Fracture Mechanics: Brittle vs Ductile, Griffith Theory, Fracture Stress and Energy, Study notes of Materials science

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KL Murty MSE 450 page 1 Fracture (7 : 1-8, 14 : 1-4) Brittle vs Ductile ⇔ Relative terms - ductile fracture implies appreciable plastic deformation prior to fracture - in general, bcc and hcp metals exhibit brittle fracture while fcc are ductile - some fcc metals can be so ductile ⇒ point fracture akin to superplastic materials Fracture in tensile testing : (Fig.7-1) and fracture characterization • strain to fracture --------------------- ductile vs brittle • mode (crystallographic) ------------ shear vs cleavage • appearance of fracture surface ----- fibrous vs granular (Figs. 7-7,8,9) ------dimpled vs faceted Fracture stress (σf) ⇒ Cohesive strength (σm) -- similar to theoretical yield strength (except here in tension) see text for derivation : σcohesivemax ˜ E π (Eq. 7-5) In a perfect brittle material, all the elastic strain energy is expended in creating the two new surfaces: σcohesivemax ˜ Eγs ao , ao is interatomic spacing (Eq. 7-8) • Stresses in a Cracked Body : presence of cracks reduces σf due to σ-concentration at the crack tip (recall MAT 201 / see section 2-15) define : c = {half crack length (interior crack)crack length of surface crack (Fig. 7-4) ρ = radius of curvature so that, σmax = σ [1 + 2 c ρ ] ~ 2 σ c ρ , here σ is the applied nominal stress ( P A ) Fracture occurs when σmax = σ cohesive max Or, σ at fracture, σf : 2 σf c ρ = Eγs ao With ρ ~ ao(sharpest crack possible), σf = Eγs 4c A c 2 c KL Murty MSE 450 page 2 Griffith Theory (of Brittle Fracture) : (7-4) A crack will propagate when the decrease in elastic strain energy is at least equal to the energy required to create new crack surface Increase in surface energy = decrease in elastic strain energy (expended in fracture) Or, UE + Us = ∆U ⇒ d∆U dc = 0. Or σf = 2Eγs πc Eq. 7-15 for a thin plate (plane stress) : as c increases, σf decreases. for thicker plate or plane strain case, σf = 2Eγs (1-ν2)πc • note these are valid for completely brittle material with no plastic deformation & γs may be altered by environment (corrosion, etc) If there is some plastic deformation, Griffith Model needs to be modified ⇒ Orowan suggested _ γt = γs + γp ,p for plastic work required to extend the crack so that σf = 2E(γs +γp) πc ˜ 2Eγp πc since γ p >> γs • Read 7-6 and 7-7 : Metallographic Aspects of Fracture / SEM Fractography • In general, cracks are formed during plastic flow ⇒ dislocation pile-ups (Fig. 7-10) lead to cracks - fracture involves 3 stages : (a) plastic deformation to produce cracks (b) crack initiation (nucleation) (c) crack propagation 3 Types of Fracture : Cleavage Quasi-cleavage Dimpled-Rupture brittle ductile flat facets dimples / tear sides dimples around inclusions around facets (microvoid coalescence) Fig. 7-7 Fig. 7-8 Fig. 7-9 KL Murty MSE 450 page 5 Crack Tip Stresses (11-3) Crack Deformation Modes (Fig. 11-3): Mode I - Opening mode • Mode II - Sliding mode Mode III - Tearing mode Stresses around the crack tip (Mode I): Eq. 11-9 / Fig. 11-2 using elasticity Plane stress σx = σ a 2r f1(θ) σy = σ a 2r f2(θ) τxy = σ a 2r f3(θ) these stresses are valid near the crack tip for a > r > ρ note : straight ahead of the crack tip (θ=0) : σx = σy = σ a 2r and τxy = 0. note - as r → 0 , σx and σy → ∞ implies there exists a zone where these elastic stress field equations are not valid indeed, at r = rc where σx , σy = σo (yield stress), yielding occurs and elasticity does not hold (plastic or process zone close to the crack tip) Irwin defines stress intensity factor K = σ πa , units of K σ (net section stress) = P (w-a)B here, mode I → KI = σ πa a w B P Now, can rewrite the stress fields in Eq. 11-9 in terms of K K is a convenient way of describing the stress distribution around a flaw: if two flaws of different geometry have the same values of K then the stress fields around them are identical In general, K = Y σ πa , where Y is a geometry factor (α in the text) Y = 1 for a crack in an infinite size body & Y = ( w πa tan πa w ) - Eq. 11-13 • can find equations for Y in ASTM standards for various geometries • KL Murty MSE 450 page 6 Now, we can relate K to G (energy release rate / crack extension force): recall G = πaσ2 E and since K = σ πa ⇒ K 2 = GE for plane stress (Eq. 11-14) & K2 = GE (1-ν2) plane strain (Eq. 11-15) Fracture Criterion : (basic concept) K-zone vs Process (or plastic) zone - near the crack-tip plastic flow - microvoids since cannot get σtip → ∞ due to plastic deformation at σ = σo → process- zone (rp) occurs when σx or σy = K 2πrp = σo or rp = 2 2 2 o K πσ K-zone Process-zone so, K-zone is the zone around the crak-tip where σ’s are scaled by K. If K-zone >> process-zone (i.e, rp is small), failure (or bond rupture) in process zone will be determined by stresses in K-zone, i.e., by K or LEFM ⇒ fracture criterion is : K = KIc ⇒ critical fracture toughness KIc is a material parameter - depends on T and strain-rate but not on geometry (for large thickness - Plane Strain Critical Fracture Toughness) - Fig. 11-7 That means in terms of fracture stress (σf) : KIc = Y σf πa • Fracture Design Basis : for a given load or stress (σ) in a material with a crack or flaw of size ‘a’ , K = Y σ πa and if K < KIc _ no failure Or, can find ac for given KIc and σ so that for a < ac no failure would occur. if ac > B (thickness) , then have the condition of Leak Before Crack - implying that the crack will pass through the wall without catastrophic brittle fracture - if ac < B, get brittle fracture ⇐ avoid / bad design KL Murty MSE 450 page 7 Plastic Zone Size : The above approach using K-fields is not valid for ductile materials where process-zone is large since rp corresponds to where σy = σo (see fig.) at rp , σy = σo = K 2πrp Or, rp = 1 2π { K σo }2. Note : as σo ⇑ rp ⇓ - approach LEFM Otherwise, we need to use EPFM or may use plasticity corrections Fig. 11-10 Before we discuss plasticity corrections, consider how to determine KIC: (11-5) • using CT, 3-point bend, or notched round specimen Plane Strain vs Plane Stress (Thick Plate) (Thin Sheet) rp = 1 6π K2 σ2o rp = 1 2π K2 σ2o (see View Graph) Relation between σο and KIc Plasticity Corrections : 1. Irwin : replace ‘a’ by aeff = a + rp so that Keff = σ π aeff = σ π (a+rp) , rp = f(K,σo) • K appears both sides ⇒ solve by iterative process 2. Dugdale : plastic zone in the form of narrow strips (Fig. 11-11) of size R= πK2 8σo , so that aeff = a + R ⇒ calculate Keff • There is a limit to the extent to which K can be adjusted •