When materials are heated or cooled, they naturally expand or contract. If this dimensional change is restrained, internal forces arise that can significantly influence structural integrity. The resulting internal force per unit area is known as thermal stress. For a bar that is fixed at both ends so it cannot elongate freely, the thermal stress is given by the relation , where is Young's modulus, the coefficient of linear thermal expansion, and the change in temperature. This calculator allows you to solve for any one of these quantities when the other three are known.
The formula derives from combining Hooke's law for linear elasticity with the definition of thermal strain. If an unconstrained bar of length experiences a temperature change , its natural length becomes . Preventing that change generates a mechanical strain . Hooke's law then states , leading directly to the stress expression above. Although the derivation is simple, understanding each component provides insights into how material properties dictate thermal response.
In practical engineering, thermal stress analysis is essential for everything from bridges and pipelines to microelectronic circuits. Consider a steel rail on a hot summer day. Steel has a typical Young's modulus of around and a linear expansion coefficient near . If the temperature rises by 40°C and the rail is constrained from expanding, the thermal stress reaches approximately 96 MPa. Such stresses can approach a significant fraction of the material's yield strength, explaining why expansion joints are crucial in railway design.
To use the calculator, enter values for any three parameters and leave the remaining field blank. The script verifies that exactly one field is empty. If the stress is blank, it multiplies modulus, expansion coefficient, and temperature change. If the modulus is blank, it divides stress by the product of coefficient and temperature change. The same rearrangement occurs to solve for the coefficient or temperature change. Units must be consistent: modulus and stress in pascals, coefficient of thermal expansion in per-degree Celsius (or per kelvin), and temperature change in degrees Celsius. Because a temperature difference is the same in kelvin, you can input data in kelvin if desired.
Thermal stress can be compressive or tensile depending on whether the material wants to expand or contract relative to its constraints. Heating a fixed bar generates compression, whereas cooling induces tension. Sign conventions vary, so this calculator returns magnitudes without sign; apply the appropriate sign based on context. The direction of stress affects whether cracks form or buckling occurs, so engineers must consider geometry and support conditions carefully when interpreting results.
Young's modulus represents the stiffness of a material in the linear elastic regime. Metals like steel and titanium have high moduli, meaning they resist deformation and consequently develop large stresses for a given strain. Polymers and rubbers have much lower moduli, so the same thermal strain produces smaller stresses. Coefficient of thermal expansion captures how much a material naturally changes length per degree of temperature change. Materials with small coefficients—such as Invar alloys—are prized for precision instruments because they resist thermal distortion.
Temperature change, though seemingly simple, can be surprisingly complex in real-world scenarios. Components may experience gradients where one side is hotter than the other, leading to bending in addition to axial stress. Rapid heating or cooling can also cause transient stresses before the entire body reaches equilibrium temperature. This calculator assumes uniform temperature change throughout the object; for nonuniform conditions, finite element analysis or more advanced heat transfer calculations are required.
The implications of thermal stress extend beyond static structures. In electronics, repeated heating and cooling cycles during operation cause solder joints to fatigue, eventually leading to failure. Turbine blades in jet engines endure intense thermal loads and must be made from materials that withstand both high temperatures and large thermal gradients. On a smaller scale, everyday objects like glass cookware can shatter if exposed to sudden temperature changes because the induced stress exceeds the material's tensile strength.
While the formula may appear straightforward, engineers often incorporate safety factors and consider nonlinear effects. For large temperature ranges, the coefficient of thermal expansion may vary with temperature, and materials can enter plastic or creep regimes where Hooke's law no longer applies. Nonetheless, the linear model remains a powerful first approximation that guides design decisions and helps identify situations requiring more detailed analysis.
Below is a table with example values for several materials, illustrating how differing moduli and expansion coefficients influence thermal stress for a 50°C temperature rise. The table assumes perfectly constrained expansion and demonstrates the wide range of stresses that can occur.
| Material | Young's Modulus (GPa) | α (10⁻⁶/°C) | Stress for ΔT=50°C (MPa) |
|---|---|---|---|
| Steel | 200 | 12 | 120 |
| Aluminum | 70 | 23 | 80.5 |
| Brass | 100 | 19 | 95 |
| Invar | 141 | 1 | 7.05 |
Notice how Invar, with its exceptionally low expansion coefficient, develops minimal stress compared with more common metals. This characteristic makes it suitable for precision applications such as clock pendulums and surveyor's tapes. Conversely, aluminum's high expansion coefficient leads to significant stress even though its modulus is lower than steel's.
In summary, thermal stress arises whenever a material's temperature changes and its expansion or contraction is restrained. By combining elastic properties with thermal expansion data, the simple relation enables quick estimates that inform engineering judgment. Use this calculator to explore how material choice and temperature swings interact, and remember that the underlying concepts tie directly to the microscopic motions of atoms vibrating about their equilibrium positions.