8.3 Linear Theory for Growing Beams and Plates
425
which compresses the bottom layer with equal and opposite force (left). Next, the
plate is released, allowing it to bend upward, generating a linear variation in bending
stress across the wall (center). Adding these two stress fields produces the total stress
distribution (right).
For cases (b) and (c), results are shown for a = ±0.01. For a > 0, the plate
bends downward with the shape being mostly concave upward, but the zero-slope
boundary condition requires the opposite curvature near x = 0 (Fig. 8.12c). As
expected, negative growth or contraction (a < 0) causes the plate to bend upward.
Since N = 0 for the roller support in case (b), the deflected shapes for a = 0.01
and a = −0.01 are mirror images of each other. In contrast, the axial force caused
by the pinned support in case (c) significantly affects the results. Tension (a < 0)
pulls the plate straighter, decreasing the magnitude of the upward deflection, while
compression (a > 0) increases the downward deflection (like pulling or pushing on
a rope).
Because κ varies with x, the stresses vary along the plate. For case (c), distributions across the wall are shown at both ends (Fig. 8.12d). Integrating Eq. (8.9) 2
shows that the moment is zero at x = L, as required by the pinned support. As
before, the stress distribution can be understood as a superposition of stretching
and bending stresses. In the linear approximation, the length of the plate remains
constant because the ends cannot move. Thus, if the beam is constrained to remain
straight, growth in the lower layer causes uniform compression in that layer,
while the nongrowing upper layer remains stress free. The compression shifts the
bending stress in the lower layer leftward at both ends (Fig. 8.12d), Without this
compression, the stress across the wall would be continuous.
The above examples show that the linear theory for growing beams and plates
can provide reasonably accurate qualitative results, even as deformations become
large. Thus, much can be learned from relatively simple solutions about fundamental
behavior. Because of the inherent variability in biological systems, especially during
development, qualitative results may suffice in many cases, and it may be tempting
to consider dispensing completely with nonlinear analyses. We must always keep
in mind, however, that the linear theory may yield misleading results for the
complex geometries and very large deformations that often characterize problems
in morphogenesis. Therefore, while linear solutions are useful for preliminary
studies and for checking whether puzzling numerical results make sense, a nonlinear
analysis should generally be the end goal.
The cantilever bending problems show that different growth patterns can produce
similar shapes (see Figs. 8.10c and 8.12a). In fact, nonlinear finite-element solutions
show that differential growth can bend the plate into a complete circle, with the
free end bending all the way around to meet the fixed end (Fig. 8.14a–c), although
the stress distributions are very different (Figs. 8.10b and 8.12b). Moreover, if the
upper layer atrophies or contracts, the shape is still circular, but the radius is smaller
(Fig. 8.14d). These examples highlight the need for additional data, e.g., stress and
strain, to determine morphogenetic mechanisms.
425
which compresses the bottom layer with equal and opposite force (left). Next, the
plate is released, allowing it to bend upward, generating a linear variation in bending
stress across the wall (center). Adding these two stress fields produces the total stress
distribution (right).
For cases (b) and (c), results are shown for a = ±0.01. For a > 0, the plate
bends downward with the shape being mostly concave upward, but the zero-slope
boundary condition requires the opposite curvature near x = 0 (Fig. 8.12c). As
expected, negative growth or contraction (a < 0) causes the plate to bend upward.
Since N = 0 for the roller support in case (b), the deflected shapes for a = 0.01
and a = −0.01 are mirror images of each other. In contrast, the axial force caused
by the pinned support in case (c) significantly affects the results. Tension (a < 0)
pulls the plate straighter, decreasing the magnitude of the upward deflection, while
compression (a > 0) increases the downward deflection (like pulling or pushing on
a rope).
Because κ varies with x, the stresses vary along the plate. For case (c), distributions across the wall are shown at both ends (Fig. 8.12d). Integrating Eq. (8.9) 2
shows that the moment is zero at x = L, as required by the pinned support. As
before, the stress distribution can be understood as a superposition of stretching
and bending stresses. In the linear approximation, the length of the plate remains
constant because the ends cannot move. Thus, if the beam is constrained to remain
straight, growth in the lower layer causes uniform compression in that layer,
while the nongrowing upper layer remains stress free. The compression shifts the
bending stress in the lower layer leftward at both ends (Fig. 8.12d), Without this
compression, the stress across the wall would be continuous.
The above examples show that the linear theory for growing beams and plates
can provide reasonably accurate qualitative results, even as deformations become
large. Thus, much can be learned from relatively simple solutions about fundamental
behavior. Because of the inherent variability in biological systems, especially during
development, qualitative results may suffice in many cases, and it may be tempting
to consider dispensing completely with nonlinear analyses. We must always keep
in mind, however, that the linear theory may yield misleading results for the
complex geometries and very large deformations that often characterize problems
in morphogenesis. Therefore, while linear solutions are useful for preliminary
studies and for checking whether puzzling numerical results make sense, a nonlinear
analysis should generally be the end goal.
The cantilever bending problems show that different growth patterns can produce
similar shapes (see Figs. 8.10c and 8.12a). In fact, nonlinear finite-element solutions
show that differential growth can bend the plate into a complete circle, with the
free end bending all the way around to meet the fixed end (Fig. 8.14a–c), although
the stress distributions are very different (Figs. 8.10b and 8.12b). Moreover, if the
upper layer atrophies or contracts, the shape is still circular, but the radius is smaller
(Fig. 8.14d). These examples highlight the need for additional data, e.g., stress and
strain, to determine morphogenetic mechanisms.