Continuum Thermo-Mechanics Fundamentals

We assume that at the continuum scales of interest here, the mass density ⇢ as well as the the momentum per unit volume ⇢u can be interpreted as well-defined averages of microscopic masses and momenta. That is, they are averages over minute spatial regions and time intervals, which can nevertheless be chosen large enough that there are negligible statistical fluctuations about those averages at a given x,t in di↵erent macroscopic realizations of the same deformation or flow process. Accordingly, u must then be interpreted as the mass averaged velocity. When studying the motion of a surface S in the continuum which is said to move with the material, we will interpret that as moving with the local velocity u. Correspondingly, then, in writing the law of conservation of, e.g., linear momentum for a region of material momentarily occupying volume V , the surface flux rate of momentum T (where T is the stress vector along the bounding surface S of V ), is understood to include contributions not just from forces acting on S but also from transfers of momentum across S associated with microscopic scale motions relative to the mass-averaged motion u.