Surface Roughness Generation Modeling: Deterministic Mechanisms for Geometric Interference, Plastic Flow, and Dynamic Stability in Precision Grinding

This report presents a deterministic modeling framework for surface roughness generation in precision grinding, shifting the focus from empirical observations to kinematic and physical causality. By analyzing the interaction between abrasive grain topography and process kinematics, it establishes the fundamental mechanisms that govern the final surface characteristics of a workpiece.

The modeling approach integrates the Geometric Envelope Mechanism with material-specific plastic flow behaviors. It quantitatively explores how successive grain trajectories overlap to form the surface profile and identifies the air-barrier-like effects of dressing parameters on active grain density. Special emphasis is placed on the transition from ideal geometric scallop heights to actual measured profiles, accounting for stochastic grain height distributions and thermomechanical softening.

Key analytical components include the derivation of the Theoretical Roughness Equation, the deterministic correction for Side Flow (Plowing), and the dynamic influence of machine tool stiffness on the effective depth of cut. By synthesizing these variables into a closed-loop predictive framework, this study provides an engineering basis for sub-micron surface integrity control.

Keywords: Surface Roughness Modeling, Deterministic Mechanism, Geometric Envelope, Active Grain Density, Plastic Flow, Grinding Kinematics.

1.1. Deterministic Mechanism of Surface Generation: Trajectory Overlap and Mathematical Definition of the Envelope

The cross-sectional profile of a surface generated in the grinding process is determined by the Envelope of the lowest paths left by the numerous abrasive grains on the rotating wheel. The trajectory z_i(x) left by an individual grain i is defined as a cycloidal function governed by the wheel radius (R_s) and velocity parameters. The final generated surface Z(x) is expressed as the following deterministic minimum value function:

Each trajectory possesses a phase difference based on the circumferential position and protrusion height of the grain. This envelope mechanism implies that as the wheel completes one revolution, trajectories shifted by the workpiece feed distance overlap to leave behind micro-valleys. Consequently, surface roughness is not a mere replication of the grain shape but a result of geometric interference determining which trajectory occupies the “lowest point.”

1.2. Mathematical Modeling and Variable Analysis of Theoretical Maximum Roughness (R_t)

By simplifying the previously defined envelope model and assuming that grains of identical height are arranged at a constant distance L, the theoretical maximum roughness (R_t) formed at the intersection of two adjacent trajectories is derived through the geometric relationship with the grain tip radius r as follows:

This equation explicitly states that surface roughness is proportional to the square of the speed ratio, providing a core guideline for process control. However, in actual machining, roughness higher than the theoretical value is often observed due to the irregularity of grain protrusion heights. This occurs because certain highly protruding grains “mask” the trajectories of adjacent grains, effectively increasing the value of L. Therefore, for high-precision modeling, the stochastic distribution characteristics of the grains must be integrated.

1.3. Numerical Correlation between Dressing Parameters and Active Grain Density (C)

The effective cutting edge density (C) is not a fixed constant but a dynamic function determined by dressing conditions. The Dressing Lead (f_d) and Dressing Depth (a_d), as the diamond dresser traverses the wheel surface, determine the density of the trajectory set by fracturing or dislodging grain tips. From a deterministic perspective, the correlation between C and f_d is defined by the following inverse model:

Ultimately, precision dressing is the act of minimizing the dressing lead by increasing the overlap ratio, which aligns the grain tips closely on the same plane to maximize the active grain density (C). This mechanism minimizes the grain spacing (L), serving as the most powerful deterministic means of controlling roughness in the surface generation model.

2.1. Material Flow Mechanism in Grinding: Plowing Effect

The geometric models previously discussed assume ideal cutting, where 100% of the material in the path of the abrasive grain is removed. However, in actual metal grinding, the large negative rake angle of the grains prevents immediate chip separation. Instead, Side Flow—a phenomenon where material is pushed to the sides of the grain—becomes dominant.

This plowing effect does not significantly alter the depth of the valleys in the previously defined envelope Z(x), but it forms plastically deformed micro-ridges (Piling-up) alongside the grain trajectories, deterministically increasing the peak heights. Consequently, measured roughness values are invariably higher than the theoretical geometric scallop heights, a discrepancy that is amplified by the material’s ductility and the applied grinding pressure.

2.2. Formulation of the Roughness Correction Equation Considering Plastic Deformation

To model the increase in roughness (ΔR_p) caused by side flow, a deterministic correction equation is introduced that combines the effective indentation depth (h_m) of the grain with the mechanical properties of the material. The final predicted roughness R_{a, total} achieves physical equilibrium as follows:

The above correction equation indicates that the roughness error increases sharply when the indentation depth (h_m) is significantly smaller than the grain tip radius (r). This provides a deterministic basis for explaining the physical limit in ultra-precision grinding: even if the theoretical geometric roughness is reduced by lowering the feed rate, elastic recovery and plastic flow make it impossible to improve surface finish beyond a certain threshold.

2.3. Dynamic Variation of Plasticity Constant due to Grinding Temperature and Thermal Softening

The side flow mechanism is governed by the variation in the material’s yield strength as a function of the grinding zone temperature (T). The plasticity constant (K_p), treated as a constant in the previous correction equation, is actually a function of flow stress dependent on temperature, and can be deterministically defined through a modification of the Johnson-Cook model:

In High-Speed Grinding (HSG) environments, shear deformation energy is rapidly converted into heat, causing the grinding zone temperature to spike. The resulting thermal softening effect lowers the material’s viscous resistance and promotes chip formation, thereby offsetting the plowing energy that drives material to the sides. Consequently, if the temperature rises above a certain critical point, a paradoxical improvement in surface finish occurs as the formation of plastic ridges (piling-up) is suppressed.

Conversely, if excessive coolant cooling keeps the temperature low while grinding loads are forced higher, the plasticity coefficient remains high while indentation depth increases, causing the surface finish to deviate drastically from theoretical values. Therefore, precise roughness modeling requires not only geometric interference analysis but also dynamic tracking of K_p(T) through the temperature gradient of the machining zone.

3.1. Stochastic Distribution of Grain Protrusion Heights and Formulation of Effective Cutting Edge Density (C_act)

In actual grinding, the cutting edge density—treated as a constant in basic models—is a dependent variable determined by the probability density function p(z) of the grain protrusion heights (z) on the wheel surface. Assuming that the grain heights follow a Gaussian distribution, the Effective Cutting Edge Density (C_act) participating in actual machining is defined for a given depth of cut (a_e) as follows:

3.2. Time-Variant Evolution Model of Tip Radius (r) due to Wheel Wear

As machining time elapses, abrasive grains undergo Attritious Wear (flattening), which alters the grain tip radius—the critical variable determining surface finish. The evolution of the effective tip radius r(t) relative to the cumulative specific material removal (V’_w) is modeled using the wear rate coefficient (γ) as follows:

When the tip radius increases due to wear, the geometric curvature of the surface becomes gentler, which might seemingly decrease the theoretical geometric roughness. However, in reality, this causes a sharp decline in the indentation parameter (the ratio of depth to radius) discussed previously. This triggers the Plowing Mechanism to become dominant, leading to a “Roughness Inversion” where the increase in ridges caused by plastic flow outweighs any geometric reduction, ultimately degrading the final surface finish.

3.3. Determination of the Deterministic Critical Point for Roughness Transition

The variation in roughness due to wheel wear transitions from an initial steady state to a stage of rapid degradation. This critical specific material removal threshold (V’_{w, crit}) is determined either by grain dislodgement (bond failure), which causes a sharp drop in C_act, or by excessive flattening of the tip radius r(t).

A larger dressing lead results in a lower initial effective cutting edge density and a higher standard deviation in the grain height distribution, thereby shortening the time to reach this critical threshold. Conversely, precision dressing compresses the height distribution into a narrower range, inducing uniform tip wear and ensuring the temporal stability of the surface finish. Therefore, a robust roughness model must be completed as a dynamic system that integrates wheel wear life (G-ratio) with material plasticity correction models.

4.1. Dynamic Interference Model: Relative Vibration and Trajectory Displacement

Surface roughness is not determined solely by the static envelope geometry discussed previously. The relative vibration (A) between the grinding wheel and the workpiece during machining superimposes dynamic displacements onto the individual grain trajectories z_i(t). These vibrations, caused by wheel unbalance or external excitation, are expressed as time-variant trajectory equations:

Here, A represents the vibration amplitude and f_v denotes the vibration frequency. This dynamic displacement forcibly injects wave-like errors into the theoretical geometric roughness. Particularly, when the vibration frequency is not an integer multiple of the wheel’s rotational frequency, a phase shift (φ) occurs with each revolution. This results in the formation of grid patterns or irregular “Chatter marks” on the surface, causing a far more significant increase in roughness than the micro-ridges formed by plastic flow discussed earlier.

4.2. Modeling Surface Degradation via System Stiffness and Process Damping

The static and dynamic stiffness (k) of the machining system dictates the amount of elastic deformation under grinding loads, which in turn fluctuates the effective depth of cut. The variation in the effective depth of cut due to the load becomes a decisive factor in altering the effective cutting edge density defined previously.

The total stiffness of the grinding system is a core parameter determining not only the static displacement between the wheel and the workpiece but also the dynamic stability. According to the above relationship, lower system stiffness leads to greater uncertainty in the effective depth of cut under load, causing irregular fluctuations in the effective cutting edge density. Consequently, the vibration displacement amplified by the frequency response function physically distorts the surface generation envelope, serving as the deterministic cause of surface “Waviness.”

Deterministic Integration of the Surface Generation Model and Engineering Implications

The surface roughness generation modeling explored in this report is not merely a list of individual variables but a process of integrating physical causalities across the entire machining system. The final surface quality is defined as the linear combination of the fundamental geometric trajectory overlap mechanism, error corrections for plastic material flow, temporal transitions due to wheel wear, and the dynamic instabilities of the system.

In conclusion, advanced surface generation modeling suggests that the entire process—from the micro-topographical design of the wheel to the analysis of thermomechanical material behavior and the control of machine tool dynamics—must be understood as a single Closed-loop system. This deterministic approach minimizes empirical trial-and-error on the shop floor and provides a robust engineering foundation for predicting and securing sub-micron precision during the initial design stages.