On the modelling of an aluminium alloy milling: 3D FEM approach/Aliuminio lydiniu frezavimo proceso modeliavimas: 3D BEM taikymas.

Asad, M. ; Mabrouki, T.

1. Introduction

Finite element (FE) modelling of machining processes had proved its effectiveness in unveiling multiphysical phenomena occurring at tool workpiece interface. These models are worthy in improving the production efficiency in terms of cutting tool geometry and optimal cutting parameters selection. FE-models are equally valuable in selecting and improving existing machine tools in aspect of their dynamic stability characteristics, to minimize the cutting vibrations [1]. Precious efforts had been started since 1970s to model the cutting process by FE methods. Initially, temperature fields in the chip and cutting tool were investigated [2]. Since then, valuable researches focusing various aspects of machining have been conducted and an exhaustive literature is now available. Nevertheless, complex nature of actual cutting phenomena and time consuming computing numerical techniques had restricted the researchers to limit their models to simplified 2D approaches with plane strain hypothesis. Later, assumption holds good for: depth of cut [a.sub.p] >> cutting feed (f) i.e. rough machining case. Whilst this hypothesis does not work well for: [a.sub.p] [approximately equal to] f i.e. for semi-finish and finish machining cases.

Under the foresaid machining cases, 3D models become inevitable to get the factual physical apprehension of ongoing processes. 3D models are also essential to realize some interesting features of cutting phenomena e.g. oblique machining [3], 3D cutting tool wear prediction [4] etc. Which otherwise are difficult to comprehend with 2D models.

In this framework, the present contribution put forwards a 3D FE approach to perform a parametric study highlighting the effects of depth of cut and cutting speed on surface and chip morphologies, for machining an aerospace grade aluminium alloy A2024-T351. Explicit approach of a FE code ABAQUS[R] (version 6.9.1) have been exploited to model the rough to finish machining operations for down cut milling. Numerical modelling and simulation work has been conceived in two successive steps. Primarily, a 3D model for rough down cut milling case, based on authors previously developed 2D model [5] was established. To validate the model, numerical results concerning chip morphology and cutting force were compared with the experimental data. Afterwards, numerical parametric study on the effects of lower [a.sub.p] values i.e. of the order of f (semi-finish and finish machining cases) and cutting speeds, on surface finish and chip morphology was conducted.

2. Three-dimensional FE model for orthogonal milling

2.1. Geometry, meshing and boundary conditions

In the present section the conceived geometry, boundary conditions, meshing, interactions and hypothesis to build a FE based 3D down-cut peripheral milling case are discussed. During the machining operation, cutting tool and workpiece come in contact. Numerically difficult to build contact and interaction definitions need special attention in developing FE based cutting models. To overcome contact complexities, the workpiece was modelled in three parts; chip, cutter path and machined part (Fig. 1). Tie-constraint algorithm (ABAQUS[R] built in algorithm) was used to assemble these parts. Once assembled, these parts behaved as a single entity "workpiece" and not as individual parts. Workpiece was meshed with thermally coupled continuum brick elements C3D8RT, to run coupled temperature-displacement calculations.

Literature study shows that whatever is the type of elements, mesh density plays a vital role to get physical results from FE based analyses. Unfortunately in the literature dealing with FE based cutting models, there is no defined criterion for an optimized mesh density. Mostly, very fine mesh (2-20 um) for complex plasticity problems is used. However, time penalty is quiet high for very fine meshes. Recently, Asad [6] in his doctoral work has performed a mesh sensitivity test for six different mesh densities for a 2D orthogonal cutting model and found an optimal mesh density for 27 x 27 [micro]m for the studied material. In the present work a mesh density of 28 x 28 x 40 [micro]m decreasing to 21 x 28 x 40 [micro]m has been conceived in the variable section of chip for down cut milling model. The cutting tool was assumed as a rigid body and was meshed with bilinear rigid quadrilateral elements R3D4.

[FIGURE 1 OMITTED]

Schematic representation of the conceived model, for 90[degrees] entering angle and 0[degrees] edge inclination angle is shown in Fig. 1. During the simulation, tool cutting edge was simultaneously orthogonal to the cutting and the feed velocities. This represents a three dimensional orthogonal cutting case.

Further, it can be seen in the figure that the workpiece is constrained with fixed boundary conditions. While tool can advance with feed velocity [V.sub.f] (feed rate f = 0.2 mm/tooth) in the negative X-axis direction and can rotate with angular velocity [omega] in the anticlockwise direction, simultaneously. A 25 mm diameter milling tool with two cutters was used in the present work. As the tool rotates and advances simultaneously, the cutter traces trochoidal path. This produces variable section chip with decreasing uncut chip thickness. To avoid the big efforts involved at lower uncut chip thickness values (with very fine mesh density), present model represents a 3D milling model with a radial depth of cut [a.sub.e] = 7.67 mm. This represents an uncut chip thickness (UCT) up to l60 [micro]m. The trochoidal path equations were used to model milling cutter path zone (cutter path/chip separation zone) and chip section geometry.

The conceived 3D cutting model employs well known Zorev's stick-slip friction model to define the frictional interaction between the chip and tool with an average friction coefficient [mu] = 0.17.

2.2. Material behaviour and chip separation model

Constitutive material modelling equations are the same as used in authors recent research work [7]. However, some necessary details are mentioned in the present paper. Jhonson and Cook (JC) equivalent stress model is employed in the model as presented by following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

While, JC shear failure model is used as a damage initiation criterion, as represented by following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where, A is the initial yield stress, B is the hardening modulus, C is the strain rate dependency coefficient, m is the thermal softening coefficient, n is the work-hardening exponent, T is the temperature at a given calculation instant, [T.sub.r] is the room temperature, [T.sub.m] is the melting temperature, [bar.[epsilon]] is the equivalent plastic strain, [??] is the plastic strain rate and [[??].sub.0] is the reference strain rate. [D.sub.1] to [D.sub.5] are the coefficients of JC material shear failure initiation criterion, p is the hydrostatic pressure, [bar.[sigma]] is the von Mises equivalent stress and p/[bar.[sigma]] is the stress triaxiality.

Damage is initiated when the scalar damage parameter [[omega].sub.0i] exceeds 1, based on Eq. (3):

[[omega].sub.0i] = [summation] [DELTA][bar.[epsilon]]/[[bar.[epsilon]].sub.0i]. (3)

Whereas, damage evolution parameter can be defined in the form a scalar stiffness degradation parameter D that can evolve linearly (Eq. (4)), used for cutter path section or exponentially (Eq. (5)), used for chip section:

D = L[[bar.[epsilon]]/[[bar.u].sub.f]] = [[bar.u]/[[bar.u].sub.f]; (4)

D = 1 - exp (-[sup.[bar.u][integral].sub.0] [bar.[sigma]]/[G.sub.f] d[bar.u]. (5)

Whereas, [DELTA][bar.[epsilon]] is equivalent plastic strain increment and [[bar.[epsilon]].sub.0i] plastic strain at damage initiation. L is characteristic length assumed to the cubic root of the integration point element volume. [G.sub.f] is fracture energy dissipation (required to open unit area of crack and is defined as a material parameter), [bar.u] is the equivalent plastic displacement and [[bar.u].sub.f] is the equivalent plastic displacement at failure expressed by following relation:

[[bar.u].sub.f] = 2[G.sub.f]/[[sigma].sub.y]. (6)

In ABAQUS[R], an element is deleted from the mesh if all of the section points at any one integration location have lost their load carrying capacity (D = 1). This is how the chip separation is realized from the workpiece. JC laws material entities and thermo-mechanical properties of the material used in the simulations are same as used in authors previous work [5]. These are specified in Table 1 and Table 2.

3. Results

In the present section numerical results concerning 3D down cut milling process of an aluminium alloy A2024-T351with the conceived 3D FE model (section 2) are discussed. Simulation results are presented in two steps. Initially, the results with 3D model for [a.sub.p] >> f (representing rough machining) are presented. The numerical results are compared with the experimental data in terms of chip morphology and cutting force. Subsequently, the results of the numerical investigations to study the effects of lower [a.sub.p] values i.e. of the order of f (representing semi-finish and finish machining) in high speed machining regime (cutting speeds [V.sub.C] = 800 and 1200 m/min) on surface finish and chip morphology are highlighted.

3.1. 3D numerical simulation for rough milling operation

Fig. 2 represents the chip morphology evolution for 3D down cut milling simulation, for cutting parameters: [a.sub.P] = 4 mm, f = 0.2 mm/tooth, [V.sub.C] = 800 m/min. It can be seen that, slightly segmented chip morphology (Fig. 2, a) is fairly comparable with the experimental one (Fig. 2, b). Big efforts are involved as UCT decreases in down cut milling case (with very fine mesh density), as already mentioned in section 2. Therefore, simulations were performed up to a radial depth of cut [a.sub.e] = 7.67 mm, corresponding to UCT = 160 [micro]n. Break line on the experimental chip figures out a chip thickness variation from 200 [micro]m up to 160 [micro]m. Evolution of the cutting force for 3D down cut milling case is depicted in Fig. 3. Numerically registered cutting force is globally comparable with the experimental one, under investigated cutting conditions of tool geometry and cutting parameters [8].

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

3.2. 3D numerical simulation for semi-finish and finish milling operations

Numerical results with the 3D model developed for rough machining, were found in reasonable correlation with the experimental ones. The model was then extended to semi-finish and finish machining cases. The global aim is to comprehend the multiphysical phenomena occurring in the vicinity of tool chip interface during semi-finish and finish machining operations, which help to generate good surface finish (as compared to rough machining) in industrial high speed machining. Therefore, in the following the effects of the variation of [a.sub.p] during high speed machining ([V.sub.C] = 800 and 1200 m/min) on chip morphology and surface texture are discussed.

Fig. 4 represents numerical simulation result on spatial displacement of nodes along Z-axis, U3 (i.e. along depth of cut [a.sub.p]) for cutting parameters: [a.sub.p] = 0.2 mm, f = 0.2 mm/tooth, [V.sub.C] = 1200 m/min. An average displacement of U3 = 0.0862 mm can be figured out. The percentage displacement of nodes along Z-axis (U3, %) comes out 43.1.

Table 3 represents U3 and %U3 when simulations were performed for other cutting parameters. It can be easily remarked that as [a.sub.p] decreases U3, % increases. This consequently, results in higher plastic strains along Z-axis.

The numerically registered values of plastic strain component along Z-axis (PE33) for various cutting parameters are shown in Table 3.

[FIGURE 4 OMITTED]

Increasing values of %U3 and PE33; as [a.sub.p] decreases suggests that, an extended and larger % volume plasticises at lower [a.sub.p] values .This results in an increase in material strength, because of the high requirement of inelastic dissipation of energy. This result is in consistence with recent research work of Liu and Melkote [9] on their study on material strengthening mechanisms and their contribution to size effect in micro cutting. They have shown in their 2D orthogonal machining numerical work that an edged radius tool widens the plasticisation zone in comparison to a sharp tool. This in turn requires higher energy dissipation, hence contributing to the size effect in micro cutting.

Frictional dissipation of energy, increases as cutting speed increases from 800 to 1200 m/min. This results in increasing the temperature leading to thermal softening. However, at these high cutting speeds strain rate hardening seems more dominant than the thermal softening phenomena, as can be deduced by the more regular and continuous chip morphology obtained at higher cutting speed (Fig. 5) in comparison with the one obtained at lower cutting speed (Fig. 2). An increase in both %U3 and PE33 values can also marked (Table 3) at higher cutting speed.

[FIGURE 5 OMITTED]

An insight observation of Fig. 2-5 and Table 3 suggests that as [a.sub.p] decreases and [V.sub.C] increases, material strengthens by higher inelastic dissipation of energy and strain rate hardening phenomena. This generates a smooth continuous chip morphology (Fig. 4), if compared with one produced with higher [a.sub.p], and lower [V.sub.C] values (Fig. 2).This in turn results in fine quality machined surface topology in high speed finishing operations, as shown in Fig. 6. This result is in good relation with findings of Mabrouki et al. [7]. They have shown in their numerical and experimental work on orthogonal machining that chip morphology dictates the quality of machined surface.

Figs. 6-8 represent the displacement of machined surface nodes along Z-axis for [a.sub.p] = 0.2, 1 and 4 mm, respectively at two UCT and cutting speed values.

Generally, it can be seen that a decrease in [a.sub.p] results in smoother machined surface textures. Conversely, numerical simulation results with high [a.sub.p] depicts comparatively rough undulated surface texture.

It can also be observed in Figs. 6-8 that, as UCT decreases for down cut milling process surface quality improves. This can be attributed to the evolution of the chip morphology during milling operation. For example, in

Fig. 2 initially a segmented chip and onward at lower UCT a continuous (non segmented) chip morphology is obtained.

In this context, Nakayama and Tamura [10] believe that, as UCT reduces, shear plane angle becomes very small leading to greater plastic energy dissipations in the workpiece subsurface, thus strengthens the material.

While, Liu and Melkote [11] consider that a decrease in secondary deformation zone temperature contributes dominantly to strengthen the material as UCT decreases. Presence of high strain gradients at lower UCT also strengthens the material [5]. This shows that, multiple phenomena strengthen the material as UCT decreases, leading to continuous chip and smoother surface texture.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

At high cutting speeds improved machined surfaces are obtained, as can be remarked in Figs. 6-8. This corresponds to smoother chips (non segmented) obtained at higher cutting speeds.

Finally, it can be stated that 3D cutting models are necessary for better comprehension of machining process, for instance for semi-finish and finish cutting operations, etc.

4. Conclusions

A 3D FE based numerical model for down cut milling process to investigate influence of cutting speed and depth of cut on chip morphology and surface finish has been developed. The prime objective is to bring comprehension of physical phenomena accompanying chip formation, which help to generate a smooth continuous (non segmented) chip morphology and better surface texture in semi-finish and finish cutting operations in high cutting speed regime.

Numerical simulation results show that the spatial displacement of nodes along Z-axis (along depth of cut) increases as depth of cut decreases (towards finish cutting). This eventually represents an extended and widened percentage of volume undergoing plastic deformation, resulting in higher dissipation of inelastic energy. The results also depict that material strain rate hardening characteristics increase the material strength at higher cutting speeds for the studied material. These strengthening phenomena help to generate a continuous chip and improved surface topology in high speed finishing operations.

Finally, the present study highlights only few of the many mutilphysical phenomena leading to high quality machined surface, during high speed semi-finish and finish machining operations. However, this contribution will allow an improvement in the existing cutting models and will help to optimize the cutting conditions. In future, effects of strain gradient hardening, tool geometries and machining conditions on 3D machined surface topology shall be focused.

Received July 09, 2012

Accepted October 10, 2013

References

[1.] Asad, M.; Mabrouki, T.; Rigal, J.F. 2010. Finite-element-based hybrid dynamic cutting model for aluminium alloy milling, Proceedings of IMechE Part b: Journal of Engineering Manufacture 224(1): 1-13. http://dx.doi.org/10.1243/09544054JEM1590.

[2.] Tay, A.O.; Stevenson, M.G.; Davis, G.V. 1974. Using the finite element method to determine temperature distributions in orthogonal machining, Proceedings of IMechE 188 (55): 627-638. http://dx.doi.org/10.1243/PIME_PROC_1974_188_074 _02.

[3.] Ceretti, E.; Lazzaroni, C.; Menegardo, L.; Altan, T. 2000. Turning simulations using a three-dimensional FEM code, Journal of Material Processing Technology 98: 99-103. http://dx.doi.org/10.1016/S0924-0136(99)00310-6.

[4.] Attanasio, A.; Ceretti, E.; Rizzuti, S.; Umbrello, D.; Micari, F. 2008. 3D finite element analysis of tool wear in machining, CIRP Annals Manufacturing Technology 57: 61-64. http://dx.doi.org/10.1016/j.cirp.2008.03.123.

[5.] Asad, M.; Mabrouki, T.; Girardin, F.; Zhang, Y.; Rigal, J.F. 2011. Towards a physical comprehension of material strengthening factors during macro to micro scale milling, Mechanika 17(1): 97-104. http://dx.doi.org/10.5755/j01.mech.17.L210.

[6.] Asad, M. 2010. Elaboration of concepts and methodologies to study peripheral down-cut milling process from macro-to-micro scales, PhD thesis, Insa Lyon France.

[7.] Mabrouki, T.; Girardin, F.; Asad, M.; Rigal, J.F. 2008. Numerical and experimental study of dry cutting for an aeronautic aluminium alloy (A2024-T351), International Journal of Machine Tools and Manufacture 481(11): 187-197.

[8.] Zaghbani, I.; Songmene, V. 2009. A force-temperature model including a constitutive law for dry high speed milling of aluminium alloys, Journal of Material Processing Technology 209(5): 2532-2544. http://dx.doi.org/10.1016/j.jmatprotec.2008.05.050.

[9.] Liu, K.; Melkote, S.N. 2007. Finite element analysis of the influence of tool edge radius on size effect in orthogonal micro-cutting process, International Journal of Mechanical Science 49(5): 650-660. http://dx.doi.org/10.1016/j.ijmecsci.2006.09.012.

[10.] Nakayama, K.; Tamura, K. 1968. Size effect in metal cutting force, Journal of Engineering for Industry 90(1): 119-126. http://dx.doi.org/10.1115/1.3604585.

[11.] Liu, K.; Melkote, S.N. 2006. Material strengthening mechanisms and their contribution to size effect in micro cutting, Journal of Manufacturing Science and Engineering 128(3): 730-738. http://dx.doi.org/10.1115/1.2193548.

M. Asad *, T. Mabrouki **

* Center of excellence in science and applied technologies, Islamabad, Pakistan, E-mail: [email protected]

** Universite de Lyon, INSA-Lyon, LaMCoS, UMR5259, F69621, France, E-mail: [email protected]

http://dx.doi.org/10.5755/j01.mech.19.5.5538

Table 1
Johnson-Cook material behaviour and damage parameters

A, Mpa   B, MPa    n       C      m   [D.sub.1]   [D.sub.2]

352       440     0.42   0.0083   1     0.13        0.13

A, Mpa   [D.sub.3]   [D.sub.4]   [D.sub.5]

352        -1.5        0.011         0

Table 2
Workpiece thermo-mechanical properties

Physical parameters                      Workpiece (A2024- T351)

Density [rho], kg/[m.sup.3]                        2700

Young's modulus E, MPa                            73000

Poisson's ratio v                                  0.33

Fracture energy [G.sub.f], N/m                     20E3

Specific heat [C.sub.p],                       0.557T+877.6
[Jkg.sup.-1] [degrees][C.sup.-1]

Expansion coefficient                      [8.910.SUP.-3]T+22.2
[[alpha].sub.d], [micro]m
[m.sup.-1] [degrees][.sup.-1]

Thermal conductivity [lambda],         25 [less than or equal to] T
W [m.sup.-1][C.sup.-1]                 [less than or equal to] 300:
                                         [lambda] = 0.247T+114.4
                                      300 [less than or equal to] T
                                    [less than or equal to] [T.sub.m]:
                                         [lambda] = 0.125T + 226

Meting temperature, [T.sub.m],                     520
[degrees]C

Room temperature, [T.sub.r],                        25
[degrees]C

Table 3
Numerical simulation results for f = 0.2 mm/tooth

[V.sub.c],     Cutting     [a.sub.p],   Avg. U3,       U3 =       PE33
m/min         operation        mm          mm      =(U3/) x
                                                   [a.sub.p]
                                                    x 100, %

800             Rough          4         0.0713        1.78        0.2
             Semi-finish       1         0.0632        6.32       0.21
               Finish         0.2        0.0833        41.6       0.533

1200            Rough          4         0.0725        1.81       0.198
             Semi-finish       1         0.0676        6.76       0.232
               Finish         0.2        0.0862        43.1       0.584