# Finite Element Analysis of Residual Stress in the Diffusion Zone of Mg/Al Alloys

Authors: Yunlong Ding; Dongying Ju
Journal: Advances in Materials Science and Engineering
 (2018)

---

## Abstract

In this study, the finite element method was applied for analyzing the effect of annealing temperatures on residual stress in the diffusion zone of AZ31 Mg and 6061 Al alloys. The microstructure and mechanical behavior of the diffusion zone were also investigated. Simulations on the annealing of the welded specimens at 200°C, 250°C, and 300°C were conducted. Moreover, experiments such as diffusion bonding and annealing, analysis of residual stress by X-ray diffraction, elemental analysis using an electron probe microanalyzer, and microstructure investigation via scanning electron microscopy were performed for further investigation of the diffusion layers. According to the results of the simulations and experiments, the diffusion layers widen with increasing annealing temperatures, and the results of the simulations are in good agreement with those of the experiments. The microstructure and elemental distribution were the most uniform and the residual stress was the least for samples annealed at 250°C. Thus, 250°C was found to be the most appropriate annealing temperature.


---

## Body

## 1. Introduction

The finite element method (FEM) has many applications in modern industry and technology because of the extensive use of computers [1–6]. This method is presently the most popular and fastest developing numerical method in aircraft, ballistic missile, automotive, shipbuilding, machine, and electrotechnics industries and is used in fields such as biomechanics, medicine, mechatronics, and materials technology. Computational methods mainly help optimize design processes [7–10].In addition, FEM is used in plastic forming and can simulate the press forming of aluminum by selecting appropriate forming parameters for the material, such as pressure force and falling speed of the punch [11].Recently, many investigations on the welding of Mg/Al alloys have been conducted using FEM, especially on the analysis of residual stress during welding, because magnesium and aluminum alloys are widely used in aerospace, automotive, machine, electrical, and chemical industries owing to their superior properties [12–14]. Further, the combination of the superior properties of magnesium and aluminum alloys provides insight into the research of lightweight vehicles.However, most studies have focused on the analysis of residual stress during butt welding, laser beam welding, or friction stir welding [15, 16]. In contrast, this study considers a different welding process, diffusion bonding. To decrease the residual stress during diffusion bonding, annealing experiments, simulations, and investigations of residual stress by X-ray diffraction (XRD) were performed. To the best of our knowledge, this is the first study focusing on diffusion bonding between magnesium and aluminum alloys. Based on the results of this study, more extensive applications of FEM and diffusion bonding can be determined, and the properties of composite materials containing magnesium and aluminum alloys can be investigated. The composite materials can contribute to the realization of lightweight components. In addition, the depletion of resources and energy will decrease, thus mitigating environmental pollution.


## 2. Materials and Methods

In this study, AZ31 magnesium alloy and 6061 aluminum alloy were used for diffusion bonding and annealing. Simulations and experiments were performed to analyze residual stress and evaluate the microstructure.


### 2.1. Theoretical Analysis

During diffusion bonding and annealing, microstructures of the alloys vary with temperature, and at the same time, thermal stress is induced. If the stress exceeds the elastic limit, plastic deformation occurs. A series of varieties do not emerge individually but interact with each other. The theory for analyzing the interaction is called metallo-thermomechanics, which is the foundation of thermal treatment analysis.When the AZ31 magnesium alloy and 6061 aluminum alloy are welded by diffusion bonding, diffusion between Mg and Al should be considered. The diffusion phenomenon can be analyzed by Fick’s law and can be expressed by the following equation:(1)∂C∂t=D∂2C∂x2,where C is the concentration of the element and D is the diffusion coefficient representing the diffusion property of the material and is a function of C. In general, if the influence of the microstructure is ignored, the diffusion equation can be expressed based on element concentration as follows:(2)C•=divD⁡grad⁡C.If the energy of the object is represented ase=g+Tη+trσεe, then the first law of thermodynamics can be written as follows:(3)ρe•−trσε•+div⁡h=0.If the Fourier law (h=kgradT) is used, plastic work and latent heat of transformation are not considered, and terms related to elastic strain and hardening coefficient are ignored, then (3) can be written as follows:(4)ρcT•−k⁡divgrad⁡T=0,where ρ is the density of the material, c is the specific heat, and k is the thermal conductivity. When the coefficient of heat transfer and the temperature of the fluid in contact with the object do not change, the boundary condition is expressed by the following equation:(5)−k⁡grad⁡T⋅n=hTT−Tω,where n is a vector whose direction is outward from the surface of the object, Tω is the temperature of the object’s environment, and hT is the coefficient of heat transfer between the object and the environment. Generally, the coefficient of heat transfer is a function of temperature T and can be obtained from the experimental value of the cooling curve [17].When plastic materials are subjected to loading, they undergo elastic or plastic deformation. Hooke’s law is applicable to three-dimensional stress and strain and can be expressed as(6)εij•=εije•+εijp•+εijT•,where εij• is the total strain rate, εije• is the elastic strain rate, εijp• is the plastic strain rate, and εijT• is the thermal strain rate. Elastic strain rate and thermal strain rate are expressed by the following equation:(7)εije•+εijT•=Eijkleσkl•+βijT•,βij=∂Eijkle∂Tσkl+αij∂αij∂TT−T0,where α is the linear coefficient of expansion and T−T0 is the temperature difference. Eijkle is a coefficient that is represented by the following equation:(8)Eijkle=12Gδikδjl+δilδjk−ν1+νδijδkl,where ν and G are Poisson’s ratio and the shear modulus, respectively. The plastic strain rate is expressed as follows:(9)εijp•=1G∧∂F∂σmnσmn•+∂F∂TT•∂F∂σij,1G∧=−1∂F/∂κ∂F/∂σklσkl,F=Fσ,T,where F is the Mises yield function.First, the definition of the mixture and the mixing rule are explained. Intermetallic compounds, such as Al3Mg2 and Mg17Al12, are formed during diffusion bonding between magnesium and aluminum alloys. It is assumed that many microstructures are present in the mixture. Further, the mixture contains N compositions whose volume fractions are ξI⁡I=1,2,…,N; the sum of the volume fractions of the compositions is 1:(10)∑I=1NξI=1.If the properties of compositionI are represented by χI, then χ denotes all the properties of N compositions. Therefore, the property χ can be expressed as (11), which is called the mixing rule [18]:(11)χ=∑I=1NχIξI,ξI⁡I=1,2,…,N.In the intermetallic compounds of Mg and Al alloys, the ratio of atomic quantity can be expressed as the following equation:(12)M1M2=C1⋅Ar2C2⋅Ar1,where M1 and M2 are the quantity of Mg and Al atoms. C1 and C2 are the atomic mass of Mg and Al. Ar1 and Ar2 are the relative molecular masses.The molar concentration of Mg is expressed as follows:(13)CB1=M1V0⋅NA,where V0 is the volume of the intermetallic compounds and NA is the Avogadro constant. The amount of substance for Mg is represented as follows:(14)n1=G1Ar1⋅g,where G1 is the weight of Mg and g is the gravitational acceleration.So the volume ratio of Mg to Al can be expressed as follows:(15)V1V2=G1⋅C2G2⋅C1,V1 and V2 are the volume of Mg and Al and G2 is the weight of Al.According to the mixing rule and the volume ratio of Mg to Al, the coefficients of simulations can be calculated. Thermal-stress coupling field of ANSYS was applied to simulate stress at different temperatures during the diffusion process and annealing process. The element type was “Coupled Field, Vector Quad 13,” and the material model was “Structural” and “Thermal.” Temperature conditions were 200°C, 250°C, 300°C, and room temperature (20°C). It was assumed that the interface could not move during the diffusion process. Therefore, a fixed boundary condition was applied to the contact surface, and then the simulations were performed. The model is shown in Figure1.Figure 1
Analysis model.The finite element mesh is shown in Figure2. The element size was 1 mm, and the total number of elements was 14,005.Figure 2
Finite element mesh.


### 2.2. Experimental

The AZ31 magnesium alloy sheets and 6061 aluminum alloy sheets were successfully welded using vacuum diffusion bonding. The joining temperature was 440°C. After vacuum diffusion bonding, the samples were annealed at 200, 250, and 300°C. After heat treatment, the samples were cooled to room temperature in an electric furnace.XRD was used to investigate the residual stress distribution of the specimens annealed at different temperatures. Based on the testing principle of residual stress, the X-ray was adjusted. Initially, the specimen was radiated, and the corresponding diffraction angle 2θ was obtained, which was later used to calculate the slope M of 2θ−sin2ψ (ψ was set as 0°, 15°, 30°, and 45°). In addition, the relationship between 2θ and sin2ψ was obtained, and the residual stress σ was calculated according to the following equation:(16)σ=K·Δ2θΔsin2ψ=K⋅M,where K is the stress constant of the XRD analysis and can be expressed as follows:(17)K=−12⋅cotθ0⋅E1+υ⋅π180,where E is the elastic modulus of the material, θ0 is the diffraction angle without stress, and υ is Poisson’s ratio [19].For the 6061 Al alloy, 2θ was set as 140°, the stress constant was −163.32 MPa/°, tube type was Cr, wavelength was kα, and the size of the collimator was φ = 0.5 mm. For the AZ31 Mg alloy, 2θ was set as 155° and stress constant was −79.14 MPa/°; tube type, wavelength, and the size of the collimator were the same as that for the 6061 Al alloy. In addition, the 2θ of Mg17Al12 and Al3Mg2 were set as 150° and 145° and their stress constants were −98.97 MPa/° and −126.22 MPa/°, respectively. Based on these values, the values of residual stress were calculated.Furthermore, the microstructure and elemental distribution of the diffusion zone were investigated by SEM and EPMA.


## 2.1. Theoretical Analysis

During diffusion bonding and annealing, microstructures of the alloys vary with temperature, and at the same time, thermal stress is induced. If the stress exceeds the elastic limit, plastic deformation occurs. A series of varieties do not emerge individually but interact with each other. The theory for analyzing the interaction is called metallo-thermomechanics, which is the foundation of thermal treatment analysis.When the AZ31 magnesium alloy and 6061 aluminum alloy are welded by diffusion bonding, diffusion between Mg and Al should be considered. The diffusion phenomenon can be analyzed by Fick’s law and can be expressed by the following equation:(1)∂C∂t=D∂2C∂x2,where C is the concentration of the element and D is the diffusion coefficient representing the diffusion property of the material and is a function of C. In general, if the influence of the microstructure is ignored, the diffusion equation can be expressed based on element concentration as follows:(2)C•=divD⁡grad⁡C.If the energy of the object is represented ase=g+Tη+trσεe, then the first law of thermodynamics can be written as follows:(3)ρe•−trσε•+div⁡h=0.If the Fourier law (h=kgradT) is used, plastic work and latent heat of transformation are not considered, and terms related to elastic strain and hardening coefficient are ignored, then (3) can be written as follows:(4)ρcT•−k⁡divgrad⁡T=0,where ρ is the density of the material, c is the specific heat, and k is the thermal conductivity. When the coefficient of heat transfer and the temperature of the fluid in contact with the object do not change, the boundary condition is expressed by the following equation:(5)−k⁡grad⁡T⋅n=hTT−Tω,where n is a vector whose direction is outward from the surface of the object, Tω is the temperature of the object’s environment, and hT is the coefficient of heat transfer between the object and the environment. Generally, the coefficient of heat transfer is a function of temperature T and can be obtained from the experimental value of the cooling curve [17].When plastic materials are subjected to loading, they undergo elastic or plastic deformation. Hooke’s law is applicable to three-dimensional stress and strain and can be expressed as(6)εij•=εije•+εijp•+εijT•,where εij• is the total strain rate, εije• is the elastic strain rate, εijp• is the plastic strain rate, and εijT• is the thermal strain rate. Elastic strain rate and thermal strain rate are expressed by the following equation:(7)εije•+εijT•=Eijkleσkl•+βijT•,βij=∂Eijkle∂Tσkl+αij∂αij∂TT−T0,where α is the linear coefficient of expansion and T−T0 is the temperature difference. Eijkle is a coefficient that is represented by the following equation:(8)Eijkle=12Gδikδjl+δilδjk−ν1+νδijδkl,where ν and G are Poisson’s ratio and the shear modulus, respectively. The plastic strain rate is expressed as follows:(9)εijp•=1G∧∂F∂σmnσmn•+∂F∂TT•∂F∂σij,1G∧=−1∂F/∂κ∂F/∂σklσkl,F=Fσ,T,where F is the Mises yield function.First, the definition of the mixture and the mixing rule are explained. Intermetallic compounds, such as Al3Mg2 and Mg17Al12, are formed during diffusion bonding between magnesium and aluminum alloys. It is assumed that many microstructures are present in the mixture. Further, the mixture contains N compositions whose volume fractions are ξI⁡I=1,2,…,N; the sum of the volume fractions of the compositions is 1:(10)∑I=1NξI=1.If the properties of compositionI are represented by χI, then χ denotes all the properties of N compositions. Therefore, the property χ can be expressed as (11), which is called the mixing rule [18]:(11)χ=∑I=1NχIξI,ξI⁡I=1,2,…,N.In the intermetallic compounds of Mg and Al alloys, the ratio of atomic quantity can be expressed as the following equation:(12)M1M2=C1⋅Ar2C2⋅Ar1,where M1 and M2 are the quantity of Mg and Al atoms. C1 and C2 are the atomic mass of Mg and Al. Ar1 and Ar2 are the relative molecular masses.The molar concentration of Mg is expressed as follows:(13)CB1=M1V0⋅NA,where V0 is the volume of the intermetallic compounds and NA is the Avogadro constant. The amount of substance for Mg is represented as follows:(14)n1=G1Ar1⋅g,where G1 is the weight of Mg and g is the gravitational acceleration.So the volume ratio of Mg to Al can be expressed as follows:(15)V1V2=G1⋅C2G2⋅C1,V1 and V2 are the volume of Mg and Al and G2 is the weight of Al.According to the mixing rule and the volume ratio of Mg to Al, the coefficients of simulations can be calculated. Thermal-stress coupling field of ANSYS was applied to simulate stress at different temperatures during the diffusion process and annealing process. The element type was “Coupled Field, Vector Quad 13,” and the material model was “Structural” and “Thermal.” Temperature conditions were 200°C, 250°C, 300°C, and room temperature (20°C). It was assumed that the interface could not move during the diffusion process. Therefore, a fixed boundary condition was applied to the contact surface, and then the simulations were performed. The model is shown in Figure1.Figure 1
Analysis model.The finite element mesh is shown in Figure2. The element size was 1 mm, and the total number of elements was 14,005.Figure 2
Finite element mesh.


## 2.2. Experimental

The AZ31 magnesium alloy sheets and 6061 aluminum alloy sheets were successfully welded using vacuum diffusion bonding. The joining temperature was 440°C. After vacuum diffusion bonding, the samples were annealed at 200, 250, and 300°C. After heat treatment, the samples were cooled to room temperature in an electric furnace.XRD was used to investigate the residual stress distribution of the specimens annealed at different temperatures. Based on the testing principle of residual stress, the X-ray was adjusted. Initially, the specimen was radiated, and the corresponding diffraction angle 2θ was obtained, which was later used to calculate the slope M of 2θ−sin2ψ (ψ was set as 0°, 15°, 30°, and 45°). In addition, the relationship between 2θ and sin2ψ was obtained, and the residual stress σ was calculated according to the following equation:(16)σ=K·Δ2θΔsin2ψ=K⋅M,where K is the stress constant of the XRD analysis and can be expressed as follows:(17)K=−12⋅cotθ0⋅E1+υ⋅π180,where E is the elastic modulus of the material, θ0 is the diffraction angle without stress, and υ is Poisson’s ratio [19].For the 6061 Al alloy, 2θ was set as 140°, the stress constant was −163.32 MPa/°, tube type was Cr, wavelength was kα, and the size of the collimator was φ = 0.5 mm. For the AZ31 Mg alloy, 2θ was set as 155° and stress constant was −79.14 MPa/°; tube type, wavelength, and the size of the collimator were the same as that for the 6061 Al alloy. In addition, the 2θ of Mg17Al12 and Al3Mg2 were set as 150° and 145° and their stress constants were −98.97 MPa/° and −126.22 MPa/°, respectively. Based on these values, the values of residual stress were calculated.Furthermore, the microstructure and elemental distribution of the diffusion zone were investigated by SEM and EPMA.


## 3. Results and Discussion

The Mises stress data were obtained after completion of the simulation. The simulation results of stress distribution are shown in Figures3(a)–3(d).Figure 3
Results of simulations: (a) without annealing and annealing at 200°C (b), 250°C (c), and 300°C (d), previously presented in [20].
(a)
(b)
(c)
(d)Figure3 shows that the stress values change with annealing temperature. Because stress is mainly concentrated at the edge of the interface, the values along the interface are the largest, which in turn causes premature failure [20].In this paper, stress distribution along the line crossing the edge of the interface was investigated (the measuring line is shown in Figure3). Based on the stress values at the nodes and the distance between the nodes, Figure 4 is obtained; this figure shows the distribution of residual stresses calculated from the simulations.Figure 4
Distribution of residual stress obtained by simulation, previously presented in [20].Residual stress is a vector, and in this study, its direction was along the axial direction of the specimens. According to material mechanics, tensile stress is positive, while compressive stress is negative. Therefore, it could be concluded from Figure4 that the value of stress near the interface was positive, that is, tensile stress, and the largest. The stress near the interface of the specimens annealed at temperatures of 200, 250, and 300°C was 66, 61, and 63 MPa, respectively. The untreated specimens exhibited a stress value of 71 MPa. However, as the distance from the interface increased, stress decreased and even became negative, that is, compressive stress. In addition, the residual stress was almost 0 MPa far away from the interface.The residual stress measured by XRD is shown in Figure5.Figure 5
Residual stress of specimens annealed at different temperatures.The stress of the untreated specimens was about 65 MPa, while that of the specimens annealed at 300°C and 200°C was 51 MPa and 59 MPa, respectively. However, when the specimens were annealed at 250°C, the stress was approximately 44 MPa. During the experiments, the diffraction peak for the (311) plane of Al appeared at a 2θ of about 139°, while that for the (104) plane of Mg appeared at a 2θ of 152°. Furthermore, the 2θ value of the intermetallic compounds near Al and Mg was 139.4° and 143°, respectively [21]. Thus, it could be concluded that the distribution of residual stress obtained from the experiments is nearly the same as that obtained from simulations.The above results showed that 250°C is the most appropriate annealing temperature. This could be further confirmed by microstructure investigation. The results of the elemental analysis are shown in Figures6 and 7. In Figure 6, the colors represent the different amounts of elements detected, and the y-axis numbers represent the levels that the different colors stand for.Figure 6
Results of elemental analysis, surface scanning: (a) without annealing and annealing at (b) 200°C, (c) 250°C, and (d) 300°C.
(a)
(b)
(c)
(d)Figure 7
Results of elemental analysis, line scanning: (a) without annealing and annealing at (b) 200°C, (c) 250°C, and (d) 300°C.
(a)
(b)
(c)
(d)As shown in Figures6 and 7, the diffusion layers widened with increasing annealing temperatures. As the temperature increased, the magnesium and aluminum contents varied in regions A, B, and C. Specifically, the untreated specimens exhibited a magnesium content ranging from 800 to 300 counts, while the aluminum content was 50–200 counts. For the specimens annealed at 200°C, the Mg content ranged from 600 to 300 counts in the direction from Mg to Al, while the Al content was 50–100 counts in the same direction. However, for the specimens annealed at 300°C, the Mg content varied from 800 to 650 counts, while the Al content varied within the range of 50–70 counts. The result for the specimens annealed at 250°C is shown in Figure 7(c). For this specimen, the magnesium content ranged from 500 to 300 counts, while the Al content was 50–70 counts. The reason for the variations in the diffusion layer width and element content was the difference in the annealing temperature. As the annealing temperature increased, the diffusion rate increased. Therefore, the diffusion layers were the widest for the specimens annealed at 300°C. However, as shown in Figure 7, the element content for the specimens annealed at 250°C was steady. This indicates that the microstructure is more uniform at 250°C.To verify the above results, the microstructures of the specimens annealed at different temperatures were investigated. The microstructures of the joints are shown in Figure8.Figure 8
SEM micrographs of joints: (a) without annealing and annealing at (b) 200°C, (c) 250°C, and (d) 300°C; layer A: Al3Mg2, layer B: Mg17Al12, and layer C: Mg-based solid solution.
(a)
(b)
(c)
(d)The diffusion layers, including layers A, B, and C, which were investigated and confirmed to be Al3Mg2, Mg17Al12, and Mg-based solid solutions, respectively, could be clearly observed. Moreover, the width of the diffusion layers increased with increasing annealing temperature. However, the microstructure of the specimens annealed at 250°C was more uniform than that of those annealed at other conditions, thus confirming the results of the elemental distribution analysis.As shown in Figure9, for specimens annealed at 250°C, the Zn content was relatively steady and was greater than that for specimens annealed at other temperatures. Although Zn is a kind of rare earth element and its content is low, it greatly affects the diffusion zone of Mg and Al alloys. Zn could retard the formation of intermetallic compounds of Mg and Al. Though intermetallic compounds were formed, they precipitated and were dispersed, thus generating precipitation strength [22]. In other words, the amount of the hard and brittle phase at the interface was small. Thus, the properties of the specimens annealed at 250°C were better than those of the specimens annealed at other temperatures.Figure 9
Elemental distribution of Zn, line scanning: (a) without annealing and annealing at (b) 200°C, (c) 250°C, and (d) 300°C.
(a)
(b)
(c)
(d)


## 4. Conclusions

The following results were obtained from the simulations and experiments:(1)
Annealing temperatures have a great effect on the microstructure and elemental distribution. The most appropriate annealing temperature for the diffusion-bonded Mg/Al alloy is 250°C.(2)
It is difficult to obtain a sufficiently high-quality diffusion zone by diffusion bonding magnesium and aluminum alloy sheets because of the formation of intermetallic compound layers. However, this study used annealing to improve the microstructure, thus achieving such a diffusion zone.(3)
The outcomes obtained by FEM are in good agreement with those of the experiments; thus, the accuracy of FEM for analyzing residual stress during annealing is reliable.



---

*Source: 1209849-2018-03-27.xml*