This summer I worked with Dr. Janice Pawloski of the Department of Physics and Engineering at Hope College to research the Temperature Distribution of Stress and Strain in Materials. Our goal was to theoretically find the most favorable strain conditions in a hollow cylinder caused by thermal stresses in various materials.

Applying a temperature change on the boundaries of a hollow cylinder causes thermal strains in that object which can lead to deformation, thermal stresses and fracturing. The behavior of a material undergoing thermal conduction is characterized by thermal conductivity, k. It provides an indication of the rate at which energy is transferred. k depends on the physical structure of the material, atomic and molecular, related to the state of the matter. For example, the thermal conductivity for ceramic is less than that of steel because heat travels slower in ceramic materials.

To find the strain we first need to find the temperature distribution in a material. Temperature distribution is how temperature varies with position in the medium. We formulated conduction equations whose solutions describe the temperature distribution of that material. We used Fourier’s Law in cylindrical coordinates as our conduction equations in 1-D:

To see the differences, we looked at three cases of material, in which k varies.

First, the cylinder is made of a homogeneous material where k = constant. After including k into the conduction equation we solved for r and applied the boundary conditions T(b=0.2) = 0 and T(c=1) = 1.

We found that the temperature distribution increases logarithmically. The values eventually level off because heat transferred per unit area decreases as the area increases. The strain also increases like the temperature distribution and illustrates that the cylinder expands as the temperature change is applied.

In the next case, the cylinder is layered, steel surrounded by ceramic. We want to keep the temperature in the steel as low as possible and therefore insulate it with ceramic. With two materials, we have two values of k and therefore two equations to solve for.

Again, after including the same boundary conditions we find that the temperature in the steel part distributes rapidly in comparison to the ceramic portion. Although this is a good temperature distribution because the steel temperature remains low, the strain behavior for this case is poor. The steel expands more for a given temperature, causing debonding and cracking of the cylinder.

In the final case, the cylinder is made up of a functionally graded material. A functionally graded material is one that varies continuously from one material composition to another as a function of position, where k is a function of r. Since the material is functionally graded, k represents a combination of steel and ceramic at a specified point. We chose k(r) = 1/(f + a r¹⁰). Again we included it into the conduction equation, solved for temperature and applied the same boundary conditions as before.

We found that the temperature distribution, similar to the second case, for steel distributes more rapidly than the ceramic. This is a good temperature distribution because the temperature value for steel remained low. The strain for the functionally graded material did not cause any debonding or cracking, unlike the second case. The temperature distributed more uniformly allowing the cylinder to expand uniformly.

Overall, we found that the strain on a material depends on the temperature distribution. Also we concluded that the temperature distribution depends on the thermal conductivity, k. the best strain and temperature distribution outcome wasfor the functionally graded material, k a function of r.