Dissemination of IT for the Promotion of Materials Science (DoITPoMS)

DoITPoMS Lecture Demonstration Packages Crack growth in inflated balloons Crack growth in inflated balloons (all content)

Crack growth in inflated balloons (all content)

Note: DoITPoMS Teaching and Learning Packages are intended to be used interactively at a computer! This print-friendly version of the LDP is provided for convenience, but does not display all the content of the LDP. For example, any video clips and answers to questions are missing. The formatting (page breaks, etc) of the printed version is unpredictable and highly dependent on your browser.

Contents

Introduction

The package provides information needed to set up and present a short practical demonstration in a lecture.  This LDP (Lecture Demonstration Package) is centred on crack propagation in inflated balloons. It would be suitable for a first year undergraduate course covering the basics of fracture.

After covering this lecture you should know:

  • Stress Fields in Internally Pressurised Cylinders
  • Griffith Treatment of the Energetics of Crack Propagation

It would be useful to look at the TLP on Brittle Fracture before you start.

(A) Scientific background

Stresses in a Thin-Walled Tube

The demonstration involves the use of 3 large pre-inflated cylindrical balloons. The hoop and axial stresses in such a balloon are dictated by the internal pressure, the wall thickness and the radius of the cylinder, as illustrated below.

Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.

Force balances in transverse and axial directions allow these two expressions to be derived.  It can be seen that the hoop stress is twice the axial stress. The actual magnitudes can be estimated by evaluating the pressure, wall thickness and radius. See also pressure measurement

Energetics of Crack Advance

The energy-based approach to crack propagation was first presented by A.A. Griffith in the 1920s. It states that unstable crack growth occurs when the stress intensity is high enough that the elastic strain energy released through crack growth is sufficient to supply the energy required to form new surfaces. The driving force for crack advance comes from elastic strain energy stored in the stressed material. The longer the crack the stress region is larger.

Flash

So the energy released when the crack extends (at both ends) by dc is the product of expression

\[U = \frac{{{{\mathop \sigma \nolimits_0 }^2}}}{{2E}}\]

and the increase in stress-free volume.

 

The strain energy release rate is the rate at which energy is absorbed by growth of the crack. It is usually used the more general equation which is given by

G ~\(\frac{{{{\mathop \sigma \nolimits_0 }^2}c}}{{2E}}\),

where σ0 is applied stress, c is half the crack length and E is Young’s modulus.
Important consequences:

  • Larger for larger cracks (flaws)
  • More energy released (per unit of crack area) when longer cracks propagate.

If the strain energy released exceeds a critical value Gc, then the crack will grow spontaneously. For a brittle material this fracture energy is simply given by 2 γ (where γ is the surface energy, with the factor of 2 arising because there are two new surfaces created when a crack forms). It can be considered as a critical strain energy release rate, Gc. It is a material property.

Gc = 2 γ.

If a component is to be subjected to a particular stress level in service, then the concept emerges of a critical flaw size, which must be present in order for fracture to occur

\[\mathop c\nolimits_* \approx \frac{{2\gamma E}}{{\pi {\sigma _0}^2}}\]

IT resources

(A) Description of the demonstration

Description of how the demonstration is carried out, including identification of critical points to ensure success, any safety issues etc.  Include also any hints about audience participation, use of volunteers etc.

You can start with slides (see at the IT resources: TWC_crack_growth.ppt).

Slide 1. Crack Growth in Balloons – Stresses in a Pressurised Tube. To describe this slide you can use the Scientific Background / Stresses in a Thin-walled Tube. It’s important to note such points as equations for deriving of hoop and axial stresses, internal pressure of tube and thickness of wall.

Slide 2. Crack Growth in Balloons – Griffith Energy Balance. If you like you can use the Scientific Background / Energetics of Crack Advance. Probably it’s better to highlight the equation for the strain energy release rate in order to make theme clear to understand. Don’t forget to mention the critical flaw size for rubber that is used in demonstration.

Experiment. Now you need the volunteer. Firstly show the fast fracture on one of balloons and poke it. Ask your volunteer to help you and hold the balloon in horizontal way.

Then the fracture behaviour may be shown on balloon with sticky tape. So put the tape on balloon with longest side to the horizontal axis. After creating the flaw make volunteer to pay attention for crack propagation, it would be seen under tape.

The last one should be carried out with tape putted on the balloon across, i.e. along vertical axis. Obviously, in this case it would explode faster. Thank your volunteer for help J So now it’s time to explain what did happen with stresses during the experiment.

Slide 3. Crack Growth in Balloons – Inhibition of Strain Energy Balance. Although there are two comparative pictures, it’s worth to note that the tape constrains the relaxation of rubber and reduces the stress-free region. Due to this point the energy release rate also decreases.

Hints

  • It’s really important to stick the tape properly. Be sure there are no gaps, bubbles and gathers.
  • It’s better to poke balloon in centre of tape in order to crack propagation extends at both ends simultaneously.
  • Look to keep volunteer quite far from needle and not too close to balloon. Because he will be badly surprised when it would explode loud. Or you can warn him to be aware.

Video of the demonstration

Academic consultant: Bill Clyne (University of Cambridge)
Content development: Ayuna Merinova
Photography and video: Steve Penney, Jess Gwynne
Web development: Lianne Sallows and David Brook

This DoITPoMS LDP was funded by the UK Centre for Materials Education and the Department of Materials Science and Metallurgy, University of Cambridge.