## Introduction

The J-integral is a useful tool for predicting crack growth in different materials, including polymers. In this article I will discuss how you can experimentally measure the critical J-integral value using simple force-displacement experiments. See my other article for more information about the J-integral theory.

## J-Integral Definition

As I discussed in my previous article, the J-integral can be calculated from the following equation:

\( J = \displaystyle\int_{\Gamma_0} \mathbf{C} \mathbf{n} \cdot \mathbf{e}_c ds.\)

Unfortunately, this equation does not provide much information about how to experimentally measure the J-integral. To understand how to measure the J-integral we need to go back to the definition of the J-integral: \(J = -dU/dA\), that is, the J-integral is the change in potential energy during crack growth. This definition directly leads to an experimental approach for measuring the J-integral. Here are the steps:

(1) Create a tension specimens with an edge crack of length *a* (see image below). Perform a tension test on this specimen to measure the force-displacement response.

(2) Then create another tension specimen with an edge crack of length *a*+*da*. Again, perform a tension test on this specimen to measure the force-displacement response.

(3) Then plot the force-displacement curves for the two tests in the same graph (see image below). The area between the curves is the released energy due to crack growth, and the J-integral can therefore be calculated from:

\( \displaystyle J(a) = – \frac{1}{B} \frac{dU}{da},\)

where *B* is the specimen width.

## J-Integral Resistance Curve

This multi-specimen approach is described in more detail in ASTM standards (e.g. D6068). Note that the standards use specific fracture specimen geometries (three-point bend specimen, and compact tension specimen) in order to get more accurate results.

One final concept of interest is the J-Resistance curve. Let *R* be the resistance to crack extension (think of it as *J _{c}*). From before we know that a crack will grow if

*J*>

*R*, but it is not clear if the crack growth will be stable or unstable. To determine that we need to look at the derivative of the

*J*<

*R*equation with respect to crack length:

This can also be visualized graphically. For material A (see figure below), once the applied J-integral values reaches J_{C} the crack will start to grow in an unstable way. Material B, on the other hand, has a non-vertical start of the resistance curve. This means that the crack tip is blunting. Once J reaches the critical value the crack growth will be stable for this material. This commonly occurs for ductile materials.

## 1 thought on “Elastic-Plastic Fracture Mechanics: J-Integral Experiments”

mingpeiHi, Dr. Jorgen,

For the experimental measurement of J-interal, does it need to make sure the crack doesn’t grow during the stress/strain curve measurement?