6.3 Equations for a Growing Bar
255
6.2 A Few Words About Growth Theories
Most mathematical theories for growth have been developed during the past few
decades. Today, the most popular theory for volumetric growth was formulated by
Rodriguez, Hoger, and McCulloch (RHM). Combining ideas on growth mechanics
originally developed by Skalak and coworkers (Skalak 1981; Skalak et al. 1982;
Tozeren and Skalak 1988) with concepts from elastoplasticity theory, these investigators formulated a general 3D theory for finite volumetric growth of pseudoelastic
soft tissues (Rodriguez et al. 1994). Although other theories for growth exist, the
fundamental ideas of RHM theory underlie most analyses of growth, remodeling,
and morphogenesis used throughout this book.
Adding volume creates new points in a body, while subtracting volume removes
points. In this respect, RHM theory violates a central tenet of continuum mechanics,
whereby each point in the undeformed body should have one and only one
corresponding point in the deformed body. The mapping r = r(R) embodies this
assumption. While the significance of this issue is still being debated, RHM theory
has been used successfully to model growth in numerous tissues.
To avoid this dilemma, some researchers have proposed alternative approaches
that incorporate growth into a mixture formulation. The theory for remodeling
discussed in the next chapter treats tissues as mixtures composed of different types
of solid constituents that are created and degrade with time (Humphrey 1999;
Humphrey and Rajagopal 2002, 2003). If the synthesis and degradation rates are
not equal, the mass of the tissue changes. If the mixture also contains fluid, volume
changes can occur by water flowing into or out of the tissue (Ateshian 2007).
Ultimately, these theories seem to provide a promising way forward, although
models based on mixture theory are generally more difficult to solve numerically
than models based on elasticity theory. Future work could lead to improved theories
for growth and remodeling, perhaps by combining various ideas.
6.3 Equations for a Growing Bar
The RHM theory for growth is based on the concept of evolving zero-stress
configurations, the same idea used in the previous chapter to simulate contraction.
One important difference is that, whereas the volume of a tissue changes relatively
little, if at all, during contraction, the same tissue may grow to many times its
original size.
The interactions between mechanics and growth can be quite complex and are not
always intuitive. To help understand fundamental behavior, this section examines
a series of relatively simple problems involving growing bars. First, however, we
discuss how the equations of elasticity theory are modified to accommodate growth.
255
6.2 A Few Words About Growth Theories
Most mathematical theories for growth have been developed during the past few
decades. Today, the most popular theory for volumetric growth was formulated by
Rodriguez, Hoger, and McCulloch (RHM). Combining ideas on growth mechanics
originally developed by Skalak and coworkers (Skalak 1981; Skalak et al. 1982;
Tozeren and Skalak 1988) with concepts from elastoplasticity theory, these investigators formulated a general 3D theory for finite volumetric growth of pseudoelastic
soft tissues (Rodriguez et al. 1994). Although other theories for growth exist, the
fundamental ideas of RHM theory underlie most analyses of growth, remodeling,
and morphogenesis used throughout this book.
Adding volume creates new points in a body, while subtracting volume removes
points. In this respect, RHM theory violates a central tenet of continuum mechanics,
whereby each point in the undeformed body should have one and only one
corresponding point in the deformed body. The mapping r = r(R) embodies this
assumption. While the significance of this issue is still being debated, RHM theory
has been used successfully to model growth in numerous tissues.
To avoid this dilemma, some researchers have proposed alternative approaches
that incorporate growth into a mixture formulation. The theory for remodeling
discussed in the next chapter treats tissues as mixtures composed of different types
of solid constituents that are created and degrade with time (Humphrey 1999;
Humphrey and Rajagopal 2002, 2003). If the synthesis and degradation rates are
not equal, the mass of the tissue changes. If the mixture also contains fluid, volume
changes can occur by water flowing into or out of the tissue (Ateshian 2007).
Ultimately, these theories seem to provide a promising way forward, although
models based on mixture theory are generally more difficult to solve numerically
than models based on elasticity theory. Future work could lead to improved theories
for growth and remodeling, perhaps by combining various ideas.
6.3 Equations for a Growing Bar
The RHM theory for growth is based on the concept of evolving zero-stress
configurations, the same idea used in the previous chapter to simulate contraction.
One important difference is that, whereas the volume of a tissue changes relatively
little, if at all, during contraction, the same tissue may grow to many times its
original size.
The interactions between mechanics and growth can be quite complex and are not
always intuitive. To help understand fundamental behavior, this section examines
a series of relatively simple problems involving growing bars. First, however, we
discuss how the equations of elasticity theory are modified to accommodate growth.