Effects of aluminum alloy constitutive models on the behavior of concrete-filled aluminum tubular stub columns under axial compression

: This study aims to examine the effects of various aluminum alloy constitutive models on the behavior of concrete-filled aluminum tubular stub columns under axial compression. The bi-linear model with hardening, the Baehre model, and the Ramberg-Osgood (R-O) model, which follow the European standard (EC9) were analyzed and compared in terms of their ability to describe the stress-strain behavior of aluminum alloy tensile coupons over the full range, and their respective application scenarios were discussed. A total of 74 sets of experimental data results were collected to examine the effects of these three models on the ultimate load of concrete-filled aluminum tubular stub columns. Furthermore, a full-scale model was constructed to analyze the effect of the hardening exponent n in the R-O model on the load-displacement curves. The results show that, apart from the bi-linear model with hardening, the other two aluminum alloy constitutive models are capable of accurately predicting the stress-strain behavior of aluminum alloys throughout the full range. The accuracy of the R-O model is significantly influenced by the calculation methods of n . The Baehre model is found to be more suitable for non-heat-treated aluminum alloys. The simulated ultimate load values obtained from the three constitutive models fall within a deviation range of ±10%, indicating their suitability for practical engineering applications. Among the three models, the R-O model exhibits the highest stability, as changes in the hardening exponent n do not affect the ultimate load but have a significant effect on the load-displacement curves beyond the ultimate load.


Introduction
Aluminum, the second largest category of metals, is the most extensively utilized metal variety following steel.Owing to its low density, it serves as a lightweight metal material primarily employed in the industry as an aluminum alloy.Aluminum alloy materials possess the characteristics of lightweight, aesthetically pleasing appearance, corrosion resistance, ease of maintenance, excellent durability, and low life cycle cost.They have extensive applications in the construction [1], packaging, electronics, and transportation sectors [2].Among the various applications of aluminum alloys, its most familiar one is serving as a building decoration material, including aluminum alloy door and window frames, glass curtain wall support systems, and aluminum alloy exterior cladding of buildings.
Currently, various countries including China, the United States, Australia, and Europe have established design specifications for aluminum alloy structures.The code for design of aluminum structures (GB50429-2007) [3] in China stipulates that the elastic modulus of aluminum alloys is E = 70000 MPa (approximately 1/3 of that of ordinary steel), the Poisson's ratio is v = 0.3, and the yield strength of commonly used 6061-T6 series aluminum alloy tubes in construction is 240 MPa, with a tensile strength of 260 MPa.Aluminum alloy 6061 can achieve the strength of ordinary steel while exhibiting excellent corrosion resistance and low density (2700 kg/m 3 ), which is only about one-third that of steel.Initially employed predominantly as a building decoration material, aluminum alloy has subsequently evolved into integral structural components.Aluminum alloy structures, serving as loadbearing components, first emerged in Europe and America, encompassing aluminum alloy bridges, large-span roofs, grid shells, and trusses, as well as support systems for glass curtain walls.Fig. 1 illustrates representative cases of domestic and international aluminum alloy structural engineering projects in recent years.In contrast to steel, the stress-strain curve of aluminum alloys lacks a distinct yield plateau and instead exhibits a continuous and smooth curve.The classical three-parameter Ramberg-Osgood model [4] (hereinafter referred to as the R-O model) has found extensive application in the mechanical analysis of aluminum alloy components [5][6][7].However, the European standard EN 1999-1-1:2007 [8] (hereinafter referred to as EC9) introduces several other constitutive models in addition to the R-O model.Based on their types, all models can be categorized into piecewise models and continuous models.Furthermore, piecewise models can be subdivided into bi-linear and three-linear models, with further categorization possible based on the existence of a hardening stage.Continuous models can be categorized as the Baehre model [9] (expressed as σ=σ(ε)) and the R-O model (expressed as ε=ε(σ)) based on different expression forms.Previous studies [10][11][12] on aluminum alloy components have consistently emphasized the stability and buckling behavior of aluminum alloy columns, influenced by the lower elastic modulus of aluminum alloys.However, to counteract the propensity of aluminum alloy tubes to buckle, the concept of a concrete-filled steel tube can be employed, wherein the aluminum alloy tube is filled with concrete to fully exploit the mechanical properties of both materials.This combination of an aluminum alloy tube and concrete is referred to as concrete-filled aluminum alloy tubular (CFAT).In 2008 and 2009, Zhou and Young [13,14] performed axial compression tests on rectangular and circular concrete-filled aluminum tubular (CFAT) stub columns, followed by numerical simulation studies [15] on circular CFAT stub columns in 2012.Zhou and Young [16,17] proposed a formula for evaluating the load-bearing capacity of CFAT stub columns, which was subsequently validated in experiments, demonstrating the formula's high accuracy.
Over the past 15 years, there has been a significant number of extensive experimentations and numerical simulations conducted on CFAT columns featuring diverse cross-sectional geometries, encompassing circular [14,[18][19][20][21][22][23][24][25][26], rectangular (including square) [13,[27][28][29][30], circular double-skin [16, 000039-3 31,32], and other configurations involving double-tube composites [17,33].Due to the poor weldability of aluminum alloy tubes and the reduced strength caused by welding, the traditional eccentric loading method of welding the column and the end plate and connecting by placing the hinge is not suitable for CFAT columns, and a reasonable and reliable connection device needs to be set up.Therefore, there are only a few reports on the research of eccentrically loaded CFAT columns [34,35].At present, researchers generally use the R-O model and its modified version [36][37][38][39] as the constitutive model of aluminum alloy for CFAT stub columns, mainly because the model can produce results consistent with the actual behavior when simulating aluminum alloy structures.Based on the experience accumulated in the study of aluminum alloy structures, the researchers applied it to the study of such composite structures.But EC9 not only defines the R-O model as the only constitutive model for aluminum alloys.Compared with aluminum alloy components, there is a significant difference in the contribution of concrete and aluminum alloy to the bearing capacity under pressure, and the interface bonding force between the two materials will produce a significant combination effect so that it cannot be simply regarded as a direct superposition of material strength.Because of the complexity of numerical analysis caused by the above factors, it is necessary to conduct further investigations to ascertain whether employing alternative classical aluminum alloy constitutive models in simulating the composite columns can yield outcomes that align with those obtained using the R-O model.
To summarize, this study initiates from the practicality of aluminum alloy constitutive models and chooses three representative models: the bi-linear model with hardening, the Baehre model, and the R-O model, within the EC9.For testing purposes, a total of eight longitudinal tensile coupons, each with a thickness of 2.5mm, were extracted from both circular and square 6061-T6 aluminum tubes.By utilizing the existing tensile test data for aluminum alloys, a comprehensive comparison and evaluation of the descriptive characteristics and accuracy of these models in describing the stress-strain relationship of aluminum alloys were conducted.Moreover, explicit utilization scenarios of diverse constitutive models were thoroughly examined.The experimental data of 74 pieces of concrete-filled aluminum tubular (CFAT) stub columns were collected from home and abroad and analyzed numerically using ABAQUS software.Although numerous studies and discussions have focused on concrete constitutive models for CFAT stub columns under axial compression, there remains a dearth of research on aluminum alloy constitutive models for these components.The research conducted by Wang et al. [22] demonstrated the suitability of Han's concrete model [40] in simulating CFAT axial compression members.Consequently, Han's concrete model was employed for simulation purposes in this study.Subsequently, the ultimate load obtained from three distinct aluminum alloy constitutive models was computed, compared, and rigorously evaluated.Furthermore, the effect of the hardening exponent n in the R-O model on the load-displacement curve of CFAT stub columns under axial compression was quantitatively assessed.These endeavors facilitate a more extensive understanding of the variances among distinct aluminum alloy constitutive models, thereby furnishing an expanded array of dependable choices for the finite element analysis and engineering design of CFAT stub columns under axial compression.

Characteristics and preliminary evaluation of the selected models
Table 1 presents a comprehensive summary of the fundamental categories, indispensable parameters, applicable strain intervals, distinctive features, and initial assessment of the three chosen constitutive models examined in this investigation.

Bi-Linear Model with Hardening
Eq. (1) presents the expression of the bi-linear model with hardening, while Fig. 2 illustrates its stress-strain relationship curve.The figure demonstrates that the bi-linear model with hardening fails to accurately depict the seamless transition of the stress-strain curve for the aluminum alloys from the elastic to the plastic stage.Moreover, the model anticipates the yield strain at the nominal yield point in advance, restricting the definition of the stress-strain relationship within a limited range of 0.5ε u to avoid overestimating the strength within the strain hardening range.

 
where f p is the conventional elastic limit of proportionality, ε p , and ε max correspond to the strain at f p and f max , E is the elastic modulus, E 1 is the hardening modulus.In the absence of a more accurate assessment of the aforementioned parameters, the following assumed values can be used: f p = nominal value of f 0.2 (f 0.2 represents the conditional yield strength corresponding to residual strain of 0.002), f max = nominal value of f 0.2 , and ε max = 0.5ε u (ε u denotes the nominal value of ultimate strain), Test curve Piecewise linear model Fig. 2. Bi-linear model with hardening  It can describe the nonlinear behavior of the σ-ε curve.
When the test curve is known, the model parameters can be determined by fitting experimental data or using empirical formulas.

Preliminary evaluation
A simple formula with a limited range of applications.
A complex formula with a limited range of applications.
A concise formula that subdivides various analytical conditions with a full range of applications.

Baehre Model
Eq. ( 2) presents the expression of the Baehre model, while Fig. 3 illustrates its stress-strain relationship curve.The elastic behavior is represented by region 1, the inelastic behavior by region 2, and the strain-hardening behavior by region 3.Each region exhibits distinct stress-strain relationships that must maintain continuity at their respective limiting points.The stress-strain relationship defined by the Baehre model is limited to the range of 0.5ε u .Within this range, it is bounded by where f e is the conventional elastic limit, f max is the tensile strength at the top point of the σ-ε curve,

R-O Model
Eq. ( 3) presents the expression of the R-O model, while Fig. 4 illustrates its stress-strain relationship curve.
where n denotes the exponent of the degree of hardening of the curve.To calculate the exponent n, it is necessary to know the second reference stress f 0.2 , and its corresponding residual strain ε o,x in addition to the conditional yield strength f 0.2 .
The selection of the second reference point (ε o,x , f x ) should be based on the specific strain range relevant to the phenomenon being investigated.For analyses focusing on the elastic deformation range, the stress-strain obtained using the 0.1% offset method can serve as the second reference point (refer to Fig. 4(a)).In this scenario, f x = f 0.1 , ε o,x =0.001, and n can be expressed in Eq. ( 4).Conversely, for analyses involving the plastic deformation range, the tensile stress at the top of the curve σ-ε can be regarded as the second reference point (refer to Fig. 4(b)).In such cases, f x = f u , ε o,x =ε u,pl , ε u,pl represent the residual strains corresponding to f u , and n can be expressed in Eq. ( 5).
  where ε u,pl = ε uf u /E, and ε u,pl ≈ ε u since f u /E is relatively small.
Reference point Based on the equations provided above, the calculations of the value of n are complicated.Industry standards generally only provide E and f 0.2 , making the application of the above model challenging.To address this issue, Steinhardt [41] proposed Eq. ( 6) in 1971, which has been widely tested and verified internationally.The value of f 0.1 can be determined by substituting Eq. ( 6) into Eq.( 4).Additionally, the value of n can be determined through fitting methods.It is a common engineering method to convert the R-O expression into a linear equation using double logarithmic coordinates and perform linear least squares fitting to determine its value.Wang et al. [42] proposed the utilization of the fast simulated annealing (FSA) algorithm to accurately determine the n value, which yields more stable results.Varying values of n e alter the shape of the R-O model curve.For instance, consider using tubes from the 6061-T6 series specified in the GB50429-2007, which have material properties, such as E = 70,000 MPa, and f 0.2 = 240 MPa.As depicted in Fig. 5, every stress-strain curve intersects at point When n e = 1, the R-O model represents linear elasticity, resulting in a straight line.For n e > 1, the R-O model exhibits a continuously changing curve with a steeper "corner" as n e increases.This indicates higher stress before point A and lower stress after point A, corresponding to a lower hardening modulus.When n e → ∞, the curve represents ideal elasticity, characterized by a bifold line without a hardening section.

Analysis of the Comparisons Between the Test Curves and the 3 Models' Predictions
An initial investigation was conducted to examine the constitutive relationship and mechanical properties of aluminum alloys via tensile coupon tests.Coupons were extracted from aluminum alloy 6061-T6 extruded tubes with circular or square cross-sections, each measuring 2.5mm in thickness.All coupons were longitudinally extracted from the aluminum alloy tubes, and their dimensions were processed by the specifications outlined in the GB/T 228.1−2010 [43].The specific shape and dimensions are shown in Fig. 6.
Tensile tests were performed utilizing the SUNS electronic universal testing machine.During the tensile deformation, strain measurements were acquired using extensometers that matched the coupon gauge length specifications.The loading was carried out under room temperature conditions with a 0.2mm/min strain rate.Fig. 7 displays the fractured coupons, revealing that the majority experienced fractures within the gauge length.Moreover, a more prominent necking phenomenon was observed at the fracture surface of coupons extracted from square tubes in comparison to those obtained from circular tubes.The bi-linear model with hardening initially slightly underpredicts the test curves within the range of 0.5ε u , but later reaches overall agreement.However, beyond the range of 0.5ε u , it increasingly deviates from the test curves, indicating that the bi-linear model with hardening cannot adequately simulate situations where large strains occur in aluminum alloys.The Baehre model slightly overpredicts the strain-hardening behavior of aluminum alloy from the yield point to 0.5ε u , and then gradually approaches the test curves again after 0.5ε u , as shown in Fig. 8 (e), (f), and (g).The strain hardening exponent n e values are recorded for the respective tensile coupons as 71.37, 59.07, and 75.67.Considering Fig. 5, this suggests that these three tensile coupons exhibit relatively weak strain-hardening behavior, and the prediction deviation in their strain-hardening behavior is significantly larger than the other seven groups.Thus, it can be concluded that the Baehre model is not suitable for accurately predicting the strain-hardening behavior of aluminum alloys with low n e values.Furthermore, based on the current predictions, the Baehre model demonstrates higher accuracy for predicting the stress-strain curves of aluminum alloys within n e in the range of (1,30].Further discussions will be provided in the subsequent sections.Among the three constitutive models, only the Baehre model and the R-O model demonstrate the capability to accurately capture the characteristic roundedness of the stress-strain curves observed in tensile tests conducted on aluminum alloys.However, it is worth noting that the R-O model displays relatively higher accuracy in this regard, while the bi-linear model with hardening fails to capture the roundedness.Currently, none of the three constitutive models mentioned can predict the subsequent descending stage after a fracture occurs in the tensile coupons.
3 Discussion on the specific spplication scenarios of models

The effects of different calculation methods for the n on the R-O model curves Table 2. Measured material properties obtained from tensile coupon tests
Note: C indicates that this tensile coupon was taken from a circular tube, and S indicates that this tensile coupon was taken from a square tube.Here n e is calculated according to Eq. ( 4) as the value of n considering elastic range analysis, and n pl is calculated according to Eq. ( 5) as the value of n considering plastic range analysis.Fig. 9 demonstrates the effects of two different calculation methods, Eq. ( 4) and Eq. ( 5), on the R-O curves.Specimens C-3, S-1, S-2, and S-3, which have the largest differences in calculated values of n, are selected to illustrate this discrepancy.As shown in Fig. 8, Eq.( 4) is applicable to the range of elastic deformation.The R-O curves obtained with the calculated values n e exhibit a higher degree of fit with the test curves before the yield point compared to the curves obtained with the calculated values n pl , and they appear more complete before the yield point.However, after the yield point, the R-O curves using the calculated values n e significantly deviate from the test curves, and their accuracy is lower than that of the R-O curves using the calculated values n pl .When the test curves reach ε u , the stress values corresponding to the R-O curves using the calculated values n e are lower by 4.8%, 15.8%, 16.2%, and 20.3% compared to those using the calculated values n pl .This observation indicates that although the R-O model is more accurate than the bi-linear model with hardening, it relies on selecting a reasonable calculation method for the n.If an appropriate calculation method for the n is not selected within the required analysis range, the accuracy of the R-O model may also be compromised.In section 2.5 of the comparative analysis, it can be reasonably inferred that the Baehre model demonstrates higher accuracy in predicting the stress-strain curve of aluminum alloys within n e in the range of [1,30].Therefore, further investigation is conducted to explore the accuracy of the Baehre model within this specific range.Due to the limited availability of experimental data on tensile coupons within this range and the proven optimal accuracy of the R-O constitutive model, the typical R-O model curve of the 6061-T6 series aluminum alloy tubes in Fig. 5 is considered to determine the range of n e values that closely match the Baehre model curve.According to the GB50429-2007, ε u is taken as 0.08 and f u as 260MPa for the 6061-T6 tubes.
Fig. 10 illustrates that the Baehre model curve falls within the range of 8 < n e < 15, which corresponds to the R-O curve.For non-heat-treated aluminum alloys, the n e values range from approximately 8 to 15, while for general heat-treated aluminum alloys, the n e values range from approximately 20 to 40.Hence, it can be concluded that the Baehre model is more suitable for predicting the stress-strain curves of non-heat-treated aluminum alloys.

Selection of simulated specimens for CFAT stub columns
The present study employs the bi-linear model with hardening, the Baehre model, and the R-O model to conduct finite element simulations on CFAT stub columns under axial compression.The objective is to investigate the effects of these three models on the accuracy of the simulations.A total of 74 CFAT stub columns selected from [14,16,18,19,26,32], are included in the analysis.The material parameters and geometric parameters of the CFAT stub columns are presented in Fig. 11 and Fig. 12 respectively.From Fig. 11, it is evident that the material parameters primarily consist of the elastic modulus E, yield strength f 0.2 , ultimate tensile strength f u , ultimate strain ε u , strain hardening exponent n of the aluminum alloy, as well as the compressive strength f cu of concrete.The average elastic modulus of the aluminum alloy in the specimens measures 68.3 GPa, slightly below the standard value of 70 GPa.Aluminum alloy tubes with yield strength and ultimate tensile strength below 350 MPa constitute 95.9% of the total specimens, indicating a predominant focus on aluminum alloy tubes with standard strength levels in current research on CFAT stub columns.The average ultimate strain value for the aluminum alloy tubes in the specimens stands at 8.4%, signifying favorable ductility characteristics.The strain hardening exponent of the aluminum alloy tubes is a computed parameter.To facilitate calculations, this study employs Eq. ( 6) mentioned earlier, revealing that the strain hardening exponent n in this sample primarily ranges from 20 to 30.
The specimens encompass different types of concrete, including ordinary, high-strength, and ultrahigh-strength concrete.Ordinary concrete accounts for 43.2%, while high-strength and ultra-highstrength concrete account for 32.4% and 24.4% respectively.To address the absence of standardized compressive strength values in the literature, this study utilizes the GB 50010-2015 [44] and formulas presented by Goode and Lam [45] to convert the prism strength f c and cylinder strength of concrete to cube strength f cu , as demonstrated in Eq. ( 7) and (8).
where α 1 is the ratio of prismatic strength to cubic strength, and α 2 is the brittle discount factor for highstrength concrete, the exact values of which can be found in [44].The specimens' numerical models are established using the ABAQUS finite element software.Since simulation methods for such specimens are well-established, [15,23] can be consulted for specifying element types, contact methods, and boundary conditions.Wang et al. [22] demonstrated that the constrained constitutive model proposed by Han [40] is suitable for simulating CFAT stub columns under axial compression.To ensure control over a single variable, the concrete constitutive model employed for the specimens in this study is Han's constitutive model.For specific details, refer to [40].Three typical constitutive models for aluminum alloy were selected for simulating and comparing the ultimate load of CFAT stub columns.Refer to [46], Eq. ( 9) and ( 10  Fig. 13 shows the comparison of the experimental and FE analysis failure modes.The experimental specimens exhibited local buckling in the middle left and upper right, whereas the failure modes obtained from the three different aluminum alloy constitutive models all showed symmetrical local buckling only in the central region, still presenting a certain degree of discrepancy from the experimental failure modes.This is mainly because the finite element simulation was conducted under ideal conditions, whereas the axial loading in the experiment could not ensure eccentric-free loading.After all, all metal products are manufactured with permissible thickness deviations and the specimens themselves may have had initial defects.A comparison of the strain nephograms obtained using three  [18] aluminum alloy constitutive models found that, except for the specimens using the R-O model, there was no significant deformation at the ends of the outer aluminum alloy tubes.However, the degree of deformation in the central part of the outer aluminum alloy tubes was as follows: the two-segment model exhibited the most significant deformation, followed by the Baehre model, with the R-O model showing the least deformation. Fig. 14 and Table 3 present a comparative analysis of the simulated ultimate load values obtained from three constitutive models and the corresponding experimental ultimate load values.The comparison reveals that the deviation in the simulated ultimate load values from the three constitutive models generally falls within a ±10% range.Both the Baehre model and the R-O model exhibit similar levels of simulation accuracy.However, the R-O model demonstrates greater stability in numerical simulations and lower variability compared to the Baehre model.Among the three constitutive models, the bi-linear model with hardening exhibits the poorest simulation accuracy and highest variability.Notably, extreme values may arise in the simulated values for large compressive strains in columns.Nevertheless, considering that the compressive strain limit state of columns in practical engineering scenarios does not reach such extreme experimental values, this model is still worth considering.Table 3 indicates that only the simulated values obtained using the Baehre model exhibit slight conservatism.The average ratio of simulated values to experimental values is 0.998, with a coefficient of variation (COV) of 0.090.For the R-O model and bi-linear model with hardening, the average ratios of simulated values to experimental values are 1.009 and 1.040 respectively, with COV of 0.079 and 0.114.Thus, the simulated values obtained from these three constitutive models fall within an acceptable range for practical engineering applications.

The effects of n e values on the N-△ curves of CFAT stub columns
As mentioned in Section 3.1, the choice of different methods for calculating the value of n significantly affects the stress-strain curve of the R-O model.It is necessary to further discuss whether this effect will affect the load-displacement curve of the CFAT axial compression stub column model simulated using the R-O model.Additionally, Fig. 12 demonstrates that all current experimental designs involve scaled specimens.Once the optimal R-O constitutive model is determined, this study refers to GB 50936-2014 [47] to establish the dimensions of the full-scale finite element model, using the following basic parameters: D=500mm, t=12mm, H=1500mm; f cu =50MPa, f 0.2 =240MPa.Varying values of n are taken as 8, 24, 40, and 100.different n e values.Assuming a synergistic behavior between the aluminum alloy tube and concrete under axial compressive loads, the strain of the aluminum tube at approximately 80 mm displacement of the CFAT stub column aligns with the dashed line in Fig. 9(a).This strain corresponds to four stress values, relative to the stress at n e =100 during that instance.The stress values for n e =8/24/40 exhibit increases of 43.7%, 10.2%, and 5.3% respectively.Likewise, at an 80mm displacement of the CFAT stub column, the load-displacement behavior, referring to the load at n e =100, exhibits a corresponding pattern.The loads for n e =8/24/40 demonstrate increases of 42.0%, 10.6%, and 4.8% respectively.This suggests that the substantial influence of various n e values on the stress-strain curves of the R-O model can be transferred to the load-displacement curves of the CFAT axial compression stub column simulated with this model.The primary effect occurs after reaching the ultimate load but has minimal effect on the ultimate load itself.This paper explores three typical constitutive models for aluminum alloys and their capabilities in predicting stress-strain curves of tensile coupons, as well as their application scenarios.Moreover, the predictive effects and influential factors of these three models on the axial compressive load capacity of CFAT stub columns are studied.The following conclusions can be drawn: (1) Only the R-O and Baehre models can accurately predict the full range of ultimate strain (ε u ).The bi-linear model with hardening is accurate only up to 0.5ε u .The R-O model is the most accurate and can reliably represent the rounded stress-strain response typical of aluminum alloys, outperforming the Baehre model.In addition, since the typical aluminum alloy constitutive curves in this paper do not take fracture strain into account, they cannot predict the decline stage.
(2) The R-O model is found to be the most accurate and adaptable, with its curve shape adjusting to changes in the hardening exponent (n).There are various methods to calculate n, and the choice should be application-specific.The R-O model's accuracy heavily depends on the right n choice, with Steinhardt's formula being the simplest for this purpose.
(3) The Baehre model tends to overestimate the behavior in the strain-hardening phase, especially for alloys with limited strain-hardening capacity.As a result, it often produces fuller curves compared to the R-O model.The article suggests that the Baehre model's curve fits within the range of 8 < n e < 15 relative to the R-O curve, making it more suitable for predicting the stress-strain curves of unheatedtreated aluminum alloys.
(4) For simulating the ultimate loads of CFAT stub columns under axial compression, the R-O model provides the most stable results, while the Baehre model is more conservative, and the bi-linear model with hardening is more prone to extreme values.For practical engineering, the simpler bi-linear model with hardening is recommended.When high accuracy is needed and stress-strain data from tensile tests are available, the R-O model is preferred.
(5) The effect of hardening exponent n on the R-O model affects the load-displacement curve of axially compressed CFAT stub columns, primarily influencing the descending stage after reaching the ultimate load but not affecting the ultimate load itself.

Fig. 1 .
Fig. 1.Recent typical aluminum alloy structure projects the nonlinear behavior of the σ-ε curve.

e 0 .
5 (ε p ) to separate the elastic and inelastic behavior, and by e 1.5 (ε e ) to distinguish the inelastic behavior from the strain-enhanced behavior.

Fig. 5 .
Fig. 5. Effect of n on R-O curves

Fig. 10 .
Fig. 10. Discussion of the Baehre model scope of application

Fig. 12 . 4 . 2
Fig. 12. Geometric parameters of CFAT stub columns ) are utilized to transform the nominal stress-strain model into a true stress-strain model suitable for finite element simulation.σ true and σ nom are the true stress and nominal stress of the aluminum alloy respectively, pl ln  and nom  are the log plastic strain and nominal strain of the aluminum alloy respectively.

Fig. 13 .
Fig. 13.Comparison of experimental and FE models

Fig. 14 .
Fig. 14.Comparison of ultimate loads between measurements and numerical analysis

Fig. 15 (
a) displays the R-O model curves for aluminum alloys with varying n e values, while Fig. 15(b) illustrates the load-displacement curves of the CFAT axial compression stub columns with

Table 1 .
Three typical selected models

Table 3 .
Comparison of ultimate loads from simulation and test