  Thermoelastic loss is an important energy dissipation mechanisms in resonant
systems. A careful analysis of the thermoelastic loss is critical to the design
of low-noise devices for high-precision applications, such as the mirrors used
for gravitational-wave detectors. In this paper, we present analytical
solutions to the thermoelastic loss due to thermoelasticity between different
materials that are in contact. We find expressions for the thermoelastic loss
of multimaterial coatings of finite substrates, and analyze its dependencies on
material properties, mirror design and operating experimental conditions. Our
results show that lower operating mirror temperature, thinner layers and higher
number of interfaces in the coating, and the choice of the first layer of the
coating that minimizes the thermal expansion mismatch with the substrate are
strategies that reduce the thermoelastic loss and, therefore, diminish the
thermal noise that limits the resolution in sensing applications. The results
presented in this paper are relevant for the development of low-noise
gravitational-wave detectors and for other experiments sensitive to energy
dissipation mechanisms when different materials are in contact.
