Formula & Calculator
Griffith's Fracture Criterion
Predicts the critical stress at which a pre-existing crack of length 2a will propagate catastrophically in a brittle material.
Interpretation
σ_f = √(2Eγ_s/(πa)). Griffith criterion relates fracture stress to crack length and surface energy. For brittle materials. Foundation of fracture mechanics.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| sigma_f | Critical (fracture) stress | MPa |
| E | Young's modulus | GPa |
| gamma_s | Surface energy of the material | J/m2 |
| a | Half the internal crack length | m |
What it means
Griffith’s fracture criterion, proposed by A.A. Griffith in 1921, is a fundamental concept in fracture mechanics. It states that a crack will propagate when the decrease in elastic energy due to crack extension is at least equal to the energy required to create new surfaces. For an elliptical crack in a brittle material, the fracture stress σ_f is given by σ_f = √(2Eγ_s/(πa)), where E is Young’s modulus, γ_s is the surface energy per unit area, and a is the crack half‑length. This criterion explains why brittle materials fail at stresses much lower than theoretical strength due to pre‑existing flaws. It laid the foundation for fracture mechanics and is still used to understand the strength of ceramics and glasses. Later developments (Irwin) introduced stress intensity factors and toughness. Understanding Griffith’s criterion is essential for material scientists and engineers dealing with brittle fracture.
Worked example
Griffith's Fracture Criterion – Two Examples
Real‑World| Parameter | Value |
|---|---|
| E | 70 GPa |
| γ_s | 1 J/m² |
| a | 0.0001 m |
| Parameter | Value |
|---|---|
| E | 400 GPa |
| γ_s | 10 J/m² |
| a | 5×10⁻⁵ m |
Common mistakes
- Griffith criterion: For brittle fracture – assumes a pre‑existing crack of length 2a (or half‑crack length a) in an infinite plate.
- Surface energy γ_s: The energy per unit area of the new surfaces created – in J/m².
- Elastic modulus E: Young’s modulus of the material.
- Fracture stress σ_f: The stress at which the crack propagates – in Pa.
- Assumptions: Linear elastic, plane strain, and the crack is sharp. For ductile materials, use more advanced models.
Applications
Griffith's fracture criterion (σ_f = √(2Eγ_s/(πa))) determines the critical stress for brittle fracture in the presence of a crack of length a, based on surface energy γ_s. This is the foundation of fracture mechanics, explaining why brittle materials fail at stresses much lower than their theoretical strength due to microscopic flaws. Engineers use this concept to assess the integrity of brittle components like glass, ceramics, and some polymers. It guides design to avoid stress concentrations and to incorporate toughness via microstructural design. By understanding Griffith's criterion, materials scientists and engineers can develop tougher materials and design against catastrophic brittle failure.
- Design of glass and ceramic components (windows, optics, insulators)
- Failure analysis of brittle materials in service
- Surface treatment and flaw control to improve strength
- Development of tough ceramics and composites
- Microstructural design for high fracture resistance
Frequently Asked Questions
Griffith's criterion predicts the fracture stress of a brittle material containing a crack. The formula is σ_f = √(2·E·γ_s / (π·a)), where E is Young's modulus, γ_s is the surface energy per unit area, and a is the half‑crack length. It is based on energy balance: crack growth occurs when the released elastic strain energy exceeds the surface energy.
It shows that the fracture stress decreases with increasing crack length, and that materials with higher fracture toughness (surface energy) are stronger. It laid the foundation for linear elastic fracture mechanics.
- Perfectly brittle material (no plastic deformation).
- Infinite plate with a central crack.
- Plane stress or plane strain conditions.
- Linear elastic material.
For plane stress, the criterion uses E; for plane strain, the surface energy is replaced by the critical strain energy release rate G_IC, and the formula becomes σ_f = √(2·E·G_IC / (π·a)) for plane stress, or with (1–ν²) for plane strain.
Since G_IC = K_IC² / E (plane stress), Griffith's criterion can be rewritten as σ_f = K_IC / (Y√(π·a)), which is the same as the fracture toughness relationship.
Ductile materials absorb significant energy through plastic deformation at the crack tip, which is not accounted for in the surface energy term. The effective energy release rate includes plastic work, leading to much higher toughness.
It is used for brittle materials like glass, ceramics, and some polymers, where plastic deformation is minimal. It also provides a theoretical basis for understanding fracture.
Orowan extended Griffith's criterion to include the plastic work per unit area (γ_p): σ_f = √(2·E·(γ_s + γ_p) / (π·a)). For ductile metals, γ_p is much larger than γ_s, so the fracture stress is much higher.
Applying it to ductile metals without accounting for plastic deformation. Also, using the wrong geometry factor for non‑central cracks.
Surface energy is typically measured by cleavage experiments or by thermodynamic calculations. For most materials, it is on the order of 1‑10 J/m². For ceramics, it is used to estimate fracture strengths.