The article focuses on the analysis of thermomechanical modeling of cutting by single abrasive micro-size grain while the process of grinding. In this research, applying the Johnson-Cook” material model, which relates the intensity of stress with strain rate, temperature and the accumulated plastic strain. Using arbitrary Lagrangian-Eulerian (ALE) approach makes it possible avoid the distortion of finite elements in the simulation of chip-formation under large deformations. The simulation allows predicting cutting force for processing workpiece of titanium alloy Ti6Al4V by single grain grinding

S. A. Voronov

Weidong Ma

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 39 
Simulation of chip-formation by a single grain of 
pyramid shape 
S. A. Voronov1, Weidong Ma2 
Bauman Moscow State Technical University, Moscow, Russia 
1Corresponding author 
E-mail: 1voronovsa@yahoo.com, 2314651368@qq.com 
(Received 29 July 2016; accepted 31 July 2016) 
Abstract. The article focuses on the analysis of thermomechanical modeling of cutting by single 
abrasive micro-size grain while the process of grinding. In this research, applying the 
Johnson-Cook” material model, which relates the intensity of stress with strain rate, temperature 
and the accumulated plastic strain. Using arbitrary Lagrangian-Eulerian (ALE) approach makes it 
possible avoid the distortion of finite elements in the simulation of chip-formation under large 
deformations. The simulation allows predicting cutting force for processing workpiece of titanium 
alloy Ti6Al4V by single grain grinding. 
Keywords: grinding, abrasive grain, FEM, approach ALE, cutting forces, chip formation. 
1. Introduction 
Grinding is a finishing machining operation, which is used to obtain the desired surface 
roughness and specified precision of shape characteristics. In the process, a large amount of 
abrasive grains, distributed on the outer cylindrical surface of the grinding wheel, play the role of 
cutting tools and remove unwanted material. Thermomechanical behavior of the material of 
workpiece is determined by its interaction with the cutting tool during the formation of chips. The 
cutting process takes place at high pressure and temperature in the zone of contact of grain and 
material [1]. The periodical grain entrance in machined material leads to the dynamical workpiece 
and tool behavior that can decrease the surface quality and the accuracy of processing. The arising 
vibration have the mechanism of forced vibrations and the regenerative mechanism due to cutting 
by each grain the surface that was generated by the previous grain [2]. The determination of cutting 
forces under the specified cutting conditions is required for the vibration analysis. It can be done 
by modeling of single grain and machined material interaction while cutting. 
Therefore, the mechanism of micro material removal with a single grain and the process of 
chip formation are necessary to investigate in details, as they are determined by the properties of 
workpiece material, cutting thickness, cutting speed, geometry of cutting edge, and etc. Simulation 
of cutting process with single grain is conducted by using “Abaqus/Explicit”, which allows to 
predict the shape of the chips, cutting forces, the surface quality, to estimate the residual stresses 
in workpiece, and determine the effect of operating parameters on the quality of the treated  
surface, in order to optimize primary operational parameters and geometry of cutting tools [3]. 
2. Description of cutting model with a single grain 
Real abrasive grain usually has shape of pyramid with flat faces. Grain enters into the material 
like a cutting wedge with a negative rake angle. In this case, the chip material easily slides along 
the rake face of the grain, after been separated from the workpiece material. In order to avoid 
infinity in the calculation of FEM model, the side four sharp edges of pyramid are chamfered 
while the geometry modeling of pyramidal shape grain. Taking into account of grain deterioration 
in grinding process, the bottom of grain is covered by spherical top with a given radius ݎ = 10 μm. 
Geometric scheme of pyramid-shaped grain is shown in Fig. 1. 
In fact, each abrasive grain on grinding wheel, entering into the material, moves along a path 
close to a circle. For simplicity, the trajectory is replaced by the path shown in Fig. 2, consisting 
of three straight segments. This designed scheme of grain cutting can reduce time of computation 
SIMULATION OF CHIP-FORMATION BY A SINGLE GRAIN OF PYRAMID SHAPE.  
S. A. VORONOV, WEIDONG MA 
40 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
and allows explore the cutting forces acting on single grain. We assume that, the process of exit 
from cutting of grain from the material in the third stage doesn't greatly change characteristics of 
the cutting forces, so the results of the third stage are not essential in simulation. Thus modeling 
is limited to two stages, which can significantly reduce the time of calculation. Besides, all the 
dimensions are in micrometers. 
 
Fig. 1. Geometric scheme of pyramidal shaped grain 
 
Fig. 2. Simulation path of process single grain cutting 
We assume that, the movement of grain is specified kinematically and doesn’t be influenced 
by the interaction between tool and workpiece material. Grain penetrates into the workpiece 
material with a constant cutting speed (specified ሬܸԦ =	5 m/s). At step-1, grain enters into the 
material with the depth, which varies linearly from 0 to ℎ௖௨, and passes along horizontal direction 
through 100 μm. Next, at step-2, the grain moves with constant speed and the depth of ℎ௖௨, and 
passes on at 200 μm. Modeling was performed for a given cutting thickness ℎ௖௨ = 40 μm. 
Zone of deformed surface, where grain penetrates into the material, is several degrees less than 
the dimensions of the part. Therefore, the actual shape and dimensions of the whole part it is of 
no importance, and the workpiece is considered as a cuboid. Due to the large temperature gradient 
and plastic deformations in the contact area, it requires a fine mesh of finite elements. Using 
uniform grid for workpiece is not efficient, besides it significantly increases the amount of 
computation. 
In the simulation approach ALE is used for finite element mesh construction with the help of 
built-in procedures of software “ABAQUS”, in order to get a normal chip formation without 
distortion of finite elements. Fig. 3 shows construction of grid. 
 
Fig. 3. Construction of grid: a) partition of blocks, b) initial mesh, c) the grinding part 
SIMULATION OF CHIP-FORMATION BY A SINGLE GRAIN OF PYRAMID SHAPE.  
S. A. VORONOV, WEIDONG MA 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 41 
3. Approach ALE for the formation of chips 
The three main approaches are widely used in the field of metal cutting FEM simulation: 
Lagrangian, Eulerian, and Arbitrary Lagrangian-Eulerian (ALE). 
Approach ALE combines peculiarity of both approaches Lagrangian and Eulerian. The ALE 
approach FE mesh is neither spatially fixed nor attached to the material. The mesh moves with the 
flow of material, determination of displacement is performed applying the Lagrangian approach, 
meanwhile, at each iteration, the grid is rebuilt, determination of velocities is performed applying 
Euler approach. 
The idea of modeling used in metal cutting consists in following way. In the zone near the 
edge of the cutting tool, Lagrange approach is used, while to analyze the areas where there is a 
free-boundary flow of chip material, Euler approach is used. This eliminates significant distortion 
of the grid elements [4]. 
Fig. 4 shows a scheme of ALE approach and the boundary conditions for workpiece in the 
simulation. Workpiece is modeled by using element of type C3D8RT, which is an 8-node 
thermally coupled brick, considering 3-D displacement and temperature, reduces integration time, 
and controls element parasitic form generation. The number of elements is 210540. The grain is 
considered as a rigid body. 
 
Fig. 4. Scheme of ALE approach and boundary conditions of workpiece 
4. Material model and properties 
The behavior of the workpiece material during deformation beyond the elasticity limit is 
considered by using thermo-elastic-plastic “Johnson-Cook” constitutive model. This model relates 
the intensity of the stress with strain rate, temperature and the accumulated plastic strain [5]: 
ߪ = ൫ܣ + ܤߝ௣௟തതതത௡൯ ∙ ቈ1 + ܥln ቆ
ߝ̅ሶ
ߝ଴ഥሶ ቇ቉ ∙ ቈ1 − ቆ
ܶ − ௥ܶ௘௙
௠ܶ௘௟௧ − ௥ܶ௘௙ቇ
௠
቉, (1)
where, ߪ – equivalent stress, ߝ௣௟തതതത – equivalent plastic deformation, ߝ̅ሶ – actual strain rate, ߝ଴ഥሶ  – basic 
effective strain rate, ߝ଴ഥሶ =	 1.0 s-1, ௥ܶ௘௙ – room temperature, ௠ܶ௘௟௧ – melting temperature of the 
material. ܣ  – characteristic coefficient, having dimension of stress, ܣ = ߪ்  (yield strength of 
material), ܤ  – characteristic coefficient, having the dimension of stress, ݊  – the exponent of 
hardening effect of plastic deformation, ܥ  – coefficient of influence of the strain rate 
(dimensionless), ݉ – material constant considering heat softening (dimensionless). 
Coefficients of model “Johnson-Cook” for material Ti6Al4V are shown in Table 1 [6]. 
Table 1. Coefficients of model “Johnson-Cook” for material Ti6Al4V 
ܣ (МPа) ܤ (МPа) ݊ ܥ ݉
1098 1092 0.93 0.014 1.1 
SIMULATION OF CHIP-FORMATION BY A SINGLE GRAIN OF PYRAMID SHAPE.  
S. A. VORONOV, WEIDONG MA 
42 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
In the simulation the deformation of the material is supposed to use model of cumulative 
damage summation ܦ: 
ܦ = 1ߝ௙ ෍∆ߝ௣
௜
௜
,	 (2)
where, ∆ߝ௣௜  – increment of effective plastic strain in FE on ݅th step of integration time. 
The value of critical accumulated strain ߝ௙ is used as the criteria of material damage, which is 
determined by the “Johnson-Cook” destruction model: 
ߝ௙ = ቂܦଵ + ܦଶ ∙ exp ቀܦଷ
݌
ߪതቁቃ ∙ ቈ1 + ܦସ ln ቆ
ߝ̅ሶ
ߝ଴ഥሶ ቇ቉ ∙ ቈ1 + ܦହ ቆ
ܶ − ௥ܶ௘௙
௠ܶ௘௟௧ − ௥ܶ௘௙ቇ቉, (3)
where, ݌ – the hydrostatic pressure, ߪത  – the equivalent von Mises stress, ܦଵ  to ܦହ  – material 
damage parameters, that show the influence of strain, strain rate and temperature on the material 
damage. 
To describe fracture of material the “Johnson-Cook” model in the package “Abaqus” is applied 
which states that, when the damage parameter of FE reaches 1, then destruction of the FE occurs. 
The parameters of the damage law in “Johnson-Cook” material model for Ti6Al4V are given in 
Table 2 [7]. 
Mechanical, physical and thermal parameters of the workpiece material are given in  
Table 3 [8]. 
Table 2. parameters of the damage law in “Johnson-Cook” material model for Ti6Al4V 
ܦଵ ܦଶ ܦଷ ܦସ ܦହ
–0.09 0.27 0.48 0.014 3.87 
Table 3. Mechanical, physical and thermal parameters of the workpiece material Ti6Al4V 
Mass density ߩ (kg/m3) 4430 Specific heat ܥ௣ (J/kg°C) 526.3 
Young’s modulus ܧ (MPa) 118 Thermal conductivity ߣ (W/m°C) 6.7 
Poisson’s ratio ߥ 0.349 Thermal expansion coefficient ߙ௅  (µm/m°C) 9.2 
௠ܶ௘௟௧ (°C) 1640 ௥ܶ௘௙ (°C) 20 
5. Finite element simulation results 
Heavily distorted elements during simulation of metal cutting can be removed from the mesh 
by setting the damage variables (SDEG), so as to reduce the possibility of termination program. 
Element deletion occurs when the degradation value (SDEG) calculated at a specified increment 
time reaches 1, ultimately element failure occurs via element deletion technique. Fig. 5 shows 
distribution of the damage variable ܦ (SDEG) in the simulation. 
The process of chip formation during the immersion of grain into material is shown in Fig. 6. 
Several pieces of material are distributed out of workpiece into air. This phenomenon is related to 
the fact that; material is separated from workpiece in the region around the edge of pyramidal 
shaped grain. Chips are formed during the extrusion of the material under the grain, and slide 
along the rake face of grain. 
The von Mises stress distribution through the simulation path is illustrated with contour lines 
in Fig. 6. It is very obvious to detect stress change during process. The maximum stress is achieved 
at the area near the end of step 1, and near the beginning of step 2, also at the region where the 
grain changes the direction of its motion. It’s interesting that, the stress decreases, along the path 
of grain immersion into the material. But if determine the temperature distribution, which is shown 
in Fig. 7, we can see that maximum temperature is reached at the contact area of grain and material. 
This coincides with the fact that, metal cutting is the thermomechanical process with high 
SIMULATION OF CHIP-FORMATION BY A SINGLE GRAIN OF PYRAMID SHAPE.  
S. A. VORONOV, WEIDONG MA 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533 43 
temperature and high stress. Besides, cutting heat will soften metal, consequently influences on 
stress distribution in workpiece. 
 
Fig. 5. Scalar element degradation parameter ܦ (SDEG) during simulation of chip-formation 
 
Fig. 6. The von Mises stress distribution during simulation of chip-formation (Semi-sectional view) 
 
Fig. 7. Temperature distribution during simulation of chip-formation (Semi-sectional view) 
 
Fig. 8. Configuration of cross-section (ܣ௖ – cross-sectional area of the contact grain and material) 
Fig. 8 shows the configuration of grain and material cross-section. Obviously, the immersion 
of grain into material, formed groove, side flow, side burrs, as shown in Fig. 9(i).  
SIMULATION OF CHIP-FORMATION BY A SINGLE GRAIN OF PYRAMID SHAPE.  
S. A. VORONOV, WEIDONG MA 
44 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCT 2016, VOL. 8. ISSN 2345-0533  
Finally, the cutting forces of chip-formation with a single grain are shown in Fig. 9. At step-1, 
cutting forces increase, when the grain moves forward, meanwhile, the cutting thickness also 
increases linearly with the position of the grain. In most cases of approximate calculation, the 
cutting force can be seen as proportional to the cutting thickness. 
At Step-2, cutting forces have some obvious waves with 4 peaks and 3 troughs, which are 
consistent with the appearance of chip separation into pieces in the simulation, as shown in Fig. 7. 
Comparing to the change of cross-sectional area of the contact grain and material (ܣ௖), as shown 
in Fig. 10, we can assume that, there is some correlation between cutting forces and ܣ௖. In this 
paper only the results of the simulated case for cutting thickness ℎ௖௨ = 40 μm are presented. The 
results are not enough to find the correlation between cutting conditions and forces. It’s necessary 
to add several simulations with different cutting thickness ℎ௖௨ in future and investigate in details. 
It will allow construct mechanistic model which can be used for vibration analysis under grinding. 
 
Fig. 9. The dependence of cutting forces (ܨ௭, ܨ௫) and cross-sectional area  
of the contact grain and material (ܣ௖) on the position of the grain 
6. Conclusions 
FEM model of metal cutting is analyzed; it confirms that approach ALE is very convenient to 
investigate chip-formation. The results of FEM simulation provide stress and temperature 
distribution, chip formation during the implantation of single grain. Configuration of cross-section 
is observed, cross-sectional area of the contact grain and material (ܣ௖)  is measured, and 
dependence of cutting forces on grain path is also given. 
References 
[1] Astakhov V. P., Shvets S. The assessment of plastic deformation in metal cutting. Journal of Materials 
Processing Technology, Vol. 146, 2004, p. 193-202. 
[2] Altintas Y., Weck M. Chatter stability in metal cutting and grinding. Annals of the CIRP, Vol. 53, 
Issue 2, 2004, p. 619-642. 
[3] Li Xuekun Modeling and Simulation of Grinding Processes Based on a Virtual Wheel Model and 
Microscopic Interaction Analysis. Dissertation, 2010, p. 4-12. 
[4] Kilicaslan Cenk Modelling and Simulation of Metal Cutting by Finite Element Method. Dissertation, 
2009, p. 22-24. 
[5] Ratchev S. M., Afazov S. M. Mathematical modelling and integration of micro-scale residual stresses 
into axisymmetric FE models of Ti6Al4V alloy in turning. CIRP Journal of Manufacturing Science 
and Technology, Vol. 4, 2011, p. 80-89. 
[6] Leseur D. R. Experimental Investigations of Material Models for Ti-6Al-4V Titanium and 2024-T3 
Aluminum. Technical Report, US Department of Transportation, Federal Aviation Administration, 
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[7] Kay Gregory Failure Modeling of Titanium 6Al-4V and Aluminum 2024-T3 with the Johnson-Cook 
Material Model. US William J. Hughes Technical Center, Washington, Vol. 9, 2003. 
[8] Bragov A., Konstantinov A. Experimental and numerical analysis of high strain rate response of 
Ti-6Al-4V titanium alloy. DYMAT International Conferences EDP Sciences, Vol. 7, 2009. 


Simulation of chip-formation by a single grain of pyramid shape

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