洛阳理工学院毕业设计论文 an element or chip-separation criterion
The application of non-linear finite element techniques to the simulation of metal forming processes resulted also in finite element modelling of metal cutting processes. More sophisticated models have been developed, mainly aiming at the determination of the residual stress and strain, the temperature distribution and the prediction of the cutting forces. The two methods employed were the Eulerian-based approach, where cutting is simulated from the steady state, avoiding, therefore, the need for a chip-separation criterion, but requiring that the shape of the chip must be known in advance; and the Lagrangian approach, where cutting can be simulated from the incipient to the steady state, allowing for the prediction of the chip geometry and the residual stresses in the workpiece. Note, however, that a chip-separation criterion must be provided, to enable the separation of the chip from the workpiece.
Chip-separation criteria, based on a geometrical consideration, have been suggested in, but see also the criteria based on the critical value of the strain energy density in and the work by Ceretti et al. who developed a cutting model by deleting elements having reached a critical value of accumulated damage.
In almost all of the finite element models developed so far, non-commercial FE codes have been employed. The application of commercial FE codes is, therefore, desirable for the industrial use of the FE method for the simulation of metal cutting processes, by enabling tool makers to optimize cutting tool design, and users to evaluate the effect of cutting process parameters on the quality of machined parts prior to expensive and time-consuming experimental testing.
In the present work, the commercial implicit finite element code MARC was employed to construct a coupled thermo-mechanical finite element model of plane-strain orthogonal metal cutting with continuous chip formation The material is modelled as elastic-plastic, while its flow stress is taken as a function of strain, strain-rate, and temperature, so as to represent the real behaviour in cutting, where strain and strain-rate values, as well as the temperature rise, are large. The entire cutting process is simulated, i.e. from the initial to the steady state, and a geometrical chip-separation criterion, based on a critical distance value at the tool tip regime of the workpiece, is implemented into the MARC code by employing the rezoning feature.
2. Finite element model
The finite element model used for the plane-strain orthogonal metal cutting simulation is based on the updated Lagrangian formulation as provided by the MARC code. The plane-strain assumption constitutes a reasonable approximation, since in real metal cutting processes the width of cut is at least five times greater than the depth of cut, therefore, the chip is produced under nearly plane strain conditions.
The dimensions and the initial finite element mesh of both the workpiece and the tool are shown in Fig.1(a).The upper part of the mesh, which constitutes the workpiece, is finer, to enable the stress, strain, strain-rate and temperature in the chip and the tool tip regime to be accurately predicted. For the dimensions of the rest a coarser mesh is sufficient, as the boundaries of the mesh do not influence the predictions. The vertical displacement of the nodes, Y, at the lower boundary of the workpiece, and the horizontal displacement of the nodes, X, at the left boundary, are zero. The workpiece consists of 1340 four-node isoparametric plane-strain quadrilateral elements and 1428 nodes. This kind of lower-order elements has been proven to be more accurate in analyzing large-strain plasticity problems, compared to higher-order elements with eight nodes.
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洛阳理工学院毕业设计论文 However , as in this element bilinear interpolation functions are used, the strains tend to be constant throughout the element, resulting in poor representation of the shear behaviour， which is the dominant mode of deformation in cutting. In order to improve the shear characteristics, alternative interpolation functions are used, by employing the assumed strain formulation provided by the MARC code.
Fig.1. Chip formation in simulated orthogonal metal cutting: (a) initial undeformed state; (b) shape of the deformed chip after a tool path of 0.48 mm; (c) shape of the deformed chip after a tool path of 1.44 mm; (d) shape of the deformed chip after a tool path of 2.58 mm.
The tool consists of 425 four-node isoparametric quadrilateral planar heat-transfer elements, as nodes. The lower half of its mesh, expected to be in contact with the chip, is modelled with a finer mesh, in order to be able to predict the temperature field developed in the tool.
2.1. Chip-separation criterion
To investigate the chip formation and its separation from the workpiece, a chip-or element-separation criterion has been implemented into the MARC finite element code. Cutting is supposed to take place at the line representing the undeformed chip thickness, therefore, separation is assumed to occur only at the nodes lying along this line. The criterion for the separation of nodes in front of the tool tip is based on a geometrical consideration. When the tool tip approaches a node within a small critical distance, that node separates from
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Fig.2. Schematic diagram of the Geometrical separation criterion: (a) before element separation, D > Dc; (b) after element separation, D≤ Dc. 洛阳理工学院毕业设计论文 the workpiece and becomes part of the chip (see Fig.2).
When the distance D, between the tool tip and the node B, becomes equal or less than the pre-defined critical value Dc, a rezoning step is conducted. The connectivity of the element E2 changes and a new node BH replaces the node B in that element. Simultaneously, the coordinates of the nodes B and BH are altered, so that the node B moves upwards along BC by small distance, whilst node BH moves downwards by a small distance along BHF. The algorithm implemented into the MARC code for the implementation of the above mentioned procedure, is shown in Fig.3(a)and is explained with the aid of Fig.3(b).Note that the rezoning feature, which has been used extensively in metal forming problems to define a new mesh when the previous one is distorted due to excess plastic deformation, constitutes a very useful tool for modifying the initial finite element mesh, in a order to model the chip formation using a commercial finite element code.
In the case of metal cutting with a continuous chip, experimental observations reveal that chip formation takes place without a crack extension in front of the cutting tool tip. Furthermore, Fig.3. Pressing: (a) the flow-chart for chip-separation is a continuous process just ahead implementing the geometrical of the tool edge and, therefore, a geometrical separation criterion in simulated separation criterion based on distance is a realistic orthogonal metal cutting; (b) a assumption. Note that comparison of the schematic illustration of the algorithm geometrical separation criterion in modelling the used for the geometrical separation cutting process to other chip-separation criteria, criterion. based on the values of effective plastic strain and strain energy density, is made in.
The value of the critical distance, which in the present work is equal to 3 um and represents 5% of the element length, is taken as small enough to ensure continuous chip formation without causing numerical instability. Note that estimation of a proper value for the critical distance and its effect on the accuracy of the results involves difficulties and, therefore, it can be only validated experimentally.
2.2. Workpiece and tool material modelling
The workpiece material used for the plane-strain orthogonal cutting simulation was mild steel with 0.18% C, modulus of elasticity E=188 G Pa, Poisson's ratio v=0.3 and coefficient of linear thermal expansion c=1.281×10^5mm/mm℃
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洛阳理工学院毕业设计论文 The material was modelled as isotropic elastic-plastic, with isotropic strain-hardening. In the cutting processes, the deformation of the material in the cutting zone takes place at elevated temperature, and high strains and strain-rates. Therefore, in order to allow for their effect on the material properties, the flow stress of the workpiece is taken as a function of strain, strain-rate and temperature using the constitutive equation taken from Ref.
(1)
(2)
where T (K) is the temperature, e the total strain, e? the total strain-rate, s the flow stress (in M Pa).
Due to the high elastic modulus of the tool material, which is tungsten carbide, the tool is considered as a perfectly rigid body and only a heat-transfer analysis is conducted on it. The physical properties of the mild steel and the carbide are tabulated in Table 1.
2.3. Friction modelling
Experimental observations revealed that the tool/chip interface may be divided into a sticking and a sliding region. Therefore, friction modelling in metal cutting must account for both situations. The friction force is modelled as a distributed tangential force Ft, along the chip/tool inter-face, given by
(3)
where, following the notation, m is the Coulomb friction coefficient, Fn the normal reaction force, Vr the relative sliding velocity between the chip and the tool and t=Vr/∣Vr∣ the tangent unit vector in the direction of the relative velocity. C is a constant representing the relative sliding velocity below which friction force starts dropping considerably to zero: in that way, sticking of the tool rake face is reproduced, by allowing variable very small slips. Table 1
Physical properties of the workpiece and tool material Material Mild steel Density (kg/m3) 7833 Thermal conductivity (W/m ℃) 54 33.5 Specific heat (J/kg ℃) 465 234 Tungsten carbide 12700 2.4. Heat transfer
Knowledge of the temperature distribution in the workpiece, chip and tool, is very important, since it has a great effect on the quality of the surface integrity of the tool wear. The main sources of heating, responsible for the high temperature rise observed in cutting processes, are the plastic work and the friction at the chip/tool interface, which are converted into heat. The rate of specific volumetric flux due to plastic work is given by the equation
（4）
where, following the notation, Wp is the rate of the plastic work, r the density, M the mechanical equivalent of heat to account for a consistent system of units and Wh the percentage of plastic
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洛阳理工学院毕业设计论文 deformation converted into heat, which usually accounts for about 90%.
The distributed heat flux generated at the interface between the chip and the tool rake face due to friction is described by
（5）
where Ft is the contact friction force and Vr the relative sliding velocity between the chip and the tool rake face. This flux is split into two equal parts, assigned to each of the contacting parts, i.e. the chip and the tool.
Machining is performed at ambient temperature (i.e. the initial temperature of both the workpiece and the tool is 20℃) while the heat losses to the environment from the free surface of the workpiece, due to convection heat transfer, are determined by the distributed heat flux
（6）
where h=17.04 W/㎡℃ is the convection heat-transfer coefficient of the workpiece material, Tw the temperature of the workpiece, To the ambient temperature, taken as 20℃.Heat transfer by radiation is considered insignificant and is not therefore taken into account.
2.5. Process parameters
The tool geometry and the cutting conditions used for the orthogonal metal cutting simulation are presented in Table 2. Table 2
Cutting conditions and tool geometry
Tool rake angle (。) 20 Tool clearance angle (。) 5 Tool edge radius 0 Undeformed chip thickness (mm) 0.27 Width of cut (mm) 3.5 Cutting speed (mm/s) 600 Coulomb friction coefficient 0.4
3. Results and discussion
The cutting tool is advanced incrementally into the workpiece from the initial position, as shown in Fig.1.The chip is formed gradually until steady state is attained, i.e. the cutting force reaches a constant value. Each time the chip-separation criterion is satisfied, the procedure of Fig.2 is followed; the shape of the deformed chip, the cutting forces, the stress, strain and strain-rate distributions in the chip and the workpiece, as well as the temperature distribution in the chip, workpiece and tool, being obtained.
In Fig.1 (b)-(d), the shape of the deformed chip after a tool advance of 0.48, 1.44 and 2.58 mm, respectively, is indicated, the last case representing the steady state of chip formation. The distortion of the initially rectangular elements reveals the expected severe shearing of the material in the chip regime. Also, from the shape of the deformed mesh at the steady state, see Fig.1 (d), cutting parameters such as the shear angle, the deformed chip thickness, the cutting ratio, the chip-tool contact length and chip curling, can be estimated. The distribution of stress, strain, strain-rate and temperature in the deformation zone, which can be hardly are virtually
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