Out-of-plane distortion prediction of large thin-walled structures induced by friction stir welding

Abstract

A numerical method is developed to predict the out-of-plane distortion for large thin-walled structures induced by friction stir welding. Considering the constraint of non-welded region in a large structure, the non-uniform stress distribution along the weld is accurately computed with a small structure. The welds in the small and large structures are divided into the beginning, middle, and end regions due to the non-uniform stress situation. By using the interpolation function, the stress components in various regions of the small structure are mapped to the corresponding locations in the large thin-walled structure. Then, the out-of-plane distortion of the large structure is computed when the clamps are released. The predicted residual stress and distortion for the large welded structure are in accordance with the experimental data.

Keywords

Welding distortion non-uniform welding stress non-welded region large thin-walled structure friction stir welding

Introduction

The friction stir welding (FSW) is developed to join the aluminium alloy to achieve good mechanical performance and low distortion, especially for large thin-walled structures in aerospace industries. The out-of-plane distortion induced by FSW is inevitable to appear for large thin-walled structures that could lead to worse manufacturing quality and even buckling failure of joined structures. The accurate prediction of welding distortion for large thin-walled structures may provide valuable insights into the manufacturing process in advance and improve their jointed quality.

The strong nonlinear mechanical behaviour due to the thermo-elastic–plastic deformation during the welding process is restricted in the vicinity of the weld. The moving source analysis method has good accuracy to predict all types of welding distortions [1], which is dependent on the element size in the numerical simulation. The structure is at least discretised by three-layer brick elements in the thickness direction to describe the warp deformation. Therefore, it is time-consuming when it is used for large thin-walled structures [2]. The inherent strain method is frequently used to compute welding distortion of large structures [3]. The plastic strain components are obtained and assumed as the constants to be applied on the long weld [4, 5]. Then, the deformation of the large welded structures may be computed directly with the elastic procedure to improve the computational efficiency. Zhang et al. [6] developed a three-dimensional (3D) applied plastic strain method, in which the six plastic strain components of each weld were calculated by using a simplified model with a shorter length of the weld. These strain components were uniformly acted on the large structure to obtain the final distortion. Actually, the stress along the weld is not uniform especially in both ends of the weld, which should be considered in the calculation of welding deformation.

The welding distortion more easily appears in large thin-walled structures because of their weak structural stiffness. In the majority of the welded region, the stress is nearly uniform. So it may be first computed with a small structure during the welding process and then is used to calculate the final distortion of the large structure. Dialami et al. [7] proposed a local–global strategy to compute the residual stresses and distortions in the FSW process. The material stirring and the heat generation induced by the rotation of pin and shoulder were solved in the local level analysis, which were equivalent as the power input for the welding analysis at the global level. Ahmed and Saha [8] studied the effect of the fixtures on the saddle shape distortion of thin aluminium plates by FSW. Michaleris et al. [9] and Moraitis and Labeas [10] found that the stress distribution is apparently discontinuous near the end–start location of the weld, which is different from the uniform stress distribution in the middle of the weld. Lee et al. [11] predicted the welding angular distortion of large structures whose mesh size has no restriction near the welding region. Zhang et al. [6] also stated that the non-welded region had constraint effect on the welded region that could cause the variation of stress and the distortion of large structures. Therefore, the constraint of the non-welded region should be involved to improve the predicting accuracy of welding distortion for large thin-walled structures.

In this paper, a prediction method is proposed to compute the out-of-plane distortion of large thin-walled structures induced by FSW. The non-uniform welding stress before releasing clamps is calculated by using the simplified small structure, in which the influences caused by two ends of the weld and the constraint of the non-welded region are included. Then, by using the coordinate transformation and the interpolation function, the stress components in various regions of small structure are mapped onto the similar large thin-walled plate to determine the final distortion when the clamps are released. The experiment of FSW is conducted to verify the rationality of the proposed method. Meanwhile, the welding distortion affected by the two ends of the weld is discussed.

Prediction model of welding distortion for large thin-walled plates

Figure 1 illustrates the schematic of the proposed method that is summarised as follows. (1) The stress before the release of clamps for the simplified small structure is accurately calculated with the moving source analysis method. Here, the non-welded region of large thin-walled structure is equivalent as the substructure and applied on the boundary of the small structure. (2) The weld and the heat affected region in the small structure are divided into the beginning, middle and end regions in terms of the non-uniform stress distribution. The dimensions of the three regions are analytically computed by the deflection along the weld when it is subjected to the unit pressure. (3) The stress components of elements in small structure are mapped to large thin-walled plate to calculate the out-of-plane distortion. The large structure near the weld is also divided into three regions. The corresponding relation of integration points of the large thin-walled plates and the small structure is calculated based on the coordinate transformation. By using the interpolation function, six stress components at the integration points in the elements of the small structure are updated and mapped to the corresponding integration points in the elements of large thin-walled structure, which are used to compute the welding distortion of large thin-walled structure when the clamps are released.

Equivalent substructure of non-welded region

The non-welded region in the large thin-walled structure has the constraint on the deformation of the welded region and the whole structure. Here, the non-welded region is equivalent to the substructure in the small structure to improve computational efficiency and shown as the gridding area in Figure 1.

The large thin-walled structure may be discretised to finite elements. The stiffness matrix of the structure is denoted as (1) where and denote the nodes in the non-welded and the welded regions as shown in Figure 1. The corresponding displacement and force vectors are written as (2)

A static condensation of the internal nodes may be done to obtain the equivalent force vector and the stiffness matrix of the substructure, which are written as (3)

The non-welded region in the large thin-walled structure is equivalent as the substructure, and the stiffness matrix is outputted by ABAQUS. Then, it is attached on the boundary of the small structure directly with tie constraint condition in the software. The interface of the substructure is fixed and shown as the gridding area in Figure 1. The stress in the small structure is computed by using the moving source analysis method.

Thermal source model of FSW

The mechanical power is directly dependent on the heat input during the FSW process. Here, the power dissipating into the environment is ignored because the fraction of lost power is very small [12]. The heat input consists of the surface heat source on the interface of shoulder and workpiece and the volume heat source induced by the friction force between the tip of pin and workpiece . The heat input is denoted as (4) where is the conversion efficiency of mechanical power, is the angular velocity and is the torque. According to Ref. [12], .

The surface heat source and the volume heat source of the pin may be given as (5) (6) where and are the radii of shoulder and pin, is the radial position corresponding to the axis of the tool, is the pin height. And is the flow stress of material, which is dependent on the initial temperature of structure , and the effective strain rate . In Ref. [13], it may be written as (7) where and are material parameters, is the Zener–Hollomon parameter and is calculated as (8) where is the activation energy and is the gas constant. In the Cartesian coordinate system, may be written as (9)

For the friction stirring welding, the cylindrical coordinate system is convenient for the theoretical solution. Meanwhile, the radial velocity and the axial velocity of the pin are very small and can be neglected. The effective strain rate may be rewritten as (10) where is the circumferential velocity of a pin. The flow stress has to be theoretically solved by the iteration method. For the engineering application, Colegrove et al. [14] provided the curves between the flow stress and two parameters and for the aluminium alloy material. The flow stress may be easily obtained when and are known. Finally, the heat source may be calculated by user subroutine DFLUX in ABAQUS.

Length determination for two ends of weld

The stress in the beginning and end regions of weld in Figure 1 is different with that in the middle region due to the effect of nonlinear geometrical constraints. The dimensions of the two regions are important parameters for the distortion prediction of large thin-walled plates. Their widths could be determined by the distance of clamps around the weld. Their lengths are related to the structural stiffness and are derived as follows.

A rectangular plate subjected to the pressure load along the weld is employed to calculate the structural stiffness and is shown in Figure 2. The deflection of the plate can be computed with the single trigonometric series [15] as (11) where is a function with respect to y only. This equation satisfies the boundary conditions and . Substituting Equation (11) into the force equilibrium equation of plate, an ordinary differential equation can be obtained as (12) where is the flexural rigidity of the plate that is calculated by , is the elastic modulus, is Poisson's ratio, and is the thickness of the plate. Both sides of Equation (12) are multiplied by and integrated with respect to from 0 to . When , , otherwise, it is equal to 0. Thus, it may be rewritten as (13)

The general solution of Equation (13) may be written as (14) where is the particular solution of Equation (13).

Equation (13) can be simplified as (15)

Then, may be given as (16)

Substituting Equations (14) and (16) into Equation (11), the deflection function can be computed as (17)

The deflection of the plate is symmetric because the pressure is a constant, and the boundary condition is symmetric. Thus, is an even function that is symmetric with respect to the y-axis. It follows that . Then, w becomes (18)

The boundary conditions of the free sides are (19)

Substituting Equation (18) into Equation (19), they may be obtained as (20)

The coefficients and are solved as (21)

Substituting Equation (21) into Equation (18), the deflection function is written as (22) where is the deflection coefficient. The structural stiffness can be described by . The variation of the deflection along the weld is the primal interest to determine the length for the beginning and end regions in Figure 1.

A large thin-walled structure with dimensions of 1000×300×6.8 mm is employed to study the correlation of the structural stiffness and the lengths of the two ends of the weld. Since and are constants, the deflection coefficient in Equation (22) may represent the variation of structural stiffness along the weld and is shown in Figure 3. Here, is computed with the increase of . When the relative error of different is less than 1% for two contiguous values of , the convergence of is determined. According to the calculation process, the convergence of is achieved when . The critical points for the middle, beginning and end regions are determined when the slope of is beyond 0.2%. Thus, the demarcation points of the beginning and end regions are at y = −367 mm and y = 367 mm. Here, the length of the two ends for this structure is defined as 150 mm.

The calculation of distortion for large thin-walled structure

The mapping process of stress components from the small structure to the large thin-walled plates may be summarised into the following steps. First, the location correlation of the element-integration points in the small structure and large structure is established with the transformation matrix of coordinate systems. Then, six stress components at the integration point in the large thin-walled structure are calculated by using the interpolation function. Finally, the weld distortion of the large structure is computed with the stress components when the clamps are released.

Coordinate transformation

The stress induced by welding process in small structure can be accurately computed one time by using the moving source method and be mapped to similar large thin-walled structures with different sizes, which may improve the computing efficiency. The global and local coordinate systems are located on the large and the small structures, respectively. The rotation matrix of the two coordinate systems is denoted as (23) where the subscripts and denote the local and the global coordinate systems, respectively. The vectors , and are illustrated the projections of unit vectors from the global coordinate axes to the local coordinate axes. The rotation matrix is orthogonal and satisfies .

In the global coordinate system, the distance of two origins can be written as , where and denote the origins of the local and global coordinate systems, respectively. The coordinates of integration points in the element of the large structure can be transformed to the local coordinate system as (24) where is the coordinate vector of integration points in the global coordinate system.

Stress components mapping of large structure

The welded region in large thin-walled plates is also divided into three parts as shown in Figure 1. The initial non-uniform stress is applied at the integration points of every element in the large structure and the welded distortion is calculated when the clamps are released. Obviously, the coordinates of integration points in the large and small structures are not consistent due to the difference of the elements’ number and size even when they are described in the same coordinate system. The linear interpolation method is used to obtain the stress mapping relation from the small structure to the large structure. For every integration point in the elements of large structure when their coordinates are described in the local coordinate system, the adjacent integration points in the small structure are searched and the stress is computed with the linear interpolation function. To avoid the hourglass problem and reduce the sensitivity of shear locking, the 3D reduced-integration brick element is chosen to discretise the structures. One element contains eight integration points.

To solve the coordinates of the integration points in the isoparametric element, the shape function of the brick element may be written as (25) where , , and are the nodal coordinates of the isoparametric element, and , , and are the corresponding coordinates of the integration point that can be solved as (26) where is consisted of the coordinates of eight integration points in the element of the large structure, and can be written as (27) When , , and are obtained, the mapping correlation of the stress components at the integration point in the element of the small and large structures may be calculated as (28) here is the stress at the integration point of the element in the large structure in the local coordinate system, and is the stress at the integration points in the corresponding element in the small structure. And is given as (29) To predict the final distortion of the large structure, the stress components should be described in the global coordinate system. The stress components at the integration points of the element in the large structure in the local coordinate system may be written as (30)

The stress can be transformed to the global coordinate system as (31) where is the stress at the integration points in the element of the large structure denoted in the global coordinate system. Then, these stress components are applied on the large thin-walled plate to predict the welding distortion. In the beginning and end regions, the number of elements in the large structure is the same as that in the small structure. Therefore, the one-to-one mapping process for elements in different structure is done for the two regions between the small and large structures. The stress components of the elements in the middle region of the weld in the small structure are uniformly applied on each element in the middle region of the large structure. The calculation of the stress mapping procedure between the large and small structures is conducted with software MATLAB. Finally, the welding distortion of the large structure may be computed in terms of the mapped stress field when clamps are released.

Model verification

Finite element model

Figure 4 shows the finite element models of the small and large structures. The small structure is discretised with element type C3D20RT and the element size is 5 mm. Three-layer elements are divided in the thickness direction. Two clamped regions during the welding process and dimensions are shown in Figure 4(a). The degree-of-freedoms (DOFs) of the elements on the bottom surface are constrained to describe the support of a backing plate in the small structure. For the large structure, the DOFs of the end surface are fixed and shown in Figure 4(b). First, the stiffness matrix for the elastic region in Figure 4(b) is calculated by software ABAQUS and is applied in the small structure as the substructure.

The material of the specimen is aluminium alloy AA2024-T3. Figure 5 illustrates the material properties at various temperatures [16, 17]. The heat transferring behaviour of the specimen and the environment, the specimen and the backing plate are involved in the simulation.

FSW experiments

The sizes of the specimen are 1000 × 300 × 6.8 mm. The experiment of FSW and the fixture locations of the workpiece are illustrated in Figure 6. The shoulder radius is 10 mm and the pin radius is 3 mm. The advancing speed is 300 mm min⁻¹, the rotation speed is 300 rev min⁻¹, and the depression depth is 0.1 mm. The force control method for the tool is used during the welding process and the constant axial force is 2019 N. Thus, the moving pressure on the shoulder/workpiece interface is 7.1 MPa. The initial temperature of the workpiece is 20°C.

The temperature of the specimen is measured with a thermal infrared imager. Its arrangement and the measurement points are located in the vicinity of the weld shown in Figure 7. The testing data of the temperature cycle are recorded when the tool advances in the workpiece from the beginning to the end.

The measurement of welding stress before releasing clamps is impossible due to the constraints of the clamps. However, the residual stress on the surface of specimen is convenient to be measured when the clamps are released. The verification of the welding stress before releasing the clamps is indirectly conducted with the comparison of the residual stress obtained from the numerical results and testing data. The locations of the measurement points are shown in Figure 8 and their interval is 10 mm. By using the X-ray diffraction (XRD) equipment, the residual stress in different regions may be measured in the transverse direction. Here, the precision of the XRD equipment is Mpa, which could ensure the accuracy for the measurement of residual stress in the structure.

The out-of-plane distortion of the plate is measured by the laser tracker device before and after the welding process. The measuring precision of the laser tracker is 0.005 mm m⁻¹. The plates are first measured in view of the influence of initial deflection. The maximal difference of measurement points on the plates ranges from 0 to 0.011 mm, which is close to the measuring precision of the laser tracker. Thus, the original deflection of plates may be ignored. The welded specimen and the laser tracker are shown in Figure 9. The surface of the platform is defined as X–Y plane. The X-axis is parallel with the weld. The thickness of the plate is subtracted from the testing data.

Discussions

Figure 10 shows the comparisons of temperature cycles at two locations. The measurement points P1 and P2 are 15 and 25 mm away from the weld in the transverse direction. At the point P1, the welding temperature obviously increases when the heat source approaches to the measurement point. Then, the temperature decreases quickly because the heat source moves away from the testing point. The maximum temperatures obtained from the numerical results and testing data are 200 and 187°C when t = 16 s. At the point P2, the maximum temperatures obtained from the numerical results and the testing data are 118 and 102°C when t = 16 s. The temperatures obtained from the numerical results are consistent with those from the testing data.

Figure 11 presents the comparison of the residual stress obtained from numerical results and testing data via XRD in different regions. The tensile residual stress appears near the location of weld. With the increase of the distance from the weld, the residual stress is converted to the compressive state. The maximal tensile stress in the middle region is larger than that in the beginning and the end regions. However, the compressive stress is smaller in the middle region. It can be inferred that the welding stress before releasing clamps is non-uniform along the weld that directly affects the out-of-plane distortion of the large thin-walled structure.

The distortion of large thin-walled plates in the longitudinal direction induced by FSW is shown in Figure 12(a), which has the convex bending, rather than the concave bending caused by fusion welding process [18]. The welding distortion in the transverse direction has concave bending, which is also in contrary with the distortion caused by the fusion welding process. The out-of-plane distortion of the plate in the Z direction is approximated to 5.2 mm. The similar distortion tendency appears in the experiment and is shown in Figure 12(b). The differences of the out-of-plane distortion of the testing data and the numerical results may be caused by ignoring the dynamic recrystallisation and recovery in the numerical analysis, which would consume a portion of strain energy.

To quantitatively compare the welding distortion, the beginning and the end positions of the weld are chosen to be the datum points. The distortion of the structure is defined as the perpendicular distance from the measurement point to the straight line linked by two datum points. The out-of-plane distortion obviously appears on the lateral edges A and B as shown in Figure 12(a). The magnitudes of distortion at the two edges of large thin-walled structure with non-uniform stress distribution in the two end regions are presented in Figure 13. The maximal error of the out-of-plane distortion is 0.43 mm on the side edge B. The maximal distortion on this edge in the testing data is 4.63 mm. Thus, the relative error of welding distortion is about 9.3%. The welding distortion obtained from the numerical simulation has good agreement with the testing data.

For the traditional methods, the identical stress components are applied on the elements in the weld to compute the out-of-plane distortion of large structures. Here, the stress components of the elements in the middle region of the small structure are applied on all the elements in the weld of the large structure. As shown in Figure 13, the out-of-plane distortion of the welded structures with the uniform stress distribution is apparently lower than that of the testing data. The stress gradient between the two end regions and the middle region results in this discrepancy of welding distortion. The stress components in the end regions are actually smaller than those in the middle region due to the difference of the structural stiffness along the welding direction.

Conclusions

An accurate method is proposed for predicting the welding distortion of large thin-walled structure indu-ced by FSW. The non-uniform welding stresses before releasing clamps are obtained by using a small structure. The constraint of the non-welded region in the large structure is equivalent as the substructure and applied on the small structure. The lengths of the end regions with non-uniform stress distribution along the weld are analytically determined according to the structural stiffness. Then the stress components are mapped on the large thin-walled structure to compute the welded distortion. The maximum error of the distortion is 9.3% based on the comparison of the experiment and analytical model. Consequently, the proposed method has good accuracy for the out-of-plane distortion prediction of large thin-walled structures by FSW. The non-uniform stress distribution in the weld and the constraint of the non-welded region must be considered to numerically predict the welded distortion for large thin-walled structures.

Footnotes

Disclosure statement

No potential conflict of interest was reported by the authors.

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