Fracture Mechanics

Patrick Carmody's picture Patrick Carmody

Hope College

Dr. Janice Pawloski

Supported by the NSF-REU

As part of the Hope College summer research program, I spent two months this summer doing research in an area of engineering known as fracture mechanics. I worked with Dr. Pawloski, professor of engineering at Hope. My specific problem was to solve for the temperature within a given body, in preparation to solve for the stresses due to a crack in the body. Funding for my work was provided by REU, in cooperation with Hope College.

Fracture mechanics is the study of how cracks propagate through materials, and how cracks in materials affect the amount of load that the material can hold. The area of fracture mechanics was first developed during the second world war, when boats known as Liberty Ships began to develops fractures, which sometimes caused the ships to break right in two. Many people, especially the government, wanted to know what was causing these cracks, and how they could be avoided and, more importantly, handled with out replacing the cracked material.

Most of fracture mechanics is focused on finding the stress intensity factor at the crack tip, which is a measure of the amount that the material is weakened by having a crack. The stress intensity factor, or SIF, is a function of the stresses and strains within the material, which in turn are dependent on the geometry and temperature of the material. The first step to solving for the SIF, is finding the temperature throughout the body, given the geometry of the material and certain boundary conditions. Then, one can solve for the stresses and strains in the material, and finally calculate the SIF. My work was centered on the first step: finding the temperature inside the material at all points.

The specific problem that I dealt with involved a cylinder, under axial loading, with a circumferential crack. The following boundary conditions were given:

dT/dz = 0 for z = 0, 0 < r < d
T = g(r) for z = 0, d < r < 1
T = f(z) for r = 1, 0 < z < oo

The functions g(r) and f(z) are the initial temperatures on the surface of the material, and inside the crack itself. The body was also assumed to be steady state, meaning that the temperatures have had time to come to equilibrium, and aren't dependent on time. It was also assumed that there was no heat generated inside the material. These assumptions lead to the differential equation del-squared(T) = 0. This equation can be solved using a sine transform method. When it is solved, two unknown equations, A(C) and B(C), appear. We know that B(C) must have the form of the integral of Tau(t) times sin(C*t) with respect to t. Knowing this, we can solve for A(C), and back substitute A(C) and B(C) into the equations from solving the DE. After substituting and solving for the unknown function Tau, we get:

                                  1
                                 /`
                                 |   
        Tau(r) = H(r) + lambda * | Tau(t) * K(t) dt
                                ,/
                                 r

Notice the equation is solved for Tau, and Tau also appears inside the integral. This type of equation is known as an integral equation, and is an analogous form of a differential equation. This integral equation does not have a closed form, so we resort to solving it by numerical means, using a computer.

In order to solve the problem numerically, a FORTRAN program was written, and a commercial routines called NAG routines were used to do the integration and solving of the integral equation. The temperature, being a function of Tau, was plotted using MATLAB, once Tau had been solved for. For the following plots, f(z) = 0, d = 0.5, and three functions were used for g(r), as noted.

Notice that at r = 1.0, the temperature is 0, as dictated by f(z). Also, between r = d and r = 1.0 at z = 0, the temperature approaches the function g(r), although numerical round-off prevented the temperature from being evaluated very close to z = 0. These two result lead me to believe that the values calculated for the temperature are accurate. The next step, then, is to use these values to calculate the stresses and strains inside the body, but that is left to the next researcher.

I would like to thank Hope College, REU, and Dr. Pawloski for their support in my research. It has been a wonderful learning experience, and I hope that this research will be continued in the future.

99carmodypm@alma.edu