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 |
Executive Summary
Section titled âExecutive SummaryâThis research provides fundamental analytical solutions for crack tip stress fields in cubic anisotropic materials, specifically single crystal Silicon (Si) and 3C-SiC, crucial for microelectromechanical systems (MEMS) reliability.
- Core Achievement: Derived closed-form analytical expressions for Mode I and Mode II crack tip singular stress fields in cubic elastic anisotropic solids using the Eshelby-Hirth-Lothe (EHL) dislocation method.
- Relevance to MEMS: The analysis focuses on standard MEMS orientations (crack face {100}, crack front <100>), providing a necessary theoretical foundation for understanding brittle fracture in these devices.
- Anisotropy Control: The singular stress fields are shown to be fundamentally controlled by the three independent elastic stiffness coefficients (C11, C12, C44) and the resulting anisotropy ratio (A).
- Singularity Confirmed: The derived solutions confirm the expected 1/sqrt(r) stress singularity near the crack tip, aligning with classical Linear Elastic Fracture Mechanics (LEFM).
- Mixed-Mode Prediction: The study applied the Erdogan-Sih maximum hoop stress criterion to mixed-mode loading, demonstrating that the predicted crack propagation angle (θ0) is significantly dependent on the materialâs elastic anisotropy, especially for crack angles (β) away from 0° and 90°.
- Validation Tool: These analytical results serve as essential benchmarks for validating complex numerical methods (e.g., Finite Element Analysis) used in micro-scale fracture simulations.
Technical Specifications
Section titled âTechnical SpecificationsâThe analysis relies on the fundamental elastic properties of the semiconductor materials studied, particularly the three independent stiffness coefficients (Cij) and the calculated anisotropy ratio (A).
| Parameter | Value | Unit | Context |
|---|---|---|---|
| Elastic Stiffness C11 | 165.7 | GPa | Single Crystal Silicon (Si) |
| Elastic Stiffness C12 | 63.9 | GPa | Single Crystal Silicon (Si) |
| Elastic Stiffness C44 | 79.6 | GPa | Single Crystal Silicon (Si) |
| Anisotropy Ratio A | 1.56 | N/A | Si (A = 2C44 / (C11 - C12)) |
| Elastic Stiffness C11 | 540 | GPa | Single Crystal 3C-SiC |
| Elastic Stiffness C12 | 180 | GPa | Single Crystal 3C-SiC |
| Elastic Stiffness C44 | 250 | GPa | Single Crystal 3C-SiC |
| Anisotropy Ratio A | 1.39 | N/A | 3C-SiC (A = 2C44 / (C11 - C12)) |
| Stress Singularity | 1/sqrt(r) | N/A | Radial dependence of crack tip stress field |
| Crack Face Orientation | {100} | N/A | Standard plane for MEMS structures |
| Crack Front Direction | <100> | N/A | Standard direction for MEMS structures |
| Maximum Hoop Stress Angle Difference | ~6.2% | Degrees | Difference in the anisotropy parameter (Ď) between Si and 3C-SiC |
Key Methodologies
Section titled âKey MethodologiesâThe study employed a classical analytical fracture mechanics approach based on dislocation theory to derive the singular stress fields.
- Fundamental Solution Selection: The analysis utilized the exact solutions for the elastic field of an isolated edge dislocation in a cubic anisotropic solid, known as the Eshelby-Hirth-Lothe (EHL) stress field.
- Crack Representation: The physical crack was modeled as a continuous distribution of isolated dislocations along the crack plane (x2 = 0, x1 less than |a|).
- Stress Cancellation (Buecknerâs Principle): The density of the dislocation distribution (Fi(x1)) was determined such that the internal stresses generated by the dislocations exactly cancel the far-field uniform applied stress (ĎA) on the crack faces.
- Singular Integral Equation: The determination of the dislocation density resulted in a singular integral equation, which was solved using the Hilbert transform method.
- Stress Field Derivation: The crack tip stress field (Ďij) was calculated by integrating the EHL stress field over the derived dislocation density function Fi(x1).
- Complex Variable Integration: The integration was performed in the complex plane using contour integration, applying the Laurent expansion and the residue theorem to handle the singularity at the crack tip.
- Fracture Criterion Analysis: The derived Mode I and Mode II stress fields were combined for mixed-mode loading, and the Erdogan-Sih criterion was applied to find the angle (θ0) where the tangential stress (Ďθθ) is maximized and shear stress (Ďrθ) is zero, predicting the crack propagation direction.
Commercial Applications
Section titled âCommercial ApplicationsâThe analytical framework developed is critical for improving the reliability and design of micro-scale devices where fracture mechanics govern failure.
- MEMS Reliability Engineering: Essential for predicting failure initiation and propagation paths in Si and 3C-SiC structural components of accelerometers, gyroscopes, and resonant sensors.
- High-Temperature/Harsh Environment Sensors: Directly applicable to 3C-SiC based MEMS used in extreme conditions (e.g., automotive, aerospace) where high stiffness and fracture resistance are paramount.
- Semiconductor Processing Optimization: Provides guidance for selecting optimal crystal orientations during bulk micromachining or thin-film deposition (CVD) to minimize stress concentrations and maximize fracture toughness.
- Micro-Actuator Design: Informing the structural design of micro-actuators and micro-grippers that rely on the precise mechanical response of anisotropic single-crystal materials under high cyclic stress.
- Quality Control and Failure Analysis: Used as a theoretical basis for interpreting experimental fracture data and validating numerical models used in the quality control of semiconductor wafers and MEMS dies.
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.