Constitutive Model for Equivalent Stress-Plastic Strain Curves Including Full-Range Strain Hardening Behavior of High-Strength Steel at Elevated Temperatures

High-strength steel has been increasingly applied to engineering structures and inevitably faces fire risks. The equivalent stress-plastic strain (σeq− εeqp) curves of steel at elevated temperatures are indispensable if a refined finite element model is used to investigate the response of steel members and structures under fire. If the tensile deformation of steel is considerable, the σeq− εeqp curves at elevated temperatures are required to consider the strain-hardening behavior during the post-necking phase. However, there is little research on the topic. Based on the engineering stress-strain curves of Q890 high-strength steel in a uniaxial tension experiment at elevated temperatures, the σeq−εeqp curves before necking are determined using theoretical formulations. An inverse method based on finite element analysis is used to determine the σeq− εeqp curves during the post-necking phase. The characteristics of σeq−εeqp curves, including the full-range strain hardening behavior at different temperatures, are discussed. An equivalent stress-plastic strain model of Q890 steel at elevated temperature is proposed, which is consistent with the σeq−εeqp curves. The constitutive model is further verified by comparing the finite element analysis and test results.

Due to the complexity of the mechanical behavior of steel structures under fire and the limit of experimental study (e.g., high cost and time consuming), numerical analysis models [1,2] have become an essential and effective means to investigate and evaluate mechanical behavior. It can also provide more response information than experiments. For numerical analysis models, the temperature-dependent stress-strain curves of in Figure 1. The steady-state test method was used, and nine elevated temperature levels were set, including 200, 300, 400, 450, 500, 550, 600, 700, and 800 • C. In the test, the specimens were firstly heated to the target temperature with a heating rate of 10 • C/min. Then the target temperature was held for 15 min to acquire a uniform temperature field in the specimens. After that, specimens were stretched under the stationary temperature with a strain rate of 0.003/min until fracture. Figure 2 shows the tensile force-elongation curves and engineering stress-strain curves of Q890 steel at elevated temperatures, and Table 1 gives the mechanical properties. A detailed introduction of the experiment and test results can be found in [14].
constitutive model for the curves is proposed and verified in Section 5. The conclusions are drawn in Section 6.

Summary of Tensile Tests of Q890 High Strength Steel at Elevated Temperatures
A series of uniaxial tensile tests have been performed on the round specimens of quenched and tempered Q890 high-strength steel at room temperature and elevated temperatures up to 800 °C by Huang et al. [14]. The geometry of round specimens is shown in Figure 1. The steady-state test method was used, and nine elevated temperature levels were set, including 200, 300, 400, 450, 500, 550, 600, 700, and 800 °C. In the test, the specimens were firstly heated to the target temperature with a heating rate of 10 °C/min. Then the target temperature was held for 15 min to acquire a uniform temperature field in the specimens. After that, specimens were stretched under the stationary temperature with a strain rate of 0.003/min until fracture. Figure 2 shows the tensile force-elongation curves and engineering stress-strain curves of Q890 steel at elevated temperatures, and Table 1 gives the mechanical properties. A detailed introduction of the experiment and test results can be found in [14].   are drawn in Section 6.

Summary of Tensile Tests of Q890 High Strength Steel at Elevated Temperatures
A series of uniaxial tensile tests have been performed on the round specimens quenched and tempered Q890 high-strength steel at room temperature and eleva temperatures up to 800 °C by Huang et al. [14]. The geometry of round specimen shown in Figure 1. The steady-state test method was used, and nine elevated temperat levels were set, including 200, 300, 400, 450, 500, 550, 600, 700, and 800 °C. In the test, specimens were firstly heated to the target temperature with a heating rate of 10 °C/m Then the target temperature was held for 15 min to acquire a uniform temperature fi in the specimens. After that, specimens were stretched under the stationary temperat with a strain rate of 0.003/min until fracture. Figure 2 shows the tensile force-elongat curves and engineering stress-strain curves of Q890 steel at elevated temperatures, a Table 1 gives the mechanical properties. A detailed introduction of the experiment a test results can be found in [14].

Procedure for Determining the Equivalent Stress-Plastic Strain Curves of Q890 Steel at Elevated Temperatures
The equivalent stress-plastic strain curves are the input data to the FE model. According to the mechanical characteristics of tensile specimens, the engineering stress-strain (σ n -ε n ) curves can be divided into the pre-necking and post-necking phases to acquire the equivalent stress-plastic strain curves of Q890 high-strength steel at elevated temperatures. Accordingly, the procedure for determining the curves is described in two sections.

Determination of Equivalent Stress-Plastic Strain Curves during the Pre-Necking Phase
It is well known that the necking of tensile specimens initiates at the ultimate tensile strength. In the pre-necking phase, the tensile specimen is under uniaxial uniform strain and stress in the gauge length. During this phase, the true stress (σ true ), true strain (ε true ) and true plastic strain (ε tp ) can be obtained based on Equations (1)-(3). It is worth noting that the equivalent stress (σ eq ) and strain (ε eq ) are equal to the true stress and strain during the pre-necking phase [27]. Consequently, equivalent plastic strain (ε eqp ) is equal to the true plastic strain (Equation (3)). The σ eq − ε eqp curves during the pre-necking phase can be calculated from the engineering stress-strain curves (Figure 2b) using Equations (1) and (3).

Determination of Equivalent Stress-Plastic Strain Curves during the Post-Necking Phase
After necking, the stress status of tensile specimens is under complex stress conditions, and the stress is not uniform across the necking cross-section [27,28]. Equations (1)-(3) are no longer suitable during this phase. That makes it impossible to obtain the post-necking σ eq − ε eqp curves directly from the engineering stress-strain response.
The Bridgman [28] and MLR [27] correction methods are usually used to obtain the post-necking strain hardening curves by correcting the average true stress corresponding to the instantaneous minimum sectional area during tension. The Bridgman correction method is inaccurate under large strain due to the assumption of uniform distribution of the equivalent strain in the minimum cross-section [25]. Moreover, it is expensive since the instantaneous curvature radius of the necking profile should be measured [25]. MLR correction method assumes that the ratio of equivalent stress to the average true stress is material-independent, including a regression equation for the ratio. Still, it needs to be verified when applied to structural steels [26] because the regression equation is based on limited test data. The diameter of the instantaneous minimum sectional area needs to be measured for the MLR correction method.
Recently, inverse methods based on the FE model have been widely used to identify steel's hardening behavior after necking [25,[29][30][31][32], providing accurate results and having a low cost. The method can modify the equivalent stress-strain curve by iterations, stopping when the FE analysis and test results are consistent. In different studies, test curves for comparison with FE analysis results are different due to the different shapes of specimens and measurement methods. Comparison of tensile force-elongation curves [29], tensile force-engineering strain curves [31,32], or engineering stress-strain curves [32] is the easiest means since only elongation in gauge length and tensile force should be measured in the tests. In this paper, the inverse method based on the FE model combined with the test data described in Section 2 is used to determine the σ eq − ε eqp curves during the post-necking phase.
The procedure for the inverse method based on the FE model to obtain σ eq − ε eqp curves are described as follows: (1) Calculate the pre-necking σ eq − ε eqp curves from the test curves with Equations (1)-(3), which is a part of the input data for the FE model described in Section 4.
(2) Establish an FE model to simulate the tensile tests of Q890 steel at different temperatures. The post-necking σ eq − ε epq curves as the input data for the FE model is determined with the following two approaches.
Approach I-Iteration procedure for strain-hardening functions (i) Use an appropriate strain-hardening function to extrapolate the post-necking σ eq − ε eqp curves. Here, the Voce model (Equation (4)) [39] and Ludwik model (Equation (5)) [40] are adopted after different strain-hardening functions for steel are tried out.
where n is the material coefficient determining the development of strain hardening; σ s and σ i are the stress at saturation and the initial yield stress, respectively.
where σ 0 is the initial yield stress of the material; K L is the hardening coefficient; n is the strain-hardening exponent.
According to the Considéré criterion, the strain-hardening functions (Equations (4) and (5)) satisfy Equation (6) at the onset of necking (the peak point of engineering stressstrain curves). The strain-hardening functions also satisfy Equation (7) since it passes the onset of necking. The parameters σ s and σ i in Equation (4) and σ 0 and K L in Equation (5) can be determined using Equations (6) and (7) by assuming the n value. In other words, the two strain-hardening functions are definite for a given n value. In the analysis, an initial n value is assumed first, and the corresponding Voce curve or Ludwik curve is taken as the input data for the FE model.
where σ u true and ε u tp are the true stress and plastic strain at the onset of necking. (ii) Conduct FE analysis based on the acquired σ eq − ε eqp curve and then get the tensile force-engineering strain curve.
(iii) Compare the tensile force-engineering strain curves obtained from the FE analysis and test. If the FE analysis curve is well consistent with the test curve, the corresponding σ eq − ε eqp curve is the desired curve. When the FE analysis curve doesn't match the test curve, set a new n value and conduct a new FE analysis again. Through iterations, the optimum n value and σ eq − ε eqp curve can be obtained, which assures the consistency between FE analysis and test results.
As an example, Figure 3 shows the σ eq − ε epq curves by extrapolation with the Ludwik model with different n values at 500 • C. Figure 4 shows the tensile force-engineering strain curves from FE analysis with the Ludwik model with different n at 500 • C after iteration. It is seen that the σ eq − ε eqp curve corresponding to n = 0.78 can be used to simulate the test response accurately. In Table 2, the strain-hardening models with parameter values for the post-necking σ eq − ε eqp curves at 20-500 • C are tabulated. at 500 °C after iteration. It is seen that the − curve corresponding to n = 0.78 can be used to simulate the test response accurately. In Table 2, the strain-hardening models with parameter values for the post-necking − curves at 20-500 °C are tabulated.   In the temperature range between 550 and 800 °C, the authors didn't find an appropriate strain-hardening function to describe the post-necking − curves. Therefore, approach II was employed to determine the post-necking − curves in this temperature range.
Approach II-Iteration procedure for equivalent stress ( Figure 5) at 500 °C after iteration. It is seen that the − curve corresponding to n = 0.78 can be used to simulate the test response accurately. In Table 2, the strain-hardening models with parameter values for the post-necking − curves at 20-500 °C are tabulated.   In the temperature range between 550 and 800 °C, the authors didn't find an appropriate strain-hardening function to describe the post-necking − curves. Therefore, approach II was employed to determine the post-necking − curves in this temperature range.
Approach II-Iteration procedure for equivalent stress ( Figure 5)  In the temperature range between 550 and 800 • C, the authors didn't find an appropriate strain-hardening function to describe the post-necking σ eq − ε eqp curves. Therefore, approach II was employed to determine the post-necking σ eq − ε eqp curves in this temperature range. Approach II-Iteration procedure for equivalent stress ( Figure 5) (i) A series of sample points of equivalent plastic strain (ε i eqp ) during the post-necking phase are chosen. The sample point is with an interval of 0.01 to consider the apparent bending of the initial descending curves when the equivalent plastic strain is less than 0.1. When the equivalent plastic strain exceeds 0.1, the sample point is with a maximum interval of 0.2, which is turned down if the calculated curve deviates from the test curve due to the excessive strain interval. The equivalent stress σ i eq at ε i eqp (initially i = 1) is assumed. (ⅰ) A series of sample points of equivalent plastic strain ( ) during the post-necking phase are chosen. The sample point is with an interval of 0.01 to consider the apparent bending of the initial descending curves when the equivalent plastic strain is less than 0.1. When the equivalent plastic strain exceeds 0.1, the sample point is with a maximum interval of 0.2, which is turned down if the calculated curve deviates from the test curve due to the excessive strain interval. The equivalent stress at (initially i = 1) is assumed. (ⅰi) FE analysis is performed with the equivalent stress-plastic strain curve, including the pre-necking and post-necking curves. The post-necking curve is the piece-wise curve ( , ) (k = 1 to i). The tensile force-engineering strain curve is obtained from FE analysis, and the endpoint of the curve is ( , ), where and are engineering strain and tensile force from FE analysis.
(ⅲ) The FE analysis result is compared with the test curve. If the endpoint ( , ) does not fall into the test curve, the equivalent stress at is modified, and FE analysis is carried out again using the modified equivalent stress-plastic strain curve. The iteration process continues until the endpoint ( , ) is located on the test curve. (ⅳ) Repeat steps ⅰ-ⅲ for the next sampling point . In this step, the post-necking equivalent stress-plastic strain curve is unchanged from point 1 to i, and only the equivalent (ii) FE analysis is performed with the equivalent stress-plastic strain curve, including the pre-necking and post-necking curves. The post-necking curve is the piece-wise curve (ε k eqp , σ k eq ) (k = 1 to i). The tensile force-engineering strain curve is obtained from FE analysis, and the endpoint of the curve is (ε i FE , F i FE ), where ε i FE and F i FE are engineering strain and tensile force from FE analysis.
(iii) The FE analysis result is compared with the test curve. If the endpoint (ε i FE , F i FE ) does not fall into the test curve, the equivalent stress σ i eq at ε i eqp is modified, and FE analysis is carried out again using the modified equivalent stress-plastic strain curve. The iteration process continues until the endpoint (ε i FE , F i FE ) is located on the test curve. (iv) Repeat steps i-iii for the next sampling point ε i+1 eqp . In this step, the post-necking equivalent stress-plastic strain curve is unchanged from point 1 to i, and only the equivalent stress σ i+1 eq is modified by iteration until the simulated curve is consistent with the test curve.

Description of FE Model
The FE model was established based on the general-purpose software ABAQUS to complete the numerical analysis task. As shown in Figure 6a, an axisymmetric model for round specimens was employed to simplify the FE model and save computational costs. linear axisymmetric quadrilateral and reduced integration element. A sufficiently refined mesh is vital to yield accurate results. Therefore, a mesh convergence study was performed to determine the appropriate mesh size. As shown in Figure 6b, five approximate global mesh sizes (1 mm, 0.5 mm, 0.25 mm, 0.125 mm, and 0.065 mm) were used in the study. The analysis results suggested that it is rational to set the approximate global size to 0.125 mm, which makes the FE result converge well. Figure 6c shows the mesh size of 0.125 mm and the necking profile before fracture.

Constitutive Model for Equivalent Stress-Plastic Strain Curves of Q890 Steel at Different Temperatures
The equivalent stress-plastic strain (σ − ε ) curves including full-range strain hardening behavior of Q890 steel at different temperatures are shown in Figure 7, which are obtained based on the above procedure. Figure 7 shows that the equivalent plastic strain at fracture differs at various temperatures, which is given in Table 3. The observation of the failure models [14] shows that the length of the necked region of the coupon increases at higher temperatures (500-800 °C), which is also reflected in the engineering stress-strain curves (Figure 2b) with larger strain at fracture. That means more reduction in sectional area and increasing equivalent strain at fracture at higher temperatures, which is consistent with the values in Table 3. Figure 2b shows that the post-necking phase of engineering stress-strain curves becomes less steep in the temperature range of 500-800 °C. Correspondingly, Figure 7 shows the notable change in the trend of equivalent stress-plastic strain curves at 450-550 °C. The σ eq − ε eqp curves reach saturation stress at temperatures ranging from 20 to 450 °C. When Q890 steel is exposed to 500-800 °C, the σ eq − ε curves monotonically increase with equivalent plastic strain. The curves at 20-500 °C are characterized by the convex curve. On the contrary, the curves at 550-800 °C consist of a convex curve at the initial stage and a next concave curve. Therefore, different formulations are required to describe the σ − ε  curves due to their discrepancy at different temperature ranges. The two end surfaces of specimens were coupled to reference points RP-1 and RP-2, respectively, with the kinematic coupling constraint. The fixed end was applied to RP-2 and the tensile displacement loading to RP-1.
The classical metal plasticity model with isotropic hardening in the ABAQUS material library was chosen to describe the behavior of steel. The σ eq − ε eq curve obtained according to Section 3 is the necessary input data for the plasticity model. The fracture is simulated when the plastic strain reaches a critical value. The σ eq is incremental with the ε eq except for the last point of the σ eq − ε eq curve. The last point is set artificially with equivalent stress close to zero (greater than zero to avoid the numerical difficulty) and equivalent plastic strain close to that of the penultimate point of the curve. That makes the σ eq − ε eq curve drops sharply at the penultimate point, and the equivalent plastic strain of the penultimate point is the strain at fracture (critical value). The equivalent plastic strain at fracture is determined by a trial-and-error method until the endpoints of the force-engineering strain curve are consistent between numerical calculation and the tensile test.
Axisymmetric element CAX4R was used in the FE model, which is the 4-node, bilinear axisymmetric quadrilateral and reduced integration element. A sufficiently refined mesh is vital to yield accurate results. Therefore, a mesh convergence study was performed to determine the appropriate mesh size. As shown in Figure 6b, five approximate global mesh sizes (1 mm, 0.5 mm, 0.25 mm, 0.125 mm, and 0.065 mm) were used in the study. The analysis results suggested that it is rational to set the approximate global size to 0.125 mm, which makes the FE result converge well. Figure 6c shows the mesh size of 0.125 mm and the necking profile before fracture.

Constitutive Model for Equivalent Stress-Plastic Strain Curves of Q890 Steel at Different Temperatures
The equivalent stress-plastic strain (σ eq − ε eqp ) curves including full-range strain hardening behavior of Q890 steel at different temperatures are shown in Figure 7, which are obtained based on the above procedure. Figure 7 shows that the equivalent plastic strain at fracture differs at various temperatures, which is given in Table 3. The observation of the failure models [14] shows that the length of the necked region of the coupon increases at higher temperatures (500-800 • C), which is also reflected in the engineering stress-strain curves (Figure 2b) with larger strain at fracture. That means more reduction in sectional area and increasing equivalent strain at fracture at higher temperatures, which is consistent with the values in Table 3. Figure 2b shows that the post-necking phase of engineering stressstrain curves becomes less steep in the temperature range of 500-800 • C. Correspondingly, Figure 7 shows the notable change in the trend of equivalent stress-plastic strain curves at 450-550 • C. The σ eq − ε eqp curves reach saturation stress at temperatures ranging from 20 to 450 • C. When Q890 steel is exposed to 500-800 • C, the σ eq − ε eqp curves monotonically increase with equivalent plastic strain. The curves at 20-500 • C are characterized by the convex curve. On the contrary, the curves at 550-800 • C consist of a convex curve at the initial stage and a next concave curve. Therefore, different formulations are required to describe the σ eq − ε eqp curves due to their discrepancy at different temperature ranges.

Constitutive Model for Equivalent Stress-Plastic Strain Curves of Q890 Steel at Different Temperatures
The equivalent stress-plastic strain (σ − ε ) curves including full-range stra hardening behavior of Q890 steel at different temperatures are shown in Figure 7, whi are obtained based on the above procedure. Figure 7 shows that the equivalent plas strain at fracture differs at various temperatures, which is given in Table 3. The observ tion of the failure models [14] shows that the length of the necked region of the coup increases at higher temperatures (500-800 °C), which is also reflected in the engineeri stress-strain curves (Figure 2b) with larger strain at fracture. That means more reducti in sectional area and increasing equivalent strain at fracture at higher temperatur which is consistent with the values in Table 3. Figure 2b shows that the post-necki phase of engineering stress-strain curves becomes less steep in the temperature range 500-800 °C. Correspondingly, Figure 7 shows the notable change in the trend of equiv lent stress-plastic strain curves at 450-550 °C. The σ eq − ε eqp curves reach saturati stress at temperatures ranging from 20 to 450 °C. When Q890 steel is exposed to 500-8 °C, the σ eq − ε curves monotonically increase with equivalent plastic strain. T curves at 20-500 °C are characterized by the convex curve. On the contrary, the curves 550-800 °C consist of a convex curve at the initial stage and a next concave curv Therefore, different formulations are required to describe the σ − ε  curves due their discrepancy at different temperature ranges.   Although the calibrated formulations based on the NIST model for elastic modulus (E T ) and 0.2% proof stress f y, T (engineering stress) of Q890 steel at elevated temperatures were provided by Huang et al. [14], equations (8) and (9) are proposed to more accurately predict the E T and f y, T , respectively. Table 4 shows the comparison of the predicted results between Huang's and present formulations for E T and f y, T . It is seen that a significant error exists at 800 • C in the predicted results by Huang's formulations and the present formulations have a slight error at different temperatures. Moreover, the current formulations have a low standard deviation, which means the scatter of the predicted results is slight.
where E 20 and f y,20 are the elastic modulus and 0.2% proof stress at room temperature, respectively. Through regression analysis, the constitutive model for the σ eq − ε eqp curves is proposed as follows: For σ eqy, T = f y, T 1 + f y, T /E T (11) where σ eqy, T is the equivalent stress at initial yielding at T • C and Equation (11) is inferred from Equation (1) by assuming the linear elasticity behavior before initial yielding; f H, T represents the strain-hardening function at T • C. The values of the parameters in Equation (10) are listed in Table 5.
For 500 • C < T ≤ 800 • C, σ eq = σ eqy, T + f H1, T ε eqp ε eqp ≤ 0.01 σ eq,0.01 + f H2, T ε eqp 0.01 < ε eqp (12) where σ eq,0.01 is the equivalent stress at ε eqp =0.01 at T • C; f H1, T and f H2, T represent the strain-hardening functions at different stages of the curve. The values of the parameters in Equations (13) and (14) are listed in Table 6.   (13) and (14).  Figure 7, the comparison of the proposed constitutive model and the σ eq − ε eqp curves obtained by the proposed procedure indicate that the proposed model is well consistent with the σ eq − ε eqp curves. Moreover, the proposed model was used for the FE analysis of the tensile test of Q890 steel at elevated temperatures. As shown in Figure 8, the tensile force-engineering strain curves from the FE model agree with the test curves well, which furtherly verifies the validity of the proposed constitutive model.

450
−89.   (13) and (14). In Figure 7, the comparison of the proposed constitutive model and the σ curves obtained by the proposed procedure indicate that the proposed model consistent with the σ − ε curves. Moreover, the proposed model was used FE analysis of the tensile test of Q890 steel at elevated temperatures. As shown in 8, the tensile force-engineering strain curves from the FE model agree with t curves well, which furtherly verifies the validity of the proposed constitutive mod

Conclusions
In the paper, a constitutive model for equivalent stress-plastic strain (σ eq − ε eqp ) curves of Q890 high-strength steel at elevated temperatures was investigated to provide a basis for the finite element analysis of the fire-resistant behavior of steel members and structures. The constitutive model includes the full-range strain hardening behavior, containing the post-necking behavior. A procedure was proposed for determining the σ eq − ε eqp curves of Q890 steel at elevated temperatures from 20 to 800 • C. As a part of the procedure, an inverse method based on the finite element analysis was utilized to obtain the post-necking equivalent stress-plastic strain curves. Two iteration approaches for the inverse method were used in two different temperature ranges, respectively. The procedure successfully obtained the σ eq − ε eqp curves.
Different trends of σ eq − ε eqp curves are found in different temperature ranges: at elevated temperatures from 20 to 450 • C, the equivalent stress approaches saturation stress at large strain; at high temperatures not less than 500 • C, the equivalent stress monotonically increases with the increasing strain. The σ eq − ε eqp curves are convex at elevated temperatures not exceeding 500 • C, but they become concave when the temperature reaches 550 • C.
Two formulations were proposed for elastic modulus and 0.2% proof stress of Q890 steel at elevated temperatures, which predicts the results more precisely than theexisting formulations.
A constitutive model containing a series of formulations was proposed to describe the σ eq − ε eqp curves including full-range strain hardening behavior of Q890 steel at 20-800 • C. The values of the parameters in the model were determined by regression analysis. The proposed constitutive model was verified by comparing the finite element analysis and test results.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.