Modelling of serrated chip formation processes using the stabilized optimal transportation meshfree method

Highlights

• Both the chip separation and the saw-toothed chip are successfully captured by the stabilized OTM method.

• The adiabatic shear band formation is driven by the thermal softening effect but under-predicted by the Johnson–Cook model.

• The over failure within adiabatic shear band is circumvented by restricting the positive stress triaxiality in fracture model.

• The simulated chip formation process has a good agreement with the experimental results.

Abstract

Numerical modelling of chip formation is important for a better understanding thus for an improvement of the high speed metal cutting process. The challenge in the modelling of chip formation lies in capturing the shear band formation, the material separation and the tool-chip interaction accurately. Mostly, some assumptions are made when modelling the material separation and the serrated morphology generation, which leads to an unrealistic prediction of the chip formation. In this work, both the serrated morphology on the chip upper surface and the material separation at the chip root are treated using a ductile fracture model. Additionally, a recently developed Galerkin type meshfree scheme, the stabilizedoptimal transportation meshfree (OTM) method is applied as a numerical solution method in combination with a material point erosion approach. This enables the modelling of material separation and serrated morphology generation of the cutting process in a more realistical and convenient way. The frictional contact force exerted by the cutting tool is imposed on the workpiece using a predictor-corrector strategy. The shear band formation is described by the thermal softening term in the Johnson–Cook plastic flow stress model. Using this model, it can be demonstrated that thermal softening is the main cause for the shear band formation. However, this phenomenon is under-predicted by the Johnson–Cook flow stress model. Additionally, it can be seen that the Johnson–Cook fracture model shows limitations in capturing the fracture on the chip upper surface. Thus, a supplementary condition for the stress triaxiality is applied. This condition allows a more accurate measurement of the chip size, like chip spacing, peak and valley. These improvements are demonstrated by comparing the calculated chip morphology, cutting force and chip formation process with experimental results.

Introduction

Metal machining is a very important manufacturing technology in the automobile, aerospace, and other major industries, where the material is removed continuously from the workpiece by a cutting tool. Modelling of the high speed metal machining has been proved to be particularly complex due to the coupled physical phenomena, such as the extremely large plastic deformation, the adiabatic shear band formation, the ductile fracture and the frictional contact. It is still a challenge to realistically predict the chip morphology, the cutting force as well as the cutting temperature by both experimental and numerical approaches [1]. For an accurate modelling of the chip formation process, the first key point is to find appropriate constitutive models that can successfully capture the related physical behavior. At the same time, it is also important to apply appropriate computational methods to cope with the topology change of the body due to large deformation and fracture. For a recent review on modelling of metal machining processes, see [2] and references therein.

The adiabatic shear band is a main cause for the serrated chip formation in high speed machining. The Johnson–Cook flow stress model [3] has been widely employed as the constitutive law to model the shear band formation in metal cutting processes, such as [4] and [5], where the temperature related term describes a softening effect and drives the shear band formation. To account for the strain softening effect in metal cutting, the modified Johnson–Cook model was developed in [6], where the strain softening term is added to the Johnson–Cook model in a multiplicative form [7]. Thus, the Johnson–Cook flow stress model is applied in this work within a finite plasticity formulation to describe the shear band formation.

Another difficulty in modelling of the serrated chip formation is related to the material separation at the chip root as well as the serrated morphology on the chip upper surface. In the literature, both of the behaviors are treated either by a large plastic deformation approach, see [8], [9], [10], or a separation layer approach, see [11], [12], [13]. In the first approach, the chip formation is considered as a solid undergoing large deformations without fracture where adaptive remeshing is used to deal with the mesh distortion. In the second approach, a separation layer of finite elements is artificially set on the workpiece where elements will be removed when the critical value of equivalent plastic strain is reached. In this case, no fracture model is implemented on the chip material so that the serrated morphology on chip upper surface is generated solely by adiabatic shearing through thermal softening. Hence, both of the above approaches can not generate a clearly serrated morphology on the chip upper surface.

In addition to the constitutive model, the proper selection of a computational method is also important for metal cutting simulations. Thefinite element method (FEM) is applied in [14] and [15] to model the chip formation based on adaptive remeshing to eliminate the deformation induced element distortion. However, the mapping of the state variables from one configuration to the next configuration is required, which leads to an inefficient computation as well as to accumulated numerical errors. As a more flexible and convenient approach for large deformation problems, meshfree methods, such as the discrete element method (DEM) and the particle finite element method (PFEM), are recently applied to metal cutting simulations, see [8], [9], [10], [16] for instance. However, appropriate force laws between particles need to be selected in DEM and the geometric boundary of the body needs to be redefined in PFEM.

Recently, theoptimal transportation meshfree (OTM) method is developed by Li et al. [17], based on the optimal transportation theory [18] and the material point sampling concept. Here a stabilized OTM algorithm, derived by Weißenfels and Wriggers [19] from the principle of virtual work, is applied to increase the accuracy. Because of its advantage in algorithmic robustness and its convenience in fracture modelling, the stabilized OTM method is applied as a meshfree solution method in this work to model the chip formation.

In this work, it is shown that the Johnson–Cook flow stress model leads to an under prediction of the thermal effect for shear band formation. Hence, a larger magnitude of the Taylor–Quinney coefficient in the temperature evolution equation has to be applied to reproduce thermal softening effects more realistically. However, this larger magnitude of the Taylor–Quinney coefficient is physically not correct. Instead of the pure plastic deformation approach and the separation layer approach in the literature, both the chip separation from the workpiece and the serrated morphology on chip upper surface are treated as ductile fracture. This enables a more realistic modelling of the chip formation. As a well-validated ductile fracture model, such as in [20] and [21], the Johnson–Cook fracture model [22] shows limitations in capturing the fracture on the chip upper surface. Thus, a supplementary condition of positive stress triaxialilty is introduced in this work to improve the performance of the Johnson–Cook fracture model. This condition allows a more accurate representation and measurement of the chip size, like chip spacing, peak and valley. By combining the stabilized OTM method with the material point erosion approach, both the chip separation from the workpiece and the fracture at the chip upper surface are successfully captured.

This paper is structured as follows: In Section 2, the physical mechanisms of chip formation process are firstly analyzed. Then, the corresponding constitutive model within the finite plasticity framework and ductile fracture model used to model the serrated chip formation are presented in Section 3. In Section 4, the frictional contact model is given. The formulation of the stabilized OTM method as well as the material point erosion approach within the OTM discretization is described in Section 5. The serrated chip formation process and the effects of constitutive parameters on the chip morphology are investigated by the numerical simulation results in Section 6, which is followed by the conclusions in Section 7.