Process analyses of friction drilling using the Smoothed Particle Galerkin method

In the friction drilling process, also named thermal drilling, flow drilling, thermo-mechanical drilling, friction stir drilling and form drilling [1], heat is generated by friction between the drill made of a hard material like tungsten carbide and the metallic workpiece. The induced heat softens the workpiece material and the drill passes through the workpiece under the combination of axial force and high speed rotation, causing the material to be squeezed and to flow up as well as down forming the upper and lower bushing. Figure 1 depicts the friction drilling process, which is not a separating process, but rather a process of joining by forming. The total height of the formed bushing is usually two to three times the thickness of the original workpiece, with the lower bushing height generally accounting for two-thirds of the total height [1]. Additionally, a thread can be formed in the produced bushing to create a detachable connection of thin-walled workpieces. By using the friction-drilled bushing, a longer thread length can be applied, which makes the connection more reliable in comparison to traditional joining methods by bolt and nuts.

Due to the increasing demand for lightweight structures, friction drilling has been more and more widely used in thin-walled sheet metals and tubular workpieces in recent years. In order to predict the process parameters of friction drilling, methods such as artificial neural network (ANN) [2], genetic algorithm (GA) [3] and fuzzy logic algorithms [4] have been proposed to predict the length and roughness of bushings with good results. Moreover, image-based modelling is a necessary tool for understanding material flow, temperatures, stresses and strains during friction drilling, which are difficult to measure by experiment [5]. The modelling of friction drilling can be used to reduce the number of experiments, because the optimal process parameters can be obtained based on accurate process predictions [6]. Until now, the majority of numerical method studies of friction drilling were based on the finite element method (FEM) [7]. However, the traditional mesh-based methods suffer from mesh distortion when dealing with problems such as friction drilling with large deformations [8]. This often leads to numerical difficulties, such as non-convergence of implicit simulations or instability of explicit simulations. However, mesh-based methods inevitably delete elements when dealing with extreme deformations and material failures, which can render it difficult to maintain a high-quality mesh or make the adaptive mesh unstable [9]. In addition, the element erosion technique leads to a loss of mass as well as energy and will affect the stress responses in friction drilling. This can also seriously affect the accuracy of the computational results and even lead to erroneous failure predictions. Therefore, the material flow and the formation of high-quality upper and lower bushings are difficult to achieve when using FEM to simulate friction drilling.

To overcome the inherent shortcomings of mesh-based methods, more than 20 meshfree methods and correction schemes have been proposed in the past few decades. The effectiveness of the meshfree methods has been demonstrated for large deformations [10], impact problems [11] and evolving cracks [12]. In contrast to the mesh-based finite element method, the meshfree methods use a set of particles to discretize the solving region and to construct an approximation function, which does not require the division of the mesh or remeshing. However, meshfree methods also have their own drawbacks. Compared with FEM, meshfree methods cause higher computational costs and the application of essential boundary conditions is more difficult.

The first meshfree method is generally considered to be the Smoothed Particle Hydrodynamics (SPH) method, developed by Gingold and Monaghan [13] and Lucy [14] in 1977. It was primarily used in the field of astrophysics, later in fluid mechanics [15] and solid mechanics [16]. However, the SPH method suffers from three instability problems [17] that seriously affect the numerical convergence, namely tensile instability, rank deficiency and material instability. Some schemes and new methods have been proposed to improve these problems, among which Belytschko et al. [18] and Liu et al. [19] developed the Element Free Galerkin (EFG) method and the reproducing kernel particle method (RKPM). However, the EFG method needs a background integral mesh. When the material is damaged, the background integral mesh needs to be re-established, so it is not recommended for failure analysis. The stabilized conforming nodal integration (SCNI) method developed by Chen et al. [20] in 2001 employs a strain smoothing stabilization for nodal integration and does not involve numerical control parameters, significantly improving accuracy and convergence rate for the meshfree method using direct nodal integration (DNI). However, the SCNI method still cannot be separated from the background mesh. A new meshfree method, the Smoothed Particle Galerkin (SPG) method, proposed by Wu et al. [21] in 2015 provides a better solution to the separation problem.

SPG is a meshfree Galerkin method using the DNI technique. However, the use of DNI is prone to the hourglass problem (zero energy). So, in order to obtain a converged numerical solution, an additional non-residual penalty–type enhancement stabilization term based on the so-called displacement smoothing theory is added. The semi-discrete equations of motion for solving explicit dynamics problems can generally be written as where \({\varvec{M}}\) is the mass matrix, \(\ddot{{\varvec{u}}}\) is the acceleration, \({{\varvec{f}}}^{ext}\) is the external force vector and \({{\varvec{f}}}^{int}\) is the internal force vector. The internal force term needs to be calculated for each particle at each time step, which is very time-consuming. Therefore, the DNI technique with displacement smoothing scheme is adopted to improve efficiency. Thus, the equation of motion to be solved can be written as where \({\widehat{{\varvec{f}}}}^{stab}\) is the stabilization term derived from the smooth displacement theory. It can be calculated by DNI where \({J}^{0}\) is the determinant of the Jacobian matrix, \({V}_{N}^{0}\) is the volume of particle \(N\), \({\widehat{{\varvec{B}}}}_{I}^{T}\) is the stabilization gradient matrix associated with the displacement smoothing function and \(\widehat{{\varvec{\sigma}}}\) is the stabilization stress. The complete derivation process and the calculation of gradient matrix as well as stabilization stress are documented by Wu et al. in [21] and [22].

To simulate the material separation as well as failure more physically and to avoid potential spurious damage growth issues with meshfree approximations used in material failure analysis, a bond-based failure mechanism is also proposed under the framework of SPG [23]. This means that each SPG particle has an influence domain of a certain size and the connection between each particle within the domain is defined as a bond. When a part reaches a user-defined failure criterion during plastic deformation, such as a limit for the effective plastic strain, the bond breaks and the material fails. This has no effect on other particles and bonds in the domain, except that there is no longer any interaction between the two particles originally connected. Therefore, the subsequent stress and strain can still be given by the material constitutive laws. This avoids loss of material, momentum and energy caused by element deletion and stress nulling in contrast to FEM.

Due to the advantages of particle methods for the simulation of large deformation in materials, more and more forming simulations have adopted particle methods in the past 20 years. Most of these studies employed relatively mature meshless methods, such as SPH, RKPM and EFG. Among them, Prakash and Cleary used the SPH method to simulate the extrusion process of cylindrical aluminium alloy billet with a diameter of 40 mm in 2006 [24]. A total of 7760 SPH particles were used in the 3D model with a spacing of 1.6 mm. The simulation results demonstrated the high plastic strain of the extrusion process well and the effects of die geometry such as the extrusion ratio and die angle on the strain as well as maximum extrusion force are predicted. Prakash et al. also performed 2D simulations of different types of forging processes for aluminium alloy A6061 using the SPH method. The material deformation and flow during forging were studied, also forging defects such as flashing as well as incomplete die filling are effectively simulated and predicted. Furthermore Reddy et al. [25] used the SPH method to simulate multi-die forging of industrial components and verified the accuracy of the SPH by comparing the results with the FEM in a uniaxial compression simulation. Bonet and Kulasegaram [26] adopted the corrected smooth particle hydrodynamics (CSPH) with an integral correction and a least-squares stabilization method to further improve the consistency and stability. The effectiveness of the CSPH method in metal forming such as axisymmetric forging and plane strain upsetting was demonstrated.

In 1998, Chen et al. [27] first applied the RKPM to large deformation and metal forming analysis. A Lagrangian RKPM with material loading path-dependent behaviour and frictional contact conditions based on a penalty method was proposed to ensure stability in large deformation analysis. This method showed to be in high agreement with the membrane analytical solution in the prediction of metal punch stretching. Its effectiveness has also been demonstrated in ring compression and upsetting analysis. Shangwu et al. [28] developed a flow formulation of rigid-plastic materials based on the RKPM approach for slightly compressible material models. The simulation results for flat rolling, compression of rods and heading of cylindrical billets were in good agreement with FEM and experimental results. Liu et al. [29] proposed a low order integration scheme to improve the computational efficiency of RKPM, which is suitable for bulk metal forming simulations such as forging and backward extrusion. Furthermore, RKPM is also used to simulate splitting rolling [30], wheel forging analysis [31] and deep drawing [32].

In 2001 Li and Belytschko [33] first demonstrated the effectiveness of the EFG method in metal forming simulations such as upsetting, rolling and extrusion processes. Xiong et al. further showed how effective the combination of EFG and slightly compressible rigid–plastic material models [34] and the combination of EFG and the boundary element method (BEM) [35] in the prediction of plane strain rolling results are. Guan et al. [36] proposed an EFG method based on the hypothesis of a visco-plastic material, which is suitable for the metal forming analysis of arbitrarily shaped dies. Yonghui et al. [37], Lu et al. [38] and Liu et al. [39] verified the agreement between the EFG method and FEM for the numerical analysis of bulk metal forming. Yuan et al. [40] and Wang et al. [41] applied the EFG method to simulate a sheet metal flexible-die forming process and the results were in agreement with FEM as well as experimental results.

Furthermore, there are some other types of meshfree methods used for forming simulations. Alfaro et al. [42] verified the applicability of the natural element method (NEM) based on \(\alpha\)-shapes node cloud scheme, which makes the treatment of essential boundary conditions more reliable, for 3D thermal simulations of extrusion forming of aluminium alloys. Filice et al. [43] and Lu et al. [44] also demonstrated the effectiveness of NEM for numerical computations of metal forming. Yoon et al. [45] proposed an accelerated meshfree method based on a stabilized conforming nodal integration method and the introduction of strain smoothing stabilization, which significantly improves the computational accuracy of metal forming. Kwon et al. [46] proposed a new meshfree method based on an elastic–plastic first-order least-squares formulation, which represents the residuals in the form of a first-order differential system using displacement and stress components as nodal unknowns. Its effectiveness was verified in numerical simulations of metal forming such as ring compression and axisymmetric forging. Hah and Youn [47] proposed an Euler meshfree method using non-uniform rational B-spline (NURBS) curves as a boundary representation. This method eliminated the need to solve convection transport equations in the traditional Euler framework and the obtained results of the rigid-plastic analysis in the bulk metal forming process were consistent with the experimental and other numerical methods.

In practical applications, the SPG method has proven its effectiveness and accuracy in large deformation simulations such as high-speed metal grinding of aluminium alloys [48], self-piercing rivet (SPR) connections of aluminium alloys [49], penetration and perforation analysis of metal targets [50] and friction drilling [51]. On this basis, Wu et al. further developed a momentum-consistent smoothed particle Galerkin (MC-SPG) method for simulating friction drilling [52] and the flow drill screw-driving process [53]. The main improvement of this method is the introduction of a new velocity smoothing algorithm that consistently satisfies the conservation of momentum to stabilise the formulation. Therefore, the method does not require any additional stabilisation term making it more efficient. A comparison of simulation results and experimental data shows that this method predicts the response of force and torque very accurately.

However, almost all simulations of friction drilling using numerical methods to date have not considered the formation of the bushing on the specimen. FEM models were used to study process parameters like axial force and torque, while the SPG method was used to study the sensitivities of the computational parameters in friction drilling processes. Therefore, in this paper, the SPG method is investigated regarding its applicability to friction drilling of HX220. The particle distance as well as the friction coefficient were varied and validated with experiments regarding axial force, temperature evolution in the specimen as well as material flow of the bushing. Based on the SPG model the input parameters sheet thickness, feed rate and rotation speed are numerically investigated.

The SPG method was used to model a friction drilling process with particle methods. For this purpose, a 3D simulation model was created in the pre-processor LS-PrePost and calculated with the LS-DYNA solver. To create the numerical models, the effective parts of the tools and the specimen were abstracted, processed and discretized. The set-up corresponds to that of the experimental investigations, consisting of the friction drill and the specimen, which are shown in Fig. 2. The specimen corresponds to a cylinder with a diameter of 10 mm and a thickness of 1.2 mm. The distance between the particles was set to 0.4 mm and at the deformation area in the centre of the specimen with a diameter of 5 mm a denser particle distance of 0.2 mm was used resulting in a total of 7350 particles. An M4 friction drill with a diameter of 3.7 mm was modelled. The contour of the friction drill with the forming lugs was simplified and assumed to be rotationally symmetric. For discretization of the friction drill, shell elements with an element edge length of 0.2 mm with a refinement of 0.05 mm at the top of the drill resulting in 3331 elements were used.

The boundary conditions were chosen based on friction drilling tests. Due to the very complex physical phenomena of the friction drilling process, several assumptions were made in the numerical model. For example, friction during friction drilling is a complex and changing condition. The friction parameters are difficult to determine experimentally and therefore the friction between the specimen and the friction drill was simplified using Coulomb's law of friction with a friction coefficient of 0.3. The contact between the specimen and the friction drill was modelled with automatic nodes to surface contact using the penalty method. The outer edge of the specimen was fixed in all rotational and translational directions, whereas the friction drill was only movable in the y-direction and rotatable around the y-axis. A rotational speed of 5800 rpm and a feed rate of 50 mm/min were used for the friction drill.

The specimen material HX220 was modelled elastic–plastic using the Johnson–Cook hardening model [54] based on literature data from Behrens et al. [55]. In the J-C hardening model, flow stress \({k}_{\mathrm{f}}\) is analytically determined by plastic strain \({\varepsilon }_{\mathrm{pl}}\), plastic strain rate \({\dot{\varepsilon }}_{\mathrm{pl}}\), and temperature \(T\) where \({\dot{\varepsilon }}_{0}\) is the reference plastic strain rate, \({T}_{\mathrm{room}}\) is the room temperature, \({T}_{\mathrm{melt}}\) is the melting temperature of the material, and \(A\), \(B\), \(C\), \(m\), \(n\) are material parameters. The parameters of the Johnson–Cook hardening model used are summarized in Table 1. Due to the material parameter \(C\) being equal to zero, the material is modelled strain rate independent. The Young’s modulus of the specimen was set to 210 GPa, the Poisson's ratio to 0.3 and the density to 7850 kg/m³. To predict the failure of the specimen, the bond failure mechanism based on the effective plastic strain at failure 0.4 was used, which was obtained by an FE simulation of the tension test in [55]. Additionally, the thermal properties of the specimen material HX220 used are heat capacity 500 J/kgK, thermal conductivity 50 W/mK and thermal expansion coefficient 15 × 10^–6 1/K. Furthermore, as a thermal boundary condition to model the heat flux in the experiment convection was applied to the specimen using a heat transfer coefficient to air of 5 W/m²K assuming free flow. The heat conduction of the material is assumed to be isotropic and all heat during friction drilling is generated by plastic deformation as well as friction between the drill and the specimen, ignoring thermal radiation at the contact surfaces due to its low influence. The friction drill was modelled as a heat conductive rigid body made of tungsten carbide with a heat capacity of 280 J/kgK and a thermal conductivity of 85 W/mK.

Since the calculation of the models requires a small time step to ensure the stability of the calculation, which will lead to a very long calculation time, a mass scaling factor of 100 and a time scaling factor of 100 were adopted to improve the computational efficiency. For the calculation, a cubic spline was used as the kernel function with an Eulerian kernel approximation that is suitable for large deformation and failure analysis of ductile materials. The normalized support size of particles in three directions of space DX, DY and DZ was set to 1.9 to ensure the stability of the calculation. The parameter SMSTEP, which gives the frequency of the numerical displacement smoothing by the number of time steps, was set to 50 to reduce the calculation. An iterative solver and the explicit calculation method with hourglass control were used for the coupled thermo-mechanical analysis. The LS-DYNA solver R12.0 with double precision was applied and the calculation took about 22 h utilising the Intel(R) Xeon(R) CPU E5-2680 v4 with 2.4 GHz and 16 cores.

Since a slight change in the process conditions of friction drilling can greatly affect the complex process behaviour [56], the SPG method was studied in this paper and qualitative comparisons were made. By varying different parameters, the applicability of the SPG method to friction drilling and the effect of the friction drilling input parameters to the quality of the results could be investigated. In Table 2, the parameter variations are depicted.

First, parameters for the applicability of the SPG method were studied. The particle distance was varied (study 1). Next, the SPG method was analysed for different friction coefficients (study 2). Both applicability parameters were studied for their effect on friction drilling regarding the axial force as well the torque of the tool and the surface temperature as well as material flow of the specimen. Further, a comparison to experimental results for validation purposes was performed (study 3). The input parameters studied in this paper are the specimen thickness (study 4), the tool feed rate (study 5) and the rotational speed of the spindle (study 6). The effects of the input parameters were compared regarding axial force, torque, temperature and the resulting bushing. All other settings in the simulations of the parameter studies were kept constant, only the corresponding parameter was changed.

The model and the main input parameters used in the simulations must be consistent with the experimental conditions for comparisons to be meaningful. Therefore, experimental tests were utilized to validate the particle simulations. The test setup was installed in a 5-axis DMU 100 Monoblock milling machine from DMG MORI, consisting of a friction drill, a specimen, a clamping and force measuring plates. An M4 friction drill and a separating agent were used for the tests. The tested rotational speed and feed rate were 5800 rpm and 50 mm/min. For testing, specimens made of the HX220 steel with a thickness of 1.2 mm were used. Since direct temperature measurement on the bushing during the test is problematic due to the setup of the experiment and the rapid temperature changes, thermocouples were welded onto the sheet located at about 4 mm from the centre to measure the surface temperature on the specimen. The axial drilling force during friction drilling was measured with a force measuring plate. Further, the contour of the bushing was optically captured by microscope images. Therefore, experiments were performed and stopped after different forming depths. Using a VR-3200 3D profilometer from Keyence, optical images as well as 3D profiles of the upper and lower bushing were taken after complete forming. The input parameters and the experimental process of friction drilling are described in detail by Behrens et al. in [55]. Important results of friction drilling, namely axial force, temperature and bushing form will be compared with the simulation in order to validate the results.