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**3 – Fracture of Materials**

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**Part of the top of an Aloha Airlines jet peeled of during the flight**

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**Liberty Ship failures involved both brittle fractures and fatigue fractures.**

Of 2700 ships built, 400 suffered hull and deck fractures. 90 were considered serious and in 20 ships fracture was essentially total with 10 ships breaking in half without warning.

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**Outline Fracture of Materials Types of Fracture Brittle Fracture**

Ductile Fracture Brittle to Ductile Transition Fracture Mechanics Stress Concentration LEFM K & G Kc Appendix - Griffith Theory

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Fracture Fracture: separation of a body into pieces due to stress, at temperatures below the melting point. Steps in fracture: Crack formation Crack propagation Depending on the ability of material to undergo plastic deformation before the fracture two fracture modes can be defined - ductile or brittle

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**Ductile fracture - most metals (not too cold):**

Extensive plastic deformation ahead of crack Crack is “stable”: resists further extension unless applied stress is increased Brittle fracture - ceramics, ice, cold metals: Relatively little plastic deformation Crack is “unstable”: propagates rapidly without increase in applied stress Ductile fracture is preferred in most applications

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**Brittle vs. Ductile Fracture**

Ductile materials - extensive plastic deformation and energy absorption (“toughness”) before fracture Brittle materials - little plastic deformation and low energy absorption before fracture

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**Brittle vs. Ductile Fracture**

Brittle fracture; (b) Shearing fracture in single crystal (c) Completely ductile (d) Ductile fracture

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**Ductile Fracture Necking, (b) Cavity Formation,**

(c) Cavity coalescence to form a crack,

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Ductile Fracture (d) Crack propagation, (e) Fracture

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**(Cap-and-cone fracture in Al)**

Ductile Fracture (Cap-and-cone fracture in Al)

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Ductile Fracture Scanning Electron Microscopy: Fractographic studies at high resolution. Spherical “dimples” correspond to micro-cavities that initiate crack formation.

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**Brittle Fracture No appreciable plastic deformation**

Crack propagation is very fast Crack propagates nearly perpendicular to the direction of the applied stress Crack often propagates by cleavage – breaking of atomic bonds along specific crystallographic planes (cleavage planes)

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Brittle Fracture Brittle fracture in a mild steel

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Brittle Fracture Transgranular fracture: Fracture cracks pass through grains. Fracture surface have faceted texture because of different orientation of cleavage planes in grains. Intergranular fracture: Fracture crack propagation is along grain boundaries (grain boundaries are weakened or embrittled by impurities segregation etc.)

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Brittle Fracture

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**Ductile-to-Brittle Transition**

Ductile-to-Brittle Transition: As temperature decreases a ductile material can become brittle Alloying usually increases the ductile-to-brittle transition temperature. FCC metals remain ductile down to very low temperatures. For ceramics, this type of transition occurs at much higher temperatures than for metals.

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**Ductile-to-Brittle Transition**

DBTT: Ductile-Brittle Transition Temperature

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**Fracture Mechanics & Fracture Toughness**

Stress Concentration Energy Release Rate Crack Tip Plasticity Fracture Toughness

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Stress Concentration

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**Stress Concentration m 2 (a/)1/2**

For a long crack oriented perpendicular to the applied stress the maximum stress near the crack is: m 2 (a/)1/2 (1) where σ is the applied external stress, a is the half-length of the crack, and ρ the radius of curvature of the crack tip. (note that a is half-length of the internal flaw, but the full length for a surface flaw).

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**Stress Concentration Factor**

The ratio of the maximum stress and the nominal applied tensile stress is denoted as the stress concentration factor, Kt, where Kt can be calculated by Equation 1. The stress concentration factor is a simple measure of the degree to which an external stress is amplified at the tip of a small crack.

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Stress Concentration The stress distribution at the crack tip in a thin plate for an elastic solid is given by: (2)

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**Stress Concentration K: Stress Intensity Factor (3)**

Unit of K: psiin, MN/(m3/2), or MPam

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Stress Concentration Substitute K into Equation (3): (4)

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**Stress Concentration Opening In-plane Shear Out-of-Plane Shear**

Fracture Modes

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**Energy Release Rate G = - dU/dA**

Energy Release Rate: In 1956, Irwin defined an energy release rate, G, which is a measure of the energy available for an increment of crack extension: G = - dU/dA Since G is obtained from the derivative of a potential, it is also called the crack extension force or crack driving force

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**Energy Release Rate Griffith Approach: G = 2a/E**

Where is the nominal stress, a is half-length of crack, E is Young’s Modulus

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**Relationship between K and G**

Mode I Plain Stress: G = KI2/E Plain Strain: G = KI2(1-2)/E

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**Plain Stress vs. Plain Strain**

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Crack Tip Plasticity Irwin Approach

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Crack Tip Plasticity Irwin Approach

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**Crack Tip Plastic Zone Shape**

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Fracture Toughness Kc: If we assume a material fails locally at some combination of stresses and strains, then crack extension must occur at a critical K value. This Kc value, which is a measure of fracture toughness, is a material constant that is independent of the size and geometry of the cracked body.

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**Fracture Toughness Measurement**

MEASURE STRENGTH WITH A CRACK OF KNOWN LENGTH SHARP CRACK LINEAR ELASTIC? SMALL PROCESS ZONE RULES ASTM STANDARDS: E-399

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**SUMMARY Types of Fracture: Brittle Fracture Ductile Fracture**

Ductile Fracture: Crack initiation, crack growth, fracture, fractography Brittle Fracture: Intergranular & Transgranular (Cleavage) Brittle to Ductile Transition: DBTT

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**SUMMARY – Cont’d Stress Concentration**

Stress distribution in front of the crack tip Stress concentration factor Linear Elastic Fracture Mechanics K: Stress Intensity Factor G: Energy Release Rate Relationship between K & G Crack tip plasticity Fracture Toughness

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**Appendix - Griffith Theory for Brittle Fracture**

For defect-free material: sth ~ E/10 is orders of magnitude higher than usually observed Where E – Young’s modulus g – specific surface energy a0 – lattice parameter sth – theoretical strength of the material

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**Energy criterion for brittle fracture (Griffith Theory)**

Energy criterion: when rate of change of the release of elastic energy with respect to crack size rate at which energy is consumed to create new surfaces, fracture occurs Onset: Where s – Griffith’s stress (the critical stress) gs – surface energy/per unit area

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Griffith Equation Griffith equation is only for glasses & ceramics – brittle fracture, no plastic deformation occurs prior to fracture Metals and polymers plastic deformation occurs before fracture modified Griffith equation (Orowan equation)

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