FINITE ELEMENT MODELING AND SIMULATION OF THE GRINDING PROCESS mô HÌNH hóa và mô PHỎNG QUÁ TRÌNH mài BẰNG PHƯƠNG PHÁP PHẦN tử hữu hạn

Abrasive wear models of the grinding process are based on the basic premise that the grinding process is essentially an abrasive process, in which abrasive grains attack a workpiece surf

FINITE ELEMENT MODELING AND SIMULATION OF THE GRINDING

PROCESS

MÔ HÌNH HÓA VÀ MÔ PHỎNG QUÁ TRÌNH MÀI BẰNG PHƯƠNG PHÁP

PHẦN TỬ HỮU HẠN

Nguyen Tuan Anh 1a , Pham Huy Hoang 2b

1Nguyen Tat Thanh University, Ho Chi Minh City, Vietnam

2Ho Chi Minh City University of Technology (VNU-HCM), Ho Chi Minh City, Vietnam

ABSTRACT

In recent years, finite element method (FEM) has become popularly used in studying the grinding process It has been applied to investigate various aspects of the grinding process In this paper, a micro-scale FEM model of the process is presented It will be shown that the FEM simulation can be used to investigate the complex phenomena underlying the grinding process

Keywords: grinding, FEM, single grit, simulation

TÓM TẮT

Trong những năm gần đây, phương pháp phần tử hữu hạn (FEM) đã trở nên phổ biến trong nghiên cứu quá trình mài Phương pháp này đã được ứng dụng để nghiên cứu các khía cạnh khác nhau của quá trình mài Trong nghiên cứu này, một mô hình phần tử hữu hạn cho quá trình mài ở mức độ micro được trình bày Nghiên cứu cho thấy, phương pháp phần tử hữu hạn có thể được áp dụng để nghiên cứu các hiện tượng phức tạp trong quá trình mài

Từ khóa: mài, FEM, hạt mài, mô phỏng

1 INTRODUCTION

Probably one of the oldest manufacturing processes, grinding is still widely used on the shop floor The popularity of grinding can be explained by its advantages over other processes: the precision it can deliver and the materials it can process However, grinding is often considered as “a deep dark mystery” due to its complicated nature of the process, affected by many factors, involving wheel, workpiece, machine, and process setting [1]

Thus, unsurprisingly, the grinding process has been the object of extensive research in the past 30 years, especially by modeling and simulation A properly-constructed model and its simulation will help facilitating the understanding of the complex physical phenomena as well as enhancing the predictions of the process performance There exist a variety of models for the grinding process On the basis of their underlying principles, the models can be divided into: abrasive wear, statistical, kinematic, and finite element method (FEM)

Abrasive wear models of the grinding process are based on the basic premise that the grinding process is essentially an abrasive process, in which abrasive grains attack a workpiece surface and remove material Thus, it is quite plausible to apply abrasive wear theories for studying grinding processes Typical models based on abrasive wear theories can

be seen in [2,3] On the other hand, the statistical type of modelling tries to predict the grinding performance based on the statistical analysis of the grinding process Generally, these models provide mathematical formulae for predicting surface roughness [4,5] The third type of models is kinematic, which is relied on the fact that surface generation in grinding is inherently a random process and a ground surface is the cumulative outcome of multiple

abrasive grain- workpiece interaction events Thus, the ground groove on the workpiece surface can be considered as an imprint of abrasive grain shapes with due adjustment for kinematic relationship between the grinding wheel and the workpiece Examples of kinematic modelling can be found in [6,7]

In recent years, finite element method (FEM) has been applied in studying the grinding process It was used to investigate various aspects of grinding such as thermal effect, structural analysis, and interaction of the grinding grit with the workpiece [8-10] The advantages of FEM are: it does not involve experimental set-up; can take into account realistic material behaviour, complicated geometric and kinematic conditions; and allows wide range of parametric study FEM models can be categorized into micro or macro-scale The micro- scale focuses on the grit-workpiece interaction while the later centers on wheel-workpiece interaction [11]

In this paper, a micro-scale FEM model and simulation of a single grit interaction with the workpiece in grinding will be presented The simulation is implemented as a fully coupled temperature-displacement analysis in which an abrasive grit moves following a predefined grinding path of the wheel, and cuts into the workpiece The simulation is accomplished using

a commercial package ABAQUS/Explicit The material removal process during the grit-workpiece interaction will be investigated in terms of force, stress, strain and temperature

2 PROBLEM DEFINITION

Figure 1 Single grit grinding: (a) schematic diagram of single grit grinding (Note that the wheel and the workpiece are drawn with different proportional ratios), (b) assumed

grit shape

Grinding is essentially a stochastic process, in which a large number of abrasive grits act as a cutting tool The grinding result is just the summation of individual grit-workpiece interaction Hence, in order to understand the grinding process, it is important to investigate the grinding mechanism of a single grit grinding

In Figure 1, the schematic diagram of single grit grinding is shown A rotating wheel at

a peripheral velocity v s cuts into the workpiece at a depth of cut a When the workpiece moves toward at constant velocity v w, the single grit attached on a rotating wheel cuts successive grooves in the workpiece The grinding action of the grit here replicates surface

up-grinding The grinding grit is represented as a cone with a tip radius R (Figure 1b) In

reality, abrasive grits have irregular shapes However, while grinding, only a tip of the grit interacts with the surface of the workpiece, hence, a sphere or a cone could be well approximated for a grinding grit [7]

The single grit grinding is modeled with the following parameters: wheel diameter d s =

250 mm, wheel velocity v s = 45 m/s, workpiece velocity v w = 0.225 m/s, velocity ratio

between wheel and workpiece 200, depth of cut a =20 µm, effective radius R =50 µm The

parameters are chosen to replicate common grinding setting [12] The workpiece material is carbon steel ANSI 1020

3 FEM MODEL

3.1 Mesh and boundary conditions

For the sake of reducing computation time, a 2D plane strain condition is assumed The

grit is modeled as a cone with round tip R of 50 µm by using a rigid surface, and will not be

deformed in the grinding action The workpiece has a length of 2.3 mm, and a height of 0.036

mm The length of the workpiece is chosen to accommodate the groove made by the grit The groove on the workpiece is modelled as an arc with radius of 125 mm (half of wheel diameter)

For meshing, the workpiece is divided into nine layers, with upper layers are meshed with finer elements as they will be loaded heavily (Figure 2) The five upper layers consist of

1440 elements of length 1.6 µm The ratio between the heights of elements in successive layers of the first five is 0.9 The remaining four layers are meshed with larger elements Two middle layers have 720 elements each, while the last two contain 360 elements each The lengths of elements in these layers are correspondingly 3.2 and 6.4 µm The height of elements in the same layer increases from the left to right, with the ratio between the leftmost and rightmost is approximately 0.5 All the elements are four-node reduced integration plane strain formulation The upper layers are meshed with smaller size elements to accommodate for the complex stress-strain behaviour in the upper part of the workpiece

For the boundary conditions, the left and right sides of the workpiece are assumed to be long in horizontal direction, thus can be constrained in that direction The bottom of the workpiece is constrained in vertical direction The grit is constrained to rotate around the

centre of the wheel at a rotational velocity of 360 rad/s, thus achieving peripheral velocity v s

of 45 m/s At the same time, the centre of the wheel moves at a translational velocity v w of 0.225 m/s

Figure 2 FEM mesh for single grit grinding: (a) the leftmost part of the model, (b) The

rightmost part of the model

3.2 Friction model

In this study, the modified Coulomb friction law is assumed for the friction condition of the tool-chip interface Let τ be the chip shear stress at a contact point along the tool-chip interface and σ the normal pressure at the same point This law states that relative motion occurs at the contact point when τ is equal to or greater than the critical friction stress τc When τ is smaller than τc there is no relative motion and the contact point is in a state of sticking The critical friction stress is determined by

where µ is the friction coefficient and τth is the threshold value related to material failure The threshold value τth should be less than the shear flow stress of the softer material

at the contact interface The friction coefficient is assumed to be 0.3, following [13]

3.3 Heat model

Temperature rise in the workpiece are due to the plastic work done in the deformation zone and the sliding friction along the tool-chip interface Let ∆Tp be the change in temperature (local temperature rise) in the workpiece induced by plastic work in a time interval ∆t ∆Tp can be calculated as follows

e p p

p

t T

C

π

σ ε ∆

ρ

where σe is the effective stress, εp is the effective plastic strain rate, c is the specific

heat, ρ is the mass density, and ηp is the percentage of plastic work that is transformed into heat ηp=90% is assumed in this simulation Similarly, local temperature rise ∆T f caused by friction along the grit-workpiece interface is determined using the following equation:

p

t T

C

τγ η ρ

∆

where τ is the frictional shear stress along the interface, γ is the slip rate, ρ is the mass

density, C p is the specific heat, ηf is the portion of friction-induced heat that goes into the workpiece, which is taken as 0.5

3.4 Chip separation criterion

The cutting action of the grit is simulated by using Johnson-Cook failure model for a chip separation criterion This criterion states that material will fail when the damage

parameter w exceed 1 The damage parameter w is defined as

∑∆εε 

=

pl f

pl

where ∆ is an increment of the equivalent plastic strain, εpl pl

f

ε is the strain at failure, and the summation is performed over all increments in the analysis When the shear failure criterion is met at an element integration point, all the stress components will be set to zero and the material point fails In this this study, the chip separation criterion is assigned to be 1,

and assumed to be independent of stress, strain rate and temperature, following [13]

3.5 Material data

There exist various flow stress data for ANSI 1020 In this study, the flow stress data employed is obtained from high-speed hot compression [14], which has been shown to give good FEM simulation results [13] Other material parameter: thermal expansion coefficient α,

the heat capacity C p, and the thermal conductivity coefficient λ is calculated or taken from [14]

4 SIMULATION RESULT AND DISCUSSION

A grinding time of 5x10-5 s is simulated The analyses were fully coupled displacement-temperature Due to very small simulated time, the heat loss due to convection at the workpiece surface is negligible, thus, only heat transfer due to conduction in the workpiece was considered in the analyses

The graph of cutting force shows 4 distinct zones (Figure 3) At the start of the process (zone I), the cutting force is rather small at approximately less than 0.2 N/mm When the grit continues cutting into the workpiece, the cutting force rises rapidly to 3.5 N/mm In the next zone (III) the cutting force keeps rising to the maximum of 5.2 N/mm, and drops fast at zone

IV, which corresponds to the moment of grit leaving the workpiece surface

Figure 3 Cutting force in single grit grinding

The above behaviour of the cutting force can be explained by the depth of penetration of the grit into the workpiece At the beginning, the penetration of the grit is still negligible, which causes a small deformation Thus, friction and elastic deformation are the only main contributions to cutting force As the wheel rotates and moves forward, the grit depth of penetration increases, consequently, causes more deformation In zone II, the cutting force now has plastic deformation as the dominant component, which rises up as the grit cuts into the workpiece In zone III, plastic deformation continues increasing to the limit value set by the chip separation criterion As dictated by the chip separation criterion, chip will be formed

in this zone, and the fluctuation of the cutting force corresponds to the moment of chip formation In zone IV, the grit leaves the workpiece surface; as a result, deformation and the cutting force values reduce

The analysis of effective plastic strain in the workpiece confirms to the above explanation Figure 4 shows the distribution of effective plastic strain in the workpiece at various stages of interaction between the grit and the workpiece At 0.010x10-3 s (zone I), there is no plastic deformation; as the grit rotates, the effective plastic strain increases to 0.254

at the simulated time of 0.026x10-3 s (zone II) The effective plastic strain reaches the limit value (chip separation criterion equals 1) at 0.048x10-3 s of simulated time (zone III) Figure 4c shows part of the workpiece being removed as chip

Figure 4 also shows that as the grit pushes into the surface and moves forward the material under the grit-workpiece contact zone is pushed up and forms a wave ahead of the grit, which has the largest plastic deformation in the workpiece The wave formation moves along the surface ahead of the grit The workpiece material is predicted to fail at elements, which locates directly under the grit and at the start of the wave formation The plastic deformation in the workpiece is limited only to a shallow depth into the subsurface of the workpiece It is also noted that the wave formation and plastic deformation in this simulation has similar characteristic as observed in experiments [15], in which a hard wedge of tool steel was indented and subsequently slid horizontally across a specimen surface of aluminium-magnesium alloy

a) b)

c)

Figure 4 Effective plastic strain in the workpiece: (a) at 0.010x10 -3 s simulated time (zone I), (b) at 0.026x10 -3 s simulated time (zone II), (c) at 0.048x10 -3 s simulated time (zone III)

Figure 5 represent the contours of Von Mises stress and mean stress distributions in the workpiece The stress distributions display a high stress concentration in the wave zone As the depth of penetration of the grit increases, the magnitude of the equivalent stress in the workpiece increases In contrast with stress distribution in machining, stress in the grinding process spreads in circled bands, rather than parallel-sided bands (Figure 5a,b) This suggests

that the chip formation process in grinding is quite different from the one in machining From

the simulation, it is proposed that shear of workpiece material is not limited only to the part that forms chip Shearing, if happens, should occur at the start of the wave formation and continues down in the workpiece subsurface, which still remains after grinding Figure 5c,d also reveal that in the grinding process, the material in the workpiece subsurface undergoes a change of stress state from high compressive to tensile one

Figure 5 Equivalent stress distribution in the workpiece: (a) at 0.010x10 -3 s simulated time (zone I), (b) at 0.048 x 10 -3 s simulated time (zone III);Pressure distribution in the workpiece: (c) at 0.010x10 -3 s simulated time (zone I), (d) At 0.048x10 -3 s simulated time (zone III)

parts of the workpiece

wave formation

Figure 6 Temperature at surface and subsurface of the workpiece at the cross section in the middle of the grinding contact zone: (a) temperature as a function of time, (b)

temperature as a function of depth in the surface

Study of heat transfer points out the extreme temperature gradients in time and space near the surface (Figure 6) The temperature at the workpiece surface rises and drops dramatically in a short duration Although the temperature at the surface reaches high value in the grinding process, the temperature at subsurface of the workpiece grows less considerably Moreover, the temperature in the workpiece will drop to approximately 75 °C immediately after grinding by the single grit As a function of the depth, the temperature decreases exponentially in the near-surface area and decreases more slowly at greater depths The maximum temperature of 400 °C at the surface is comparable with the grinding temperature

of 530 °C, estimated using the method proposed by Shaw [1] The depth of the workpiece affected by heat flux seems to be shallower than reported values [16]

The high gradients of temperature in the grinding zone can be explained by the large deformation when the grit cuts into the workpiece Due to the high speed associated with grinding, the deformation occurs as an adiabatic process, which causes the temperature rising substantially However, as the deformation is limited to a small volume of material, the dissipated energy is not large and is quickly conducted to surrounding material

CONCLUSION

A micro-scale FEM model of the single grit grinding is presented, and the simulation is carried out using commercial finite element code It is shown that FEM can be used to simulated high-speed material removal processes such as grinding The simulation allows studying the behaviour of cutting forces, stress, strain and temperature the grinding process Information such as stress, strain, temperature, difficult to obtain by analytical or experimental approaches, can be assessed with high resolution in terms of time and space The analysis of FEM results suggests that the chip formation in the grinding process should be dramatically distinct from other machining processes

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AUTHOR INFORMATION

1 Nguyen Tuan Anh, Nguyen Tat Thanh University, ntanh@ntt.edu.vn, 0989619024

2 Pham Huy Hoang, Ho Chi Minh City University of Technology (VNU-HCM),

phhoang@hcmut.edu.vn, 0989166420