Application of newly proposed hardening laws for structural steel rods

Chia sẻ: _ _ | Ngày: | Loại File: PDF | Số trang:11

Thêm vào BST

Báo xấu

4
lượt xem 1
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

In civil engineering, distinct mechanical properties and behaviors of structural steel rods necessitate a novel approach to material modeling. This study extends the application of recently proposed strain-hardening laws, originally developed for automotive sheet metals, to several structural steel rods (CB240-T and CB300-T). Standard uniaxial tensile tests are conducted for each examined material to obtain experimental stress strain data.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Application of newly proposed hardening laws for structural steel rods

Vietnam Journal of Mechanics, Vol. 46, No. 2 (2024), pp. 93 – 103 DOI: https:/ /doi.org/10.15625/0866-7136/20106 APPLICATION OF NEWLY PROPOSED HARDENING LAWS FOR STRUCTURAL STEEL RODS Van Nam Nguyen 1 , Duy Triet Doan 2 , Nhat-Phi Doan 1,∗ 1 Department of Civil Engineering, Industrial University of Ho Chi Minh City, Vietnam 2 Department of Mechanical Dynamics, Vinh Long University of Technology Education, Vinh Long, Vietnam ∗ E-mail: doannhatphi@iuh.edu.vn Received: 07 February 2024 / Revised: 10 April 2024 / Accepted: 23 April 2024 Published online: 04 June 2024 Abstract. In civil engineering, distinct mechanical properties and behaviors of structural steel rods necessitate a novel approach to material modeling. This study extends the ap- plication of recently proposed strain-hardening laws, originally developed for automo- tive sheet metals, to several structural steel rods (CB240-T and CB300-T). Standard uniax- ial tensile tests are conducted for each examined material to obtain experimental stress- strain data. Various curve ﬁtting methods are then employed to reﬁne the parameters of the strain-hardening laws, enabling accurate representation of the steel rods mechanical behavior. Subsequently, these laws are implemented in Abaqus software for numerical simulation of uniaxial tensile tests, facilitating the analyses of material response under uniaxial tensile loading condition. Compared to the measured data, the predicted force- displacement curves are in good agreement with the measurements until the tail of the curves. The comparisons verify the ability and potential of the examined hardening law for studying the post-necking behavior of structural steels. The outcomes provide a frame- work for more precise characterization of structural steel materials. Keywords: structure steel rods, hardening law, post-necking, ﬁnite element analysis, uni- axial tensile test. 1. INTRODUCTION Structural steel rods are widely used in building constructions and civil engineer- ing due to their high strength and toughness. Plastic deformations induce the so-called hardening behavior of steel rods that signiﬁcantly affects material properties [1, 2]. Un- derstanding this mechanical behavior plays a crucial role for ensuring the safety and
94 Van Nam Nguyen, Duy Triet Doan, Nhat-Phi Doan reliability of structures. Therefore, accurate modeling of the hardening behavior of steel rods is mandated to evaluate the collapse resistant capacity of structures. Traditionally, material models for structural steel have relied on simple linear elastic or isotropic hard- ening laws to simulate their behavior under stress. However, these models often fail to capture the nonlinear and post-necking behaviors exhibited by steel rods, particularly under extreme loading conditions [3, 4]. Descriptions of strain hardening behavior of steel materials have been investigated for many years. The hardening responses are commonly characterized by true stress- strain curves obtained from uniaxial tensile tests. Furthermore, the captured hardening behavior is then reproduced mathematically by a hardening law. Many formulas have been proposed to describe the hardening behavior of different steel materials, for ex- ample, Ramberg and Osgood [5], Hollomon [6], Swift [7], Ludwigson [8]. Within each formula, application of the hardening law for a wide variety of materials is a challenge, although excellent results have been provided for speciﬁc materials. These formulas have been extended in different ways to broaden their application in practical use [9–11]. These extended formulas require more parameters to reproduce the hardening behavior of the investigated materials. The act always increases the number of parameters involved to the hardening law formulation, that raises difﬁculty in calibrating these parameters. Recently, several formulas have been proposed by one of the authors to provide a better description of the hardening behavior of automotive sheet metals [12, 13]. Ben- eﬁts of these proposed formulas, such as high ﬂexibility within a requirement of four parameters were demonstrated in previous studies for automotive aluminum alloy and steel sheets [14, 15]. It is worth noticing that the chemical components of structural steels differ signiﬁcantly from those of automotive materials. In detail, the percentage of car- bon in the former is extremely higher than those of the latter. The difference leads to phase-transform phenomenon which are frequently observed structural steels. Due to the occurrence of phase transformations, the stress-strain curves of structural steels are conventionally divided into different stages of deformation [16]. That makes describing the hardening behavior of structural steels by a single function more difﬁcult, in compar- ing to automotive steels. This study examines the potential of newly proposed hardening laws for several structural steel rods including CB240-T and CB340-T materials. The rest of the paper is structured as follows. Section 2 presents in detail the uniaxial tensile tests that were conducted to achieve the experimental hardening behaviors of the tested materials. For- mulations of the examined hardening laws are revisited in Section 3. Parameters of these hardening laws are then identiﬁed by a common curve ﬁtting method. Section 4 validates the usefulness of the identiﬁed hardening laws in simulating the uniaxial tensile tests for
Application of newly proposed hardening laws for structural steel rods 95 all investigated materials. Section 5 summarizes and discusses the work’s perspectives and limitations. 2. EXPERIMENT The tested materials in this study are structural steel rods CB240-T and CB300-T 2 with diameters of 6 mm and 8 mm, respectively. These materials are widely used in civil engineering and construction. Uniaxial tensile tests are conducted following the Viet- 45 namese standard TCVN 1651-1:2018been proposed experimental tests, a specimen which Recently, several formulas have [17]. During by one of the authors to provide a better 46 description of the hardening behavior of automotive sheet metals [12-13]. Benefits of these proposed 47 is prepared with an flexibility withinofrequirement of four parametersconstant crosshead previous formulas, such as high initial length a 250 mm is pulled with a were demonstrated in speed 48 of 3 mm/min until failure. The loadssteel sheets [14-15]. It is worth noticing that by a load- studies for automotive aluminum alloy and acting on the specimen are recorded the chemical 49 cell, while thestructural steels differ significantly from those is recorded. Fig. 1 shows force- components of displacement of an initial gauge length of automotive materials. In detail, the 50 displacement curves obtained from three tests conducted of the latter. The difference leads to percentage of carbon in the former is extremely higher than those for each investigated material 51 to verify the repeatability. For both materials,observed structural steels. Due to the curves and phase-transform phenomenon which are frequently the derived force-displacement occurrence of 52 phase transformations, the stress-strain curves of structural steels are conventionally divided into 53 maximum forcesdeformation [16]. That makesuntil the tail of the curves, where presents a different stages of are in high agreements describing the hardening behavior of structural steels 54 moderate difference in the datain comparing to automotive steels. Test 3 data (for CB300-T) by a single function more difficult, of Test 2 data (for CB240-T) and 55 comparingstudy examines samples. Thus, the curve obtained from the ﬁrst teststructural ex- This to other test the potential of newly proposed hardening laws for several of each steel 56 amined material is used to calculate the stress-strain curves, which are reported in Fig. 2. rods including CB240-T and CB340-T materials. The rest of the paper is structured as follows. Section 57 It is seen that the the uniaxial tensile approximately 8.3 MPa at 2.5 mm of displacement for 2 presents in detail yield points are tests that were conducted to achieve the experimental hardening 58 behaviors of the tested materials. Formulations of the examined hardening laws are revisited in Section CB240-T and 13.2 MPa at 2.8 mm of displacement for CB300-T, respectively. Prior to this 59 point is the elastic region. Following then identifiedthe a common curve fitting method. Section 4 3. Parameters of these hardening laws are this point is by hardening phase until the CB240-T 60 validates the usefulness of the identified hardening laws in simulating the uniaxial tensile tests for all 61 experiencesmaterials. Section 5 summarizes and discusses at approximately 42.7 mm displace- investigated maximum stresses of around 11.7 MPa the work’s perspectives and limitations. ment, and 21.7 MPa at 47.8 mm displacement for CB300-T, in line with the maximum forces. Beyond this threshold, the mechanical behavior transitions into the post-necking 62 2. EXPERIMENT 15 25 (a) CB240-T (b) CB300-T 12 20 9 15 Force (kN) Force (kN) Test 1 Test 1 6 10 Test 2 Test 2 Test 3 Test 3 3 5 Max. force Max. force 0 0 0 10 20 30 40 50 0 10 20 30 40 50 60 63 Displacement (mm) Displacement (mm) 64 Fig. 1. Force-displacement curves obtained from three uniaxial tensile tests for two tested materials (a) Fig. 1. Force-displacement curves obtained from three uniaxial tensile tests for 65 CB240-T and (b) CB300-T two tested materials (a) CB240-T and (b) CB300-T
96 Van Nam Nguyen, Duy Triet Doan, Nhat-Phi Doan region, which enables the rapid decrease of stress over a short relative displacement until 3 the fractures occur. 600 600 (a) CB240-T (b) CB300-T 500 500 400 400 Stress (MPa) Stress (MPa) 300 300 200 200 Engineering Engineering 100 TRUE 100 TRUE 0 0 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 66 Strain Strain 67 Fig. 2. Stress-strain curves of the tested materials obtained from the uniaxial tensile test (a) CB240-T and Fig. 2. Stress-strain curves of the tested materials obtained from the uniaxial tensile test 68 (b) CB300-T (a) CB240-T and (b) CB300-T 69 70 The tested materials incurve for arematerial is constructed by elongating the sample and The stress-strain this study a structural steel rods CB240-T and CB300-T with diameters 71 of 6 recordingmm, stress variation with strain until theused in civil engineering and construction. mm and 8 the respectively. These materials are widely sample fractures. It is often presumed 72 Uniaxial tensile tests are conducted following the Vietnamese standard TCVN 1651-1:2018 [17]. During that the cross-sectional area of the material remains constant throughout the entire defor- 73 experimental tests, a specimen which is prepared with an initial length of 250 mm is pulled with a 74 constant crosshead speed of 3 mm/min assumption The loads acting on the specimen are recorded dur- mation process. However, this until failure. is inaccurate as the actual area decreases by 75 a loadcell, while the displacement of an initialand plastic deformation. The curve originating from ing deformation due to both elastic gauge length is recorded. Fig. 1 shows force-displacement 76 curves obtainedcross-sectionalconducted for each length is termed the ‘engineering’ stress-strain the initial from three tests area and gauge investigated material to verify the repeatability. For 77 bothcurve, also known as nominal stress-strainand maximumcontinuous line in Fig. 2), until materials, the derived force-displacement curves curve (i.e., forces are in high agreements while 78 the tail of the curves, where presentsthe instantaneous cross-sectional areadata (for CB240-T) and the curve originating from a moderate difference in the data of Test 2 and length is termed 79 Test 3 data (for CB300-T) comparing to other test samples. Thus, the curve obtained from the first test 80 of each examined material is used to calculate theline in Fig. curves, which are reported in Fig. 2.deter- the ‘true’ stress-strain curve (i.e., dash stress-strain 2). The detailed procedure to It is 81 seenminethe yield points stress-strain curves and true stress-strain curvesfor CB240-T and in the that engineering are approximately 8.3 MPa at 2.5mm of displacement can be found 13.2 82 MPa at 2.8mm of displacement [18]. An obvious observation is that the true stress-strain curve Abaqus software manual for CB300-T, respectively. Prior to this point is the elastic region. 83 Following this point is the hardening phase until the CB240-T experiences maximum stresses ofmaterial consistently maintains or increases its value. This is attributed to the fact that the around 84 11.7does not weaken. The decrease in the engineering MPa atis an illusion created for CB300- MPa at approximately 42.7mm displacement, and 21.7 stress 47.8mm displacement because the 85 T, in line with the maximum forces. Beyond this threshold, the mechanical behavior transitions into the engineering stress does not consider the decreasing cross-sectional area of the sample. 86 post-necking region, which enables the rapid decrease of stress over a short relative displacement until 87 the fractures occur. 3. CONSTITUTIVE MODEL 88 The stress-strain curve for a material is constructed by elongating the sample and recording the 89 stress variation with strain until the sample fractures. It is often presumed that the cross-sectional area 90 3.1. Hardening models of the material remains constant throughout the entire deformation process. However, this assumption 91 is inaccurate as the actual area decreases during deformation due to both elastic and plastic deformation. 92 The curve originating from the initial cross-sectional area and gauge length is termedis the model pro- The most widely used hardening constitutive for steel materials the ‘engineering’ 93 stress-strainby Swift [5] of which the formulation is expressed as posed curve, also known as nominal stress-strain curve (i.e., continuous line in Fig. 2), while the 94 curve originating from the instantaneous cross-sectional area and length is termed the ‘true’ stress-strain 95 curve (i.e., dash line in Fig. 2). The detailed procedure to determine engineering stress-strain curves and 96 true stress-strain curves can be found in the Abaqus software manual [18]. An obvious observation is 97 that the true stress-strain curve consistently maintains or increases its value. This is attributed to the fact 98 that the material does not weaken. The decrease in the engineering stress is an illusion created because 99 the engineering stress does not consider the decreasing cross-sectional area of the sample.
Application of newly proposed hardening laws for structural steel rods 97 Swift: σ = c 1 ( c 2 + ε ) c3 , (1) where σ is the ﬂow stress, ε is the equivalent plastic strain, and c1 ∼ c3 are material parameters that are needed to be identiﬁed. Although this model has been widely used for many steel materials, it seems to be lack of the ﬂexibility to reproduce the entire stress- strain relationship for many sheet metals, especially in the post-necking regimes [19– 22]. Recently, Pham and Kim [12] proposed a multiplicative hardening law (labelled by “Model 2”) for sheet metals expressed as the follows Model 2: σ = c1 + c2 (1 − exp (−c2 ε)) (0.002 + ε)c4 . (2) Furthermore, improvements of the Swift hardening law were proposed [11]. Based on these models (labelled by “Model 3” and “Model 4”), the ﬂow stress can be calculated as the follows Model 3: σ = c1 (2 − exp (−c2 ε) + c3 ε)c4 , (3)   ¯ c2 ε Model 4: σ = c1 1 + . (4)   1/c4  4 ¯ 1 + ( c3 ε ) These proposed hardening laws contain four parameters. Their capacity in captur- ing the hardening behavior of several automotive sheet metals has been proved in the previous study [11]. 3.2. Parameter identiﬁcation The most widely used method for parameter identiﬁcation is numerical ﬁtting. The method is available in calculation packages such as Excel, Matlab, etc. In this calibration method, a cost function is constructed based on the difference between the experimental data and the hardening law’s predictions as follows N pre 2 ∑ exp f = σi − σi , (5) i =1 exp pre where σi and σi denote the experimental and predicted stresses, N denotes the num- ber of total data. An optimization algorithm, such as general gradient decent, is applied to determine parameters by minimizing the cost function. The goodness of the identiﬁed hardening law is estimated by the coefﬁcient of determination, of which formulation is expressed as follows exp pre 2 2 ∑ σi − σi R = 1− exp 2 , (6) ∑ σi − µσ
5 138 Table 1. Identification of hardening law’s parameters and their evaluation 98 Van Nam Nguyen, Duy Triet Doan, Nhat-Phi Doan Model 𝑐1 𝑐2 𝑐3 𝑐4 𝑓 𝑅2 781.13 0.0315 0.2651 - 84.9 0.99993 where µσ is the mean value of all experimental stresses. The aforementioned ﬁtting Swift MPa MPa2 method is applied to identify parameters of all considering hardening laws for two tested materials. 316.16 589.12 61.65 0.630 387.9 0.99967 Model 2 MPa MPa 2 MPa CB240-T 3.3. Comparison 312.73 9.03 15.18 0.333 84.0 0.99993 Model 3 MPa MPa2 Table 1 reports parameters of the identiﬁed hardening laws for both two materials 311.98 2 9.75 3.74 0.594 101.4 0.99991 Model function and R coefﬁcient. In addition, Fig. 3(a) and Fig. 4(a) il- along with the cost 4 MPa MPa2 lustrate the ﬂow curves of these materials predicted by the examined hardening laws in 840.11 0.0278 0.290 - 232.2 0.99982 Swift comparison to the experimental data. Moreover, Fig. 3(b) and Fig. 4(b)MPa2 the extrap- depict MPa olation of these hardening laws to an631.92 305.71 extensive strain range. 45.38 0.602 618.4 0.99953 Model 2 MPa MPa MPa2 Table 1. Identiﬁcation of hardening law’s parameters and their evaluation CB300-T 300.0 10.32 4.18 0.601 31.3 0.99998 Model 3 MPa MPa2 2 Model c1 c2 c3 c4 f R 299.26 9.63 4.37 0.760 33.3 0.99998 Swift Model 4 781.13 MPa 0.0315 0.2651 MPa - 0.99993 84.9 MPa22 MPa 2 Model 2 316.16 MPa 589.12 61.65 0.630 387.9 MPa 0.99967 CB240-T 139 According to Table 1, all MPa 15.18 give 84.0 MPa2 Model 3 312.73 considering hardening laws0.333 good approximations for the 9.03 0.99993 140 experimental data where the coefficient of determination, 𝑅 2 is always higher than 0.999. In the case of Model 4 311.98 MPa 9.75 3.74 0.594 101.4 MPa 2 0.99991 141 CB240-T materials, the highest cost function of 387.9 MPa2 and the lowest R2 are observed by Swift 2 142 model. Whereas the highest cost function of 618.4 MPa2 and the lowest 232.2 MPa Swift 840.11 MPa 0.0278 0.290 - R2 for CB300-T0.99982 is material 2 143 observed in the case of2Model 2. Furthermore, Model 3 and Model0.602 extremely good cost functions CB300-T Model 305.71 MPa 631.92 45.38 4 yield 618.4 MPa 0.99953 144 and coefficient of determination for both two materials, 4.18 Model 3 300.0 MPa 10.32 0.601 31.3 MPa2 especially compared to those of the0.99998 others. The 145 comparison indicates 4 299.26 MPa these two hardening models for 33.3 MPa2 the hardening Model the flexibility of 9.63 4.37 0.760 reproducing 0.99998 146 behaviors of steel materials. 550 850 (a) (b) 500 750 450 650 Stress (MPa) Stress (MPa) 400 Experiment 550 Swift Swift 350 Model 2 450 Model 2 Model 3 Model 3 300 Model 4 350 Model 4 250 250 0 0.1 0.2 0.3 0 0.2 0.4 0.6 0.8 Effective plastic strain Effective plastic strain 147 148 Fig. 3. Identiﬁed hardening laws compared to the experimental data data of CB240-T material (a) Fit- Fig. 3. Identified hardening laws compared to the experimental of CB240-T material (a) Fitting results 149 and (b) Extrapolation to a large strain range Extrapolation to a large strain range ting results and (b) According to Table 1, all considering hardening laws give good approximations for the experimental data where the coefﬁcient of determination, R2 is always higher than
Application of newly proposed hardening laws for structural steel rods 99 0.999. In the case of CB240-T materials, the highest cost function of 387.9 MPa2 and the lowest R2 are observed by Swift model. Whereas the highest cost function of 618.4 MPa2 and the lowest R2 for CB300-T material is observed in the case of Model 2. Furthermore, Model 3 and Model 4 yield extremely good cost functions and coefﬁcient of determina- tion for both two materials, especially compared to those of the others. The comparison indicates the ﬂexibility of these two hardening models for reproducing the hardening behaviors of steel materials. 6 600 850 (a) (b) 550 750 500 650 450 Stress (MPa) Stress (MPa) 550 400 Experiment Swift 450 Swift 350 Model 2 Model 2 Model 3 350 Model 3 300 Model 4 Model 4 250 250 0 0.1 0.2 0.3 0 0.2 0.4 0.6 0.8 150 Effective plastic strain Effective plastic strain 151 Fig. 4. Identified hardening laws compared to the experimental data of CB300-T material (a) Fitting results 152 4. Identiﬁed hardening laws compared to the experimental data of CB300-T material Fig. (b) Extrapolation to a large strain range 153 As seen in Fig. 3a and Fig. 4a, all Extrapolation to a large strain range (a) Fitting results (b) of these hardening laws present excellent approximations for 154 the experimental data up to around 0.25 of effective plastic strain, for both CB240-T and CB300-T 155 materials. However, their extrapolations to large strain ranges (i.e., beyond 0.3 of effective plastic strain) 156 seen in Fig. 3(a) and Fig. 4(a),in Fig. 3b and Fig. 4b. The stresses atpresent excellent approx- As show significant deviations as shown all of these hardening laws the transition points are 157 predicted at around 569 MPa for CB240-T and 589 MPa for CB300-T, respectively. Model 2 always imationsgives the highest predictions in datastrain to around 0.25 of effective plastic strain, for both 158 for the experimental large up ranges; Model 4 gives the lowest predictions. The forecasts CB240-Tof Model 3 and Swift are close together. A slight their extrapolations to large strain ranges (i.e., 159 and CB300-T materials. However, deviation is observed where the prediction of Model 3 160 is foremost linear in the extensive strain ranges, while, those of Swift shows a nonlinear curvature. The beyond 0.3 of effective plastic strain) show signiﬁcant deviations asofshown in Fig. 3(b) 161 observation is explainable from the formulation of Model 3 where the contribution the linear term and Fig. (i.e. 𝑐3 𝜀̅)The stresses the non-linear term (i.e. exp⁡(−𝑐2 are predicted at around 569 MPa for 162 4(b). exceeds those of at the transition points 𝜀̅)) in the large strain ranges. CB240-T and 589 MPa for CB300-T, respectively. Model 2 always gives the highest pre- 163 dictions in large strain ranges; Model 4 gives the lowest predictions. The forecasts of Model 3 and Swift are close together. A ELEMENT ANALYSIS 164 4. FINITE slight deviation is observed where the prediction of Model 3 is foremost linear in the extensive strain ranges, while, those of Swift shows 165 a nonlinear curvature. The observation is explainable from the formulation of Model 3 166 Fig. 5. Mesh generation in these simulations of CB300-T specimen 167 A finite element model linear term (i.e. c3 ε) exceeds [18] to simulate the uniaxial where the contribution of the is developed in Abaqus/Explicit softwarethose of the non-linear term (i.e. exp (−c2 ε)) in the large strain ranges. elements with reduced integration (C3D8R) with a total 168 tensile tests for both materials. Eight-node solid 169 of 5168 elements are used to model the steel rods. The steel rods have a length of 250mm with a diameter 170 of 6mm for CB240-T and 8mm for CB300-T, respectively. Figure 5 shows mesh generation on the FE 171 specimen. The maximum mesh size is not exceeded 2mm. During simulations, one end of the rod is 172 fixed and a constant velocity, FINITE to the constant crosshead speed of 3 mm/min in experiments, 4. replicating ELEMENT ANALYSIS 173 is applied to the another end. Mises yield function [18] is coupled with the identified hardening laws 174 ﬁnite element modelto describe the plastic deformation of the testedsoftware [18] to simulate the A (obtained from Section 3) is developed in Abaqus/Explicit materials. uniaxial tensile tests for both materials. Eight-node solid elements with reduced integra- tion (C3D8R) with a total of 5168 elements are used to model the steel rods. The steel rods
152 (b) Extrapolation to a large strain range 153 As seen in Fig. 3a and Fig. 4a, all of these hardening laws present excellent approximations for 154 the experimental data up to around 0.25 of effective plastic strain, for both CB240-T and CB300-T 155 materials. However, their extrapolations to large strain ranges (i.e., beyond 0.3 of effective plastic strain) 156 show significant deviations as shown in Fig. 3b and Fig. 4b. The stresses at the transition points are 157 predicted at around 569 MPa for CB240-T and 589 MPa for CB300-T, respectively. Model 2 always 158 gives 100 highest predictions in large strain ranges; Model 4 gives the lowest predictions. The forecasts the Van Nam Nguyen, Duy Triet Doan, Nhat-Phi Doan 159 of Model 3 and Swift are close together. A slight deviation is observed where the prediction of Model 3 160 is foremost a length the250 mm with a diameter of 6mm for CB240-T anda8nonlinear CB300-T, re- have linear in of extensive strain ranges, while, those of Swift shows mm for curvature. The 161 observation is explainable from the formulation of Model 3specimen. The maximumthe linear term spectively. Fig. 5 shows mesh generation on the FE where the contribution of mesh size is 162 (i.e. 𝑐3 𝜀̅) exceeds those of the non-linear term (i.e.one end 2 𝜀̅)) inrod is ﬁxed and a constant veloc- not exceeded 2 mm. During simulations, exp⁡(−𝑐of the the large strain ranges. 163 ity, replicating to the constant crosshead speed of 3 mm/min in experiments, is applied to the another end. Mises yield function [18] is coupled with the identiﬁed hardening laws (obtained from Section 3) to describe the plastic deformation of the tested materials. 164 4. FINITE ELEMENT ANALYSIS 165 166 Fig. 5. Mesh generation in these simulations of CB300-T specimen Fig. 5. Mesh generation in these simulations of CB300-T specimen 7 167 A finite element model is developed in Abaqus/Explicit software [18] to simulate the uniaxial 168 tensile tests for both materials. Eight-node solid elements with reduced integration (C3D8R) with a total 169 15 30 of 5168 elements are CB240-T model the steel rods. The steel rods have a length of 250mm with a diameter used to (a) 170 of 6mm for CB240-T and 8mm for CB300-T, respectively. FigureCB300-T mesh generation on the FE (b) 5 shows 171 specimen. The maximum mesh size is not exceeded 2mm. 12 25 During simulations, one end of the rod is 172 fixed and a constant velocity, replicating to the constant crosshead speed of 3 mm/min in experiments, 20 Axial Load (kN) 173 is applied to the another end. Mises yield function [18] is coupled with the identified hardening laws 9 Axial Load (kN) 174 (obtained from Section 3) to describe the plastic deformation of the tested materials. 15 Experiment Experiment 6 Swift Swift Model 2 10 Model 2 3 Model 3 Model 3 5 Model 4 Model 4 0 0 0 20 40 0 20 40 60 175 Crosshead displacement (mm) Crosshead displacement (mm) 176 Fig. 6. Comparison between the predicted forces based on the examined hardening laws and the measured Fig. 6. Comparison between the predicted forces based on the examined hardening laws and 177 data (a) CB240-T and (b) CB300-T the measured data (a) CB240-T and (b) CB300-T 178 After simulations, the predicted axial forces based on these hardening laws are compared to the 179 experimentalsimulations, the predicted axial forces based onin this hardening lawsthe measured After data and reported in Figure 6. The vertical line these figure indicates are com- 180 crosshead displacement of the maximum loading force measured in the experiments (i.e., 11.7 MPa at pared to the experimental data and reported in Fig. 6. The vertical line in this ﬁgure 181 42.7mm displacement for CB240-T, and 21.7 MPa at 47.8mm displacement for CB300-T). As seen in 182 Figure 6, allthethese hardening laws yield good predictions for axial loads up to the maximum force. It indicates of measured crosshead displacement of the maximum loading force measured 183 in the experiments (i.e., 11.7 MPa at 42.7 mm displacement for CB240-T, and 21.7 MPa at can be said that all hardening laws adopted in this study are perfect applicable to the hardening behavior 184 the structural steel rods. Slight CB300-T). As seen in Fig. theall ofof these curves (i.e., post-necking 47.8 mm displacement for differences are observed at 6, tails these hardening laws yield 185 behavior), where the for axial loads up to2the maximum force. It can be said the two materials. good predictions predictions of Model overestimate the experimental data of that all hard- 186 Swift and Model 3 give similar predictions during the entire loading the hardening which are more or ening laws adopted in this study are perfect applicable to forces process, behavior the 187 less comparable to the measured data of CB240-T and CB-300-T, respectively. Model 4 provides structural steel rods. Slight differences are observed at the tails of these curves (i.e., post- 188 excellent prediction for the experimental curves of the tested materials, especially for CB240-T rod. 189 Comparisons clarify the potential predictionsof Model 32and Model 4 inthe experimental data necking behavior), where the application of Model overestimate reproducing the hardening 190 behaviortwo materials. Swift and Model 3 give similar predictions during the entire load- of the of structural steels. ing forces process, which are more or less comparable to the measured data of CB240-T 191 and CB-300-T, respectively. Model 4 provides excellent prediction for the experimental 192 5. CONCLUSIONS 193 This study examined the ability of four hardening laws in reproducing the hardening behaviors of 194 two steel rods. These hardening laws include a well-established model proposed by Swift and three 195 newly proposed models which were initially introduced for automotive sheet metals. It is seen that all
Application of newly proposed hardening laws for structural steel rods 101 curves of the tested materials, especially for CB240-T rod. Comparisons clarify the po- tential application of Model 3 and Model 4 in reproducing the hardening behavior of structural steels. 5. CONCLUSIONS This study examined the ability of four hardening laws in reproducing the harden- ing behaviors of two steel rods. These hardening laws include a well-established model proposed by Swift and three newly proposed models which were initially introduced for automotive sheet metals. It is seen that all of these models are able to capture well the ex- perimental data obtained from uniaxial tensile tests of CB240-T and CB300-T steel rods. Comparison between simulated and measured loading forces during the testes pointed out that the applications of Model 4 for the tested materials are very promising, while predictions of Model 3 are mostly identical to those of Swift model in the current exami- nations. Future studies on their applications for structural steels are deserved for further investigations. However, the predictions of Model 2 seem to overestimate the experi- mental data of two materials. Care should be taken in applying this model for different structural steel materials, especially for predictions in the large strain ranges. Use of an advanced calibration method may improve its accuracy of post-necking prediction. Although the promising results and insights are provided by this study, several lim- itations and assumptions can be drawn as following: - The study focused on limited materials with two speciﬁc types of structural steel rods (CB240-T and CB300-T). This requires additional calibration and validation on other steel grades and structural steel plates to cover the structural steel in civil engineering; - Uniaxial tensile tests were used to evaluate the performance of the hardening laws. While this provides valuable data, variation of loading conditions such as compression, combined loading, fatigue loading or cyclic loading on structural rods are often more complex and must be taken into account in reality; - The Von-Mises model used in the analysis assumes the material homogeneity and the strain isotropy, which may not always be the case in practical scenarios. Hetero- geneities in material properties, caused by factors such as manufacturing processes or structural defects, have the potential to affect the accuracy of the predictions; - The mesh size being used is assumed to be sufﬁcient to capture the deformation of the material in the large deformation zone (post-necking behavior); - The analyses do not account for the strain rate sensitivity of structural steel rods, where the material mechanical properties can vary depending on the rate at which it is deformed.
102 Van Nam Nguyen, Duy Triet Doan, Nhat-Phi Doan DECLARATION OF COMPETING INTEREST The authors declare that they have no known competing ﬁnancial interests or per- sonal relationships that could have appeared to inﬂuence the work reported in this paper. ACKNOWLEDGMENT The support of this research by Industrial University of Ho Chi Minh City and Vinh Long University of Technology Education is gratefully acknowledged. The authors also thank Dr. Quoc Tuan Pham - Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden for his support with the computer code of the new strain-hardening laws for automotive sheet metals. REFERENCES ´ [1] M. B. Jabłonska, K. Kowalczyk, M. Tkocz, R. Chulist, K. Rodak, I. Bednarczyk, and A. Ci- ´ chanski. The effect of severe plastic deformation on the IF steel properties, evolution of structure and crystallographic texture after dual rolls equal channel extrusion deformation. Archives of Civil and Mechanical Engineering, 21, (2021). https:/ /doi.org/10.1007/s43452-021- 00303-6. [2] A. A. Shah and Y. Ribakov. Recent trends in steel ﬁbered high-strength concrete. Materials & Design, 32, (2011), pp. 4122–4151. https://doi.org/10.1016/j.matdes.2011.03.030. [3] S. L. Chan. Non-linear behavior and design of steel structures. Journal of Constructional Steel Research, 57, (2001), pp. 1217–1231. https://doi.org/10.1016/s0143-974x(01)00050-5. [4] Z.-J. Zhang, B.-S. Chen, R. Bai, and Y.-P. Liu. Non-linear behavior and design of steel struc- tures: Review and outlook. Buildings, 13, (2023). https:/ /doi.org/10.3390/buildings13082111. [5] W. Ramberg and W. R. Osgood. Description of stress-strain curves by three parameters. Tech- nical report, National Advisory Committee for Aeronautics, (1943). [6] J. H. Hollomon. Tensile deformation. Transactions of AIME, 162, (1945), pp. 268–290. [7] H. W. Swift. Plastic instability under plane stress. Journal of the Mechanics and Physics of Solids, 1, (1952), pp. 1–18. https://doi.org/10.1016/0022-5096(52)90002-1. [8] D. C. Ludwigson. Modiﬁed stress-strain relation for FCC metals and alloys. Metallurgical Transactions, 2, (1971), pp. 2825–2828. https:/ /doi.org/10.1007/bf02813258. [9] N. Y. Golovina. The nonlinear stress-strain curve model as a solution of the fourth or- der differential equation. International Journal of Pressure Vessels and Piping, 189, (2021). https://doi.org/10.1016/j.ijpvp.2020.104258. [10] A. Lavakumar, S. S. Sarangi, V. Chilla, D. Narsimhachary, and R. K. Ray. A “new” empirical equation to describe the strain hardening behavior of steels and other metallic materials. Materials Science and Engineering: A, 802, (2021). https://doi.org/10.1016/j.msea.2020.140641. [11] T. Li, J. Zheng, and Z. Chen. Description of full-range strain hardening behavior of steels. SpringerPlus, 5, (2016). https://doi.org/10.1186/s40064-016-2998-3. [12] Q. T. Pham and Y. S. Kim. Identiﬁcation of the plastic deformation characteristics of AL5052- O sheet based on the non-associated ﬂow rule. Metals and Materials International, 23, (2017), pp. 254–263. https:/ /doi.org/10.1007/s12540-017-6378-5.
Application of newly proposed hardening laws for structural steel rods 103 [13] Q. T. Pham and Y.-S. Kim. Evaluation on ﬂexibility of phenomenological hardening law for automotive sheet metals. Metals, 12, (2022). https://doi.org/10.3390/met12040578. [14] Q. T. Pham, H. S. Le, A. T. Nguyen, X. Xiao, Y.-S. Kim, V. D. Nguyen, H. S. Tran, and X. Van Tran. A machine learning–based methodology for identiﬁcation of the plastic ﬂow in aluminum sheets during incremental sheet forming processes. The International Journal of Ad- vanced Manufacturing Technology, 120, (2022), pp. 3559–3584. https:/ /doi.org/10.1007/s00170- 022-08698-z. [15] Q. T. Pham, T. Nguyen-Thoi, J. Ha, and Y.-S. Kim. Hybrid ﬁtting-numerical method for determining strain-hardening behavior of sheet metals. Mechanics of Materials, 161, (2021). https:/ /doi.org/10.1016/j.mechmat.2021.104031. [16] J. Vetrone, J. E. Obregon, E. J. Indacochea, and D. Ozevin. The characterization of deforma- tion stage of metals using acoustic emission combined with nonlinear ultrasonics. Measure- ment, 178, (2021). https:/ /doi.org/10.1016/j.measurement.2021.109407. [17] TCVN 1651-1:2018. Steel for the reinforcement of concrete - Part 1. Vietnamese National Stan- dard, Ministry of Construction. [18] Dassault Syst` mes®. Abaqus analysis user’s guide, (2023). e ˜ [19] M. Korgesaar. The effect of low stress triaxialities and deformation paths on ductile fracture simulations of large shell structures. Marine Structures, 63, (2019), pp. 45–64. https:/ /doi.org/10.1016/j.marstruc.2018.08.004. ¨ [20] S. Marth, H.-A. H¨ ggblad, M. Oldenburg, and R. Ostlund. Post necking characterisation for a sheet metal materials using full ﬁeld measurement. Journal of Materials Processing Technology, 238, (2016), pp. 315–324. https:/ /doi.org/10.1016/j.jmatprotec.2016.07.036. [21] S.-J. Park, B. C. Cerik, and J. Choung. Comparative study on ductile fracture prediction of high-tensile strength marine structural steels. Ships and Offshore Structures, 15, (2020), pp. S208–S219. https:/ /doi.org/10.1080/17445302.2020.1743552. [22] Z. Yao and W. Wang. Full-range strain-hardening behavior of structural steels: Experimental identiﬁcation and numerical simulation. Journal of Constructional Steel Research, 194, (2022). https:/ /doi.org/10.1016/j.jcsr.2022.107329.