260
6 Growth
For numerical calculations, it is advantageous to choose the unloaded ZSS as
the initial configuration. Then, we can set u i = 0 and G i = 1 (no growth) for all
points in the body at t = 0. Applied loads are then ramped up from zero during the
solution process. This approach sidesteps any convergence issues related to sudden
initial jumps in loading.
Growth begins with the first few cell divisions in the egg. Thus, some growth
has likely occurred prior to the chosen initial configuration. This is not a problem,
however, as we can just interpret the growth computed for t > 0 as additional
growth. On the other hand, the initial configuration likely contains stress and,
therefore, is not actually a zero-stress state. To make matters worse, determining
a truly stress-free state is a difficult, if not impossible, task in mechanobiology.
Fortunately, although initial stresses may significantly affect results near the onset
of a calculation, these “start-up conditions” generally become less important after a
sufficiently long period of time.
6.4 Fundamental Growth Mechanics
To build insight into the fundamental mechanics of growth, we consider again
growth of a rectangular bar, which can serve as a simple model for an actin fiber
or a row of epithelial cells. For convenience, let (X 1 , X 2 , X 3 ) = (X, Y, Z) and
(x 1 , x 2 , x 3 ) = (x, y, z), with X and x representing the longitudinal coordinates in
the initial zero-stress state and current state, respectively.
6.4.1 Growth of an Unconstrained Bar
If an unconstrained, unloaded bar grows uniformly, a simple equilibrium analysis
shows that the bar remains stress-free. As we will see, depending on the specific
growth pattern, differential growth, i.e., nonuniform growth, can generate stress in
the bar.
Example 6.1 An unloaded bar of initial length L 0 grows nonuniformly in the axial
direction according to the relation
G x (X, t) = 1 +
a + b(X/L 0 )
2
(1 − e
−βt ),
(6.15)
where a, b, and β are constants. This relation satisfies the initial condition
G x (X, 0) = 1. If the bar remains free of external constraints, determine the length
of the bar as a function of time.
6 Growth
For numerical calculations, it is advantageous to choose the unloaded ZSS as
the initial configuration. Then, we can set u i = 0 and G i = 1 (no growth) for all
points in the body at t = 0. Applied loads are then ramped up from zero during the
solution process. This approach sidesteps any convergence issues related to sudden
initial jumps in loading.
Growth begins with the first few cell divisions in the egg. Thus, some growth
has likely occurred prior to the chosen initial configuration. This is not a problem,
however, as we can just interpret the growth computed for t > 0 as additional
growth. On the other hand, the initial configuration likely contains stress and,
therefore, is not actually a zero-stress state. To make matters worse, determining
a truly stress-free state is a difficult, if not impossible, task in mechanobiology.
Fortunately, although initial stresses may significantly affect results near the onset
of a calculation, these “start-up conditions” generally become less important after a
sufficiently long period of time.
6.4 Fundamental Growth Mechanics
To build insight into the fundamental mechanics of growth, we consider again
growth of a rectangular bar, which can serve as a simple model for an actin fiber
or a row of epithelial cells. For convenience, let (X 1 , X 2 , X 3 ) = (X, Y, Z) and
(x 1 , x 2 , x 3 ) = (x, y, z), with X and x representing the longitudinal coordinates in
the initial zero-stress state and current state, respectively.
6.4.1 Growth of an Unconstrained Bar
If an unconstrained, unloaded bar grows uniformly, a simple equilibrium analysis
shows that the bar remains stress-free. As we will see, depending on the specific
growth pattern, differential growth, i.e., nonuniform growth, can generate stress in
the bar.
Example 6.1 An unloaded bar of initial length L 0 grows nonuniformly in the axial
direction according to the relation
G x (X, t) = 1 +
a + b(X/L 0 )
2
(1 − e
−βt ),
(6.15)
where a, b, and β are constants. This relation satisfies the initial condition
G x (X, 0) = 1. If the bar remains free of external constraints, determine the length
of the bar as a function of time.