NSEHHUhlig Corrosion Lab 2271 October 23 2012 recap Griffiths vs Irwin amp Orowan but much larger Energy Release Rate Precracked Purely elastic Loaded with weight P ID: 933288
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Slide1
FRACTURE MECHANICS II
Sara Ferry NSE|H.H.Uhlig Corrosion Lab 22.71|October 23, 2012
recap:
Slide2Griffiths vs. Irwin &
Orowan(but much larger!)
Slide3Energy 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:
Slide4Energy release rate dU
1. Only the weight does work on the bodyThis work is stored as elastic energyCrack doesn’t changeObtain expression for change in U and integrate2.Crack increases in area, reducing elastic energy
Weight does no work
Displacement held fixed
1+2.
dU
= P
dΔ
– G dA
Slide5Fracture 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
PdΔ
=
dU
+
ΓdA
Elastic processes
Inelastic processes
…and
fracture criterion
:
The crack grows if energy release rate
G = fracture energy Γ.
(fracture energy resists crack growth)
Slide6How can we measure fracture energy?
Look it up: results from previous experimentsPerform a fracture test yourselfUse a computer simulation (not standard) For common materials, you can expect to measure a
fracture energy of …
10 J/m
2
in
glass
50 J/m
2
in ceramics
103 J/m2 in polymers 10
4 J/m2 in aluminum
105 J/m2 in
steel**heat treating the steel can drastically change its fracture energy!
Slide7Potential Energy
Consider elastic body + weight P as a combined system Π
=
U – PΔ
Analogous to Gibbs free energy
.
Subbing in expression for
dU
:
dΠ = -ΔdP –
GdA, such that Π
= Π(P, A)
Slide8Linear Elasticity
There is a linear relationship between applied force P and displacement Δ. Elastic energy U =
P
Δ
/2
Potential energy
Π
= -
UEnergy 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)
Slide9Compliance
Linearly elastic bodyLinear relationship between load and displacementC = compliance, is a function of crack areaOpposite of stiffness (stiffness: rigidity; resistance to deformation when force is applied)
Δ
= C(A)P
When
A
increases, compliance
C
increases.
More compliance = more elastic energy stored when load is fixed
More compliance = less elastic energy stored when displacement is fixed
Slide10Applications of Fracture Mechanics
1. Measure fracture energyIf other constants known except σc and ΓLoad 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
σ
c
if other values known (may have to experimentally determine Γand measure a
first) without carrying out fracture experimentUse to compare the critical load for various crack sizes (what is maximum allowable crack size)
3. Estimate flaw sizeMeasure σc
, Γ, and βa = ΓE/πσ2
4. Knowing the material that will be used, expected stresses, and typical flaw sizes, design a structure to minimize likelihood of fracture