The aim of our contribution is to draw attention to the fact, which so far has rarely absorbed theoreticians studying continuous media, that a moving surface may carry not only disturbances, but also physical properties different from those of the surrounding media. Here, we present a unified view of the theory of non-relativistic thermodynamics incorporating phenomena with singularities. These singularities will present as discontinuous functions or their derivatives, and in the form of the discontinuity in respect to the Lebesgue measure of physical quantities. A special focus exists on the fracture at interfaces. Current topics include the role of thermal residual or processing induced stresses, the detailed role of plasticity, and geometric effects on interface crack driving forces. We model a situation of this kind by the movement of a surface separating two well-behaved material media, while attributing to the surface the physical properties of a phase change. For a proper understanding of interfacial (transport) processes, one needs to be familiar with the basic geometrical description of a surface, we present here.