Fracture Mechanics Laboratory

Introduction

The student should, by now, be familiar with the concept of strength and in particular the yield strength and ultimate tensile strength. Unfortunately, materials and structures in the real world often contain cracks and when loaded will fail at stresses well below the tensile strength. The lower strength of solids containing cracks arises from the concentration of stress at the crack tip. To determine the stress required to propagate a crack, Griffith [1] used a thermodynamic approach in which he states: crack growth can occur if the energy required to form an additional crack of length da can just be delivered by the system. This may sound a little vague at first until we recognize that there are two energy sources available to drive the crack forward in stresses body. Firstly, under conditions where the applied load is free to move (such as in dead weight loaded test), the external load can supply the work by moving (recall work done = force x distance). Secondly, under the conditions described previously and in the case where the ends of the sample are fixed (such as the stress that might result from thermal contraction) the work to drive the crack must come from the energy stored within the solid, i.e.. the elastic strain energy. Clearly, for a given material there is a critical strain energy release rate at which crack growth will occur.

One of the objectives of this laboratory exercise is the measurement of the strain energy release rate (G). Now,

where F is the work performed by the external load, U the elastic energy contained in the plate, P the applied load, B the plate thickness, u the displacement and a the crack length. The deformations are assumed to be elastic. As long as there is no crack growth, the deflection is proportional to the load (Hooke’s Law) i.e.

where C is the compliance (the inverse of stiffness). As the crack gets longer, the compliance increases, or the solid behaves like a less stiff spring. The elastic energy contained within the plate is simply the area under the load deflection plot. Since the material is assumed to be linear elastic - hence linear elastic fracture mechanics - the elastic energy is simply

Thus from equation (1) we obtain

The terms with dP/da cancel, which means the G is independent of whether or not the load is constant and is always equal to the derivative of the elastic energy. It should be apparent from equation (4) that the strain energy release rate can be determined from a measurement of (i) the load P at which the crack propagates and (ii) the rate of change of compliance with crack length for the length of crack tested, i.e.. the slope of the tangent to a graph of compliance versus crack length at the crack length tested.

Experiment

You are provided with 6 double cantilever beams of a polymer in which a blunt crack has been introduced. Measure the crack length from the centre of the points of loading to the crack tip. After placing the sample into the testing machine, measure the load deflection curve up to an applied load of 200N, unload and repeat the measurement 3 times for each crack length. Determine the compliance from the inverse slope of the load (y) - deflection (x) plot.

Plot a graph of compliance vs. crack length.

A final sample is supplied in which there is a sharp crack produced by fatigue. measure the length of the crack and then test this sample to failure, making a note of the load (P) at which the crack extends. Determine the strain energy release rate.

Report

Your write up should consist of two parts. Firstly, tables and graphs of the appropriate measurements and calculations together with an estimate of the errors involved. Secondly, write an outline of the effect of sample thickness on the measured strain energy release rate and show how the strain energy release rate is related to the fracture toughness (K) in a linear elastic solid.