A very readable introduction to the basic principles of continuum mechanics, with a focus on clearly explaining how models of solid deformation and fluid flow can be derived from assumptions concerning conserved quantities and constitutive laws. This is an excellent book for introducing tensor calculus and kinematics, although it doesn't go into any of the differential geometry required for unusual geometries, and it doesn't give any practical examples of solving the equations that it formulates. (Review by Cameron Hall).
Applied Solid Mechanics by P. Howell, G. Kozyreff and J. Ockendon (2009).
Another excellent book that introduces the equations of elasticity, and then clearly describes mathematical techniques used to construct analytical or approximate solutions to the equations of elasticity in a wide range of common and useful situations (antiplane strain, plane strain, torsion, simple rods and shells, simple contact problems etc.). While there isn't much about numerical methods, it's a wide-ranging and practical guide to exploiting simplifications that can be made if certain assumptions hold. (Review by Cameron Hall).
