Engineering Materials I - Fracture & Fat

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Fracture Mechanics Discipline concerned with the behavior of materials containing cracks or other small flaws. Flaw: Small pores (holes), inclusions, or microcracks. Does not refer to atomic-level defects like vacancies or dislocations. Allows quantification of relationships between material properties, stress level, crack-producing flaws, and crack propagation mechanisms. Stress Concentration Measured fracture strengths are lower than theoretical values due to flaws/cracks. Flaws/cracks always exist at the surface and within the interior of a material. Applied stress may be amplified/concentrated at the crack tip, depending on crack orientation and geometry. Stress Raisers: Flaws that amplify applied stress. Stress profile across an internal crack: Localized stress diminishes with distance from crack tip. Far from crack tip, stress equals nominal stress $\sigma_0$. Fracture Toughness Maximum stress a material can withstand with existing flaws of a certain size and geometry. Measures the ability of a material with a flaw to withstand an applied load (does not require high strain rate). Resistance to brittle fracture when a crack is present. Fracture Toughness Test: Apply tensile stress to a specimen with a known flaw; stress is intensified at the flaw. Stress Intensity Factor (K) Formula: $K = f \sigma \sqrt{\pi a}$ $f$: Geometry factor for specimen and flaw ($f=1.0$ for infinite width, $f=1.12$ for small single-edge notch). $\sigma$: Applied stress. $a$: Flaw size (crack length). Critical Stress Intensity Factor ($K_c$): The value of $K$ that causes a flaw to grow and cause failure. This is defined as the fracture toughness. Units: psi $\sqrt{\text{in.}}$ or MPa $\sqrt{\text{m}}$. Plain Strain Fracture Toughness ($K_{Ic}$) Fracture toughness depends on specimen thickness. As thickness increases, $K_c$ decreases to a constant value, known as $K_{Ic}$. $K_{Ic}$ is the property normally reported for a material and does not depend on sample thickness. Factors Affecting Crack Growth Larger flaws reduce the permitted stress. Manufacturing techniques aim to reduce flaw size. Ability to deform: Ductile materials deform at crack tip, blunting it and reducing stress intensity. Brittle materials (ceramics, polymers) have lower toughness. Thicker, more rigid pieces have lower fracture toughness than thin materials. Rate of application of the load: Increasing load rate (e.g., impact test) reduces fracture toughness. Temperature: Increasing temperature normally increases fracture toughness. Grain size: Small grain size improves fracture toughness. More point defects/dislocations reduce toughness. Stress-induced transformations: In certain ceramics, these lead to compressive stresses, increasing toughness. Importance & Applications of Fracture Mechanics Allows designing/selecting materials accounting for inevitable flaws. Three variables: Material property ($K_c$ or $K_{Ic}$), stress ($\sigma$), flaw size ($a$). If two are known, the third can be determined. Applications: Selection of a Material: Given max flaw size ($a$) and applied stress ($\sigma$), select material with sufficient $K_c$. Design of a Component: Given max flaw size ($a$) and material ($K_c$), calculate max stress component can withstand. Design of Manufacturing/Testing Method: Given material, stress, and component size, calculate max tolerable flaw size. Design NDT to detect flaws larger than this. Brittle Fracture Cracks/imperfections limit ceramic's ability to withstand tensile stress due to stress concentration. Actual stress at crack tip: $\sigma_{actual} \approx 2\sigma \sqrt{a/r}$ $a$: Crack length. $r$: Radius of crack tip. For very thin cracks ($r$ small) or long cracks ($a$ large), $\sigma_{actual}/\sigma$ becomes large (stress intensification). If $\sigma_{actual}$ exceeds yield strength, crack grows and causes failure, even if nominal $\sigma$ is small. Fatigue Lowering of strength or failure of a material due to repetitive stress (may be above or below yield strength). Common in load-bearing components (cars, airplanes, etc.) subjected to repetitive stresses. Stresses include tension, compression, bending, vibration, thermal expansion/contraction. Often occurs below yield strength, but over sufficient cycles, causes failure. Stages of Fatigue Crack Initiation/Nucleation: Tiny crack initiates after some loading, often at surface defects (scratches, pits, sharp corners, inclusions, grain boundaries). Crack Propagation: Crack gradually propagates as load continues to cycle. Sudden Fracture: Occurs when remaining cross-section is too small to support load. Fatigue in Engineering Materials For fatigue to occur, at least part of the stress must be tensile. Mainly concerns metallic and polymeric materials. Ceramics: Not usually considered for fatigue due to low fracture toughness (fail by brittle fracture first). Polymeric materials: Show fatigue, but mechanism differs from metals. Heating near crack tips due to repetitive stress relates fatigue to creep. Composites: Reinforcing phases degrade, elastic modulus decreases, weakening observed before fracture. Characteristics of a Fatigue Failure Fracture surface is typically smooth near the origin, becoming rougher and fibrous as crack grows. Microscopic and macroscopic examinations reveal: Beach marks (clamshell marks): Formed when load changes during service or loading is intermittent (allowing oxidation). Striations: Finer scale marks showing crack tip position after each cycle. Beach marks suggest fatigue, but their absence does not rule it out. Fatigue Test (Rotating Cantilever Beam) Method to measure material's resistance to fatigue. Cylindrical specimen mounted in a motor-driven chuck with a weight on the opposite end. Rotation causes a point on the specimen to experience a complete sinusoidal stress cycle (max tensile $\rightarrow$ zero $\rightarrow$ max compressive). Maximum stress: $\pm \sigma = \frac{32M}{\pi d^3} = \frac{16FL}{\pi d^3} = 5.09 \frac{FL}{d^3}$ $M$: Bending moment. $F$: Load. $L$: Distance from force to support. $d$: Specimen diameter. S-N Curves (Wöhler Curves) Plot of stress amplitude ($S$) vs. number of cycles to failure ($N$) from fatigue tests. Used to determine fatigue life and maximum allowable loads. Endurance Limit: Stress below which fatigue failure will not occur (50% probability). Preferred for design. Fatigue Life: Number of cycles a component survives at a particular stress. Fatigue Strength: Maximum stress for which fatigue will not occur within a specific number of cycles. Endurance Ratio: $\frac{\text{endurance limit}}{\text{tensile strength}} \approx 0.5$ (for steels). Factors Affecting Fatigue Life Mean Stress: Increasing mean stress level decreases fatigue life. Surface Effects: Most fatigue cracks originate at surface stress amplification sites. Fatigue life is sensitive to surface condition and configuration. Design Factors: Notches or geometrical discontinuities (grooves, holes, keyways, threads) act as stress raisers. Rounded fillets improve fatigue lifetime. Effect of Temperature: Increasing temperature decreases fatigue life and endurance limit. Cyclical temperature changes cause thermal fatigue due to non-uniform expansion/contraction. High-frequency stresses can heat polymers, reducing fatigue life.