  The multiplicative decomposition model is widely employed for predicting
residual stresses and morphologies of biological tissues due to growth.
However, it relies on the assumption that the tissue is initially in a
stress-free state, which conflicts with the observations that any growth state
of tissue is under a significant level of residual stresses that helps to
maintain its ideal mechanical conditions. Here, we propose a modified
multiplicative decomposition model where the initial state (or reference
configuration) of biological tissue is endowed with residual stress instead of
being stress-free. Releasing theoretically the initial residual stress, the
initially stressed state is first transmitted into a virtual stress-free state,
resulting in an initial elastic deformation. The initial virtual stress-free
state subsequently grows to another counterpart with a growth deformation and
the latter is further integrated into its natural configuration with an
excessive elastic deformation that ensures tissue compatibility. With this
decomposition, the total deformation may be expressed as the product of elastic
deformation, growth deformation and initial elastic deformation, while the
corresponding free energy density depends on the initial residual stress and
the total deformation. We address three key issues: explicit expression of the
free energy density,predetermination of the initial elastic deformation, and
initial residual stress. Finally, we consider a tubular organ to demonstrate
the effects of the proposed initial residual stress on stress distribution and
on shape formation through an incremental stability analysis. Our results
suggest that the initial residual stress exerts a major influence on the growth
stress and the morphology of tissues.
