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Elastic-Plastic Fracture Mechanics: J-Integral Experiments


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 Jc). 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 JC 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.


More to explore

LLDPE Material Modeling

Demonstration of how you can use the PolyUMod Material Database model for LLDPE. This is an excellent model for generic LLDPE.

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

  1. Hi, 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?

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