psto-cp-o103 4, Fracture Criteria 4.1 Kasa Failure Criterion From previous analysis, it is clear that when stresses at the crack tip exceed yield (which always happens for engineering materials), plasticity results. However, if the redistribution of stress has a minimal effect on the crack tip elastic stress field, then the K approach to defining the stress field is still of sufficient accuracy for engineering applications. Thus, if plasticity is minimal, then a LEFM approach is justified, Of importance to practical applications is the critical stress and strain state at the crack tip zone, which, when attained, causes the crack to propagate in a brittle, catastrophic manner. The most dangerous situation occurs when a crack is in a high-energy but constrained field that permits only slight plastic deformation at the crack tip. Expressed another way, the amount of energy absorbed in plastic deformation is reduced to a minimum extent and much more energy is thus available for fracture, i.e. crack propagation. This critical state can be described by a critical stress intensity factor K, K=K, (4) which may imply either a low stress acting on long crack or a small crack suffering a high stress. It is important to note the different meaning of the two sides of the above equation. The left hand side represents the driving force of the crack, which depends ‘on the applied loads and the geometry of the components. The right hand side of equation (4.1) signifies the materials’ resistance to fracture, which is an environment and load rate dependent material property. Laboratory testing indicates that the fracture toughness value depends on the thickness B of the specimen tested. The plane strain fracture toughness of the materials is a material property (denoted as Kic, where subscript I denotes mode I loading). Under plane strain condition, since the crack tip plastic zone is small in relation to the component thickness, plastic contraction in the through thickness direction is suppressed by the surrounding elastic material. Tensile stresses are set up in the thickness direction of the plastic zone so that the stress state is triaxial, giving rise to constrained plastic deformation. Table 4.1 lists some typical values of plane strain fracture toughness. As before, the suffix I refers to the tensile opening mode of crack extension, whilst II and III symbolise shear and anti-plane tear modes, respectively. When the plastic zone is large compared with the component's thickness, the triaxiality may be relaxed and the through thickness stresses normal to the plane of the component will be negligible. In this case, the fracture toughness may vary with the specimen thickness, B, The form of variation of Ke with specimen thickness is schematically shown in Fig. 4.1. Beyond a certain thickness, a state of plane strain prevails (see Chapter 3) and the toughness reaches asymptotic value. If the thickness of the specimen is reduced, more energy will be dissipated as a result of plastic deformation near the specimen surface which is under plane stress condition. There 46psto-cpn03 seems to exist an optimum thickness where the toughness reaches its highest level, see Fig slant shear fracture lips 0 4 a thin plate plane stress Ke thick plate plane strain Kc | I 5 ickness specimen Fig.4.1 Effect of thickness on fracture toughness In order to achieve plane strain conditions at the elastic-plastic interface, the plastic zone must be small compared to the specimen thickness, crack length, and width of ligament: w B “s (4.2) 30° 50 “4 According to the ASTM standard, the following requirements must be satisfied a,B(W-a)>2. 5K ? (43) Os ‘which is equivalent to setting the plasticity constraint factor to be V3 47psto-cp-o103 Table 4.1 Typical values of fracture toughness Material Young's modulus E (GPa) Steels 210 medium carbon pressure vessel high strength alloy AFC 77 stainless Aluminium alloys 72 2024 T8 7075 16 7178 16 Titanium alloys 108 Ti-6AL4V (high yield) Comparative data Concrete 45 Ice 9 Epoxy 23 Boron fibre 441 Carbon fibre 250-390 Boron/epoxy composite 220-340 ERP 70-200 GERP 38 Yield stress oy (MPa) 260 470 1460 1530 420 540 560 1060 ‘1100 80 85 30-60 3000 2200-2700 725-1730 300-1400 100-300 ‘+ notat room temperature! 4.2 Residual Strength and Critical Crack Size Toughness Ke (MPay/m) 208 98 3 7 30 2B 38 0214 02" 053 46 32-45 20-60 Thickness requirement 25 (Kye) (em) 108 489.6 nu 74 104 79 42 12.6 31 Since the severity of a cracked component is characterised by stress intensity factor, K, and failure will occur when K = Ke, the residual strength of a cracked component is, 48 co)psto-cpn03 where Y is a geometry correction factor. Note that the stress is the gross stress on the section on which the function a is defined, where residual strength implies a net section condition. In the case of plane strain K,=K,c. It is conservative to assume that K, = Kc if the detailed stress state is not known. The size of the crack at this stress is called the ‘critical crack size’. This is normally difficult to solve in closed form as Y(a) is normally a complicated function of crack length and component geometry. Nevertheless, it can be solved numerically through iteration or, if the value of Y varies slowly with crack size, e.g. for a relatively small crack in a wide panel, an approximate value may be used. The critical crack size that a component can tolerate for a given load is If Ke) +z 7 wall 45) The above two equations provide the basis for fracture mechanics based design methodologies. It should be pointed that equation (4.4) is valid only when linear fracture mechanics is applicable, that is the net section stress level is far below the material's yield stress. Otherwise the component will fail in a different mode: plastic collapse. Consider a centre cracked panel with a finite width W, the absolute highest load carrying capability is bounded by the plastic collapse strength: the stress level over the entire section exceeds the yield or ultimate tensile strength of the material. It is easy to show that the nominal stress at collapse is 46) When this happens, the plastic deformation becomes unbounded and fracture will occur, regardless of the fracture toughness. Therefore there are two possible failure modes: brittle fracture and plastic collapse. Should the fracture stress «7, be higher than the stress causing failure by collapse, then collapse will prevail, As a result, the actual residual strength is the lowest of o and © ,_- Considering a centre cracked panel, there are three situations in which a plastic collapse failure would prevail: (1) the toughness is very high; (2) the crack is very small; and (3) the width Wis very small. A sketch is shown in Fig.4.2, The intersection of the two curves is given by W-2a a7 On W Tra (seca W) In the short crack regime, the exact transition from one mechanism to the other is not clear, but a plausible engineering approximation is the ‘tangent’ rule: drawing a tangent line passing through the ultimate tensile strength point. More accurate prediction can be achieved by using elasto-plastic fracture mechanics methods 49