Numerical and Experimental Investigation on Dynamic Behavior in Turning Process of Thin-walled Workpieces made of 42CrMo4 Steel Alloy


 Machining thin-walled parts is generally cumbersome due to their low structural rigidity. Thus, to better understand the dynamic behavior of thin-walled parts during machining, various engineers and researchers in the field of metal cutting employ the Finite Element Method (FEM) due to its ability to highlight the physics involved in chip formation and the range of force generated in the cutting zone. The results of numerical simulations are evaluated using comparison with experimental data. In this paper, we study the effect of feed rate as well as the thickness of the wall part made of 42CrMo4 steel alloy on the cutting forces and workpiece displacements both experimentally and numerically during roughing and finishing turning process. The numerical study is based on the development of a three-dimensional (3D) Finite Element Model (FEM) in Abaqus/Explicit frame. In the model, the workpiece material is governed by a behavior law of Johnson-Cook. The detachment of the chip is simulated by a ductile fracture law also of Johnson-Cook. Numerical and experimental results show that the cutting forces and the quality of the machined surface depend not only on the choice of cutting parameters but also on the dynamic behavior of thin-walled parts due to their low rigidity and low structural damping during of the machining operation. Indeed, cutting forces are proportional to the feed rate and inversely proportional to the thickness of the part. The largest displacements recorded on the part are mainly along the direction of the tangential component of the cutting force. The flexibility of the part generates instability in the cutting process, but the frequencies of the vibrations are higher than the frequency of rotation of the part.


Introduction
In the machining field by cutting tool, the cutting process investigations and its optimization are still very delicate and financially costly. This is mainly due to the complexity of the multi-physical phenomena governing the tool-matter interaction [1]. Thus, recourse to the use of mathematical modeling tools (analytical, numerical, etc.) for the prediction of cutting tool/matter behavior becomes essential [2][3][4]. These prediction tools are only relevant if they are rigorously constructed and fed with reliable experimental data reflecting the reality of this cutting process [5,6]. The models generally developed in the literature have as objective functions the cutting phenomena (vibrations, chatter, temperature, forces, wear, etc.) where the characteristics of the machined part are linked to its integrity and to its resistance to fatigue (roughness, residual stresses, topography, hardness, microstructure, etc.) [7][8][9][10]. The study of the dynamic behavior of machined parts necessarily requires consideration of the aforementioned phenomena. Thus, Mehdi et al. [11,12] proposed a dynamic cutting force model for turning process of a thin walled workpiece in which dynamic deformations are different from those of massive ones. Due to the diversity of thin walled workpieces in the manufacture industry, and in order to have a global model of the dynamic behavior of a variety workpieces, the study have been oriented to the characterization of dynamic behavior of thin walled tubular parts. The model takes into account the damping due to interference between the tool flank and the machined workpiece surface. The different tests carried out clearly show the effect of cutting damping on cutting forces and the stability of cuttingprocess. Lorong et al. [13] presented an experiment accompanied with full time domain simulations featuring strong chatter and bringing forward the impact of the damping on the instability onset.
Gerasimenko et al. [14] presented an experimental investigation completed with a numerical model of a straight turning operation on a thin-walled structure (tube). There work reveals the instabilities of the quasi-steady cutting under variable conditions, due to the structure's mass and compliance variation.
Recently, Sahraoui et al., [15,16] presented an analysis of a turning process stability using an analytical model supported and validated with experimental tests results of roughing and finishing operations conducted on AU4G1 thin-walled tubular workpieces with different thickness values. They showed the important effects of the dynamic behavior of the thin walled workpieces on the stability criteria, which cannot be ignored in machining process planning and cutting parameters selection. They justified the influence of an additional structural damping on chatter suppression in a machining of thin walled workpieces.
On the other hand, the finite element simulation methods have been used more in research related to the study of cutting processes (turning, milling or drilling). Given the diversity of the results offered by these simulation methods as well as the low implementation costs compared to those relating to experimental studies, the researchers developed several simulation models in order to study the cutting parameters. These models are closely linked to the material properties of the machined parts as well as to the phenomena which accompany the tool and the workpiece interaction. Several results have been obtained, such as the prediction of chip formation and the study of its morphology [17,18]. Likewise, other models have been developed to predict cutting forces and heat transfers during machining [19][20][21][22] or the influence of these parameters on the morphology of the chip [23]. This paper deals with the dynamic behavior of thin walled workpieces made of steel alloy 42CrMo4 during roughing and finishing turning process. To better understand the different damage observed during turning of these workpieces, a numerical study, supported and validated with experimental tests results, is proposed. The numerical study is based on the development of a three-dimensional (3D) Finite Element Model (FEM) in Abaqus/Explicit frame in order to predict the cutting forces in radial, tangential and axial directions and the impact of the machining parameters as well as the thickness of the wall part on the cutting forces and workpiece vibrations. In the model, the workpiece material is governed by a behavior law of Johnson-Cook. The detachment of the chip is simulated by a ductile fracture law also of Johnson-Cook.
The rest of the paper is organized as follows. The developed finite element model is presented in the next section. The experimental procedure is exposed in section 3. Experimental and numerical results are discussed in section 4. The paper is concluded with the main observations and potential future work.

Behavior law
The major characteristic of the chip formation process is the presence of intense deformations with high rates of deformation. Thus our simulation model is developed on the explicit version of the Abaqus finite element analysis software which supports the resolution of dynamic non-linear problems with this kind of conditions. In this study, the workpieces are made of 42CrMo4 steel materials which are characterized, in the case of a traction load, by an elastic behavior, a plastic behavior and a phase of damage. During this last phase, the mechanical properties of the material deteriorate. Analyzes of experimental tests, made by researchers, have shown the formation of micro-cracks and micro-cavities which leads to rupture. Therefore, the study of cutting processes requires the characterization of the actual behavior of the material through laws that describe the behavior and damage of the material.
The mechanical behavior at deformation of the machined workpiece can be described by a thermoviscoplastic model which takes into account high deformation rates, inelastic deformations and the effects linked to the variation in temperature. The behavioral law Johnsons Cook has been implemented for flow stress of the workpiece (Eq. 1). This law is based on the value of the equivalent plastic deformation of an element. The damage occurs when the damage parameter reaches the value 1.
where  is the equivalent plastic stress, pl  is the equivalent plastic strain, & pl is the equivalent plastic strain rate, 0 & is the reference equivalent plastic strain rate, A is the limit of elasticity of the material, B and n are the coefficients related to the work hardening, C is the sensitivity coefficient to deformation rate and m is the temperature sensitivity coefficient. is the temperature ratio given by Eq. 2.
where T is the workpiece temperature, room T is the room temperature and m T is the melting temperature. In our study, ) m  is equal to one because only the mechanical behavior of the process has been considered in the model.

Friction law
The interactions between the chip and the cutting tool result in friction between two surfaces. With Abaqus, there is a "master" surface and a slave surface (geometry or mesh). The essential criterion in this contact is that the nodes relating to the slave surface do not penetrate into the "master" surface. In addition, normal vectors are calculated for each node. In the case of digital cutting simulations, the tool is defined as master and the workpiece is the slave.
The most used friction laws are that of Coulomb and that of Tresca. In the case of Coulomb's law, the two tangential and normal stresses exerted on the contact surfaces are linked by the following relation (Eq. 3) in the case of a rigid tool and the deformable part: Where  represents the friction coefficient.
The Tresca model imposes a constant friction threshold and the slip limit is independent of the normal stress. In this case, the friction law is given by Eq. 4: where % m is the Tresca coefficient ( 0 ) and e  is the elastic limit of the material of the workpiece.
In this simulation, the Coulomb model has been adopted with a friction coefficient value of 0.8.

Damage law
The contact zone between cutting edge and machined part is characterized by significant plastic deformations where the laws of the mechanics of the media contained are not applicable. For a more realistic simulation, it is necessary to use a damage model. Figure 1 shows the result of the tensile test applied on metallic materials. In the damage phase, the stress tensor can be translated by the following relation describing the law of elasticity: Where %  represents the tensor of effective stress, E % represents the tensor of elasticity,  is the total strain and p  the viscoplastic strain. The term D represents the damage variable which can vary according to a linear or exponential law.
The damage process is described by three phases: initiation of the damage at a deformation level, an evolution of the damage accompanied by increased loading and rupture. The initiation phase of the damage can be characterized by the following relationship (Eq. 6): Where p   is the increment of the equivalent plastic deformation and i  is the deformation equivalent to the rupture of the material. The damage is initiated if  takes the value 1. This term is calculated at each increment and for each mesh element.
A linear evolution of the damage is described by the following expression (Eq. 7): An exponential evolution of the damage is described by the following expression (Eq. 8): Where u represents the equivalent plastic displacement, f u is the fracture displacement, y  is the flow stress and f G is the rate of energy restitution of the material.
Where  is the Poisson coefficient of the material, E is its Young's modulus and , Ci K is the factor describing the breaking strength (in mode I (crack) or II (shear)) [23].
The models of damage are varied and they essentially describe the plastic deformation equivalent to rupture. These models are generally based on constants relating to each type of material which can be identified from experimental tests or numerical simulations. In our simulation, we have chosen to use the ductile damage and the shear damage criteria which describe the equivalent deformation of the material at break. The shear criterion is suitable for breaks due to the location of the shear band. The model supposes that the plastic strain equivalent at the beginning of the damage is function of the report of shear stress and the speed of deformation. The ductile criterion of the rupture is suitable for the ruptures due to the growth and the propagation of the cracks. The model supposes that the plastic strain equivalent at the beginning of the damage is function of the three axial of the stresses and the rate of strain.

Modeling of cutting process
The finite element simulation is based on the mesh of the regions studied into elements. The distribution, the density of the mesh and the dimensions of the elements are specified according to the problem studied. In the case of the cutting process, the mesh can never maintain its initial state. Changes are always present and the mesh risks the problem of distortion which leads to errors and to stopping of the calculation. The choice of the type of the elements, the automatic correction of the mesh (adaptive mesh) and the increase in the density of the localized mesh (dimensions of the elements) are presented as techniques which help to eliminate this problem. The simulation is developed via Abaqus/Explicit.
The "Dynamic Explicit" module which supports nonlinear effects and large deformations has been exploited. The thermal effects are not taken into account within the framework of this simulation. In fact, we are mainly interested in studying the cutting forces as a function of the wokpieces wall thickness and the variation of the feed.

Modeling of cutting system
In the simulation model, we have opted for a cantilever mounting system (Fig. 2). This choice is justified by the fact that the length to outside diameter ratio of the workpiece is less than 2. The three jaws are modeled as a rigid body with a reference point which controls the imposed boundary conditions. The elements of mesh chosen are specific to the rigid type and they are also of triangular type R3D3 (linear) for Explicit resolution. The density of the mesh at the level of the surface in contact with the part respects that adopted for the part. Table 2 gives the considered cutting parameters for marching 42CrMo4 steel parts considered in numerical and experimental tests. The boundary conditions are imposed on the tool and the three jaws through their reference points. The tool has a constant feed rate along the z axis. The jaws turn at a constant angular speed around the feed rate axis. The rotational movement is transmitted to the workpiece through the jaws contact surface. For the contact properties, the part is considered as "slave", the jaws and the cutting tool are considered as "master". The type of contact chosen is general "General contact" with two types of properties: The tool and workpiece contact is with friction translated by the introduction of a coefficient of friction µ. The jaws and workpiece contact is non-slip to transmit the rotational movement of the jaws towards the workpiece.
Wall thickness ep (mm) 6 6 6 4.5 4. With Abaqus the components, which are considered as a rigid body, are transformed into a rigid shell whose mesh is possible. The selected elements are specific to the rigid type (Discrete Rigid Element) and they are of quadratic type R3D4 (linear) for Explicit resolution. The mesh density is important at the nose of the tool in order to respect the tool / part contact as much as possible. For this rigid body, one associated a point of reference which controls all the possible movements (Fig. 3). Complementary angle of the cutting edge direction: ψr (°) -5 -5

Modeling of machined parts
The CAD models of the machined parts were developed with Abaqus software. The parts are thin tubes with a length "L" equal to 200 mm and an internal diameter "D" equal to 100 mm. The wall thicknesses "ep" depend on the machining operation. Thus, for a roughing operation, the chosen thicknesses are 3, 4.5 and 6 mm. For a finishing operation, the wall thickness is 1.5 mm. The material properties taken into account in the simulation are shown in Table 2.
The workpieces are considered to be deformable and meshed with linear tetrahedral elements C3D4.
The highest mesh density is located in the part-cutting tool contact zone. The distortion control is activated in order to reduce the mesh errors during the tool and workpiece contact caused by intense deformations. For this type of element the adaptive mesh (ALE) cannot be applied.

Experimental procedure
The experimental tests were conducted with roughing and finishing operation using tubular parts with 200 mm length of and an internal diameter of 100 mm. The workpieces material is the steel alloy 42CrMo4 which is used in mechanics for parts of different sizes (shafts, racks, crankshafts, gears, etc.) due to its good mechanical characteristics. For the roughing operation, the wall part thicknesses "ep" are 3, 4.5 and 6 mm. For a finishing operation, the wall thickness is 1.5 mm. Table 2 and Table3 give the cutting process parameters and the cutting tool parameters used during the roughing and finishing operations. In total we have twelve experimental tests: Nine tests for the roughing operation and three tests for the finishing operation). The experiments consist in measuring firstly the cutting force components x F , y F , z F respectively along radial, tangential and axial directions and secondly the tangential workpiece vibrations along the Y direction. The experimental setup is illustrated in Figure 4.
The measurement of the cutting forces components along the three directions X, Y and Z is conducted using tri axial force piezoelectric dynamometer which can measure the cutting forces components signals in three directions during active cutting process. The workpiece displacement along the Y direction is measured using a capacitive sensor without contact fixed at a distance of 0.5 mm to the free end of the workpiece. The engine lathe is equipped with a 20 kW DC motor. The rotation speed is controlled by an electronic regulator. The signals received from the dynamometer and the sensor, are acquired by a data acquisition system and analyzed using specific software installed on a laptop. The experimental results of the twelve performed tests (Table 2) are summarized in Table 5 From these results we can conclude that the cutting forces and the quality of the machined surface depend not only on the choice of cutting parameters but also on the dynamic behavior of thin-walled parts due to their low rigidity and low structural damping during of the machining operation. Figure 5 shows the variation of tangential cutting force and of the radial displacement of the wall of the part respectively for the roughing test "R-3" (Fig. 5-a) and for the finishing test "F-2" (Fig. 5-b). In roughing, the cutting process is accompanied by low amplitude vibrations while in finishing the vibrations are intense.   Fz_exp. Those obtained using the analytical model, presented in [16], are denoted respectively Fx_sim, Fy_sim and Fz_sim. Finally, those, obtained using the FEM, are denoted respectively Fx_fem, Fy_fem and Fz_fem. In the case of roughing operations (Fig. 6 a- During finishing operations ( Fig. 6-d), we note that there is agreement between the cutting forces measured experimentally and those simulated analytically. In the case of test F-2, a difference of 18% is recorded between the axial force measured experimentally and that simulated analytically. However, we can notice that the average values of the radial, tangential and axial cutting forces simulated using the FEM are lower than those measured experimentally. The FEM was used to predict the workpiece wall displacements x U , y U , z U respectively along radial, tangential and axial directions at the free end of the workpiece. The workpiece wall vibrations can be simulated by imposing a coupling constraint between a reference point (RP-1) located on the axis of revolution and the circumferential internal surface of the free end of the workpiece (Fig. 7).  Simulated of radial deformation at the free end of the workpiece for roughing tests R-2 and R-5 and finishing test F-2 Using the analytical stability lobes model, described in [16], figure 10 shows comparison of the vibratory state of the roughing and finishing conducted tests with the stability lobes. For a workpiece with 6 mm thick, corresponding to roughing tests R-1, R-2 and R-3 ( Fig. 10-a), it can be noted that for a depth of cut of 1.5 mm and for the two feeds per tooth 0.2 and 0.3 mm/rev, the cutting process is stable for a spindle speed of 625 rpm. For the three values of the feed rate 0.2, 0.3 and 0.4 mm/rev, a zone of stability is located for a depth of cut equal to 1.3 mm and a spindle speed equal to 630 rpm. It is noted that for the roughing experimental tests, the values retained for the spindle speed (450.7 rpm) and the depth of cut (1.5 mm) are located in unstable zones.
In the case of the roughing operation of workpiece with 4.5 mm thick ( Fig. 10-b) corresponding to the three roughing tests R-4, R-5 and R-6, the cutting depth allowing a stable zone is limited to 1.4 mm with a spindle speed of 500 rpm and a feed rate equal to 0.2 mm/rev. The choice of a cutting depth equal to 1 mm is possible for the two feeds per revolution 0.2 and 0.3 mm/rev and a spindle speed of 500 rpm.
For the three values of the feed rate 0.2, 0.3 and 0.4 mm/rev, a depth of cut equal to 0.8 mm is located in the stable zone with a spindle speed equal to 500 rpm.
In the case of the roughing operation of workpiece with 3 mm thick (Fig. 10-c) corresponding to the three roughing tests R-7, R-8 and R-9, in order to migrate to the stable zones, it is necessary to increase the spindle speed and reduce the depth of cut. To maintain low spindle speeds, the depth of cut should be less than 1 mm. This value is reached for the feed 0.2 mm/rev and a spindle speed between 410 and 415 rpm.
In the case of the finishing operation of workpiece with 1.5 mm thick ( Fig. 10-d) corresponding to the three finishing tests F-1, F-2 and F-3, in order to migrate to stable zones, it is necessary to increase the spindle speed and decrease the depth of cut. For a depth of cut equal to 0.5 mm, the stable zones are located for a spindle speed greater than 1000 rpm.  From figure 12, it can be seen that, in roughing operation of parts with 3 mm thick, the stress increases when the feed rate increases. And for parts of 4.5 and 6 mm thickness, the stress is not too much influenced by the feed rate. For the finishing operation, the stresses reach their maximum values at the cutting edge contact of the tool and the workpiece. And we can also distinguish the primary and secondary shear zones.

Figure 11.
FFT spectra of radial vibrations of the workpiece wall during experimental roughing and finishing operations.

Conclusion
This work gives focuses on the dynamic behavior of thin-walled workpieces made of 42CrMo4 steel alloyduring turning process. The impact of feed rate as well as the thickness of the wall part on the cutting forces and workpiece displacements has been investigated both experimentally and numerically during roughing and finishing operations. The proposed FEM, using Abaqus/Explicit software, is based on the behavior law of Johnson-Cook for the workpiece material. The detachment of the chip is simulated by a ductile fracture law of Johnson-Cook. The simulated cutting forces and the wall workpiece displacements are validated according to a series of experiments performed in the same cutting conditions. In the model, the cutting tool is considered to be a rigid body (to simplify the model and reduce calculation time) and animated with a forward movement at constant speed. However, the workpiece, animated with rotational movement, is considered as a deformable body, in order to most simulate its rotation and its dynamic behavior.
From this investigation, the following conclusions can be drawn.
1. Cutting forces and the quality of the machined surface depend not only on the choice of the feed rate but also on the dynamic behavior of thin-walled parts due to their low rigidity and low structural damping during of the machining operation. 5. The numerical model results can be improved by integrating the effect of temperature, tool wear and the variation of the specific cutting energy during the turning operation.