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Problems
for Chapter 9
Modeling
Material Failure
9.4. Energy
Methods in Fracture Mechanics
9.4.1.
The figure
shows a double-cantilever beam fracture specimen that is loaded by applying
moments to the ends of the beams. Define
the compliance of the solid as 
9.4.1.1.
Derive an
expression relating the crack tip energy release rate to compliance and
dimensions of the specimen.
9.4.1.2.
Use the result
of 9.4.1.1 to calculate the crack tip stress intensity factors for the
specimen in terms of M and relevant
geometric and material properties
9.4.2.
The figure
shows a thin film with thickness h,
Youngs modulus E and Poisson’s
ratio . To test the interface between film and
substrate, a delaminated region of the film with width 2a is subjected to a vertical force P. By idealizing the
delaminated region as a beam, and using the method of compliance, estimate
the crack tip energy release rate in terms of P and relevant material and geometric properties.
9.4.3.
The figure
shows a thin rectangular strip of material with height 2h and out-of-plane thickness B. The material can be idealized as a linear
elastic solid with Young’s modulus E and
Poisson’s ratio .
The strip is damaged by an array of widely-spaced cracks with length 2a and spacing L>>a. It is loaded
by a uniform tensile traction t
acting on the top and bottom surface, which induces a displacement .
9.4.3.1.
Write down an
expression for the compliance of the undamaged strip (i.e. with no cracks)
9.4.3.2.
Write down a
relationship between the compliance of the strip and the crack tip energy
release rate.
9.4.3.3.
Estimate the
crack tip energy release rate using the energy release rate for an isolated
crack in an infinite solid. Hence, find
an expression for the compliance of the cracked strip, in terms of relevant
geometric and material parameters.
9.4.4.
The figure
shows a double cantilever beam specimen that is loaded by forces applied to
the ends of the beams. Evaluate the J
integral around the path shown to calculate the crack tip energy release
rate. You can use elementary beam
theory to estimate the strain energy density, stress, and displacement in the
two cantilevers. The solution must, of
course, be independent of b.
9.4.5.
Use the J
integral, together with the solution for the stress and displacement field
near the tip of a crack given in Section 9.3.1, to calculate the relationship
between the crack tip energy release rate and the stress intensity factor for
a Mode I crack.
9.4.6.
The figure
shows a thin film with thickness h,
thermal expansion coefficient ,
Young’s modulus and Poisson’s ratio on a large substrate with thermal expansion
coefficient . The film is initially perfectly bonded to
the substrate and stress free. The
system is then heated, inducing a thermal stress in the film. As a result, the film delaminates from the
substrate, as shown in the figure.
9.4.6.1.
Calculate the
state of stress a distance d>>h
ahead of the advancing crack tip
9.4.6.2.
Assume that the
film is stress free a distance d>>h
behind the crack tip. By directly
calculating the change in energy of the system as the crack advances, find an
expression for the crack tip energy release rate
9.4.6.3.
Check your
answer to 9.4.5.2 by evaluating the J integral around the path indicated in
the figure.
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