Preamble

In this chapter we address fundamental aspects of continuum modelling in respect of kinematics, mass conservation, balances of linear and rotational momentum, and balance of energy.

After considering the role of mass density in modelling the presence of ‘matter’, we discuss the manner in which the detailed macroscopic distortion of any material body can be monitored. This is markedly different for solids and fluids, but in both cases it is possible to motivate the notion of material point and thereby establish basic kinematic concepts such as deformation, motion, and velocity. The formal (axiomatic) approach to kinematics is outlined for comparison. Mass conservation is motivated for solids and postulated to hold in general. Dynamical considerations are first addressed for a body as a whole. In addition to tractions on boundaries, the possibility of surface and body couples is considered. Global balances of linear and rotational momentum are postulated and applied to rigid bodies both to emphasise their often-neglected status as a special case of material continua and to develop familiarity with notation, concepts and basic manipulations. Local forms of balance are derived in standard fashion by postulating balances for matter in arbitrary subregions of the region instantaneously occupied by the body, invoking a transport theorem, and then establishing the existence of stress and couple stress tensors and a heat flux vector. It is these local forms of balance that can be derived directly from molecular considerations using the weighting function methodology to be introduced in Chapter 4.