Introduction

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

1.1. Kinematics of processes 25

1.2. Constitutive relations of elastoplastic finite deformation processes 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 finite 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 correct implementation of the mathematical model 54

2.5. Analysis of model behavior under small deformations 57

2.6. Modeling the finite element process of material separation 58

2.7. Construction of a model for introducing a rigid wedge into a semi-infinite elastic-plastic body 60

2.8. Mechanism for taking friction into account in the cutting model 62

Chapter 3. Mathematical modeling of the cutting process . 65

3.1. Free cutting process 65

3.2. Factors influencing the chip formation process 68

3.3. Boundary conditions during modeling 70

3.4. Finite element implementation of the cutting process 74

3.5. Simulation of steady cutting conditions 75

3.6. Iterative process at step 77

3.7. Justification for choosing 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 the work

destruction of metal under such extreme conditions that are usually not encountered either when testing materials or in other technological processes. The cutting process can be studied using idealized physical models using mathematical analysis. Before starting to analyze 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 scheme cutting is a rectangular (orthogonal) cutting, when the cutting edge is perpendicular to the cutting speed vector, and an oblique cutting scheme, when a certain angle of inclination of the cutting edge is specified

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 into both 4 and 3 types. According to this, 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 a build-up on the tool.

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. Thieme, when cutting structural materials under any conditions, four types of chips are formed: elemental, joint, drain and fracture. Elemental, joint and joint chips are called shear chips because their formation is associated with shear stresses. Fracture chips are sometimes called pull-out chips because their formation is associated with tensile stresses. Appearance All listed types of chips are shown in Fig. 3.

Rice. 3. Types of chips according to Thieme’s classification.

Figure 3a shows the formation of elemental chips, consisting of individual “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, during the cutting process, periodic destruction of the cut layer occurs.

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

In the figure Sv is the formation of drain chips. The main feature is its continuity (continuity). If there are no obstacles in the path of the drain chips, then they flow down in a continuous ribbon, curling into a flat or helical spiral until part of the chips breaks off under the influence of 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 surface is polished as a result of friction against the front surface of the tool. Its opposite surface 2 is called the free surface (side) of the chips. It is covered with small notches and has a velvety appearance at high cutting speeds. The chips come into contact with the front surface of the tool within the contact area, the width of which is designated 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 layer being cut.

In Figure 3g - the formation of fracture chips, consisting of individual, unconnected pieces of various shapes and sizes. The formation of fracture chips is accompanied by fine metal dust. Fracture surface tp may be located below the cutting surface, as a result of which the latter is covered with traces of pieces of chips broken out of it.

Introduction 9

According to what is stated in, the type of chip largely depends on the type and mechanical properties of the material being processed. When cutting plastic materials the formation of the first three types of chips is possible: elemental, jointed and draining. As the hardness and strength of the material being processed increases, the drain chips become jointed and then elemental. When processing brittle materials, either elemental chips or, less commonly, fracture chips are formed. As the hardness of a material, such as cast iron, increases, elemental chips turn into fracture chips.

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

The type of chip is influenced by the feed (thickness of the cut layer) and cutting speed. The depth of cut (width of the cut layer) has virtually no effect on the type of chip. An increase in feed (thickness of the cut layer) leads, when cutting plastic materials, to a consistent transition from continuous chips to jointed and elemental ones. When cutting brittle materials with increasing feed, elemental chips turn into fracture chips.

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

Introduction 10

growth, as the cutting speed increases, the chips change from elemental to jointed and then confluent. However, when processing some heat-resistant steels and alloys, titanium alloys, increasing the cutting speed, on the contrary, turns the drain chips into elemental ones. The physical cause of this phenomenon has not yet been fully elucidated. An increase in cutting speed when processing brittle materials is accompanied by the transition of fracture chips into elemental chips with a decrease in the size of individual elements and strengthening of the bond between them.

Given the geometric parameters of tools and cutting modes used in production, the main types of chips when cutting plastic materials are often drain chips and less commonly joint 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 sufficiently studied. The reason is the complexity in the mathematical description of both the process of large elastoplastic deformations and the process of material separation.

The shape and type of cutter in production depends primarily on the area of ​​application: on lathes, rotary, turret, planing and slotting machines, automatic and semi-automatic lathes and special machines. Cutters used in modern mechanical engineering are classified by design (solid, composite, prefabricated, holder, adjustable), by type of processing (through, scoring, cutting, boring, shaped, threaded), by the nature of processing (roughing, finishing, for fine turning), by installation relative to the part (radial, tangential, right, left), by the cross-sectional shape of the rod (rectangular, square, round), by material

Introduction

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

The relative position of the working part and the body is different for different types 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 supporting plane of the body, for boring cutters with a round body - along the axis of the body or below it. The body of the cutting tools in the cutting zone has a slightly higher height to increase strength and rigidity.

Many cutter designs as a whole and their individual structural elements have been standardized. To unify the designs and connecting dimensions of the tool holders, the following series of rod sections, mm, were 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 (aspect ratio H:H=1.6 is used for semi-finishing and finishing, and H:H=1.25 for roughing).

The All-Russian product classifier provides for 8 subgroups of cutters with 39 types in them. About 60 standards have been published on the design of cutters and technical specifications. In addition, 150 standard sizes of high-speed steel plates for all types of cutters, about 500 standard sizes of carbide brazed inserts, 32 types of multifaceted non-regrindable 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.

Basic geometric parameters of the cutter, taking into account the above.

Purpose of the back angle A- reduce friction of the rear surface on the workpiece and ensure unhindered movement of the cutter along the surface being processed.

Introduction12

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

As the clearance angle increases, the sharpening angle decreases and thereby the strength of the blade decreases, the roughness of the machined surface increases, and heat dissipation into the cutter body worsens.

As the clearance angle decreases, friction against the machined surface increases, which leads to an increase in cutting forces, cutter wear 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 must 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 layer being cut.

The reference books provide average values ​​of the optimal angles, A confirmed by the results of industrial tests. Recommended values ​​of the back angles of the incisors are given in Table 1.

Introduction13

Purpose of the front angle U- reduce the deformation of the cut layer and facilitate chip flow.

The influence of the rake angle on cutting conditions: increasing the angle at facilitates the cutting process, reducing cutting forces. However, in this case, the strength of the cutting wedge decreases and heat dissipation into the cutter body deteriorates. Decrease angle U increases the durability of incisors, including dimensional stability.

Rice. 6. Shape of the front surface of the incisors: a - flat with a chamfer; b - curved with chamfer

The size of the rake angle and the shape of the rake 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 curved (with or without chamfers) forms of the front surface are used (Fig. 1.16).

The flat rake surface is used for cutters of all types of tool materials, while the blade is sharpened with a hardening chamfer for

angle UV-^~5 - for high speed steel cutters and Uf =-5..-25 . for cutters made of carbide alloys, all types of ceramics and synthetic superhard materials.

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

Introduction

To be used with a flat rake surface, no 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 produce chips that break (cast iron, bronze). With a small cut thickness, comparable to the radius of rounding of the cutting edge, the value of the rake angle has virtually no effect on the cutting process, since the deformation of the cut layer and its transformation into chips is carried out by the rounded edge of the radius. In this case, rake angles for all types of tool materials are accepted within the range of 0...5 0. The size of the rake angle significantly affects the durability of the incisors.

Purpose of the main angle in plan - change the ratio between the width b and thickness A cutting at constant depth of cut t and submission S.

Decrease angle increases the strength of the cutter tip, improves heat dissipation, increases tool life, but increases cutting forces Pz And, Rat increases

spinning and friction against the surface being treated creates conditions for vibration. When increasing The chips become thicker and break better.

Cutter designs, especially those with mechanical fastening of carbide inserts, provide a range of angle values#>: 90, 75, 63, 60, 50, 45, 35, 30, 20, 10, which allows you to select the angle , most appropriate for specific conditions.

The process of material separation depends on the shape of the cutter. According to cutting, metal is separated; one might expect 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 of the 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, while Kick, based on similar studies, came to the opposite conclusions. With the help of more advanced microphotography techniques, it was shown that metal cutting is based on the process of plastic flow. As a rule, under normal conditions an advanced crack does not form; it can occur only under certain conditions.

According to the presence of plastic deformations extending far ahead of the cutter, it was established by observing the chip formation process 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 different orientations of the crystallographic planes of individual grains of the metal being processed. A periodic concentration of shear strains is observed at the final boundary of the chip formation zone, as a result of which the process of plastic deformation periodically loses stability and the outer boundary of the plastic zone receives local distortions, and characteristic teeth are formed on the outer boundary of the chip.

T^- \ : " G

Introduction

Rice. 7. Contour of the chip formation zone established by studying free cutting using filming.

Rice. 8. Microphotograph of the chip formation zone when cutting steel at low speed. The microphotograph shows 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 probability distribution of plastic deformations within the chip formation zone.

Accurate determination of the stressed and deformed state of the plastic zone using the plastic mechanics method is very difficult. The boundaries of the plastic region are not given and are themselves subject to determination. 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 based on experimental studies. The experimental methods are most fully described 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 stated:

Visual observation method. The side of the sample subjected to free cutting is polished or a large square mesh is applied to it. When cutting at low speed, the distortion of the mesh, 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 is

Introduction17

completely 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.

High-speed filming method. 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 the application of a precise square dividing mesh with cell sizes of 0.05 - 0.15 mm. The dividing mesh is applied in various ways: by rolling with printing ink, etching, vacuum spraying, screen printing, scratching, etc. The most accurate and in a simple way 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 “instantly” stop the cutting process, in which the cutter is removed from under the chips by a strong spring or the energy of the explosion of a powder charge. Using an instrumental microscope, the dimensions of the cells of the dividing mesh, distorted as a result of deformation, are measured on the resulting chip root. Using the device mathematical theory plasticity, the size of the distorted dividing grid can be used to determine the type of deformed state, the size and shape of the deformation zone, the intensity of deformation at various points of the deformation zone and other parameters that quantitatively characterize the chip formation process.

Metallographic method. The root of the chip obtained using a device for “instant” cutting stop is cut out, its side is thoroughly polished, and then etched with an appropriate reagent. The resulting microsection of the chip root is examined under a microscope at a magnification of 25-200 times or a microphotography is taken. Change of structure

Introduction

chips and deformation zones in comparison with the structure of an undeformed material, the direction of the deformation texture make it possible to establish the boundaries of the deformation zone and judge the deformation processes occurring 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, measuring the microhardness of the chip root gives an indirect idea of ​​the intensity of deformation in various volumes of the deformation zone. To do this, using the PMT-3 device, microhardness is measured at various points of the chip root and isoscleres (lines of constant hardness) are constructed, with the help of which the magnitude of tangential stresses in the deformation zone can be determined.

Polarization-optical method, or the photoelasticity method is based on the fact that transparent isotropic bodies, when exposed to external forces, become anisotropic, and if they are viewed in polarized light, the interference pattern allows one to determine the magnitude and sign of the acting stresses. The polarization optical method for determining stresses in the deformation zone has limited use for the following reasons. Transparent materials used for cutting have completely different physical and mechanical properties than technical metals - steel and cast iron. The method gives accurate values ​​of normal and shear stresses only in the elastic region. Therefore, using the polarization-optical method, it is possible to obtain only a qualitative and approximate idea of ​​the stress distribution in the deformation zone.

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

Introduction 19

From the surface of a sample cut from a machined part, very thin layers of material are sequentially removed and the deformation of the sample is measured using strain gauges. Changing the dimensions of the sample leads to the fact that under the influence of residual stresses it becomes unbalanced and deforms. From the measured deformations one can judge the magnitude and sign of the residual stresses.

Based on the above, we can draw a conclusion about the complexity and limited applicability of experimental methods in the field of studying processes and patterns 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 to use the experimental base only at the stage of confirming the mathematical model. Currently, a number of methods are used to calculate cutting forces, which are not confirmed by experiments, but derived from them.

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

p, = c P f p sy K P

Where WedG - coefficient that takes into account the influence on the strength of some permanent conditions; *R- cutting depth; $^,- longitudinal feed; TOR- generalized cutting coefficient; xyz- exponents.

Introduction 20

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

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

voltage qF = 0.285^, where &To- actual final tensile strength. On this basis, A.A. Rosenberg obtained another formula for calculating the main component of the cutting force:

(90-y)"cos/

-- їїдГ + 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 calculations, it is necessary to determine the parameters TOA And$k experimentally, which is very labor-intensive. According to numerous experiments, it was revealed that when replacing a curved shear line with a straight line, the angle U close to 45, and therefore the formula will take the form:

dcos U

Pz = - "- r + sin^

tg arccos

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

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

r max = gir"x ~ b) reaches a certain value (for each material).

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

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

Introduction23

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

The criteria for the strength of materials that unequally resist tension and compression include the criterion of Schleicher, Balandin, Mirolyubov, Yagna. Disadvantages include difficulty of application and poor experimental validation.

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

Analysis of the above approaches to describing the theory of separation allows us to highlight the following characteristic features:

Existing approaches to describing destruction processes are acceptable at the stage of the beginning of the destruction process and when solving problems in a first approximation.

The process model should be based on a description of the physics of the cutting process, rather than 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 clearly provide information

Introduction

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

Based on the above, main goal of the work is to create a mathematical model of separation that 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 chips and workpieces and to study the patterns of the chip removal process.

In the first chapter The dissertation outlines a mathematical model of finite deformation and 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 constructed. An analysis of the mechanisms of friction and destruction is provided in relation to the finite element model. Comprehensive testing of the resulting 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 mechanism for modeling the process and its finite element implementation are described in detail. A comparative analysis of the obtained data with experimental studies is carried out, 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 computer science" (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 computer science" ( Tula, 2003), at the scientific and practical conference “Young Scientists of the Russian Center” (Tula, 2003).

Constitutive relations of elastoplastic finite deformation processes

To individualize the points of the environment, an arbitrary coordinate system 0 is derived for the initial t - About a fixed, so-called calculated configuration (KQ), with the help of which each particle is assigned a triple of numbers (J,2,3) “assigned” to this particle and unchanged throughout the entire movement. The system 0 introduced in the reference configuration, together with the basis, =-r (/ = 1,2,3) is called a fixed Lagrangian coordinate system. Note that the coordinates of particles at the initial moment of time in the reference system 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 Eulerian.

As is known, the occurrence of stress in the body is generated by the deformation of material fibers, i.e. changing 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 to indicate measures for their description. It should be noted that this representation is not unambiguous and several approaches to describing the environment can be indicated, in which the division of motion into portable “quasi-solid” and relative “deformation” is carried out in various ways. In particular, in a number of works, deformation motion is understood as the motion of the neighborhood of a material particle in relation to the underlying Lagrangian basis ek; In the works, movement in relation to a rigid basis is considered as a deformation movement, the translational movement of which is determined by the rotation tensor connecting the main axes of the left and right measures of distortion. 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 rate is defined as the relative velocity of the particle relative to the rigid orthogonal trihedron of the vortex basis, the rotation of which is specified by the vortex tensor Q. It should be noted that in the general case of motion of the medium, the main axes of the tensor W pass through different material fibers. However, as shown in , for processes of simple and quasi-simple loading in the real range of deformations, the study of deformation motion in a vortex basis seems very satisfactory. At the same time, when constructing relations describing 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 coupled with the measure of stress through the expression of elementary work. 2) rotation of a material element as an absolutely rigid body should not lead to a change in the measures of deformation 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 must be preserved. The last requirement is highly desirable.

As the analysis shows, the use of the above measures to describe the process of finite deformation, as a rule, leads either to insufficient correctness in the description of deformation or to a very complex procedure for their calculation.

Invariants are used to determine the curvature and twist of the trajectory

tensors W ", which are nth-order Jaumann derivatives of the strain rate deviator, as shown in. They can be determined by known value metric tensor and derivatives of its components at the considered moment of time. Consequently, the value of curvature and twist, in contrast to the second and third invariants of the functional measure of deformation H, do not depend on the nature of the change in the metric over the entire interval. The relationship of the general postulate of isotropy in the form (1.21) is the starting point for constructing specific models of finitely deformable bodies and their experimental substantiation. It seems natural to generalize the known relationships for small deformations by moving to the proposed measures of deformation and loading. Note that since in problems of studying the process of deformation of a medium, as a rule, the velocity formulation 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. In this case, the velocities of the deformation and loading vectors correspond to the relative derivatives of tensors and deviators in the sense of Jaumann.

Construction of a model for introducing 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 capable of qualitatively describing the course of the process. It is more acceptable to use numerical methods based on the variational principles of Lagrange and Jourdain. Existing approximate methods for solving boundary value problems in the mechanics of a deformable solid are described in sufficient detail in monographs.

In accordance with the basic concept of FEM, the entire volume of the deformable medium is divided into a finite number of elements in contact with each other at nodal points; the combined motion of these elements models the motion of a deformable medium. Moreover, within each element, the system of characteristics describing 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 nodes.

The use of a simplex element significantly simplifies the procedure for constructing a finite element representation of relation (2.5), since it allows the use of simpler operations of one-point integration over the volume of the element. At the same time, since the requirements of completeness and continuity are satisfied for the selected approximation, the necessary degree of adequacy of the finite element model to a “continuous system” - a 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 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 method of sequential loading has a certain physical meaning, since at each step the reaction of the system to an increment of load is considered as it occurs in the actual process. Therefore, the method allows us to obtain much more information about the behavior of a body than just the magnitude of displacements under a given load system. Because naturally we obtain a complete set of solutions corresponding various parts load, then it becomes possible to study intermediate states for stability and, if necessary, make appropriate modifications to the procedure to determine branching points and find possible continuations of the process.

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

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 from experiments on uniaxial tension of samples of the material under study form a specific type of constitutive relations, using, in accordance with the requirements of clause 1.2, any of the most common methods of approximating the experimental curve. When solving a problem, a certain type of plasticity theory is assumed to be unchanged for the entire volume under study throughout the entire process. The fairness of the choice is subsequently assessed by the curvature of the deformation trajectory, calculated at the most characteristic points of the body. This approach was used when studying models technological processes finite deformation of tubular samples in regimes of simple or close external loading. In accordance with the chosen procedure of step-by-step integration, the entire loading interval with respect to parameter t is divided into a number of fairly small stages (steps). In the future, the solution to the problem for a typical step is constructed using the following algorithm. 1. For the region configuration newly determined based on the results of the previous step, the metric characteristics of the deformed space are calculated. At the first step, the configuration of the region coincides with the configuration determined at t = O. 2. The elastic-plastic characteristics of the material are determined for each element in accordance with the stress-strain state corresponding to the end of the previous step. 3. A local matrix of stiffness and element force vector is formed. 4. Kinematic boundary conditions on the contact surfaces are specified. For an arbitrary contact surface shape, a well-known procedure for transition to a local coordinate system is used. 5. A global system stiffness matrix and the corresponding force vector are formed. 6. The system of algebraic equations is solved, the vector column of velocities of nodal movements is determined. 7. The characteristics of the instantaneous stress-strain state are determined, the tensors of the strain rate W, vortex C1, and the rate of change of volume 0 are calculated, the curvature of the deformation trajectory X is calculated 8. The velocity fields of the stress and strain tensors are integrated, and a new configuration of the region is determined. The type of stress-strain state, zones of elastic and plastic deformation are determined. 9. The achieved level of external forces is determined. 10. The fulfillment of equilibrium conditions is monitored and residual vectors are calculated. When implementing a scheme without clarifying iterations, the transition is carried out immediately to step 1.

Factors influencing the chip formation process

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

According to , only four factors have a direct influence on the chip formation process during rectangular cutting: the angle of action, the rake angle of the tool, the cutting speed and the properties of the material. All other factors influence indirectly. To identify these dependencies, the process of free rectangular cutting of material on a flat surface was selected. The workpiece is divided into two parts by the line of intended division GA, the top layer is the future chip, the thickness of the removed layer is o, the remaining workpiece is thick h. Point M is the maximum point of reaching the tip of the cutter during penetration, the path traversed by the cutter is S. The width of the sample is finite and equal to b. Let's consider a model of the cutting process (Fig. 3.1.) Assuming that at the initial moment of time the sample is undeformed, intact, without cuts. A workpiece consisting of two surfaces connected by a very thin layer of AG, 8 .a thick, where a is the thickness of the chips being removed. AG - estimated dividing line (Fig. 3.1.). When the cutter moves, contact occurs along the two surfaces of the cutting tool. At the initial moment of time, no destruction occurs - the cutter is introduced without destruction. Elastic-plastic isotropic material is used as the main material. The calculations considered both ductile (the ability of a material to undergo 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, which eliminates the occurrence of stagnation on the front surface. 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 to both numerically and experimentally study the cutting process of the cutting layer, which is 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 step-by-step loading of the sample, in the case of cutting - by small movement of the cutter in the direction of the sample. The problem is solved by kinematically specifying the movement on the cutter, because the cutting speed is known, but the cutting force is unknown and is a determinable quantity. To solve this problem, a specialized software package Wind2D, capable of solving three problems - providing results confirming the validity of the calculations obtained, calculating test problems to justify the validity of the constructed model, and having the ability to design and solve a technological problem.

To solve these problems, a model was chosen for the modular construction of the complex, which included 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. The mathematical model may not be unique. In the original design there are three of them for two different types of elements. Each task also represents a module associated with a mathematical model with three procedures and with the shell with one procedure for calling the module, thus, the integration of a new module is reduced to entering four lines into the project and recompiling. The high-level language Borland Delphi 6.0 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 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 ones to model are dynamic boundaries; if you trace the process of separation by nodes, then when the destruction criterion is reached in a 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 із (Fig. 3.10). Next, from 1 and the node goes to C and then to C. The node assigned to A p immediately or after several steps falls on the surface of the cutter and goes to C, where it can be detached for two reasons: reaching the detachment criterion, or upon reaching point B, if the chipbreaker is determined when solving this problem. Next, the node moves to G9 if the node in front of it is already unpinned.

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

As mentioned earlier, the work used a step-by-step loading method, the essence of which is to divide the entire path of the wedge 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 necessary for an accurate description of 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 K becomes 0.99. Some criteria may not be used in a number of tasks; all criteria are described below (Fig. Evil): 1. Prohibition of material penetration into the cutter body - achieved by checking all nodes from I\L 9"! 12 at the intersection of the front cutting surface boundary. 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 reduction in the step size is determined. The step is recalculated. 2. Elements that have passed the yield point at this step are identified, and a reduction factor for the step is determined so that only a few elements “pass” the limit. The step is recalculated. 3. Nodes from a certain area belonging to the dividing line GA are identified that exceed the value of the destruction criterion at this step. A reduction factor for the step is determined so that only one node exceeds the value of the failure criterion. The step is recalculated. Chapter 3. Mathematical modeling of the cutting process 4. Prohibition of material penetration into the cutter body through the rear cutting surface for units from A 6, if this boundary is not secured. 5. For nodes 1 8, the detachment condition and the transition to the center at point B can be specified if the condition used in the calculation with a chipbreaker is selected. 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 recalculated. 7. The minimum step size reduction factor is determined, and if it is less than 0.99, then the step is recalculated, otherwise the transition to the next conditions occurs. 8. The first step is considered frictionless. After 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 preserved in separate entry. If the step is calculated with friction, then it is checked whether the direction of movement of the nodes on which the friction force acts has changed. If it has changed, then these units are rigidly fixed to the front cutting surface. The step is recalculated. 9. If the transition to the next step is carried out, and not recalculation, then the nodes approaching the front cutting surface are secured - TRANSITION OF NODES FROM 12 K 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 unit 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 a node belonging to AG is identified that exceeds the value of the destruction criterion at this step by an acceptable (small) value. Enabling the separation mechanism: instead of one node, two are created, one belonging to - and, the other 1 from; renumbering of body nodes using a special algorithm. Move to the next step.

The final implementation of 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 arises when using a small number of steps in the calculation, and very rarely when using a large number of steps at the same plunge depth. However, this criterion does not allow the nodes to “fall” inside the cutter, leading to incorrect results. According to (9), the nodes are fixed at the stage of transition to the next step, and not during several recalculations.

The implementation of criterion (2) consists of 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 pure cutting process, without the influence of a sharp increase in temperature on the interaction surface and in a sample in which flush chips are formed, without the formation of a built-up surface on the cutting surface, a cutting speed of about 0.33 mm/sec is required. Taking this speed as the maximum, we find that to advance the cutter by 1 mm, it is necessary to calculate 30 steps (subject to a time interval of 0.1 - which ensures the best stability of the process). When calculating using a test model, when introducing a cutter by 1 mm, taking into account the use of previously described criteria and without taking into account friction, 190 steps were obtained instead of 30. This is due to a decrease in the advance step size. However, due to the fact that the process is iterative, 419 steps were actually counted. This discrepancy is caused by too large a 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 was obtained - 344. A further increase in the number to 150 leads to an increase in the number of calculated steps to 390, and therefore 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 penetration, with an uneven division of the mesh with the number of elements 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.

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

BULLETIN OF TOMSK STATE UNIVERSITY

2009 Mathematics and Mechanics No. 2(6)

MECHANICS

A.N. Shipachev, S.A. Zelepugin

NUMERICAL SIMULATION OF PROCESSES

HIGH SPEED ORTHOGONAL CUTTING OF METALS1

The processes of high-speed orthogonal cutting of metals using the finite element method were numerically studied within the framework of an elastoplastic model of the medium in the cutting speed range of 1 – 200 m/s. The limiting value of the specific energy of shear strains was used as a criterion for chip separation. The need to use an additional criterion for chip formation has been identified, for which a limiting value of the specific volume of microdamage has been proposed.

Key words: high-speed cutting, numerical modeling, 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 friction of chips on the front surface of the cutter and friction of the rear surface of the tool on the cutting surface, occurring under conditions of high pressures and sliding speeds. The mechanical energy expended in this case transforms into thermal energy, which in turn has a great influence on the patterns of deformation of the cut layer, cutting forces, wear and durability of the tool.

Products of modern mechanical engineering are characterized by the use of high-strength and difficult-to-process materials, a sharp increase in 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 directions Such improvement is high-speed processing.

In the scientific literature, theoretical and experimental studies of the processes of high-speed cutting of materials are extremely insufficiently presented. There are individual examples of experimental and theoretical studies of the influence of temperature on the strength characteristics of a material during 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 to more fully take into account the properties of materials, thermal and inertial effects led to the work being carried out with the financial support of 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), Ministry of Education and Science of the Russian Federation within the framework of the AVTsP “Development of the scientific potential of higher education” (project 2.1.1/5993).

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

–  –  –

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

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

Modeling of the process of chip separation from the workpiece was carried out using the criterion of destruction of the calculated elements of the workpiece, and an approach similar to simulation modeling of the destruction of erosion-type material was used. The limiting value of the specific shear strain energy Esh was used as a fracture criterion—the 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 energy of shear deformations depends on the interaction conditions and is specified by the function of the initial impact velocity:

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

Calculation results Calculations were carried out for cutting speeds from 1 to 200 m/s. Dimensions of the working part of the tool: length of the top edge 1.25 mm, side edge 3.5 mm, rake angle 6°, back angle 6°. The processed steel plate had a thickness of 5 mm, a length of 50 mm, and a cutting depth of 1 mm. The material of the workpiece 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 the 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 Mie – Grüneisen form.

The results of calculating the process of chip formation when the cutter moves at a speed of 10 m/s are presented in Fig. 1. From the calculations it follows that the cutting process is accompanied by intense plastic deformation of the workpiece being processed in the vicinity of the tip of the cutter, which, when chips are formed, 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 required small time step used in the calculations, ensure the stability of the calculation in the event of significant deformation,

–  –  –

Rice. 1. Shape of the chip, workpiece and working part of the cutting tool at times 1.9 ms (a) and 3.8 ms (b) when the cutter moves at a speed of 10 m/s Numerical modeling of high-speed orthogonal cutting processes 113 until the separation criterion is met shavings. At cutting speeds of 10 m/s and below, areas appear in the sample where the chip separation criterion is not triggered in a timely manner (Fig. 1, a), which indicates the need to use 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.

In Fig. Figure 2 shows the fields of temperature (in K) and specific energy of shear deformations (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 energy of shear deformations, which indicates that a 1520

–  –  –

Rice. 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 Numerical modeling of high-speed orthogonal cutting processes 115 Conclusion The processes of high-speed orthogonal cutting of metals were numerically studied by the finite element method within the framework of an elastoplastic model environment in the cutting speed range 1 – 200 m/s.

Based on the obtained calculation results, it was established that the nature of the distribution of lines of the level of specific energy of shear strains and temperatures at ultra-high cutting speeds is the same as at cutting speeds of the order of 1 m/s, and qualitative differences in the mode may 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 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 microdamage - 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: Publishing house 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 spacecraft protection elements // Space Research. 2008. T. 46. No. 6. P. 559 – 570.

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

8. Ivanova O.V., Zelepugin S.A. Condition for joint deformation of 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 // Izvestia RAS. MTT. 1999. No. 5. P. 173 – 188.

10. Zelepugin S.A., Shpakov S.S. Destruction of a two-layer barrier boron carbide - titanium alloy under 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 to study the orthogonal cutting of metals with an STM tool, taking into account destruction and temperature effects. // Superhard materials. 1995. No. 5. P. 33 – 38.

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V 0 z. H/L 1 (wide plate), where N- thickness, L- length of the workpiece. The problem was solved on a moving adaptive Lagrangian-Eulerian mesh using the finite element method with splitting and using explicit-implicit equation integration schemes...

In the work, using the finite element method, a three-dimensional simulation of the unsteady process of cutting an elastoviscoplastic plate (workpiece) with an absolutely rigid cutter moving at a constant speed was carried out V 0 at different inclinations of the cutter face a (Fig. 1). The simulation was carried out based on a coupled thermomechanical model of an elastoviscoplastic material. A comparison is given of 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 when changing the geometry of the workpiece and cutting tool, speed and depth of cut, as well as the properties of the material being processed. The thickness of the workpiece was varied in the direction of the axis z. The stressed state changed from plane-stressed I = H/L 1 (wide plate), where N- thickness, L- length of the workpiece. The problem was solved on a moving adaptive Lagrangian-Eulerian grid using the finite element method with splitting and using explicit-implicit equation integration schemes. It is shown that numerical modeling of the problem in a three-dimensional formulation makes it possible to study cutting processes with the formation of continuous chips, as well as with the destruction of chips 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. The dependences of the force acting on the cutter for different geometric and physical parameters of the problem were obtained. It is shown that quasi-monotonic and oscillating modes are possible and their physical explanation is given.

BULLETIN OF TOMSK STATE UNIVERSITY Mathematics and mechanics

MECHANICS

A.N. Shipachev, S.A. Zelepugin

NUMERICAL SIMULATION OF HIGH-SPEED ORTHOGONAL CUTTING OF METAL PROCESSES1

The processes of high-speed orthogonal cutting of metals using the finite element method were numerically studied within the framework of an elastoplastic model of the medium in the cutting speed range of 1 - 200 m/s. The limiting value of the specific energy of shear strains was used as a criterion for chip separation. The need to use an additional criterion for chip formation has been identified, for which a limiting value of the specific volume of microdamage has been proposed.

Key words: high-speed cutting, numerical modeling, 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 friction of chips on the front surface of the cutter and friction of the rear surface of the tool on the cutting surface, occurring under conditions of high pressures and sliding speeds. The mechanical energy expended in this case transforms into thermal energy, which in turn has a great influence on the patterns of deformation of the cut layer, cutting forces, wear and durability of the tool.

Products of modern mechanical engineering are characterized by the use of high-strength and difficult-to-process materials, a sharp increase in 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 extremely insufficiently presented. There are individual examples of experimental and theoretical studies of the influence of temperature on the strength characteristics of a material during 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 to more fully take into account the properties of materials, thermal and inertial effects led to

1 The work was carried out with the financial support of 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 AVTsP “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 elastoplastic model of the medium.

Numerical calculations use a model of a damaged medium, characterized by the possibility of the initiation and development of cracks in it. The total volume of the medium is made up of its undamaged part, which occupies the volume of the liquid and is characterized by density pc, as well as the cracks occupying the volume of the liquid, in which the density is assumed to be zero. The average density of the medium is related to the entered parameters by the relation p = pc (Zhs / Zh). The degree of damage to the medium is characterized by the specific volume of cracks V/ = Ж//(Ж р).

The system of equations describing the unsteady adiabatic (both during elastic and plastic deformation) motion of a compressible medium consists of the equations of continuity, motion, energy:

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

Modeling of “several” fractures is carried out using a kinetic model of active type fracture:

When creating the model, it was assumed that the material contains potential sources of destruction with an effective specific volume V:, on which cracks (or pores) form and grow when the tensile pressure Рc exceeds a certain critical value P = Р)У\/(У\ + V/ ), which decreases as the resulting microdamages grow. The constants VI, V2, Pk, K/ were selected by comparing the results of calculations and experiments on recording the velocity of the back 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 intact substance is considered a function of the specific volume and specific internal energy and is determined over the entire range of loading conditions.

Formulation of the problem

Shu(ri) = 0;

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

^ = | - я§п (Рс) к7 (Рс | - Р*)(У2 + У7),

if Rs< -Р* или (Рс >P* and Y^ > 0).

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

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

Modeling of the process of chip separation from the workpiece was carried out using the criterion of destruction of the calculated elements of the workpiece, and an approach similar to simulation modeling of the destruction of erosion-type material was used. The limiting value of the specific energy of shear deformations Esh was used as a criterion for destruction - a criterion for chip separation. The current value of this energy is calculated using the formula:

The critical value of the specific energy of shear deformations depends on the interaction conditions and is set 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 calculations, and the parameters of neighboring cells are adjusted taking into account conservation laws. The adjustment consists of removing the mass of the destroyed element from the masses of the nodes that belonged to this element. If in this case the mass of any calculation unit becomes

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

Calculation results

Calculations were carried out for cutting speeds from 1 to 200 m/s. Dimensions of the working part of the tool: length of the top edge 1.25 mm, side edge 3.5 mm, rake angle 6°, back angle 6°. The processed steel plate had a thickness of 5 mm, a length of 50 mm, and a cutting depth of 1 mm. The material of the workpiece being processed 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: 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 the 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 Mie-Grüneisen form.

The results of calculating the process of chip formation when the cutter moves at a speed of 10 m/s are presented in Fig. 1. From the calculations it follows that the cutting process is accompanied by intense plastic deformation of the workpiece being processed in the vicinity of the tip of the cutter, which, when chips are formed, 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 required small time step used in the calculations, ensure the stability of the calculation in the event of significant deformation,

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

until the chip separation criterion is met. At cutting speeds of 10 m/s and below, areas appear in the sample where the chip separation criterion is not triggered in a timely manner (Fig. 1, a), which indicates the need to use 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.

In Fig. Figure 2 shows the fields of temperature (in K) and specific energy of shear deformations (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 energy of shear deformations, which indicates that

Rice. 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

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

The calculation results presented in Fig. 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 strains or temperature, therefore, in calculations, the limiting value of the specific volume of microdamages can be used (independently or additionally) in calculations as a criterion for chip separation.

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. 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

The processes of high-speed orthogonal cutting of metals using the finite element method were numerically studied within the framework of an elastoplastic model of the medium in the cutting speed range of 1 - 200 m/s.

Based on the obtained calculation results, it was established that the nature of the distribution of lines of the level of specific energy of shear strains and temperatures at ultra-high cutting speeds is the same as at cutting speeds of the order of 1 m/s, and qualitative differences in the mode may 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 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 microdamage - 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: Publishing house 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. and others. Experimental and theoretical study of the collision of a group of particles with spacecraft protection elements // Space Research. 2008. T. 46. No. 6. P. 559 - 570.

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

8. Ivanova O.V., Zelepugin S.A. Condition for joint deformation of 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 // Izvestia RAS. MTT. 1999. No. 5. P. 173 - 188.

10. Zelepugin S.A., Shpakov S.S. Destruction of a two-layer barrier boron carbide - titanium alloy under 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 to study the orthogonal cutting of metals with an STM tool, taking into account destruction and temperature effects. Superhard Materials. 1995. No. 5. P. 33 - 38.

SHIPACHEV Alexander Nikolaevich - 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 Mechanics of Deformable Solids of the Faculty of Physics and Technology of Tomsk State University, Senior Researcher of the Department of Structural Macrokinetics of the Tomsk Scientific Center SB RAS. Email: [email protected], [email protected]

SOLID MECHANICS<3 2008

© 2008 V.N. KUKUDZHANOV, A.L. LEVITIN

NUMERICAL SIMULATION OF CUTTING PROCESSES OF ELASTOVISCOPLASTIC MATERIALS IN THREE-DIMENSIONAL FORMULATION

In this work, a three-dimensional simulation of the unsteady process of cutting an elastoviscoplastic plate (workpiece) with 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 simulation was carried out based on a coupled thermomechanical model of an elastoviscoplastic material. A comparison is given of 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 when changing the geometry of the workpiece and cutting tool, speed and depth of cut, as well as the properties of the material being processed. The thickness of the workpiece was varied in the direction of the z axis. The stress state changed from plane stress 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 using the finite element method with splitting and using explicit-implicit equation integration schemes. It is shown that numerical modeling of the problem in a three-dimensional formulation makes it possible to study cutting processes with the formation of continuous chips, as well as with the destruction of chips 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. The dependences of the force acting on the cutter for different geometric and physical parameters of the problem were obtained. It is shown that quasi-monotonic and oscillating modes are possible and their physical explanation is given.

1. Introduction. Cutting processes play an important role in the processing of hard-to-deform materials on turning and milling machines. Machining is the main cost-determining operation in the manufacture of complex profile parts from hard-to-deform materials, such as titanium-aluminum and molybdenum alloys. When cutting them, chips are formed, which can break into separate pieces (chips), which leads to a non-smooth 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 processed material during high-speed cutting is extremely difficult. An alternative is numerical modeling of the process, which allows one 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 destruction is important for efficient cutting. Mathematics

Clinical 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.

An exact solution to these processes has not yet been obtained, although research has been undertaken since the mid-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 formation, destruction and fragmentation of chips during thermomechanical softening of the material.

In the last few years, thanks to numerical modeling, certain advances have been made in the study of these processes. Research has been carried out on the influence of the cutting angle, thermomechanical properties of the part and cutter, and the mechanism of destruction on the formation and destruction of chips. 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 along a predetermined interface. All these limitations did not allow us to study cutting in full, and in some cases led to an incorrect understanding of the mechanism of the process itself.

Moreover, as experimental studies show recent years, at high strain rates e > 105-106 s-1, many materials exhibit an anomalous temperature dependence associated with a restructuring of the mechanism of dislocation motion. 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 big problems during high-speed cutting. These problems have not been studied at all in the literature to date. Modeling a high-speed process requires the development of models that take into account the complex dependencies of the viscoplastic behavior of materials and, first of all, taking into account damage and destruction with the formation of cracks and fragmentation of particles and pieces of deformable material. To take into account all the listed

8 Solid Mechanics, No. 3

These effects require not only complex thermophysical models, but also modern computational methods that make it possible to calculate large deformations that do not allow extreme distortions of the mesh and take into account the destruction and appearance of discontinuities 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 formulation of the problem is accepted. In fig. Figure 1 shows the region and boundary conditions in the cutting plane. In the direction perpendicular to the plane, the workpiece has a finite thickness I = H/b (b is the length of the workpiece), which varied over a wide range. The spatial arrangement allows freedom of movement of the processed material from the cutting plane and a smoother chip exit, which provides more favorable cutting conditions.

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

ryi/yg = ; (2.1)

Hooke's law with temperature stresses

yO;/yg = k1 - еы - "М) (2.2) heat influx equations йй

pSe y- = K 0,.. - (3 X + 2ts)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 law of plastic flow

ep = Хй^/о; (2.4)

and plasticity conditions

L, Еы, X;, 9) = Оу (]Еы, X;, 0)< 0 (2.5)

where A] are the invariants of the stress tensor, E; - plastic deformation tensor. The evolutionary equations for internal variables have the form

yX /yg = yLk, Xk, 9) (2.6)

2.3 Material model. The work adopts a thermoelastic-viscoplastic model of the Mises type - a plasticity model with a yield stress in the form of a multiplicative relationship (2.7), including strain and viscoplastic hardening and thermal softening:

ou (ep, ¿*,9) = [a + b (ep)"]

where оу is the yield stress, ер1 is the intensity of plastic deformation, 0 is the relative temperature referred to the melting temperature 0т: " 0<0*

(0 - 0*) / (0t - 0*), 0*<0<0т

The material of the part is assumed to be homogeneous. The calculations used the relatively soft material A12024-T3 (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, t = 1.7, 9* = 300 K, 9t = 775 K, v = 0.9) and harder 42CrMo4 (E = 202 GPa, V = 0.3, A = 612 MPa, B = 436 MPa, n = 0.15, e0 = 5.77 ■ 10-4, C = 0.008, t = 1.46, 9* = 300 K, 9t = 600 K, v = 0.9). A comparison is made of the adiabatic cutting process with the solution of a complete thermomechanical problem.

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

intensity of plastic deformations e:

e = [yx + y2exp (y311/12)][ 1 + y41n (yor/y0)](1 + y59) (2.8)

where is th. - material constants determined from experiment.

If the destruction criterion is met in a Lagrangian cell, then the connections between the nodes in such cells are released and the stresses either relax to zero, or the resistance is maintained only with respect to compression. Lagrangian nodal masses, when destroyed, turn into independent particles, carrying away mass, momentum and energy, moving as a rigid whole and not interacting with undestroyed particles. A detailed overview of these algorithms is given in. In this work, fracture is determined by the achievement of a critical intensity of plastic deformation e and the fracture surface is not specified in advance. In the above calculations

e p = 1.0, the cutter speed was assumed to be 2 m/s and 20 m/s.

2.5. Method of integrating equations. To integrate the reduced coupled system of thermoplasticity equations (2.1)-(2.8), it is advisable to apply the splitting method developed in the work. The splitting scheme for elastoplastic equations consists in splitting the complete process into a predictor - a thermoelastic process, in

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

pdb/dr = a]

d aL = « - a§ «9) pSei9/yg = K.9ts - (3X + 2ts)a90ei

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ASTASHEV V.K., RAZINKIN A.V. - 2008