G=-dU/dA change in potential energy per unit increase of area Crack extension occurs when G=work to fracture

G=G_c=2γ_s+γ_p

G_c is a measure of fracture toughness

1. brittle fracture is reversible under the right circumstances
2. whether it occurs or not is governed by balancing stored elastic energy with work of fracture

a0 is found using U_E and U_S

1. must be a perfectly clean environment (vacuum)
2. only elastic deformation/not plastic

What is the fundamental flaw in linear elastic fracture mechanics?

Why is it okay to continue to use LEFM?

the calculations assume elastic behavior, however, for a crack to grow, there must be some  plastic flow ahead of the tip

Ultimately, the contribution from the nonelastic bits are a small fraction of the total equation

What is the principal of superposition

What are the two reasons for why it can be used

1. all stresses are elastic
2. all stresses are small

it suggests you can add two stresses at right angles to each other to produce a 2D hydrostatic tension

R=2γ_S+γ_P

it is known as the resistance to crack growth and is hte maount of energy needed to make a crack grow

A-crack area

B-depth/thickness

2A-crack surface area

k_t-stress concentration factor σ_max/σ₀

K-stress intensity factor, the extent to which the crack intensifies any stress below the failure stress

K_IC-the critical stress intensity factor or fracture toughness (minimum stress before propagation begins)

describe crack modes 1, 2, 3

What are the equations for total stress with plane stress and plane strain

1-opening

2-shearing front/back

3-shearing side/side

EG=K_I²+K_II²+(1+ν)K_III² [stress]

EG=(1-ν²)K_I²+(1-ν²)K_II²+(1+ν)K_III² [strain]

Total G is always dominated by Mode I

1. voids nucleate (usually on second phase particles) and grow most rapidly in the center of the sample where triaxial stress state exists
2. Voids grow and coalesce to produce a circular internal crack which grows
3. the sample fails by shear in plane stress at the outer regions of the sample

where void nucleation is difficult (ie pure metals) much more ductility is observed and the sample can thin almost to a point before failure occurs

• this cleavage fracture surface is characterized by a planar inter-granular crack which changes plane by the formation of discrete steps
• facets correspond to the individual grains and in single crystals an entire slip plane can consist of one facet
• eventually these facets converge and eventually disappear int eh direction of crack growth
• they form at grain boundaries where the cleavage palne in one grain is not parallel to the plane in teh adjacent grain
  

• high yield stress
• reduced slip systems (HCP and BCC metals, low temperature)
• high constraint (plane strain) and rapid deformation

what is the relationship between grain size the failure type?

what is the significance of grain size?

small grains>>yielding precedes failure

large grains>>yielding and failure at any point

small grain size increases both toughness and strength therefore it is one of the best strengthening mechanisms

microcracks

these exist to act as stress concentrators

these are limited to the size of a grain because they do not have the energy to cross a grain boundary

1. high yileld stress - large amount of stored elastic energy
2. large grain size - large build up of stress from pile-ups
3. coarse carbides - can crack
4. deep notches - constraint
5. thick specimens (plane strain)

• Crack Tip Opening Displacement (CTOD)
• J-Integral

It allows for one to understand elastic-plastic effects of small components and involves looking at the curvature of a crack tip

curvature is related to the fracture toughness of a material

What are J-Integrals

What are its three main stages?

the rate of energy absorbed per unit area as the crack grows

J=-dU/dA

1-crack blunting

2- fracture initiation

3-steady state crack growth

R(Brittle) - these lines are flat and R is not effected by crack length

R(ductile) - shape of Sqrt(a) because as crack length increases, often the plastic zone increases as well

G v. a increases for load control

G v. a decreases for strain control fixed grips

What are the conditions for fast fracture?

stable crack growth?

dG/da>dR/da

dG/da=dR/da

1. deformation occuring in teh plastic zone greatly increases R, the work to propagate crack
2. for small plastic zone size, LEFM can be applied to ductile failure

a way of calculating zone size by suggesting there is a virtual crack tip that extends the distance of the actual tip placing it at the center of the plastic zone

based on the equation for K_IC and simply involves the rearrangment of variables to find r_y

σ=K_I/Sqrt[2πr_y], r_y=a+Δa and r_p=Δa+r_y

Use a yield criterion (such as von Mises)

plug in principal stresses then plot r_p v. angle

thin specimens tend to have higher values of K_IC and exhibit plane stress

thick specimens tend to have lower values of K_IC and exhibit plane strain