2.2.1. Establishment of Axial Drilling Force Analytical Model
To intuitively characterize the correlation between uncut layer deformation and axial drilling force, the deformation behavior was amplified and schematically illustrated in
Figure 3. In the present study, a layer-wise analytical model for hole-exit delamination defects was established. Specifically, the axial drilling force
FN was simplified as a concentrated force, whereas the interlaminar bonding force
q was idealized as a uniformly distributed load. The key drilling process parameters and their corresponding symbols involved in the theoretical modeling are summarized in
Table 1.
Notably, the bending deformation of the workpiece domain to be machined necessitates the interlaminar bonding force between adjacent laminates to maintain a state of mechanical equilibrium. In accordance with the Griffith fracture criterion, Mode I cracks will be initiated when the downward tensile stress induced by deformation exceeds the interlaminar bonding strength of the composite. With the continuous propagation and interconnection of these cracks, delamination defects are ultimately formed at the hole exit.
It is widely acknowledged that axial drilling force dominates the initiation of exit delamination during composite drilling. When axial thrust exceeds the critical threshold, interlaminar cracks nucleate and eventually evolve into hole-exit delamination failure. Therefore, developing an analytical axial force model carries vital theoretical and engineering value, which can provide quantitative guidance for process optimization and delamination suppression of composite materials. In this work, an axial drilling force analytical model is established based on cutting-grinding mechanics and Usui’s grinding theory, combined with the geometric and material removal characteristics of brazed diamond abrasive drills.
As illustrated in
Figure 4, the material removal behavior of brazed diamond drills drilling 2.5D woven C/SiC ceramic matrix composites is analogous to surface grinding. Hence, the grinding mechanical framework proposed by Eiji Usui [
38] is adopted as the theoretical foundation to derive the cutting force formula.
In the force modeling shown in
Figure 5, two geometric assumptions are adopted for the brazed diamond abrasive grains: each single abrasive grain has a fixed cone half-angle
γ, and abrasives are uniformly distributed over the drill bit surface.
Let
ρ be the generatrix length of the conical cutting zone engaged with the workpiece. The infinitesimal micro-element area OAB shown in
Figure 6 can be expressed as [
39]:
where
dA denotes differential area of microelement OAB;
ρ denotes effective generatrix length participating in cutting;
γ denotes abrasive cone half-angle;
φ denotes circumferential angle between the microelement and cutting direction. The total axial force is derived by integrating the infinitesimal force
dp acting on each microelement OAB (see
Figure 6).
Two types of normal stress act on abrasive microelements, with opposite mechanical effects:
- (1)
σ: tensile/extrusion normal stress induced by drill cutting, which drives crack propagation (positive tensile stress);
- (2)
σ0: uniform external pre-compressive stress loaded by the fixture, which counteracts the cutting tensile stress and produces crack-tip closure (negative compressive stress).
The term σtotal = σ + σ0 is a unified algebraic superposition formula, where σ0 carries a negative sign in actual calculation to reflect compressive action. Increasing the magnitude of applied pre-compression enlarges the absolute value of negative σ0, reduces the net tensile equivalent stress σtotal, and thereby lowers the resulting axial drilling force. This algebraic definition unifies the mathematical derivation and physical suppression mechanism, eliminating apparent contradiction between formula form and experimental trends.
The superimposed equivalent normal stress on micro-area
dA is written as
=
σ +
σ0. The infinitesimal cutting force borne by the microelement is:
where
σ denotes tensile contact stress generated by drilling cutting;
σ0 denotes external pre-compressive stress applied on the workpiece, the core adjustable control parameter of this work.
Orthogonal decomposition of Equation (2) yields the vertical component of infinitesimal force:
Integrating over the full circumferential range [
,
] gives the vertical downward force generated by one single active abrasive grain:
where
Fn denotes vertical load contributed by one cutting abrasive grain.
Let
j denote the total number of abrasives simultaneously engaged in cutting, the resultant total axial drilling force reads:
In actual drilling processes, the maximum feed per revolution is limited by the exposed height of diamond abrasives, which are constrained by the geometric and kinematic features of brazed diamond drills. An excessively large feed will cause ineffective cutting or abrasive breakage. Under this constraint, the geometric relation among feed rate, spindle speed and effective cutting generatrix length is:
where
f denotes feed rate, and
v denotes spindle rotation speed.
Substitute Equation (6) into Equation (5) to eliminate
ρ:
Mathematically, Equation (7) presents a linear correlation between FN and the algebraic sum σ + σ0. Physically, σ0 is compressive stress with negative algebraic value. When the magnitude of applied pre-compression rises, increases, the net tensile value of σ + σ0 decreases, and the predicted axial drilling force FN reduces accordingly. This theoretical deduction is fully consistent with the later experimental observation that elevated pre-compressive stress suppresses axial drilling load.
During drilling, as the drill approaches the hole exit, the bottom laminate bears axial load and tends to separate from the base material along the hole edge. This delamination initiation occurs at the critical state where axial drilling force balances interlaminar bonding strength. At this critical position, axial load induces downward displacement X of the loaded ply and triggers elastic deformation in the drilling-affected zone.
The mechanical work done by axial drilling force is converted into elastic strain energy stored in the deformed material and Mode I crack propagation energy. Based on energy conservation law, the energy balance relation at critical delamination onset is established as:
where
GI denotes Mode I strain energy release rate per unit crack area;
dA denotes incremental crack expansion area;
FN denotes axial drilling force;
dX denotes infinitesimal downward displacement of the loaded laminate.
The incremental crack area is approximated as:
where
a denotes initial crack radius;
da denotes infinitesimal crack propagation increment.
The elastic strain energy stored in the circular deformed region beneath the drill is expressed as [
34]:
where
M denotes unit bending stiffness of composite laminate, calculated via thin-plate bending theory:
where
E denotes elastic modulus,
ν denotes Poisson’s ratio and
h denotes single ply thickness.
The downward ply displacement induced by axial drilling force is derived as [
40]:
where
X denotes vertical displacement of the bottom ply under axial thrust.
2.2.2. Correlation Between Strain Energy Release Rate and Delamination Criteria
Built on the energy conversion model derived in
Section 2.2.1, which quantifies the relation between axial drilling force and Mode I crack growth energy, this subsection further establishes the quantitative linkage among strain energy release rate
GI, adjustable external pre-compressive stress
σ0, and delamination initiation criterion. The explicit inclusion of controllable
σ0 in the revised theoretical formulas strengthens the logical consistency between theoretical derivation and subsequent experimental characterization, and clarifies the physical mechanism of pre-compression assisted delamination suppression. The core goal is to form a quantitative judgment standard for hole-exit delamination nucleation and expansion, providing theoretical support for optimizing delamination suppression strategies by regulating
σ0.
Substitute displacement Equation (12) into energy balance Equation (8) and rearrange to solve for
GI:
Further substitute the axial force Formula (7) containing core variable
σ0 into Equation (13):
Mathematically, Equation (14) shows GI is positively correlated with the square of σ + σ0. From physical stress superposition, σ0 is negative compressive stress offsetting positive cutting tensile stress σ. Increasing the magnitude of applied pre-compression reduces the net tensile value of σ + σ0, which decreases both axial force FN and Mode I strain energy release rate GI. This interpretation reconciles the mathematical formulation with the physical suppression mechanism observed in tests.
Equation (14) directly incorporates the external pre-compressive stress σ0, the key research parameter of this work. This quantitative relation bridges machining mechanics (axial load regulated by σ0) and linear elastic fracture mechanics (Mode I crack driving energy GI), enabling quantitative evaluation of how adjusting pre-compression alters GI, critical fracture threshold GIC, and delamination initiation risk.
During C/SiC drilling, the machining zone is dominated by vertical axial load, while radial force components can be neglected, satisfying plane-strain conditions. According to Griffith energy release rate criterion, interlaminar crack propagation initiates once Mode I energy release rate exceeds the material intrinsic critical value
GIC:
Combined with Equation (14):
As interpreted from the physical meaning of Equation (16), raising the magnitude of external pre-compressive stress σ0 counteracts cutting tensile stress, lowers the net equivalent σ + σ0, and reduces GI to a value below GIC, thereby effectively restraining interlaminar crack nucleation and hole-exit delamination. The revised delamination analytical model identifies three feasible approaches to mitigate exit damage: optimizing processing parameters (feed rate f and spindle speed v), reducing axial drilling thrust FN by increasing pre-compressive stress σ0, and improving the material inherent critical energy release rate GIC. The complete theoretical derivation with explicit σ0 term can quantitatively predict the degree to which tuning pre-compression reduces GI, narrows the gap between GI and GIC, and correspondingly decreases the measured delamination factor. Guided by this unified mechanical logic, the adjustable pre-compression loading method offers a quantifiable and effective solution for suppressing hole-exit delamination.
Fracture mechanics experiments prove that for a fixed material under definite boundary constraints,
GIC is a constant critical threshold; crack initiation and expansion will occur once
GI surpasses
GIC. The critical Mode I strain energy release rate
GIC has different mathematical expressions under plane-strain and plane-stress states:
In this work, the hole-exit region bears vertical z-direction axial drilling force superimposed on uniform in-plane pre-compressive stress
σ0. The combined load distributes evenly along the
z-axis and acts uniformly on all cross-sections perpendicular to the drilling direction, conforming to the plane-strain assumption adopted above. Therefore, the critical energy release rate is simplified as:
At this stage, the quantitative linkage among critical fracture threshold GIC, material fracture toughness KIC, and adjustable external pre-compressive stress σ0 is fully established. Combining Equation (14) and Equation (18), it is concluded that increasing the magnitude of applied σ0 significantly reduces the crack driving energy GI and widens the safety margin between GI and GIC. This theoretically verifies that applying mechanical pre-compression to generate crack-tip closure stress field is an effective technical route to improve the equivalent fracture resistance of C/SiC composites and suppress drilling-induced delamination.