Fracture mechanics fundamentals Stress at a notch Stress at a crack Stress intensity factors Fracture mechanics based design
Failure modes Failure can occur in a number of modes: - plastic deformation - (brittle) fracture - instability (static & dynamic Failure mode will depend on load conditions, material, temperature etc The course focuses on fracture
Example fracture of a finger
Fracture details A crack normally propagates in mode I (perpendicular to the load) Possible reasons for branching: - altered load direction - anisotropy (path of least resistance) - influence of free edge
Stress at a notch!! The elastic stress at the notch edge is defined by the stress concentration factor:! max = K t!! nom! At an elliptic hole in a large plate, the maximum stress is! max = (1+2a/b)!! nom! When b"0, (the case of a crack) we get! max =! (result of elastic mtrl)" and K t =! (useless measure)#! We need something else to charactarize the state of stress! " nom K t = 3 " = 3#" nom " = $" nom % K t = 1+2 a ( ' * & b) 2b $ nom 2a K t " # då b " 0 $ nom 2a " nom $ nom $ nom
Crack tip blunting!! Metals: plastic deformation crack tip opening displacement (CTOD) can be used as a (ambiguous) measure of crack loading! Polymers: crazing! Ceramics (e.g. concrete): damage zones
Loading of a crack! Three loading modes: " mode I opening " mode II shearing " mode III tearing! In fatigue mode I loading dominates: " often (quasi-)uniaxial loading " shear driven growth more difficult the longer the crack " the crack grows in mode I! Exceptions: " small crack growth " contact fatigue " anisotropic or inhomogenous material
Stresses ahead of a crack tip! In mode I, stresses ahead of a crack can be expressed as! xx = K I /($2%r) cos(#/2)[1 " sin(#/2)sin(3#/2)]+ #! yy = K I /($2%r) cos(#/2)[1 " sin(#/2)sin(3#/2)]+ #! xy = K I /($2%r) " cos(#/2)sin(#/2)cos(3#/2)]+ #! K I depends on loading and geometry! The expression is only valid close to the crack tip! Similar expressions are valid for loading in modes II and III
Stress intensity factor! The elastic stress field at a crack tip is characterized by K I#! The stress intensity factor K represents how fast the stress goes to infinity at the crack tip! For modes II and III we get similar results (K II and K III )! For the stress directly ahead of the crack tip, we get! xx =! yy = K I /$(2$x)#! The dimension of the stress intensity factor is [N"m/m 2 ] or [MPa"m]! Do not confuse the stress intensity factor K with the stress concentration factor k t [ ].! Some presumptions for K to be valid are: " linear elastic material " isotropic material " ideal crack geometry " loading in one mode
How to estimate K! K can be derived from the stress distribution for instance in the following manner: " analytical match K to the stress distribution " FE-simulations employ asymptotic formulas (typically the crack face displacement, %=$, gives the best results) " handbooks of solved cases, K described as K =!$($a) f where " is nominal stress and f a load geometry-factor
Superposition of K! Superposition can be used to derive K under combined or two (or more) loads! Demands: " same geometry " load in the same mode
Fracture mechanics based design! If K is assumed to reflect the stress at the crack tip, we can assume fracture to occur at a certain magnitude of K K I = K IC! K IC is called the fracture toughness! K IC is valid at plane strain (small plastic zone size)! Criterion for plane strain: a, t, (W-a) & 2.5(K IC /! y ) 2 " must be fulfilled for LEFM to be valid (note that dimensions must agree!)
Fracture mechanics design! Determine allowable crack size: " Specify location of the crack " Determine stress intensity factor K a.f.o. {a,!,f(a)}# " from handbooks " from FE-simulations " Find a c that yields K I = K IC " Assure that LEFM is valid a, t, (b-a), h & 2.5(K IC /! y ) 2# " If not, apply corrections " If true, apply safety factors! Determine allowable stress: " Specify crack location " Specify allowable crack size " Determine stress intensity factor K a.f.o. {a,!,f(a)}# " from handbooks " from FE-simulations " Find! that yields K I = K IC " Assure that LEFM is valid a, t, (W-a), h & 2.5(K IC /! y ) 2# " If not, apply corrections " If true, apply safety factors
Plastic zone size! The elastic stress is theoretically infinite at the crack tip, but in reality plasticity will occur! The plastic zone must not be so large that it obscures the K-field
Plastic zone size continued! The plastic zone must not extend to a free edge! Ensure that the distance towards the free edge is sufficient! Also ensure a small plastic size as compared to the crack length a, (b a), h! 4"(K/" 0 ) 2 /!! Additionally you can assure plane strain conditions (ASTM standard conditions) t, a, (b a), h! 2.5"(K/" 0 ) 2! This is the most usual criterion for static LEFM! Conservative or nonconservative to not follow these?
Plane strain vs plane stress! If plane strain conditions are not ensured the fracture toughness will depend on the component geometry
Estimation of K! For cases when K is unknown, it can be estimated from known cases
Advanced estimation of K! Example: " a hole in the middle of a crack will have a small influence if the crack is long " for a large hole and small crack, the crack will just experience the nearby stress i.e. a two-sided crack will be similar to a one-sided crack " There is no influence of a stress parallel to a long crack " At the hole this stress will give an effect. Why?
Leak before break
Fracture toughness! Fracture toughness is mainly a material parameter, but dependent on factors as: " geometry (plane stress, plane strain) " ductility " temperature " crack size! Optional methods if LEFM is not valid: " Irwin correction (crack length correction, see book) " J-integral (energy measure)
Fracture toughness, continued
Fracture toughness, continued
Some practical comments! LEFM in static and dynamic loading! Final fracture criterion (and limits of validity) in static and dynamic loading! Plane stress plane strain (also in the same component)! Environmental influence! Elastic vs elastoplastic modelling! Overloads (final fracture vs crack growth)! FE simulations " Derivation of K " J-integral " asymptotic formulas " Crack modelling " mesh " crack face contacts! Handbook solutions " Choice of standard case " approximations " superpositions " Derivation of nominal stresses! Testing and scatter