Trends with Materials Heat treatment will cause embrittlement Cast Iron 4140 Steel Q&T UTS = 1550 MPa UTS = 950 MPa More tempering provides Lower strength More ductile failure Larger % carbon in steel Higher ductile-to-brittle transition Lower energy absorption More brittle From: Dowling, Callister

Intro to Fracture Mechanics Recall our discussion of Theoretical Strength … Cohesive strength was estimated to be ~E/4 to E/8 Way too high for most materials We rationalized that overprediction was due to flaws Recall our discussion of Material Properties ... Some materials failed in a brittle manner, while some were ductile Plastic deformation was introduced We will now: Attempt to understand “quantitatively” the effect of defects Tie in to our discussion of ductile vs. brittle behavior Understand why brittle materials are more effected by these defects Understand how ductile materials absorb some energy

Silver Bridge

Statistics Completed 19 May 1928 Connects Huntington, VA to Middleport, OH Charleston, VA to Dayton, OH Major east-west connection for US Route 35 Two lanes of automobile traffic 1750 feet in length Steel Eyebar Suspension Bridge Aluminum Paint (“Silver Bridge”) A major east-west connection for US Route 35 connecting Charleston, WV and major cities in Ohio

Ohio River

4:58 PM December 15, 1967

Disaster Second most deadly U.S. bridge disaster 64 hit the water 18 rescued 46 dead or missing 31 vehicles on the bridge

Wreckage

Wreckage

Wreckage

What’s Different About Silver Bridge? First “eyebar” suspension bridge in the U.S. First bridge that used high-strength, heat- treated carbon steel High Risk: new structure on a new scale, using new materials.

Silver Bridge Collapse Collapse, Wearne, P. TV Books, NY 1999

Source

Cause of Failure Bridge Design? Eyebar Manufacturing Quality? Material Choice?

Bridge Design at Fault? Partially! Steel Eyebar Suspension Suspended “Bicycle Chain” Weakest Link, No Redundancy Cable Suspension has hundreds of links Partially!

Failed Eyebar

Failure Evidence John Bennet, US Bureau of Standards “The Ohio River there is very heavily traveled so the U.S. Corps of Engineers had taken all the debris and just piled it on the shore – it was a terrific mess.” “Fortunately, each piece had been photographed as they took it out.”

Failure Evidence

Photograph of Failed Eyebar 330 John Bennet, US Bureau of Standards “Looking at it, the fractures on the two sides were completely different. “One side was very straight, almost like a saw cut. “The other side was extensively deformed, the metal bent and the paint chipped off.

Eyebar Loading 12” wide 2” thickness 6 12

Eyebar 330 Failure Sketch 12” wide 2” thickness 6 12 1/8” corroded crack Brittle Appearance 12” wide 2” thickness 6 12 Ductile Appearance

Crack on Eyebar 330

Conditions of Failure Crack formed by original forging operation Quenched and tempered steel Stress corrosion cracking 32oF ambient temperature

Wireless Sensors for Bridges

Fracture Mechanics Timeline Origins of modern fracture mechanics date to 1920 A.A. Griffith energy balance approach All energy that is created must be used George Irwin defines G in 1950’s H.M. Westergaard published solutions using a stress based approach in 1939 (Irwin, 1958) Use K, which characterizes the magnitude of the stresses in the vicinity of a crack, then can use a universal stress field equation for different geometries Irwin proposes R-curve in 1960 Jim Rice introduces J-integral in 1968 Important parameters G : strain energy release rate K: stress intensity factor Different but similar to Kt R: crack growth resistance J : J-integral

Crack Propagation Cracks propagate due to sharpness of crack tip A plastic material deforms at the tip, “blunting” the crack. deformed region brittle Energy balance on the crack Elastic strain energy- energy stored in material as it is elastically deformed this energy is released when the crack propagates creation of new surfaces requires energy plastic

Uncracked plate elastic energy Griffith Approach applied Uncracked plate elastic energy 2a Introduce a crack which Reduces elastic energy by: Increases total surface energy by: l For a crack to grow, the energy provided by new surfaces must equal that lost by elastic relief 2w applied DUel2a = DG thickness, t

Griffith Approach l a thickness, t applied Minimum criterion for stable crack growth: Strain energy goes into surface energy a 2a 2a l 2w applied thickness, t

Plastic Energy Term l a thickness, t applied Previous derivation is for purely elastic material Most metals and polymers experience some plastic deformation Strain energy (U) goes into surface energy (G) & plastic energy (P) Orowan introduced plastic deformation energy, p Materials which exhibit plastic deformation absorb much more energy, removing it from the crack tip a 2a 2a l 2w applied thickness, t

When Does a Crack Propagate? Crack propagates if above critical stress where E = modulus of elasticity s = specific surface energy a = one half length of internal crack Kc = sc/s0 For ductile => replace gs by gs + gp where gp is plastic deformation energy i.e., sm > sc or Kt > Kc

Energy Release Rate, G l a thickness, t applied Irwin chose to define a term, G, that characterizes the energy per unit crack area required to extend the crack: a 2a Change in crack area 2a l 2w Comparing to the previous expression, we see that: Works well when plastic zone is small (“fraction of crack dimensions”) applied thickness, t

Experimentally Measuring G F x Load cracked sample in elastic range a Fix grips at given displacement Load a + Da DU t Allow crack to grow length a a + a a Unload sample Compare energy under the curves Displacement G is the energy per unit crack area needed to extend a crack Experimentally measure combinations of stress and crack size at fracture to determine GIc F x

Stress Based Crack Analysis Westergaard, Irwin analyzed fracture of cracked components using elastic-based stress theory Defined three basic modes for crack loading From: Socie

Stress Intensity Factor, K specimen geometry flaw size operating stresses Critical parameter is now based on: Stress Flaw Size Specimen geometry included using “correction factor” crack shape specimen size and shape type of loading (i.e. tensile, bending, etc)

Stress Intensity Factor, K Stress Fields (near crack tip) Note: as r  0, stress fields   Plane stress - too thin to support stress through thickness Plane strain - so thick that constrains strain through thickness

Design Example: Aircraft Wing • Material has Kc = 26 MPa-m0.5 • Two designs to consider... Design A --largest flaw is 9 mm --failure stress = 112 MPa Design B --use same material --largest flaw is 4 mm --failure stress = ? • Use... • Key point: Y and Kc are the same in both designs. 9 mm 112 MPa 4 mm --Result: Answer: • Reducing flaw size pays off!

Design Against Crack Growth • Crack growth condition: K ≥ Kc = • Largest, most stressed cracks grow first! --Result 1: Max. flaw size dictates design stress. s amax no fracture --Result 2: Design stress dictates max. flaw size. amax s no fracture

Stress Intensity Factor, K Displacement Fields Displacement in x-direction Displacement in y-direction

Fracture Toughness, KC Critical value of K specimen geometry (sometimes f(a/W)) flaw size Fracture strength From: Callister Fracture occurs when stress exceeds critical value, c Geometric correction includes ratio between crack length and specimen width Empirical mathematical expressions Y calibration curves Units are MPam or psiin Measure of material’s resistance to brittle fracture But some materials undergo large plastic deformations prior to failure….

New Concepts & Terms Ductile-brittle transition Stress Concentration Factor, Kt Inglis approach Strain Energy Release Rate, G Elastic strain energy, U Surface energy, s Plastic deformation energy p Griffith & Irwin approaches Experimental measures of G Stress Intensity Factor, K Mode I, II, III Fracture toughness Plain strain fracture toughness Plastic zone size Plane strain fracture toughness testing Minimum dimensions Relation to G Designing with K Critical crack size Leak before break Fracture Process Ductile vs. brittle Crack formation Crack propagation Stable Unstable Ductile Fracture Cup-and-cone Shear vs. fibrous regions Brittle Fracture Transgranular (cleavage) Intergranular Chevron and fan-like patterns Impact Testing Charpy and Izod Falling Weight Maximum load Energy absorption Ductile-brittle transition