498
8 Morphogenesis
downward bending corresponds to positive curvature.) Growth during the loading
period causes plastic-like deformation that remains after the load is removed.
For a = b = 1, the plate undergoes an HR response that depends on the stress
distribution. During the time the bending moment is applied, the axial stress is
approximately linearly distributed across the thickness, being tensile on the convex
(upper) side and compressive on the concave (lower) side (Fig. 8.75b, t = 0.1).
These stresses cause the target stresses to initially move in the opposite directions,
i.e., negative on top and positive on the bottom (Fig. 8.75c).
When the load is removed, the distribution becomes roughly parabolic to satisfy
the conditions of zero net force and moment throughout the plate (Fig. 8.75b, t = 1).
Notably, the stresses at the top and bottom are approximately equal and, therefore,
cannot simultaneously approach their respective target stresses, which continue to
diverge (Fig. 8.75c). The negative target stress induces the upper surface to grow,
while the positive target stress induces the lower surface to contract (Fig. 8.75d),
causing the plate to continue to bend downward at an increasing rate (Fig. 8.75a). In
fact, the plate bends all the way around until its ends meet (Fig. 8.75a, inset). This
unbounded behavior is like that for the problem of stretching two bars in series with
unequal stiffnesses (Fig. 8.73c, d).
Experiments have confirmed this predicted behavior. Kremnyov et al. (2012)
cultured sections of the blastocoel roof dissected from frog embryos. The explants
were artificially bent for a few minutes and then released. Bending increased in the
unloaded samples over the next few hours.
8.9.5 HR: Sea Urchin Gastrulation
In Sect. 8.4.3, we investigated the problem of sea urchin gastrulation. With the blastula modeled as a spherical shell, two invagination mechanisms were considered:
apical contraction and matrix swelling. Both generated a local dimple but with
limited depth (Fig. 8.22). Here, we examine the possibility that hyper-restoration
produces a deeper invagination following initial contraction of the apex.
The previous sea urchin model is extended in two main ways. First, for axisymmetric deformation, we assume that HR is governed by the evolution equations
(Taber 2009)
˙
G θ = a(σ θ − σ θ0 )G θ
˙
G φ = a(σ φ − σ φ0 )G φ
(8.84)
and
˙
σ θ0 = −b(σ θ − σ θ0 )
˙
σ φ0 = −b(σ φ − σ φ0 ),
(8.85)
8 Morphogenesis
downward bending corresponds to positive curvature.) Growth during the loading
period causes plastic-like deformation that remains after the load is removed.
For a = b = 1, the plate undergoes an HR response that depends on the stress
distribution. During the time the bending moment is applied, the axial stress is
approximately linearly distributed across the thickness, being tensile on the convex
(upper) side and compressive on the concave (lower) side (Fig. 8.75b, t = 0.1).
These stresses cause the target stresses to initially move in the opposite directions,
i.e., negative on top and positive on the bottom (Fig. 8.75c).
When the load is removed, the distribution becomes roughly parabolic to satisfy
the conditions of zero net force and moment throughout the plate (Fig. 8.75b, t = 1).
Notably, the stresses at the top and bottom are approximately equal and, therefore,
cannot simultaneously approach their respective target stresses, which continue to
diverge (Fig. 8.75c). The negative target stress induces the upper surface to grow,
while the positive target stress induces the lower surface to contract (Fig. 8.75d),
causing the plate to continue to bend downward at an increasing rate (Fig. 8.75a). In
fact, the plate bends all the way around until its ends meet (Fig. 8.75a, inset). This
unbounded behavior is like that for the problem of stretching two bars in series with
unequal stiffnesses (Fig. 8.73c, d).
Experiments have confirmed this predicted behavior. Kremnyov et al. (2012)
cultured sections of the blastocoel roof dissected from frog embryos. The explants
were artificially bent for a few minutes and then released. Bending increased in the
unloaded samples over the next few hours.
8.9.5 HR: Sea Urchin Gastrulation
In Sect. 8.4.3, we investigated the problem of sea urchin gastrulation. With the blastula modeled as a spherical shell, two invagination mechanisms were considered:
apical contraction and matrix swelling. Both generated a local dimple but with
limited depth (Fig. 8.22). Here, we examine the possibility that hyper-restoration
produces a deeper invagination following initial contraction of the apex.
The previous sea urchin model is extended in two main ways. First, for axisymmetric deformation, we assume that HR is governed by the evolution equations
(Taber 2009)
˙
G θ = a(σ θ − σ θ0 )G θ
˙
G φ = a(σ φ − σ φ0 )G φ
(8.84)
and
˙
σ θ0 = −b(σ θ − σ θ0 )
˙
σ φ0 = −b(σ φ − σ φ0 ),
(8.85)