is the length before the change of temperature and is the length after the change of temperature. For most solids, thermal expansion is proportional to the change in temperature: Thus, the change in either the strain or temperature can be estimated by: where is the difference of the temperature between the two recorded strains, measured in degrees Fahrenheit, degrees Rankine, degrees Celsius, or kelvin, and is the linear coefficient of thermal expansion in "per degree Fahrenheit", "per degree Rankine", "per degree Celsius", or "per kelvin", denoted by , , , or , respectively. In the field of continuum mechanics, thermal expansion and its effects are treated as eigenstrain and eigenstress. Area expansion The area thermal expansion coefficient relates the change in a material's area dimensions to a change in temperature. It is the fractional change in area per degree of temperature change. Ignoring pressure, we may write: where is some area of interest on the object, and is the rate of change of that area per unit change in temperature. The change in the area can be estimated as: This equation works well as long as the area expansion coefficient does not change much over the change in temperature , and the fractional change in area is small . If either of these conditions does not hold, the equation must be integrated. Volume expansion For a solid, we can ignore the effects of pressure on the material, and the volumetric (or cubical) thermal expansion coefficient can be written: where is the volume of the material, and is the rate of change of that volume with temperature. This means that the volume of a material changes by some fixed fractional amount. For example, a steel block with a volume of 1 cubic meter might expand to 1.002 cubic meters when the temperature is raised by 50 K. This is an expansion of 0.2%. If we had a block of steel with a volume of 2 cubic meters, then under the same conditions, it would expand to 2.004 cubic meters, again an expansion of 0.2%. The volumetric expansion coefficient would be 0.2% for 50 K, or 0.004% K−1. If we already know the expansion coefficient, then we can calculate the change in volume where is the fractional change in volume (e.g., 0.002) and is the change in temperature (50 °C). The above example assumes that the expansion coefficient did not change as