ENB331 Lecture Materials and Manufacturing 2
Griffith Critical Stress
A balance between surface energy and strain energy
E = Modulus of elasticity
γs= Specific surface energy
a = Half the internal crack length
Fracture Toughness (K )
KICs a measure of a materials resistance to brittle
fracture when a crack is present
KIC Material selection
σ = Design stress
a = Allowable flaw size or NDT flaw detection
Stress Distribution around crack tip in a thin infinite plate
Generalized Hooke’s Law
A uniaxial stress gives rise to a triaxial state of strain
Total strains (Generalized Hooke’s Law)
1 Week 5 Tuesday, 27 August 2013 ENB331 Lecture Materials and Manufacturing 2
Effect of material thickness on local stresses and strains
If a plate is very thin, only local stresses σ and σ (biaxial In thicker sections, crack root is constrained by material
tension) are induced. No localized tensile stress along on either side and thus the strain along z-axis is zero (ε =z
the z-axis (σz= 0) and is called Plane Stress as one of the 0). This is called Plain Strain as one of the principal
principal stresses is zero. From equation 6, the local strains is zero. From equation 6:
stresses cause a negative strain at the crack root.
Negative strain means thinning of the plate (free
contraction) near the crack tip due to the high value of σ
and σ y
The constrained material induced stress σ and zhis
creates triaxial tension.
Effect of material thickness on fracture toughness
Fracture toughness depends on volume of a material and thus the thickness. Thinner materials exhibit higher
toughness (due to only biaxial tension) and this is known as plane stress fracture toughness. Thicker materials have
lower limit of material toughness (due to triaxial tension) and is known as plane strain fracture toughness
Elastic Stress Equations: As r goes to 0 (at crack tip), local stresses σx,ynd to
infinity and such high stress magnitudes do not occur in
structures. Irwin argued that once local stresses starts to
exceed the yield stress of the material, plastic
deformation occurs at the crack tip and thus stress
singularity cannot exist.
Approximations were done to determine the amount of plastic deformation at the crack tip using plastic zone.
Irwin assumed circular shape of the plastic zone. Only situation along x-axis (θ=0) for σ will be ynalysed.
Plastic Zone, ry
2 Week 5 Tuesday, 27 August 2013 ENB331 Lecture Materials and Manufacturing 2
Fracture Mode Transition
Full plane stress occurs if the calculated size of the
plastic zone is approximately the same as the thickness
of the specimen, B. Predominant plane strain is expected
when the calculated plastic zone size is approximately
1/10 of the thickness. Plane strain fracture toughness
(KIC remained constant with thickness and thus shows
Strength vs Fracture toughness
Yield strength – Stress level that produces permanent plastic deformation
Fracture strength – stress level that causes failure
Fracture toughness – stress-intensity level at the crack tip that causes failure.
Typically as materials get stronger, the fracture toughness decreases.
How to improve both toughness and strength
Microstructure refinement (grain size reduction) improves toughness and strength.
Smaller dislocation pile-ups at grain boundaries
Less chance for cracks to form
Grain growth in steel is retarded by addition of vanadium, titanium, niobium, aluminium and to a lesser extent by
nickel. Vanadium (as little as 0.1% is the most potent grain-refining element by forming finely dispersed carbides and
nitrides in steel.
Fracture Toughness Measurements
The most common Fracture Toughness test configurations are: Compact tension, Single Edge Notch Bend (SENB) and
W = width of specimen
B = thickness of specimen
D = diameter for round specimen
The position and orientation of