As an active research area in biomechanics, growth mechanics studies mechanical aspects of growth (and resorption) of biological tissues [1]-[5]. While the classical theory of mechanics has provided useful tools in such studies, the physiological process of growth in turn presents special challenges to the classical theory of continuum mechanics. For solids, the kinematical theory in continuum mechanics is based on a deformation function which maps a fixed reference configuration of the body to the current configuration, and the constitutive equation relates the stress in the current configuration to certain kinematical quantities defined through the reference configuration. These and some other fundamental concepts of continuum mechanics are not adequate to describe growth, especially surface growth where a growth body may undergo changes in topology.

In this talk, we address some of these issues, and develop a growth theory using consistent and clean kinematical and constitutive equations. The kinematics of a growing body is described by the velocity field and a growth rate function defined on the region occupied by the body at the current time. Also defined on this region is the mass density field. An equation for the balance of mass is derived with these functions. To describe surface growths as well as volume growths, the growth rate function may be singular on a surface in the current configuration, while the velocity field and mass density field may suffer jump discontinuities across the surface. They are weak solutions of the equation of mass balance.

For constitutive equations, we discard the standard practice of using a fixed reference configuration, and use the current configuration as the reference. In contrast to the constitutive equations for an elastic body in the classical continuum mechanics, the response function of a growing elastic body in the present theory gives the Cauchy stress when the body in the current grown/deformed configuration is subjected to an imaginary further elastic deformation.  In particular, it gives the Cauchy stress at the current configuration when this imaginary deformation is the identity deformation. The form of such a response function changes constantly with growth and deformation even when the intrinsic mechanical properties of the material remain unchanged. The evolution equation for the response function is derived by considering a sequence of growth/deformation processes. Again, this evolution equation may admit weak solutions for surface growths.

The balance of mass, the evolution equation for the response function, and the usual balance of linear momentum equation, along with appropriate initial and boundary conditions, can be solved to yield the velocity field, the mass density, the stress field, and the response function.  In particular, one can find residual stresses due to growth when the body is subjected to zero applied load.