The linking of solid mechanics with thermodynamics through the mathematics of differentials

Engineers are familiar with the concept of differentials but tend to think of these in an informal sense, that is, as infinitesimally small quantities. It is argued in this paper that differentials should be presented to engineering students in a more formal way. Differentials can be formalised by defining then as functions on tangent spaces. This enables their numerical evaluation and makes clear the nature of their behaviour when restricted to tangent subspaces. It is shown that through a proper understanding of differentials the physics of many processes can be more fully appreciated. Inexact and exact differentials and their link to path dependency are examined in the context of state spaces. The work differential is considered and it is shown how knowledge of the state space combined with a simple test for path dependency provides real insight. Differentials allow the unification of solution methodologies for thermodynamics and solid mechanics. Work differentials and state spaces associated with elasticity and plasticity theory are examined in the paper. Reversibility in a one-dimensional state space provides for the explicit formulation of work and is applicable to elasticity. The use of monotonicity in replace of reversibility is examined for plasticity theory. The idea of conservative and non-conservative forces in mechanics is also considered in the context of state spaces and differentials. Examples are presented in the paper to illustrate the wide applicability of differentials across subject boundaries.