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Modelling and Simulation in Engineering
Volume 2011, Article ID 469262, 17 pages
http://dx.doi.org/10.1155/2011/469262
Research Article

Slicing Cuts on Food Materials Using Robotic-Controlled Razor Blade

1Department of Mechanical and Industrial Engineering, University of Minnesota, Duluth, MN 55812, USA
2Food Processing Technology Division, ATAS Lab, Georgia Tech Research Institute, Atlanta, GA 30332, USA

Received 27 May 2011; Revised 10 August 2011; Accepted 17 August 2011

Academic Editor: Xiaosheng Gao

Copyright © 2011 Debao Zhou and Gary McMurray. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Cutting operations using blades can arise in a number of industries, for example, food processing industry, in which cheese, fruit and vegetable, even meat, are involved. Certain questions will rise during these works, such as “why pressing-and-slicing cuts use less force than pressing-only cuts” and “how is the influence of the blade cutting-edge on force”. To answer these questions, this research developed a mathematical expression of the cutting stress tensor. Based on the analysis of the stress tensor on the contact surface, the influence of the blade edge-shape and slicing angle on the resultant cutting force were formulated and discussed. These formulations were further verified using experimental results by robotic cutting of potatoes. Through studying the change of the cutting force, the optimal slicing angle can be obtained in terms of maximum feeding distance and minimum cutting force. Based on the blade sharpness properties and the specific materials, the required cutting force can be predicted. These formulation and experimental results explained the basic theory of blade cutting fracture and further provided the support to optimize the cutting mechanism design and to develop the force control algorithms for the automation of blade cutting operations.

1. Introduction

Food cutting, such as potato or cheese cuts, is different from metal cuts because of the material deformability and shape-variability. Cutting mechanics formulation is constantly a hot research topic since it can provide useful information for cutting operations. In the literature, mainly two methods have been documented, that is, energy formulation method and stress tensor distribution method.

Metal cutting creates plastically deformed offcuts which permanently store energy, while the food offcuts permanently store almost no energy. Thus, many researchers have formulated the food cutting problems using energy method, most notably, using the fracture toughness concept [1, 2]. Using energy concepts, Atkins et al. [3, 4] have been able to explain why the cutting fracture requires smaller force when pressing and slicing compared to pressing only. Kamyab et al. [5] formulated the stress and force distribution in cheese cutting.

Stress distribution formulation can provide an alternative explanation. Two methods can be applied: potential function method and superposition method. In the potential function method [6, 7], the strain and stress tensor are expressed in terms of the space derivatives of certain airy functions in the form of biharmonic equations. Using this method, the closed form expression of the stress distribution generated by a point force acting in either normal or tangential direction to the boundary of the semi-infinite body was derived in [8], the solution to a load applied normal to an infinite half-space was given by [9, 10], and the stress distribution generated by a tangential force applied to a surface was referred as Cerruti problems, and its solution was obtained using reciprocal theorem as summarized in [1113]. Based on the derivation of the integral of the point force airy functions, Love [14] provided the integral for a rectangular area with constant normal pressure. Using the same method, the solution for first-order polynomial load applied to a rectangular surface has been completed by [15], to a triangular region has also been given by [16, 17], and to a circular area by [18]. The second method is to formulate the internal stress distribution, for example, using Hertz contact mechanics [19]. Yoshihara and Matsumoto [20] studied the shearing properties of wood using the stress analysis method. Lucas used fracture mechanics to explain the function of teeth in food cutting based on the detailed investigation of dental structure [21].

Blade sharpness is another factor that affects the cutting forces. Contact between the cutting object and the blade is an area, which can be shown from the microstructure of a knife [22]. Blade sharpness also directly influences the cutting moments and the grip forces applied by an operator such as the research performed by McGorry et al. [23]. Szabo et al. developed a procedure to establish knife-steeling schedules based on increased force due to knife dullness from repetitive use to minimize operator exertions and physical stress associated with work-related musculoskeletal disorders [24].

In this paper, mainly applied to the hard and crispy materials such as cheeses and potatoes, we will focus on using shear stresses at fracture to describe the cutting mechanism and blade sharpness. Firstly, the distribution of the external cutting force on the cutting-contact area is described in Section 2.1. From Sections 2.2 to 2.3, cutting force influence factors, such as the shape of the blade cutting edge, slicing angle, and so forth, are discussed. In Section 2.4, the relative sharpness factor concept built in [25] is generalized to be applicable to the pressing and slicing cuts. In Section 3, computation results are provided to illustrate the stress distribution during cutting. The computation results provide several purposes: (1) to show the stress distribution and to verify the analytical model; (2) to find out where the maximum stress is and to use them to evaluate the fracture location; (3) to understand the influence of the blade shape (cutting force shape) and slicing angle to the fracture force; (4) to figure out the fracture mode of blade cuts. Experimental results were provided in Section 4 to verify the formulation. Justification about the application of this algorithm on nonlinear materials are provided in Section 5. Conclusions were drawn in Section 6.

2. Modeling of the Cutting Interaction

2.1. Simplified Model of Pressing and Slicing Cuts on Hard-Crispy Materials

The illustration of potato cutting is shown in Figure 1. The modeling and experiments will focus on using the blade shown in Figure 1(a) to realize the robot-controlled cuts shown in Figure 1(b). The microstructure photo of the edge of a blade is shown in Figure 2(a) with the measured width of the blade cutting edge. For that particular razor blade, the edge of this blade is 850 nm with flat surface. The range of the edge width of a brand new blade is around 500 nm to 1250 nm with the shape shown in Figure 2(b).

fig1
Figure 1: Razor blade cutting.
fig2
Figure 2: Microstructure of the edge of the razor blades used in the modeling and experiments.

The microstructure of potatoes showed that the homogeneity of the particles is 500 times bigger than the blade edge width. Since the target problem is about cutting fracture, the focused problem is the stress distribution at the contact area at the moment just before the initiation of the cutting fracture.

At that moment, enough pressure has been built up beneath the blade, and the blade has a firm and solid contact with the target. Thus, the cutting interaction, such as the potato cutting shown in Figure 1, is simplified as a belt-area force acting on the surface of a semi-infinite body as illustrated in Figure 3(b), where the interaction between the blade and the target (Figure 3(a)) is simplified as a belt-area force, and the target is simplified as a semi-infinite body. Given the contact length between the blade and cutting material is 𝑙 and the width of the blade edge is 2𝑎, the contact area between the blade and the material is belt-like, that is, (𝑙/2, 𝑙/2) in 𝑥 direction and [𝑎, 𝑎] in 𝑦 direction, where point 𝑜 in frame 𝑜𝑥𝑦𝑧 is at the center of the belt area, 𝑜𝑥𝑦 plane is on the contact surface, 𝑥-axis is the blade length direction, and 𝑧-axis point into the material as shown in Figure 3(b). Normally, the width of a brand new blade (2𝑎) ranges from 500 nm to 1250 nm, and the contact length is normally bigger than one centimeter, roughly 𝑙/𝑎>104, that is, the contact length in 𝑥 direction is much bigger than that in 𝑦 direction. Thus, 𝑙 can be considered as infinite compared to 𝑎.

fig3
Figure 3: Simplified cutting model.

For the symbolic expression in this paper, capital 𝑃 is used to express a cutting force and small 𝑝 is used to express a pressure variable, while subscript and superscript 𝑝 of a variable is used to express the variable is either point-force or point-pressure related. Subscript and superscript 𝑛 stands a variable in normal or 𝑧 direction and subscript and superscript 𝑡 stands a variable in tangential or 𝑥 direction. For example, 𝑃𝑝 is an ideal point force acting on the contact surface of a semi-infinite body with normal component 𝑃𝑝𝑛 and tangential component 𝑃𝑝𝑡, while 𝑝𝑛 and 𝑝𝑡 are the cutting pressure components in the 𝑧 direction and 𝑥 direction, respectively. Subscript and superscript 𝑙 is used to express the variable is line-pressure related.

Force 𝑃 has two components: normal force 𝑃𝑛 and tangential force 𝑃𝑡 as shown in Figure 4. Then there are𝑃𝑛=𝑃cos𝛼,𝑃𝑡=𝑃sin𝛼,(1) where 𝛼 is defined as the slicing angle. If we assume the pressure does not change along a line parallel to the 𝑥-axis, the line-pressures on a line parallel to 𝑦-axis are 𝑝𝑙𝑛 in normal direction and 𝑝𝑙𝑡 in tangential direction, there are𝑃𝑛=𝑝𝑙𝑛𝑙,𝑃𝑡=𝑝𝑙𝑡𝑙.(2) The area pressures 𝑝𝑛 and 𝑝𝑡 can be used to calculate line pressures 𝑝𝑙𝑛 and 𝑝𝑙𝑡 and total force 𝑃𝑛 and 𝑃𝑡 as follows: 𝑃𝑛=𝑙/2𝑙/2𝑎𝑎𝑝𝑛𝑑𝑠𝑑𝑣,𝑃𝑡=𝑙/2𝑙/2𝑎𝑎𝑝𝑡𝑝𝑑𝑠𝑑𝑣,𝑙𝑛=𝑎𝑎𝑝𝑛𝑑𝑣,𝑝𝑙𝑡=𝑎𝑎𝑝𝑡𝑑𝑣.(3)

469262.fig.004
Figure 4: Cutting force acting on a semi-infinite body with slicing angle 𝛼.

Since the dimension of the blade edge is around 1000 nm (1 um), there is no force sensor in such small size to measure the stress distribution. According to the edge shape of a razor blade, the cutting force distributions in both tangential and normal direction to the cutting surface have to be assumed. In this paper, they are assumed to be trapezoid as shown in Figure 5 and their mathematical expressions are summarized in Table 1.

tab1
Table 1: Pressure distribution along a line in 𝑦-axis direction.
fig5
Figure 5: Force distribution profile along 𝑦-axis at 𝑧=0, that is on the contact surface.

There are 𝑝𝑙𝑛=𝑎𝑎𝑝𝑛𝑑𝑣=𝑞𝑛𝑝(𝑎+𝑤),𝑙𝑡=𝑎𝑎𝑝𝑡𝑑𝑣=𝑞𝑡𝑃(𝑎+𝑤),𝑛=𝑃cos𝛼=𝑝𝑙𝑛𝑙=𝑞𝑛𝑃(𝑎+𝑤)𝑙,𝑡=𝑃sin𝛼=𝑝𝑙𝑡𝑙=𝑞𝑡(𝑎+𝑤)𝑙.(4) Then there are𝑞𝑛=𝑃cos𝛼(𝑎+𝑤)𝑙,𝑞𝑡=𝑃sin𝛼.(𝑎+𝑤)𝑙(5)

Other simplifications in this paper are as follows. (1) Relaxation and creep are ignored since we consider the instantaneous cuts. (2) The cutting force intensity is zero at the edge of the contact area. (3) The offcuts move away from the knife and no friction force acts on the side of the blade; the contact area between the blade edge and the material does not change once full contact is established.

2.2. Modeling of the Stress Distribution in the Cutting Materials

The modeling starts from a stress field generated by a point force on a semi-infinite solid body. Superposition method is used to obtain the stress field generated by a belt-shape area force on a semi-infinite solid body. The procedure is Table 2. In the last row of Table 2, the functions 𝑓1, 𝑓2, 𝑓3, 𝑓4, 𝑓5, and 𝑓6 can be explicitly expressed using (A.8) to (A.10) and (A.12) to (A.14) using the variables 𝑦,𝑧,𝑎,𝑤. The details are shown in the appendix.

tab2
Table 2: Area external force.

Although the modeling method uses a standard superposition which can be found in any material mechanics textbook, it is first time in the literature that the stress distribution due to a belt-shaped area force was obtained in a closed and manageable form.

Substituting 𝑞𝑛 and 𝑞𝑡 in (5) into (*) in Table 2, the stress tensor at point 𝐴(𝑥,𝑦,𝑧) generated by cutting force 𝑃 is obtained as[𝜎]=𝜎𝑥𝜏𝑥𝑦𝜏𝑥𝑧𝜏𝑥𝑦𝜎𝑦𝜏𝑦𝑧𝜏𝑥𝑧𝜏𝑦𝑧𝜎𝑧=𝑃𝑓𝑙(𝑎+𝑤)1cos𝛼𝑓4sin𝛼𝑓5𝑓sin𝛼4sin𝛼𝑓2cos𝛼𝑓6𝑓cos𝛼5sin𝛼𝑓6cos𝛼𝑓3.cos𝛼(6)

The principle stresses 𝜎1, 𝜎2, 𝜎3 can then be obtained through solving [𝜎]𝐱=𝜆𝐱,(7) where the solutions to 𝜆 are the principle stresses 𝜎1, 𝜎2, and 𝜎3; the solutions to 𝐱 (3 by 1 vector) express the direction of the principle stresses. The maximum shear stress (𝜏𝑖𝑗) can then be obtained as𝜏𝑖𝑗=12𝜎𝑖𝜎𝑗,(8) where 𝑖 and 𝑗 are one of the numbers 1, 2, or 3.

2.3. Fracture and Initialization during Blade Cutting

Tresca’s failure criterion [26] was adopted to identify fracture initialization during blade cutting. This criterion can be expressed as𝜏max𝜏=Maxabs12𝜏,abs23𝜏,abs13𝜏𝑢𝐾𝑠,(9) where 𝜏𝑢 is the ultimate shear strength, 𝐾𝑠 is the ratio of the fracture initialization force over continuous cutting force, the function abs(#) returns the absolute value of the expression #, and the function Max(𝑥,𝑦,𝑧) returns the biggest value among 𝑥, 𝑦, and 𝑧.

During the cutting process, there are two sets of fracture: fracture initialization and postfracture. Prior to material fracture, all of the stresses in the material are due to deformation. At the moment of fracture initialization, (9) can be applied with 𝐾𝑠=1. After the fracture initiation, postfracture criterion will apply. During postfracture, the initial failure criterion (𝐾𝑠=1) is no longer valid. The stress concentration factor needs be determined experimentally using𝐾𝑠=𝜏𝑢𝜏𝑜=𝑃𝑢𝑃𝑐,(10) where 𝜏𝑢 and 𝜏𝑜 are the ultimate shear stress at fracture initialization (material property) and the ultimate stress during postfracture (determined experimentally), respectively, and 𝑃𝑐 is the cutting force during continuous cutting (post-fracture), 𝑃𝑢 is the cutting force at the moment of fracture initiation. From (6)–(9), there is𝜏𝑖𝑗=𝑃𝑓𝑙(𝑎+𝑤)7𝑖𝑗(𝑦,𝑧,𝑎,𝑤,𝛼),(11) where function 𝑓7𝑖𝑗(#) expresses a function of variable #. 𝑓7𝑖𝑗(#) can be explicitly expressed using (A.8) to (A.10) and (A.12) to (A.14). At the moment of fracture initiation, the required force is obtained by substituting (11) into (9), 𝑃𝑢=𝜏𝑢𝑙(𝑎+𝑤)𝐾𝑠𝑓8𝑦𝑢,𝑧𝑢,𝑃,𝑎,𝑤,𝛼𝑛𝑢=𝑃𝑢cos𝛼,𝑃𝑡𝑢=𝑃𝑢sin𝛼,(12) where 𝑃𝑢 is the cutting force during cutting fracture and (𝑦𝑢,𝑧𝑢) represents the 𝑦 and 𝑧 coordinates of the fracture location, and function 𝑓8(#) can be explicitly expressed using (9) and (11). 𝑃𝑛𝑢 is the normal cutting force and 𝑃𝑡𝑢 is the tangential cutting force.

2.4. Shape of Blade Edge: Blade Relative Sharpness Factor

In general, “sharpness of a blade” is an approximate measurement of the magnitude of the applied force 𝑃𝑢 in different cuts by keeping the cutting material (𝜏𝑢) and cutting manners (𝑙 and 𝛼) unchanged. It may be possible to explicitly express the relationship between the cutting force and the factors, 𝑎, 𝑤, 𝑙, 𝛼, and (𝑦𝑢,𝑧𝑢) using (12). However, since the magnitude of 𝑤 and 𝑎 is in several hundred nanometers, it will not be practical to put a microscope in a workshop to measure the width for each blade in order to estimate its sharpness. So a relative sharpness factor is defined as follows:𝜂=(𝑎+𝑤)𝐾𝑠𝑓8𝑦𝑢,𝑧𝑢.,𝑎,𝑤,𝛼(13) When setting 𝐾𝑢=𝑙𝜏𝑢, there is 𝑃𝑢=𝑙𝜏𝑢𝜂=𝐾𝑢𝜂.(14) Let 𝑃𝑢0 be the cutting force of the sharpest knife, 𝑃𝑢𝑓 be the cutting force of the dullest knife, and 𝑃𝑢 be any other cutting force. In the sharpest case, such as a brand new blade from factory, the blade relative sharpness factor is defined as 𝜂0=𝑃𝑢0𝐾𝑢.(15) The relative sharpness of certain blade can be expressed as𝜂=𝜂0𝑃𝑢𝑃𝑢0.(16) In the dullest case (in which the maximum allowable force has to be applied in order to realize cutting), there is𝜂𝑓=𝜂0𝑃𝑢𝑓𝑃𝑢0.(17) Then for convenience, another parameter, knife relative sharpness level, 𝜅 is defined as𝜅=int(𝑛1)𝜂𝜂0𝜂𝑓𝜂0+1,(18) where 𝑛 is a user-defined integer which is used to distinguish the sharpness level of a blade and int(#) is a function to round the number # to the nearest integer. The function int(#) forces the knife sharpness level value to an integer. Substituting (15)–(17) into (18), there is𝑃𝜅=int(𝑛1)𝑢𝑃𝑢0𝑃𝑢𝑓𝑃𝑢0+1.(19) It can be easily seen that the knife relative sharpness level can be determined using the cutting forces at different conditions. Since the cutting forces are measureable, a knife can be categorized into 𝑛-level sharpness starting from level 1 by defining 𝜂0=1 in the sharpest case.

3. Simulation Analysis

In the computation, the change of the stresses with the parameters 𝑙 (contact length), 𝑎 (half width of the blade cutting edge), 𝑤 (load shape), 𝑦 and 𝑧 (different locations in material) are discussed, and the stress distributions in the material under different conditions have been visualized for further parametric study. By changing the cutting force profile, the influence from blade sharpness can be observed and the fracture modes are obtained. In the calculation, the coordinate values or distance data are normalized by 𝑎 (half width of the blade edge), the stresses generated by normal and tangential forces are normalized by the maximum pressure 𝑞𝑛 and 𝑞𝑡, respectively, (or set 𝑎 = 1 mm, 𝑞𝑛 = 1 pa, and 𝑞𝑡 = 1 pa).

3.1. Stress Variation with Depth

The results shown in this section are to illustrate the change of the stress distribution with the depth (𝑧) in the material at different 𝑦 locations. The results were obtained using (*) in Table 2 by assuming 𝑤=0.8𝑎, constant pressure, 𝑞𝑛 = 1 pa, and 𝑞𝑡 = 1 pa. The results of the stresses 𝜎𝑥, 𝜎𝑦, 𝜎𝑧, 𝜏𝑦𝑧, 𝜏𝑦𝑥, 𝜏𝑥𝑧 have been calculated and only the results for stress 𝜏𝑦𝑧 are shown in Figure 7. In Figure 7, each curve represents the stresses change with 𝑧 (from 0 to 3𝑎 or 20𝑎) on the same 𝑦 coordinates (𝑦 from 2𝑎 to 2𝑎).

It is observed from the results of 𝜎𝑥, 𝜎𝑦, 𝜎𝑧, 𝜏𝑦𝑧, 𝜏𝑦𝑥, 𝜏𝑥𝑧 that at the contact surface, the maximum stress may not exceed the cutting stress from the blade and the stress will decrease dramatically as 𝑧 increases. At the location where the 𝑧 coordinate (depth) is about twice of the half width of the cutting blade (at 𝑧=2𝑎), the force decreases roughly to half of the maximum stress. In general, the magnitude of the width of a sharp blade is about 1 um and the size of the inhomogeneity in potatoes and cheese is bigger than 0.1 mm (100 um). This means that at the depth there are inhomogeneity, the stress already reduced to very small value.

Moreover, this paper is only interested in the stress to generate the fracture. From the calculation, it is found when slicing angle is smaller than 10°, the fracture happens at the depth of 𝑧=0.37𝑎 or at about 𝑧 = 0.06–0.32 um; when slicing angle is bigger than 10°, the fracture happens at the contact surface or at the depth of 𝑧 = 0 um. Thus, in the microscale sense, the material can be considered as homogenous although in macrosense, the cutting material may be anisotropic or heterogeneous. When we only consider the stress to generate the cutting fracture, the material can be considered as homogenous.

Here we define the zone with at least tenth of the applied maximum stress from the blade as the effective zone. Since the depth of the effective zone in the cutting modeling is much smaller than the size of the fiber, or inhomogeneous, or anisotropic zone of the cutting materials, although the material itself is inhomogeneous or anisotropic, only in the cutting effective zone can the cutting modeling still be approximated as homogenous material.

3.2. Change of the Stress Distribution with Cutting Width

This calculation is to show the variation of the stresses with different 𝑤. The results are shown in Figure 8 where the abscissa is y coordinate with 2𝑎𝑦2𝑎 and the ordinate is the stress magnitude at different 𝑧. In Figure 8, each curve represents the stress change with different 𝑧 coordinates between 0 and 𝑎 with step size at 0.2𝑎. The 𝑧 values have been marked on each curve. For clarity, the different stress components are drawn with different line styles.

The subplots in Figure 8 show the stresses under three different external force distribution profiles, 𝑤=𝑎 or rectangle-shaped profile is shown in Figure 8(a), 𝑤=0.85𝑎 or a trapezoid shaped profile is in Figure 8(b) and 𝑤=0 or a triangle-shaped profile is in Figure 8(c). It is observed that all the stress components, except 𝜏𝑦𝑧, have the maximum magnitude when 𝑧=0. They then decrease as 𝑧 increases. The maximum 𝜏𝑦𝑧 happens inside the material at 𝑧=0.37𝑎. Since 𝑎 is very small (in the magnitude of 10−9 m), the location of maximum 𝜏𝑦𝑧 is very close to the surface. The maximum values of 𝜎𝑦, 𝜎𝑧 and 𝜏𝑥𝑧 are in the same magnitude of 𝑞𝑛 or 𝑞𝑡, respectively.

3.3. Fracture Force via Blade Shape

The knife shape can be defined using the parameters 𝑙, 𝑎, and 𝑤. If 𝑙 and 𝑎 increase, the maximum pressure will decrease when the total external force keeps unchanged. The influence of the edge shape can be roughly expressed using w (please refer to Figure 5). Using the expression in (6), the stress distributions for the different values of the 𝑤 on the blade-material contact plane 𝑜𝑥𝑦 are obtained and shown in Figure 9. From Figure 9, it is observed that by keeping the external force unchanged, when the force distribution changes from constant intensity to linear intensity, the maximum magnitude of all the stresses increases. This leads to an increase in the magnitude of the maximum shear stress. Thus, by assuming 𝜏𝑢 is constant, the external force to realize cutting fracture will decrease. Or in order to realize cutting on the same material with the same constraints, the blade with the linear edge shape will use less force than any other blades. It will be the sharpest one.

3.4. Fracture Force via Slicing Angle

In this section, the knife shape and the external force intensity profile are fixed with 𝑤=0.85𝑎. The obtained stress distribution is shown in Figure 8(b). It is observed that there are four possible locations with maximum stress, and they are summarized in Table 3.

tab3
Table 3: Locations and values of the possible maximum stresses.

The changes of the corresponding stresses distribution, the principle stresses and the maximum shear stress, with slicing angle, are shown in Figures 6(a), 6(b), and 6(c), respectively. In Figure 6, only the stresses at location (ii) (𝑦/𝑎,𝑧/𝑎) = (.85,.00) are illustrated. Similar results at locations (i), (iii), and (iv) can be obtained accordingly. The results in Figure 6(a) are obtained using (6) by assuming 𝑃/𝑙=1, that is, 𝑝𝑙𝑛 = 1 when α = 0° and 𝑝𝑙𝑡 = 1 when α = 90°. The results in Figures 6(b) and 6(c) are obtained using (7) and (8), respectively. Using Tresca’s fracture criterion [26], if 𝜏𝑢 is generated by (𝜏31)max, according to (9) by setting 𝐾𝑠=1, the cutting force 𝑃𝑢 is obtained using (12) as shown in Figure 6(d). Using the same method, the maximum shear stresses and the corresponding external cutting forces at locations (i), (iii), and (iv) can also be obtained. The results are shown in Figure 10(a).

fig9
Figure 6: Stresses and external forces change with slicing angle at location (ii).
fig6
Figure 7: Stress tensor changes along 𝑧 for different 𝑦.
fig7
Figure 8: Comparison of the normalized tensor for different cutting force profiles.
469262.fig.008
Figure 9: Stresses change with 𝑤 on the blade-material contact plane 𝑜𝑥𝑦 with same cutting force.
fig10
Figure 10: Maximum shear stresses and required minimum cutting force.

Since the external force 𝑃𝑢 is obtained using the same ultimate shear stress 𝜏𝑢, the smallest one among all the 𝑃𝑢 at the four locations from (i) to (iv) will initiate cutting fracture. For clarity, the largest maximum shear stress and its required smallest external force are redrawn in Figure 10(b). It can be observed from Figure 10(b) that when 𝛼 is from 0° to 10°, the shear stress at location (iv) will initialize fracture, and when 𝛼 is from 10° to 90°, the shear stress at location (ii) will initialize fracture.

The total cutting force and its tangential and normal components are shown in Figure 11. From Figure 11, it can be seen that the fracture at slicing angle from 0° to 10° is due to 𝜏𝑦𝑧 which is just beneath the surface (𝑧=0.37𝑎 or roughly 𝑧370 nm). By considering the stress direction, it can be seen that it is the mode II fracture, edge-sliding fracture [1]. The fracture from 10° to 90° is due to 𝜏𝑥𝑦 and 𝜏𝑥𝑧 which is just on the surface (𝑧=0). It is the mode III fracture, out-of-plane tearing [1]. From the external force profile shown in Figure 11, the influence of the slicing angle can be clearly seen. The required force to cut by pressing-only is far larger than the force required by pressing-and-slicing cuts. Note also is that force 𝑃𝑡𝑢 does not change a lot from 10° to 90° in the mode III fracture and its value is just the total force when the slicing angle is 90°.

469262.fig.0011
Figure 11: Total cutting force and the force components in 𝑥 direction (𝑃𝑡𝑢) and 𝑧 direction (𝑃𝑛𝑢).

4. Experimental Verification

Raw Russet Baking Potatoes, purchased from a local Kroger Store in Atlanta Ga, USA, were selected as the testing materials in the experiments. The potatoes were purchased in fresh. They are firm and smooth without dark spots, green areas, mold, or cuts. They are in the size of 125 mm–150 mm long and about 75 mm in diameter. The moisture content is about 4 g water per 5 g fresh potato solid. Moisture content of the sample was determined by drying thin potato slabs (10 mm by 20 mm by 45 mm) at 70°C under vacuum for 24 hours. The experiments were carried out in an air conditioning controlled room at temperature about 22°C and moisture about 70%. Once a fresh potato was peeled and cut to a roughly 20 mm by 40 mm × 100 mm brick, the experiments were performed without delay. The experimental setup is shown in Figure 12. The system consists of an ABB robot IRB 140 [27] for motion generation, speed control and distance measurement, and an ATI Force/Torque sensor ISA F/T-16 Mini40 [28] for force measurement. The robot is commanded to move at speed of 0.5 mm/s for cutting. The force data are saved on a central computer. The forces during the cut of potatoes are shown in Figure 13.

469262.fig.0012
Figure 12: Experiment setup.
469262.fig.0013
Figure 13: Cutting force changes with contact length.
4.1. Evaluating the Influence of Slicing Angle and Cutting Fracture Modes

When changing the slicing angle, the same shapes of the cutting force changing profiles have been obtained. The average data with standard deviation less than 0.2 lb from 5 sets of cuts using different slicing angles on potatoes are summarized in Figure 14. According to the maximum external force when 𝛼 is near 90°, where the tear mode dominates, the ultimate shear stress (𝜏𝑢) can be estimated as 0.18 lb. Then, using this 𝜏𝑢 and the procedure to obtain 𝑃𝑡𝑢 and 𝑃𝑛𝑢 in Figure 11, the theoretical fracture forces can be estimated using (12) by assuming the same 𝑎, 𝑤, 𝐾𝑠, 𝑦𝑢, and 𝑧𝑢 since only the slicing angle (𝛼) changed during different trails. Both of the experimental data and the theoretically estimated data are shown in Figure 14. Good match is observed.

469262.fig.0014
Figure 14: Experimental data and theoretical estimation of the cutting forces during potato cuts.
4.2. Evaluating Relative Blade Sharpness Factor

These experiments were used to demonstrate how to implement the sharpness definition in this paper. In the experiments, blades with different sharpness were used to cut the same material to study the effect of blade sharpness. The materials (potatoes) were in rectangular shape with the same width (13 mm) and thickness (30 mm) to make sure the same contact length of each cut. The blades were prepared such that the sharpest blade and the dullest blade defined in (15)–(17) were included.

The procedure of the blade preparation is as follows. (1) Choose a brand-new blade (industrial-grade single-edge razor blade standard, 0.2286 mm thick) as the sharpest one (Blade A), define its relative sharpness 𝜂0=1 and obtain its potato cutting forces; (2) use the sharpest blade (Blade A) to manually cut cardboard 30 times to form Blade B and obtain its potato cutting forces; (3) use Blade B to cut through a piece of pine wood 50 times to form Blade C and obtain its potato cutting forces; (4) blade C is dulled on aluminum by rubbing 30 times to form Blade D and obtain its potato cutting forces; (5) the dullest knife is obtained by dulling Blade D on sand paper (aluminum oxide cloth sanding sheet 80 grit) by 10 times and then obtain its potato cutting forces; (6) according to the cutting forces of the sharpest blade (Blade A) and the dullest blade (Blade E) to get 𝜂𝑓; (7) set sharpness level (𝑛) of Blade E is 5 (𝑛=5) and according to (18) to specify the sharpness level κ to which another blade belongs. Part of the obtained force results are shown in Figure 15, and the blade sharpness level results are summarized in Table 4, where the first 4 columns show the fracture forces, Column 5 is the average fracture force, relative blade sharpness level 𝜂 and relative blade sharpness level κ are calculated based on (13) and (18). From the last column in Table 4, it can be seen that blades B, C, D are in the 2nd, 3rd, and 4th sharpness levels, respectively.

tab4
Table 4: Blade relative sharpness level.
469262.fig.0015
Figure 15: Cutting results using different sharp blades (only blades A, D, E are shown).

By using this method, any blade can be assigned to a certain sharpness level. When the cutting force reaches certain level by which the knife is decided to be dull, the knife needs to be resharpened or disposed.

5. Justification

It can be seen that the computation results from the model agree well with the potato cutting results. Noted is that the results are from the cuts of hard-crispy material. The modeling method should applicable.

Then the question is how about large deformation nonlinear materials. The mathematical model is clearly not applicable to nonlinear materials, such as meat. However, our everyday experience told us that the cuts on meat follow the rule: slicing is easier that pressing. Here, the author just provides a weak explanation of the cuts on meat: Type II fracture happens at the depth (𝑧) about 0.37 times of half of the blade width or about 𝑧 = 185 nm if 2𝑎 = 1000 nm. It is almost at the contact surface when you compare the whole thickness of the cutting material. Type III fracture happens at the contact surface, that is, 𝑧 = 0. In this small distance or on the contact surface, the inhomogeneity, inelasticity, and anisotropy of the cutting material will not affect the stress distribution since the react force/pressure will balance the external force statically based on Newton’s third law of motion. From Figure 8, it has been checked that the stress distribution in these locations is balanced by the external force. Thus, at the moment of cutting initialization, although the model will not be valid at bigger 𝑧 for big deformation nonlinear materials, the model would be still valid at 𝑧=0 and 𝑧0.37𝑎. Since we only care about the stresses to generate the possible fracture (the results in Figure 7 are just to visualize the stress distribution yielded by the model), the stress when 𝑧>0.37𝑎 is not within our consideration and does not relate to our conclusion. Thus, the model is applicable to large deformation nonlinear materials only when 𝑧0.37𝑎. We further constrain our application range for hard-crispy materials within 𝑧0.37𝑎, that is, not all 𝑧.

6. Conclusions

In order to understand the question: why pressing-and-slicing cuts use less force than pressing-only cuts during food cutting, such as potatoes, this paper formulated and studied the change of the stress distribution with various influence factors. The cutting interaction was modeled as a belt-area force acting on the surface of a semi-infinite body. The cutting force was assumed to be in certain profile based on the observed shape of blade cutting edges. The closed form expression of the cutting interaction has been developed using the direct integration method. Compared to the knowledge in the current literature, the improvements of the modeling are twofolds: (1) this work originally expressed the cutting force (force at cutting fracture) and its change with slicing angle using mathematical equations. (2) for the first time, this work mathematically expressed the blade sharpness with blade parameters and used relative sharpness concepts based on mathematical descriptions (not based on experience) to describe blade sharpness.

The computation results were used to predict the maximum stress locations. The relationships between the applied force and slicing angle, blade edge shape, blade edge width, contact length, and the fracture force and material property were discussed. Experiments have been performed to validate the formulations. The following conclusions are drawn. (1)The model is only applied to the moment of cutting fracture initialization. For elastic, homogeneous and isotropic material, the model is always valid. Since we are only interested in the cutting fracture, even for hard-crispy materials, the applicable range in material is only 𝑧0.37𝑎. (2)During the cut with slicing angle smaller than 10°, or pressing-only or mainly pressing cuts, blade cutting is a type II fracture due to the shear stress 𝜏𝑦𝑧. With slicing angle bigger than 10°, or called pressing-and-slicing cuts, blade cutting is a type III fracture due to the shear stress 𝜏𝑥𝑦 and 𝜏𝑥𝑧. Type III fracture uses considerable less force than type II fracture. This answered why pressing-and-slicing cuts use less force than pressing-only cuts. However, cuts with bigger slicing angle will make the cutting feeding less. In order to keep work efficiency, an optimal slicing angle should be selected to satisfy both minimum cutting force and maximum cut feeding speed. (3)The shape of blade cutting edge determines the distribution of the cutting force on the contact surface between the blade and the cutting material. The cutting force distribution profile determines the sharpness of a blade. By using the relative sharpness factor concept, blade sharpness can be quantified based on the obtained forces from sharpest blades and dullest blades. (4)It can be observed from both the computation and experimental results that the cutting force (𝑃𝑢) is proportional to the contact length (𝑙). This obvious observation agrees with those from everyday life. (5)Edge shape and edge width have the combined influence on fracture force. For the same edge width, external force is proportional to the maximum force intensity, which the edge shape can generate. (6)Based on material properties, the knife sharpness properties and the interaction between the blade and the material, the required force to realize certain cuts can be predicted. This observation provides the principle to optimize the cutting mechanism design and the force control algorithm design for the automation of the cutting operations.

Regarding the future work, the stress intensity factors 𝐾II and 𝐾III will be analyzed and quantified. The influence of the relative moving speed will also be investigated. On the long term, the research work will try to understand and model the behavior of interesting biomaterials or hybrid materials during robot-controlled cuts. Then robots can be intelligent to adapt themselves for any deformation of material and any varieties of material’s structure.

Appendix

A. Derivation of Stress Distribution dut to Various Forces

A.1. Stress Distribution due to Point Force

(1) Normal Point Force Only
When there is only normal point force 𝑃𝑝𝑡, for material with Poisson’s ratio 𝜇, the normal stress 𝜎 and the shear stress 𝜏 at point 𝐴(𝑥,𝑦,𝑧) are given as 𝜎𝑛𝑝=𝜎𝑥𝑛𝑝𝜎𝑦𝑛𝑝𝜎𝑧𝑛𝑝𝜏𝑛𝑝𝑥𝑦𝜏𝑛𝑝𝑦𝑧𝜏𝑛𝑝𝑥𝑧𝑇,(A.1a) where 𝜎𝑥𝑛𝑝=(12𝜇)𝑅𝑟𝑅+𝑧32𝑧𝑅3𝑥2𝑟2𝑧+(12𝜇)𝑅𝑅𝑦𝑅+𝑧2𝑟2𝑃𝑝𝑛2𝜋𝑅2,𝜎(A.1b)𝑦𝑛𝑝=(12𝜇)𝑅𝑟𝑅+𝑧32𝑧𝑅3𝑦2𝑟2𝑧+(12𝜇)𝑅𝑅𝑥𝑅+𝑧2𝑟2𝑃𝑝𝑛2𝜋𝑅2,𝜏(A.1c)𝑛𝑝𝑥𝑦=(12𝜇)𝑅𝑟𝑅+𝑧32𝑧𝑅3𝑧(12𝜇)𝑅𝑅𝑅+𝑧𝑥𝑦𝑟2𝑃𝑝𝑛2𝜋𝑅2,𝜎(A.1d)𝑧𝑛𝑝=3𝑧3𝑅3𝑃𝑝𝑛2𝜋𝑅2𝜏,(A.1e)𝑛𝑝𝑦𝑧=3𝑦𝑧2𝑅3𝑃𝑝𝑛2𝜋𝑅2𝜏,(A.1f)𝑛𝑝𝑥𝑧=3𝑥𝑧2𝑅3𝑃𝑝𝑛2𝜋𝑅2.(A.1g)The superscript 𝑛𝑝 (stands for normal direction and on a point) in (A.1a) to (A.1g) represents the stresses generated by external force 𝑃𝑝𝑛.

(2) Tangential Point Force Only
When there is only tangential point force 𝑃𝑝𝑡, the stress distribution is known as the Cerruti solution and is given in [8] as𝜎𝑡𝑝=𝜎𝑥𝑡𝑝𝜎𝑦𝑡𝑝𝜎𝑧𝑡𝑝𝜏𝑡𝑝𝑧𝑥𝑦𝜏𝑡𝑝𝑦𝑧𝜏𝑡𝑝𝑥𝑧,(A.2a) where 𝜎𝑥𝑡𝑝=𝑥(12𝜇)(𝑅+𝑧)2𝑅2𝑦22𝑅𝑦2𝑅+𝑧3𝑥2𝑅2𝑃𝑝𝑡2𝜋𝑅3,𝜎(A.2b)𝑦𝑡𝑝=𝑥(12𝜇)(𝑅+𝑧)23𝑅2𝑥22𝑅𝑥2𝑅+𝑧3𝑦2𝑅2𝑃𝑝𝑡2𝜋𝑅3𝜏,(A.2c)𝑡𝑝𝑥𝑧=𝑦(12𝜇)(𝑅+𝑧)2𝑅2+𝑥2+2𝑅𝑥2𝑅+𝑧3𝑥2𝑅2𝑃𝑝𝑡2𝜋𝑅3𝜎,(A.2d)𝑧𝑡𝑝=3𝑥𝑧2𝑅2𝑃𝑝𝑡2𝜋𝑅3𝜏,(A.2e)𝑡𝑝𝑥𝑦=3𝑥𝑦𝑧𝑅2𝑃𝑝𝑡2𝜋𝑅3𝜏,(A.2f)𝑡𝑝𝑦𝑧=3𝑥2𝑧𝑅2𝑃𝑝𝑡2𝜋𝑅3.(A.2g)The superscript 𝑡𝑝 in (A.2a) to (A.2g) represents the stresses generated by external force 𝑃𝑝𝑡.

A.2. Stress Distribution due to Line Force

(1) Normal Line Force Only
The closed form solution of the stresses at any point (𝑥,𝑦,𝑧) in the semi-infinite body generated by line force acting along (𝑥1,𝑥2) are shown in (A.3).𝜎𝑥𝑛𝑙𝜎𝑦𝑛𝑙𝜎𝑧𝑛𝑙𝜏𝑛𝑙𝑥𝑦𝜏𝑛𝑙𝑦𝑧𝜏𝑛𝑙𝑥𝑧=𝑝𝑙𝑛(𝑥𝑠)2𝜋𝑟2𝑧𝑦(12𝜇)+2+𝑧22𝑅2𝑧2+2𝜇(𝑥𝑠)4+2𝑦4+3𝑦2𝑧2+𝑧4+(𝑥𝑠)23𝑦2+2𝑧2𝑦2+𝑧2𝑅3𝑝𝑙𝑛(𝑥𝑠)2𝜋12𝜇𝑟2+𝑧𝑦2𝑅3𝑦2+𝑧2𝑧2(𝑥𝑠)2𝑦2+(32𝜇)𝑦4+(12𝜇)2𝑦2+𝑧2𝑧2𝑟2𝑅𝑦2+𝑧22𝑝𝑙𝑛(𝑥𝑠)𝑧33𝑅2(𝑥𝑠)2𝑦2𝜋2+𝑧22𝑅3𝑝𝑙𝑛𝑦2𝜋𝑟2𝑧(1+2𝜇)+2(1𝜇)𝑟2+(12𝜇)𝑧2𝑅3𝑝𝑙𝑛(𝑥𝑠)𝑦𝑧23𝑅2(𝑥𝑠)2𝑦2𝜋2+𝑧22𝑅3𝑝𝑙𝑛𝑧22𝜋𝑅3||||||||||||||||||||||||𝑠=𝑥2𝑠=𝑥1(A.3)When 𝑥1 and 𝑥2, the stress tensor is 𝜎𝑥𝑛𝑙𝜎𝑦𝑛𝑙𝜎𝑧𝑛𝑙𝜏𝑛𝑙𝑥𝑦𝜏𝑛𝑙𝑦𝑧𝜏𝑛𝑙𝑥𝑧=2𝑝𝑙𝑛𝑧𝜋𝑦2+𝑧2𝑦𝜇2𝑦2+𝑧2𝑧2𝑦2+𝑧20𝑦𝑧𝑦2+𝑧20.(A.4)

(2) Tangential Line Force Only
The closed-form expression of the stress distribution by a tangential force on line segment [𝑥1,𝑥2] is given in (A.5). 𝜎𝑥𝑡𝑙𝜎𝑦𝑡𝑙𝜎𝑧𝑡𝑙𝜏𝑡𝑙𝑥𝑦𝜏𝑡𝑙𝑦𝑧𝜏𝑡𝑙𝑥𝑧=𝑝𝑙𝑡𝑦2𝜋2+𝑧2𝑅3+(12𝜇)𝑦2+3𝑧2𝑧2𝑅(12𝜇)𝑦2𝑧(𝑧+𝑅)2𝑦(12𝜇)2+𝑧2𝑧2𝑓𝜎(𝑧+𝑅)𝑦𝑝𝑙𝑡𝑧22𝜋𝑅3𝑓𝜏𝑥𝑦𝑝𝑙𝑡𝑦𝑧2𝜋𝑅3𝑝𝑙𝑡𝑧𝑦2𝜋2+𝑧2(𝑥𝑠)3𝑅3||||||||||||||||𝑠=𝑥2𝑠=𝑥1,(A.5)where𝑓𝜎𝑦=𝑝𝑙𝑡2𝜋𝑟42(12𝜇)𝑦2𝑧(12𝜇)(𝑥𝑠)2+𝑦21𝑧+𝑅3𝑧2𝑦+2𝜇2𝑧2𝑠4+8𝑢𝑥𝑦2+𝑧24𝑧2𝑥𝑠3+2𝑢2𝑦4+𝑦2𝑧2𝑧4+6𝑥2𝑦2𝑧2+𝑧26𝑥2𝑦2+𝑧2𝑠2+4𝜇𝑥2𝑦4+𝑦2𝑧2𝑧4+2𝑥2𝑦2𝑧22𝑧2𝑥22𝑥2𝑦2+𝑧2𝑠+𝑧2𝑥4+𝑥2𝑦2+𝑧2𝑦22𝑦2+𝑧2𝑥+2𝜇22𝑦4+𝑦2𝑧2𝑧4+𝑥4𝑦2𝑧2+𝑦2𝑦2+𝑧22,𝑓𝜏𝑥𝑦=𝑝𝑙𝑡𝑦2𝜋2(2𝜇1)(𝑠𝑥)𝑧(𝑥𝑠)2+𝑦22𝑅2(2𝜇1)(𝑥𝑠)(𝑥𝑠)2+𝑦22(4𝜇1)(𝑥𝑠)(𝑥𝑠)2+𝑦2𝑧2+𝑥𝑠𝑅4+(4𝜇1)(𝑥𝑠)𝑅2𝑧2+(𝑥𝑠)(𝑥𝑠)2+𝑦2𝑦2𝜇2+𝑧2(𝑥𝑠)2+𝑦2𝑦2+𝑧2𝑅2.(A.6)
As 𝑥1 and 𝑥2 go to and , respectively, that is, a line force acting on the boundary of a semi-infinite body, the stress distribution changes to 𝜎𝑥𝑡𝑙𝜎𝑦𝑡𝑙𝜎𝑧𝑡𝑙𝜏𝑡𝑙𝑥𝑦𝜏𝑡𝑙𝑦𝑧𝜏𝑡𝑙𝑥𝑧=𝑝𝑙𝑡𝜋𝑦2+𝑧2.000𝑦0𝑧(A.7)

A.3. Stress Distribution due to Area Force

(1) Normal Area Force Only
The stresses generated by the normal force intensity between [𝑎,𝑤] are 𝑓11𝑛=𝜎𝑥𝑛1=2𝜇𝜋𝑞𝑛𝑧𝑎𝑤(𝑦+𝑎)𝑇+2Γ||||𝑓𝑣=𝑤,𝑣=𝑎,21𝑛=𝜎𝑦𝑛1=1𝜋𝑞𝑛𝑧𝑎𝑤𝑎(𝑦𝑡)𝑣+𝑡𝑦+𝑦2+𝑧2(𝑦𝑣)2+𝑧2𝑧+(𝑎+𝑦)𝑇+2Γ||||𝑓𝑣=𝑤,𝑣=𝑎,31𝑛=𝜎𝑧𝑛1=1𝜋𝑞𝑛𝑧𝑎𝑤𝑎(𝑣𝑦)+𝑣𝑦+𝑦2+𝑧2(𝑦𝑣)2+𝑧2||||𝑓+(𝑎+𝑦)𝑇𝑣=𝑤,𝑣=𝑎,4𝑛=𝜏𝑛1𝑥𝑦𝑓=0,51𝑛=𝜏𝑛1𝑦𝑧=𝑧𝜋𝑞𝑣𝑎𝑤𝑧(𝑎+𝑣)(𝑣𝑦)2+𝑧2||||,𝑓𝑇𝑣=𝑤𝑣=𝑎61𝑛=𝜏𝑛1𝑥𝑧=0,(A.8) where 𝑇=tan1[(𝑦𝑣)/𝑧], Γ=ln((𝑣𝑦)2+𝑧2) and ||{𝑓(𝑣)}𝑣=𝑤𝑣=𝑎=𝑓(𝑤)𝑓(𝑎).
The stresses generated by the normal force intensity between (𝑤,𝑤) are obtained as 𝑓12𝑛=𝜎𝑥𝑛2=2𝑞𝑛𝜇𝜋𝑇||||𝑓𝑣=𝑤,𝑣=𝑤,22𝑛=𝜎𝑦𝑛2=𝑞𝑛𝜋(𝑦𝑣)𝑧(𝑦𝑣)2+𝑧2||||𝑓+𝑇𝑣=𝑤,𝑣=𝑤,32𝑛=𝜎𝑧𝑛2=𝑞𝑛𝜋(𝑣𝑦)𝑧(𝑣𝑦)2+𝑧2||||𝑓+𝑇𝑣=𝑤,𝑣=𝑤,42𝑛=𝜏𝑛2𝑥𝑦𝑓=0,52𝑛=𝜏𝑛2𝑦𝑧=𝑞𝑛𝑧2𝜋(𝑣𝑦)2+𝑧2||||𝑓𝑣=𝑤,𝑣=𝑤,62𝑛=𝜏𝑛2𝑥𝑧=0.(A.9)
The stresses generated by the normal force intensity between [𝑤,𝑎] are obtained as 𝑓13𝑛=𝜎𝑥𝑛3=2𝜇𝜋𝑞𝑛𝑧𝑤𝑎(𝑦𝑎)𝑇+2Γ||||𝑓𝑣=𝑎,𝑣=𝑤,23𝑛=𝜎𝑦𝑛3=1𝜋𝑞𝑛𝑧𝑤𝑎𝑎(𝑦𝑣)+𝑣𝑦+𝑦2+𝑧2(𝑦𝑣)2+𝑧2𝑧+(𝑦𝑎)𝑇+2Γ||||𝑓𝑣=𝑎,𝑣=𝑤,33𝑛=𝜎𝑧𝑛3=1𝜋𝑞𝑛𝑧𝑤𝑎𝑎(𝑣𝑦)+𝑣𝑦+𝑦2+𝑧2(𝑦𝑣)2+𝑧2||||𝑓+(𝑦𝑎)𝑇𝑣=𝑎,𝑣=𝑤,43𝑛=𝜏𝑛3𝑥𝑦𝑓=0,53𝑛=𝜏𝑛3𝑦𝑧=𝑧𝜋𝑞𝑛𝑤𝑎𝑧(𝑎+𝑣)(𝑣𝑦)2+𝑧2||||𝑓𝑇𝑣=𝑎,𝑣=𝑤,63𝑛=𝜏𝑛3𝑥𝑧=0.(A.10)
The stress tensor generated by normal force can then be expressed as 𝜎𝑛𝑥𝜏𝑛𝑥𝑦𝜏𝑛𝑥𝑧𝜏𝑛𝑥𝑦𝜎𝑛𝑦𝜏𝑛𝑦𝑧𝜏𝑛𝑥𝑧𝜏𝑛𝑦𝑧𝜎𝑛𝑧=3𝑖=1𝜎𝑥3𝑛𝑖𝑖=1𝜏3𝑛𝑖𝑥𝑦𝑖=1𝜏3𝑛𝑖𝑥𝑧𝑖=1𝜏3𝑛𝑖𝑥𝑦𝑖=1𝜎𝑦3𝑛𝑖𝑖=1𝜏3𝑛𝑖𝑦𝑧𝑖=1𝜏3𝑛𝑖𝑥𝑧𝑖=1𝜏3𝑛𝑖𝑦𝑧𝑖=1𝜎𝑦𝑛𝑖=3𝑖=1𝑓31𝑖𝑛𝑖=1𝑓34𝑖𝑛𝑖=1𝑓36𝑖𝑛𝑖=1𝑓34𝑖𝑛𝑖=1𝑓32𝑖𝑛𝑖=1𝑓35𝑖𝑛𝑖=1𝑓36𝑖𝑛𝑖=1𝑓35𝑖𝑛𝑖=1𝑓3𝑖𝑛.(A.11)

(2) Tangential Area Force Only
The stresses generated by the tangential force intensity between [𝑎,𝑤] are obtained as 𝑓11𝑡=𝜎𝑡𝑥𝑓=0,21𝑡=𝜎𝑡𝑦𝑓=0,31𝑡=𝜎𝑡𝑧𝑓=0,41𝑡=𝜏𝑡𝑥𝑦=1𝜋𝑞𝑡𝑣𝑎𝑤𝑡𝑧𝑇+(𝑎+𝑦)2Γ||||𝑓𝑣=𝑤,𝑣=𝑎,51𝑡=𝜏𝑡𝑦𝑧𝑓=0,61𝑡=𝜏𝑡𝑥𝑧=1𝜋𝑞𝑡𝑧𝑎𝑤(𝑎+𝑦)𝑇+2Γ||||𝑣=𝑤.𝑣=𝑎.(A.12)
The stresses generated by the tangential force intensity between (𝑤,𝑤) are obtained as 𝑓12𝑡=𝜎𝑡𝑥𝑓=0,22𝑡=𝜎𝑡𝑦𝑓=0,32𝑡=𝜎𝑡𝑧𝑓=0,42𝑡=𝜏𝑡𝑥𝑦=𝑞𝑡Γ||||𝑓2𝜋𝑣=𝑤,𝑣=𝑤,52𝑡=𝜏𝑡𝑦𝑧𝑓=0,62𝑡=𝜏𝑡𝑥𝑧=𝑞𝑡𝜋𝑇||||𝑣=𝑤.𝑣=𝑤.(A.13)
The stresses generated by the tangential force intensity between [𝑤,𝑎] are obtained as 𝑓13𝑡=𝜎𝑡𝑥𝑓=0,23𝑡=𝜎𝑡𝑦𝑓=0,33𝑡=𝜎𝑡𝑧𝑓=0,43𝑡=𝜏𝑡𝑥𝑦=1𝜋𝑞𝑡𝑤𝑎𝑣𝑧𝑇+(𝑎+𝑦)2Γ||||𝑓𝑣=𝑎,𝑣=𝑤,53𝑡=𝜏𝑡𝑦𝑧𝑓=0,63𝑡=𝜏𝑡𝑥𝑧=1𝜋𝑞𝑡𝑧𝑤𝑎(𝑎+𝑦)𝑇+2Γ||||𝑣=𝑎.𝑣=𝑤.(A.14)
The final stress tensor generated by tangential force can then be expressed as 𝜎𝑡𝑥𝜏𝑡𝑥𝑦𝜏𝑡𝑥𝑧𝜏𝑡𝑥𝑦𝜎𝑡𝑦𝜏𝑡𝑦𝑧𝜏𝑡𝑥𝑧𝜏𝑡𝑦𝑧𝜎𝑡𝑧=3𝑖=1𝜎𝑥3𝑡𝑖𝑖=1𝜏3𝑡𝑖𝑥𝑦𝑖=1𝜏3𝑡𝑖𝑥𝑧𝑖=1𝜏3𝑡𝑖𝑥𝑦𝑖=1𝜎𝑦3𝑡𝑖𝑖=1𝜏3𝑡𝑖𝑦𝑧𝑖=1𝜏3𝑡𝑖𝑥𝑧𝑖=1𝜏3𝑡𝑖𝑦𝑧𝑖=1𝜎𝑦𝑡𝑖=3𝑖=1𝑓31𝑖𝑡𝑖=1𝑓34𝑖𝑡𝑖=1𝑓36𝑖𝑡𝑖=1𝑓34𝑖𝑡𝑖=1𝑓32𝑖𝑡𝑖=1𝑓35𝑖𝑡𝑖=1𝑓36𝑖𝑡𝑖=1𝑓35𝑖𝑡𝑖=1𝑓3𝑖𝑡.(A.15)

(3) Force with Both Normal and Tangential Components
Using (A.11) and (A.15), the stress distribution generated by both 𝑞𝑛 and 𝑞𝑡 is obtained as 𝜎𝑥𝜏𝑥𝑦𝜏𝑥𝑧𝜏𝑥𝑦𝜎𝑦𝜏𝑦𝑧𝜏𝑥𝑧𝜏𝑦𝑧𝜎𝑧=𝜎𝑛𝑥𝜏𝑛𝑥𝑦𝜏𝑛𝑥𝑧𝜏𝑛𝑥𝑦𝜎𝑛𝑦𝜏𝑛𝑦𝑧𝜏𝑛𝑥𝑧𝜏𝑛𝑦𝑧𝜎𝑛𝑧+𝜎𝑡𝑥𝜏𝑡𝑥𝑦𝜏𝑡𝑥𝑧𝜏𝑡𝑥𝑦𝜎𝑡𝑦𝜏𝑡𝑦𝑧𝜏𝑡𝑥𝑧𝜏𝑡𝑦𝑧𝜎𝑡𝑧=3𝑖=1𝑓1𝑖𝑛+𝑓1𝑖𝑡3𝑖=1𝑓4𝑖𝑛+𝑓4𝑖𝑡3𝑖=1𝑓6𝑖𝑛+𝑓6𝑖𝑡3𝑖=1𝑓4𝑖𝑛+𝑓4𝑖𝑡3𝑖=1𝑓2𝑖𝑛+𝑓2𝑖𝑡3𝑖=1𝑓5𝑖𝑛+𝑓5𝑖𝑡3𝑖=1𝑓6𝑖𝑛+𝑓6𝑖𝑡3𝑖=1𝑓5𝑖𝑛+𝑓5𝑖𝑡3𝑖=1𝑓3𝑖𝑛+𝑓3𝑖𝑡.(A.16) Equation (A.16) can be rewritten as 𝜎𝑥𝜏𝑥𝑦𝜏𝑥𝑧𝜏𝑥𝑦𝜎𝑦𝜏𝑦𝑧𝜏𝑥𝑧𝜏𝑦𝑧𝜎𝑧=𝑞𝑛𝑓1(𝑦,𝑧,𝑎,𝑤)𝑞𝑡𝑓4(𝑦,𝑧,𝑎,𝑤)𝑞𝑡𝑓5𝑞(𝑦,𝑧,𝑎,𝑤)𝑡𝑓4(𝑦,𝑧,𝑎,𝑤)𝑞𝑛𝑓2(𝑦,𝑧,𝑎,𝑤)𝑞𝑛𝑓6(𝑞𝑦,𝑧,𝑎,𝑤)𝑡𝑓5(𝑦,𝑧,𝑎,𝑤)𝑞𝑛𝑓6(𝑦,𝑧,𝑎,𝑤)𝑞𝑛𝑓3,(𝑦,𝑧,𝑎,𝑤)(A.17) where the functions 𝑓1, 𝑓2, 𝑓3, 𝑓4, 𝑓5, and 𝑓6 can be explicitly expressed using (A.8) to (A.10) and (A.12) to (A.14) using the variables 𝑦,𝑧,𝑎,𝑤.

Symbols

𝑎:Half of the width of the blade edge (see Figure 3)
𝐴(𝑥,𝑦,𝑧):A point in the material with coordinate (𝑥,𝑦,𝑧) in frame 𝑜𝑥𝑦𝑧 (see Figure 3)
𝐾𝑠:Ratio of the fracture initialization force over continuous cutting force
𝑙:Contact length between the blade and the material (see Figure 3)
𝑝𝑛 and 𝑝𝑡:Cutting pressure components in the 𝑧 (normal) direction and 𝑥 (tangential) direction, respectively (see Table 2)
𝑝𝑙𝑛 and 𝑝𝑙𝑡:Normal and tangential components of line distributed cutting force
𝑃:Total cutting force (see Figure 4 and Table 2)
𝑃𝑐:Force during continuous cutting fracture
𝑃𝑢:Required force to initialize cutting fracture
𝑃𝑛 and 𝑃𝑡:Normal and tangential component of total applied external force, respectively
𝑃𝑛𝑢 and 𝑃𝑡𝑢:Normal and tangential component of total cutting force during fracture, respectively
𝑃𝑝𝑛 and 𝑃𝑝𝑡:Normal and tangential component of a point applied external force, respectively
𝑞𝑛 and 𝑞𝑡:Maximum of the area force intensity between [𝑤,𝑤] in the normal and tangential direction, respectively
𝑟:Distance between point 𝑜 and point 𝐵 (see Figure 3(b))
𝑅:Distance between point 𝐴 and point 𝐵 (see Figure 3(b))
𝑤:Width of the top side of the trapezoid profile of force intensity distribution (see Figure 5)
𝛼:Slicing angel, that is, the angle between 𝑃 and 𝑃𝑛 (see Figures 3(b) and 2)
𝜎 and 𝜏:𝜎 is the tensile stress or just an expression of any of the stress components and 𝜏 is only the shear stress
𝜏𝑖𝑗:Maximum shear stress, 𝑖 or 𝑗=1, 2, or 3
𝜏max:Maximum of the maximum shear stresses 𝜏𝑖𝑗, 𝑖 or 𝑗=1, 2, or 3
𝜏𝑢:Ultimate shear stress
𝜇:Poisson’s ratio
𝜂:Blade relative sharpness factor
𝜂0:Relative sharpness factor of the sharpest blade
𝜂𝑓:Relative sharpness factor of the dullest blade
𝜅:Blade relative sharpness level.

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