# Crack tip opening displacement

Crack tip opening displacement (CTOD) is the displacement at the original crack tip and the 90° intercept.

LEFM (Linear elastic fracture mechanics) is not valid. This assumption was proven when a unique relationship between CTOD and the J integral was established.[1]

## Relationship between K and CTOD

CTOD can be expressed in terms of K as ${\displaystyle {\frac {K^{2}}{\sigma _{y}Em}}}$[2] m=1 for plane stress and 2 for plane strain

## Relationship between J-integral and CTOD

For Linear Elastic Conditions, the relationship between CTOD and G is given by an equation[3] between CTOD and J in the limit of Small-Scale Yielding.

## Relationship between G and CTOD

Since J = G for linear elastic material behavior, these equations also describe the relationship between the G and CTOD also.

## Testing

A CTOD test is usually done on materials that undergoes plastic deformation prior to failure. The testing material more or less resembles the original one, although dimensions can be reduced proportionally. Loading is done to resemble the expected load. More than 3 tests are done to minimize any experimental deviations. The dimensions of the testing material must maintained proportionality. The specimen is placed on the work table and a notch is created exactly at the centre. The crack should be generated such that the defect length is about half the depth. The load applied on the specimen is generally a three-point bending load. A strain gauge is used to measure the crack opening. Crack tip plastically deforms until a critical point after which a cleavage crack is initiated that may lead to either partial or complete failure. The critical load and strain gauge measurements at the load are noted and a graph is plotted. The crack tip opening can be calculated from the length of the crack and opening at the mouth of the notch. According to the material used, the fracture can be brittle or ductile which can be concluded from the graph.

### Laboratory measurement

Early experiments used a flat, paddle-shaped gauge that was inserted into the crack; as the crack opens, the paddle gauge rotates and an electronic signal is sent to an x–y plotter. This method was inaccurate, however, because it was difficult to reach the crack tip with the paddle gauge. Today, the displacement V at the crack mouth is measured and the CTOD is inferred by assuming that the specimen halves are rigid and rotate about a hinge point.[4]

## Application

Examination of fractured test specimens led to the observation that the crack faces had moved apart prior to fracture, due to the blunting of an initially sharp crack by plastic deformation. The degree of crack blunting increased in proportion to the toughness of the material.[5] This observation led to considering the opening at the crack tip as a measure of fracture toughness. This parameter became known as CTOD. Irwin later postulated that crack-tip plasticity makes the crack behave as if it were slightly longer. Thus, estimation of CTOD can be done by solving for the displacement at the physical crack tip.

## Use as a design parameter

CTOD is a single parameter that accommodates cracktip plasticity. It is easy to measure when compared with techniques such as J integral. It is a fracture parameter that has more physical meaning than the rest.

However, the equivalence of CTOD and J integral is proven only for non-linear materials, but not for plastic materials. It is hard to expand the concept of CTOD for large deformations. It is easier to calculate J-integral in case of a design process using finite element method techniques.

## References

1. ^
2. ^ Anderson 2005, pp. 104-105.
3. ^ Anderson 2005, pp. 120-122.
4. ^ B E Amstutz, M A Sutton, D S Dawicke"An Experimental study of CTOD for mode I/mode II stable crack growth in thin aluminium specimens", ASTM Special 1995
5. ^ J C Newman, M A James, U Zerbst, "Engineering Fracture mechanics", Elsevier 2003