Fracture Behavior. Section
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1 Section 6 Fracture Behavior In January 1943 the one-day old Liberty Ship, SS Schenectady, had just completed successful sea trials and returned to harbor in calm cool weather when... "Without warning and with a report which was heard for at least a mile, the deck and sides of the vessel fractured just aft of the bridge superstructure. The fracture extended almost instantaneously to the turn of the bilge port and starboard. The deck side shell, longitudinal bulkhead and bottom girders fractured. Only the bottom plating held. The vessel jack- knifed and the center portion rose so that no water entered. The bow and stern settled into the silt of the river bottom." Approximately 2700 ships were built from 1942 to the end of WWII, using prefabricated all welded construction. At the start of the program 30% of Liberty Ships suffered catastrophic fracture. What factors attributed to the failures?.
2 Fracture Mechanics How to design something to withstand: Griffith s energy relation English aeronautical engineer, A.A.Griffith, recognized that theoretical predictions of the stresses at the tip of a crack approach infinity (thus any material with a crack would fail). He developed a approach to predict failure where he assumed that the growth of a crack requires creation of surface energy (which is the same as the surface tension of the material, γ, times the total crack surface area created). A necessary condition for crack growth is that the negative change in potential energy is greater than or equal to the change in the surface energy generated with the creation of a new crack: dπ = du1 du3 du 4 2γdA 52
3 For a center notched panel, with crack length of 2a, Griffith Equation is 2 πσ a 2γ = E (2γ ) E σ = πa 1/ 2 This only works well for materials. For other materials (i.e. plastics, metals, composites) other energy absorbing components are added to the equation, such as plastic work W pl and the strain energy release rate, G, is defined to include requirements for all kinds of energy, including surface energy. 1/ 2 (2γ + ) E GE σ = pl = πw a πa 1/ 2 2 πσ a So for this case (center notched panel), G = E The strain energy release rate, G, can be understood as the. The units of G are. For failure, G c, the critical strain energy release rate (also fracture energy) is defined as: G c 2 πσ f a = where σ f is the applied fracture stress. E Gc can be interpreted as. The fracture energy, G c, is a, while G is a which is a function of applied loads and geometry. Modes of Loading There are three ways of applying force to enable a crack to propagate: 53
4 Stress Intensity Factor Another approach to fracture mechanics is the Stress Intensity Factor (K I, K II, and K III ) which, for the center notched panel is: K I = σ πa Combining this with the equation for G above, we see that K and G are related as: K = EG K c = EGc this is actually only the case for plane stress (thin samples) For (thick samples compared to the plastic zone size) K and G are related as: EG K = where E and ν are the Young s modulus and the Poisson s ratio respectively. 2 1 ν The stress intensity factor, K, is a and is a measure of the stress singularity at the crack tip. It depends on. The critical stress intensity factor, K C, also called the, is a. When K K then. I IC Note, crack tip creates a 1/sqrt(r) singularity. The Yield stress prevents the stress from approaching infinity. How is the plastic zone size, r y, determined? In general, K crack length. C 1/ 2 = Yσ Ca where Y is a geometrical factor that corrects for the sample shape and 54
5 Mode I Mode II Mode III 55
6 Stress Intensity Factor for various geometries Infinite Plate with a Center Through Crack under Tension Infinite Plate with a Hole and Symmetric Double Through Cracks under Tension Semi-infinite Plate with an Edge Through Crack under Tension Infinite Stripe with a Center Through Crack under Tension Infinite Stripe with an Edge Through Cracks under Tension Infinite Stripe with Symmetric Double Through Cracks under Tension 56
7 Stress σ=p/a σ y Stress Intensity Factor (K I ) (K IC ) Stress intensity factor is to as stress is to. 57
8 Experimental Fracture Mechanics Example double cantilever beam DCB testing to evaluate interlaminar fracture toughness DCB specimen was first introduced to evaluate adhesives Energy release rate is determined by: h G 3δ = 2ba P b a o L 58
9 Data Reduction P U δ = C(a)P δ 2 U = 1 2 Pδ = 1 2 C δ Fixed Grips BG = Π a = U δ a = δ 1 δ 2 C 2 dc da = 1 2( P) 2 dc da Eq. (1) x M a P d 2 u y dx 2 = M EI = P(a x) EI du u y dx = 0 y = 0 at x=0 u y = P EI 2 x3 6 a x2 u y (a) = 4Pa 3 I = 1 since EBh 3 12 Bh 3 δ = 2 u y = 8P a EB h 3 Eq. (2) 59
10 Eq. (2) δ =... C = δ P = 8 a EB h 3 E = 8P a δb h 3 Eq. (1) G(C(a))=... G = 12a2 EB 2 h 3 ( P ) 2 G = 12a2 δ EB 2 h 3 C 2 = 3δ 2 Eh 3 16a 4 G = 3δP 2aB Cube root of specimen compliance plotted versus crack length G = 3δP 2ba a* = a + Δ Correction for both shear deformation and root rotation by adding a length Δ to a, where Δ can be determined experimentally by plotting C 1/3 60
11 Typical load-displacement curve for virgin and healed reference specimens (8H satin weave) Virgin C A Healed δ (mm) B: D DCPD & catalyst injected E A: Crack propagation commences ahead of precrack B: Loading of virgin is completed and catalyzed DCPD is injected into delamination C: Crack propagation commences for the healed specimen D: The crack has propagated through the entire healed region E: Further loading creates a new virgin crack ahead of previously healed region a (mm) Specimen No AVG. 61
12 Mode II delamination testing End-Notched Flexure (ENF) Test Mixed-Mode Bending (MMB) Test 62
13 Other Experimental Methods for Fracture Behavior Tensile Measurements Impact Testing o Izod, Charpy, Drop Tower Fatigue Testing o Governed by Paris law da m = AΔK dn ΔK = K max K min where A and m are material dependent parameters. In all cases, the properties measured are highly dependent on. 63
14 There are a number of mechanisms by which polymers and composites fail. One of the common modes (especially in rubber toughened systems) is. As seen below, fibril strands span the craze (ahead of the crack) absorbing energy. 64
15 W Flag Instrumented Tup Velocity gate H v 0 x(t) Support Specimen
16 Example Problem: In the compliance calibration of an edge cracked fracture toughness testpiece of a tough composite material, it was observed that a load of 100 kn produced a displacement between the loading pins of mm when the crack length was 24.5 mm and mm when the crack length was 25.5 mm. The fracture load of an identical testpiece, containing a crack of length 25.0 mm is 158 kn. Calculate the critical strain energy release rate, G IC, and the plane-strain fracture toughness, K IC, of the material. All test pieces were 25 mm thick. The following elastic constants are given for the material: ν=0.3 and E=70 GPa.
17 Example Problem: t Substrate P Laminate a The mode I energy release rate, G I, for a crack (blister) growing from internal pressure, P, is given as 2 2 ( 1 ) G I P a a where E is the effective Young s modulus, is the effective 3 E 32t Poisson s ratio, a is the crack length, and t is the laminate thickness. Determine whether crack growth will be stable or unstable for a monotonically increasing pressure loading.
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