62a1⁝ Research Note - Physically Based Simulation of Surface Micro-Geometry Analysis of Contact Mechanics, Elastoplastic Deformation, and Rendering Algorithms

Report Summary

Report prepared for my project Forensic-Traces-Generator (https://github.com/pawel-dubiel/Forensic-Traces-Generator), this report tries to investigate a real-time pipeline for generating forensic toolmarks-scratches, grooves, and abrasive wear-from a digital tool model. The approach combines tool “kernels” for contact geometry, fast contact-patch parameterization, and an elastoplastic height-field deformation model. Displaced material is redistributed with volume-conserving rules to form pile-up ridges, and elastic recovery (springback) is modeled with spatial relaxation filters tied to contact scale and material stiffness. Striations are synthesized by convolving tool edge profiles along the tool path and adding multi-scale roughness. The pipeline produces mechanically consistent, visually realistic traces at interactive rates and sets up future validation against measured toolmarks and comparison workflows.

1. Introduction: The Convergence of Tribology and Computer Graphics

The accurate representation of surface defects, specifically scratches and abrasive wear, constitutes a fundamental challenge in the fields of computer graphics, computational mechanics, and virtual prototyping. Historically, the simulation of such phenomena has been bifurcated into two distinct domains: the phenomenological approach of texture mapping in computer graphics, which prioritizes visual plausibility at the expense of physical accuracy, and the rigorous finite element analysis (FEA) of tribology, which prioritizes mechanical precision but incurs computational costs prohibitive for real-time application. The research presented herein analyzes a unified “method” that bridges this divide -a physically based simulation pipeline that synthesizes contact patch parameterization, elastoplastic deformation algorithms, and procedural micro-geometry generation to render surface scratches that are visually realistic, haptically perceptible, and mechanically grounded.

This document provides examination of the mathematical and algorithmic foundations of this method. It explores the utilization of “tool kernels” as geometric definitions of the indenter, the adaptation of the Fiala contact patch model for high-frequency tool interactions, and the application of cellular automata to model the conservation of volume during pile-up formation. Furthermore, it analyzes the constitutive laws governing strain hardening and elastic recovery (springback), and the convolution-based techniques used to generate high-frequency striations. By integrating disparate research from tire mechanics, materials science, and digital signal processing, this analysis establishes a cohesive framework for understanding the genesis and rendering of surface damage.

1.1 The Limitations of Traditional Texturing

In traditional computer graphics, defects are often modeled using bidirectional reflectance distribution functions (BRDFs) or static normal maps. While effective for macroscopic roughness, these techniques fail to capture the dynamic nature of scratch formation. A scratch is not merely a change in color or surface normal; it is a physical event involving the displacement of material. As noted in foundational research on scratch rendering, individually visible scratches lie in a representation scale between the BRDF and the texture map.1 They possess a distinct 3D micro-geometry-a groove flanked by pile-up ridges-that causes anisotropic reflection, shadowing, and masking effects that flat textures cannot reproduce.

The necessity for a physically based model arises from the need to simulate the process of scratching rather than just the result. By parameterizing the scratching tool, the penetration forces, and the material properties (hardness, elasticity), one can derive the micro-geometry procedurally.2 This allows for the simulation of complex interactions, such as intersecting scratches where a new groove cuts through the work-hardened pile-up of a previous scratch, or the “stick-slip” behavior observed in polymers.3

1.2 Overview of the Simulation Pipeline

The method under investigation operates as a sequential pipeline, transforming input forces and tool geometries into a deformed height field. The core stages of this pipeline include:

1. Contact Detection and Patch Extraction: Utilizing simplified contact models, such as the Fiala tire model or Hertzian theory, to estimate the contact area and pressure distribution based on the tool kernel.4
2. Plastic Displacement and Pile-up: Calculating the volume of material displaced by the indenter and redistributing it to the scratch margins using volume conservation principles and cellular automata logic.6
3. Elastic Recovery (Springback): Modeling the post-load relaxation of the material, where the groove depth decreases and the pile-up retracts due to stored elastic energy, often implemented via spatial filtering.9
4. Striation Generation: Convolving the tool’s edge profile with the scratch path and modulating it with fractal noise to simulate the interaction of microscopic asperities and debris.12

This report is structured to analyze each of these stages in depth, providing the relevant mathematical formulations and citing the supporting literature that validates the approach.

2. Contact Mechanics and the Tool Kernel

The initiation of any scratch simulation requires a robust definition of the contact interface. The “tool kernel” serves as the fundamental unit of interaction -a mathematical description of the indenter’s cross-sectional geometry that determines the boundary conditions for material displacement.

2.1 The Concept of the Tool Kernel

In the context of this simulation method, a “kernel” is not merely a computational function but a geometric profile $K(x,y)$ representing the shape of the scratching tip. This profile is swept along the scratch trajectory $C(t)$ to define the raw volume of interaction. The choice of kernel geometry is critical, as it dictates the stress concentration and the flow regime of the displaced material.

Common indenter geometries referenced in tribological literature and utilized in this method include:

• Berkovich Indenter: A three-sided pyramid with a centerline-to-face angle of 65.3°. It is favored for hardness testing because its area-to-depth ratio is similar to that of a Vickers indenter, but it avoids the “chisel edge” problem of four-sided pyramids.15

• Cube Corner Indenter: A three-sided pyramid with mutually orthogonal faces (90° corner). This sharper geometry produces higher stresses for a given load, displacing more volume and promoting fracture or cutting rather than simple ploughing.15

• Spherical/Conical Indenters: Axisymmetric tools that transition from elastic Hertzian contact to plastic flow. Spherical indenters are unique in that the effective attack angle increases with penetration depth, allowing for the simulation of the transition from elastic to plastic to cutting regimes.18

The tool kernel is effectively a height map of the tool tip, $Z_{tool}(x,y)$. During the simulation, this kernel is transformed (translated and rotated) to the tool’s position $P(t)$ and orientation. The intersection of this transformed kernel with the terrain surface $H(x,y)$ defines the Contact Patch.

2.2 Adaptation of the Fiala Model for Contact Patches

A distinct feature of the method is the utilization of the Fiala tire model algorithms for determining contact patch parameters. While originally developed for vehicle dynamics to predict tire forces on uneven roads, the mathematical abstraction of the Fiala model -specifically its “brush model” approach -is highly applicable to tool-surface interactions.4

The Fiala model treats the contact interface as a collection of independent elastic elements (bristles) that deform under load. In the context of a scratch simulation:

• Vertical Force ($F_z$): Analogous to the tire load, derived from the interpenetration volume or depth.
• Lateral Force ($F_y$): Analogous to the cornering force, derived from the deflection of the bristles as the tool slides.
• Contact Patch Shape: The model provides analytical methods to extract the contact length and width based on the penetration of a rigid body (the tool) into a deformable medium.4

The utility of the Fiala model lies in its efficiency. It reduces the complex 3D contact problem to a set of physically meaningful parameters (e.g., contact length $l_c$, contact width $w_c$, and pressure distribution $p(x)$) without requiring a full finite element mesh solution. This allows the simulation to run in real-time or near-real-time, updating the contact patch shape dynamically as the tool encounters irregularities in the surface height field.20

For a simplified parabolic pressure distribution typical of Fiala/Hertzian models, the pressure $p$ at a distance $r$ from the contact center is given by:

$p(r)=p_{max}{\left(1-{\left(\frac{r}{a}\right)}^2\right)}^{1/2}$

Where $a$ is the contact radius. This pressure field drives the subsequent plastic deformation algorithms.

2.3 Hertzian Contact Theory and the Elastic Limit

Before permanent deformation (scratching) occurs, the contact is elastic. The method incorporates Hertzian contact theory to model this initial phase and to determine the threshold for plasticity. For a spherical indenter of radius $R$ interacting with a flat surface, the contact radius $a$ and the mutual approach $\delta$ (indentation depth) are related to the normal load $P$ by:

$a={\left(\frac{3PR}{4E^{\ast }}\right)}^{1/3}$

$\delta =\frac{a^2}{R}={\left(\frac{9P^2}{16R(E^{\ast }{)}^2}\right)}^{1/3}$

Where $E^{\ast }$ is the reduced modulus of elasticity, defined as:

$\frac{1}{E^{\ast }}=\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}$

Here, $E_1,E_2$ are the Young’s moduli and ${\nu }_1,{\nu }_2$ are the Poisson’s ratios of the tool and the surface, respectively.5

While Hertzian theory assumes purely elastic behavior, real scratches involve extensive plasticity. The simulation method must detect when the contact pressure exceeds the material’s yield strength. The transition from Hertzian (elastic) to fully plastic contact is often modeled using the Meyer’s Law relationship or by modifying the pressure distribution to flatten at the hardness value $H$.22

2.4 Elastoplastic Transition and Hardness

The relationship between the applied force and the resulting scratch depth is governed by the material’s hardness. The method utilizes Meyer’s Law, an empirical relation widely used in indentation testing:

$P=k{d}^n$

Where $P$ is the load, $d$ is the chordal diameter of the indentation, and $k$ and $n$ are material constants. The Meyer index $n$ typically ranges from 2 (for fully plastic materials) to 2.5, reflecting the strain hardening capacity.19

It is crucial to distinguish between static indentation hardness and Scratch Hardness ($H_s$). Scratching involves lateral motion, which alters the stress field and the effective constraint of the material. The scratch hardness is defined as:

$H_s=\frac{qP}{w^2}$

Where $P$ is the normal load, $w$ is the scratch width, and $q$ is a geometric factor (often $q\approx 8/\pi$ for a viscoelastic-plastic material).16

The simulation method likely allows for the input of specific hardness parameters (Vickers, Brinell, or Meyer) to calibrate the depth of the tool kernel penetration. Specifically, the “displaced volume” discussed in the next section is a direct function of this load-depth relationship. Research indicates that different indenter shapes (e.g., Berkovich vs. Cube Corner) displace the same volume of material for a given load, provided the material is isotropic, which simplifies the simulation by allowing volume displacement to be calculated primarily from load and hardness, independent of complex tool geometry at the macro scale.15

2.5 Geometric Characteristics of Tool Kernels

Indenter Type	Geometry	Effective Cone Angle	Application in Simulation	Reference
Berkovich	3-sided Pyramid	70.3°	General hardness, anisotropic pile-up	15
Vickers	4-sided Pyramid	70.3°	Standard hardness, square symmetry	23
Cube Corner	3-sided Pyramid	42.28°	High sharpness, fracture initiation	17
Knoop	Elongated Pyramid	Asymmetric	Anisotropic hardness testing	28
Spherical	Sphere (Radius $R$)	Variable (Depth dependent)	Elastic-plastic transition, worn tools	19

The “sharpness” of the kernel is a critical parameter. A sharper tool (lower effective angle) induces a higher degree of plastic flow and cutting, whereas a blunter tool favors elastic deformation and sink-in. The simulation must account for this by modulating the ratio of displaced volume to residual groove volume based on the kernel’s sharpness.31

3. Plastic Displacement and Pile-up Algorithms

The defining visual characteristic of a scratch is not just the furrow, but the ridges of material that form along its sides. This phenomenon, known as pile-up, is the result of plastic flow where the material displaced by the indenter is pushed upward and outward. The simulation method models this process using a Volume Conservation approach implemented via Cellular Automata.

3.1 The Principle of Volume Conservation

In a plastic indentation event, assuming no material is removed (i.e., no cutting chips are detached), the mass of the material must be conserved. The volume of the depression ($V_{-}$) must equal the volume of the pile-up ($V_{+}$).

$V_{-}=V_{+}$

$V_{displaced}={\int }_{A_{contact}}(Z_{original}-Z_{new})dA$

This conservation law is the cornerstone of the simulation’s validity. Unlike simple texture painting, where a dark line is drawn to represent a scratch, this method physically lowers the height field in the contact zone and must raise it elsewhere to maintain mass balance.7 This creates the characteristic “rings” or lobes of material seen in real profilometry data.6

However, strict volume conservation is an idealization. In reality, densification (in porous materials) or elastic compression can result in a loss of apparent volume. The simulation likely includes a compressibility parameter $\beta$ to account for this:

$V_{pile-up}=\beta \cdot V_{displaced}$

Where $\beta =1.0$ for perfectly plastic incompressible flow (metals) and $\beta <1.0$ for compressible materials (foams, porous soils).35

3.2 Cellular Automata for Material Flow

To distribute the displaced volume dynamically, the method employs a Cellular Automata (CA) or finite-difference approach on the height field grid. The height field acts as a grid of cells, where each cell state contains the terrain height $h(x,y)$.

The algorithm proceeds as follows:

1. Excavation: The tool kernel is applied to the grid. For every cell $(i,j)$ within the contact patch, if $h_{tool}(i,j)<h_{grid}(i,j)$, the difference $\Delta h$ is calculated.

2. Accumulation: The total volume removed $\Delta V=\sum \Delta h\cdot {\text{Area}}_{cell}$ is stored in a “sediment” buffer or carried by the tool as a “bow wave.”

3. Distribution (The “Ring” Algorithm): The accumulated volume is distributed to the neighbors of the contact patch. This distribution is often modeled as a diffusion process or a “sandpile” toppling model.

   • Toppling Rule: If the slope between adjacent cells exceeds the material’s angle of repose (friction angle), material moves from the higher cell to the lower cell.8

   • Ring Expansion: The material is pushed radially outward, forming concentric rings of deformation. The amplitude of the ring decays with distance from the contact center, often following a Gaussian or exponential decay function.38

This CA approach allows the simulation to handle complex behaviors such as the intersection of scratches. When a tool crosses an existing pile-up ridge, the CA rules naturally handle the “bulldozing” of the existing material, pushing it into the old groove or adding it to the new bow wave.39

3.3 The Role of Strain Hardening in Pile-up

The morphology of the pile-up -specifically whether it forms a sharp ridge or a diffuse swell -is governed by the material’s work-hardening exponent ($n$).

• Low Work-Hardening ($n\approx 0$): Materials like fully work-hardened metals or amorphous polymers exhibit significant pile-up. The plastic zone is localized near the indenter faces, forcing material to flow up the tool. The simulation models this with a “tight” distribution kernel, placing displaced volume immediately adjacent to the groove.16

• High Work-Hardening ($n>0.3$): Annealed metals distribute the plastic strain over a larger volume. The material tends to “sink in” rather than pile up, as the deformation spreads radially. The simulation handles this by using a wide, diffuse distribution kernel, resulting in lower, broader ridges or even a net depression (sink-in).22

The Pile-up Parameter ($h_{pile-up}/h_{indent}$) is often used to quantify this effect. Research correlates this parameter with the ratio of Yield Strength to Young’s Modulus (${\sigma }_y/E$).16 The simulation likely uses this ratio to dynamically adjust the spread of the CA distribution function.

3.4 The “Ring” Phenomenon in Detail

Ring In the context of indentation and impact physics, deformation often occurs in annular zones.

• Inner Ring: The immediate contact zone, undergoing severe plastic deformation and hydrostatic compression.
• Outer Ring: The pile-up zone, where material is uplifted.
• Far-Field Ring: The elastic zone, where the surface may bulge slightly due to long-range stress fields.38

In the simulation, the “Ring Distribution Algorithm” explicitly manages these zones. It ensures that material is not just randomly scattered but deposited in a coherent wave front. This is particularly important for high-speed scratching, where inertial effects (modeled by the Fiala/CA dynamics) can cause the pile-up to detach or form periodic ridges (chatter marks).44

4. Elastoplastic Response and Springback

A physically accurate scratch simulation cannot assume that the surface remains in its maximum deformed state. Upon the removal of the tool (unloading), the material undergoes Elastic Recovery, commonly referred to as Springback. This phenomenon reduces the depth of the groove and alters the profile of the pile-up.

4.1 The Physics of Elastic Recovery

Springback is the release of stored elastic strain energy. While the plastic deformation is permanent, the elastic component of the total strain is reversible.

${ϵ}_{total}={ϵ}_{plastic}+{ϵ}_{elastic}$

When the load is removed, ${ϵ}_{elastic}$ returns to zero (or a residual stress state). This manifests as a shallowing of the scratch groove.45

The magnitude of springback depends on the material’s $E/H$ ratio.

• Metals: Typically exhibit 10-20% depth recovery.

• Polymers: Can exhibit 50-80% recovery due to viscoelasticity and entropy elasticity.36

The simulation models the final depth $h_f$ as a function of the maximum depth $h_{max}$:

$h_f=h_{max}(1-R_e)$

Where $R_e$ is the elastic recovery ratio. This ratio is not constant; it depends on the scratch depth. Shallow scratches (which are mostly elastic) may recover almost completely ($R_e\to 1$), while deep scratches (mostly plastic) recover less ($R_e\to 0$).10

4.2 Spatial Filtering and Characteristic Lengths

The simulation implements springback not as a global scalar operation but as a Spatial Filter applied to the height field. The elastic recovery is nonlocal; the relaxation of one point on the surface affects its neighbors.

The method applies a low-pass filter (e.g., Gaussian or Box filter) to the difference map between the original and deformed surfaces. The width of this filter is determined by the Characteristic Length ($\lambda$) of the material’s deformation field.11

$\lambda \propto \frac{E}{{\sigma }_y}\times w_{contact}$

A large characteristic length (stiff, high-yield material) implies that the elastic recovery is spread over a wide area, leading to a smooth, gentle rebound. A small characteristic length (soft or brittle material) implies localized recovery.48

4.3 Viscoelasticity and Time-Dependent Recovery

For polymer simulations, the method may incorporate time-dependent springback (viscoelastic recovery). The depth of the scratch is a function of time $t$ after the tool passes:

$h(t)=h_{\infty }+(h_{max}-h_{\infty })e^{-t/\tau }$

Where $\tau$ is the relaxation time constant. This allows the simulation to render “self-healing” behaviors observed in automotive clear coats, where fine scratches disappear over minutes or hours.50

4.4 Simulation of Residual Stresses

The springback process also leaves Residual Stresses in the material. While the height field simulation typically creates a geometric surface, advanced implementations may store a “stress map” as an auxiliary texture channel. This residual stress affects the hardness of the material for subsequent scratches. If a second scratch crosses a region of high residual compressive stress (pile-up), the apparent hardness is higher, and the tool penetration is reduced. This captures the “work hardening” effect in a multi-pass simulation.41

5. Micro-Geometry and Striation Generation

While the tool kernel and pile-up define the macroscopic shape of the scratch, the visual realism -specifically the anisotropic glare -relies on the Micro-Geometry. This consists of fine, parallel grooves (striations) within the main furrow, caused by the roughness of the tool edge.

5.1 Tool Edge Profiles and Convolution

Real tools are never perfectly smooth. The cutting edge possesses microscopic irregularities, chips, and grain structures. The simulation models this by defining a Tool Error File (TEF) or a high-resolution 1D profile of the tool edge.52

The generation of striations is mathematically modeled as a Convolution of this tool profile with the scratch path.

$H_{micro}(u,v)=T_{profile}(v)\ast P_{path}(u)$

Where $u$ is the coordinate along the scratch path and $v$ is the coordinate perpendicular to it. As the tool moves, it “extrudes” this profile along the trajectory. This ensures that the striations are continuous and follow the curves of the scratch, a feature that distinguishes them from simple noise textures.13

5.2 Fractal Noise Integration

To simulate the stochastic nature of abrasion -such as debris tearing, grain pull-out, and tool wear -the method integrates Fractal Noise (e.g., Perlin noise, Simplex noise) into the tool profile.

$T_{effective}(v)=T_{base}(v)+{\sum }_i{A}_i\text{Noise}(f_{i}v)$

The noise is often parameterized by the material’s grain size. Large grains (as in some cast metals) introduce low-frequency, high-amplitude noise (rough, jagged striations), while fine grains (glass, polished steel) introduce high-frequency, low-amplitude noise.53

5.3 The Attack Angle and Ploughing-Cutting Transition

The nature of the striations is heavily dependent on the Attack Angle of the abrasive particle.

• Ploughing Regime (Low Angle): The tool pushes material aside. Striations are smooth and continuous.

• Cutting Regime (High Angle): The tool shears material off. Striations are sharp, deep, and may exhibit fracture patterns.

There exists a Critical Attack Angle (typically around 60-70° for metals) where the mechanism transitions from ploughing to cutting.55 The simulation monitors the local slope of the tool kernel relative to the surface. If the effective attack angle exceeds the critical threshold, the “sharpness” of the generated striations is increased, and the volume conservation rule is modified to account for material removal (chip formation) rather than just displacement.57

6. Rendering and Haptic Feedback

The final stage of the pipeline is the translation of the simulated height field into sensory outputs: visual rendering and haptic force feedback.

6.1 BRDFs and Normal Mapping

The micro-geometry generated by the simulation is too fine to be represented by the geometric mesh of the object. Instead, it is converted into texture maps:

1. Height Map: The direct output of the simulation.

2. Normal Map: Calculated from the gradient of the height map ($\nabla H$).

   $N(x,y)=\text{normalize}\left(-\frac{\partial H}{\partial x},-\frac{\partial H}{\partial y},1\right)$

3. Anisotropy Map: The direction of the striations (tangent vectors) is stored to drive anisotropic BRDFs (e.g., Ward, GGX Anisotropic). This ensures that the scratch highlights elongate perpendicular to the scratch direction, a key visual cue for metallic surfaces.1

Recent advances allow for the rendering of “glints” -sparkling highlights caused by individual micro-facets within the scratch. The simulation generates a distribution of micro-facet normals derived from the fractal noise, allowing for the physically based rendering of these glint effects.58

6.2 Haptic Force Feedback Models

For applications involving haptic devices (e.g., surgical simulators, texture painting tools), the method calculates the force vector $F_{haptic}$ returned to the user.

$F_{haptic}=F_{normal}+F_{friction}+F_{texture}$

• Normal Force: Proportional to the penetration depth and material hardness ($P=k{d}^n$).

• Friction: Based on the adhesion and ploughing components ($F_L=\mu F_N+F_{plough}$).

• Texture (Vibration): A high-frequency force component derived from the striation profile.

6.3 Sharpness-Dependent Haptic Rendering

The “feel” of the scratch depends on the Sharpness of the tool and the features. A sharp tool on a hard surface generates high-frequency vibrations (“crisp” feeling), while a blunt tool on a soft surface generates low-frequency resistance (“dull” or “gummy” feeling).

The simulation uses a Sharpness-Dependent Filter to modulate the texture force:

$F_{texture}(t)=\text{HighPass}(H(x,y))\times S_{tool}$

Where $S_{tool}$ is a sharpness coefficient. This models the psychophysical perception of roughness, which is closely linked to the spectral content of the vibration signal.60

7. Computational Implementation and Optimization

Implementing this “method” requires efficient data structures and algorithms to run at interactive rates.

7.1 Height Fields vs. Volumetric Meshes

The method predominantly utilizes Height Fields ($2.5D$ grids) rather than full volumetric meshes (tetrahedral FEM).

• Advantages: Height fields are memory efficient ($O(N^2)$ vs $O(N^3)$), easy to modify on the GPU, and natively support texture mapping.

• Limitations: They cannot represent true overhangs or complex 3D fracture debris. However, for surface scratches where depth $\ll$ width, this approximation is valid.62

7.2 GPU Acceleration

The core algorithms -Fiala contact patch, Cellular Automata update, Convolution -are highly parallelizable. They are typically implemented as Compute Shaders (CUDA, OpenCL, or HLSL).

• Tool Kernel: Stored as a texture.

• CA Update: A fragment shader pass that reads the current height and neighbor heights to compute flow.

• Springback: A separable Gaussian blur pass.64

This GPU-based approach allows for the simulation of millions of grid points in real-time, enabling applications in video games and interactive design tools where the user can scratch a surface and immediately see and feel the result.

8. Conclusion

The “method” for researching and simulating surface scratches is a sophisticated synthesis of multiple disciplines. It rejects the static, purely visual approach of traditional texturing in favor of a dynamic, physical model. By understanding the Contact Mechanics of the tool kernel (Hertzian/Fiala), the Plastic Flow of the material (Cellular Automata/Volume Conservation), the Elastic Recovery (Springback/Spatial Filtering), and the Micro-Geometry (Convolution/Fractal Noise), one can simulate the complete lifecycle of a surface defect.

This physically based approach ensures that the simulated scratches are not just visually convincing but also mechanically consistent. They respect the conservation of mass (pile-up), respond to material properties (hardness, elasticity), and exhibit the complex optical and haptic behaviors observed in the real world. As graphics hardware continues to evolve, such physically based defect simulation is poised to become the standard for high-fidelity digital twins and immersive virtual environments.

9. Data and Comparison Tables

Table 1: Comparison of Indenter Geometries and Resulting Pile-up

Indenter Type	Geometric Angle	Contact Shape	Pile-up Tendency	Primary Mechanism
Sphere	Variable	Circular	Sink-in (Elastic)	Hertzian Elasticity → Plasticity
Berkovich	65.3°	Triangular	Moderate (Anisotropic)	Plastic Flow / Ploughing
Vickers	68°	Square	Moderate (Symmetric)	Plastic Flow
Cube Corner	35.26°	Triangular	High (Distinct Lobes)	Cutting / Fracture
Cone	Variable ($\theta$)	Circular	Depends on $\theta$	Ploughing ($>{\theta }_c$) / Cutting ($<{\theta }_c$)

Table 2: Influence of Material Properties on Simulation Parameters

Material Property	Simulation Parameter	Effect on Scratch Morphology
Hardness ($H$)	Penetration Depth ($d$)	Determines depth for given Force ($P=k{d}^n$). Higher $H$ $\to$ Shallower scratch.
Elastic Modulus ($E$)	Springback Ratio ($R_e$)	Determines depth recovery. Higher $E$ $\to$ Less recovery (deeper final scratch).
Yield Strength (${\sigma }_y$)	Characteristic Length ($\lambda$)	Determines width of pile-up/recovery zone. Higher ${\sigma }_y$ $\to$ Wider features.
Work Hardening ($n$)	Pile-up Distribution	Low $n$ $\to$ Sharp pile-up ridges. High $n$ $\to$ Sink-in / Diffuse pile-up.
Grain Size	Fractal Noise Frequency	Coarse grains $\to$ Low freq noise. Fine grains $\to$ High freq noise.

Sources:

1. A Physically-Based Model for Rendering Realistic Scratches - Johns Hopkins Computer Science https://www.cs.jhu.edu/~misha/ReadingSeminar/Papers/Bosch04.pdf

2. A Physically-Based Model for Rendering Realistic Scratches | Request PDF - ResearchGate https://www.researchgate.net/publication/220505935_A_Physically-Based_Model_for_Rendering_Realistic_Scratches

3. THE DEVELOPMENT OF SCRATCH TEST METHODOLOGY AND CHARACTERIZATION OF SURFACE DAMAGE OF POLYPROPYLENE - CORE https://core.ac.uk/download/pdf/4268652.pdf

4. Contact Patch Parameter Extraction - Altair Product Documentation https://help.altair.com/hwdesktop/hwx/topics/motionview/fiala_tire_contact_patch_param_ext_r.htm

5. (PDF) Approximate Analytical Model for Hertzian Elliptical Contact Problems https://www.researchgate.net/publication/239403129_Approximate_Analytical_Model_for_Hertzian_Elliptical_Contact_Problems

6. Nonlinear finite element simulations of ice forces on offshore structures - DiVA portal http://www.diva-portal.org/smash/get/diva2:999067/FULLTEXT01.pdf

7. Length-scale-dependent stress relief mechanisms in indium at high homologous temperatures - NIH https://pmc.ncbi.nlm.nih.gov/articles/PMC8045576/

8. cellular automata and lattice boltzmann techniques: an approach to model and simulate complex systems https://cui.unige.ch/~chopard/CA/acsProof.pdf

9. Clearing the springback hurdle - The Fabricator https://www.thefabricator.com/thefabricator/article/bending/clearing-the-springback-hurdle

10. Researches and Simulation of Elastic Recovery Phenomena during Roller Burnishing Process of Macro-Asperities of Surface - NIH https://pmc.ncbi.nlm.nih.gov/articles/PMC7700643/

11. 16th LS-DYNA Forum 2022 - TIB https://edocs.tib.eu/files/e01mr23/1831275864.pdf

12. Generation and Reproduction of Random Rough Surfaces - MDPI https://www.mdpi.com/2075-4442/13/11/506

13. Recognition of Scratches and Abrasions on Metal Surfaces Using a Classifier Based on a Convolutional Neural Network - MDPI https://www.mdpi.com/2075-4701/11/4/549

14. Retrieval of Striated Toolmarks using Convolutional Neural Networks - ResearchGate https://www.researchgate.net/publication/318647366_Retrieval_of_Striated_Toolmarks_using_Convolutional_Neural_Networks

15. Volume, Side-Area, and Force Direction of Berkovich and Cubecorner Indenters, Novel Important Insights - Scirp.org. https://www.scirp.org/journal/paperinformation?paperid=113407

16. Mechanical analysis of the scratch test on elastic perfectly plastic materials with the three-dimensional finite element modeling | Request PDF - ResearchGate https://www.researchgate.net/publication/223436308_Mechanical_analysis_of_the_scratch_test_on_elastic_perfectly_plastic_materials_with_the_three-dimensional_finite_element_modeling

17. Force–displacement curves extracted from MD simulations: (a) 4.1 nm… - ResearchGate https://www.researchgate.net/figure/Force-displacement-curves-extracted-from-MD-simulations-a-41nm-tungsten-atom-tip-b_fig4_259319129

18. A New Analytical Model for Estimation of Scratch‐Induced Damage in Brittle Solids https://www.researchgate.net/publication/227824525_A_New_Analytical_Model_for_Estimation_of_Scratch-Induced_Damage_in_Brittle_Solids

19. INDENTATION CONTACT MECHANICS https://indentpt.com/pdf/IndentationContactMechanics_En_Revised2020.pdf

20. Simulating Real-time Tyre Contact Patch Dynamics via Nanopiezoelectric Sensor Arrays. Adapting agricultural R&D for Racing Physics (MATLAB Source Included) : r/F1Technical - Reddit https://www.reddit.com/r/F1Technical/comments/1pi9hkc/simulating_realtime_tyre_contact_patch_dynamics/

21. New Algorithms for Calculating Hertzian Stresses, Deformations, and Contact Zone Parameters http://notes-application.abcelectronique.com/166/166-47934.pdf

22. Strain Hardening From Elastic-Perfectly Plastic to Perfectly Elastic Indentation Single Asperity Contact - Frontiers https://www.frontiersin.org/journals/mechanical-engineering/articles/10.3389/fmech.2020.00060/full

23. hardness testing | Total Materia https://www.totalmateria.com/en-us/articles/hardness-testing/

24. Scratch hardness at a small scale: Experimental methods and correlation to nanoindentation hardness | Request PDF - ResearchGate https://www.researchgate.net/publication/353003091_Scratch_hardness_at_a_small_scale_Experimental_methods_and_correlation_to_nanoindentation_hardness

25. Volume, Side-Area, and Force Direction of Berkovich and Cubecorner Indenters, Novel Important Insights - ResearchGate https://www.researchgate.net/publication/356497759_Volume_Side-Area_and_Force_Direction_of_Berkovich_and_Cubecorner_Indenters_Novel_Important_Insights

26. MULTI-SCALE INDENTATION HARDNESS TESTING; A CORRELATION AND MODEL A Thesis by DAMON WADE BENNETT Submitted - OAKTrust - Texas A&M University https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/ETD-TAMU-2008-08-32/Damon%20Bennett%20Thesis.pdf

27. Scratch Durability of Automotive Clear Coatings: A Quantitative, Reliable and Robust Methodology https://www.paint.org/wp-content/uploads/2021/09/AUG00_Scratch-Durability-of-Auto-Clear-Coatings.pdf

28. SYNC Diffraction Distributions, Image Analysis with Reference to Sharpness Determination of Abrasive Powders - Microtrac https://www.microtrac.com/files/86148/abrasives.pdf

29. Image Analysis with Reference to Sharpness Determination of Abrasive Powders - Sync Diffraction Distributions - AZoM https://www.azom.com/article.aspx?ArticleID=18965

30. Relationship between scratch hardness and yield strength of elastic perfectly plastic materials using finite element analysis - Cambridge University Press & Assessment https://www.cambridge.org/core/journals/journal-of-materials-research/article/relationship-between-scratch-hardness-and-yield-strength-of-elastic-perfectly-plastic-materials-using-finite-element-analysis/C329767EE4B3FC702D229A428F994F60

31. Abrasive Sensitivity of Martensitic and a Multi-Phase Steels under Different Abrasive Conditions - PMC - NIH https://pmc.ncbi.nlm.nih.gov/articles/PMC8000481/

32. An Investigation into the Densification-Affected Deformation and Fracture in Fused Silica under Contact Sliding - MDPI https://www.mdpi.com/2072-666X/13/7/1106

33. Micro and Nano Mechanics of Materials Response during Instrumented Frictional Sliding - DSpace@MIT https://dspace.mit.edu/bitstream/handle/1721.1/36207/76904424-MIT.pdf?sequence=2

34. Pile-up geometry (not to scale) and notation | Download Scientific https://www.researchgate.net/figure/Pile-up-geometry-not-to-scale-and-notation_fig3_282389064

35. DOCTOR OF PHILOSOPHY Laser-Driven Acceleration of Ions: Biological Effects at Ultra-High Dose Rates Gwynne, Deborah - Queen’s University Belfast https://pureadmin.qub.ac.uk/ws/files/159598007/D_Gwynne_PhD_v4.pdf

36. Scratch on Polymer Materials Using AFM Tip-Based Approach: A Review - PMC - NIH https://pmc.ncbi.nlm.nih.gov/articles/PMC6835326/

37. Cellular automaton-based simulation of bulk stacking and recovery https://www.repositorio.ufop.br/bitstreams/ee40e717-47cf-4bf1-bdef-98828f226e4f/download

38. Applying the analytic theory of colliding ring galaxies | Monthly Notices of the Royal Astronomical Society | Oxford Academic https://academic.oup.com/mnras/article/403/3/1516/1050053

39. A multiscale model of terrain dynamics for real-time earthmoving simulation - DiVA portal http://www.diva-portal.org/smash/get/diva2:1485087/FULLTEXT02.pdf

40. Cellular automata model of gravity-driven granular flows | Request PDF - ResearchGate https://www.researchgate.net/publication/226079540_Cellular_automata_model_of_gravity-driven_granular_flows

41. Sliding Contact at Plastically Graded Surfaces and Applications to Surface Design - DSpace@MIT https://dspace.mit.edu/bitstream/handle/1721.1/39260/173619801-MIT.pdf?sequence=2

42. Cellular automaton modeling of dynamic recrystallization of Ni–Cr–Mo-based C276 superalloy during hot compression | Journal of Materials Research | Cambridge Core https://www.cambridge.org/core/journals/journal-of-materials-research/article/cellular-automaton-modeling-of-dynamic-recrystallization-of-nicrmobased-c276-superalloy-during-hot-compression/E02C38BF8C167D2D608C190F82E82EDD

43. Impact cratering https://geosci.uchicago.edu/~kite/doc/Melosh_ch_6.pdf

44. Molecular Dynamics Study on Tip-Based Nanomachining: A Review - PMC https://pmc.ncbi.nlm.nih.gov/articles/PMC7561650/

45. Research on Determining Elastic–Plastic Constitutive Parameters of Materials from Load Depth Curves Based on Nanoindentation Technology - PMC - NIH https://pmc.ncbi.nlm.nih.gov/articles/PMC10223561/

46. Friction Mechanisms of γ-Al2O3 Under Nanoindentation–Scratch Coupling: A Deep Neural Network Potential Approach | The Journal of Physical Chemistry C - ACS Publications https://pubs.acs.org/doi/10.1021/acs.jpcc.5c04118

47. Experimental and Numerical Investigation on the Effect of Scratch Direction on Material Removal and Friction Characteristic in BK7 Scratching - MDPI https://www.mdpi.com/1996-1944/13/8/1842

48. Explicit Dynamics Analysis Guide - Ansys Help https://ansyshelp.ansys.com/public/Views/Secured/corp/v251/en/pdf/Ansys_Explicit_Dynamics_Analysis_Guide.pdf

49. Morse Shifted GirifalcoWeizer 1959LowCutoff Na MO_707981543254_004 MO_707981543254 · Interatomic Potentials and Force Fields - OpenKIM https://openkim.org/id/Morse_Shifted_GirifalcoWeizer_1959LowCutoff_Na__MO_707981543254_004

50. Effects of Interlaminar Failure on the Scratch Damage of Automotive Coatings: Cohesive Zone Modeling - MDPI https://www.mdpi.com/2073-4360/15/3/737

51. Buckling failure mode in the scratch test; (a) pile-up ahead of the… - ResearchGate https://www.researchgate.net/figure/Buckling-failure-mode-in-the-scratch-test-a-pile-up-ahead-of-the-moving-indenter-and_fig1_223744633

52. PRECISION ENGINEERING CENTER https://pec.ncsu.edu/wp-content/uploads/sites/10/2021/08/PEC2005I.pdf

53. Maya Creative 帮助 | Texturing with procedural noise shaders - Autodesk Help https://help.autodesk.com/view/MAYACRE/CHS/?guid=LookdevX_LookdevX_for_Maya_Tutorial_Files_Noises_Noises_html

54. Value Noise and Procedural Patterns - Scratchapixel https://www.scratchapixel.com/lessons/procedural-generation-virtual-worlds/procedural-patterns-noise-part-1/simple-pattern-examples.html

55. Influence of the attack angle on the scratch testing of an aluminium alloy by cones: Experimental and numerical studies | Request PDF - ResearchGate https://www.researchgate.net/publication/248464835_Influence_of_the_attack_angle_on_the_scratch_testing_of_an_aluminium_alloy_by_cones_Experimental_and_numerical_studies

56. Experimental, numerical and analytical studies of abrasive wear: correlation between wear mechanisms and friction coefficient - Comptes Rendus de l’Académie des Sciences https://comptes-rendus.academie-sciences.fr/mecanique/item/10.1016/j.crme.2005.09.005.pdf

57. Material removal factor (fab) - Sci-Hub https://2024.sci-hub.se/6363/233e4f110f12432d44bf146802373c9d/franco2017.pdf

58. Rendering Specular Microgeometry with Wave Optics - UCSB Computer Science https://sites.cs.ucsb.edu/~lingqi/publications/paper_glints3.pdf

59. Scratch-based Reflection Art via Differentiable Rendering - Beibei Wang https://wangningbei.github.io/2023/DiffGlints_files/paper_diffGlints_compressed.pdf

60. Earthquake Safety Training through Virtual Drills - UMass Boston CS https://www.cs.umb.edu/~craigyu/project_pages/vr17_projectpage/vr2017earthquake_files/vr2017earthquake.pdf

61. Streamlining Haptic Design with Micro-Collision Haptic Map Generated by Stable Diffusion https://www.mdpi.com/2076-3417/15/13/7174

62. ShapeFlow: Learnable Deformations Among 3D Shapes - NeurIPS https://proceedings.neurips.cc/paper/2020/file/6f1d0705c91c2145201df18a1a0c7345-Paper.pdf

63. Real Time Physics Class Notes - GitHub Pages https://matthias-research.github.io/pages/publications/realtimeCoursenotes.pdf

64. Automated classification of behavioural and electrophysiological data in Neuroscience - White Rose eTheses Online https://etheses.whiterose.ac.uk/id/eprint/20932/1/thesis_final.pdf

65. The Granule-In-Cell Method for Simulating Sand–Water Mixtures - ResearchGate https://www.researchgate.net/publication/398342310_The_Granule-In-Cell_Method_for_Simulating_Sand-Water_Mixtures**

62a⁝ Generowanie śladów