Stress intensity factor, abbreviated as \(K\) or \(SIF\), is used in fracture mechanics to predict the stress state near the tip of a crack or notch in a material subjected to an applied load or residual stresses. It quantifies the stress concentration at the crack tip and is a key factor in determining whether the crack will propagate and cause failure.
K characterizes crack tip stress, it describes the intensity of the stress singularity at the crack tip. In an ideal, perfectly sharp crack within a linear elastic material, the stress theoretically approaches infinity at the crack tip. The stress intensity factor provides a measure of this elevated stress.
Depends on loading, geometry, and crack size, the value of the stress intensity factor () is not a material property alone. It has to do with: aplied stress, the magnitude of the external load acting on the material), Crack size, typically represented by the crack length (for an edge crack) or half the crack length (for a central crack). Geometry factor, a dimensionless factor that accounts for the specific geometry of the component and the crack configuration. This factor is often found in handbooks or determined through numerical analysis.
Failure criterion is used to predict fracture in brittle materials and to understand crack growth in materials exhibiting small-scale yielding at the crack tip. Failure occurs when the \(K\) reaches a critical value known as the fracture toughness ( or ) of the material. Fracture toughness is a material property that represents the material's resistance to crack propagation.
Stress Intensity Factor Mode Types - Modes of cracking can be subjected to different types of loading, which are categorized into three basic modes:
Mode I - Stress Intensity Factor formula |
||
|
\( K_1 \;=\; Y_I \cdot \sigma \cdot \sqrt{ \pi \cdot a } \) (Mode I) \( Y_I \;=\; \dfrac{ K_1 }{ \sigma \cdot \sqrt{ \pi \cdot a } }\) \( \sigma \;=\; \dfrac{ K_1 }{ Y_I \cdot \sqrt{ \pi \cdot a } }\) \( a \;=\; \dfrac{ K_1 }{ Y_I \cdot \sigma \cdot \sqrt{ \pi } }\) |
||
| Symbol | English | Metric |
| \( K_1 \) = Stress Intensity Factor for Mode I Loading | \(lbf-ft\) | \(J\) |
| \( Y_I \) = Geometric Factor (Stress Intensity Factor Coefficient) | \( dimensionless \) | \( dimensionless \) |
| \( \sigma \) (Greek symbol sigma) = Applied Stress | \(lbf \;/\; in^2\) | \(Pa\) |
| \(\large{ \pi }\) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( a \) = Crack Size (Half Length of the Crack) | \(in\) | \( mm \) |
Mode I (Opening) - Tensile stress is applied perpendicular to the crack plane, causing the crack surfaces to pull apart. The stress intensity factor for this mode is denoted as \(K_1\). This is the most common mode considered in engineering design.
- Mode I fracture toughness describes the resistance of a material to crack propagation under tensile loading or opening mode.
- This mode occurs when the crack propagates perpendicular to the direction of the applied tensile stress.
- It is commonly measured using standard fracture mechanics tests, such as the single-edge notched bend (SENB) test or compact tension (CT) test.
Mode II - Stress Intensity Factor Formula |
||
|
\( K_2 \;=\; Y_{II} \cdot \tau \cdot \sqrt{ \pi \cdot a } \) (Mode II) \( Y_I \;=\; \dfrac{ K_2 }{ \sigma \cdot \sqrt{ \pi \cdot a } }\) \( \sigma \;=\; \dfrac{ K_2 }{ Y_I \cdot \sqrt{ \pi \cdot a } }\) \( a \;=\; \dfrac{ K_2 }{ Y_I \cdot \sigma \cdot \sqrt{ \pi } }\) |
||
| Symbol | English | Metric |
| \( K_2 \) = Stress Intensity Factor for Mode I Loading | \(lbf-ft\) | \(J\) |
| \( Y_{II} \) = Geometric Factor (Stress Intensity Factor Coefficient for Mode II) | \( dimensionless \) | \( dimensionless \) |
| \( \tau \) (Greek symbol tau) = Applied Shear Stress | \(lbf \;/\; in^2\) | \(Pa\) |
| \(\large{ \pi }\) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( a \) = Crack Size (Half Length of the Crack) | \(in\) | \( mm \) |
Mode II (Sliding) - Shear stress is applied parallel to the crack plane and perpendicular to the crack front, causing the crack surfaces to slide relative to each other in an in-plane direction. The stress intensity factor for this mode is denoted as \(K_2\).
- Mode II fracture toughness relates to resistance against crack propagation under shear loading or sliding mode.
- In this mode, the crack propagates parallel to the direction of the applied shear stress.
- Mode II fracture toughness is crucial in materials subjected to shear loading, such as in adhesive joints or in materials with high interlaminar shear strength in composites.
- Testing methods for Mode II fracture toughness include the end-notched flexure (ENF) test and the asymmetric four-point bend test.
Mode III - Stress Intensity Factor Formula |
||
|
\( K_3 \;=\; Y_{III} \cdot \tau \cdot \sqrt{ \pi \cdot a } \) (Mode III) \( Y_I \;=\; \dfrac{ K_3 }{ \sigma \cdot \sqrt{ \pi \cdot a } }\) \( \sigma \;=\; \dfrac{ K_3 }{ Y_I \cdot \sqrt{ \pi \cdot a } }\) \( a \;=\; \dfrac{ K_3 }{ Y_I \cdot \sigma \cdot \sqrt{ \pi } }\) |
||
| Symbol | English | Metric |
| \( K_3 \) = Stress Intensity Factor for Mode I Loading | \(lbf-ft\) | \(J\) |
| \( Y_{III} \) = Geometric Factor (Stress Intensity Factor Coefficient for Mode III) | \( dimensionless \) | \( dimensionless \) |
| \( \tau \) (Greek symbol tau) = Applied Shear Stress | \(lbf \;/\; in^2\) | \(Pa\) |
| \(\large{ \pi }\) = Pi | \(3.141 592 653 ...\) | \(3.141 592 653 ...\) |
| \( a \) = Crack Size (Half Length of the Crack) | \(in\) | \( mm \) |
Mode III (Tearing) - Shear stress is applied parallel to both the crack plane and the crack front, causing the crack surfaces to slide relative to each other in an out-of-plane direction. The stress intensity factor for this mode is denoted as \(K_3\).
- Mode III fracture toughness involves resistance to crack propagation under tearing or tearing mode.
- In this mode, the crack propagates perpendicular to the direction of the applied shear stress.
- Mode III fracture toughness is less commonly studied compared to Modes I and II, but it's relevant in certain material configurations and loading conditions.
- Testing methods for Mode III fracture toughness are less standardized compared to Modes I and II.
