 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.

(c) A.F. Bower, 2008
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