Analyses of Shear Angle in Orthogonal Cutting of Pine Wood Analiza

• The determination of energy effects for wood machining processes, such as cutting power and cutting forces, is very useful in designing of manufacture process of wooden products. A more accurate prediction of cutting forces requires a correct determination of the shear angle value, which can be determined using various models. In this article, shear angle values for an orthogonal linear cutting process of pine wood are determined. The pine wood analysed was represented by two groups of samples with different moisture content levels, 12 % and 20 %. Three different models were used to determine the shear angle values: the Merchant model, which is based on the rake angle and the angle of friction, model based on chip compression ratios and Atkins model based on material properties (elements of fracture mechanics). The values obtained have been analysed for comparison. Results showed that the values of the shearing angles determined from the chip compression ratios turned out to be higher than the values from Merchant equation. The shear angles determined from the Atkins model are, as expected, lower than those determined from the Merchant model. Furthermore, the shear angle values for moisture content of 20 % are higher than for moisture content of 12 %.


INTRODUCTION 1. UVOD
The determination of energy effects for wood machining processes is a frequently studied issue and the results are very useful for the design of machining processes (Nasir and Cool, 2020;Atanasov and Kovatchev, 2019;Atanasov, 2021). Most of the analytical modelling works aim at producing equations that can determine cutting forces, without any experimental work. This approach is useful since other parameters can be derived by cutting forces, and analysis of tool wear, surface integrity and workpiece quality can be carried out. The problem involved in the determination of the cutting effects ends up in determining a suitable relationship between the shear angle F c , the rake angle g f and the friction coefficient m between a chip and the rake plane A γ (Markopoulos, 2013). Attempting to find a unique shear angle relationship has long attracted scientific and practical interest. Ståhl et al. (2012) report that more than 50 relationships for determining shear angle can be found in the literature in the last 100 years. On the other hand, Markopoulos (2013) cites 13 such functions in his book.
A numerical model for orthogonal cutting using the material point method was applied by Nairn (2016) to woodcutting using a bench plane. The cutting process was modelled by accounting for surface energy associated with wood fracture toughness for crack growth parallel to the grain. The simulations were verified by comparison to an analytical model and then used to conduct virtual experiments on wood planing.
Experimental works compared to modelling of the wood cutting process is a larger group. The strain associated with orthogonal cutting with and against the grain of hinoki (Chamaecyparis obtusa) was measured by Matsuda et al. (2019). A digital image correlation (DIC) method has been used in a number of published works to measure strain distribution (Matsuda et al., 2018, 2019; Ohtani and Iida, 2015; Radmanović et al., 2017).
The values of shear angles determined based on the Atkins model (Atkins, 2005) for sawing wood on a frame sawing machine, a band sawing machine and a circular sawing machine have been published in the papers of Orlowski et al. (2013Orlowski et al. ( , 2014. The goal of this work was to demonstrate how the method of determination of shear angle can affect the obtained values while pine wood (Pinus sylvestris L.) is cut in the orthogonal process. The new features in this paper is that the experimental values of shear angles, determined in a function of the chip compression ratio (Grzesik, 2017(Grzesik, , 2018Ståhl et al., 2012), are compared to the values calculated from the Merchant equation (Merchant, 1945) and the values of shear angles computed based on the Atkins model (Atkins, 2005), which incorporate realistic effects of cut material properties.

THEORETICAL BACKGROUND 2. TEORIJSKE OSNOVE
Orthogonal cutting represents a two-dimensional mechanical problem with no side curling of the chip considered. It represents only a trickle of machining processes. Nevertheless, it is widely used in theoretical and experimental work due to its simplicity. The case of free orthogonal cutting occurs when (Grzesik, 2016(Grzesik, , 2017ISO 3002-4, 1984): · cutting speed v c is perpendicular to the cutting edge, · the cutting edge S is rectilinear, · tool cutting edge angle is κ r = 90°, and inclination angle is λ s = 0°, · the width of the cut layer (sample width) W s is much greater than the uncut thickness h, · the length of the active cutting edge is greater than the cutting width.

Prediction of shear angle with Merchant model 2.1. Predviđanje kuta smicanja Merchantovim modelom
The engineering approach to the description of plastic deformation in the cutting zone is simplified, including: replacing curvilinear boundaries by straight lines (shear zone is in the shape of the fan) proposed by Zorev (1966), parallel-sided shear band inclined at shear angle (Oxley, 1989), and one shear plane proposed by Merchant (1945) as presented in Figure 1. Although this 2D single shear plane model is criticised, it is usually discussed in machining handbooks due to Figure 1 Geometry of chip forming zone with one shear plane. Legend: A a -flank face, A g -rake face, h -uncut chip thickness, h ch -chip thickness, F c -shear angle, a f -side flank (clearance) angle, b f -side wedge angle, g f -rake angle Slika 1. Geometrija zone nastanka strugotine s jednom ravninom smicanja (A a -leđna površina, A g -prsna površina, h -debljina neodvojene strugotine, h ch -debljina strugotine, F c -kut smicanja, a f -leđni kut, b f -kut klina, g f -prsni kut its simplicity and it is the basis for calculating several process parameters, e.g. cutting forces (Grzesik, 2017;Markopoulos, 2013). Merchant (1945) proposed to calculate the shear angle Φ c considering the rake angle γ f of and the angle of friction β m . The latter depends on the coefficient of friction m between the rake face and the chip. The relationship by Merchant is as follows: The coefficients of friction are going to be taken on the basis of the measured cutting forces.

Shear angle as a function of chip compression ratio 2.2. Kut smicanja kao funkcija omjera kompresije strugotine
Knowledge of the chip compression ratio λ h that takes place enables the shear angle Φ c to be computed with the use of the relationship (Merchant, 1945 (2) The chip compression ratio λ h could be calculated as follows (Merchant, 1945;Ståhl et al., 2012): To determine the chip thickness h ch (Figure 2), the chips were collected after each test and immediately measured with a calliper. The digital calliper by GEDORE company was applied (GEDORE Werkzeugfabrik GmbH & Co. KG, Remscheid, Germany).

Prediction of shear angle with Atkins model 2.3. Predviđanje kuta smicanja primjenom Atkinsova modela
Atkins (2003) stated that sensible material-dependent predictions for Φ c are obtained when proper magnitudes of separation work are incorporated in the minimisation. According to Atkins (2003) for least cutting force F c, the shear angle Φ c satisfies: in which is the parameter which makes Φ c material dependent, where: R is specific work of sur-face separation/formation (fracture toughness), and τ γ is the shear yield stress (along the shear plane). The nonlinear equation (4) (further called the Atkins Model -AM) can be solved with Newton's method (Wanat 1994), which is one of the iterative methods, in which the roots can be found. The cutting process is a simple and exceptional way for simultaneously measuring toughness and shear yield stresses (Atkins, 2005). Cutting forces can be expressed as a linear regression function (Atkins, 2003

Materijali
Scots pine (Pinus sylvestris L.) was used to prepare the samples. One log was randomly chosen among others in the sawmill yard. From the middle part of the 4 m long log, rectangular samples with dimensions W = 60 mm × H = 60 mm × L = 600 mm (width × height ×length, respectively) were cut. Then, ten prepared samples were dried and conditioned under laboratory conditions assuring constant air temperature of 20 °C and relative humidity of 65 % for three months. They were divided into two group -one  Tool side rake angle (tool-in machine system) γ f was equal to 15°, and tool wedge angle was b f = 55°.

REZULTATI I RASPRAVA
The obtained experimental results of cutting forces F c in orthogonal cutting process on the sliding microtome in longitudinal direction to pine wood fibre (direction 90°−0° according to Kivimaa (1950)) at two levels of MC equal to 12 % and 20 % are presented in Figure 3.
On the assumption that the cutting model includes work of separation (fracture toughness) in addi-  Figure 3.
The results of computations of R and τ γ together with determined average friction coefficients m between the rake face and the chip are presented in Table 1.
Cutting forces for pine wood of 12 % MC in a function of the uncut chip thickness are slightly higher than in the case of cutting pine of 20 % MC (Figure 3), and this phenomenon could be caused by differences in friction coefficient, which is smaller for pine wood of 20 % MC. Moreover, as the values of the uncut chip thickness increases, the differences in force values decrease. The obtained values of friction coefficients (Table 1) are in agreement with the statement that intermediate moisture content may range from 0.5 to 0.7 (Simpson and TenWolde, 1999).
The shear angle Φ c , considering the rake angle γ  Figure 4a and 4b as f_MER.
Some collected pine chips from the orthogonal cutting on the sliding microtome in longitudinal direction to pine wood fibres are shown in Figure 5.
Measured with callipers, the chip thickness values allowed the determination of the chip compression ratio Table 1 Fracture toughness R, shear yield stresses τ γ and average friction coefficients m between rake face and chip for pine wood in a function of MC Tablica  In prediction of the shear angles with the Atkins model (Eq. 4), which makes Φ c material dependent, the values of fracture toughness R and shear yield stresses τ γ , and values of friction coefficients m presented in Table 1 were applied in numerical computations. The results of these calculations are shown in Figure 4a

ZAKLJUČAK
The novel experimental analyses of shear angles proposed here provide a unique opportunity to assess the effect of the method of determining shear angles on their values.
For pine wood (Pinus sylvestris L.) with a moisture content of MC = 20 %, lower values of the friction coefficient m between the chip and the rake face were observed during orthogonal cutting along the wood fibre than for a moisture content level of MC = 12 %. The difference is equal to 14.2 %. As a consequence of this difference, lower values of the cutting force F c were recorded over the entire range of variations in the uncut chip thickness.
The values of the shearing angles determined from the chip compression ratios turned out to be higher than the values from Merchant equation. This can be explained by the measurement error resulting from the use of digital callipers.
The shear angles determined from the Atkins model are, as expected, lower than those determined from the Merchant model. Furthermore, the shear angle values for 20 % MC are higher than for moisture content 12 % MC.
The area of shear angle analysis for wood machining processes requires deeper analysis at both modelling and experimental verification levels. Accurate determination of the shear angle for wood machining processes would allow a fairly accurate prediction of the energy demand for these cutting processes, resulting in a reduction of material waste.