Abstract

This two-part paper is aimed at developing a microstructure-based mechanistic modeling framework to predict the cutting forces and acoustic emissions (AEs) generated during bone sawing. The modeling framework is aimed at the sub-radius cutting condition that dominates chip-formation mechanics during the bone sawing process. Part 1 of this paper deals specifically with the sawing experiments and modeling of the cutting/thrust forces. The model explicitly accounts for key microstructural constituents of the bovine bone (viz., osteon, interstitial matrix, lamellar bone, and woven bone). The cutting and thrust forces are decomposed into their shearing and ploughing components. Microstructure-specific shear stress values critical to the model calculations are estimated using micro-scale orthogonal cutting tests. This approach of estimating the microstructure-specific shear stress overcomes a critical shortcoming in the literature related to high-strain rate characterization of natural composites, where the separation of the individual constituents is difficult. The six model coefficients are calibrated over a range of clinically relevant depth-of-cuts (DOCs) using pure haversian regions (comprising of osteon and interstitial matrix), and pure plexiform regions (comprising of lamellar bone and woven bone). The calibrated model is then used to make predictions in the transition region between the Haversian and plexiform bone, which is characterized by gradient structures involving varying percentages of osteon, interstitial matrix, lamellar bone, and woven bone. The mean absolute percentage error in the force predictions is under 10% for both the cutting and thrust forces. The reality of spatially varied properties in the cortical bone limits the universal use of microstructure-specific shear stress values reported here. Fundamental advancements in the literature associated with both high-strain rate bone mechanics and machining are needed to address this critical limitation.

1 Introduction

Microstructure-based machining models have recently emerged as the preferred approach for process planning efforts involving diverse material systems such as carbon/glass fiber composites, polymer nanocomposites, and metal alloys [13]. The distinguishing aspect of this approach is the fact that rather than treating the material system as homogenous [4,5], it explicitly models the key microstructural constituents relevant to the machining outcomes. As a result, the model-predictive capabilities span a wide range of microstructural variations in the material [13]. While this approach appears ideal for modeling the microstructural heterogeneity in bone [6], it has received little attention in the literature related to bone machining [711].

Bovine cortical bone has been used extensively in both experimental and modeling investigations [711]. This is primarily due to its ease of access and similarities with human bone both in terms of structure and mechanical properties [6,12,13]. Bovine cortical bone has four key microstructural constituents that influence its machining outcomes, viz., osteon, interstitial matrix, woven bone, and lamellar bone [6]. Recent experimental studies have highlighted the unique role played by these constituents in dictating the machining responses of bovine bone [14]. These experimental findings are in stark contrast to the state of literature related to the bone machining models spanning analytical [15,16], mechanistic [8,11,17,18], and finite element [1922] approaches, where the bone workpiece is treated to be homogeneous. This assumption is primarily made because unlike man-made composites where the individual phases can be separated for characterization purposes [13], naturally occurring bio-composites such as bone pose a challenge in extracting properties/coefficients relevant to its microstructural components [12].

In light of the current state-of-literature, this two-part paper is aimed at developing a microstructure-based mechanistic modeling framework to predict the cutting forces and acoustic emissions (AEs) generated while cutting with a bone-sawtooth. Both these machining responses are chosen specifically given their proven use in process-planning efforts [23,24]. The modeling framework is aimed at the sub-radius cutting condition that dominates chip-formation mechanics during the bone sawing process [8]. Part 1 of this paper deals specifically with the sawing experiments and modeling of the cutting/thrust forces. Part 2 deals with the analysis of the AE signatures collected from the sawing experiments, followed by the development of a model to predict the AE energy rate. The cutting force model of James et al. [8,11] is the closest work in literature that maps to this effort, with the critical difference being that the work of James et al. [8,11] does not deal with the microstructural heterogeneity or the acoustic emissions encountered during bone-sawing.

The remainder of this Part 1 paper is divided into the following sections: Sec. 2 deals with the specifics of the bone sawing experiment. Section 3 discusses the development of the mechanistic cutting force model, including the extraction of the microstructure-specific shear stress values. Next, Sec. 4 deals with the model calibration and validation. Finally, Sec. 5 presents the specific conclusions that can be drawn from this work.

2 Bone Sawing Experiment

This section will first discuss the specimen preparation, followed by a description of the experimental setup and sawing conditions. Finally, cutting force and thrust force trends are presented along with the evidence for shear-based cutting at the depth-of-cuts (DOCs) used in this study.

2.1 Specimen Preparation and Microstructure.

All experiments were performed on bovine femoral cortical bone harvested from a ∼30-month-old cow. The sample orientation was chosen such that the cutting experiments generated a slot transverse to the femoral longitudinal/osteonal axis, as depicted in Fig. 1(a). In addition to being clinically relevant, this cutting direction was chosen because it resulted in statistically significant microstructural variation along the length of cut, which is a critical requirement for the model development effort (Figs. 1(b)1(d)) [15,2527].

Fig. 1
Bone cutting direction and microstructural variation along length of cut: (a) schematic illustration of bone cutting direction, (b) pure Haversian region showing osteon and interstitial matrix components, (c) pure plexiform region shown lamellar and woven bone components, (d) transition region showing all four microstructural constituents, and (e) digital identification of characteristic structures in the transition region in (d) (Note: Scale = 100 µm in (b), (c), and (d))
Fig. 1
Bone cutting direction and microstructural variation along length of cut: (a) schematic illustration of bone cutting direction, (b) pure Haversian region showing osteon and interstitial matrix components, (c) pure plexiform region shown lamellar and woven bone components, (d) transition region showing all four microstructural constituents, and (e) digital identification of characteristic structures in the transition region in (d) (Note: Scale = 100 µm in (b), (c), and (d))
Close modal

The mid-section of the femoral bone shaft (i.e., mid-diaphysis) was sectioned using a Mar-Med bone saw, under constant water irrigation. After harvesting, the samples were covered with phosphate-buffered saline (PBS) soaked gauze and frozen at −20 °C for storage. They were thawed and then saturated with PBS to keep the sample moist during the sawing experiment [14].

In general, the anterior and posterior quadrants depicted in Fig. 1(a) are dominated by pure plexiform and pure Haversian regions, respectively (over 50%), while the medial and lateral quadrants contain gradient microstructures that are a blend between the two regions. [14,28]. For this study, three samples were harvested such that they contained clearly demarcated pure Haversian and pure plexiform regions along with a transition region between the two (Fig. 1(a)).

Figure 1(b) depicts the characteristic cross section of the pure Haversian regions used in this study. The Haversian region is composed of unidirectional fiber-like structures called osteons that are embedded in an interstitial matrix. On average these osteons are 3–5 mm long and elliptically shaped with a major and minor diameter of 190 μm and 140 μm, respectively [14]. Hole-like resorption cavities can also be found where the bone has eroded during the formation or remodeling process [29,30]. The pure plexiform region shown in Fig. 1(c) contains alternating layers of woven and lamellar bone that are each ∼70–50 μm thick and ∼ 500 μm long, giving it the appearance of a composite laminate [14]. The transitional region between the pure Haversian and plexiform regions of the bone is characterized by gradient microstructures involving all four of the microstructural components, viz., osteon, interstitial matrix, woven bone, and lamellar bone (Figs. 1(d) and 1(e)).

2.2 Experimental Setup and Cutting Conditions.

This section will present the details about the tool, the experimental setup, and associated cutting conditions that generated the cutting force and acoustic emissions data used for the model development work presented in this two-part paper.

2.2.1 Tool.

For all cutting experiments, a single tooth isolated from a KM-256 Brasseler USA oscillating stainless steel saw blade was used [8,11]. The tooth rake and clearance angles were approximately −4 and +30 deg, respectively, with a measured edge radius of ∼14.6 μm and tool width of 740 μm. The tooth was also examined post-cutting to confirm that wear was not a factor while interpreting the results.

2.2.2 Experimental Testbed.

In order to generate cutting conditions similar to those encountered in power bone sawing, James et al. [8,11] designed the testbed seen in Fig. 2(a). The studies reported in this paper were conducted on this same testbed. While the detailed design and working principles of the testbed can be found in Ref. [11], its unique DOC control mechanism is briefly discussed here, for purposes of continuity. As seen in Fig. 2(a), the apparatus relies on a rotary table to impart the cutting velocity to the bone sample. The sawtooth is mounted at the point of tangency to the sample. As the sample-carrying table rotates, a power-screw is used to drive a self-locking wedge that in-turn advances the tool into the workpiece [8,11]. For every rotation of the rotary table, the tool advances by a specific constant DOC, as dictated by the wedge. The use of different wedges results in different nominal DOC values [8,11]. It should be noted that the design feature responsible for a straight-line, constant DOC engagement between the sawtooth and the bone sample relies on the relative rotary motion between the bone sample holder and the driving motor. This design feature of the testbed is described in detail in Ref [11] and is not repeated here for purposes of brevity.

Fig. 2
High-speed linear sawing testbed and associated sensing: (a) testbed [11] and (b) cartoon depicting details present in the inset in (a) showing force and acoustic emission sensor mounting and rotation of sample. Note: The straight-line, constant depth control mechanism related to (a) is detailed in Ref. [11].
Fig. 2
High-speed linear sawing testbed and associated sensing: (a) testbed [11] and (b) cartoon depicting details present in the inset in (a) showing force and acoustic emission sensor mounting and rotation of sample. Note: The straight-line, constant depth control mechanism related to (a) is detailed in Ref. [11].
Close modal

For the work reported here, three distinct modifications were done to the original setup of James et al. [8,11]. These include

  1. Friction reduction: Molybdenum disulfide lubricant was applied on the wedge contact surfaces to reduce friction in the testbed and thereby improve its depth-of-cut repeatability.

  2. Sensors: A Kistler 9251A three-axis load-cell and a Physical Acoustics Nano 30 sensor were both incorporated into the tool-holder to measure the cutting forces and acoustic emissions, respectively. Figure 2(b) shows the relative positioning of the sensors with respect to the tool-holder. As seen, the load-cell is positioned underneath the tool-holder, whereas the AE sensor was mounted to contact the tool-holder. For the AE sensor, contact was ensured using a spring-loaded mounting plate and vacuum grease [31]. A common sampling rate of 2 MHz was implemented for both sensors to synchronize the data collection. It should be noted that the higher sampling rate was dictated by the needs of the AE data collection. The AE data was first collected using a Physical Acoustics 2/4/6 voltage pre-amplifier (60 db gain setting) and then filtered using a band-pass filter between 100 and 800 KH based on the response frequency range of the AE sensor.

  3. External calibration: Given the sensitivity of the cutting forces and AE to the changes in the DOC, the linear depth-of-cut imposed by each wedge was externally calibrated using a combination of high-speed imaging and a micrometer calibration slide. This allowed for visibility into the specific DOC taken by the tool for every pass. Unlike the work of James et al. [8,11] where only one nominal DOC value was chosen for each wedge, the calibration procedure implemented here confirmed that a single wedge can in fact impose a range of specific DOCs that are centered around its nominal value. The three wedges used in this study resulted in datasets, corresponding to the DOC ranges of 1.8–3.8 μm, 4.0–5.4 μm, and 5.6–9.4 μm, respectively, with ∼ 25 slots cut in each of the three ranges. For the remainder of this two-part paper, these datasets will be referred to as Data set 1 (low DOC), Data set 2 (medium DOC), and Data set 3 (high DOC), respectively.

2.2.3 Experimental Conditions.

During the sawing procedure, 10 mm long slotting experiments were conducted using the setup in Fig. 2. The experiments were conducted at a cutting velocity of 3700 mm/s, with multiple DOC values in the range of 1.8–9.4 μm as mentioned above. The cutting velocity and DOC values selected here are representative of past studies involving bone sawing [4,8,11]. The three bone samples harvested, as described in Sec. 2.1, were each used to generate the cutting force and acoustic emission data related to Data set 1 (1.8–3.8 μm DOC range), Data set 2 (4.0–5.4 μm DOC range), and Data set 3 (5.6–9.4 μm DOC range), respectively. Given the lack of discernable tool-wear, the same sawtooth was used for all the tests. The bone samples were oriented such that the DOC was first initiated in the pure Haversian region (Fig. 1(b)). The tool then continued into the transition region (Fig. 1(d)) before finally exiting the workpiece after cutting through the pure plexiform region (Fig. 1(c)). Table 1 summarizes the sawing conditions. Both cutting force and acoustic emission signals were collected during the entire duration of the cut. The data from the pure Haversian and pure plexiform regions was used to calibrate the model coefficients, whereas the data from the transition region was used for model validation (refer Sec. 4 ahead).

Table 1

Summary of experimental conditions

Workpiece• Bovine cortical bone samples polished perpendicular to osteon axis
• Model calibration: pure Haversian and pure plexiform regions
• Model validation: transition regions
Cutting tool• Bone sawtooth—Stainless steel
• 14.6 μm edge radius (rE)
• −4 deg rake angle (αnom)
• 30 deg clearance angle
• 740 μm wide
Cutting speed• 3700 mm/s
Depth of cut (DOC)• Data set 1 (Low DOC): 1.8–3.8 μm
• Data set 2 (Medium DOC): 4.0–5.4 μm
• Data set 3 (High DOC): 5.6–9.4 μm
Workpiece• Bovine cortical bone samples polished perpendicular to osteon axis
• Model calibration: pure Haversian and pure plexiform regions
• Model validation: transition regions
Cutting tool• Bone sawtooth—Stainless steel
• 14.6 μm edge radius (rE)
• −4 deg rake angle (αnom)
• 30 deg clearance angle
• 740 μm wide
Cutting speed• 3700 mm/s
Depth of cut (DOC)• Data set 1 (Low DOC): 1.8–3.8 μm
• Data set 2 (Medium DOC): 4.0–5.4 μm
• Data set 3 (High DOC): 5.6–9.4 μm

2.3 Cutting and Thrust Force Trends.

Figure 3(a) depicts the characteristic trends seen in the cutting force and the thrust force as the sawtooth cuts from the pure Haversian region (on the extreme left) into the pure plexiform region on the extreme right, at a DOC of 4.7 μm. As seen in the data, for the sub-radius cutting condition encountered in bone-sawing, the thrust force is a dominant component of the resultant cutting force. This observation is in line with associated literature [8,11,32]. The thrust forces are seen to steadily increase as the underlying microstructure transitions from a combination of osteon and interstitial matrix seen in the pure Haversian region (Fig. 1(b)) to a combination of woven and lamellar bone seen in the plexiform region (Fig. 1(c)). In contrast, the cutting force does not appear to be as sensitive to the microstructural variation as the thrust force. In general, the trend seen here of a rising resultant cutting force, as the tool moves from pure Haversian regions to pure plexiform regions, is comparable with the finding in the micro-milling work of Conward and Samuel [14]. This trend is attributed to the distinctly different properties of the underlying microstructural components [14]. The effect of the microstructures encountered in the transition region is also captured in the signals as seen by the gradual rise, especially in the thrust force.

Fig. 3
Cutting and thrust force trends with depth-of-cut: (a) example of cutting force signal variations attributed to different bone types at 4.7 μm depth-of-cut (Scale = 120 µm), (b) experimental data from pure Haversian and plexiform regions, (b-1) pure Haversian region cutting force, (b-2) pure Haversian region thrust force, (b-3) pure plexiform region cutting force, and (b-4) pure plexiform region thrust force (Note: Figs. (b-1)–(b-4) also have an overlay of the calibrated model predictions, discussed ahead in Sec. 4).
Fig. 3
Cutting and thrust force trends with depth-of-cut: (a) example of cutting force signal variations attributed to different bone types at 4.7 μm depth-of-cut (Scale = 120 µm), (b) experimental data from pure Haversian and plexiform regions, (b-1) pure Haversian region cutting force, (b-2) pure Haversian region thrust force, (b-3) pure plexiform region cutting force, and (b-4) pure plexiform region thrust force (Note: Figs. (b-1)–(b-4) also have an overlay of the calibrated model predictions, discussed ahead in Sec. 4).
Close modal

Figures 3(b) (1–4) depict the trends seen in the average thrust and cutting force values obtained from the pure Haversian and pure plexiform regions, as a function of the depth-of-cut. As seen, the data are binned into three separate ranges of DOCs, Data set 1 (1.8–3.8 μm DOC range), Data set 2 (4.0–5.4 μm DOC range), and Data set 3 (5.6–9.4 μm DOC range), corresponding to the three wedges used in this study. In addition to the raw experimental data, Figs. 3(b) (1–4) also have an overlay of the calibrated model predictions, which will be discussed ahead in Sec. 4.

Chip morphologies were examined to ascertain that shear is the dominant failure mode encountered within the range of DOCs used in this study. This is critical for justifying the underlying assumptions made by the mechanistic models developed for both cutting forces as well as AE predictions. Figures 4(a) and 4(b) show representative images of the chips that are formed at DOCs of 1.8 μm and 9.4 μm, respectively. The semi-continuous chip with thickness values that are not disproportionately higher than the DOC value confirm shear-dominated failure. This finding is in line with the results of past researchers [4,14,15,27] who observed similar shear-dominant chip morphologies at depths of cut in the (2–18 μm).

Fig. 4
Chip morphologies indicating shear-dominant cutting: (a) pure Haversian region, DOC 1.8 mm and (b) pure plexiform region, DOC 9.4 mm (Scale bar = 300 mm)
Fig. 4
Chip morphologies indicating shear-dominant cutting: (a) pure Haversian region, DOC 1.8 mm and (b) pure plexiform region, DOC 9.4 mm (Scale bar = 300 mm)
Close modal

3 Cutting Force Model Development

The model presented in Part 1 of this paper will be focused on predicting the magnitude of the cutting force and thrust force, as a function of the specific microstructural constituents and depth-of-cuts encountered by the bone-sawtooth. The pure Haversian region only comprised two components, viz., osteon, and interstitial matrix. Similarly, the pure plexiform region comprised only two components, viz., woven and lamellar bone. Given the distribution of the microstructural constituents discussed in Sec. 2.1, the microstructure-based mechanistic model will be first calibrated on the pure Haversian and pure plexiform regions (Figs. 1(b) and 1(c)). The model predictions will then be validated on the transition region (Fig. 1(d)) that has varying area fractions of all four of the microstructural constituents. Part 2 of this paper will use this same approach for calibrating and validating the model for predicting the AE energy rate.

3.1 Model Assumptions.

The following four assumptions are made for modeling the cutting and thrust force magnitudes encountered in bone sawing:

Assumption 1:

The sawtooth slot cutting is approximated as a two-dimensional process implying that it can be modeled solely by the in-plane cutting and thrust force components. This assumption is in line with prior models dealing with bone sawing [8,11] and was also confirmed by the relatively low magnitude of the out-of-plane forces collected by the three-axis load-cell in Fig. 2.

Assumption 2:

Both shearing and ploughing phenomena are assumed to be contributors to the cutting and thrust forces. This assumption is captured in the framework shown in Fig. 5 that depicts the uncut chip thickness (tO) being smaller than the tool edge radius (rE). A penetration depth (tP), as indicated by Point A in Fig. 5, separates the two cutting mechanisms. The material above this point is plastically deformed by a shear zone to result in chip removal, whereas the material below this point is “ploughed” by the cutting edge due to elastic-plastic deformation [33,34].

Assumption 3:

Based on prior work in literature [3538], the penetration depth (tP) for bone sawing is assumed to be 1/10th of the edge radius (Note: The edge radius is ∼15 μm in this study). This assumption is made here because of the inconsistencies in the analytical solutions used to predict the stagnation angle (θ) [39]. It is supported by the semi-continuous chip morphologies in Fig. 4 that suggest that sawing DOC is always greater than the assumed penetration depth of ∼1.5 μm.

Assumption 4:

Osteons, interstitial matrix, woven, and lamellar bone are the key microstructural constituents that contribute to the cutting forces in bovine bone [14]. Larger void structures such as resorption cavities and Haversian canals are not assumed to influence the cutting force magnitudes and as such are neglected from the area fraction calculations.

Fig. 5
Schematic illustration of shearing and ploughing during sub-radius orthogonal cutting
Fig. 5
Schematic illustration of shearing and ploughing during sub-radius orthogonal cutting
Close modal

3.2 Rounded Cutting-Edge Model Calculations.

The mechanistic model presented in this paper builds on the rounded cutting-edge models in literature dealing with sub-radius cutting [3,8,11,32,40,41]. Given the 2D framework outlined in Fig. 5, the tool width (w) is discretized into infinitesimally small elemental slices. The cutting and thrust forces are decomposed into their shearing and ploughing components as
wdFCTotal=wdFCs+wdFCp
(1)
and
wdFTTotal=wdFTs+wdFTp
(2)
where
  • dFCTotal and dFTTotal are the elemental cutting and thrust forces, respectively;

  • dFCs and dFTs are the shearing components of the elemental cutting and thrust forces, respectively; and

  • dFCp and dFTp are the ploughing components of elemental cutting and thrust forces, respectively.

As shown in Eqs. (1) and (2), the total cutting and thrust force encountered by the sawtooth can be obtained by summing the elemental cutting and thrust forces, over the entire tool width w.

Once the penetration depth tp is determined as per Assumption 3, the stagnation angle can be found as
θ=cos1{rEtPrE}
(3)
where rE is the edge radius of the tool. Using the calculated stagnation angle θ (Eq. (3)) and the nominal rake angle of the tool αnom (Fig. 5), the effective rake angle αeff (Fig. 5) can be calculated as [30,31]
αeff=tan1{(c*tOrE1)tan(αnom)sec(αnom)+sinθc*tOrE1+cosθ}
when c*tOrE>1+sin(αnom); and
αeff=tan1{(2c*tOrE)c*tOrEsinθc*tOrE1+cosθ}
when
c*tOrE1+sin(αnom)
(4)

In Eq. (4), the variables tO and rE have previously been described as uncut chip thickness and tool edge radius, respectively. In line with prior literature, the constant c describing the tool-chip contact length is assumed to have a value of 2 [40,41].

The normal and friction forces acting on the line AB in Fig. 5 are associated only with chip removal along the effective rake face and can be described as
dFCs=dFnscosαeff+dFfssinαeff
(5)
dFTs=dFnssinαeff+dFfscosαeff
(6)
where
  • dFCs and dFTs are the shearing components of the cutting and thrust forces, respectively, as described in Eqs. (1)(2);

  • dFns and dFfs are the normal and friction shearing forces, respectively; and

  • αeff is the effective rake angle described in Eq. (4).

Similarly, the ploughing forces act along the projected line AC in Fig. 5 and can be expressed in relation to the stagnation angle θ as
dFCp=dFnpsinθ+dFfpcosθ
(7)
dFTp=dFnpcosθdFfpsinθ
(8)
where
  • dFCp and dFTp are the ploughing components of cutting and thrust forces, respectively, described in Eqs. (1)(2);

  • dFnp and dFfp are the normal and friction ploughing forces, respectively; and

  • θ is the stagnation angle described in Eq. (3).

3.3 Modeling Microstructure-Specific Normal and Friction Coefficients.

In line with the mechanistic cutting force models in the literature, the normal and friction forces are assumed to be proportional to the uncut chip area [3,8,11,23,30,31]. While machining a heterogenous material like bone, these forces will depend on the relative percentage of each of the microstructural constituents encountered by the cutting edge. If Kns and Kfs are the specific normal and friction shearing pressures, respectively, then the shearing forces dFns and dFfs can be modeled as
dFns=Kns*dAs=Kns*tsdw
(9)
dFfs=Kfs*dAs=Kfs*tsdw
(10)
where the shearing area dAs is represented by the product of the uncut shear chip thickness ts and the elemental thickness dw of the tool width slice.

3.3.1 Shearing Pressures.

The work of Vogler et al. [3] showed that microstructure-specific normal and friction pressure terms can be calibrated through cutting studies involving each of the individual constituents. For example, to model the cutting forces in ductile iron, whose microstructure comprises of a heterogenous combination of ferrite, graphite, and pearlite phases, the phase-specific coefficients are estimated from cutting studies performed separately on pure ferrite, pure graphite, and pure pearlite alloys. Natural composites such as bovine bone pose a challenge to this approach since its microstructural constituents (Fig. 1) cannot be easily separated. Therefore, in this model, Kns and Kfs are expressed in terms of the microstructure-specific transverse shear stress term τi(Trans), whose estimation is discussed ahead in Sec. 4. The specific expressions take the form
Kns=Gn*[i(τi(Trans)*Ai)]
(11)
Kfs=Gf*[i(τi(Trans)*Ai)]
(12)
where Ai is the area fraction corresponding to constituent i and Gn and Gf are normal and friction coefficients, respectively. In the sawing tests described in Sec. 2, both the nominal rake angle of the tool as well as the cutting velocity are kept constant. Therefore, the coefficients Gn and Gf only change with the uncut chip thickness tO, and are expressed as
Gn=a0(tO)a1
(13)
Gf=b0(tO)b1
(14)
where tO is the depth of cut and a0, a1, b0, and b1 are constants calibrated using the experimental data. It should be noted that the form of Eqns. 13 ad 14 are based on prior mechanistic models in literature [3,8,11,23,30,31].

3.3.2 Ploughing Pressures.

If Knp and Kfp are the specific normal and friction ploughing pressures, respectively, then similar to Eqs. (9)(14), the ploughing forces (dFnp and dFfp) can be modeled as
dFnp=KnpdAp=Knptpdw=[Hn*i(τi(Trans)*Ai)]*tpdw
(15)
dFfp=KfpdAp=Kfptpdw=[Hf*i(τi(Trans)*Ai)]*tpdw
(16)
where the ploughing area dAp, is determined by the penetration depth tp and the elemental thickness, dw, of the tool width slice. Similar to the approach laid out in Sec. 3.3.1, the specific normal and friction pressures during ploughing are also expressed in terms τi(Trans) and Ai, and the normal and friction coefficients Hn and Hf, respectively. The ploughing coefficients are expressed as
Hn=c0(tO)c1
(17)
Hf=d0(tO)d1
(18)
where the calibrated coefficient parameters are c0, c1, d0, and d1. It should be noted here that the use of microstructure-specific shear stress in the calibrated ploughing force expressions is based on prevailing theories in elastic-plastic sliding cylindrical indentation [42].

3.4 Calculation Sequence.

Equations (1)(18) provide the basis for predicting the cutting and thrust forces encountered in bone sawing. These calculations hinge on the estimation of three categories of inputs, viz.: (1) microstructure-specific transverse shear stress (τi(Trans)) values of the four microstructural components, viz., osteon, interstitial matrix, woven bone, and lamellar bone, (2) shearing pressure coefficients a0, a1, b0, and b1, and (3) ploughing pressure coefficients c0, c1, d0, and d1. Once these are estimated, the calculation proceeds as follows:

  • Step 1: Use Eqs. (9)(18) to estimate the normal and friction forces resulting from the shearing and ploughing mechanisms;

  • Step 2: Once the normal and friction forces are determined, calculate the shearing and ploughing components of cutting and thrust force using Eqs. (5)(8); and

  • Step 3: Use Eqs. (1) and (2) to estimate the total cutting and thrust force.

Sections 3.5 and 4 ahead deal with the estimation of microstructure-specific transverse shear stresses and the shearing/ploughing coefficients, respectively.

3.5 Extracting Microstructure-Specific Transverse Shear Stresses.

The transverse direction of the sawing operation necessitates the extraction of the microstructure-specific transverse shear stress values, τi(Trans), for the osteon, interstitial matrix, lamellar, and woven bone. This was done using shear-dominant micro-scale orthogonal cutting tests combined with high-speed imaging protocols that enabled spatio-temporal insight into the specific microstructural constituents being cut.

As part of this effort, a separate set of “sharp tool” micro-scale orthogonal cutting experiments were conducted on pure Haversian and pure plexiform regions. The cutting was done in the “across” direction, where the tool width spans the length of the columnar structures (Fig. 6) such as osteons in the pure Haversian region (Fig. 1(b)) and lamellar bone in the pure plexiform region (Fig. 1(c)). This cutting direction was chosen for two specific reasons. First, cutting in this direction ensured that the tool width would separately encounter each of the microstructural constituents of interest [43]. This could not be ensured while cutting in the transverse or longitudinal direction as the tool width would cut through multiple microstructural constituents encountered in the cross-section. Second, this direction provided the high-speed camera with a clear view of the microstructural constituent being cut, which is needed to extract both the constituent-specific cutting force data and associated shear angle values.

Fig. 6
Orthogonal machining to extract τi(Across): (a) setup schematic, (b) “Across” cutting direction showing tool edge parallel to osteonal axis, and (c) high-speed camera view of osteon cutting
Fig. 6
Orthogonal machining to extract τi(Across): (a) setup schematic, (b) “Across” cutting direction showing tool edge parallel to osteonal axis, and (c) high-speed camera view of osteon cutting
Close modal

The orthogonal experiments allowed for the extraction of the microstructure-specific shear stress values in the “across” direction, τi(Across). These values were then transformed to their transverse equivalent, τi(Trans), needed for the model calculations (Eqs. (11)(12), (15)(16)). The remainder of this section first presents the details of the experiments conducted to obtain the τi(Across) estimates, followed by the stepwise procedure to transform those values to their transverse equivalent, τi(Trans).

3.5.1 Experimental Testbed.

Figure 6 shows a schematic of the experimental setup along with one of the high-speed images showing the tool edge cutting through an osteon. The experiments were performed on a three-axis micro-machining center (MikrotoolsTM DT-110, Singapore). A stationary tool mount was designed for orthogonal machining with an embedded Kistler 9256C1 dynamometer [44]. The tool remained stationary while the cutting operation was performed by moving the workpiece at a cutting velocity of 13.3 mm/s, the maximum linear speed of the precision motion stages. While this cutting velocity is significantly smaller than the value of 3700 mm/s seen in the sawing tests, the calibration process used to obtain the shear and ploughing coefficients (refer Sec. 4 ahead) accounts for the difference in strain rates.

The cutting tool was a high-precision ground tungsten carbide with an edge radius of 1 μm, clearance angle of +10 deg, and rake angle of +20 deg. The depth of cut was maintained at 10 μm. For this orthogonal cut, the ratio of the depth-of-cut to the edge radius of the tool was maintained at a value of 10, to ensure shear-dominant cutting. This is unlike the bone sawing experiments (Fig. 2) where this ratio is only 0.67 even at the highest depth-of-cut. Furthermore, in order to satisfy the conditions of plane-strain for the orthogonal cutting analysis [45], the width of the cutting edge (7 mm) was chosen to be considerably larger than the specimen width (0.5 ± 0.1 mm). A Phantom V.7.3 high-speed camera (5000 frames/s) was implemented into the experimental setup to observe the cutting process.

Given that the high-speed camera was deployed during the experiment, the measured cutting and thrust force values at any time instant can be spatially correlated to specific microstructural constituents (i) being cut in the workpiece (i.e., FCortho(i) and FTortho(i)). In addition to this, the high-speed images can also be used to estimate the corresponding microstructure-specific shear angle value (Φi). Knowing the width (p) and depth of cut (q) taken during the orthogonal cut, the microstructure-specific shear stress in the “across” direction can be calculated for each of the four constituents, viz., osteon, interstitial matrix, woven bone, and lamellar bone as [15,35]
τi(Across)={FCortho(i)cosΦiFTortho(i)sinΦip*qsinΦi}

3.5.2 Microstructural Area Fractions.

The samples used for these experiments were harvested separately from a similar mid-diaphyseal location as the sawing test samples. Table 2 shows the area fractions calculated for the “across” and “transverse” directions for the bone samples used in the micro-scale orthogonal studies. The method used to extract the area fractions based on the shapes of the microstructures is similar to that of Alam et al. [19]. These values take into account the (i) representative 2D area corresponding to the geometry of the cut (5 mm long, 10 μm depth of cut and 500 μm wide); (ii) the number of osteons per unit length of cut (nine osteons per 2 mm length); (iii) average diameter of the osteon (∼188 μm); (iv) average number of lamellar and woven regions per unit length (2–3 woven regions per 2 mm length); and (v) average length of the woven region (420 μm) and lamellar region (630 μm).

Table 2

Microscopy-based area fraction mapping between transverse and across directions

Material 1: Pure haversian region
PhaseAcross direction
Ai(Across)
Transverse direction
Ai(Trans)
Inter. matrix0.150.3
Osteon0.850.7
Material 2: Pure plexiform region
PhaseAcross direction
Ai(Across)
Transverse direction
Ai(Trans)
Woven0.50.6
Lamellar0.50.4
Material 1: Pure haversian region
PhaseAcross direction
Ai(Across)
Transverse direction
Ai(Trans)
Inter. matrix0.150.3
Osteon0.850.7
Material 2: Pure plexiform region
PhaseAcross direction
Ai(Across)
Transverse direction
Ai(Trans)
Woven0.50.6
Lamellar0.50.4

Note: These area fractions are specific to the samples used for the orthogonal studies.

3.5.3 Estimating τi(Across).

The following stepwise procedure is used to estimate the microstructure-specific shear stress values in the “across” direction (τi(Across)):

  1. Perform orthogonal tests in the “across” cutting direction while collecting synchronized force and high-speed camera data;

  2. Identify cutting and thrust force values for each microstructural constituent (FCortho(i) and FTortho(i));

  3. Using high-speed camera images, estimate the constituent-specific shear angles (Φi); and

  4. Using Eq. (17), calculate microstructure-specific shear stresses in the “across” direction, τi(Across).

Table 3 outlines the values for (Φi) and τi(Across) estimated from the orthogonal cutting of the pure Haversian and pure plexiform regions. Now, the effective average shear stresses for both the pure Haversian and pure plexiform regions can be calculated using the Voigt rule of mixtures [46] as (iτi(Across)Ai), where Ai is the area fraction corresponding to constituent i. As shown in Table 3 (gray-colored box), these values are 126.1 MPa and 135.5 MPa, respectively, for the pure Haversian and pure plexiform regions.

Table 3

Microstructure-specific shear stress in the “across” direction

Material 1: Pure haversian region
PhaseShear angle Φiτi(Across)Area fraction
Ai(Across)
τi(Across)
*
Ai(Across)
Inter. matrix23 deg121 MPa0.1518.15 MPa
Osteon22 deg127 MPa0.85107.95 MPa
Effective average shear stress—pure haversian region (iτi(Across)Ai(Across))126.1 MPa
Material 2: Pure plexiform region
PhaseShear angle Φiτi(Across)Area fraction
Ai(Across)
τi(Across)
*
Ai(Across)
Woven24 deg146 MPa0.573 MPa
Lamellar22 deg125 MPa0.562.5 MPa
Effective average shear stress—pure plexiform region (iτi(Across)Ai(Across))135.5 MPa
Material 1: Pure haversian region
PhaseShear angle Φiτi(Across)Area fraction
Ai(Across)
τi(Across)
*
Ai(Across)
Inter. matrix23 deg121 MPa0.1518.15 MPa
Osteon22 deg127 MPa0.85107.95 MPa
Effective average shear stress—pure haversian region (iτi(Across)Ai(Across))126.1 MPa
Material 2: Pure plexiform region
PhaseShear angle Φiτi(Across)Area fraction
Ai(Across)
τi(Across)
*
Ai(Across)
Woven24 deg146 MPa0.573 MPa
Lamellar22 deg125 MPa0.562.5 MPa
Effective average shear stress—pure plexiform region (iτi(Across)Ai(Across))135.5 MPa

3.5.4 Estimating τi(Trans).

Given that the bone sawing tests reported in Sec. 2 were performed in the transverse cutting direction (see Fig. 1), it is necessary to transform the microstructure-specific shear stresses, τi(Across), obtained in Table 3, to their equivalent values in the transverse direction, i.e., τi(Trans). This is done using the following two-stage calculation, the results of which are presented in Table 4.

  • Experimental literature related to bovine bone identifies the effective average shear stress in the transverse direction to be twice that of the corresponding value in the across direction [27,43]. Based on this, the effective average shear stress in the transverse direction is calculated to be 252.2 MPa and 271 MPa, respectively, for the pure Haversian and pure plexiform regions. It should be noted here that these numerical values are obtained by doubling the (iτi(Across)Ai(Across)) values in Table 3.

  • Interstitial matrix and woven microstructures can be assumed to be isotropic as per Refs. [4749]. Therefore, the Voigt rule of mixtures [46] can now be used to calculate the microstructure-specific shear stresses in the transverse direction for the anisotropic osteon and lamellar phases.

Table 4

Microstructure-specific shear stress in the transverse direction

Material 1: Pure haversian region
Pure haversian region: Effective average transverse shear stress252.2 MPaa
PhaseArea fraction
Ai(Trans)
τi(Trans)
Inter. matrix0.3121 MPab
Osteon0.7308.1 MPac
Material 2: Pure plexiform region
Pure plexiform region: Effective average transverse shear stress271 MPaa
PhaseArea fraction
Ai(Trans)
τi(Trans)
Woven0.6146 MPab
Lamellar0.4458.5 MPac
Material 1: Pure haversian region
Pure haversian region: Effective average transverse shear stress252.2 MPaa
PhaseArea fraction
Ai(Trans)
τi(Trans)
Inter. matrix0.3121 MPab
Osteon0.7308.1 MPac
Material 2: Pure plexiform region
Pure plexiform region: Effective average transverse shear stress271 MPaa
PhaseArea fraction
Ai(Trans)
τi(Trans)
Woven0.6146 MPab
Lamellar0.4458.5 MPac
a

Twice the (iτi(Across)Ai(Across)) value in Table 3.

b

Isotropic phase, τi(Trans) = τi(Across) value in Table 3.

c

Anisotropic phase, τi(Trans) calculated using Voigt rule of mixtures using data in Table 2.

As seen in Table 4, the τi(Trans) values for interstitial matrix, osteon, woven and lamellar bone are 121 MPa, 308.1 MPa, 146 MPa, and 458.5 MPa, respectively. These values will now be used for the mechanistic model calculations.

4 Model Calibration and Validation

As stated in Sec. 2 and 3, the data from the pure Haversian and pure plexiform regions was used to calibrate the model coefficients, whereas the data from the transition region was used for model validation.

4.1 Estimating Calibration Coefficients.

The three bone samples used for Data sets 1–3 (Sec. 1) were digitally analyzed to estimate the area fractions corresponding to pure Haversian and pure plexiform regions. The method used to map the shapes of the microstructures was similar to that of Li et al. [28]. For the pure Haversian regions used for the sawing study, the area fractions in the transverse direction were seen to be 0.47 for osteons and 0.53 for the interstitial matrix. For the pure plexiform regions in the sawing study, the corresponding area fraction values were 0.45 for the lamellar region and 0.55 for the woven region. It should be noted that these values differ from those seen in the samples used for extracting the microstructure-specific shear stress Table 2), which is expected given the regional differences seen in bovine bone.

Once the area fractions were calculated, the shearing pressure coefficients a0, a1, b0, and b1; and ploughing pressure coefficients c0, c1, d0, and d1 were obtained by least square error minimization over the experimental range of depth of cuts for the pure Haversian and plexiform regions (Table 5). Constraints were placed on a0, b0, c0, and d0 to be greater than 0 to ensure expected positive scalar values of specific normal and friction shearing and ploughing pressures. Additionally, constraints were placed on a1, b1, c1, and d1, to be less than 0 to maintain the expected specific pressure trend of decreasing values with increasing depth of cut, i.e., size effect behavior [50]. The calibrated coefficient variables can be found in Table 5. Figures 3(b)(1–4) show an overlay of the calibrated model and the experimental values seen for the pure Haversian and plexiform regions. The calibrated model predictions for the cutting forces are under 6.5% and 8.2%, respectively, for the pure Haversian and plexiform regions. For the thrust force, the calibrated model predictions are within 7.2% and 4%, respectively, for the pure Haversian and plexiform regions.

Table 5

Calibrated shear and ploughing coefficients for pure Haversian and plexiform regions

Shearinga0a1b0b1
Pure haversian region1.04−0.581.54E-5−2.13
Pure plexiform region3.26−0.335.63−3.91E-06
Ploughingc0c1d0d1
Pure haversian region2.37E-2−1.092.40E-13−1.38E-4
Pure plexiform region0.03−1.103.05E-14−3.10
Shearinga0a1b0b1
Pure haversian region1.04−0.581.54E-5−2.13
Pure plexiform region3.26−0.335.63−3.91E-06
Ploughingc0c1d0d1
Pure haversian region2.37E-2−1.092.40E-13−1.38E-4
Pure plexiform region0.03−1.103.05E-14−3.10

4.2 Model Validation.

The transition regions present in the three bone samples used to generate the cutting and thrust force data related to Data set 1 (1.8–3.8 μm DOC range), Data set 2 (4.0–5.4 μm DOC range), and Data set 3 (5.6–9.4 μm DOC range) were characterized for their respective area fractions of the four microstructural constituents. The same microstructure mapping technique stated in Sec. 4.1 was used. Per the procedure of Li et al. [28], it is difficult to distinguish the difference between interstitial matrix and woven areas. In such cases, the interstitial matrix area was chosen based on the area surrounding osteons and resorption cavities, which are characteristic to Haversian regions. An example of such a transition region map can be found in Figs. 1(d)1(e) and the respective area fractions of microstructures within each of the three maps can be found in Table 6.

Table 6

Area fractions of microstructures within each transition region

Data setDOC (μm)Area percentages
OsteonMatrixLamellarWoven
11.8–3.817%19%30%34%
24.0–5.418%27%30%25%
35.6–9.417%39%22%22%
Data setDOC (μm)Area percentages
OsteonMatrixLamellarWoven
11.8–3.817%19%30%34%
24.0–5.418%27%30%25%
35.6–9.417%39%22%22%

At any given depth-of-cut value, the following six steps are used to validate the model on the three transition regions:

  • Step 1: The calibrated coefficient variables from Table 5 are used to calculate the normal and friction coefficients for shearing and ploughing using Eqs. (13)(14) and Eqs. (17)(18), respectively.

  • Step 2: Using the microstructure-specific shear stress values in the transverse direction τi(Trans), from Table 4 and the area fractions in Table 6 the shearing pressures Kns and Kfs are obtained using Eqs. (11)(12). The shearing forces Fns and Ffs are then calculated using Eqs. (9)(10). The width value used is the sawtooth width of 740 μm.

  • Step 3: Similarly, to Step 2, the ploughing forces Fnp and Ffp are then calculated using Eqs. (15)(16).

  • Step 4: Using the effective rake angle (Eq. (4)) with the normal and friction shearing forces (Fns and Ffs) from Step 2, the shearing components of cutting and thrust force (FCs and FTs) can be computed using Eqs. (5) and (6).

  • Step 5: Using the stagnation angle (Eq. (3)) along with the normal and friction ploughing forces (Fnp and Ffp) from Step 3, the ploughing components of cutting and thrust force (i.e., FCp and FTp) can be computed using Eqs. (7) and (8).

  • Step 6: Given the shearing and ploughing components of cutting and thrust force, the total cutting and thrust forces (FCTotal and FTTotal) can be estimated using Eqs. (1)(2), respectively.

The validation results are presented in Fig. 7. As seen in Fig. 7, the predicted forces were generally found in good agreement with the experimental forces with the mean absolute percentage errors in predictions being under 10% for both the cutting and thrust forces. As shown in Part 2 of this paper, this validated cutting force model can also be used to predict acoustic emission trends observed during the sawing process.

Fig. 7
Experimental and predicted forces for transition regions: (a) cutting force, (b) thrust force, and (c) prediction errors across data sets 1–3
Fig. 7
Experimental and predicted forces for transition regions: (a) cutting force, (b) thrust force, and (c) prediction errors across data sets 1–3
Close modal

4.3 Model Limitations.

While this paper presents a mechanistic model for bone cutting that considers the microstructural phases, the following limitations exist in the current modeling framework to predict the cutting and thrust forces.

  • The model calculations are hinged on the estimation of the microstructure-specific shear stress values. However, mechanical properties of bovine bone vary based on diaphyseal regional location [51]. This prevents the universal use of microstructure-specific shear stress values reported here. The current model calculations would require a recharacterization of these values at different regions of this bone.

  • The current cutting force model does not consider cutting velocity effects. Although the strain rate sensitivity of bone has shown to be statistically insignificant within certain ranges of cutting velocity [8], mechanistic models are typically calibrated across a wide range of conditions, i.e., changes in cutting velocity, rake angle, and depth of cut. It would be worthwhile to extend the model, taking into consideration such variances.

  • Translation of these calculations to other bone machining operations such as drilling should be considered.

Both fundamental enhancements in high-strain rate bone mechanics literature, as well as machining literature, are needed to address these limitations.

5 Conclusions

The following specific conclusions can be inferred from this work:

  • A new mechanistic model has been developed for the prediction of cutting forces encountered in gradient microstructures encountered during sawing of bovine cortical bone. The model explicitly accounts for key microstructural constituents (viz., osteon, interstitial matrix, lamellar bone, and woven bone) while also decomposing the resultant cutting forces into their shearing and ploughing components.

  • Microstructure-specific shear stress values critical to the cutting force model calculation are estimated using micro-scale orthogonal cutting tests during which the cutting forces are synchronized with high-speed images. This approach of estimating the microstructure-specific shear stress overcomes a critical shortcoming in the literature related to high-strain rate characterization of natural composites, where the separation of the individual constituents is difficult.

  • The six model coefficients are first calibrated over pure Haversian and plexiform bone regions, using a range of clinically relevant depth of cuts. The calibrated model is then used to make predictions in the transition region between the Haversian and plexiform bone, which is characterized by gradient structures involving varying percentages of osteons, interstitial matrix, lamellar bone, and woven bone. The mean absolute percentage error in the cutting force predictions is under 10% for both the cutting and thrust forces.

  • The cutting force predictions by the model have been noted to be critically dependent on the estimation of the microstructure-specific shear stress. The reality of spatially varied properties in the cortical bone implies that the model is currently limited by its need for cross section-specific estimation of these stress values.

Acknowledgment

Funding support is acknowledged from the US National Science Foundation CAREER grant (CMMI 13-51275) and internal funds from Rensselaer Polytechnic Institute. Professor Thomas James (Rose-Hulman Institute of Technology) is acknowledged for providing the bone sawing apparatus.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

Nomenclature

     
  • c =

    constant describing the tool-chip contact length

  •  
  • i =

    microstructural constituent (i.e., osteon, interstitial matrix, lamellar bone, and woven bone)

  •  
  • p =

    width of the orthogonal cutting test sample

  •  
  • w =

    tool width (bone sawtooth)

  •  
  • q =

    depth of cut of the orthogonal cutting test

  •  
  • rE =

    edge radius of the tool

  •  
  • tp =

    penetration depth

  •  
  • ts =

    uncut shear chip thickness

  •  
  • t0 =

    nominal depth of cut = tp + ts

  •  
  • Ai =

    area fraction corresponding to constituent i

  •  
  • As =

    area of shear

  •  
  • FCTotal =

    total cutting force (bone sawtooth)

  •  
  • FCp =

    ploughing component of cutting force FCTotal

  •  
  • FCs =

    shearing component of cutting force FCTotal

  •  
  • Fnp, Ffp =

    normal and friction ploughing force associated with penetration depth

  •  
  • Fns, Ffs =

    normal and friction shearing force associated with chip removal along the effective rake face

  •  
  • FTTotal =

    total thrust force (bone sawtooth)

  •  
  • FTp =

    ploughing component of thrust force FTTotal

  •  
  • FTs =

    shearing component of thrust force FTTotal

  •  
  • Gf =

    friction coefficient associated with Kfs

  •  
  • Gn =

    normal coefficient associated with Kns

  •  
  • Hf =

    friction coefficient associated with Kfp

  •  
  • Hn =

    normal coefficient associated with Knp

  •  
  • a0, a1, b0, b1 =

    shearing pressure calibration coefficients

  •  
  • c0, c1, d0, d1 =

    ploughing pressure calibration coefficients

  •  
  • d(x) =

    elemental quantity x

  •  
  • FC(i), FT(i) =

    the orthogonal cutting and thrust force specific to microstructural constituent i

  •  
  • Knp, Kfp =

    specific normal and frictional ploughing pressure

  •  
  • Kns, Kfs =

    specific normal and frictional shearing pressure

  •  
  • αeff =

    effective rake angle

  •  
  • αnom =

    nominal rake angle of the tool

  •  
  • θ =

    stagnation angle

  •  
  • τi(Across), τi(Trans) =

    microstructure-specific shear stress in the “across” and “transverse” direction

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