FRACTURE MECHANICS IISara Ferry NSE|H.H.Uhlig Corrosion Lab

22.71|October 23, 2012

recap:

Griffiths vs. Irwin & Orowan

(but much larger!)

Energy Release Rate

• Precracked • Purely elastic• Loaded with weight P• Δ describes

displacement of weight

• A describes crack area

P

Elastic energy in the body:

How to find the form of U?

1. Solve boundary value problems using elasticity theory2. Measure experimentally

arbitrary body:

Energy release rate dU1. • Only the weight does work on the body• This work is stored as elastic energy• Crack doesn’t change• Obtain expression for change in U and integrate

2.• Crack increases in area, reducing elastic energy• Weight does no work• Displacement held fixed

1+2.

dU = P d – G dAΔ

Fracture Energy • Most of the work done by the weight is stored as U: only some of the

work goes toward inelastic processes (fracture, plastic deformation)• If small scale yielding condition applies, can still find U as if the body

was purely elastic

Pd = Δ dU + dAΓElastic processes

Inelastic processes

…and fracture criterion: The crack grows if energy release rate G = fracture energy Γ.(fracture energy resists crack growth)

How can we measure fracture energy?• Look it up: results from previous experiments• Perform a fracture test yourself• Use a computer simulation (not standard)

For common materials, you can expect to measure a fracture energy of …

10 J/m2 in glass50 J/m2 in ceramics103 J/m2 in polymers104 J/m2 in aluminum105 J/m2 in steel*

*heat treating the steel can drastically change its fracture energy!

Potential Energy

Consider elastic body + weight P as a combined system = Π U – PΔ

• Analogous to Gibbs free energy.

Subbing in expression for dU: d = - dP – GdAΠ Δ , such that = (Π Π P, A)

Linear ElasticityThere is a linear relationship between applied force P and displacement . Δ

Elastic energy U = PΔ/2Potential energy = -Π U

Energy release rate load fixed

Finding G:• Look it up; there are handbooks for elasticity solutions for many

situations (G is specific to the configuration of the system) • Measure C using multiple identical specimens that have different crack

areas. Obtain C(A), find U, and then find G (expression on next slide)• Elasticity boundary-value problem (finite element program)

Compliance• Linearly elastic body• Linear relationship between load and displacement• C = compliance, is a function of crack area• Opposite of stiffness (stiffness: rigidity; resistance to deformation

when force is applied)

= C(A)PΔ

When A increases, compliance C increases.More compliance = more elastic energy stored when load is fixedMore compliance = less elastic energy stored when displacement is fixed

Applications of Fracture Mechanics1. Measure fracture energy• If other constants known except σc and Γ• Load a precracked sample, record critical stress, solve for Γ• Compare fracture energies of materials and study ways to improve fracture resistance

2. Predict critical load• Solve for σc if other values known (may have to experimentally determine Γand measure a first) without carrying out fracture experiment• Use to compare the critical load for various crack sizes (what is maximum allowable crack

size)

3. Estimate flaw size• Measure σc , , and Γ β• a = E/Γ πσ2

4. Knowing the material that will be used, expected stresses, and typical flaw sizes, design a structure to minimize likelihood of fracture