Prediction of cyclic large plasticity for prestrained structural steel using only tensile coupon tests

Abstract

Cold-formed steel members, which experience complicated prestrain histories, are frequently applied in structural engineering. This paper aims to predict cyclic plasticity of structural steels with tensile and compressive prestrain. Monotonic and cyclic tests on hourglass specimens with tensile and compressive prestrain are conducted, and compared with numerical simulations using the Chaboche model. Two approaches are taken in the simulation. The first requires only the monotonic tensile test data from the prestrained steels, and the second requires both the monotonic tensile test data from the virgin steel and the prestrain histories. The first approach slightly overestimates the compressive stress for specimens with tensile prestrain, while the second approach is able to accurately predict the cyclic plasticity in specimens with tensile and compressive prestrain.

Keywords

cyclic plasticity / prestrain / Chaboche model / mild steel

Introduction

Most of the models employed in steel structural engineering are based on the postulation that the plasticity behavior under tension and compression are identical. One of the reasons is that in practice only monotonic tensile coupon test data are available. In a preceding study [1], a method to calibrate the model parameters of classical cyclic plasticity models from only monotonic tensile coupon tests has been proposed and validated with favorable accuracy for virgin structural steels. This paper aims to validate the applicability of the method to prestrained steels.

In this paper, a classical plasticity model, the Chaboche model [2,3], is first introduced. Next, a method previously proposed by authors to modify the true stress-true strain relationship of the material data after the initiation of necking, which is necessary to accurately model the plastic behavior at large plastic strain ranges, is generally reviewed. Two approaches to simulate the cyclic plastic behavior of prestrained steels are then presented. The first approach only requires the monotonic tensile test data from the prestrained steels. The second requires both the monotonic tensile coupon test data of the virgin steel and the prestrain histories. To validate the two approaches, monotonic and cyclic tests are carried out using smooth round bars and hourglass specimens. Finally, finite element (FE) analyses are carried out using the Chaboche model and are compared with the experimental results. The first approach, which doesn’t require the prestrain histories of the specimens, slightly overestimates the compressive stress for the specimens with tensile prestrain. The second approach is capable of simulating cyclic plasticity of structural steels with both tensile and compressive prestrain.

Introduction of Chaboche model

General review of plasticity models

A large number of rate-independent cyclic plasticity models have been proposed for metals, and a comprehensive review of the field can be found in the literature [4]. Herein, a brief review of a series of commonly applied two-surface models is conducted. Mróz [5] and Iwan [6] have proposed a multisurface model, which utilizes a family of surfaces in the stress space. However, the evolution rules of the models are too complicated to apply in practice. Dafalias and Popov [7,8], and Kreig [9] have proposed a two-surface model, which utilizes only two surfaces, that is a yield surface and a bounding surface, to describe the nonlinear plastic behaviors. The yield surface defines the yield condition, and the bounding surface defines the limiting stress state. Armstrong and Frederick [10] introduced a relaxation term into the evolution rule of a single backstress, and proposed another approach to describe cyclic plasticity. To describe the nonlinear behavior of different metal, Chaboche et al. [2,3] generalized the Armstrong and Frederick evolution rule with a single backstress by defining multiple independent backstresses. This generalized Armstrong-Frederick model, commonly called the Chaboche model, has been modified and applied to simulate the behavior of various materials.

The backstresses describe kinematic hardening (KH) and the size of yield surface describes isotropic hardening (IH). The Chaboche model employed in this paper is a combined hardening model with both IH and KH components. In ABAQUS [11] this model is called the “combined model.”

The Chaboche model consists of the following three components necessary to describe cyclic plasticity of metals.

1) A yield function which defines the yield condition.

2) A flow rule which defines the evolution rule of plastic strain.

3) A hardening rule which specifies the post-yielding behaviors.

The Mises yield function is adopted since it is generally accepted for metals. The yield function is given as

(1)f=\sqrt{{\displaystyle \frac{\mathrm{3}}{\mathrm{2}}}(\mathit{S}-\mathit{\alpha }):(\mathit{S}-\mathit{\alpha })}-({\sigma }_{\mathrm{y}\mathrm{0}}+R)=\mathrm{0},

where S and α are the deviatoric stress and the total backstress, σ_y0 and R are the initial yield stress and IH component, respectively. The initial values of α and R are zero.

The widely accepted associated flow rule for metals, which assumes that the flow rule is associated with the yield function, is used in the present study.

(2)\mathrm{d}{\mathit{ϵ}}_{\mathrm{p}}=d\lambda {\displaystyle \frac{\partial f}{\partial \mathit{\sigma }}},

where dϵ_p is incremental plastic strain, dλ is the plastic multiplier equal to the incremental equivalent plastic strain, dϵ_eq, for a Mises material. The hardening rules for the KH and IH components of the Chaboche model are detailed in the following.

Constitutive equations of Chaboche model

Kinematic hardening rule of Chaboche model

Most metal hardening rules are developed from a linear rule called Prager’s hardening rule [12]. However, the linear KH rule is too simple to predict nonlinear plasticity of materials, since the hardening behavior of most metals is not linear except for large plastic strain [13]. Thus, a relaxation term named dynamic recovery has been superimposed to introduce nonlinearity into the IH component by Armstrong and Frederick [10] as

(3)d\mathit{\alpha }={\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{C}_{\mathrm{0}}d{\mathit{ϵ}}_p-\gamma \cdot \mathit{\alpha }\cdot d{ϵ}_{eq},

where C₀, and γ are model parameters.

The generalized Armstrong-Frederick model, i.e., the Chaboche model, employs a combination of several backstresses. Each backstress follows the evolution rule shown in Eq. (3) and covers a specific strain range. The following generalized model gives a much better description of cyclic plasticity for various metals.

(4)\mathit{\alpha }={\displaystyle \sum _i^n{\mathit{\alpha }}_i};d{\mathit{\alpha }}_i={\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}{C}_{i}d{\mathit{ϵ}}_p-{\gamma }_i\cdot {\mathit{\alpha }}_i\cdot d{ϵ}_{\mathrm{e}\mathrm{q}},

Here n is the total number of backstresses, α_i is the i th backstress, C_i and γ_i are model parameters corresponding to the i th backstress, and α is the overall backstress. The initial values of all the backstresses are zero.

Isotropic hardening rule of Chaboche model

The IH rule of the Chaboche model, proposed by Zaverl and Lee [14], is defined as

(5)dR=k(Q_{\infty }-R)d{ϵ}_{eq},

where R and dR are the change and incremental change in the size of the yield surface, k is a model parameter to describe the IH rate, and Q_∞ is the maximum change in the size of the yield surface. The initial value of R is zero. Integration of Eq. (5) yields,

(6){\sigma }_{\mathrm{y}}={\sigma }_{\mathrm{y}\mathrm{0}}+R={\sigma }_{\mathrm{y}\mathrm{0}}+Q_{\infty }(\mathrm{1}-e^{-k\cdot {ϵ}_{\mathrm{e}\mathrm{q}}}).

The Chaboche model under a uniaxial stress state is illustrated in Fig. 1. The model in the figure has three backstresses. The evolution rule of the first backstress, α₁, is linear, and those of the other two, α₂ and α₃, are nonlinear. A backstress with a linear evolution rule is often necessary to describe cyclic plasticity of metals at large plastic strain ranges. The figure also indicates that both the KH component, α, and the IH component, R, contribute to the increase in the post-yielding stress under monotonic loading.

Modification of true stress-true strain data after necking

A review of methods to modify true stress-true strain data after necking

The true stress-true strain before initiation of necking is given by the logarithmic stress-logarithmic strain obtained from a tensile coupon test, based on the assumption of uniform elongation along the gage length and a uniaxial stress state. The true stress σ is obtained by

(7)\sigma =s\cdot (\mathrm{1}+e),

where s is the engineering stress and e the engineering strain. The true strain ϵ can be obtained by,

(8)ϵ=\ln (\mathrm{1}+e).

However, the true stress-true strain after necking initiates must be modified, since the stress state is no longer uniaxial but triaxial, as illustrated in Fig. 2. Necking occurs at the peak of the engineering stress-engineering curve, where the criterion of the following condition is satisfied,

(9)\sigma ={\displaystyle \frac{d\sigma }{dϵ}}.

Despite of this defect, one can still apply Eqs. (7) and (8) to computing the entire true stress-true strain relationship. This approach is named as the simple determination (SD) method.

Several approaches [13,15,16] have been proposed to modify the true stress-true strain after initiation of necking. Bridgman [13] first proposed a theoretical model for round bars under tension. However, the method is difficult to apply in practice, since it is difficult to measure the radius of curvature required by the method, with favorable accuracy. The other shortcoming of the method is that it is not applicable to flat bars due to the complicated profile of the neck.

Zhang and Li [15] proposed an approach to simulate the load-extension of a tensile coupon by iteration of FE analyses. The method works well for both round and flat bars, and it just requires the load-extension data of a tensile coupon test. However, the method is difficult to apply in practice, since a user defined FE program and a large number of FE analyses are required.

Ling [16] proposed a weighted average (WA) method where the upper and lower bounds of the hardening rate after necking occurs were postulated. Since the hardening modulus, dσ/dϵ, of most metals decreases as strain increases, the true stress at the initiation of necking is assumed to be an upper bound of the hardening modulus, and a power law is postulated to be a lower bound. However, the proposed lower bound is sometimes not a lower bound for structural steels. Therefore, a modified weighted average (MWA) method is proposed in this paper.

Modified weighted factor method proposed by the authors

A modified weighted factor method was proposed [17] based on three assumptions as illustrated in Fig. 3, which are found in test results of mild steels.

Assumption 1: The hardening modulus, dσ/dϵ, after necking is smaller than that at the point where necking initiates. This is in accordance with the assumption proposed by Ling [16]. Therefore, the true stress when necking initiates is an upper bound of the hardening modulus after necking.

Assumption 2: The hardening modulus, dσ/dϵ, after necking is larger than zero. This implies that zero is a lower bound of the hardening modulus after necking occurs.

Assumption 3: The true stress-true strain curve after necking is almost linear. This implies that the hardening modulus after necking is approximately a constant. This assumption is proved to be valid for many steels at large plastic strain ranges from classical hydrostatic tensile coupon tests conducted by Bridgman [13].

According to Assumption 1 through 3, the hardening modulus after the initiation of necking is a constant between zero and stress at which necking initiates, α_neck. By introducing a weighting factor, w, which takes a value between zero and one, the hardening rate after necking can be given as,

(10){\displaystyle \frac{d\sigma }{dϵ}}=(\mathrm{1}-w)\cdot \mathrm{0}+w\cdot {\sigma }_{neck}=w{\sigma }_{neck},

and the corresponding true stress-true strain relationship is given as,

(11)\sigma ={\sigma }_{neck}+w\cdot {\sigma }_{neck}(ϵ-{ϵ}_{neck}),

where ϵ_neck is the true strain at initiation of necking. In the MWA method, the true stress-true strain relationship after necking is expressed by Eq. (11) with an optimal weighted average factor, w, determined from the steps detailed in the literature [17].

Comparison of different modification methods

Monotonic tensile coupon tests

Monotonic tensile coupon tests were carried out to obtain the true stress-true strain using the MWA method. Two smooth round coupons as illustrated in Fig. 4 were tested with the setup shown in Fig. 5. At the middle length, the minimum diameter was measured using a three-dimensional laser displacement meter and the strain using two strain gauges. The strain calculated from the change in diameter matched the values from the strain gauges until the initiation of necking.

Numerical simulation of necking in tensile coupon tests

Numerical simulations of the coupon tests were carried out using the different true stress-true strain data obtained by the three methods (SD method, WA method, MWA method) introduced in section 3.2. The axisymmetric FE model shown in Fig. 6 was employed. Since a perfect model will elongate without necking, a tapered model is adopted. The slop along the length of the FE model is 0.33%, which doesn’t significantly affect the result. The nonlinear isotropic hardening plasticity model is employed to simulate the nonlinear behavior of metals under monotonic loading. The plasticity model only requires the input of a series of tabulated true stress-true strain data. Therefore, it is not necessary to calibrate model parameters of the plasticity model. Three FE analyses were carried out with the inputted true stress-true strain data obtained from the SD method, the WA method and the MWA method respectively. At the initiation of necking, the true stress, σ_neck, is 558 MPa and the true strain, ϵ_neck, is 0.20. The weighted factors, w, of the WA method is 0 and the MWA method 0.425. In this case, the optimal hardening rate after the initiation of necking for the WA method is just the assumed lower bound.

Comparison of experimental and numerical results

The load-change in diameter curves using the input data from the three methods are shown in Fig. 7 along with the experimental curve. While the SD and WA method both overestimates the load after necking occurs, the MWA method is able to accurately follow the experimental result. The error of the SD and MA method become larger as strain increases.

Calibration of model parameters for plasticity model from monotonic true stress-true strain data

Relationship between monotonic and cyclic true stress-true strain curves

The previous study [1] has found that the yield stress under compression in every cycle is close to the initial yield stress σ_y0 as illustrated in Fig. 8. Under the assumption that these values are equal, the size of the subsequent yield surface, {\sigma }_{y0}+R, can be obtained using the monotonic coupon test result

(12){\sigma }_{mono}={\sigma }_{\mathrm{y}\mathrm{0}}+\mathrm{2}R,

where σ_mono is the post-yield uniaxial stress of the monotonic true stress-true strain curve. From Eq. (1), in a uniaxial stress state, the following equation holds,

(13){\sigma }_{mono}-\alpha ={\sigma }_{y\mathrm{0}}+R,

where, a, is the uniaxial backstress. From Eqs. (12), (13), it is clear that the assumption stated above indicates that the IH component is equal to the KH component in the true stress-true strain obtained from the monotonic coupon test. This assumption coincides with the principle previously proposed by Kuhlmann-Wilsdorf and Laird [18] based on Cottrell’s [19] treatment of hysteresis loops, and has also been found to be reliable based on tests on single crystal copper and low carbon steel under small strains [20,21]. The applicability of the assumption in large strain ranges till fracture of virgin structural steels has been validated by the authors in the previous study [1], and this paper aims to validate it for structural steels with either tensile or compressive prestrain.

Calibration procedure

ABAQUS is adopted to calibrate the model parameters of the Chaboche model. The ones related with the KH component, Ci and γi, are calibrated to the monotonic true stress-true strain data input in tabular form (yield stress, equivalent plastic strain). The IH component, R, can be obtained according to Eq. (12), from true stress-true strain data of monotonic tensile coupon tests. The model parameters related with the IH component, Q_{\infty } and k, can be obtained through the input data in tabular form (post-yield stress, equivalent plastic strain).

Verification of calibration method for plasticity model parameters using hourglass specimens

Experiment

Test program

Thirteen hourglass specimens with the configuration shown in Fig. 9 were manufactured from a 65 mm thick Japanese Industrial Standard SS400 steel plate. An 8 mm long portion of the middle of the specimen is designed to have a uniform cross section, in order to produce a uniaxial stress state. To avoid the influence of impurities at the middle thickness of the steel plate, the specimens were cut from the location shown in Fig. 10.

As shown in Fig. 11, the specimen is screwed in and fixed at the two ends. The experiment is displacement controlled by an extension meter attached to the specimen which measures the elongation within a gage length of 30 mm. The minimum diameter of the specimen was also measured by a three dimensional digital laser displacement meter. Since the mechanical properties of the steel in the direction of the plate thickness and the direction orthogonal to it are slightly different due to the manufacturing process of the plate, the originally circular cross section of the hourglass specimens deform slightly into an ellipse under large deformation. Thus, to measure the average change in diameter, the diameter in a direction 45 degree from the direction of the plate thickness was measured.

In this paper, 5 of the 13 hourglass specimens [1] were investigated. KA01 was monotonically pulled to fracture, to check the stress state at the minimum cross section. The loading histories for specimens KA08 to KA11 include two steps. The specimens were first prestrained under either tension (KA08, KA09) or compression (KA10, KA11), and then loaded to fracture under either monotonic tension (KA08, KA10) or cyclic loading (KA09, KA11) approximately a month later. The loading programs for KA08 to KA11 are listed in Table 1, and the cyclic loading history for KA09 and KA11 is illustrated in Fig. 12.

Test results

The test results of KA08 to KA11 during the first loading step are not shown in this paper due to limited space. The load-deformation curves of the coupons with tensile prestrain (KA08, KA09) at the second loading step are shown in Figs. 13(a), (b), and those with compressive prestrain (KA10, KA11) are shown in Figs. 13(c), (d). The following observations can be made from the test results.

1) Figs. 13(a), (b) indicate that, for the specimens with tensile prestrain, the initial tensile yield stress is higher than that of the virgin material, while the initial reversal compressive yield stresses are approximately the same as the initial yield stress of the virgin material.

2) Figs. 13(c), (d) indicate that, for the specimens with compressive prestrain, the initial reversal tensile and compressive yield stresses are approximately the same as the initial yield stress of the virgin material.

3) Figs. 13(a) to (d) indicate that yield plateau phenomenon disappears in the specimens with compressive prestrain, but still appears in the specimens with tensile prestrain.

Commonly, monotonic tensile coupon tests are employed to determine material properties of prestrained steels in structural engineering, and the initial tensile yield stress is of great interest in practice. Based on the first and the second observations, the initial tensile yield stress of material with tensile prestrain will increase due to strain hardening, while that of the material with compressive prestrain will almost be unaffected. The reason is that Bauschinger effect occurs during the tensile test of the materials with compressive prestrain, and offset the increased stress caused by the strain hardening.

Numerical simulation

FE modeling

Numerical simulations using ABAQUS were carried out to simulate the hourglass specimens. Taking advantage of the axis-symmetry and symmetry across the mid-section, axisymmetric half models meshed with CAX8 elements as shown in Fig. 14, are used to model the hourglass specimens. The partial model is a finely meshed model of the bottom 15 mm of the whole model. Symmetric boundary conditions were applied to the symmetric edges, and the displacement is enforced at the top edge. Since the simulation results of the whole model and the partial model give almost identical load-deformation curves, the partial model was employed in the numerical simulation.

Plasticity model and calibration of model parameters

The main purpose is to accurately simulate the load-deformation curves of KA09 and KA11 at the second loading steps. Two approaches are taken in the simulation. The first approach uses plasticity model parameters calibrated from tensile coupon test results of the prestrained materials, while the second approach uses those calibrated from the tensile coupon test result of the original material along with the prestrain histories of the materials. In practice, the first approach is more commonly employed, since the coupon test results of virgin materials as well as the prestrain histories are often unknown.

For the first approach, the load-change in diameter curve of the hourglass specimen, KA01, is almost the same as that of the coupon test as shown in Fig. 15. Therefore, the test results of KA08 and KA10 at the second loading step can be deemed as the monotonic tensile coupon test results of the prestrained materials. For the second approach, the aforementioned coupon test results were used. The model parameters of the Chaboche model using the two approaches were listed in Table 2. The γ₁ of the virgin material and the prestrained materials are equal to 0, indicating that there is a backstress with a linear evolution rule for each material. The calibrated yield stress of the material with tensile prestrain is much higher than that of the virgin material, while the calibrated yield stress of the material with compressive prestrain has a value close to that of the virgin material.

Comparison of experimental and numerical results

The experimental results at the second loading step and the corresponding numerical results are compared in Fig. 13. Both the first and the second approach can well simulate the monotonic tensile tests of the materials with tensile and compressive prestrain well, as shown in Figs. 13(a), (c). The first approach slightly overestimates the compressive stress of the steel with tensile prestrain as shown in Fig. 13(b), while accurately predicts the forces for the steel with compressive prestrain, as shown in Fig. 13(d). The tensile prestrain results in an increase in the initial tensile yield force due to strain hardening, while the Bauschinger effect offsets the increase in the yield force resulted by the compressive prestrain. The initial tensile yield force of the material with compressive prestrain is thus close to that of the virgin material. Therefore, the calibration method of the Chaboche model gives favorable evaluation of the cyclic plasticity of materials with compressive prestrain using tensile coupon test results, while overestimates the isotropic hardening of the materials with tensile prestrain. For the second approach, the numerical results compare well with the experimental results for both materials with tensile and compressive prestrain.

Conclusions

An approach to calibrate model parameters of prestrained structural steels only from monotonic tensile coupon tests was validated by experimental and numerical studies in this paper, where the Chaboche model was employed in the simulation. Two approaches of different data sources were adopted in the calibration of the model parameters, where the first one is obtained from the tensile coupon test data of prestrained steels, and the second one is from that of the virgin material. The following conclusions were drawn.

1) The tensile prestrain results in increase in initial tensile yield stress due to strain hardening, while compressive prestrain results in small change in the initial tensile yield stress. The main reason is that the Bauschinger effect occurs during the tensile test of material with compressive prestrain, and offsets the increase in the yield stress resulted by strain hardening.

2) The first approach predicts the cyclic force-displacement behaviors well for structural steels with compressive prestrain, while slightly overestimates the compressive force for structural steels with tensile prestrain. This is due to that the calibration method of the plasticity model parameters overestimates the isotropic hardening of materials with tensile prestrain, and the Chaboche model assumes the same initial tensile and compressive yield stress.

3) The second approach predicts the cyclic plasticity of the two prestrained steels well, while the prestrain history and material properties of virgin materials are required and simulated in advance.

Based on the conclusions, the calibration method of the plasticity model parameters using the second approach is found to be more accurate if the prestrain histories are known. The calibration method with the first approach can also predict the cyclic plasticity of structural steels with acceptable accuracy.

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