Dimensional analysis and inertial effects in transient thermal stress analysis

Thermoelasticity or thermally-induced stress is an important scientific problem across many industries. These problems become even more challenging as devices become smaller and smaller, and are required to withstand large temperature gradients in small volumes. Non-dimensionalization or dimensional analysis for the transient form of Navier's equation of elasticity coupled with the constitutive relationship that relates stress to strain and temperature reveals a dimensionless parameter for the characteristic ratio of inertial forces to thermoelastic forces. This ratio, referred to as the thermal inertial mass effect (T.I.M.E.) ratio, quantifies the relative importance of these two forces in a medium. As this ratio increases, the relative importance of inertial forces increases, and the medium's resistance to movement increases. Transient numerical modelling for a simplified model problem demonstrates the effects of varying this characteristic ratio on the thermoelastic stress and deformation in a medium.