Advances in Materials Science and Engineering

Volume 2018, Article ID 1209849, 8 pages

https://doi.org/10.1155/2018/1209849

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

Correspondence should be addressed to Dongying Ju; pj.ca.tis@ujyd

Received 11 September 2017; Revised 9 November 2017; Accepted 25 December 2017; Published 27 March 2018

Academic Editor: Yuanshi Li

Copyright © 2018 Yunlong Ding and Dongying Ju. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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:where is the concentration of the element and is the diffusion coefficient representing the diffusion property of the material and is a function of . In general, if the influence of the microstructure is ignored, the diffusion equation can be expressed based on element concentration as follows:

If the energy of the object is represented as , then the first law of thermodynamics can be written as follows:

If the Fourier law () 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:where *ρ* is the density of the material, is the specific heat, and 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:where is a vector whose direction is outward from the surface of the object, is the temperature of the object’s environment, and is the coefficient of heat transfer between the object and the environment. Generally, the coefficient of heat transfer is a function of temperature 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 aswhere is the total strain rate, is the elastic strain rate, is the plastic strain rate, and is the thermal strain rate. Elastic strain rate and thermal strain rate are expressed by the following equation:where is the linear coefficient of expansion and is the temperature difference. is a coefficient that is represented by the following equation:where and are Poisson’s ratio and the shear modulus, respectively. The plastic strain rate is expressed as follows:where is the Mises yield function.

First, the definition of the mixture and the mixing rule are explained. Intermetallic compounds, such as Al_{3}Mg_{2} and Mg_{17}Al_{12}, 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 ; the sum of the volume fractions of the compositions is 1:

If the properties of composition are represented by , then denotes all the properties of compositions. Therefore, the property can be expressed as (11), which is called the mixing rule [18]:

In the intermetallic compounds of Mg and Al alloys, the ratio of atomic quantity can be expressed as the following equation:where and are the quantity of Mg and Al atoms. and are the atomic mass of Mg and Al. and are the relative molecular masses.

The molar concentration of Mg is expressed as follows:where is the volume of the intermetallic compounds and is the Avogadro constant. The amount of substance for Mg is represented as follows:where *G*_{1} is the weight of Mg and is the gravitational acceleration.

So the volume ratio of Mg to Al can be expressed as follows: and are the volume of Mg and Al and *G*_{2} 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 Figure 1.