# How do we calculate a crack?

The ways to calculate a crack, how it begins, how it grows, and if it stops, are all part of the field called fracture mechanics. There are many, many ways to try to calculate a crack. In fact, people are still fighting about which way is the right way. The right way really depends on what kind of material and what type of structure you're talking about. Here we'll discuss a version of fracture mechanics which can be applied to large concrete structures, like dams.

First some general ideas in fracture mechanics.

### Forces

There has to be some sort of force acting on a material to make it fracture. There are three major forces to consider:

• temperature variation -- this can be a big problem in concrete. As you learned earlier in the section on concrete, when concrete is mixed it gets very hot and expands. When it tries to cool off, the concrete wants to contract. If it can't contract (because it is connected to another wall or rock) then cracks form.
• chemical reaction -- a reaction between the cement and the aggregate in the concrete which makes the concrete "grow"
• live load -- the general pulling or pushing on a material. In a dam, loads could be caused by settling or sliding of the foundation or abutments, ice, silt, waves, earthquakes, or even bombs or asteroids!
Find out more on the forces on arch, buttress, gravity, and embankment dams.

### Modes

These forces are simplified into three different "modes", basically three ways the forces can act and what kind of fracture they cause:

• Mode I: the forces are perpendicular to the crack (the crack is horizontal and the forces are vertical), pulling the crack open. This is referred to as the opening mode.

What would happen if both of the forces were pushing down on the crack? Nothing. This would close the crack.

• Mode II: the forces are parallel to the crack. One force is pushing the top half of the crack back and the other is pulling the bottom half of the crack forward, both along the same line. This creates a shear crack: the crack is sliding along itself. It is called in-plane shear beacuse the forces are not causing the material to move out of its original plane.

In this case, what would happen if both the forces were moving in the same direction, both forward or both backward? This would not cause the crack to grow, since all of the material would be moving in the same direction.

• Mode III: the forces are perpendicular to the crack (the crack is in front-back direction, the forces are pulling left and right). This causes the material to separate and slide along itself, moving out of its original plane (which is why its called out-of-plane shear). The forces could also be pushing left and right and the same effect would occur. But the forces have to be moving in opposite directions in order to grow the crack.

You can try to create each of these modes yourself!

More than one of these modes can happen at once, which causes "mixed-mode" cracking. But we'll just concentrate on one mode, Mode I, to learn some of the basics of fracture mechanics.

Mode I cracking happens in concrete dams. If there is an existing crack on the upstream side, water can enter and open the crack:

### KI and KIc

How do we describe what's happening during Mode I cracking? To help us do this, there is something called a stress intensity factor, which has the symbol KI.

KI is basically a measure of the likelihood that the crack will grow when the opening forces are being applied. Think of KI as as the crack's money: if the crack has enough money, then it can do what it wants, which is grow. How much money the crack has, KI is based on the load on the structure and the initial size of the crack.

Now KIc is a material's resistance to crack growth. KIc tells you when the material will allow a crack to grow. This is a property which can be measured in a laboratory. We'll leave it at that for now.

### Energy

Another way to think about cracks is in terms of energy. What kind of energy and where does it come from, you say? When a material is being pulled, like this one, the material gets some energy. When a dam is being pushed on by the reservoir, the dam must do work to hold the water back. This work is the stored energy in the dam.
Cracks use up some of this stored energy when they grow in a dam.

Where do cracks start in dams?