478
8 Morphogenesis
smallest value of ¯
P cr , which is the actual critical load. The corresponding curve
defines the buckling mode, with n being the number of bumps in the mode shape.
The plots show that the critical load increases and the wavelength decreases (n
increases) with increasing k (actually α). This behavior is consistent with that of
Example 8.6 when additional constraints are added along the length of the beam (see
Fig. 8.59d). In addition, the critical load for a finite beam generally differs little from
that for a beam of infinite length, especially for larger values of k, suggesting that
the boundaries have relatively little effect for buckling of a beam on a foundation.
The reason is that the influence of the boundary conditions becomes more and
more localized to narrow regions near the ends of the beam as k increases (e.g.,
see Fig. 8.17d).
In arteries, collagen fibers appear to be buckled in compression. For this reason, as
mentioned near the end of Sect. 7.6.1, many investigators assume that collagen fibers
sustain negligible stress in compression. However, the present analysis suggests that,
depending on the stiffness of surrounding tissue, buckled collagen fiber bundles
could contribute significant stress in unloaded arteries.
As a final remark, we note that springs oriented perpendicular to the long
axis of the beam do not exert shear stresses. Thus, in this idealized model, if
an unloaded beam grows longer without axial constraint, it would remain stress
free. In contrast, underlying tissue could exert shear that constrains growth-induced
extension, thereby generating axial compression that can buckle the beam even
without other loads or constraints. Thus, a better model for a tissue foundation
would be a combination of springs both normal and parallel to the beam axis.
8.6.5 Folding of the Cerebral Cortex
As the brain tube matures, the walls of the cerebral hemispheres progressively
thicken to create brain tissue composed of a relatively thin outer layer (cerebral
cortex) surrounding a laminated core. Neurons divide in the subcortical layers and
migrate along radial glial fibers to the cortex. In most large mammals, the cortex
eventually develops a convoluted surface consisting of gyri (outward folds) and
sulci (inward folds). Folding allows more cortical tissue to fit inside the skull. This
process is critically important because the cerebral cortex is the site of higher brain
functions, such as thought, speech, and memory. Abnormal folding is associated
with various neurological disorders, including epilepsy, autism, and schizophrenia.
Folding Mechanisms Cortical folding occurs in two steps. First, primary folds
form in patterns that are consistent across individuals within a given species. Then,
secondary folds with more variable patterns appear. Folding is driven by forces
intrinsic to the brain; the constraint provided by the skull has relatively little effect
(Welker et al. 1990; Fernandez et al. 2016; Garcia et al. 2018).
8 Morphogenesis
smallest value of ¯
P cr , which is the actual critical load. The corresponding curve
defines the buckling mode, with n being the number of bumps in the mode shape.
The plots show that the critical load increases and the wavelength decreases (n
increases) with increasing k (actually α). This behavior is consistent with that of
Example 8.6 when additional constraints are added along the length of the beam (see
Fig. 8.59d). In addition, the critical load for a finite beam generally differs little from
that for a beam of infinite length, especially for larger values of k, suggesting that
the boundaries have relatively little effect for buckling of a beam on a foundation.
The reason is that the influence of the boundary conditions becomes more and
more localized to narrow regions near the ends of the beam as k increases (e.g.,
see Fig. 8.17d).
In arteries, collagen fibers appear to be buckled in compression. For this reason, as
mentioned near the end of Sect. 7.6.1, many investigators assume that collagen fibers
sustain negligible stress in compression. However, the present analysis suggests that,
depending on the stiffness of surrounding tissue, buckled collagen fiber bundles
could contribute significant stress in unloaded arteries.
As a final remark, we note that springs oriented perpendicular to the long
axis of the beam do not exert shear stresses. Thus, in this idealized model, if
an unloaded beam grows longer without axial constraint, it would remain stress
free. In contrast, underlying tissue could exert shear that constrains growth-induced
extension, thereby generating axial compression that can buckle the beam even
without other loads or constraints. Thus, a better model for a tissue foundation
would be a combination of springs both normal and parallel to the beam axis.
8.6.5 Folding of the Cerebral Cortex
As the brain tube matures, the walls of the cerebral hemispheres progressively
thicken to create brain tissue composed of a relatively thin outer layer (cerebral
cortex) surrounding a laminated core. Neurons divide in the subcortical layers and
migrate along radial glial fibers to the cortex. In most large mammals, the cortex
eventually develops a convoluted surface consisting of gyri (outward folds) and
sulci (inward folds). Folding allows more cortical tissue to fit inside the skull. This
process is critically important because the cerebral cortex is the site of higher brain
functions, such as thought, speech, and memory. Abnormal folding is associated
with various neurological disorders, including epilepsy, autism, and schizophrenia.
Folding Mechanisms Cortical folding occurs in two steps. First, primary folds
form in patterns that are consistent across individuals within a given species. Then,
secondary folds with more variable patterns appear. Folding is driven by forces
intrinsic to the brain; the constraint provided by the skull has relatively little effect
(Welker et al. 1990; Fernandez et al. 2016; Garcia et al. 2018).