Numerical and Analytical Calculations for Modeling and Designing Drilling Wicks or Rotary Cutters Based of Functionally Graded Materials

1Laboratory of Material Sciences, Department of Physics, Faculty of Sciences, University of Yaoundé 1, P.O. BOX 812, Yaoundé, Cameroon 2Department of Mine, Mineral Processing and Environment, School of Geology and Mining Engineering, University of Ngaoundéré, P.O. BOX 115, Meiganga, Cameroon 3Douala Institute of Technology, P.O. BOX, 1623 Douala, Cameroon 4Department of Physics, Faculty of Sciences, University of Douala, P.O. BOX 24157, Douala, Cameroon


Introduction
In exploration and drilling activities, a large majority of mining companies are often confronted with the rapid wear and premature deterioration of drilling wicks, cutter, and strawberries.This problem is mainly due first to the tenacity of media to drill, but then also to the alternation of different backgrounds and the diversity of characteristics they have.This requires for drilling tools, especially strawberries and wicks, performance adapted to the requirements of the media.The design and construction of such tools are therefore an important and constant challenge.It has long been that use is made of composite materials such as tungsten-carbides, tungsten-copper, or titanium, due to their very high resistance, to manufacture most of the cutting and drilling devices [1].However devices with a variety of properties to meet the above requirements may be possible by using Functionally Graded Materials (FGM) which have already helped in solving such problems in many applications [2].FGM are else purposely graded with respect to their mechanical respective elastic properties or variation of these properties is present since the density or the elasticity varies depending on materials composition [1].We must remember that many structural components meet very different operating conditions and, therefore, require materials adapted for performances that vary with location within the component.For example, a kitchen knife needs to be hard just to its cutting edge; however, the material from which it is produced must be particularly strong and hard.Similarly the body of a gear has to be solid, so that its surface should be hard and resistant to wear.The body of a turbine blade must be strong, hard, and resistant to creep, while its outer surface must be refractory and resistant to oxidation [3].It is therefore believed that even the hollow cylinders, built from FGM although appropriate, can provide solutions to applications with different characteristics and requirements which they are subject to.If one thinks of a drilling wick as a rotating hollow cylinder, one can assume that its walls are built of FGM.Optimizing the design of such a cylinder requires an understanding of the elastoplastic behavior of the material [4].Many numerical and analytical works have been done on this subject, especially on uniform rotating discs, to predict the stress field, deformations, and rupture, to avoid the risk of damage when the device is subjected to different loading conditions [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].In 2005, Ahmet N. Eraslan et al. [4] obtained in the context of the small deformation theory analytical solutions on flat deformations in elastic tubes operably under pressure grades.In 2011, Hasan Callioglu et al. [5] analyzed the state of the stresses on the rotating annular disc made of Functionally Graded Materials (FGM).Similary in 2011, Hassani et al. [6] obtained distributions of stress and strain components of rotating hyperbolic disks with nonuniform material properties subjected to a power form thermoelastic loading under different boundary conditions by semiexact methods of Liao's homotopy analysis method.In 2012, Roberta Sburlati [7] obtained analytical solutions for thick wall cylinders subjected to internal and external pressure, where the entire wall was made of FGM.In 2013 and 2014 Nejad et al. [8,9] presented a closed-form analytical solution in terms of hypergeometric functions to elastic analysis of exponentially functionally graded stationary discs subjected to internal and external pressures.In 2016 Yildirim [10] investigated the exact elastic response of a rotating disk having a continuously varying hyperbolic thickness profile under different boundary conditions.Both convergenthyperbolic and divergent-hyperbolic disk profiles together with uniform profile are all studied.Power-law grading is used for material gradation pattern.In 2017, Wenlong Shen et al. [11] investigated a new simple mathematical method to predict rock stress around a noncircular tunnel and the method is calibrated and validated with a numerical model.It can be found that the tunnel shapes and polar angles affect the applicable zone of the theoretical model signiïňĄcantly and the applicable zone of a rectangular tunnel was obtained using this method.Recently in 2018, E. R. Atangana Nkene et al. [12] studied analytically and numerically strain fields, stress fields, and displacements in a rotating hollow cylinder, whose walls were completely made of Functionally Graded Materials (FGM).Their results showed that tangential stress, tangential strain, and displacements are higher at the inner surface and internal radial pressure strongly affects the radial stresses and radial strain.
However, to the best of our knowledge, theoretical and numerical research on deformation states and displacement in a hollow inhomogeneous rotating cylinder subjected to an axial load, as it is the case for strawberries and drill wicks, remain unaddressed.In this work we use numerical and analytical calculations for modeling and designing drilling wicks or rotary cutters based of Functionally Graded Materials.
The paper is organized as follows.Modelisation of a drilling device made of FGM is firstly presented.Analytical equations obtained from modelisation are presented and solved in Section 3. Section 4 presents numerical and analytical results, and then the work is concluded in Section 5.

Modelisation of a Drilling Device Made of FGM.
Let us consider a drilling device as a hollow FGM circular cylinder with length L and subjected to the action of a uniform axial loading   .The inner radius and the outer radius of the hollow cylinder are, respectively,  and , as shown in Figure 1.
In this paper we assume that the material is linearly elastic and isotropic and its Young's Modulus depends only on the radial direction.According to Horgan we assume that elastic modulus E and density  are given by a power law form function [15]: with  0 and  0 are reference values of  and, respectively, ,  is the outer radius,  is the radial coordinate, and the exponent  can be a positive or a negative real number reflecting the degree of nonuniformity of the material.Let us consider as in [8][9][10][11][12][13][14] that, due to the axisymmetric load condition of the drilling device, the circumferential component of the displacement is zero, and the radial component  depends on the radial distance  only correspondingly; the radial and circumferential components of the stresses   and   are also -dependent only.We suppose that the length L of the drilling device is very high compared to its radius.

Stress and Strain Analysis.
The second Newton's Law applied to system presented in Figure 1 gives the following equations: By considering a drilling device an axisymmetric problem, (2) becomes where  is the angular velocity of cylinder,  is the body force of the material per unit volume, and   is external mechanical load.By taking account of the small deformations and due to the rotational symmetry, the strain-displacement relations become By applying Hookes' Law to system (see Figure 1), relations of stress-strain are given by [12] where ] is Poisson's Coefficient Substituting (5) into (3), the equation of displacement is given by [12]   2    2 + ( + 1) This equation can be written as The next section is devoted to solving analytically and numerically (7).

Analytical and Numerical Solution
3.1.Analytical Solution.Analytical solution of ( 7) is given as in [12] by where and  1 and  2 are the constants of integration.In the case of homogenous material, (8) becomes By combining ( 8) and ( 4), we obtain the components of strain tensor as in [12]: By combining (10) and ( 5), we obtain the components of stress tensor as in [12]: The Boundary Conditions.In this part we use the mechanical boundary conditions to determine the constants  1 and  2 .
It can be expressed as By substituting ( 12) into (11), the constants are given by where In the case of homogenous drilling device, (13) becomes In order to determine the constant   , we use the expression of the Z component the force   given by where As it is the case in drilling devices or drilling systems, we suppose that the rotating hollow cylinder is subjected to a hydraulic force   .Let assume that this force is constant.By solving ( 16) we obtain where  = a (inner radius) and  = b (outer radius).

Numerical Solution.
In the next part, we now study the influence of the force   and an inhomogeneous parameter  on the stress, strain, and displacement in the hollow cylinder.We use therefore the constants given by [ Numerical Schemes.Let consider (7): By applying the boundary conditions (7) can be transformed as where  1 =  and  2 = / =   = (,  1 ,  2 ).
Equation ( 22) is transformed into a set of differential equations that are solved using the shooting method and the fourth-order Runge-Kutta algorithm.The analytical and numerical solutions obtained are thus presented.

Numerical and Analytical Results
In Figures 2, 3, 4, 5, 6, and 7, the numerical results are plotted in green triangle and analytical results are plotted in red, blue, and black colors, respectively, for Fz = 10 kN, 50 kN, and 100 kN.In these figures we observe the perfect correlation between analytical results and numerical results.There is no gap between these curves.Then, in this section, comments and analysis made for analytical results are valid for numerical results.
Figure 2 presents the variation of the displacement along the radial direction of the inhomogeneous hollow cylinder submitted to an axial loading.
In Figure 2(a) we see that, for n=0.5, the displacement increases gradually with the force   .Figure 2(b) shows that, for n=-0.5, the displacement decreases gradually with increasing the force   and has its lowest value when   reaches its highest value.For n=0.5 (greater than zero), the highest value of   gives the highest value of the displacement.In addition, according to the boundary conditions, the radial displacements are zero at the inner and outer surfaces.From r=0.08 m to r=0.1 m, the displacement increases gradually and reaches the maximum values 10 -9 m and 7.5 x 10 -9 m at r=0.9, respectively, for n=0.5 and n=-0.5 and then decreases to zero.Moreover the displacements increase when the inhomogeneous parameter n increases.They have lower value for n=-0.5 when compared with the results for n=0,5.
Figure 3 presents the variation of radial strains along the radial direction of an inhomogeneous hollow cylinder submitted to an axial loading.We see that the radial strain decreases linearly along the radial direction.All radial strains are higher at the inner surface.
We observe that the force   does not much affect the radial strain, in particular when  ≈ ( + )/2.In Figure 3(a), for  = [0.8− 0.85] radial strains increase when   increases and for  = [0.95− 1] radial strains decrease when   increases.In Figure 3(b) for  = [0.8− 0.85] radial strains decrease when   increases and for  = [0.95−1]radial strains increase when   increases.Moreover radial strains increase when the inhomogeneous parameter n increases.They have lower value for n=-0.5 when compared with the results for n=0,5.
Figure 4 illustrates the variation of the tangential strain along the radial direction of the inhomogeneous hollow cylinder.We can observe that tangential strains   are zero at the inner and outer surfaces.From r= 0.08 m to r= 0.1 m, tangential strains gradually increase and reach the maximum values 11.5 x 10 -8 and 8.5 x 10 -8 at r= 0.9, respectively, for n= 0.5 and n= -0.5 and then decrease to zero.Moreover they increase when the inhomogeneous parameter n increases, and have lower value for n= -0.5 when compared with the results for n= 0.5.We can see that, in Figure 4(a), for n= 0.5, tangential strains   increase gradually with the force   and in Figure 4(b), for n= -0.5, they decrease gradually with   .For n= -0.5 (less than zero), the highest value of   gives the lowest value of strain, and for n= 0.5 (greater than zero) the highest value of   gives the highest value of tangential strain.
We note in Figure 5 that the vertical stresses vary linearly along the radial direction of the cylinder.For n=-0.5, the vertical stresses decrease slowly with radius (r) and for n=0.5, vertical stresses increase slowly in radial direction.For n=-0.5, the maximum values of vertical stresses are 8.6 MPa, 4.2 MPa, and 1 MPa, respectively, for   =100 kN, 50 kN, and 10 kN.For n=0.5, the maximum values of vertical stresses are 10 MPa, 5 MPa, and 1 MPa, respectively, for   =100 kN, 50 kN, and 10 kN.For n= -0.5 and n= 0.5, the vertical stresses increase with the force   .In addition the inhomogeneous parameter n does not much affect the vertical stresses.
As can be seen in Figures 6 and 7, radial and tangential stresses decrease linearly along the radial direction.They are higher at the inner surface and vary with parallel curves in the radial direction of the cylinder.In addition the radial and tangential stresses decrease when the inhomogeneous parameter  increases and gradually increase when the force    increases.Moreover the radial stresses curves save the parallelism when   and the inhomogeneous parameter n change.

Conclusion
In this paper, plane stresses, strains, and displacements along the radial direction of an inhomogeneous hollow rotating cylinder under axial loading have been analyzed.Analytical and numerical results obtained lead us to conclude the following: (i) The stresses, strains, and displacement obtained from analytical and numerical solutions are in good correlation.Analytical curves and numerical curves are merged.There is no gap between these curves.This confirms the veracity of numerical and analytical results.
(ii) The force   strongly affects the radial and tangential stresses.
(iii) Vertical stresses do not much change in the radial direction.
(iv) The magnitudes of radial stresses are higher than those of tangential stresses.
(v) Tangential stresses, radial stresses, tangential strain, and displacements are higher at the inner surface.
(vi) The variation of the mechanical and elastic properties, due to the graduation of the Functional Graded Material (FGM), strongly affects the distribution of tangential and radial stresses.
The results obtained in this work are certainly of great importance, as they can contribute to the modeling and designing of drilling strawberry and wicks or rotary cutters.

Figure 1 :
Figure 1: A drilling device made of FGM subjected to the action of a uniform axial load   .