Shurygin Kurenkov Numerical modeling of cutting processes. Modeling of the metal cutting process by the finite element method Vinogradov yuriy valerievich. Construction of a model for the introduction of a rigid wedge into a semi-infinite elastic-plastic body

02.06.2020

Introduction

Chapter 1. General formulation of the problem of elastic-plastic deformation 25

1.1. Process kinematics 25

1.2. Constitutive relations of the processes of elastic-plastic finite deformation 32

1.3. Statement of the problem of finite elastoplastic deformation 38

1.4. Setting up the separation process 42

Chapter 2 Numerical modeling of final forming processes 44

2.1. Numerical formulation of problem 44

2.2. Method of integration of resolving relations 50

2.3. Algorithms for solving boundary value problems of elastic-plasticity 51

2.4. Checking the correctness of the implementation of the mathematical model 54

2.5. Analysis of model behavior under small deformations 57

2.6. Modeling the process of finite element material separation 58

2.7. Building a model for the introduction of a rigid wedge into a semi-infinite elastic-plastic body 60

2.8. Mechanism of accounting for friction in the cutting model 62

Chapter 3 Mathematical modeling of the cutting process . 65

3.1. Free cutting process 65

3.2. Factors affecting chip formation 68

3.3. Boundary Conditions in Simulation 70

3.4. Finite element implementation of the cutting process 74

3.5. Simulation of steady state cutting 75

3.6. Iterative process at step 77

3.7. Justification of the choice of the calculation step and the number of finite elements 80

3.8. Comparison of experimentally found and calculated values of cutting forces 83

Bibliography

Introduction to work

destruction of metal under such extreme conditions, which are usually not encountered either in the testing of materials or in other technological processes. The cutting process can be studied on idealized physical models using mathematical analysis. Before proceeding to the analysis of physical models of the cutting process, it is advisable to familiarize yourself with modern ideas about the structure of metals and the mechanism of their plastic flow and destruction.

The simplest cutting scheme is rectangular (orthogonal) cutting, when the cutting edge is perpendicular to the cutting speed vector and oblique cutting scheme, when a certain angle of inclination of the cutting edge is set.

edges I.

Rice. 1. (a) Scheme of rectangular cutting (b) Scheme of oblique cutting.

The nature of chip formation for the considered cases is approximately the same. Various authors divide the process of chip formation both into 4 and 3 types. Accordingly, there are three main types of chip formation shown in Fig. 2: a) intermittent, including periodic separation of chip elements in the form of small segments; b) continuous chip formation; c) continuous with the formation of build-up on the instrument.

Introduction

According to another concept, back in 1870, I. A. Time proposed a classification of the types of chips formed during cutting various materials. According to the classification of I. A. Time, when cutting structural materials in any conditions, four types of chips are formed: elemental, articular, drain and fracture. Elemental, jointed and drain chips are called shear chips, since their formation is associated with shear stresses. Fracture chips are sometimes called breakaway chips because their formation is associated with tensile stresses. Appearance of all the listed types of chips is shown in Fig. 3.

Rice. 3. Types of chips according to the classification of Time.

Figure 3a shows the formation of elemental chips, consisting of separate "elements" of approximately the same shape, not connected or weakly connected to each other. border tp, separating the formed chip element from the cut layer is called the shearing surface.

Introduction8

Physically, it is a surface along which, in the process of cutting, the destruction of the cut layer periodically occurs.

Figure 36 shows the formation of jointed chips. It is not divided into separate parts. The chipping surface has only just begun to appear, but it does not penetrate the chips through the entire thickness. Therefore, the shavings consist, as it were, of separate joints, without breaking the connection between them.

In Figure 3v - the formation of drain chips. The main feature is its continuity (continuity). If there are no obstacles in the way of the drain chips, then it comes off as a continuous tape, curling into a flat or helical spiral, until part of the chip breaks off under its own weight. The surface of the chip 1 - adjacent to the front surface of the tool, is called the contact surface. It is relatively smooth and high speeds cutting is polished as a result of friction on the front surface of the tool. Its opposite surface 2 is called the free surface (side) of the chip. It is covered with small notches and has a velvety appearance at high cutting speeds. The chips are in contact with the front surface of the tool within the contact area, the width of which is indicated by C, and the length is equal to the working length of the main blade. Depending on the type and properties of the material being processed and the cutting speed, the width of the contact area is 1.5–6 times greater than the thickness of the cut layer.

Figure 3g shows the formation of a fracture chip, consisting of separate, unrelated pieces of various shapes and sizes. The formation of fracture chips is accompanied by fine metal dust. Destruction surface tp can be located below the cutting surface, as a result of which the latter is covered with traces of chips broken out of it.

Introduction 9

According to what is stated in, the type of chip depends largely on the type and mechanical properties of the material being processed. When cutting ductile materials, the formation of the first three types of chips is possible: elemental, articular and drain. As the hardness and strength of the material being processed increase, the drain chip turns into a joint chip, and then into an element chip. When processing brittle materials, either elemental chips or, more rarely, fracture chips are formed. With an increase in the hardness of a material, such as cast iron, elemental chips turn into fracture chips.

Of the geometric parameters of the tool, the chip type is most strongly affected by the rake angle and the angle of inclination of the main blade. When processing ductile materials, the influence of these angles is fundamentally the same: as they increase, the elemental chip turns into a jointed one, and then into a drain one. When cutting brittle materials at large rake angles, fracture chips can form, which, as the rake angle decreases, becomes elemental. As the angle of inclination of the main blade increases, the chips gradually turn into elemental chips.

Chip type is influenced by feed (thickness of the cut layer) and cutting speed. The depth of cut (width of the cut layer) has practically no effect on the type of chip. An increase in feed (thickness of the cut layer) leads, when cutting ductile materials, to a consistent transition from drain chips to articular and elemental chips. When cutting brittle materials, with an increase in feed, elemental chips turn into fracture chips.

The most difficult effect on chip type is cutting speed. When cutting most carbon and alloy structural steels, if we exclude the zone of cutting speeds at which na-

Introduction 10

growth, as the cutting speed increases, the chip from the elemental becomes articular, and then drain. However, when processing some heat-resistant steels and alloys, titanium alloys, an increase in cutting speed, on the contrary, turns a drain chip into an elemental one. Physical reason this phenomenon has not yet been fully elucidated. An increase in cutting speed in the processing of brittle materials is accompanied by the transition of a fracture chip into an elemental chip with a decrease in the size of individual elements and strengthening of the bond between them.

With the geometrical parameters of tools and cutting conditions used in production, the main types of chips when cutting plastic materials are more often drain chips and less often jointed chips. The main type of chips when cutting brittle materials is elemental chips. The formation of elemental chips during cutting of both ductile and brittle materials has not been studied enough. The reason is the complexity in the mathematical description of both the process of large elastic-plastic deformations and the process of material separation.

The shape and type of cutter in production depends primarily on the field of application: on lathes, carousels, turrets, planers and slotters, automatic and semi-automatic lathes and special machines. The cutters used in modern mechanical engineering are classified by design (solid, composite, prefabricated, holding, adjustable), by type of processing (through, cutting, cutting, boring, shaped, threaded), by the nature of processing (roughing, finishing, for fine turning), according to the installation relative to the part (radial, tangential, right, left), according to the shape of the rod section (rectangular, square, round), according to the material

Introduction

barrel parts (from high-speed steel, from hard alloy, from ceramics, from superhard materials), by the presence of chip crushing devices.

The mutual arrangement of the working part and the body is different for different types of cutters: for turning cutters, the tip of the cutter is usually located at the level of the upper plane of the body, for planers - at the level of the support plane of the body, for boring cutters with a body of circular cross section - along the axis of the body or below it. The body of cut-off cutters in the cutting zone has a slightly higher height - to increase strength and rigidity.

Both many designs of incisors as a whole and their individual ones are standardized structural elements. To unify the designs and connecting dimensions of the tool holders, the following series of rod sections, mm, was adopted: square with side a = 4, 6, 8, 10, 12, 16, 20, 25, 32, 40 mm; rectangular 16x10; 20x12; 20x16; 25x16; 25x20; 32x20; 21x25; 40x25;40x32;50x32; 50x40; 63x50 (the aspect ratio H:B=1.6 is used for semi-finishing and finishing, and H:B=1.25 for roughing).

The All-Russian classifier of products provides for 8 subgroups of incisors with 39 types in them. About 60 standards have been published on the design of cutters and specifications. In addition, 150 standard sizes of high-speed steel inserts for all types of cutters, about 500 standard sizes of brazed carbide inserts, 32 types of multifaceted non-regrinding inserts (over 130 standard sizes) have been standardized. In the simplest cases, the cutter is modeled as an absolutely rigid wedge, without taking into account many geometric parameters.

The main geometric parameters of the cutter, taking into account the above.

Appointment of the rear corner a- reduce the friction of the rear surface on the workpiece and ensure unhindered movement of the cutter along the workpiece.

Introduction12

The influence of the clearance angle on the cutting conditions is due to the fact that the normal force of elastic recovery of the cutting surface and the friction force act on the cutting edge from the side of the workpiece.

With an increase in the back angle, the angle of sharpening decreases and thereby the strength of the blade decreases, the roughness of the machined surface increases, and heat removal to the body of the cutter deteriorates.

With a decrease in the clearance angle, friction on the machined surface increases, which leads to an increase in cutting forces, wear of the cutter increases, heat generation at the contact increases, although heat transfer conditions improve, and the thickness of the plastically deformable layer on the machined surface increases. Under such contradictory conditions, there should be an optimum for the value of the clearance angle, depending on the physical and mechanical properties of the material being processed, the material of the cutting blade, and the parameters of the cut layer.

The handbooks give the average values of the optimal values of the angles, a confirmed by the results of industrial tests. The recommended values for the back angles of the incisors are given in Table 1.

Introduction13

Appointment of the front angle At- reduce the deformation of the cut layer and facilitate chip flow.

Effect of Rake Angle on Cutting Conditions: Increasing the Rake Angle at facilitates the cutting process by reducing cutting forces. However, in this case, the strength of the cutting wedge decreases and the heat removal to the cutter body deteriorates. Angle reduction At increases the resistance of cutters, including dimensional.

Rice. 6. The shape of the front surface of the incisors: a - flat with a chamfer; b - curvilinear with a chamfer

The value of the rake angle and the shape of the front surface are greatly influenced not only by the physical and mechanical properties of the material being processed, but also by the properties of the tool material. Flat and curvilinear (with or without chamfers) forms of the front surface are used (Fig. 1.16).

A flat front surface is used for cutters of all types of tool materials, while a hardening chamfer is sharpened at the blade under

corner UV-^~5 - for high speed steel cutters and Atf =-5..-25 . for carbide cutters, all types of ceramics and synthetic superhard materials.

For work in difficult conditions (cutting with impacts, with an uneven allowance, when processing hard and hardened steels), when using hard and brittle cutting materials (mineral ceramics, superhard synthetic materials, hard alloys with a low cobalt content), cutters can be

Introduction

to be cut with a flat front surface, without a chamfer with a negative rake angle.

Cutters made of high-speed steel and hard alloys with a flat front surface without a chamfer with ^ = 8..15 are used for processing brittle materials that give fracture chips (cast iron, bronze). With a small cut thickness comparable to the cutting edge rounding radius, the rake angle has practically no effect on the cutting process, since the cut layer is deformed and turned into chips by a rounded radius edge. In this case, the front angles for all types of tool materials are accepted within 0...5 0 . The value of the rake angle significantly affects the durability of the cutters.

Appointment of the main angle in the plan - change the ratio between the width b and thickness a cut at constant depth of cut t and filing S.

Angle reduction increases tool tip strength, improves heat dissipation, increases tool life but increases cutting forces Pz and, Rat increases

squeezing and friction on the treated surface creates conditions for the occurrence of vibrations. With an increase the chips become thicker and break better.

Cutter designs, especially with mechanical fastening carbide inserts, provide a number of angle values#>: 90, 75, 63, 60, 50, 45, 35, 30, 20, 10, which allows you to choose the angle which best suits the given conditions.

The process of separation of the material depends on the shape of the cutter. According to cutting, metal separation occurs, it could be expected that this process includes destruction with the formation and development of cracks. Initially, this idea of the cutting process was generally accepted, but later doubts were expressed about the presence of a crack in front cutting tool.

Malloch and Rulix were among the first to master microphotography of the chip formation zone and observed cracks in front of the cutter, Kick, on the basis of similar studies, came to the opposite conclusions. With the help of more advanced microphotography techniques, it was shown that the cutting of metals is based on the process of plastic flow. As a rule, under normal conditions, a leading crack does not form; it can occur only under certain conditions.

According to the presence of plastic deformations propagating far ahead of the cutter, it was established by observing the process of chip formation under a microscope at very low cutting speeds of the order V- 0,002 m/min. This is also evidenced by the results of a metallographic study of grain deformation in the chip formation zone (Fig. 7). It should be noted that observations of the chip formation process under a microscope showed the instability of the plastic deformation process in the chip formation zone. The initial boundary of the chip formation zone changes its position due to the different orientation of the crystallographic planes of individual grains of the metal being processed. There is a periodic concentration of shear deformations at the final boundary of the chip formation zone, as a result of which the plastic deformation process periodically loses stability and the outer boundary of the plastic zone receives local distortions, and characteristic teeth form on the outer boundary of the chip.

T^- \ : " G

Introduction

Rice. 7. The contour of the chip formation zone, established by studying free cutting with the help of filming.

Rice. 8. Micrograph of the chip formation zone when cutting steel at low speed. The micrograph outlines the initial and final boundaries of the chip formation zone. (100x magnification)

Thus, we can only talk about the average probable position of the boundaries of the chip formation zone and the average probable distribution of plastic deformations inside the chip formation zone.

The exact determination of the stressed and deformed state of the plastic zone by the method of plastic mechanics presents great difficulties. The boundaries of the plastic region are not given and are themselves to be determined. The stress components in the plastic region change disproportionately to each other, i.e. plastic deformations of the cut layer do not apply to the case of simple loading.

All modern methods calculations for cutting operations are built on the basis of experimental studies. The most complete experimental methods are presented in. When studying the process of chip formation, the size and shape of the deformation zone, various experimental methods are used. According to V.F. Bobrov, the following classification is presented:

visual observation method. The lateral side of the sample subjected to free cutting is polished or a large square grid is applied to it. When cutting at a low speed, the distortion of the grid, tarnishing and wrinkling of the polished surface of the sample can be used to judge the size and shape of the deformation zone and form an external idea of how the cut layer after

Introduction17

gradually turns into shavings. The method is suitable for cutting at very low speeds, not exceeding 0.2 - 0.3 m/min, and gives only a qualitative idea of the chip formation process.

The method of high-speed filming. It gives good results when shooting at a frequency of about 10,000 frames per second and allows you to find out the features of the chip formation process at practically used cutting speeds.

Dividing grid method. It is based on applying an accurate square dividing grid with cell sizes of 0.05 - 0.15 mm. The dividing grid is applied in various ways: rolling with printing ink, etching, spraying in a vacuum, screen printing, scratching, etc. The most accurate and simplest method is scratching with a diamond indenter on a PMTZ device for measuring microhardness or on a universal microscope. To obtain an undistorted deformation zone corresponding to a certain stage of chip formation, special devices are used to "instantaneously" stop the cutting process, in which the cutter is pulled out from under the chip by a strong spring or powder charge explosion energy. On the resulting chip root, using an instrumental microscope, the dimensions of the cells of the dividing grid distorted as a result of deformation are measured. Using the apparatus of the mathematical theory of plasticity, it is possible to determine the type of the deformed state, the size and shape of the deformation zone, the intensity of deformation at various points of the deformation zone, and other parameters quantitatively characterizing the chip formation process by the size of the distorted dividing grid.

metallographic method. The root of the chip obtained with the help of a device for "instant" cutting stop is cut out, its side is carefully polished, and then etched with the appropriate reagent. The resulting microsection of the chip root is examined under a microscope at a magnification of 25-200 times or a micrograph is taken. Structure change

Introduction

chips and deformation zones in comparison with the structure of an undeformed material, the direction of the deformation texture makes it possible to establish the boundaries of the deformation zone and judge the deformation processes that took place in it.

Method for measuring microhardness. Since there is an unambiguous relationship between the degree of plastic deformation and the hardness of the deformed material, the measurement of the microhardness of the chip root gives an indirect idea of the intensity of deformation in various volumes of the deformation zone. For this, microhardness is measured on the PMT-3 device at various points of the chip root and isoscleres (lines of constant hardness) are built, with the help of which it is possible to determine the magnitude of shear stresses in the deformation zone.

Polarization-optical method, or the photoelasticity method is based on the fact that transparent isotropic bodies become anisotropic under the action of external forces, and if they are considered in polarized light, then the interference pattern makes it possible to determine the magnitude and sign of the acting stresses. The polarization-optical method for determining stresses in the deformation zone is of limited use for the following reasons. Transparent materials used in cutting have completely different physical and mechanical properties than technical metals- steels and cast irons. The method gives exact values of normal and shear stresses only in the elastic region. Therefore, using the polarization-optical method, only a qualitative and approximate idea of the stress distribution in the deformation zone can be obtained.

Mechanical and radiographic methods used to study the state of the surface layer lying under the treated surface. The mechanical method developed by N. N. Davidenkov is used to determine the stresses of the first kind, which are balanced in the region of the body, which is larger than the size of the crystal grain. The method is with

Introduction 19

surfaces of the sample cut from the machined part, very thin layers of material are sequentially removed and strain gauges are used to measure the deformation of the sample. Changing the dimensions of the sample leads to the fact that under the action of residual stresses it becomes unbalanced and deformed. Based on the measured strains, one can judge the magnitude and sign of the residual stresses.

Based on the foregoing, we can conclude that the complexity and limited applicability of experimental methods in the field of studying processes and regularities in cutting processes, due to their high cost, large measurement errors and scarcity of measured parameters.

There is a need to write mathematical models that can replace experimental research in the field of metal cutting, and experimental base used only at the stage of confirmation of the mathematical model. Currently, a number of methods are used to calculate cutting forces that are not confirmed by experiments, but derived from them.

An analysis of the known formulas for determining the forces and cutting temperatures was carried out in the work, according to which the first formulas were obtained in the form of empirical degrees of dependencies for calculating the main components of the cutting forces of the form:

p, = c P f p sy K P

where WedG - coefficient taking into account the influence on the strength of some permanent conditions; *R- cutting depth; $^,- longitudinal feed; ToR- generalized cutting factor; xyz- exponents.

Introduction 20

The main disadvantage of this formula is the lack of a pronounced physical connection with mathematical models known in cutting. The second disadvantage is the large number of experimental coefficients.

According to , the generalization of experimental data made it possible to establish that the average tangent acts on the front surface of the tool

voltage qF = 0.285^ , where &to is the actual ultimate tensile strength. On this basis, A.A. Rozenberg obtained another formula for calculating the main component of the cutting force:

(90-y)"cos/

-- їїdG + Sin/

Pz=0.28SKab(2.05Ka-0,55)

2250QK Qm5(9Q - Y) "

where Kommersant- width of the cut layer.

The disadvantage of this formula is that for each specific

in the case of force calculation, parameter definition is required Toa and$k experimentally, which is very laborious. According to numerous experiments, it was found that when replacing the curved shear line with a straight line, the angle At is close to 45, and therefore the formula will take the form:

dcos At

Pz = - "- r + sin^

tg arccos

According to experiments, the criterion cannot be applied as a universal one applicable to any stressed states. However, it is used as a base in engineering calculations.

Criterion of the greatest tangential stresses. This criterion was proposed by Tresca to describe the plasticity condition, however, it can also be used as a strength criterion for brittle materials. Failure occurs when the greatest shear stress

r max = gіr "x ~ b) reaches some specific value (for each material of its own).

For aluminum alloys This criterion, when comparing the experimental data with the calculated ones, gave an acceptable result. For other materials, there are no such data; accordingly, the applicability of this criterion cannot be either confirmed or refuted.

There are also energy criteria. One of these is the Huber-Mises-Genka hypothesis, according to which destruction occurs / when the specific energy of shape change reaches a certain limit value.

Introduction23

cheniya. This criterion has received satisfactory experimental confirmation for various structural metals and alloys. The difficulty of applying this criterion lies in the experimental determination of the limiting value.

The criteria for the strength of materials unequally resistant to tension and compression include the Schleicher, Balandin, Mirolyubov, Yagn criterion. The disadvantages include the complexity of application and poor confirmation by experimental verification.

It should be noted that there is no single concept for the destruction mechanisms, as well as a universal criterion of destruction, by which it would be possible to unambiguously judge the process of destruction. AT this moment we can talk about good theoretical development of only a set of special cases and attempts to generalize them. Practical application in engineering calculations of most of the modern fracture models is not yet available.

An analysis of the above approaches to the description of separation theory allows us to identify the following characteristic features:

The existing approaches to the description of the destruction processes are acceptable at the stage of the beginning of the destruction process and when solving problems in the first approximation.

The process model should be based on the description of the physics of the cutting process, and not on statistical experimental data.

Instead of the relations of the linear theory of elasticity, it is necessary to use physically nonlinear relations that take into account changes in the shape and volume of the body under large deformations.

Experimental methods can unambiguously provide information

Introduction

information about the mechanical behavior of the material in a given range of temperatures and parameters of the cutting process.

Based on the above, the main purpose of the work is the creation of a mathematical model of separation, which allows, on the basis of universal constitutive relations, to consider all stages of the process, starting from the stage of elastic deformation and ending with the stage of separation of the chip and workpiece, and to investigate the patterns of the chip removal process.

In the first chapter dissertation presents a mathematical model of finite deformation, the main hypotheses of the fracture model. The problem of orthogonal cutting is posed.

In the second chapter within the framework of the theory described in the first chapter, a finite element model of the cutting process is built. An analysis of the mechanisms of friction and destruction is given in relation to the finite element model. Comprehensive testing of the obtained algorithms is carried out.

In the third chapter the physical and mathematical formulation of the technological problem of removing chips from a sample is described. The process modeling mechanism and its finite element implementation are described in detail. Held comparative analysis obtained data from experimental studies, conclusions are drawn on the applicability of the model.

The main provisions and results of the work were reported at the All-Russian Scientific Conference " Contemporary Issues Mathematics, Mechanics and Informatics "(Tula, 2002), as well as at the winter school on continuum mechanics (Perm, 2003), at the international scientific conference "Modern problems of mathematics, mechanics and informatics" (g. . Tula, 2003), at the scientific and practical conference "Young scientists of the center of Russia" (Tula, 2003).

Constitutive Relationships for the Processes of Elastic-Plastic Finite Deformation

To individualize the points of the medium, for the initial t - About a fixed, so-called calculated, configuration (KQ ), an arbitrary coordinate system 0 is derived, with the help of which each particle is assigned a triple of numbers (J,2,3) "assigned" to this particle and unchanged during the entire duration of the movement. The system 0 introduced in the reference configuration, together with the basis, =-r (/ = 1,2,3) is called the fixed Lagrangian coordinate system. Note that the coordinates of particles at the initial moment of time in the frame of reference can be chosen as material coordinates. It should be noted that when considering the processes of deformation of a medium with properties dependent on the history of deformation, regardless of the material or spatial variables used, two coordinate systems are used - one of Lagrangian and Euler.

As you know, the occurrence of stresses in the body is generated by the deformation of material fibers, i.e. change in their lengths and relative positions, therefore the main problem solved in the geometrically nonlinear theory of deformations is to divide the motion of the medium into translational and "purely deformational" and indicate the measures for their description. It should be noted that such a representation is not unambiguous and several approaches to the description of the medium can be indicated, in which the division of motion into a portable "quasi-rigid" and a relative "deformation" one is carried out in various ways. In particular, in a number of papers, deformation motion is understood as the motion of the neighborhood of a material particle with respect to the movable Lagrangian basis ek; in the papers, as a deformation movement, the movement is considered relative to a rigid basis, the translational movement of which is determined by the rotation tensor, which connects the main axes of the left and right distortion measures. In this work, the division of the motion of the neighborhood of a material particle M (Fig. 1.1) into translational and deformed is based on the natural representation of the velocity gradient in the form of a symmetric and antisymmetric part. In this case, the deformation velocity is defined as the relative velocity of the particle relative to the rigid orthogonal trihedron of the vortex basis, whose rotation is specified by the vortex tensor Q. It should be noted that in the general case of the medium motion, the main axes of the tensor W pass through different material fibers. However, as shown in , for the processes of simple and quasi-simple loading in the real range of deformations, the study of the deformation motion in the vortex basis seems to be very satisfactory. At the same time, when constructing relationships that describe the process of finite deformation of a medium, the choice of measures must satisfy a number of natural criteria: 1) the measure of deformation must be conjugate with the measure of stress through the expression of elementary work. 2) the rotation of the material element as absolutely solid body should not lead to a change in the deformation measures and their time derivatives - a property of material objectivity. 3) when differentiating measures, the property of symmetry and the condition for separating the processes of shape change and volume change should be preserved. The last requirement is highly desirable.

As the analysis shows, the use of the above measures to describe the process of final deformation, as a rule, leads either to insufficient correctness in the description of deformation or to a very complicated procedure for calculating them.

To determine the curvature and twists of the trajectory, the invariants are used

tensors W ", which are the nth order Jaumann derivatives of the strain rate deviator, as shown in and the third invariant of the functional measure of deformation H do not depend on the nature of the change in the metric over the entire interval.The relation of the general postulate of isotropy in the form (1.21) is the starting point for the construction of specific models of finitely deformable bodies and their experimental justification.It seems natural to generalize the known relations for small deformations by passing to the proposed measures of deformation and loading Note that since in the problems of studying the process of deformation of a medium, as a rule, the velocity statement is used, then all relations will be formed in the rates of change of scalar and tensor parameters that describe the behavior of the medium. At the same time, the relative (in the sense of Jaumann) derivatives of tensors and deviators correspond to the velocities of the strain and loading vectors.

Construction of a model for the introduction of a rigid wedge into a semi-infinite elastic-plastic body

Currently, there are no analytical methods for solving problems associated with separation operations. The sliding line method is widely used for operations such as wedge insertion or chip removal. However, the solutions obtained using this method are not able to qualitatively describe the course of the process. More acceptable is the use of numerical methods based on the variational principles of Lagrange and Jourdain. Existing approximate methods for solving boundary value problems of mechanics of a deformable solid body are described in sufficient detail in monographs.

In accordance with the basic concept of the FEM, the entire volume of the deformable medium is divided into a finite number of elements that are in contact with each other at nodal points; the combined motion of these elements simulates the motion of a deformable medium. At the same time, within each element, the system of characteristics describing the movement is approximated by one or another system of functions determined by the type of the selected element. In this case, the main unknowns are the displacements of the element's nodal points.

The use of a simplex element greatly simplifies the procedure for constructing a finite element representation of relation (2.5), since it allows one to use simpler operations of one-point integration over the volume of an element. At the same time, since the requirements of completeness and continuity are satisfied for the chosen approximation, the necessary degree of adequacy of the finite element model to a "continuous system" - deformable body is achieved by simply increasing the number of finite elements with a corresponding decrease in their sizes. A large number of elements requires a large amount of memory and even more time spent on processing this information, a small number does not provide a high-quality solution. Determining the optimal number of elements is one of the primary tasks in calculations.

Unlike other methods used, the sequential loading method has a certain physical meaning, since at each step the response of the system to the load increment is considered as it takes place in the actual process. Therefore, the method makes it possible to obtain much more information about the behavior of the body than just the magnitude of displacements for a given system of loads. Since a complete set of solutions corresponding to different parts of the load is obtained in a natural way, it becomes possible to study intermediate states for stability and, if necessary, to make appropriate modifications to the procedure to determine branch points and find possible continuations of the process.

The preliminary stage of the algorithm is the approximation of the study area for the time t = 0 by finite elements. The configuration of the region corresponding to the initial moment is considered known, while the body can be either in a "natural" state or have prestresses due, for example, to the previous stage of processing.

Next, based on the expected nature of the deformation process, the type of particular theory of plasticity is selected (section 1.2). The processed data of experiments on uniaxial tension of samples of the studied material form specific view constitutive relations, using, in accordance with the requirements of Section 1.2, any one of the most common methods for approximating the experimental curve. When solving the problem, a certain type of plasticity theory is assumed to be unchanged for the entire volume under study throughout the entire process. The validity of the choice is subsequently evaluated by the curvature of the deformation trajectory, calculated at the most characteristic points of the body. This approach was used in the study of models technological processes finite deformation of tubular specimens in modes of simple or close to it external loading. In accordance with the chosen step-by-step integration procedure, the entire loading interval with respect to the parameter t is divided into a number of sufficiently small stages (steps). In what follows, the solution of the problem for a typical step is constructed according to the following algorithm. 1. For the area newly determined from the results of the previous step of the configuration, the metric specifications deformed space Chapter 2. Numerical modeling of the process of finite form change 53 space. At the first step, the configuration of the region coincides with the configuration determined at t = O. 2. The elastoplastic characteristics of the material for each element are determined in accordance with the stress-strain state corresponding to the end of the previous step. 3. A local stiffness matrix and force vector of the element is formed. 4. Kinematic boundary conditions are set on the contact surfaces. With an arbitrary form of the contact surface, the well-known procedure for the transition to the local coordinate system is used. 5. The global stiffness matrix of the system and the corresponding force vector are formed. 6. The system of algebraic equations is solved, the vector column of velocities of nodal displacements is determined. 7. The characteristics of the instantaneous stress-strain state are determined, the tensors of the strain rate W, the vortex C1, the rate of change of volume 0 are calculated, the curvature of the deformation path X 8 is calculated. The velocity fields of the stress and strain tensors are integrated, a new configuration of the region is determined. The type of the stress-strain state, the zone of elastic and plastic deformation is determined. 9. The achieved level of external forces is determined. 10. The control of fulfillment of the equilibrium conditions is carried out, the residual vectors are calculated. When the scheme is implemented without refining iterations, the transition to step 1 is carried out immediately.

Factors affecting the chip formation process

The process of chip formation when cutting metals is a plastic deformation, with the possible destruction of the cut layer, as a result of which the cut layer turns into chips. The process of chip formation largely determines the cutting process: the magnitude of the cutting force, the amount of heat generated, the accuracy and quality of the resulting surface, tool wear. Some factors have a direct impact on the process of chip formation, others - indirectly, through those factors that affect directly. Almost all factors indirectly influence, and this causes a whole chain of interrelated phenomena.

According to , only four factors have a direct impact on the chip formation process in rectangular cutting: the angle of action, the rake angle of the tool, cutting speed and material properties. All other factors influence indirectly. To identify these dependencies, the process of free rectangular cutting of material on a flat surface was chosen. The workpiece is divided into two parts by the line of the proposed separation GA, the top layer is the future chip, the thickness of the layer being removed is o, the remaining workpiece is thick h. Point M - the maximum point of reaching the tip of the cutter during insertion, the path traveled by the cutter - S. The width of the sample is finite and equal to b. Consider the model of the cutting process (Fig. 3.1.) Considering that at the initial moment of time the sample is undeformed, intact, without cuts. A workpiece of two surfaces connected by a very thin layer of AG, 8 .a thick, where a is the thickness of the chip being removed. AG - the proposed dividing line (Fig. 3.1.). When the cutter moves, contact occurs on two surfaces of the cutting tool. At the initial moment of time, the destruction does not occur - the introduction of the cutter without destruction. An elastic-plastic isotropic material is used as the main material. The calculations considered both ductile (the ability of a material to obtain large residual deformations without breaking) and brittle (the ability of a material to break without noticeable plastic deformation) materials. The basis was a low-speed cutting mode, in which, according to the rule, the occurrence of stagnant phenomena on the front surface is excluded. Another feature is the low heat generation during the cutting process, which does not affect the change in the physical characteristics of the material and, consequently, the cutting process and the value of the cutting forces. Thus, it becomes possible both numerically and experimentally to study the cutting process of a cutting layer not complicated by additional phenomena.

In accordance with Chapter 2, the finite element process of solving a quasi-static cutting problem is carried out by stepwise loading of the sample, in the case of cutting, by a small movement of the cutter in the direction of the sample. The problem is solved by the kinematic task of moving on the cutter, because the cutting speed is known, and the cutting force is unknown and is a determined quantity. To solve this problem, a specialized software package Wind2D, capable of solving three tasks - to provide results confirming the validity of the calculations obtained, to calculate test problems to justify the validity of the constructed model, to have the ability to design and solve a technological problem.

To solve these problems, a model of modular construction of the complex was chosen, including a common shell as a unifying element capable of managing the connection of various modules. The only deeply integrated module was the results visualization block. The remaining modules are divided into two categories: problems and mathematical models. Not the uniqueness of the mathematical model is allowed. In the original project, there are three for two different types of elements. Each task also represents a module associated with mathematical model three procedures and with a shell one module call procedure, so the integration of a new module comes down to inserting four lines into the project and recompiling. The Borland Delphi 6.0 high-level language was chosen as an implementation tool, which has everything necessary to solve the task in a limited time. In each task, it is possible to use either automatically constructed finite element meshes, or use specially prepared ones using the AnSYS 5.5.3 package and saved in a text format. All boundaries can be divided into two types: dynamic (where nodes change from step to step) and static (constant throughout the calculation). The most difficult in modeling are dynamic boundaries, if we trace the process of separation by nodes, then when the destruction criterion is reached in the node belonging to the boundary Ol, the connection between the elements to which this node belongs is broken by duplicating the node - adding a new number for the elements lying below the dividing line. One node is assigned to J- and, and the other 1 iz (Fig. 3.10). Then from 1 and the node goes to C and then to C. The node assigned to A p immediately or after several steps hits the surface of the incisor and goes to C, where it can be detached for two reasons: reaching the detachment criterion, or upon reaching point B, if a chipbreaker is defined when solving a given task. Next, the node goes to G9 if the node in front of it is already detached.

Comparison of experimentally found and calculated values of cutting forces

As mentioned earlier, the work uses a step-by-step loading method, the essence of which is to divide the entire path of the wedge advance into small segments of equal length. To increase the accuracy and speed of calculations, instead of ultra-small steps, an iterative method was used to reduce the step size required to accurately describe the contact problem when using the finite element method. Both geometric conditions for nodes and deformation conditions for finite elements are checked.

The process is based on checking all the criteria and determining the smallest step reduction factor, after which the step is recalculated and so on until it becomes K 0.99. Some of the criteria in a number of tasks may not be involved, all the criteria are described below (Fig. ZLO): 1. Prohibition of penetration of material into the body of the cutter - is achieved by checking all the nodes from i \L 9 "! 12 to the intersection of the boundary of the front cutting surface. Assuming the movement to be linear in a step, the point of contact between the surface and the node is found and the coefficient of step size reduction is determined. The step is being recalculated. 2. Elements that have passed the yield point at a given step are identified, a reduction factor for the step is determined so that only a few elements "pass" the limit. The step is being recalculated. 3. Nodes from a certain area belonging to the GA section line are detected, which exceeded the value of the destruction criterion at this step. A step reduction factor is determined so that only one node exceeds the failure criterion value. The step is being recalculated. Chapter 3. Mathematical modeling of the cutting process 4. Prohibition of penetration of material into the body of the cutter through the rear cutting surface for nodes from A 6, if this boundary is not fixed. 5. For nodes 1 8, the detachment condition and the transition to the CC at point B can be set if the condition is selected to be used in the calculation of the chipbreaker. 6. If the deformation in at least one element is exceeded by more than 25%, the step size is reduced to the limit of 25% deformation. The step is being recalculated. 7. The minimum step reduction factor is determined, and if it is less than 0.99, then the step is recalculated, otherwise the transition to the next conditions. 8. The first step is considered frictionless. After the calculation, the directions of movement of the nodes belonging to A 8 and C are found, friction is added and the step is recalculated, the direction of the friction force is stored in a separate record. If the step is calculated with friction, then it is checked whether the direction of movement of the nodes, which are affected by the friction force, has changed. If it has changed, then these nodes are rigidly fixed on the front cutting surface. The step is being recalculated. 9. If the transition to the next step is carried out, and not recalculation, then the knots approaching the front cutting surface are fixed - TRANSITION OF KNOTS FROM i 12 TO A 8 10. If the transition to the next step is carried out, and not recalculation, then for nodes belonging to 1 8, the cutting forces are calculated, and if they are negative, then the assembly is checked for the possibility of detachment, i.e. detachment is carried out only if it is the top one. 11. If the transition to the next step is carried out, and not recalculation, then the node belonging to AG is detected, which exceeded the value of the destruction criterion at this step by an acceptable (small) value. Enabling the separation mechanism: instead of one node, two nodes are created, one belonging to and, the other 1 іz; renumbering of body nodes according to a special algorithm. Go to the next step.

The final implementation of the criteria (1-11) differs both in complexity and in the probability of their occurrence and the real contribution to improving the calculation results. Criterion (1) often occurs when using a small number of steps in the calculation, and very rarely when a large number of steps are used at the same depth of cut. However, this criterion does not allow nodes to "fall through" into the incisor, leading to incorrect results. According to the criterion (9), nodes are fixed at the stage of transition to the next step, and not with several recalculations.

The implementation of criterion (2) consists in comparing the old and new stress intensity values for all elements and determining the element with the maximum intensity value. This criterion makes it possible to increase the step size and thereby not only increase the calculation speed, but also reduce the error resulting from the mass transition of elements from the elastic zone to the plastic one. Similarly with criterion (4).

To study a clean cutting process, without the influence of a sharp increase in temperature on the interaction surface and in the sample, in which a continuous chip is formed, without the formation of build-up on the cutting surface, a cutting speed of the order of 0.33 mm / s is required. Taking this speed as the maximum, we get that to advance the cutter by 1 mm, it is necessary to calculate 30 steps (assuming a time interval of 0.1 - which provides the best stability of the process). When calculating, using a test model, with the introduction of a cutter by 1 mm, taking into account the use of the previously described criteria and without taking into account friction, instead of 30 steps, 190 were obtained. This is due to a decrease in the value of the advance step. However, due to the fact that the process is iterative, 419 steps were actually calculated. This discrepancy is caused by a too large step size, which leads to a multiple decrease in the step size due to the iterative nature of the criteria. So. with an initial increase in the number of steps to 100 instead of 30, the calculated number of steps is 344. A further increase in the number to 150 leads to an increase in the number of calculated steps to 390, and hence an increase in the calculation time. Based on this, it can be assumed that the optimal number of steps, when modeling the chip removal process, is 100 steps per 1 mm of infeed, with an uneven grid partition with a number of elements of 600-1200. At the same time, the real number of steps, without taking into account friction, will be at least 340 per 1 mm, and taking into account friction, at least 600 steps.

Solid Body Mechanics<3 2008

NUMERICAL SIMULATION OF CUTTING PROCESSES OF ELASTIC-VISCO-PLASTIC MATERIALS IN A THREE-DIMENSIONAL STATEMENT

In this paper, a three-dimensional simulation of the unsteady process of cutting an elastic-viscous-plastic plate (workpiece) by an absolutely rigid cutter moving at a constant speed V0 at various inclinations of the cutter face a (Fig. 1) was carried out using the finite element method. The modeling was carried out on the basis of a coupled thermomechanical model of an elastic-viscous-plastic material. A comparison is made between the adiabatic cutting process and the mode, taking into account the thermal conductivity of the workpiece material. A parametric study of the cutting process was carried out with a change in the geometry of the workpiece and cutting tool, speed and depth of cut, as well as the properties of the material being processed. The size of the workpiece thickness in the direction of the z axis was varied. The stress state changed from plane stressed H = H/L< 1 (тонкая пластина) до плоскодеформируе-мого H >1 (wide plate), where H is the thickness, L is the length of the workpiece. The problem was solved on a moving adaptive Lagrangian-Eulerian grid by the finite element method with splitting and using explicit-implicit schemes for integrating equations. It is shown that numerical simulation of the problem in a three-dimensional formulation makes it possible to study cutting processes with the formation of a continuous chip, as well as with the destruction of the chip into separate pieces. The mechanism of this phenomenon in the case of orthogonal cutting (a = 0) can be explained by thermal softening with the formation of adiabatic shear bands without involving damage models. When cutting with a sharper cutter (angle a is large), it is necessary to use a coupled model of thermal and structural softening. Dependences of the force acting on the cutter are obtained for different geometric and physical parameters of the problem. It is shown that quasi-monotonous and oscillating regimes are possible and their physical explanation is given.

1. Introduction. Cutting processes play an important role in the processing of difficult-to-deform materials on turning and milling machines. Machining is the main price-forming operation in the manufacture of complex profile parts from hard-to-deform materials, such as titanium-aluminum and molybdenum alloys. When they are cut, chips are formed, which can break down into separate pieces (chips), which leads to an uneven surface of the cut material and highly uneven pressure on the cutter. Experimental determination of the parameters of the temperature and stress-strain states of the material being processed during high-speed cutting is extremely difficult. An alternative is the numerical simulation of the process, which makes it possible to explain the main features of the process and study the cutting mechanism in detail. A fundamental understanding of the mechanism of chip formation and breakage is essential for efficient cutting. Mathematics

The mechanical modeling of the cutting process requires taking into account large deformations, strain rates and heating due to the dissipation of plastic deformation, leading to thermal softening and destruction of the material.

The exact solution of these processes has not yet been obtained, although research has been undertaken since the middle of the 20th century. The first works were based on the simplest rigid-plastic calculation scheme. However, the results obtained on the basis of rigid-plastic analysis could not satisfy either material processors or theorists, since this model did not provide answers to the questions posed. In the literature, there is no solution to this problem in a spatial formulation, taking into account the nonlinear effects of the formation, destruction, and fragmentation of chips during thermomechanical softening of the material.

In the last few years, thanks to numerical simulations, certain advances have been made in the study of these processes. The influence of the cutting angle, thermomechanical properties of the workpiece and cutter, and the fracture mechanism on the formation and destruction of chips were studied. However, in most works, the cutting process was considered under significant restrictions: a two-dimensional formulation of the problem (plane deformation) was adopted; the influence of the initial stage of the unsteady process on the force acting on the cutter was not considered; destruction was assumed to occur according to a predetermined interface. All these limitations did not allow to study cutting in full, and in some cases led to a misunderstanding of the mechanism of the process itself.

Moreover, experimental studies show recent years, at high strain rates е > 105–106 s–1, many materials exhibit an anomalous temperature dependence associated with a rearrangement of the dislocation motion mechanism. The thermal fluctuation mechanism is replaced by the phonon resistance mechanism, as a result of which the dependence of the material resistance on temperature becomes directly opposite: with increasing temperature, the strengthening of the material increases. Such effects can lead to great trouble in high speed cutting. These problems have not been studied in the literature so far. Simulation of a high-speed process requires the development of models that take into account the complex dependences of the viscoplastic behavior of materials and, first of all, take into account damage and destruction with the formation of cracks and fragmentation of particles and pieces of a deformable material. To take into account all the

8 Solid State Mechanics, No. 3

effects, not only complex thermophysical models are required, but also modern computational methods that make it possible to calculate large deformations that do not allow limiting mesh distortions and take into account the destruction and the appearance of discontinuity in the material. The problems under consideration require a huge amount of computation. It is necessary to develop high-speed algorithms for solving elastoviscoplastic equations with internal variables.

2. Statement of the problem. 2.1. Geometry. A three-dimensional statement of the problem is accepted. In FIG. 1 shows the area and boundary conditions in the cutting plane. In the direction perpendicular to the plane, the workpiece has a finite thickness H = H/L (L is the length of the workpiece), which varied over a wide range. Spatial setting allows freedom of movement of the workpiece material from the cutting plane and smoother chip exit, which provides more favorable cutting conditions.

2.2 Basic equations. The complete coupled system of thermoelasticity-viscoplasticity equations consists of the momentum conservation equation

piu/ir = ; (2.1)

Hooke's law with temperature stresses

(2.2) heat influx equation dj

pSe d- \u003d K 0, .. - (3 X + 2c) a0 ° e „■ + ko; p (2.3)

where Ce is the heat capacity, K is the thermal conductivity coefficient, k is the Queenie-Taylor coefficient, which takes into account the heating of the material due to plastic dissipation.

We also have the associated plastic flow law

ep = xi^/yo; (2.4)

and plasticity conditions

A, EE, X;, 9) = Oy (]EE, X;, 0)< 0 (2.5)

where λ] are stress tensor invariants, E; - plastic strain tensor. The evolution equations for internal variables have the form

dX / yz = yLk, Xk, 9) (2.6)

2.3 Material model. In this work, a Mises-type thermoelastic-viscoplastic model is adopted - a plasticity model with a yield strength in the form of a multiplicative dependence (2.7), including deformation and viscoplastic hardening and thermal softening:

oy(ep, ¿*,9) = [a + b(ep)"]

where oy is the yield strength, ep1 is the intensity of plastic deformations, 0 is the relative temperature referred to the melting point 0m: "0<0*

(0 - 0*) / (0m - 0*), 0*<0<0т

The material of the part is assumed to be homogeneous. The relatively soft material A12024-T3 was used in the calculations (elastic constants: E = 73 GPa, V = 0.33; plastic constants: A = 369 MPa, B = 684 MPa, n = 0.73, e0 = 5.77 × 10-4, C = 0.0083, m = 1.7; ■ 10-4, C = 0.008, m = 1.46, 9* = 300 K, 9m = 600 K, v = 0.9). The adiabatic cutting process is compared with the solution of the complete thermomechanical problem.

2.4. Destruction. The material fracture model is based on the Minchen-Sack continuum approach, based on the modeling of fracture zones by discrete particles. The critical value is taken as the failure criterion

plastic strain intensity ep:

ep = [dx + d2exp (d311/12)][ 1 + d41n (dp/d0)](1 + d59) (2.8)

where i. - constants of the material, determined from the experiment.

If the failure criterion is satisfied in the Lagrangian cell, then the bonds between the nodes in such cells are released and the stresses either relax to zero, or the resistance is preserved only with respect to compression. Lagrangian nodal masses upon destruction turn into independent particles that carry away mass, momentum and energy, move as a rigid whole and do not interact with undestroyed particles. A detailed overview of these algorithms is given in. In the present work, the fracture is determined by the achievement of the critical intensity of plastic deformation ep, and the fracture surface is not predetermined. In the above calculations

e p = 1.0, the speed of the cutter was taken equal to 2 m/s and 20 m/s.

2.5. Equation integration method. To integrate the reduced coupled system of thermoplasticity equations (2.1)-(2.8), it is advisable to apply the splitting method developed in . The splitting scheme of elastic-plastic equations consists in splitting the complete process into a predictor - a thermoelastic process, in

where ep = 0 and all operators associated with plastic deformation vanish, and the corrector - at which the total strain rate е = 0. At the predictor stage, system (2.1)-(2.6) with respect to the variables denoted by a tilde will take the form

ryb/yz = a]

y aL \u003d "- a§"9) pSei9 / yg \u003d K.9ts - (3X + 2ts) a90eu

For further reading of the article, you must purchase the full text. Articles are sent in the format

V. K. Astashev, A. V. Razinkin - 2008

"MECHANICS UDC: 539.3 A.N. Shipachev, S.A. Zelepugin NUMERICAL SIMULATION OF PROCESSES OF HIGH-SPEED ORTHOGONAL...»

BULLETIN OF TOMSK STATE UNIVERSITY

2009 Mathematics and Mechanics No. 2(6)

MECHANICS

A.N. Shipachev, S.A. Zelepugin

NUMERICAL SIMULATION OF PROCESSES

FOR HIGH SPEED ORTHOGONAL CUTTING OF METALS1

The processes of high-speed orthogonal cutting of metals by the finite element method within the framework of an elastic-plastic model of the medium in the cutting speed range of 1–200 m/s are numerically studied. As a criterion for chip separation, we used limit value specific energy of shear deformations. The necessity of using an additional chip formation criterion is revealed, which is proposed limit value specific volume of microdamages.

Key words: high-speed cutting, numerical simulation, finite element method.

From a physical point of view, the process of cutting materials is a process of intense plastic deformation and destruction, accompanied by chip friction on the front surface of the cutter and friction of the back surface of the tool on the cutting surface, occurring under conditions of high pressures and sliding speeds. The mechanical energy expended in this process is converted into thermal energy, which in turn has a great influence on the patterns of deformation of the cut layer, cutting forces, wear and tool life.

The products of modern mechanical engineering are characterized by the use of high-strength and hard-to-cut materials, a sharp increase in the requirements for accuracy and quality of products, and a significant complication of the structural forms of machine parts obtained by cutting. Therefore, the machining process requires constant improvement. Currently, one of the most promising areas for such improvement is high-speed processing.

In the scientific literature, theoretical and experimental studies of the processes of high-speed cutting of materials are presented extremely insufficiently. There are separate examples of experimental and theoretical studies of the effect of temperature on the strength characteristics of a material in the process of high-speed cutting. In theoretical terms, the problem of cutting materials has received the greatest development in the creation of a number of analytical models of orthogonal cutting. However, the complexity of the problem and the need for a more complete account of the properties of materials, thermal and inertial effects led to the work. 08-99059), Ministry of Education and Science of the Russian Federation within the framework of the AVCP "Development of the scientific potential of higher education" (project 2.1.1/5993).

110 A.N. Shipachev, S.A. Zelepugin the use of numerical methods, of which, in relation to the problem under consideration, the finite element method is most widely used.

– – –

is calculated using the equation of state of the Mie – Grüneisen type, in which the coefficients are selected on the basis of the constants a and b of the Hugoniot shock adiabat.

The constitutive relations connect the components of the stress deviator and strain rate tensor and use the Jaumann derivative. The Mises condition is used to describe plastic flow. The dependences of the strength characteristics of the medium (shear modulus G and dynamic yield strength) on temperature and the level of damage to the material are taken into account.

The simulation of the process of chip separation from the workpiece was carried out using the criterion for the destruction of the design elements of the workpiece, while using an approach similar to simulation modeling of the destruction of an erosion-type material. The limiting value of the specific energy of shear deformations Esh was used as a fracture criterion—chip separation criterion.

The current value of this energy is calculated using the formula:

D Esh = Sij ij (5) dt The critical value of the specific shear strain energy depends on the interaction conditions and is given by the function initial speed stroke:

c Esh = ash + bsh 0, (6) c where ash, bsh are material constants. When Esh Esh is in the calculation cell, this cell is considered destroyed and removed from further calculation, and the parameters of neighboring cells are adjusted taking into account conservation laws. The correction consists in removing the mass of the destroyed element from the masses of the nodes that belonged to this element. If at the same time the mass of any calculation node becomes zero, then this node is considered destroyed and is also removed from further calculation.

Calculation results Calculations were carried out for cutting speeds from 1 to 200 m/s. The dimensions of the working part of the tool: the length of the upper edge is 1.25 mm, the side is 3.5 mm, the front angle is 6°, the back angle is 6°. The steel plate being machined had a thickness of 5 mm, a length of 50 mm, and a cutting depth of 1 mm. The workpiece material is St3 steel, the material of the working part of the tool is a dense modification of boron nitride.

The following values of the workpiece material constants were used: 0 = 7850 kg/m3, a = 4400 m/s, b = 1.55, G0 = 79 GPa, 0 = 1.01 GPa, V1 = 9.2 10–6 m3/kg, V2 = 5.7 10–7 m3/kg, Kf = 0.54 m s/kg, Pk = –1.5 GPa, ash = 7 104 J/kg, bsh = 1.6 103 m/s. The material of the working part of the tool is characterized by constants 0 = 3400 kg/m3, K1 = 410 GPa, K2 = K3 = 0, 0 = 0, G0 = 330 GPa, where K1, K2, K3 are the constants of the equation of state in the form of Mie – Gruneisen.

The results of the calculation of the process of chip formation during the movement of the cutter at a speed of 10 m/s are shown in fig. 1. It follows from the calculations that the cutting process is accompanied by severe plastic deformation of the workpiece in the vicinity of the cutter tip, which, during the formation of chips, leads to a strong distortion of the original shape of the design elements located along the cutting line. In this work, linear triangular elements are used, which, with the necessary small time step used in the calculations, ensure the stability of the calculation with their significant deformation,

– – –

Rice. Fig. 1. The shape of the chip, workpiece and working part of the cutting tool at times of 1.9 ms (a) and 3.8 ms (b) when the cutter moves at a speed of 10 m/s Numerical simulation of high-speed orthogonal cutting 113 up to the fulfillment of the separation criterion shavings. At cutting speeds of 10 m/s and lower, areas appear in the sample where the criterion for chip separation does not work in time (Fig. 1, a), which indicates the need to apply either an additional criterion, or replace the used criterion with a new one.

Additionally, the need to adjust the chip formation criterion is indicated by the shape of the chip surface.

On fig. 2 shows the fields of temperature (in K) and specific shear energy (in kJ/kg) at a cutting speed of 25 m/s at a time of 1.4 ms after the start of cutting. Calculations show that the temperature field is almost identical to the field of specific shear strain energy, which indicates that a 1520

– – –

Rice. Figure 3. Fields of the specific volume of microdamages (in cm3/g) at a time of 1.4 ms when the cutter moves at a speed of 25 m/s. environments in the range of cutting speeds 1 – 200 m/s.

Based on the results of the calculations, it was found that the nature of the distribution of lines of the specific energy level of shear deformations and temperatures at ultrahigh cutting speeds is the same as at cutting speeds of the order of 1 m/s, and qualitative differences in the mode can arise due to the melting of the workpiece material, which occurs only in a narrow layer in contact with the tool, and also due to the degradation of the strength properties of the material of the working part of the tool.

A process parameter has been identified - the specific volume of microdamages - the limiting value of which can be used as an additional or independent criterion for chip formation.

LITERATURE

1. Petrushin S.I. Optimal design of the working part of cutting tools // Tomsk: Tom. Polytechnic University, 2008. 195 p.

2. Sutter G., Ranc N. Temperature fields in a chip during high-speed orthogonal cutting – An experimental investigation // Int. J. Machine Tools & Manufacture. 2007 No. 47. P. 1507 - 1517.

3. Miguelez H., Zaera R., Rusinek A., Moufki A. and Molinari A. Numerical modeling of orthogonal cutting: Influence of cutting conditions and separation criterion, J. Phys. 2006.V.IV. no. 134.

4. Hortig C., Svendsen B. Simulation of chip formation during high-speed cutting // J. Materials Processing Technology. 2007 No. 186. P. 66 - 76.

5. Campbell C.E., Bendersky L.A., Boettinger W.J., Ivester R. Microstructural characterization of AlT651 chips and work pieces produced by high-speed machining // Materials Science and Engineering A. 2006. No. 430. P. 15 - 26.

6. Zelepugin S.A., Konyaev A.A., Sidorov V.N. and others. Experimental and theoretical study of the collision of a group of particles with protection elements of spacecraft // space research. 2008. V. 46. No. 6. S. 559 – 570.

7. Zelepugin S.A., Zelepugin A.S. Modeling the destruction of obstacles during high-speed impact of a group of bodies // Chemical Physics. 2008. V. 27. No. 3. S. 71 – 76.

8. Ivanova O.V., Zelepugin S.A. The condition of joint deformation of the mixture components during shock-wave compaction // Bulletin of TSU. Mathematics and mechanics. 2009. No. 1(5).

9. Kanel G.I., Razorenov S.V., Utkin A.V., Fortov V.E. Studies of the mechanical properties of materials under shock-wave loading // Izvestiya RAN. MTT. 1999. No. 5. S. 173 - 188.

10. Zelepugin S.A., Shpakov S.S. Destruction of a two-layer barrier boron carbide - titanium alloy at high-speed impact // Izv. universities. Physics. 2008. No. 8/2. pp. 166 - 173.

11. Gorelsky V.A., Zelepugin S.A. Application of the finite element method for the study of orthogonal cutting of metals with an STM tool, taking into account destruction and temperature effects // Superhard Materials. 1995. No. 5. S. 33 - 38.

INFORMATION ABOUT AUTHORS:

SHIPACHEV Alexander Nikolaevich – post-graduate student of the Faculty of Physics and Technology of Tomsk state university. Email: [email protected] ZELEPUGIN Sergey Alekseevich – Doctor of Physical and Mathematical Sciences, Professor of the Department of Deformable Solid Mechanics of the Faculty of Physics and Technology of Tomsk State University, Senior Researcher of the Department of Structural Macrokinetics of the Tomsk Scientific Center of the Siberian Branch of the Russian Academy of Sciences. Email: [email protected], [email protected] The article was accepted for publication on May 19, 2009.

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BULLETIN OF TOMSK STATE UNIVERSITY Mathematics and mechanics

MECHANICS

A.N. Shipachev, S.A. Zelepugin

NUMERICAL SIMULATION OF HIGH-SPEED ORTHOGONAL CUTTING OF METALS1

The processes of high-speed orthogonal cutting of metals by the finite element method are numerically studied within the framework of an elastic-plastic model of the medium in the cutting speed range of 1 - 200 m/s. The limiting value of the specific energy of shear deformations was used as a criterion for chip separation. The necessity of using an additional criterion for chip formation is revealed, as which the limiting value of the specific volume of microdamages is proposed.

Key words: high-speed cutting, numerical simulation, finite element method.

1 The work was financially supported by the Russian Foundation for Basic Research (projects 07-08-00037, 08-08-12055), the Russian Foundation for Basic Research and the Administration of the Tomsk Region (project 09-08-99059), the Ministry of Education and Science of the Russian Federation within the framework of the AVCP "Development of the scientific potential of higher education "(project 2.1.1/5993).

the use of numerical methods, of which, in relation to the problem under consideration, the finite element method is most widely used.

In this work, the processes of high-speed cutting of metals are studied numerically by the finite element method in a two-dimensional plane-strain formulation within the framework of an elastic-plastic model of a medium.

In numerical calculations, a model of a damaged medium is used, which is characterized by the possibility of nucleation and development of cracks in it. The total volume of the medium W consists of its undamaged part, which occupies the volume Wc and is characterized by the density pc, as well as the cracks occupying the volume W/, in which the density is assumed to be zero. The average density of the medium is related to the introduced parameters by the relation p = pc (Ws /W). The degree of damage to the medium is characterized by the specific volume of cracks V/ = W//(W p).

The system of equations describing the non-stationary adiabatic (both with elastic and plastic deformation) motion of a compressible medium consists of the equations of continuity, motion, energy:

where p - density, r - time, u - velocity vector with components u, cmy = - (P + Q)5jj + Bu - components of the stress tensor, E - specific internal energy, - components of the strain rate tensor, P = Pc (p /pc) - average pressure, Pc - pressure in the solid component (intact part) of the substance, 2 - artificial viscosity, Bu - components of the stress deviator.

Modeling of "tear-off" fractures is carried out using a kinetic model of active type fracture:

When creating the model, it was assumed that the material contains potential fracture sites with an effective specific volume V:, on which cracks (or pores) form and grow when the tensile pressure Pc exceeds a certain critical value P = P)U\/(U\ + V/ ), which decreases with the growth of the formed microdamages. The constants VI, V2, Pk, K/ were selected by comparing the results of calculations and experiments on registering the speed of the rear surface when the sample was loaded with plane compression pulses. The same set of material constants is used to calculate both the growth and collapse of cracks or pores, depending on the sign of Pc.

The pressure in an undamaged substance is considered to be a function of the specific volume and specific internal energy, and over the entire range of loading conditions,

Formulation of the problem

Shu(ri) = 0 ;

0 if |Рс |< Р* или (Рс >P* and Y^ = 0),

^=| - n§n (Ps) k7 (Ps | - P *) (Y2 + Y7),

if Rs< -Р* или (Рс >P* and Y^ > 0).

It is calculated using the equation of state of the Mie - Gruneisen type, in which the coefficients are selected on the basis of the constants a and b of the Hugoniot shock adiabat.

The constitutive relations connect the components of the stress deviator and strain rate tensor and use the Jaumann derivative. The Mises condition is used to describe plastic flow. The dependences of the strength characteristics of the medium (shear modulus G and dynamic yield strength o) on temperature and the level of damage to the material are taken into account.

The critical value of the specific energy of shear deformations depends on the interaction conditions and is given by the function of the initial impact velocity:

Esh = ash + bsh U0 , (6)

where ash, bsh are material constants. When Esh > Esch in a computational cell, this cell is considered destroyed and is removed from further calculation, and the parameters of neighboring cells are corrected taking into account conservation laws. The correction consists in removing the mass of the destroyed element from the masses of the nodes that belonged to this element. If at the same time the mass of any calculated node becomes

turns zero, then this node is considered destroyed and is also removed from further calculation.

Calculation results

The calculations were carried out for cutting speeds from 1 to 200 m/s. The dimensions of the working part of the tool: the length of the upper edge is 1.25 mm, the side is 3.5 mm, the front angle is 6°, the back angle is 6°. The steel plate being processed had a thickness of 5 mm, a length of 50 mm, and a cutting depth of 1 mm. The workpiece material is St3 steel, the material of the working part of the tool is a dense modification of boron nitride. The following values of the constants of the workpiece material were used: p0 = 7850 kg/m3, a = 4400 m/s, b = 1.55, G0 = 79 GPa, o0 = 1.01 GPa, V = 9.2-10"6 m3/kg, V2 = 5.7-10-7 m3/kg, K= 0.54 m-s/kg, Pk = -1.5 GPa, ash = 7-104 J/kg, bsh = 1.6 -10 m/s The material of the working part of the tool is characterized by constants p0 = 3400 kg/m3, K1 = 410 GPa, K2 = K3 = 0, y0 = 0, G0 = 330 GPa, where K1, K2, K3 are the constants of the equation of state in Mi-Gruneisen form.

Rice. Fig. 1. The shape of the chip, workpiece and working part of the cutting tool at the times of 1.9 ms (a) and 3.8 ms (b) when the cutter moves at a speed of 10 m/s

up to the fulfillment of the chip separation criterion. At cutting speeds of 10 m/s and lower, areas appear in the sample where the criterion for chip separation does not work in time (Fig. 1, a), which indicates the need to apply either an additional criterion, or replace the used criterion with a new one. Additionally, the need to adjust the chip formation criterion is indicated by the shape of the chip surface.

Rice. Fig. 2. Fields and isolines of temperature (a) and specific energy of shear deformations (b) at a time of 1.4 ms when the cutter moves at a speed of 25 m/s

temperature regime in high-speed cutting is determined mainly by plastic deformation of the workpiece material. In this case, the maximum temperatures in the chip do not exceed 740 K, in the workpiece -640 K. In the process of cutting, significantly higher temperatures arise in the cutter (Fig. 2, a), which can lead to degradation of its strength properties.

The calculation results presented in Figs. 3 show that gradient changes in the specific volume of microdamages in front of the cutter are much more pronounced than changes in the energy of shear deformations or temperature, therefore, in calculations, the limiting value of the specific volume of microdamages can be used (independently or additionally) as a chip separation criterion.

0,1201 0,1101 0,1001 0,0901 0,0801 0,0701 0,0601 0,0501 0,0401 0,0301 0,0201 0,0101

Rice. Fig. 3. Fields of the specific volume of microdamages (in cm/g) at a time of 1.4 ms when the cutter moves at a speed of 25 m/s

Conclusion

A process parameter was identified - the specific volume of microdamages - the limiting value of which can be used as an additional or independent criterion for chip formation.

LITERATURE

1. Petrushin S.I. Optimal design of the working part of cutting tools // Tomsk: Tom. Polytechnic University, 2008. 195 p.

2. Sutter G., Ranc N. Temperature fields in a chip during high-speed orthogonal cutting - An experimental investigation // Int. J. Machine Tools & Manufacture. 2007 No. 47. P. 1507 - 1517.

3. Miguelez H., Zaera R., Rusinek A., Moufki A. and Molinari A. Numerical modeling of orthogonal cutting: Influence of cutting conditions and separation criterion, J. Phys. 2006.V.IV. no. 134. P. 417-422.

4. Hortig C., Svendsen B. Simulation of chip formation during high-speed cutting // J. Materials Processing Technology. 2007 No. 186. P. 66 - 76.

5. Campbell C.E., Bendersky L.A., Boettinger W.J., Ivester R. Microstructural characterization of Al-7075-T651 chips and work pieces produced by high-speed machining // Materials Science and Engineering A. 2006. No. 430. P. 15 - 26.

6. Zelepugin S.A., Konyaev A.A., Sidorov V.N. et al. Experimental and theoretical study of the collision of a group of particles with protection elements of spacecraft // Space Research. 2008. V. 46. No. 6. S. 559 - 570.

7. Zelepugin S.A., Zelepugin A.S. Modeling the destruction of obstacles during high-speed impact of a group of bodies // Chemical Physics. 2008. V. 27. No. 3. S. 71 - 76.

8. Ivanova O.V., Zelepugin S.A. The condition of joint deformation of the mixture components during shock-wave compaction // Bulletin of TSU. Mathematics and mechanics. 2009. No. 1(5). pp. 54 - 61.

9. Kanel G.I., Razorenov S.V., Utkin A.V., Fortov V.E. Studies of the mechanical properties of materials under shock-wave loading // Izvestiya RAN. MTT. 1999. No. 5. S. 173 - 188.

10. Zelepugin, S.A. and Shpakov, S.S., Destruction of a Two-Layer Boron Carbide-Titanium Alloy Barrier under High-Speed Impact, Izv. universities. Physics. 2008. No. 8/2. pp. 166 - 173.

SHIPACHEV Alexander Nikolaevich - post-graduate student of the Faculty of Physics and Technology of Tomsk State University. Email: [email protected]

ZELEPUGIN Sergey Alekseevich - Doctor of Physical and Mathematical Sciences, Professor of the Department of Deformable Solid Mechanics of the Faculty of Physics and Technology of Tomsk State University, Senior Researcher of the Department of Structural Macrokinetics of the Tomsk Scientific Center of the Siberian Branch of the Russian Academy of Sciences. Email: [email protected], [email protected]

V 0 z. H/L 1 (wide plate), where H- thickness, L- workpiece length. The problem was solved on a moving adaptive Lagrangian-Eulerian grid by the finite element method with splitting and using explicit-implicit schemes for integrating equations ...

In this paper, a three-dimensional simulation of the unsteady process of cutting an elastic-viscous-plastic plate (workpiece) by an absolutely rigid cutter moving at a constant speed was carried out using the finite element method. V 0 at various inclinations of the edge of the cutter a (Fig. 1). The modeling was carried out on the basis of a coupled thermomechanical model of an elastic-viscous-plastic material. A comparison is made between the adiabatic cutting process and the mode, taking into account the thermal conductivity of the workpiece material. A parametric study of the cutting process was carried out with a change in the geometry of the workpiece and cutting tool, speed and depth of cut, as well as the properties of the material being processed. The size of the workpiece thickness in the direction of the axis was varied z. The stressed state changed from plane stress R = H/L 1 (wide plate), where H- thickness, L- workpiece length. The problem was solved on a moving adaptive Lagrangian-Eulerian grid by the finite element method with splitting and using explicit-implicit schemes for integrating equations. It is shown that numerical simulation of the problem in a three-dimensional formulation makes it possible to study cutting processes with the formation of a continuous chip, as well as with the destruction of the chip into separate pieces. The mechanism of this phenomenon in the case of orthogonal cutting (a = 0) can be explained by thermal softening with the formation of adiabatic shear bands without involving damage models. When cutting with a sharper cutter (angle a is large), it is necessary to use a coupled model of thermal and structural softening. Dependences of the force acting on the cutter are obtained for different geometric and physical parameters of the problem. It is shown that quasi-monotonous and oscillating regimes are possible and their physical explanation is given.

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Constitutive Relationships for the Processes of Elastic-Plastic Finite Deformation

Construction of a model for the introduction of a rigid wedge into a semi-infinite elastic-plastic body

Factors affecting the chip formation process

Comparison of experimentally found and calculated values ​​of cutting forces

"MECHANICS UDC: 539.3 A.N. Shipachev, S.A. Zelepugin NUMERICAL SIMULATION OF PROCESSES OF HIGH-SPEED ORTHOGONAL...»

BULLETIN OF TOMSK STATE UNIVERSITY

MECHANICS

NUMERICAL SIMULATION OF PROCESSES

FOR HIGH SPEED ORTHOGONAL CUTTING OF METALS1

LITERATURE

INFORMATION ABOUT AUTHORS:

Comparison of experimentally found and calculated values of cutting forces