There are two types of temperature-related material coefficients: one is related to the mechanical properties of the material and the other is associated with heat conduction. The former includes factors such as E, G, v, a, while the latter consists of C (specific thermal capacity), ρ (density) and k (thermal conductivity).
These coefficients are not constant, but vary with temperature. However, when the temperature is not high, its average values are often treated as constants. In situations of high temperature or significant variation, it is essential to consider its variations with temperature.
1. Relationship between elastic coefficients and temperature
The modulus of elasticity E and the shear modulus G of metals decrease with increasing temperature, while Poisson's ratio v changes little with temperature. Measurements of E and G with temperature can be made statically or dynamically.
The static method involves testing in a high-temperature furnace using load, while the dynamic method uses vibrational or ultrasonic pulse techniques.
The vibrational method allows the test sample to undergo elastic vibration in the high-temperature furnace, with the elastic constants determined by frequency measurement.
The ultrasonic method involves applying ultrasonic waves to the test sample, and E, G and v are determined by measuring the propagation speed of the waves.
2. Relationship between Heat Coefficient and Temperature
The heat coefficient of metallic materials generally exhibits a linear relationship with temperature. The coefficient of linear expansion α tends to increase linearly as the temperature increases, while the thermal conductivity k decreases as the temperature increases, and the specific heat capacity increases with temperature.
The slope of the line or curvature of the curve representing the relationship between heat coefficient and temperature, measured by experimental tests, reveals how the heat coefficient of the specific material changes with temperature.
For example, the variation in carbon steel's heat coefficient with temperature is represented in the following graph, derived from various data sources.
3. Thermal fatigue of materials
As the temperature of ductile materials increases, they will not fail immediately, even if the stress they are subjected to exceeds the yield strength. However, even if the stress level is low, if considerable temperature changes are repeated, they will eventually fail due to fatigue, resulting in cracks. This phenomenon is known as thermal fatigue.
Consider a test bar fixed at both ends, subjected to repeated heat cycles between the highest and lowest temperatures, as illustrated in the following diagram.
Suppose at the beginning of the experiment the rod is fixed at the highest temperature and then cooled to generate tensile stress, OAF represents a line of stress change. If reheated, the stress-strain curve initially moves parallel to OA downward, yielding to a stress lower than the tensile force of the cooling cycle, eventually reaching point E.
If kept at the highest temperature for some time, the tension relaxes, resulting in a decrease in the compressive stress, reaching point E'. If cooling is resumed, it rises along E'F', reaching point F' at the lowest temperature.
As pressure relaxation does not occur at the lower temperature, if reheating begins, the curve drops along F'E”, reaching point E” at the higher temperature. Due to stress relaxation, the stress reduces and moves to point E”', if cooling is resumed, it follows the E”'F” curve reaching point F” at the lowest temperature.
If this cooling and heating cycle is repeated, the stress-strain curve traces a hysteresis loop each time, the associated recovery plastic deformation being the cause of thermal fatigue. The maximum and minimum temperatures of the thermal cycle, the average temperature, the residence time at the maximum temperature, the repetition speed and the elastic-plastic properties of the material are factors that affect thermal fatigue.
The intensity of thermal fatigue refers to the relationship between the plastic deformation of a cycle ε P and the number of repetitions N to reach failure. According to the empirical Manson-Coffin formula:
Here, ε f denotes the elongation at the point of failure of the material during a static tensile test at the average temperature of a thermal cycle.
The above-mentioned description only refers to the unidirectional thermal stress fatigue of a material. However, thermal fatigue in real structures is multidirectional and constitutes a specialized field of study.
