Towards a physical comprehension of material strengthening factors during macro to microscale milling

Worthy scientific researches have been made in past for the physical comprehension of the size effect phenomenon in the domain of micro cutting processes [1-3]. The term size effect in machining is usually attributed to define the nonlinear increase in specific cutting energy (SCE) when uncut chip thickness (h) decreases to few microns. Numerous investigations have shown that there are multiple factors that can increase material strength and contribute to the size effect in micro machining operations. From material point of view, Backer et al. [1] attributed the size effect to the reduction in material imperfections when deformation takes place on small volume. While, LarsenBasse and Oxley [2] highlighted the importance of the increase in strain rate in primary shear zone with a decrease in uncut chip thickness, as the primary cause in increasing material strength. Dinesh et al. [3] explained the increase in hardness of metallic materials with the decrease in deformation depth, as the consequence of the strong dependence of flow stress on strain gradient (SG) in the deformation zone. Based on their work Joshi and Melkote [4] had presented an analytical model for orthogonal cutting incorporating SG effects in material constitutive law. It have also been reported in previous researches that the cutting tool edge radius is the major cause of size effect [5, 6]. In this regard Nakayama and Tamura [6] believe that, as h reduces to micrometric level, tool edge radius becomes comparable or some times greater than chip thickness. Under these conditions shear plane angle becomes very small leading to greater plastic energy dissipations in the workpiece subsurface. Liu and Melkote [7] have recently shown in their numerical work that plastic shear zone generated by an edged radius tool is more expanded and widened when compared with the one produced by a sharp tool. This in turn requires higher energy dissipation, hence contributing to the size effect. Other researchers [8, 9] believe in the existence of ploughing forces associated with the frictional rubbing and ploughing mechanism as the main reason of increase in SCE with decrease in uncut chip thickness. Some researches had attributed the size effect to an increase in shear strength of the workpiece material due to a decrease in tool-chip interface temperature as h decreases [10, 11]. In this context, Liu and Melkote [10] showed in their recent micro cutting simulation work that the decrease in secondary deformation zone temperature contributes dominantly to the size effect as h decreases. Nevertheless, in literature most of the studies concerning tool-workpiece interaction [12, 13] are based on mechanistic modelling approach, which can not explain satisfactorily the physics of size effect phenomenon, as h decreases from macro to micro level. Only a few studies have been made using FE method [10, 11]. In this framework the present contribution put forwards a numerical approach based on FE method to study the dominance of strain hardening characteristics, in increasing material strength. The case of down-cut milling process where the h decreases from macro to micro level dimensions was treated. The phenomenological reason explaining on the one hand that how material strain hardening characteristics influence the size effect and on the other hand the nonlinear increase in SCE for a particular h value has been addressed. Moreover, to study the contribution of SG on the size effect under high cutting speeds, modified JohnsonCook (JC) expression of the equivalent stress [14] (incorporating SG-based plasticity approach) has been formulated in ABAQUS/EXPLICIT via its user subroutine VUMAT. Milling experiments have been carried out to compare the numerical results of SCE and chip morphology.


Introduction
Worthy scientific researches have been made in past for the physical comprehension of the size effect phenomenon in the domain of micro cutting processes [1][2][3].The term size effect in machining is usually attributed to define the nonlinear increase in specific cutting energy (SCE) when uncut chip thickness (h) decreases to few microns.Numerous investigations have shown that there are multiple factors that can increase material strength and contribute to the size effect in micro machining operations.From material point of view, Backer et al. [1] attributed the size effect to the reduction in material imperfections when deformation takes place on small volume.While, Larsen-Basse and Oxley [2] highlighted the importance of the increase in strain rate in primary shear zone with a decrease in uncut chip thickness, as the primary cause in increasing material strength.Dinesh et al. [3] explained the increase in hardness of metallic materials with the decrease in deformation depth, as the consequence of the strong dependence of flow stress on strain gradient (SG) in the deformation zone.Based on their work Joshi and Melkote [4] had presented an analytical model for orthogonal cutting incorporating SG effects in material constitutive law.
It have also been reported in previous researches that the cutting tool edge radius is the major cause of size effect [5,6].In this regard Nakayama and Tamura [6] believe that, as h reduces to micrometric level, tool edge radius becomes comparable or some times greater than chip thickness.Under these conditions shear plane angle becomes very small leading to greater plastic energy dissipations in the workpiece subsurface.Liu and Melkote [7] have recently shown in their numerical work that plastic shear zone generated by an edged radius tool is more expanded and widened when compared with the one produced by a sharp tool.This in turn requires higher energy dissipation, hence contributing to the size effect.Other researchers [8,9] believe in the existence of ploughing forces associated with the frictional rubbing and ploughing mechanism as the main reason of increase in SCE with decrease in uncut chip thickness.Some researches had attributed the size effect to an increase in shear strength of the workpiece material due to a decrease in tool-chip interface temperature as h decreases [10,11].In this context, Liu and Melkote [10] showed in their recent micro cutting simulation work that the decrease in secondary deformation zone temperature contributes dominantly to the size effect as h decreases.
Nevertheless, in literature most of the studies concerning tool-workpiece interaction [12,13] are based on mechanistic modelling approach, which can not explain satisfactorily the physics of size effect phenomenon, as h decreases from macro to micro level.Only a few studies have been made using FE method [10,11].In this framework the present contribution put forwards a numerical approach based on FE method to study the dominance of strain hardening characteristics, in increasing material strength.The case of down-cut milling process where the h decreases from macro to micro level dimensions was treated.The phenomenological reason explaining on the one hand that how material strain hardening characteristics influence the size effect and on the other hand the nonlinear increase in SCE for a particular h value has been addressed.Moreover, to study the contribution of SG on the size effect under high cutting speeds, modified Johnson-Cook (JC) expression of the equivalent stress [14] (incorporating SG-based plasticity approach) has been formulated in ABAQUS ® /EXPLICIT via its user subroutine VUMAT.Milling experiments have been carried out to compare the numerical results of SCE and chip morphology.

Geometrical model and hypothesis
A FE based numerical model for 2D orthogonal down cut peripheral milling case for an aluminium alloy A2024-T351 has been conceived in ABAQUS ® /EXPLICIT software (version 6.7.1).To simplify the problem of cutting to 2D case many assumptions have been made and which can be summarised as follows.
• Helix angle for the used insert is small, λ s = 9°.This angle does not affect too much on F z (force component along tool axis of rotation).For the studied cutting speeds, F z was noted less than 10% of the total applied force tot F , and can be ignored.Therefore, axial depth of cut (workpiece thickness, a P = 4 mm) may be assumed to be constant.
• As diameter of milling tool (D T = 22 mm) is greater than workpiece deformation area (chip and cutter path zone), therefore, the deformed area can be considered as a case of orthogonal machining process as shown in Fig. 1, which demonstrates a schematic representation of the 2D down-cut peripheral milling model with mesh densities (optimal mesh 15 -40 µm) at various h values and toolworkpiece boundary conditions.The pre-cited assumptions have also been adopted by Xinmin et al. [14] in building their numerical model for orthogonal micro milling process simulation.The conceived FE model is based on quadrilateral continuum elements CPE4RT with which it was possible to perform a coupled temperature-displacement calcula-tions [15].A relatively lower value of feed rate f when compared with the axial cutting depth a P , had allowed to perform plane strain calculations.The geometrical model concerning the present study is presented in (Fig. 1).The thickness of cutter path band was assumed to the order of tool edge radius (R n = 20 µm) [16].For the present study, a macro milling tool with the diameter of D T = 22 mm was used.Its geometry is exactly the same as that had been used in experimentation (shown later in section 3) with a rake angle γ 0 = 30° and a flank angle α 0 =11°.The tool has been modelled as rigid body and all the boundary conditions were applied to its centre of rotation so that it can advance with feed velocity V f (feed rate f = 0.2 mm/tooth) in negative x-axis direction.It can also rotate in anti-clockwise direction with an angular velocity ω r whereas the rest degrees of freedom have been blocked for tool motion.As the tool rotates and advances simultaneously, the cutter traces a trochoidal path.This produces a chip of variable section in milling process.Trochoidal path set given by Eq. ( 1) was used to model milling-cutter path (chip separation zone) and chip section geometry.
where P x(i) and P y(i) are x and y coordinates of the ith tooth of the milling tool (i = 1, 2, …, z t , where z t is the total number of tool teeth) and t is the cutting time.
In the present study, Zorev's stick-slip friction model [17] being one of the most commonly used approximations for frictional contact between the chip and tool for a critical shear stress of 203 MPa has been used.Whereas an average friction coefficient 0.17; as measured by Ni et al. [18] on pin-on-disk tests using a high temperature tribometer for diamond-like carbon coatings against aluminum alloys, has been assumed in all simulations.

Material behaviour law
The constitutive material model equations are the same as those used in previous research works [14,19].However, some details have to be mentioned in the present contribution.Indeed, the material behaviour model without considering SG is the one proposed by JC [20] presented by the expression of the equivalent stress (Eq.( 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, ε is the equivalent plastic strain, ε is the plastic strain rate and 0 ε is the reference strain rate.
In the proposed numerical model, chip formation is realised in two steps.The first step concerns the damage initiation, whereas the second one concerns damage evolution.

Damage initiation step:
The JC shear failure model (Eq.( 3)) is used as a damage initiation criterion.
where D 1 , …, D 5 are the coefficients of JC material shear failure initiation criterion, p is the hydrostatic pressure, σ is the von Mises equivalent stress and p σ is the stress triaxiality.
Damage is initiated when the scalar damage parameter 0i ω exceeds 1, based on equation (4).
where ε Δ is the equivalent plastic strain increment and 0i ε plastic strain at damage initiation.Damage evolution step: Hillerborg et al. [21] fracture energy proposal is used in ABAQUS ® /EXPLICIT to reduce mesh dependency by creating a stress-displacement response after damage initiation.The fracture energy G f required to open unit area of crack is defined as a material parameter.With this approach, the material softening response after damage initiation is characterized by a stressdisplacement response which requires the definition of a characteristic length L assumed to the square root of the integration point element.According to this law, the damage evolution law can be specified in terms of fracture energy dissipation G f .The law could be defined in the form a scalar stiffness degradation parameter D that can evolve linearly (Eq.( 5)) used for cutter path section or exponentially (Eq.( 6)) used for chip section.
Where u is the equivalent plastic displacement and f u is the equivalent plastic displacement at failure given by: In ABAQUS ® , an element is removed from the mesh if all of the section points at any one integration location have lost their load-carrying capacity (D = 1).Thus chip detachment is realised from the workpiece.JC laws material entities used in the numerical model are specified in Table 1 [22].Thermo-mechanical properties of the material are given in Table 2 [23,24].
Table 1 Johnson-Cook material behaviour and damage parameters for A2024-T351 [22] A, MPa 352 440 0.42 0.0083 1 0.13 0.13 -1.5 0.011 0 Taylor factor M t 3.06 [26] Poisson's Ratio ν 0.33 °C 25 In order to consider the SG-strengthening effects on the size effect phenomenon in machining, Xinmin et al. [14] proposed a new expression of the equivalent stress based on SG-plasticity theory.The framework of their proposed constitutive equation is expressed as where ref σ = JC σ and η (effective plastic strain gradient) which is inversely proportional to the length of the primary shear zone, L P, representing the fundamental length scale governing the size effect when L P reaches macro level σ = JC σ .The constitutive material model for orthogonal machining is mainly based on the Taylor dislocation density model [25] which gives the shear flow τ in terms of the total dislocation density ρ Τ by Eq. ( 9).
The total dislocation density ρ Τ characterising the material hardening is the sum of two densities as flows: where s ρ characterises the Statistically Stored Dislocations (SSD), which is determined by the material test in the absence of strain gradient consideration according to the following equation ( ) The flow stress σ is related to the shear flow stress τ by: Where M t is the Taylor factor which acts as isotropic interpretation of the crystalline anisotropy at the continuum level.For FCC as well as for BCC metals that slip on {1 1 0} planes M t is taken as 3.06 [26] and 3 for an isotropic solid [27].Whereas g ρ con- cerns the Geometrically Necessary Dislocations (GND), which are required for compatible deformation of various parts of the nonuniformly deformed material.It is these GNDs and their effects on the flow stress that are considered in the present study.The flow stress σ in terms of dislocation densities can be expressed as: According to Ashby [28] the total density of dislocations T ρ , which is given by the sum of statistically stored and geometrically necessary dislocations, is a special case of the following equation To properly estimate T ρ , Ashby [28] has proposed that the exponent χ (geometric factor defining the density of geometrically necessary dislocations (GNDs)) should be less than or equal to 1. Joshi and Melkote [4] believe that, due to large strain gradients commonly met in machining, a lower value of χ is reasonable to introduce in Eq. (13).So, in the general case the material flow stress can be given by: Substituting Eq. ( 11) into Eq.( 14), the flow stress can be written as (Eq.( 15) ( ) The density of geometrically necessary dislocations g ρ is related to the gradient of plastic strain by g r b η ρ = [28,29].Where r is the Nye factor introduced by Arsenlis and Parks [30] to reflect the effect of crystallography on the distribution of GNDs, and r is around 1.90 for FCC polycrystals, 1.85 and 1.93 for bending and torsion, respectively [31].Xinmin et al. [14] have considered its value as 2 for the case of machining.As discussed above that σ JC is selected to be σ ref the constitutive equation turns to be as follows ( ) where JC σ is the JC equivalent stress and y σ is the yield stress.
From the works of Joshi and Melkote [4], the SG coefficient is obtained through the dislocation analysis of primary shear zone for micro scale machining, as follows where L P is the length of the primary shear zone which can be calculated by ( ) with φ is the primary shear angle (deg).
Finally, the constitutive equation can be expressed as: Since the strain gradient η is the inverse of L P , it will decrease with the increase of h.This suggests that, when size variable L P reaches to macro level, the flow stress calculated using Eq. ( 18) will be equal to that given by JC model (Eq.( 2)).
In order to take into account the influence of SGstrengthening on the size effect phenomenon during downcut peripheral milling simulations, the SG-based equivalent stress must be employed (Eq.( 18)).Nevertheless, this theory is not available in the adopted FE analysis software.It is therefore necessary to implement the Eq. ( 18) with damage initiation (Eqs.( 3) and ( 4)) and damage evolution models (Eqs.( 5) -( 7)) in the form of a user-defined stress update algorithm known as subroutine VUMAT formulated in ABAQUS ® /EXPLICIT.After validating the precited user routine, the constitutive equation including SG (Eq.( 18)) was exploited.

Milling experiments
Down-cut peripheral milling experiments for a fixed f = 0.2 mm/tooth and different V C : 200, 400 and 600 m/min were performed on DMG, 3AXIS machining centre.
A Mitsubishi® milling tool (D T = 22 mm), refer-enced 223WA20SA, with coated carbide insert referenced AOMT123608PEER-M was used.The geometry of the insert (Fig. 2, a) was measured by using an optical measuring device.This has helped to generate a 2D geometry of the insert (Fig. 2, b).This insert profile was afterwards used in the FE model of milling (Fig. 1).While, workpiece is an aluminium alloy plate A2024-T351 with the thickness of 4 mm fixed on a standard dynamometer Kistler® 9257A.Fig. 5 shows the force diagram for orthogonal down-cut milling case.From the decomposition of forces, cutting force F c acting tangentially to cutting speed V C (m/min) can be calculated by Eq. ( 19).
Eq. ( 20) was then used to calculate the SCE at various h values and cutting speeds.Afterwards these were compared with the corresponding numerical results.

Results and analysis
This section discusses the contribution of various material strengthening factors in capturing size effect for orthogonal down-cut milling process.Numerical simulations at different cutting speeds and a fixed feed rate have been performed.

Temperature effect on material strengthening
To reveal the influence of temperature on the size effect, the evolution of the maximum secondary shear zone temperature calculated at various h, (without considering SG) for various cutting speeds has been presented in Fig. 4. It can be seen that, the lower the uncut chip thickness value, the lower the temperature of the secondary shear zone.However, for a given cutting speed; during the decrease from macro to micro dimensions, the temperature decrease (≈ 30°C) is not significant enough to cause any considerable contribution in increasing material strengthening and influencing the size effect.4.2.Strain rate hardening effect on material strengthening Fig. 5 shows the plots of SCE for various h values at different V C without considering SG.The partial capture of the size effect observed along the curves traced for all V C is apparent.In addition, by increasing V C up to 800 m/min a relatively higher capture of the size effect can be observed.This allows to further investigate the possible existence of a material strengthening mechanism other than SG-strengthening, tool edge radius effect (kept constant in this study) and temperature strengthening effects (not significant for the studied material as shown in subsection 4.1).Therefore, numerical results without considering SG concerning chip morphology were closely analysed at toolchip interface for various h values.It can be seen from Fig. 6, which represents the simulated rake face-chip contact length L C evolution at V C = 200 m/min.L C initially decreases with a decrease in h.Afterwards, L C starts to increase as h value further decreases toward micron level.A similar trend was observed when numerical simulations were performed for higher V C (Fig. 7).In fact primary shear angle decreases as h decreases [32], further because of strain rate hardening, material strengthens.Under these conditions the chip has the tendency to straighten up rather than bend, so L C increases.Nevertheless, L C values were higher at analogous h values for higher V C .This nonlinear increase in L C at small h values implies that a higher energy is dissipated during frictional interaction at toolworkpiece interface, which yields to higher SCE.This provides an explanation for the partial capture of the size effect even without SG-strengthening as it was demonstrated in Fig. 5.This trend of increase in L C is consistent with the results of Liu and Melkote [7], when a sharp tool is replaced by an edged radius tool.It can be seen in Fig. 7 that for all cutting speeds rake face-chip contact lengths initially decrease with the decrease in h value.Afterwards, L C starts to increase as h value further decreases to micro level.As, tool edge radius was unchanged for all cutting simulations, therefore this increase in tool-chip contact lengths at smaller h values, resulting in higher SCE can be attributed to highly strain rate dependent properties of the studied material.Simultaneously, it is noticeable that length scale (in term of uncut chip thickness) at which L C starts to increase; for lower h values, varies with cutting speeds.Indeed, when V C evolves from 200 to 400, 600 and 800 m/min, length scale increases from 44 to 46 µm, 48 and 60 µm, respectively.Kountanya [32] has noted this length scale value as 42.1 µm for V C = 56.4m/min in his research work on the same material studied in the present paper.
Insight analysis of both Fig. 7 and the result of Kountanya [32] show that the length scale increases with the increase in cutting speed, however this increase is remarkable for higher cutting speeds (≥ 800 m/min).This is consistent with the findings of Liu and Melkote [7], though they have attributed this increase in the length scale to the temperature drop in secondary shear zone for a strain rate insensitive aluminium alloy Al5083-H116.It can be deduced from above discussion that, independent of material properties the higher the cutting speed, the higher the nonlinear length scale (in term of uncut chip thickness).Though the reasons of this increase could be different; either temperature drop in secondary shear zone for a strain rate insensitive material [7] or strain hardening properties of a strain rate sensitive material.

Strain gradient effect on material strengthening
Further, to study the contribution of SGstrengthening on the size effect for strain rate sensitive material, cutting simulations considering SG-model (Eq.( 18)) were performed for various V C .Fig. 8 presents the SCE plots with and with-out considering SG-effects for various h values at different cutting speeds.It can be noted that the SCE values computed in the case of a simulation considering SG are closer to experimental ones than that calculated without considering SG.This proves that the modified JC law (Eq.( 18)) allows a good estimation of the cutting force during the variation of the h from macro to micro level.Moreover, for lower h, the latter law permits to capture the nonlinear increase in SCE whose evolution is more pronounced than that calculated without considering SG.However, it is interesting to underline that, the results shown in Fig. 8 depict that even at large h (e.g.h = = 200 µm), the difference between the models (with and without considering SG) is still very large, which seems not to be easily explained by microstructural effects.By plugging in numbers into Eq.( 18), one finds that the effect of the SG vanishes at unphysically large values of L P (roughly 1 m).Nevertheless, this difference (at h = 200 µm) becomes negligible if compared with the one obtained at very small values of h (where specific cutting energy increases exponentially [10,15]) around 0.25R n = 5µm or even lesser, while in present study the simulation have been run up to 22 µm (≈ R n ).This helps to infer that physical significance of this model (Eq.( 18)) prevails mainly at h values of the order of R n and below.From stress point of view, it can be observed that by introducing SG-effects in the material model, and when V C is equal to 200 m/min, the maximum von Mises stresses have been increased from 569 to 610 MPa.Whereas, for a higher V C of 800 m/min, maximum stresses were increased from 586 to 653 MPa.The increments in stress magnitude for both cutting speeds are comparable, approximately.This suggests that SG-hardening is the dominant phenomenon for material strengthening at high V C and lower h values for a strain rate dependant material.To fully capture the size effect for micro cutting operations, SG-based strengthening mechanism is inevitable.

Conclusions
The present study proposes a physical comprehension of material strengthening factors that contribute to the size effect in micro-cutting operations.Orthogonal down-cut milling case for a strain rate sensitive aluminium alloy material A2024-T351 has been investigated.The important conclusions of this study could be underlined as: 1.During down-cut milling, tool-chip contact length L C decreases with the reduction of uncut chip thickness h until it reaches a certain value.After that, this contact length increases as h decreases to micro dimensions.This increase, in L C , implies that the well-known minimum cutting chip thickness is reached and a higher energy is dissipated during frictional interaction at tool-chip interface, resulting in higher specific cutting energy.
2. Similar trend concerning tool-chip contact length variation regarding uncut chip thickness was observed for all studied cutting speeds (V C ).Nevertheless, in the micro-level the length scale at which the lengths of rake face-chip contact start to increase proportionally to V C .The higher the V C , the higher the length scales.
3. The implementation of a modified Johnsoncook material model via a user subroutine VUMAT in the commercial software ABAQUS ® /EXPLICIT has allowed to analyze accurately the contribution of the strain gradient-based hardening on the size effect phenomenon at micro cutting level.
4. The increments in the maximum von Mises stress magnitudes using strain gradient-based plasticity model, for various V C , are more or less the same.This suggests that strain gradient hardening is the dominant phenomenon for material strengthening at high V C and lower uncut chip thickness for the studied material.
5. Specific cutting energy values obtained by numerical simulations, using strain gradient -based plasticity model, were quite close to the experimental ones.This shows that, to fully capture the size effect during micro cutting operations, strain gradient-strengthening mechanism can not be ignored; even at high cutting speeds and for strain rate dependent materials.
Finally, this contribution permits to a close multiscale physical understanding of the role of various strengthening factors contributing to the size effect.Potentially, this will allow improving the existing cutting models and help to capture events happening at micro-levels.M. Asad, T. Mabrouki, F. Girardin, Y. Zhang, J.-F.Rigal

Fig. 1
Fig. 1 Tool-workpiece geometrical model and boundary conditions

Fig. 2 Fig. 3
Fig. 2 Coated carbide insert used in experimentation a-3D view b-Optical device generated 2D profile

Fig. 4
Fig. 4 Maximum secondary shear zone temperatures vs uncut chip thickness for various V C (w/o SG)

Fig. 8 SCE
Fig. 8 SCE evolution vs h values at different V C : a -V C = = 200 m/min, b -V C = 400 m/min, c -V C = = 600 m/min, d -V C = 800 m/min (here experimental force were not registered)

Fig. 9
Fig. 9 Von Mises stress plots at h = 22 µm and V C = = 200 m/min a) w/o SG b) with SG

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. Asad, T. Mabrouki, F. Girardin,Y.Zhang, J.-F.Rigal TOWARDS A PHYSICAL COMPREHENSION OF MATERIAL STRENGTHENING FACTORS DURING MACRO TO MICRO-SCALE MILLING S u m m a r y