Fig 1.
Growth process of a molluskan shell surface depicted through the steps of volume growth of mantle tissue, morphoelastic deformation, and shell surface growth via secretion and calcification. The calcified region of the shell is indicated in yellow.
Fig 2.
Local coordinate system on the surface patch.
s1 is the direction of surface growth due to mantle extension, s2 is the direction of volume growth along the mantle margin, and s3 = s1 × s2 is the normal to the local surface patch. The calcified shell edge after time τ1 forms the generating curve Γτ1 for the time step, [τ1, τ2]; the leading edge of the grown and deformed mantle strip then forms the generating curve, Γτ2 for the next time step.
Fig 3.
The observed deformation gradient, F, is composed of an incompatible growth component, Fg, and another incompatible, but elastic component, Fe, which restores compatibility of F.
Fig 4.
Mathematical model of molluskan shell growth over (tk, tk+1] through a sequence of surface growth, volume growth, secretion and calcification.
Beginning with a calcified mantle current configuration , growth over (tk, tk+1] is modelled via the following sequence of steps: (1) Surface growth—leading surface
is displaced due to mantle extension by v1Δt s1 to a new reference surface
defining a strip of the mantle in its reference configuration
. (2) Volume (morphoelastic) growth—
is imposed on the mantle strip
over Δt = tk+1 − tk, driving its nonlinear deformation into the current configuration
. (3) Shell growth occurs by secretion on the mantle strip
, followed by (4) calcification of the mantle strip
at tk+1. During the volume growth of the mantle strip from its reference configuration
to its current configuration
, boundary conditions are applied on the trailing surface (highlighted in red) and the lateral surfaces (highlighted in yellow) of the mantle strip. The front surface of the mantle strip is highlighted in green.
Fig 5.
Space-time discretization in the finite element framework.
Evolution of the mollusk shell surface through surface growth and morphoelastic volume growth of mantle strips followed by their calcification. Also see S1 Movie for the time evolution of a representative molluskan shell surface through the accretive growth of 20 mantle strips.
Fig 6.
Parameters controlling the morphology of shell ornamentations.
The reference generating curve, Γ0, with its curvature, κ along s2; the active mantle width, δs; volume growth strain increment, δg.
Fig 7.
Effect of incremental active mantle widths on mantle deformation for fixed δg and κ.
Increasing the active mantle width δ s = |v1 s1 − vc|Δt over Γ0 leads to morphologies bearing a similarity with the ornamentations on Clinocardium nuttallii (upper inset) and Tridacna squamosa (lower inset). Dirichlet boundary conditions, u = 0, are applied on the trailing surface (boundary) and traction-free Neumann conditions, PN = 0, are applied on the remaining surfaces (boundaries). See Fig 4 for location of the trailing surface, front surface and the lateral surfaces. Also see S3 Movie for a morphology that is similar to the case δs = δs*, and bears comparison to the ornamentation on members of the class Bivalvia.
Fig 8.
Scaling study of mode number n versus the incremental active mantle width δs.
Shown is the dependence of the mode number of the deformed mantle strip on the incremental active mantle width, obtained from the finite element model (FE) and the buckling analysis of a plate.
Fig 9.
Effect of volume growth strain increment.
δg = ε2 Δt on the amplitude and mode numbers of the deformed mantle. The observed high mode number morphologies are similar to the ornamentations observed in bivalves like Clinocardium nuttallii (inset). Dirichlet boundary conditions, u = 0, are applied on the trailing surface (boundary) and traction-free Neumann conditions, PN = 0, are applied on the remaining surfaces (boundaries). See Fig 4 for location of the trailing surface, front surface and the lateral surfaces.
Fig 10.
The influence of the geometry of reference curves on antimarginal ornamentation.
For fixed active mantle width, the amplitudes of crests in the deformed configurations are magnified if they are compatible, and attenuated if they are incompatible, respectively, with the reference generating curve. These reference curves bear similarity to the shape of the mantle surface (highlighted in red) found in (a) Pterynotus phyllopterus, (b) Nucella freycineti, (c) Normal Bolinus brandaris and (d) Abnormal Bolinus brandaris (insets). Dirichlet boundary conditions, u = 0, are applied on the trailing surface (boundary) and the lateral surfaces (boundaries) of the mantle, which are perpendicular to s2, and traction-free Neumann conditions, PN = 0, are applied on the front surface (boundary). The underlying spatial discretization (mesh) is also shown on the model geometries. Also see S2–S5 Movies for the evolution of mantle deformation and accretive growth over planar, arc, and closed circular geometries of the reference curves.
Fig 11.
Progression of curvature-compatible ornamentation with volume growth strain increments, δg.
The deformed mantles show marginally preferred localization around points of high curvature and thereafter the amplitude increases with volume growth strain increments. Some of these shapes with different amplitudes can be observed in the shells of the species Bolinus brandaris (bottom row of inset images). Dirichlet boundary conditions, u = 0, are applied on the trailing surface (boundary) and traction-free Neumann conditions, PN = 0, are applied on the remaining surfaces (boundaries).
Fig 12.
Influence of reference curvature singularities, and of smooth curvatures.
Mild reference curvature singularities leave virtually no visible trace on mantle deformation following volume growth. However, strong reference curvature singularities promote compatible crests, and remain visible as mild singularities in the deformed mantle lip. Smoothly varying reference curvature also replicates the trend of favoring compatible crests. Dirichlet boundary conditions, u = 0, are applied on the trailing surface (boundary) and traction-free Neumann conditions, PN = 0, are applied on the remaining surfaces (boundaries).
Fig 13.
Hierarchical ornamentation arising from temporally varying surface growth, δs volume growth strains, δg ε2.
In the Hexaplex cichoreum image shown in the inset, three levels of ornamentation hierarchy are shown: primary (red) as a low mode bifurcation from a flat surface, secondary ornamentation as a second mode bifurcation (magenta) and tertiary ornamentation mode as a third mode (blue). The corresponding first, second and third modes are traced on the mantle edge of the computational model. The dotted white line indicates the location of the fixed calcification front between (t1, t3]. Dirichlet boundary conditions, u = 0, are applied on the trailing surface (boundary) and the lateral surfaces (boundaries) of the mantle, which are perpendicular to s2, and traction-free Neumann conditions, PN = 0, are applied on the front surface (boundary). Inset image of Hexaplex cichoreum modified from source [37]. Original images licensed under the Creative Commons Attribution-Share Alike License.
Fig 14.
Spatially varying volume growth.
We impose volume growth strain increments that vary along the ξ1 direction that is the curvilinear coordinate defining s1, with an increasing gradient toward the leading edge, as shown in (a). The result appears in (b), displaying large, negative Gaussian curvature, mimicking the strongly backward arching morphology observed in a number of shell species, for example as seen also in (c) Hexaplex chicoreum. Dirichlet boundary conditions, u = 0, are applied on the trailing surface (boundary) and the lateral surfaces (boundaries) of the mantle, which are perpendicular to s1, and traction-free Neumann conditions, PN = 0, are applied on the front surface (boundary). Also see S6 Movie for the evolution of deformation leading to a morphology with strongly negative Gaussian curvature.
Fig 15.
Phase diagram representing the effect of the growth and geometric parameters—growth strain increment (δg), active mantle width (δg), and curvature of the reference curve (κ), on the morphology of shell ornamentations.