Simulation of Fracture Performance of Die-Cast A356 Aluminum Alloy Based on Modified Mohr–Coulomb Model

Abstract

The fracture performance and damage prediction of die-cast materials are critical to guarantee the safe application of die-cast structural components in lightweight vehicles. Monotonic loading experiments were conducted on different shapes of die-cast aluminum alloy A356 specimens. Finite element simulation models of the A356 monotonic loading experiments were established, and the stress state of the specimens during the loading process was analyzed. The Modified Mohr–Coulomb (MMC) failure model of A356 was fitted by the failure strain under different stress states. Finally, the established MMC failure model was verified by a uniaxial compression experiment and bending experiment. The results show that the MMC failure model can be applied to the prediction of the fracture behavior of A356.

1. Introduction

In recent years, the wide usage of advanced lightweight materials such as high-strength steels, magnesium alloys and aluminum alloys has been leading to an evolution in manufacturing methods for components and parts in the automotive industry [1]. As a typical metal forming technology, the die-casting process can satisfy various requirements of structural parts, particularly for complex shapes with high production efficiency and molding accuracy [2]. The die-casting process is used to produce automotive body and many chassis components, such as suspension systems, transmission systems, wheels and brake systems. As an essential part of the safety design and operation for whole vehicles, the safety evaluation of die-cast components must be taken seriously [3].

The strength and fracture properties of die castings need to be accurately characterized to ensure the safety of each component before the whole vehicle safety evaluation by crash testing and simulation. Finite element simulation plays an important role in whole safety analysis [4]. Numerical simulation provides the capability to evaluate the performance and reliability of components, thus optimizing the entire vehicle design process. For example, Huichi et al. [5] established a coupled gas–liquid–solid multiphase flow model through ANSYS software to analyze and control the quenching deformation of lightweight cast aluminum alloy wheels. Kong et al. [6] investigated the effects of microdefects and the stress state on the failure behavior of cast aluminum alloy A356 by experimental and numerical methods. Considering the distribution of porosity within the die-cast components, Zhang et al. [7] investigated the failure characteristics of the die-cast aluminum alloy Aural-2 by experimental and numerical methods. They proposed a probability-dependent damage model to represent the behavior of the fracture under a wide range of stress states, and found a significant dependence of the initiating fracture strain on the stress triaxiality. Up to now, the simulation of die-cast structural parts has mostly focused on the simulation of the casting process and the effects of casting defects on the service performance of the parts. From the viewpoint of die-cast components application, high-stress conditions are inevitable. The fracture behavior of cast aluminum materials must especially be characterized through a reasonable failure model to ensure the simulation accuracy of the whole vehicle crash.

When assessing safety performance, the fracture moment and fracture location are important indicators for evaluating the reliability of structures and components. Traditional fracture mechanics, represented by the stress intensity factor criterion, J-integral criterion, etc., cannot explain the fracture characteristics of uncracked solid materials under the assumption of preexisting initial cracks. At the microscopic level, the material failure process is the process of formation, growth and the coalescence of voids within the material [8]. Most studies [9,10,11] have suggested that void growth and aggregation-type microscopic fracture mechanisms can accurately illustrate the macroscopic fracture behavior under monotonic loading conditions. According to the interaction of plasticity and damage criteria, the currently applied damage mechanics models can be divided into two groups: coupled models and uncoupled models [6,12,13]. Among the coupled models, the Gurson–Tvergaard–Needleman (GTN) model considers the formation and growth process of voids, and has been widely applied in various applications [14,15]. In this model, material damage occurs once the void volume fraction of the material reaches a limiting value during deformation. However, the complicated parameter calibration process is the main reason that limits the further application of GTN models when compared to uncoupled damage models [16]. When using uncoupled damage models, the failure criterion is usually characterized by the failure strain. Experimentally obtaining a functional relationship between the stress state and the failure strain greatly simplifies the process of calibrating the unknown parameters [9,17,18,19,20,21].

In uncoupled damage models, the stress state is the critical factor affecting material fracture. In the damage models proposed by researchers, the stress state considers hydrostatic pressure and stress invariants, which are usually expressed as the stress triaxiality and the Lode angle. Considering the effect of the stress state on fracture behavior, Bao and Wierzbicki [9] conducted several sets of experiments on aluminum alloy 2024, including shear, tensile and compression, and simulated the experiments numerically through ABAQUS software. It was found that the fracture strain and the stress triaxiality are strongly correlated. For medium- and high-strength steels, Barsoum [22] investigated the relationship between failure strain, stress triaxiality, and Lode parameters. It was found that the ductility of steel could not be fully characterized by triaxiality alone. Khan et al. [8] conducted sets of experiments on aluminum alloy 2024 and established a phenomenological fracture criterion by considering stress invariants. According to the experimental results, there is a non-monotonic functional relationship between fracture strength and the stress triaxiality. Considering the hydrostatic pressure, stress invariants and Lode angle parameters, Bai and Wierzbicki [18,23] proposed the MMC failure model based on the Mohr–Coulomb equation, and applied the MMC model to successfully simulate the failure modes of aluminum and TRIP steel. Talebi-Ghadikolaee [1] conducted uniaxial, notched, plane strain and plane shear experiments for 6061 aluminum alloy and calibrated the MMC model based on the experiment results. The calibrated MMC failure model can more accurately simulate the failure behavior of aluminum alloy in the U-bending experiment. From previous studies [1,6,18,19,23,24], the failure criterion for predicting the fracture behavior of metals can be characterized by the stress triaxiality and Lode angle parameter.

In order to accurately simulate and predict the failure behavior of die-cast materials and ensure the reliability of die-cast structural parts in whole-vehicle applications, this paper conducted an investigation on the fracture behavior of die-cast aluminum alloy A356. By combining finite element simulation and experimental methods, the MMC failure model of A356 was established and the model parameters were calibrated. The purpose of this paper is to investigate the performance of the MMC failure model in predicting the fracture performance of A356 and to verify the feasibility of its application in structural safety simulation.

2. Materials and Experiment Procedure

Sample parts in the size of 300 mm × 200 mm were taken from the subframe, as shown in Figure 1. The sample sheets were first cut from the subframe and then milled to the geometry specified in the test standard. The milling process was performed with carbide tools at the appropriate speed and feed rate. The final process was a finishing process to ensure that the roughness of the milled surface was less than Ra 3.2 µm. After milling, the surface of the gauge section was ground with 2000# SiC paper and then slightly polished. The machined specimens were measured by a Coordinate Measuring Machine to ensure dimensional consistency. A fresh sheet of SiC paper was used for each sample, and the polished surfaces were measured with the surface roughness measuring instrument to ensure that the roughness was below Ra 1.6 µm.

In this study, the A356 aluminum alloy was experimented on after being milled to standard dimensions. The chemical composition of the A356 material used in this study is shown in

Table 1. Chemical composition of the A356 (wt.%).

Si	Fe	Mn	Zn	Mg	Cu	Ti
7	0.6	0.35	0.35	0.3	0.25	0.25

. The standards referenced for the experiments are shown in

Table 2. The experiment standard referenced in this study.

Material	Test	Reference Standard [25,26]	Repeat Time	Strain Rate
A356	Static tensile	GB/T 228.1-2010	3	0.001/s
	Dynamic tensile	GB/T 30069.2-2016	3	100/s, 500/s
	R5 notch	GB/T 228.1-2010	3	0.001/s
	R20 notch	GB/T 228.1-2010	3	0.001/s
	D10 center hole	GB/T 228.1-2010	3	0.001/s
	45° shear	GB/T 228.1-2010	3	0.001/s

, and the thickness of all specimens was 5 mm. Quasi-static and dynamic experiments were conducted on the CMT5205 electronic test system and the Zwick-HTM-6020 high-speed test system, respectively. Three replicate tests were performed on the specimens in each set of experiments to ensure the reliability of the experiment results. Digital Image Correlation (DIC) technology was applied in all experiments to record and analyze the strain during the experiments.

To investigate the relationship between the stress triaxiality and plastic strain at failure, we designed specimens with different geometric shapes: static tensile, dynamic tensile, 45° shear, R5 notch, R20 notch, and D10 center hole specimens, as shown in Figure 2. During the loading process, the triaxiality of each specimen was maintained until the material failure behavior occurred. The stress states were different for each specimen, allowing us to perform a more comprehensive analysis.

3. Material Failure Theory Model

The stress state is usually represented by the stress tensor. In addition, uncoupled damage mechanics models typically adopt a stress invariant to represent the stress state. In fracture mechanics, most fracture criteria are characterized by the failure strain, which is functionally related to the stress invariant. Consequently, the stress state is related to the damage state by the failure model to describe the accumulation of material damage and the occurrence of the ultimate fracture. The stress triaxiality η can be defined in terms of the hydrostatic stress ${\sigma }_m$ and the equivalent stress $\overline{\sigma }$, represented by the following equation:

$$\eta =\frac{{\sigma }_m}{\overline{\sigma }}$$

(1)

where, ${\sigma }_m=\frac{1}{3}\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)$; $\overline{\sigma }=\sqrt{\frac{1}{2}\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]}$; ${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$ represent the three principal stresses with the relationship ${\sigma }_1$ > ${\sigma }_2$ > ${\sigma }_3$.

The Lode angle θ is a significant measure of the stress state and can be defined in terms of stress invariants as follows:

$$\theta =\frac{1}{3}\left(\arccos \left(\frac{3\sqrt{3}}{2}\frac{J_3}{J_2^{\frac{3}{2}}}\right)\right)$$

(2)

where J₂ and J₃ are the second and third deviatoric stress invariants, respectively. the Lode angle can be normalized to the Lode angle parameter as:

$$\overline{\theta }=1-\frac{6\theta }{\pi }$$

(3)

The value of the Lode angle is taken within a specified range, denoted as (0, $\frac{\pi }{3}$). Therefore, the value range of the Lode angle parameter is (−1, 1).

Bai and Wierzbicki [23] established the MMC fracture model for the prediction of metal fracture based on the Mohr–Coulomb damage criterion. This model enables the calculation of the equivalent plastic strain at failure through the incorporation of stress triaxiality and the Lode angle parameter.

$${\overline{\epsilon }}_p^f={\left\{\frac{A}{c_2}\times \left[c_{\theta }^s+\frac{\sqrt{3}}{2-\sqrt{3}}\left(1-c_{\theta }^s\right)\left(\sec \left(\frac{\overline{\theta }\pi }{6}\right)-1\right)\right]\times \left[\sqrt{\frac{1+c_1^2}{3}}\cos \left(\frac{\overline{\theta }\pi }{6}\right)+c_1\left(\eta +\frac{1}{3}\sin \left(\frac{\overline{\theta }\pi }{6}\right)\right)\right]\right\}}^{-\frac{1}{n}}$$

(4)

where ${\overline{\epsilon }}_p^f$ is the equivalent plastic strain at failure, A is an equation constant, and n is the hardening exponent. In addition, c₁, c₂ and $c_{\theta }^s$ are material constants that need to be obtained by fitting the experimental results.

The damage evolution model matched with the MMC criterion is the GISSMO model. In the GISSMO model [27,28], the damage condition of material is characterized by the nonlinear damage accumulation factor D $\left(0<D\le 1\right)$:

$$\Delta D=\frac{k}{{\overline{\epsilon }}_p^f}{D}^{\frac{k-1}{k}}\Delta {\epsilon }_p$$

(5)

where k is the damage accumulation exponent; Δ${\epsilon }_p$ is the equivalent plastic strain increment.

Furthermore, the GISSMO model introduces a parameter F $\left(0<F\le 1\right)$ to quantify material instability, which reflects the effect of instability on material softening. In the GISSMO model, parameter F is associated with the equivalent plastic strain through a nonlinear equation similar to Equation (5):

$$\Delta F=\frac{k}{{\epsilon }_{crit}\left(\eta \right)}{F}^{\left(1-\frac{1}{k}\right)}\Delta {\epsilon }_p$$

(6)

where ${\epsilon }_{crit}$ is the critical equivalent plastic strain at instability. When the magnitude of the instability parameter F reaches 1, the current damage accumulation factor D is stored as the critical damage parameter D_crit. From this moment, the stress tensor is affected by the damage parameter and is recalculated according to the following equation:

$${\sigma }^{*}=\tilde{\sigma }\left[1-{\left(\frac{D-D_{crit}}{1-D_{crit}}\right)}^m\right]$$

(7)

where $\tilde{\sigma }$ is the stress without considering the material softening instability; m is the stress recession exponent.

4. Numerical Model

In this paper, a three-dimensional finite element method (FEM) model was established with the same geometry as the experimental specimen. The model was meshed using Hypermesh pre-processing software, and the element type was adopted as an eight-node hexahedral element. The element size of the model for each specimen was approximately 2 mm. The boundary conditions of the FEM model were set up with the clamping side completely fixed and the loading side loaded. In order to be as consistent as possible with the experimental environment, the five degrees of freedom at the loading side were constrained except for the moving degrees of freedom in the loading direction, and the same velocity conditions as in the experiment were applied in the loading direction. Then, the numerical simulations in this paper were all calculated using the LS-DYNA solver. The computational results were read by HyperView and HyperGraph post-processing software for the analysis of the simulation data. Figure 3 shows the finite element models for the different experimental specimens.

The experimental results were used to analyze the equivalent plastic strain at failure, whereas the FEM simulation results were used to obtain the variation of the stress state during the loading of different specimens. By combining the experimental and FEM simulation results, the parameters of the MMC model were fitted using a Python script. The GISSMO model was applied to simulate instability and eventual material fracture.

5. Results and Discussion

A356 is an aluminum–silicon (Al-Si) system casting alloy, which has a certain mechanical strength along with castability, weldability and corrosion resistance. Meanwhile, A356 aluminum alloy is a hypoeutectic alloy, whose mechanical characteristics are affected by the morphology of eutectic silicon particles. Casting defects, shrinkage, eutectic Si morphology and other factors affect the final service performance of the alloy [29]. According to previous research, the tensile strength of A356 has been reported to be between 150 and 320 MPa, the proof strength to be between 100 and 230 MPa, and the elongation at fracture to be typically in the range of 2 to 8% [30,31,32,33]. The engineering curve and the transformed real stress–strain curve are shown in Figure 4. According to the experimental results in this paper, the proof strength of the experimental material A356 is approximately 224 MPa, the tensile strength is approximately 330 MPa, and the elongation at fracture is approximately 7%. In terms of the data, the mechanical experimental results of A356 used in this paper are more consistent with the results of previous studies.

Figure 5 shows the dynamic tensile experimental results of A356. From the data curves, we obtained the proof strength and the ultimate strength at different strain rates. At the low strain rate of 0.001/s, the proof strength was 224 MPa. When the strain rate was increased to 500/s, the proof strength reached 262 MPa. Similarly to the proof strength, the flow stress also increased with the increase of the strain rate. In addition, the ultimate strength was 332 MPa at the low strain rate of 0.001/s and 418 MPa when the strain rate was increased to 500/s, showing a significant strain-rate strengthening effect. The results indicate that A356 alloy is strain-rate sensitive, which is consistent with previous studies on other aluminum alloys [24,34]. In addition, similarly to the trend in proof strength, the true strain at fracture increased with increasing strain rate, as shown in Figure 5a. When the strain rate was 0.001/s, the fracture strain was approximately 0.067, and at the strain rate of 500/s, the true strain at fracture increased to 0.096. This was also reported in studies of aluminum alloys 3003, 5052, etc. [24,35]. Based on previous studies of the dynamic properties of aluminum alloys, the effect of strain rate variation on the strength and flow behavior of the material is strongly related to the work hardening.

Parameters related to the stress state, such as the stress triaxiality and the Lode angle parameter, were calibrated by the DIC technique and FEM analysis. First, in order to clarify the change of stress triaxiality of the specimen during loading, we assigned a real strain hardening curve to the integral points in the FEM model. Then, the numerical analysis was performed to extract the stress triaxiality, Lode angle parameter and equivalent plastic strain data during the loading process by observing the elements at the crack initiation location in the specimen. With the DIC technique, we could obtain the equivalent plastic strains for different specimens at fracture. Thus, the equivalent fracture strains corresponding to the different stress-state-related parameters were obtained.

As shown in Figure 6, we obtained the stress triaxiality curves of the elements at the location of the least cross-sectional area of each specimen during tensile loading through the FEM simulation results. During the steady loading stage of the specimen, the triaxiality was maintained near a constant value. Therefore, the average triaxiality value of each specimen in the stable loading stage was selected as the stress triaxiality of the specimen for the process of analyzing the FEM results.

After experimental and FEM simulation analysis, we obtained the average triaxiality during loading and the equivalent plastic strain at the onset of fracture for each specimen, as shown in

Table 3. Experiment fracture strain at different stress triaxiality.

Specimen	Stress Triaxiality	Equivalent Plastic Strain at Fracture
Uniaxial tensile	0.33	0.065
R5 notch	0.5	0.081
R20 notch	0.42	0.12
D10 center hole	0.37	0.067
45°shear	0.25	0.13

. In agreement with the previous research literature on the fracture phenomena of metals [36,37], the average triaxiality of the tensile specimens was approximately 0.33. Since hydrostatic stresses were introduced into the notched region, the stress triaxiality value at the smallest cross section was 0.5 for the R5 specimen, and similar findings were reported in the previous literature [24].

In the MMC equation, the parameters to be determined were: k, n, $c_1$, $c_2$ and $c_{\theta }^s$. For the plane strain condition, $c_{\theta }^s$ = 1. The MMC model could thus be simplified as follows:

$${\overline{\epsilon }}_p^f={\left\{\frac{A}{c_2}\times \left[\sqrt{\frac{1+c_1^2}{3}}\cos \left(\frac{\overline{\theta }\pi }{6}\right)+c_1\left(\eta +\frac{1}{3}\sin \left(\frac{\overline{\theta }\pi }{6}\right)\right)\right]\right\}}^{-\frac{1}{n}}$$

(8)

According to the stress triaxiality and the equivalent plastic strain at fracture in Table 3, unknown parameters in the MMC equation were fitted by a Python script. The MMC fracture surface obtained after fitting is shown in Figure 7.

In the GISSMO failure model, the nonlinear damage accumulation exponent k, the equivalent plastic strain when instability deformation occurs ${\epsilon }_{crit}$, and the stress recession exponent m are unknown parameters. A parameter inversion method [38] in LS_OPT was used to obtain the value of each parameter in the GISSMO failure model based on the force–displacement data. The mean square error of the load–displacement data in the parameter inversion was less than 3%.

Figure 8 shows the comparison of force–displacement curves and failure modes between the simulation calculation and experiment results for five specimens. As shown in Figure 8a, according to the force–displacement curve, there was no softening stage before material fracture, and similarly no significant necking was observed before material fracture during the simulation and experiment. In Figure 8b,c, the minimum cross-sectional positions of the R5 and R20 notched specimens are in the middle of the specimen. The stress concentration phenomenon caused the specimen to fracture first at the edge of the notch. The fracture displacement was about 0.55 mm and 0.75 mm. During the tensile deformation process, the specimen first experienced elastic–plastic deformation and fracture occurred instantaneously at the maximum strength. Similarly to the static tensile experiment, after the tensile force reached the maximum value, the material almost did not soften and no significant necking occured; instead, fracture occured directly. Figure 8d shows the force–displacement curve of the D10 center hole specimen. The fracture was first generated at the internal hole wall of the middle section, and then extended along the hole wall normal to the axial load until the complete fracture. Figure 8e shows a sheared 45° specimen with fracture starting at the arc edge of the least cross section. With the tensile deformation, the crack at the edge propagated into the matrix and eventually led to a complete fracture. Compared with the notched specimen, the tensile displacement of the shear specimen was larger, about 0.92 mm. A significant stress concentration was observed before the fracture under all test conditions. The comparison of the FEM results with the experimental results is given in

Table 4. Comparison of FEM and experiment results of force–displacement curves.

Experiment	Displacement at Fracture (mm)			Ultimate Force (N)
	Experiment	FEM	Error	Experiment	FEM	Error
Static tensile	2.76	2.8	1.5%	15,545	15,645	0.64%
R5 notch	0.55	0.546	0.72%	18,083	18,201	0.65%
R20 notch	0.755	0.756	0.13%	16,673	16,554	0.71%
D10 center hole	0.536	0.511	4.6%	16,317	16,349	0.196%
45° shear	0.923	0.946	2.49%	1768.36	1770.72	0.13%

, wherein the maximum error was successfully controlled within 5%.

Through the failure curves of different experiment groups, the die-cast aluminum A356 showed a weak necking phenomenon before fracture and a short cumulative stage of failure, indicating a possible brittle fracture characteristic. We observed the same results through FEM simulation, as shown in Figure 6. In the plastic stage before fracture, the triaxiality remained basically consistent, and there was no obvious necking phenomenon at the fracture position. In previous studies, Moussa [29] and Jiang [39] reported that A356 has the characteristic of brittle fracture. The characteristic of an insignificant necking phenomenon is related to the casting process and microstructure of A356. Defects such as inclusions, porosity, and shrinkage produced by casting can lead to stress concentrations within the material. These casting defects will accelerate the formation and expansion of cracks. Under the action of stress, the cracks will expand in a very short period of time, leading to fracture of the material. In addition, A356 is a hypoeutectic Al–Si alloy, and coarse eutectic silicon particles can lead to stress concentration phenomena, thus increasing the possibility of material fracture [29]. A reduction of casting defects, refinement and modification of eutectic silicon by improving the casting process can improve the ductility of A356 [40,41].

In numerical calculations, the element size effect must be considered, and is shown in Figure 9. The material in the finer elements showed a more delayed fail than that in the coarser elements. An element dependence factor which scales the failure strain of the material according to the element size was thus calculated. The element dependence factor $\alpha \left(scal{e}_{el}\right)$ was defined as a function of the relative element size, and the relative element size was defined as:

$$scal{e}_{el}=\frac{Siz{e}_{el}}{Siz{e}_{ref}}$$

(9)

where $Siz{e}_{el}$ is the reference element size, which is usually the element size used in the material test calibration model; $Siz{e}_{ref}$ is the actual size of each element applied in the whole model.

According to the strain hardening curve of A356 (Figure 5a), the strain rate had a significant effect on the failure pattern. Using the failure strain value in the quasi-static condition as the reference failure strain, a strain rate correction factor was calculated to correct the failure strain at different strain rates using the following equation:

$$\beta \left(\dot{\epsilon }\right)=\frac{\dot{\epsilon }}{{\dot{\epsilon }}_{ref}}$$

(10)

where ${\dot{\epsilon }}_{ref}$ is the reference strain rate, and $\dot{\epsilon }$ is the strain rate with a different condition.

After introducing the strain rate correction factor, the FEM results under different strain rates are shown in Figure 10. The established FEM model can simulate the experimental results well, and the errors of the numerical results are all below 5%.

To verify the simulation reliability of fracture strains obtained by the MMC equation, bending specimens with the theoretical stress triaxiality of 0.66 and compression specimens with the theoretical stress triaxiality of −0.33 were thereafter our focus. The FEM model with the same conditions as the experiment was also established in the simulation environment. The fracture strains were obtained by the MMC equation and input to the material model for simulation. A comparison of the simulation results with the experimental results is shown in Figure 11. The force–displacement data of the numerical simulation are in good agreement with the experiment results, indicating that the MMC failure model is capable of simulating the failure behavior of die-cast aluminum alloy A356.

6. Conclusions

The fracture behavior of cast aluminum material A356 under monotonic loading was investigated. Uniaxial tensile, R5 notched tensile, R20 notched tensile, D10 center hole tensile and 45° shear experiments were conducted to calibrate the failure strains under different stress states and to establish the MMC failure model. By comparing the experimental and FEM results, the following conclusions were obtained:

(1) A significant strain rate effect of A356 aluminum alloy was observed through dynamic property experiments. In the results of monotonic loading experiments at different strain rates, the flow stress exhibited a significant increase with increasing strain rate. Furthermore, both the proof stress and fracture strain exhibited significant enhancements with higher strain rates.

(2) From the experimental and FEM results of uniaxial tensile and notched specimen tensile strength, it can be concluded that A356 fracture occurs at the maximum strength. Unlike other metallic materials, A356 does not experience significant necking during the failure stage.

(3) By analyzing the experimental and FEM results of uniaxial compression and bending conditions, it can be determined that the ultimate loads and fracture displacements obtained from the numerical results with the MMC equation are in agreement with the experimental results. The study indicates that the MMC failure model is capable of simulating the failure behavior of die-cast aluminum alloy A356.