proposed algorithm is validated by several fatigue tests of an aluminum alloy under the accelerated random load spectra. In the validation process, two kinds of panels with different geometries and sizes are used to calculate the stress intensity factor, critical crack length, and crack propagation life. The simulated and experimental findings indicate that when the aluminum alloy is in a low plasticity state, the crack propagation life exhibits a linear relationship with the acceleration factor. When the aluminum alloy is in a high plasticity state, this study proposes an empirical formula to calculate the equivalent stress intensity factor and crack propagation life. The normalized empirical formula is independent of the geometry and size of different samples, although the fracture processes are different in the two kinds of panels used in our study. Overall, the numerical method proposed in this paper can be applied to predict the fatigue crack propagation life for the random spectrum of large samples based on the results of the simulated accelerated crack propagation process and the accelerated fatigue tests of small samples to reduce the cost and time of the testing.

Research on influence mechanism of prefabricated hole on life extension of compressor impeller

The life of compressor impeller is very short under extreme working conditions, so it is necessary to explore the life extension method and mechanism. Since the crack initiation location cannot be accurately predicted in advance, the hole drilling strategy of prefabricated hole will determine the remaining life of impeller after cracks appear. This article presents a strategy of prefabricated hole to extend the fatigue crack propagation life (FCPL) of compressor impeller. The stress distribution of impeller during operation is simulated by finite element method, and the crack initiation location is determined. The Forman-Newman-de Koning model is used to investigate the effects of hole drilling strategy and rotational speed on FCPL, and the three-dimensional crack propagation process is simulated. In addition, this paper investigates the main phenomena and influencing mechanism for the extension of FCPL due to prefabricated hole. It is found that the prefabricated hole can attract the crack propagation path, and make the crack pass through the hole more easily. At this time, the crack propagation driving force will decrease sharply, and the crack propagation rate will decrease accordingly, thus extends the FCPL. Even if it cannot pass through the hole, the optimal hole drilling strategy can extend life by 104.23%. The life extension effect of prefabricated hole increases with the increase of speed. The design in this paper can reduce the economic cost to a great extent, and provides a new idea for the life extension problem of complex models.

Study on fatigue damage tolerance of rail steel materials using peridynamics

With railways transitioning into the maintenance era, this study proposes a method to ensure structural safety and control maintenance costs. We develop a Peridynamics (PD) fatigue crack growth prediction model to determine the safe crack threshold for rail steel material under cyclic loading, known as the fatigue damage tolerance size. By conducting fatigue tests, we obtain the PD parameters for the rail steel and rigorously validate the model's reliability. Using both the compact tensile (CT) specimen model and a transient wheel-rail contact model, we analyze factors influencing crack propagation and fatigue damage tolerance. The results reveal that higher cyclic loads and smaller stress ratios result in shorter fatigue life and smaller damage tolerance sizes. Additionally, larger initial crack angles, faster train speeds, and heavier axle loads reduce fatigue crack propagation life and damage tolerance sizes. Ultimately, based on damage tolerance analysis, it is recommended to limit crack propagation in U71Mn rail steel within a 3.8-millimeter safety range for reliable operation.

Notch structural stress theory: Part II predicting total fatigue lives of notched structures

Notch fatigue problems are among the most common topics for the reliability analysis and structural design of mechanical engineering components. In previous work, we proposed the Notch Structural Stress (NSS) method based on fracture mechanics for reflecting the notch effect on fatigue crack propagation lives. However, like other regular fatigue assessing procedures, NSS also cannot accurately predict the fatigue life of notched structures with slower stress gradients due to the inconsistent slope of their S-N curves. This paper presents the Theory of Notch Structural Stress (TNSS) for predicting total fatigue lives of notched structures, which overcomes the life prediction limitations caused by slope differences in S-N curves and constructs a life-predicting frame suitable for different notch types even including plain and sharp notches. Compared with the often-used stress-concentration-assessing procedures, TNSS provides a more effective consolidation of the S-N data from different notch types, mostly since it takes both Initiation and Propagation state resistance into account at the same time.

Analytical study of CFRP strengthening steel plates with a semi-elliptical surface crack

Flaws and defects are unavoidable in the manufacturing process of steel members. Surface cracks are a common type of defect in steel structures that can introduce fatigue failure under cyclic loading. In this study, stress analysis of carbon fiber reinforced polymer (CFRP) repaired specimens with a semi-elliptical surface crack was conducted by theoretical analysis. The analysis was used to calculate stress intensity factors at crack tips. Finite element analysis was carried out to verify the proposed equations considering CFRP strengthening. Both single-sided and double-sided strengthening methods were considered. Thereafter, a modified Paris Law model was adopted to predict fatigue crack propagation life. Several parameters, which are crack aspect ratio, crack depth ratio, layers of CFRP, CFRP width and stress range are considered to analyze their influence on fatigue life. It was found that the augmentation of crack aspect ratio, layers of CFRP and CFRP width can significantly increase fatigue crack propagation life of repaired specimens. Furthermore, single-sided repairing on cracked side is better than double-sided repairing.

Fatigue response of open hole plates: A finite element simulation investigating the influence of dynamic and static cold expansion processes

Cold Expansion (CE) techniques are extensively used in the aeronautical industry to enhance the fatigue life of open-hole plates. However, the availability of accurate Finite Element (FE) models to simulate the fatigue behavior of this process, particularly Dynamic Cold Expansion (DCE), is limited. This study introduces two novel methods for predicting the fatigue response of DCE and Static Cold Expansion (SCE) open-hole plates. The first method directly estimates the total fatigue life using Continuum Damage Mechanics (CDM) and the Theory of Critical Distance (TCD). The second method separates the prediction of fatigue crack initiation life and propagation life, incorporating CDM, TCD, and the Extended Finite Element Method (XFEM). Moreover, FE models are developed to simulate residual stress, stress under external cyclic loads, and fatigue crack propagation behavior for both DCE and SCE specimens. The proposed methods are evaluated, compared, and the mechanisms behind fatigue life enhancement and fatigue crack propagation modes in CE specimens are discussed. It is found that the prediction accuracy is enhanced by considering stress distributions along the thickness direction and improving the Line Method (LM) in TCD through the introduction of a novel CE parameter. The results demonstrate that both methods achieve good predictive performance, with an average error index within ±30%. Furthermore, it is observed that both DCE and SCE processes primarily improve the fatigue crack initiation life of open-hole plates, with the percentage of crack initiation fatigue life increasing as the expansion size increases. The majority of the fatigue crack propagation life in CE specimens is concentrated in the initial stages of crack propagation. In addition, the effects of DCE and SCE processes on reducing the fatigue crack propagation rate are more pronounced along the thickness direction compared to the width direction, leading to distinct crack propagation modes between CE and non-cold expansion (NCE) specimens.

Effect of double cracks merge on fatigue crack propagation life of engine heat shield

To prevent overheating of the airframe, an aero-engine is often designed to play a role in shielding heat radiation heat shield. Since the thermal protection system used in this paper is active, it is subject to a cold gas pressure load on the outside and a thermal radiation load from the combustion chamber on the inside. Therefore, the working environment of the heat shield is very harsh. Taking the heat shield of an aero-engine as the research object, the finite element method (FEM) is used to analyze the thermal-mechanical coupling (TMC) of the actual working process of the heat shield, and the mechanical quantities such as the thermal stresses and mechanical stresses are comprehensively analyzed. According to the finite element simulation results, two stress concentrations are obtained near the connecting lug, and the stress concentrations are set as the initial crack insertion locations. Using the theory related to linear elastic fracture mechanics (LEFM) and the Forman-Newman-de Koning (FNK) model, the fatigue crack propagation life (FCPL) and its law of the merged double cracks of the engine heat shield are calculated. The results show that the FCPL of the heat shield decreases nonlinearly and the critical crack length decreases linearly under different working conditions. This law applies not only to single cracks but also to merged double cracks. In terms of FCPL, the merged double cracks are lower than the single crack with a maximum life reduction rate of 70.9%.

Studying a Repair Method of LY12 Aluminum Alloy Plate

The difference in the strengthening effect of different thickness reinforcement plates on an LY12 aluminum alloy central defect plate and the load distribution between the reinforcement plate and central defect plate in the reinforcement plate structure were studied. Through a method of equivalent transverse load in a small interval [a + Δa], the equivalent cyclic load of the central defect plate in the reinforced plate structure under different crack lengths was calculated, and then the distribution of the internal load of the reinforced plate structure with a different thickness with the crack propagation was solved. The secondary development program of the finite element software Abaqus was written in Python language, so that Abaqus software can solve the problem of high cycle fatigue. The finite element simulation of the different thickness-reinforced plate structure is carried out by this program. Through the output data, the equivalent cyclic load of the central defect plate in the reinforced plate structure under different crack lengths is calculated. Through three different fitting methods, the mathematical relationship between the equivalent cyclic load Δσ and the crack length a at both ends of the central defect plate in the reinforced plate structure is described. Based on the mathematical relationship and the finite element output data, the fatigue crack propagation life of the reinforced plate structures with different thicknesses is calculated. It is found that, under the same crack conditions, with the increase in the thickness of the reinforced plate, the bearing load of the cracked plate decreases and the life of the cracked plate increases. With the expansion of the crack, the bearing load ratio of the reinforced plate increases. The simulation method is compared with the experimental results to verify its effectiveness.

Numerical simulation and experimental verification of fatigue crack propagation in high-strength bolts based on fracture mechanics.

To investigate the fatigue crack propagation behavior of high-strength bolts for high-speed train brake discs, the fatigue crack propagation of high-strength bolts with initial defects under various load ratios was numerically simulated and experimentally verified based on fracture mechanics in this paper. Firstly, the fracture mechanics model of a three-dimensional hexahedral mesh with initial root defects was established using ABAQUS-FRANC3D interactive technology. Then the stress intensity factor (SIF) of the crack front was calculated by the stress superposition of the crack surface to simulate the coupling effect of preload and axial cyclic load. Based on it, fatigue crack propagation was simulated. Finally, the corresponding fatigue experiments on prefabricated crack bolts were carried out. The results show that mode I cracks dominate in the process of crack propagation. The stable crack propagation zones of the fractured high-strength bolts all show a semi-elliptical cross-section. The SIF of the crack front decreases with the increase of the load ratio, thus making the crack propagation life increase with the increase of the load ratio. The experimental outcomes are in great agreement with the simulation results, which verify that the numerical simulation method can effectively and accurately evaluate the fatigue life of high-strength bolts with initial defects and provides an effective means for predicting the fatigue crack propagation life of the same type high-strength bolts in engineering applications.

Fatigue crack propagation behavior in Ti−6Al−4V alloy with surface gradient structure fabricated by high-energy shot peening

A gradient structure was successfully fabricated on the surface of Ti−6Al−4V alloy by high-energy shot peening (HESP), and its effect on fatigue crack propagation was investigated. Optical microscope, scanning electron microscope, transmission electron microscope and X-ray diffractometer were used to characterize the microstructure and residual stress evolution during HESP. The results show that a gradient nanostructure with residual compressive stress layer of 220 μm in depth is formed. The generation of gradient nanostructures improves the strength−ductility combination of the alloy. Maximum residual compressive stress is generated on the subsurface, which gradually increases with the increase of shot peening time. HESP treatment effectively reduces the crack propagation rate and increases the fatigue crack propagation life. The residual compressive stress reduces the effective stress intensity factor range at crack tip, thereby generating the crack closure effect and delaying the crack propagation. At the same time, the synergy effects of increase in grain boundaries, decrease in effective slip length and the plastic zone at the crack tip caused by the refinement of the surface grains can increase the crack propagation resistance as well.

Fatigue Crack Extension and Life Calculation Based on Finite Element Analysis

With compact tension specimen static stress analysis and fatigue life of crack as the research object, through the three-dimensional modeling software to establish compact tension specimen, using the finite element analysis software to sample statics analysis, get the maximum deformation and maximum stress, and the compact tensile samples prefabricated crack, using the fatigue crack propagation analysis software, The cycle times and a-N curves were obtained. This method can be used to calculate the fatigue crack propagation life and visualize the crack propagation process, which is convenient to analyze and master the fatigue mechanism.

Fatigue Crack Growth Behavior of Different Zones in an Overmatched Welded Joint Made with D32 Marine Structural Steel

Applying fracture mechanics theory to heterogeneous welded joints might lead to an uncertain assessment of fatigue crack propagation behavior and, consequently, an inaccurate estimation of the cyclic loading capacity and fatigue life of welded structures. Combining experimental testing and analytical equations of the marine overmatched welded joints of D32 marine structural steel provided a view of the influence of strength heterogeneity on fatigue crack growth (FCG) behavior under constant cyclic loading. FCG testing was conducted using compact tension specimens under different stress ratios. The effect of residual stress on the FCG behaviors of the heat-affected zones (HAZs) and fusion zones (FZs) of the compact tension (CT) specimens was examined in the overmatched welded joints. Subsequently, the welding residual stresses were removed by post-welding heat treatment (PWHT) to focus the FCGR assessment on the microstructural effect. The results indicated that the FCG rates (FCGRs) of the FZ and HAZ materials obviously varied in as-welded and stress-relieved states. The existence of residual stress in the overmatched welded joints led to a decrease in FCG rates and prolonged the fatigue crack propagation life for the FZs and HAZs. Moreover, the FCGR increased in the base metal (BM), HAZ, and FZ with the increase in the stress ratio. The FCG curves of these materials were fitted to correct the stress ratios using the NASGRO equation. Finally, an analytical analysis of the FCGR based on the NASGRO equation revealed the relationship between different stress ratios for different materials.

Study on Fatigue Behavior of Orthotropic Steel Bridge Deck that Considers Corrosion Effects

As a widely used deck system of long-span bridges, the U rib–diaphragm connection in orthotropic steel bridge deck (OSBD) is prone to fatigue cracking under repeated vehicle loads, and the corrosion medium will accelerate the fatigue cracking. To study the influence of corrosion on the fatigue properties of bridges and the fatigue crack propagation law, the three-dimensional (3D) corrosion morphologies of typical OBSD components under different mass loss rates (w) were obtained by the cellular automata (CA) corrosion model. The, the CA model could reflect the electrochemical metal (Me) corrosion. Then, based on AutoCAD, Rhinoceros, and ABAQUS, the solid model of a U rib–diaphragm connection with surface pits was obtained in ABAQUS. Finally, a multiscale finite-element (FE) model of a long-span bridge with millimeter-scale corrosion pits and cracks was established. The stress intensity factors at crack tips and dynamic fatigue crack propagation at the U rib–diaphragm connection were studied by the extended finite-element method (XFEM), and the difference in fatigue crack propagation life and cumulative energy release rate under different wwere investigated. The results showed that compared with the method of uniform thickness reduction to characterize corrosion effects, the ultimate bearing capacity that was obtained by the CA corrosion model was more consistent with the experimental results. Compared with when the corrosion effects were not considered, when the wof the U rib was 6%, 8%, and 10%, the peak stress intensity factor (KI) of the crack initiation at the weld toe of the U rib increased by 32%, 49%, and 86% and the fatigue growth life of the U rib–diaphragm was reduced by 6.5%, 18.9%, and 50%, respectively. According to the cumulative energy release rate of the crack initiation at the weld toe of the U rib, the crack was a composite crack that was dominated by the opening model crack, supplemented by the in-plane shear model crack and the antiplane shear model crack.

Fatigue crack propagation model and life prediction for pantographs on High-Speed trains under different service environments

The fatigue crack growth of a weld toe on the upper arm of a pantograph was investigated to predict the fatigue crack propagation life of the pantograph under normal and corrosive conditions. First, samples were subjected to four-point bending tests and simulations under the two conditions, and Paris law parameters were obtained. Second, constant-amplitude loads were determined through the equivalent treatment of variable-amplitude loads. Based on the Paris model, crack propagation was simulated using the finite element method for the two environments, and results showed a relationship between the crack length a and the stress intensity factor KI. The service life of the pantograph was predicted using the Paris law. Results indicated that the pantograph could run about 958,000 km and 7,660 km in normal and corrosive environments, respectively, after the development of its fatigue crack from a surface crack into a through-thickness crack.

Influence of 3D printing on compressor impeller fatigue crack propagation life

Taking a compressor impeller with a pressure ratio less than 3 as the research object, the influence of 3D printing on the fatigue crack propagation life of the compressor impeller is analyzed under the combined action of 3D printing residual stress and external load. The impeller is printed by selective laser melting (SLM) technology. Finite Element Method (FEM) is used to simulate the 3D printing process of the impeller, and the residual stress, residual deformation and other mechanical quantities are comprehensively analyzed. The numerical simulation results are in good agreement with the crack location in 3D printing experiment. The heat treatment and numerical simulation of the printed impeller significantly reduce the residual stress. According to the experimental results and the magnitude and direction of the first principal stress, the position and orientation of the macroscopic crack are determined. The FEM of linear elastic fracture mechanics (LEFM) and the Forman-Newman-de Koning (FNK) model are used to give the propagation law and fatigue crack propagation life of the cracked compressor impeller under different working conditions. The results show that the crack initiation position is affected by the maximum residual stress of the printed impeller after heat treatment. When only the external load is applied, the stress intensity factor (SIF) of the initial crack front varies with the rotational speed as a quadratic power. When the external load and 3D printing residual stress act at the same time, the law will no longer be satisfied. The general trend of the crack propagation path is along the radial direction. The speed has a linear influence on the critical crack size and a nonlinear influence on the fatigue crack propagation life. The research work in this paper can provide a basis for damage tolerance analysis of 3D printing compressor impeller, and also provide a reference for maintenance work.

Fatigue Crack Propagation Study of Bridge Steel Q345qD Based on XFEM Considering the Influence of the Stress Ratio

In the past, fatigue cracks have appeared in the orthotropic steel decks of bridges shortly after they opened to traffic. Previous studies have shown that high tensile welding residual stress exists in welded joints of steel bridges, which significantly changes the average stress and stress ratio of the joints. However, traditional fatigue crack propagation (FCP) calculations based on the Paris equation do not consider the influence of the stress ratio. Steel Q345qD is a common material used in bridges. Therefore, it is meaningful to study the influence of the stress ratio on the FCP life of steel Q345qD. In this paper, an FCP equation based on the energy release rate considering the influence of the stress ratio is first derived and named the da/dN-ΔG-R equation. Next, three material parameters in the equation are determined based on the results from tests of steel Q345qD under different stress ratios. Then, a user subroutine based on the extended finite element method (XFEM) that considers the influence of the stress ratio is defined and the effects of mesh size are analyzed. Finally, the effects of the stress ratio on FCP are discussed and the adaptability of the da/dN-ΔG-R equation is verified by comparing the values obtained from the equation with experimental results. The results show that: with a 95% guarantee rate, three material parameters in the da/dN-ΔG-R equation are: log(C) = −10.71, m = 2.780, and γ = 0.957; in the numerical simulation, a mesh size of 1 mm is more appropriate than other mesh sizes as it shows better accuracy and efficiency; under the same energy release rate range, the crack growth rate decreases as the stress ratio increases; under the same loading amplitude and cycles, the fatigue life decreases as the stress ratio increases; and finally, the numerical results considering the influence of stress ratio based on the da/dN-ΔG-R equation are close to the test results, while the results without considering the stress ratio based on the Paris equation are inaccurate.

The influence of bending loading on surface fatigue crack growth life

Engineering structures often serve under complex fatigue loading and the most common of which are tension and bending loading, or their combination. Fatigue cracks always initiate as surface and other three dimensional (3D) cracks and most of the fatigue life of structures spends on 3D crack growth, but fatigue crack propagation curves of materials are widely obtained using through-the-thickness specimens. In this work, a 3D fracture mechanics based method is developed to evaluate surface crack propagation life under combined tension and bending loadings by considering the plasticity induced fatigue crack closure. The method is validated firstly against available experimental data of surface fatigue cracked plates under the combined loadings, and then applied to predict fatigue crack propagation life of surface cracked railway axles under rotating bending. It is shown that the method can make prediction of life within 0.80 to 1.08 of the test life, and the predicted shape evolution of surface crack agrees well with experimental observation. With a modification of short crack closure being made and an equivalent initial crack being assumed, the proposed method can be extended to fatigue life prediction. It is found that the influence of bending loading on fatigue life increasing for larger initial cracks, especially in the submillimeter range. This fracture mechanics based method can serve as a unified way to calculate fatigue crack growth life as well as fatigue life of practical structures under both tension and bending loadings by use of the material crack growth curves obtained with straight-through cracked standard specimens.

Fatigue Tests and Analysis on Welded Joints of Weathering Steel.

To investigate the fatigue performance of vertical web stiffener to deck plate welded joints in weathering steel box girders, six specimens of the weathering steel (WS) Q345qNH, four specimens of WS Q420qNH, and four specimens of the plain carbon steel (CS) Q345q for comparison were tested by a vibratory fatigue testing machine, considering different steel grades, yield strengths, stiffener plate thicknesses, and weld types. The fatigue strength was evaluated based on S-N curves and the crack propagation was analyzed by linear elastic fracture mechanics (LEFM). The results show that the fatigue crack of the welded joints was initiated from the end weld toe of the deck plate and subsequently propagated both along the thickness of the deck plate and in the direction perpendicular to the stiffener plate. The fatigue crack initiation and propagation life of WS Q345qNH specimens were longer than those of CS Q345q specimens. The fatigue crack propagation life of WS Q345qNH specimens was longer than that of WS Q420qNH specimens, while the initiation life bore little relationship to the yield strength. Increasing the stiffener plate thickness effectively delayed crack initiation and slowed down its propagation. Compared with fillet welds, full penetration welds extended the fatigue crack propagation life, while no significant improvement was implied for the initiation life. The WS and CS specimens could be classified as having the same fatigue strengths by nominal stress, hot spot stress, and effective notch stress approaches, which were FAT 50, FAT 100, and FAT 225, respectively. Meanwhile, their material constants for LEFM were relatively close to each other.