  In this paper we formulate a geometric theory of the mechanics of growing
solids. Bulk growth is modeled by a material manifold with an evolving metric.
Time dependence of metric represents the evolution of the stress-free (natural)
configuration of the body in response to changes in mass density and "shape".
We show that time dependency of material metric will affect the energy balance
and the entropy production inequality; both the energy balance and the entropy
production inequality have to be modified. We then obtain the governing
equations covariantly by postulating invariance of energy balance under
time-dependent spatial diffeomorphisms. We use the principle of maximum entropy
production in deriving an evolution equation for the material metric. In the
case of isotropic growth, we find those growth distributions that do not result
in residual stresses. We then look at Lagrangian field theory of growing
elastic solids. We will use the Lagrange-d'Alembert's principle with Rayleigh's
dissipation functions to derive all the governing equations. We make an
explicit connection between our geometric theory and the conventional
multiplicative decomposition of deformation gradient
$\mathbf{F}=\mathbf{F}_e\mathbf{F}_g$ into growth and elastic parts. We
linearize the nonlinear theory and derive a linearized theory of growth
mechanics. Finally, we obtain the stress-free growth distributions in the
linearized theory.
