Analysis of Mode I and II Crack tip Stress Fields for MEMS Structures with Cubic Elastic Anisotropy
At a Glance
Section titled âAt a Glanceâ| Metadata | Details |
|---|---|
| Publication Date | 2022-11-15 |
| Journal | Journal of the Society of Materials Science Japan |
| Authors | Fuma KATO, Takumi NAKAHARA, Toshiyuki Toriyama |
| Institutions | Ritsumeikan University |
| Citations | 1 |
| Analysis | Full AI Review Included |
Technical Documentation & Analysis: Anisotropic Fracture Mechanics in Cubic Materials
Section titled âTechnical Documentation & Analysis: Anisotropic Fracture Mechanics in Cubic MaterialsâExecutive Summary
Section titled âExecutive SummaryâThis research provides a critical analytical framework for understanding Mode I and Mode II crack tip stress fields in cubic elastic anisotropic solids, specifically Si and 3C-SiC, which are foundational materials for MEMS mechanical structures.
- Core Achievement: Derivation of closed-form singular stress fields near crack tips using the rigorous analytical solutions for isolated dislocations (Eshelby-Hirth-Lothe theory) in anisotropic media.
- Application Relevance: Essential for predicting brittle fracture mechanisms and optimizing the reliability of micro-scale mechanical sensors and piezoresistive elements.
- Anisotropy Dependence: The analysis confirms that the three independent elastic stiffness coefficients (C11, C12, C44) and the Anisotropy Ratio (A) fundamentally govern the crack tip stress distribution.
- Key Finding: Stress components, particularly the normal stress ($\sigma_{22}$), show significant deviation from predictions based on simplified isotropic models, necessitating anisotropic modeling for accurate design.
- Methodological Rigor: The approach validates the use of complex variable integration and residue theorems to solve singular integral equations related to dislocation distribution on the crack surface.
- 6CCVD Value Proposition: This research directly supports the need for ultra-high-stiffness, precisely oriented single-crystal materials. 6CCVDâs MPCVD Single Crystal Diamond (SCD) offers superior mechanical properties and customizable crystal orientation control, extending this analysis to the ultimate cubic material.
Technical Specifications
Section titled âTechnical SpecificationsâThe following data points were extracted from the analysis of Si and 3C-SiC, highlighting the material parameters critical for anisotropic fracture modeling.
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Si Elastic Stiffness (C11) | 165.7 | GPa | Cubic Anisotropic Solid |
| Si Elastic Stiffness (C12) | 63.9 | GPa | Cubic Anisotropic Solid |
| Si Elastic Stiffness (C44) | 79.6 | GPa | Cubic Anisotropic Solid |
| Si Anisotropy Ratio (A) | 1.56 | Dimensionless | A = 2C44 / (C11 - C12) |
| 3C-SiC Elastic Stiffness (C11) | 540 | GPa | Cubic Anisotropic Solid |
| 3C-SiC Elastic Stiffness (C12) | 180 | GPa | Cubic Anisotropic Solid |
| 3C-SiC Elastic Stiffness (C44) | 250 | GPa | Cubic Anisotropic Solid |
| 3C-SiC Anisotropy Ratio (A) | 1.39 | Dimensionless | A = 2C44 / (C11 - C12) |
| Crack Tip Stress Singularity | 1 / sqrt(r) | N/A | Characteristic radial dependence for Mode I and II fields |
| Critical Crystal Orientations | {100}, <100>, <110> | N/A | Standard orientations for MEMS test structures |
Key Methodologies
Section titled âKey MethodologiesâThe analytical derivation of the crack tip singular stress fields relies on established principles of anisotropic elasticity and dislocation theory:
- Fundamental Solution Selection: The stress field of an isolated edge dislocation in a cubic anisotropic solid (the EHL field) is used as the basic building block for the solution.
- Crack Modeling via Dislocation Distribution: The physical crack is modeled by distributing isolated dislocations along the crack surface (x2 = 0, x1 < |a|).
- Stress Cancellation: The internal stresses generated by the distributed dislocations are used to cancel the uniform external stress ($\sigma_{A}$) applied at infinity, satisfying the boundary condition that the crack surface is traction-free.
- Singular Integral Equation: This process results in a singular integral equation for the dislocation density function F(x1), which is solved using the Hilbert transform method.
- Complex Variable Integration: The stress components ($\sigma_{ij}$) near the crack tip are derived by integrating the EHL stress field, utilizing complex variable theory, Laurent expansion, and the Cauchy residue theorem to handle the singularity at the crack tip.
- Mixed Mode Analysis: The derived Mode I and Mode II solutions are combined to analyze the maximum hoop stress direction ($\theta_{0}$) based on the Erdogan-Sih criterion, demonstrating its dependence on the materialâs cubic anisotropy.
6CCVD Solutions & Capabilities
Section titled â6CCVD Solutions & CapabilitiesâThe rigorous analysis of anisotropic fracture mechanics in Si and SiC highlights the critical need for materials with predictable, high-performance mechanical properties and precise crystal orientation control. 6CCVDâs MPCVD Diamond (SCD and PCD) is the ideal material platform to replicate, validate, and extend this research, offering mechanical performance far exceeding Si or SiC.
Applicable Materials for Advanced Fracture Studies
Section titled âApplicable Materials for Advanced Fracture Studiesâ| Research Requirement | 6CCVD Material Recommendation | Technical Advantage |
|---|---|---|
| Ultimate Stiffness & Toughness | Optical Grade Single Crystal Diamond (SCD) | Diamond possesses the highest elastic modulus and fracture toughness of any known material, enabling the design of ultra-durable MEMS structures where Si/SiC fail. |
| Precise Anisotropy Control | Custom SCD Wafers (Specific Orientation) | We provide SCD wafers with precise crystal cuts (e.g., (100), (110), (111)) necessary to study the orientation-dependent fracture mechanics analyzed in this paper, where diamondâs Anisotropy Ratio (A $\approx$ 1.2) is highly relevant. |
| Piezoresistive MEMS Elements | Heavy Boron-Doped Diamond (BDD) | For applications requiring integrated sensing (as noted in the paperâs introduction), our BDD films offer stable, high-temperature piezoresistive performance coupled with diamondâs superior mechanical stability. |
| Large-Area Mechanical Structures | Polycrystalline Diamond (PCD) Plates | We offer PCD plates and wafers up to 125 mm in diameter, suitable for large-scale mechanical testing or industrial MEMS production, with thicknesses ranging from 0.1 ”m to 500 ”m. |
Customization Potential for Fracture Mechanics Experiments
Section titled âCustomization Potential for Fracture Mechanics ExperimentsâThe complexity of anisotropic fracture requires highly customized test specimens. 6CCVD provides the necessary engineering capabilities:
- Custom Dimensions and Substrates: We supply SCD and PCD plates in custom dimensions, including substrates up to 10 mm thick, allowing researchers to define specific crack geometries and loading conditions.
- Ultra-Precision Polishing: To ensure that surface defects do not interfere with the intrinsic fracture mechanics being studied, our SCD material is polished to an ultra-low roughness of Ra < 1 nm. Inch-size PCD can achieve Ra < 5 nm.
- Integrated Metalization Services: For applying strain gauges, electrical contacts, or defining specific loading points, 6CCVD offers in-house metalization capabilities, including Au, Pt, Pd, Ti, W, and Cu layers, directly integrated onto the diamond surface.
- Thickness Control: We guarantee precise thickness control for both SCD and PCD films, critical for thin-film fracture studies (0.1 ”m to 500 ”m range).
Engineering Support
Section titled âEngineering Supportâ6CCVDâs in-house PhD team specializes in the growth and characterization of MPCVD diamond. We offer expert consultation to assist engineers and scientists in:
- Material Selection: Choosing the optimal diamond grade (SCD, PCD, or BDD) based on required stiffness, thermal management, and electrical properties for [Anisotropic Fracture Mechanics] projects.
- Orientation Specification: Defining the precise crystal orientation and cut necessary to validate or extend the anisotropic stress field analysis presented in this research.
- Design for Diamond: Providing technical guidance on integrating diamond into complex MEMS designs, leveraging its unique mechanical and thermal advantages over Si and SiC.
For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.
View Original Abstract
This paper addresses analysis of mode I and II crack tip singular stress fields in single crystal silicon and 3C-SiC used as MEMS mechanical structures. Fundamental solutions of anisotropic elastic fields of isolated dislocations obtained by Eshelby, Hirth and Lothe were used to derive the closed form crack tip singular stress fields. It was concluded that three elastic stiffness coefficients of diamond lattice structure have important role to control the crack tip singular stress fields, and maximum hoop stress for mixed mode proposed by Erdogan and Sih.