Keywords

1 Introduction

The advancement of integrated die-cast conception and lightweight design in new energy vehicles has propelled the use of die-cast aluminum alloy [1], with an increasing number of die-cast parts found in automobiles. The transition from tiny and basic die-cast parts to massive structural die-cast elements has already been applied [2]. Hence, die-cast parts become intricate and varied, with unique structural attributes. Throughout the car manufacturing process, numerous structural holes are needed for machining, coating, and final assembly during the car production process. These structural holes, referred to as structural defects, significantly affect the strength and stiffness of die-cast parts used in automobiles. The implementation of accurate material constitutive relation is essential for part performance analysis, which is a critical stage in simulation. Given the intricacy of parts, higher simulation accuracy is demanded. The effects of structural defects on parts were disregarded in the investigation of mechanical performance; therefore, to overcome this limitation, calculation accuracy improvements are needed.

J–C constitutive model is a succinct empirical model that explains the link between temperature, strain rate, and stress and strain [3,4,5]. The J–C model is commonly used to predict the mechanical performances of materials [6, 7]. However, the accuracy of the empirical models, which are derived from curve fitting, is limited. Structural features and constitutive models with defect rates have not been investigated thoroughly yet but are urgently needed to understand the mechanical behavior of parts with structural features. Therefore, an agreement on the constitutive model that takes structural features into account needs to be reached.

JDA1b an acronym for Shanghai Jiao Da Aluminum 1#, is a high-strength, exceptionally ductile, non-heat-treated die-cast aluminum alloy [8]. It has been utilized in numerous automotive components. Compared to traditional die-cast aluminum alloys, the alloy does not experience rectification or dimensional instability. This study aims to obtain the defect rate effects and create a constitutive model with defect rates of die-cast aluminum alloys.

2 Materials and Experiments

2.1 Materials and Specimens

The cast and specimen are presented in Fig. 1. Figure 1a shows the JDA1b alloy ingots. A TOYO BD-350-V5 die-casting machine was employed for manufacturing the castings of JDA1b alloy (Fig. 1b). The casting pressure was 75 MPa. The square thin plates with a thickness of 3.0 mm were obtained after removing the runner on the castings as shown in Fig. 1c and d. Specimens for this study were cut from the square plates using the computerized numerical control wire-cut electric discharge technique. The specimens were polished to achieve a smooth surface. Figure 1e displays several specimens that were created in compliance with Chinese standard GBT 2281–2021. There are two types of specimens: those without a hole and those with a circular hole in the center. The dimensions are displayed in Fig. 1f. Here, the structural features are supposed to be circular holes with varying sizes in the center and the defect rate (denoted with λ) is the ratio of the maximum projected area of the circular hole along the stretching direction to the contour area of the cross-section in the specimen’s gauge. Since the specimen's thickness is homogeneous, the defect rate λ may be written as ρ/b, where ρ is the radius of holes and b = 7.5 mm is the specimen's half-width (Table 1).

Fig. 1
6 close-ups titled from a to f are as follows. A stack of metal bars, a metal sheet with vertical ridges and an attachment, a stack of metal sheets, a sheet with 6 structure outlines and a ruler at the top, 7 structures with increasing hole sizes, and a metal structure with metal flow direction.

Cast and specimen of the JDA1b alloy

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Table 1 Chemical compositions of the JDA1b aluminum alloy (wt.%)
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2.2 Experimental Apparatus and Method

A total of eight groups of specimens (five specimens per group, 40 specimens in total) were tested at room temperature (25 ℃) at a tensile rate of 1 mm/min using a Zwick Roell 8506 universal electrical testing machine (100 kN). The quasi-static tensile experiments were performed in accordance with the international standard ASTM E8/E8M-16a. The gauge length strain was measured using the extensometer. The Zwick Roell information processing systems were utilized to obtain the true stress–strain curves. When calculating stresses, the reference area is the cross-sectional area of the specimen’s gauge that is devoid of holes. Five specimens were repeatedly tested for each defect rate and took the average value as the experimental outcome.

3 Experimental Result

The true stress–strain curves for the JDA1b alloy per defect rate are shown in Fig. 2, suggesting these curves are stable within each group and have comparable hardening characteristics. The results of each experimental group are thereafter represented by the average curves, which are indicated by solid curves. Strong nonlinearity exists in the work hardening. The stresses rose swiftly in the early stages of deformation due to the generation and multiplication of dislocations. After a quick rise, the stresses turn to a slow increase. In the later stage of deformation, the stress steadily stabilizes. Figure 2i superimposed the average true stress–strain curves at various defect rates, ranging from 0 to 46.67%. Defect rates have a noticeable impact on stress. The stresses under the same strain decrease as defect rates rise. The yield strength, ultimate tensile strength, and elongation all decrease as the defect rate rises. Several fractured specimens are displayed in Fig. 2i, and all specimens were fractured at the smallest cross-section.

Fig. 2
9 line graphs titled from a to i plot sigma versus epsilon. A, test 1 (0.05, 275). B, test 2 (0.04, 265). C, test 3 (0.03, 240). D, test 4 (0.020, 200). E, test 5 (0.005, 120). F, test 1 (0.015, 165). G, average (0.010, 135). H, test 4 (0.015, 138). I, lambda = 0% (0.10, 305). Values are estimated.

True stress–strain curves of JDA1 alloy at defect rates of a 0, b 6.67%, c 13.33%, d 20.00%, e 26.67%, f 30.00%, g 40.00%, h 46.67%, and i average curves

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Fig. 3
A line graph with scatterplots plots concave downward ascending trends for sigma versus epsilon. J C model (0.00, 165), (0.05, 275), (0.10, 310). Proposed model (0.00, 158), (0.05, 280), (0.10, 310). Experiment (0.00, 155), (0.05, 280), (0.10, 310). Values are estimated.

Comparison of true stress–plastic strain data among J–C model, proposed model, and experimental data

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4 Constitutive Model

The J–C model is expressed as:

$$ \sigma = (A + B\varepsilon^{n} )(1 + C\ln \dot{\varepsilon }^{*} )(1 - T^{*m} ) $$
(1)

where ε is the plastic strain; A, B, C, n, and m are undetermined parameters; \(\dot{\varepsilon }\) is the plastic strain rat, \(\dot{\varepsilon }_{0}\) is the reference plastic strain rate; \(\dot{\varepsilon }^{*} = \dot{\varepsilon }/\dot{\varepsilon }_{0}\); T * = (T-Tr)/(Tm-T), where Tr is the room temperature; and Tm is the melting temperature. The current work focuses on strain-strengthening effects, hence the last two terms in Eq. (1) are ignored.

The true stress–plastic strain data of intact specimens acquired from the quasi-static uniaxial tensile test are shown in Fig. 3. The yield stress is 163.25 MPa, or A = 163.25 MPa. Strain strengthening coefficients, B = 392.93 MPa and n = 0.41345, are calculated by the method from references [9, 10]. The stress of JDA1b stabilized eventually. However, the stress did not converge in the J–C model. This indicates that precise flow stress predictions are not possible using the original J–C model. Therefore, to better predict stress, convergence must be considered in constitutive models.

The proposed constitutive model is expressed as

$$ \sigma = \left[ {Q - \frac{Q - A}{{1 + \left( {{\varepsilon \mathord{\left/ {\vphantom {\varepsilon {\varepsilon_{0} }}} \right. \kern-0pt} {\varepsilon_{0} }}} \right)^{p} }}} \right](1 + C\ln \dot{\varepsilon }^{*} )(1 - T^{*m} ){(}1 - \lambda^{k} {)} $$
(2)

where Q is the tensile stress limit value; ε0 is the critical strain; λ presents the defect rate of specimens with a circular hole in the center; λ = ρ/b, p and k are unknown coefficients, and the remaining parameters the same as in the J–C model. The parameters Q, A, and ε0 show clear physical significance, and their values have been previously discussed in our work [11]. Here, Q = 366.70 MPa, ε0 = 0.03425, p = 0.94521, and k = 1.52873.

The proposed new constitutive model introduces the (1-λk) term to represent the influence of defects and uses stress limit value and critical strain to describe the stress–strain curve. The convergence of the stress is represented by the inverse proportional function \(Q - \frac{Q - A}{{1 + \left( {{\varepsilon \mathord{\left/ {\vphantom {\varepsilon {\varepsilon_{0} }}} \right. \kern-0pt} {\varepsilon_{0} }}} \right)^{p} }}\) term. The predicted curve of the proposed model is displayed in Fig. 3. The J–C model’s form is changed by the proposed model. Based on the comparisons, the proposed model fits the tendency of the stress better and shows better predictability than the J–C model.

5 Results and Discussion

5.1 Effect of Defect Rates on Strength and Elongation

The Effect of defect rates on strength and elongation are summarized in Fig. 4. Here, YS stands for yield strength, UTS stands for ultimate tensile strength, and EL stands for elongation. As defect rates rise, the YS, UTS, and EL all nonlinearly decline. When defect rates are less than 13.33%, the EL fluctuates widely, from 7.9 to 14.7%, but when defect rates are more than 13.33%, the EL fluctuates narrowly. Furthermore, YS fluctuates quite little.

Fig. 4
A column chart with error bars plots Y S U T S and E L on the vertical axes versus lambda. Y S (0, 160), (13.33, 157), (46.67, 110). U T S (6.67, 295), (33.33, 185), (40, 160). E L (0, 0.125), (26.67, 0.028), (40, 0.002), (46.67, 0.002). Values are estimated.

Effect of defect rates on strength and elongation

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5.2 Analysis of Constitutive Model Accuracy

The J–C model can be used to express the defect rate effect by multiplying the defect term (1-λk) behind Eq. (1). The proposed model and the J–C model’s projected stresses were compared with the results of experiments. As can be seen in Fig. 5, the J–C model's prediction accuracy is not optimal for all strains. Only a weak correlation between the expected and experimental stresses. When the defect rates are higher than 13.33%, the predicted values exhibit a significant deviation from the experimental results. The J–C model does not adequately capture the trend of the stresses, thus it appears unreasonable. From Fig. 6, for all defect rates, the true stress–strain curves predicted by the proposed model show a consistent trend with the experimental curves. The results demonstrate that, over the whole range of strain and defect rates, the proposed model outperforms the J–C model in describing the deformation behavior of die-cast aluminum alloys.

Fig. 5
A scatterplot plots concave downward ascending trends for sigma versus epsilon. Lambda = 0% J C model (0.10, 310). Lambda = 6.67% experiment (0.05, 275). Lambda = 13.33% J C model (0.00, 157). Lambda = 20% experiment (0.00, 150). Lambda = 26.67% J C model (0.00, 145). Lambda = 33.33% experiment (0.00, 135). Values are estimated.

Comparisons between experimental stresses and predicted stresses by the J–C model for the JDA1b alloy at different defect rates

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Fig. 6
A scatterplot plots concave downward ascending trends for sigma versus epsilon. Lambda = 0% proposed model (0.10, 317). Lambda = 6.67% experiment ((0.05, 275). Lambda = 20% proposed model (0.00, 150). Lambda = 40% experiment (0.00, 121). Lambda = 46.67% proposed model (0.00, 110). Values are estimated.

Comparisons between experimental stresses and predicted stresses by the proposed model for the JDA1b alloy at different defect rates

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The accuracy of the two constitutive models is assessed using the average absolute relative error (AARE) between the experimental data and the predicted results. The AARE values for the J–C model and the proposed model are 2.66~12.54% and 0.21~0.36%, respectively. Compared to the J–C model, the proposed model's AARE is significantly lower. When the defect rates rose, the J–C model’s error increased gradually. The primary reason for this inaccuracy was the J–C model’s failure to converge. The suggested model's predicted values were more closely correlated with the experimental values, and its accuracy was noticeably higher than the J–C model’s accuracy. Thereby resolving the J–C model’s poor prediction accuracy problem.

5.3 Validation with a Die-Cast Part

To verify the proposed constitutive model, we created a JDA1b die-cast part with a complicated cross-section and conducted tensile tests on it. The die-cast part is numerically simulated using the proposed constitutive model with and without defect rates. The experimental setups and three-dimensional model are shown in Fig. 7. Figure 7a presents the tensile experimental setup. A unique fixture is designed to clamp the parts. The die-cast part’s cross-sectional shape and dimensions are displayed in Fig. 7b. The part's hole measures 7 mm in diameter, while the side and bottom walls have widths of 32 mm and 67 mm, respectively. As a result, the defect rates of the entire side wall and bottom wall are 21.87% and 10.77%, respectively. Figure 7c displays a three-dimensional finite element model.

Fig. 7
A close-up and a diagram of a metallic structure with arrows indicating fixtures at the top and bottom are titled a. The measurement of a tray-like structure, its left, and top sides is titled b. A diagram of a metallic structure with multiple upward-pointing arrows at the top is titled c.

Tensile experiments and simulations of the JDA1b die-cast part

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The deformation in tests and simulations is displayed in Fig. 8. Figure 8a compares the parts before and after tests. The largest displacement, measuring 9 mm in tests, is found at the top of the part. The displacement simulated by the proposed model with defect rates is 9.06 mm, which is near to the displacements obtained experimentally. As shown in Fig. 8b, the applied maximum tensile load is roughly 33 kN. The simulation's fracture form and position are consistent with the test’s (see Fig. 8c). However, the fracture does not occur, and the constitutive model without defect rates only simulates a displacement of 5.37 mm (see Fig. 8d). Therefore, applying a constitutive model that considers the defect rate will produce more realistic simulation results. The weakening phenomenon of holes to the mechanical performance is reflected in the proposed constitutive model.

Fig. 8
A, Photos of deformed and undeformed structures labeled experiment. B, a load graph plots force versus displacement. C, stress, and displacement with color gradients labeled constitutive model with defect rates. D, stress, and displacement color gradient labeled constitutive model without defect rates.

Experiment and simulation of the JDA1b die-cast part

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6 Conclusions

The mechanical performances of die-cast aluminum alloys with structural defects were investigated experimentally and numerically. A constitutive model of die-cast aluminum alloy with defect rates was proposed and verified. One can infer the following conclusions.

  1. (1)

    The structural features of the JDA1b alloys are connected to their tensile mechanical performances. Yield strength, ultimate tensile strength, and elongation all decrease with increasing defect rates. Performance deterioration can be severe even with a small structural defect.

  2. (2)

    The stress limit value, critical strain, and defect rate are incorporated in the proposed constitutive model to describe the stress–strain curves, enabling more accurate stress prediction at different defect rates. The predicted results of the proposed model show extreme consistency with experimental results and exhibit superior predictability than the J–C model. The physical significance is reflected in the parameters of the proposed model. Simulation results will be more realistic if a constitutive model accounting for the defect rate is used.

  3. (3)

    The experimental and numerical research on the JDA1b die-cast part confirms that the proposed constitutive model is accurate. This indicates that the plastic deformation behavior of die-cast aluminum alloy can be accurately described by the proposed constitutive model with defect rates.

This work offers suggestions for investigating the mechanical behavior of alloys with structural features. It facilitates accurate numerical simulations and studies relevant to engineering products besides die-cast aluminum parts.