Mechanical Overview of Mollusk Shell Formation and Structure

Table of Contents

Abstract

For mollusks to thrive in ecosystems swarming with predators and harsh environmental conditions, their shells have acquired specific characteristics aiming to minimize the mechanical energy from external and internal stresses. This research focuses on four such structural adaptations: coiling, ribbing, spines, and shell microarchitecture. Each of these structures has its own geometric models and mechanical analyses and offers a unique contribution to the optimization of mollusks’ physical properties. The coiling of the shell responds to mechanical stresses from the soft body of the mollusk and its external environment. Both ribbing and the hierarchical microarchitecture of the shell maximize its structural strength, whereas the formation of spines acts as a defence mechanism against predation while also minimizing interior stresses. The molluscan shell characteristics are all ultimately a testament to its ability to protect its soft body, dissipate mechanical stresses and protect itself from shell failure.

Introduction 

Mollusks are vulnerable creatures without their shells. An effective layer of protection is what stands between them and many predators who would readily consume their soft bodies. This protective layer also needs to adapt to the mechanical stresses that the shell is subject to, as much from its own soft body as to the environmental conditions of its habitat. To form this coat of protection, they make use of biomineralization, the process by which an organism creates and accumulates minerals (Lowenstam, 1989), to form layers of calcite and aragonite (Johnson et al., 2019). 

The principal organ responsible for shell formation is the mantle, a tissue that adheres to the inner aspect of the shell opening, gradually secreting the calcium carbonate and matrix proteins (the proteins responsible for support) forming the shell (Frisch, 2021) (Johnson et al., 2019). Thus, any deformations in the mantle, often resulting from physical stress caused by growth in the tissue, strongly affect shell formation. The new shape of the calcified shell will then affect the conformation of the mantle during the next growth step, carrying on the deformation to all subsequent shell layers.

Mollusk shells are typically composed of three layers: the periostracum, the prismatic layer, and the nacreous layer (Fig. 1) (Boro et al., 2012). 

Fig. 1 The different layers that make up the shell: the mantle, the prismatic layer, and the nacreous layer (Adapted from Erlich et al., 2018).

Depending on the species, however, the separation between these layers is not always clear-cut. For example, some Uniondae mussels have been observed to possess two layered periostracums (Ponder et al., 2019) and the prismatic and nacreous layers in Nautilus pompilius are better described as one continuous layer (Schoeppler et al., 2019). Nevertheless, shell composition does vary depending on the area studied. The exterior periostracum remains formed of organic material, while the inner layers are mostly inorganic (albeit containing a small percentage of organic material) (Ponder et al., 2012).

As shown by the tremendous geometric diversity (Fig. 2), mollusks have adjusted the growth of the ribbings, the spines, and the overall shape of their shells to respond to the shape of their soft body, their habitat, predators, or any other constraints they face.

This necessity to grow side by side with the organism comes with a variety of different coiling types, such as planispiral, bivalve, helicospiral and irregular coiling (Chirat et al., 2021). These structural strength issues have also been partially addressed through the formation of ribs. Indeed, ribs are essential as they help reinforce the structure of the hull. Ribs also make it possible to balance the compression and tension forces endured by the shell. As for the threat of shell-crushing predators, some mollusks addressed this threat by developing spiny defences covering their shell surfaces. The molluscan shell can resist such powerful stresses thanks to its complex hierarchical structure. Both the macroscopic shell features and the microarchitecture of the shell’s material are responsible for its incredible strength while also providing adequate space for the soft body. Each feature’s mechanical model of formation will be investigated separately to shed light on exactly how mollusks’ shields evolved to become so strong. 

Fig. 2 Variation in shell morphologies of marine gastropods (Bunge, 2004).

Shell Coiling 

Early geometric model of shell coiling

Raup’s early research on shell coiling (Raup, 1965) created a starting framework for models that were more recently developed. Raup’s model considered 4 variables: the shape of the cross-section of the shell (S), the rate of cross-sectional size increase per revolution (W), the ratio between the inner and outer margin measured relative to the coiling axis (D) and the rate of growth along the coiling axis (T). The parameter S was kept as a simple circle, while the other parameters were varied to obtain the shapes shown in Figure 3. Most morphologies were generated by this model. As W increases, the shape produced goes from a planispiral shell to that of a bivalve, common in brachiopods and pelecypods (James et al., 1992; J. Pojeta, 1978). As D increases, the planispiral form increases in number of revolutions, a common form in most ammonoids (Klug et al., 2015). Furthermore, as T increases, the shape tends toward a helicospiral shell, common in gastropods (Raup, 1966).

Fig. 3 Computer-generated shell shapes with varying expansion rates (W), distance from coiling axis (D) and translation rate (T) (Adapted from Raup, 1966).

While Raup’s model created a good basis for most shell forms, many heteromorph ammonites such as Nipponites (Fig. 4) cannot be explained by his model (Chirat et al., 2021). Additionally, his model gives a mostly geometric viewpoint (Raup, 1966), leaving space for further research on the mechanical reasons for shell chirality. 

Fig. 4 Shells of Nipponites mirabilis (Adapted from Chirat et al. 2021).

Energy minimizers for chiral shells

Subsequent research provides a theoretical model explaining that irregular coiling in heteromorph ammonites is caused by a ‘mechanically induced twist of the soft body’ (Chirat et al., 2021). This, coupled with a difference in growth rate between the soft body and outer shell, creates mechanical stresses resulting in the deformation of some ammonites. The soft body growth is modeled by ventral and dorsal elastic rods, where the ventral rod grows at a quicker rate than the dorsal rod. These rods are contained and restricted to the outer shell (Fig. 5), which results in three shell types: planispiral, helicospiral and meandering (Chirat et al., 2021).

Fig. 5 Difference in growth rate between the inner soft body (visualized by dorsal and ventral rod) and the outer shell induces mechanical stresses relieved by a twist of the axis of rotation (Adapted from Chirat et al., 2021).

The reference body shape, representing the growth of the inner body free of any restrictive forces by the outer shell, is assumed to be a planar logarithmic spiral. The planispiral shell is obtained when the twist of the inner body is zero. Both the helicospiral and meandering shell are obtained by a mismatch between the inner body and the outer shell growth rates, resulting in the twist of the inner rods that deviates their axis of rotation. (Fig. 6) The twist of the helicospiral shell is constant whereas the meandering shell twist is oscillatory (Chirat et al., 2021).

Fig. 6 Three shell morphologies resulting from no twist (planispiral), constant twist (helicospiral) and oscillatory twist (meandering) (Adapted from Chirat et al., 2021).

The most likely morphology is set to be determined by the shape that will minimize the body’s mechanical energy, including the contribution of stretching, bending, and twisting. In Chirat et al. ’s model, for a given body and shell growth rate, the twist was varied, and the total energy (ε) was calculated to determine which shape minimizes mechanical energy. The energy is written as a function of twisting:  ε = ε(d0, d1, d2), where d0 is a constant twist, while d1 and d2 are the parameters describing an oscillatory twist. This results in the energy of a planispiral shell εp = ε(0, 0, 0), the energy of a helicospiral shell εh = ε(d0, 0, 0) and the energy of a meandering shell εm = ε(0, d1, d2). The resistance of the body to stretching (K1), bending (K2) and twisting (K3) are also defined. Figure 7 demonstrates that for given values of K1, K2 and K3, the shell shape that will be the energy minimizer is either planispiral (in green), helicospiral (in blue) or meandering (in red). 

Fig. 7 Shell form (planispiral in green, helicospiral in blue and meandering in red) that minimizes mechanical energy for values of K1, K2 and K3 (Adapted from Chirat et al., 2021).

These results demonstrate how various types of coiling could be favored to minimize the mechanical stresses on the body and shell of ammonites and gastropods. Their model also supports shells that demonstrate irregular coiling (Fig. 8). One event that triggers these changes in coiling is the arrival of sexual maturity, as the body chamber needs to accommodate the mature reproductive organs (Collins & Ward, 2010). This change in the mechanical stresses caused by the change in growth of the body will result in the transition from one shell twist to another. The general pattern observed during this stage is a transition from planispiral, to meandering, to helicospiral. 

Fig. 8 Modeling of D. nebrascense and N. malagasyene by varying shell twisting from planispiral (green), to meandering (blue) and to helicospiral (red) (Adapted from Chirat et al., 2021).

Effects of stability and available space for soft body on shell shape

While the mechanical forces between the inner body and outer shell of the mollusk are potential factors that dictate shell coiling (Chirat et al., 2021), many other forces are involved. Research has also examined how postural stability and available space for a soft body can restrict coiling in snails (Noshita et al., 2012). Ideally, a shell will maximize both the volume available for soft body and postural stability. Coiling affects both aspects, as a low spiral and high whorl overlap results in high postural stability, whereas a high spiral and reduced overlap increases the available space for a soft body. In their research (Noshita et al., 2012), the moment of force caused by gravitational force acting on the shell’s center of gravity was represented by the amount of postural stability, whereas the available space for soft body was represented by ‘the ratio of internal shell space relative to the volume of shell required to enclose the space’(Noshita et al., 2012). Computer simulations determined that there is an ideal value of whorl expansion rate, shell radius (T) and aperture inclination (△) that allow for postural stability and sufficient available space as shown in the cluster in Fig. 9.

Fig. 9 Plot showing aperture inclinations (△) and shell radius ( Γ ) of shell specimens from different habitats (Adapted from Noshita et al., 2012).

Hydrodynamic effects on shell morphology

Research has also shown that coiling may be constrained by external forces caused by the organism’s habitat. Notably, the effect of the moment of gravity on freshwater snails is lessened due to buoyancy, allowing for more significant variation in forms while retaining postural stability (Noshita et al., 2012). The link between shell morphology and life mode was also explored in Westermann (Westermann, 1996) and Jacobs’s research (Jacobs, 1992; Ritterbush & Bottjer, 2012). Their hydrodynamic experiments with various ammonite morphotypes support the proposal of a link between shell shape and life modes (Fig. 10). The link was established using three different types of shells; serpenticones, spherocones and oxycones. Each of these shells varies in morphology, with the degree of exposure of the umbilicus being the main feature affecting the drag level. This morphological characteristic dictates the life modes of these respective shell types, including plankton, vertical migrant and nekton life modes. Serpenticonic shells demonstrated high levels of drag due to the exposure of the umbilicus, reducing swimming capabilities and leading Westerman to classify them as plankton. Spherocones were determined to be most efficient at low speeds due to their inflated shape and covered umbilicus, leading to the classification as vertical migrants. Oxycones’ covered umbilicus and flattened shape allow for the minimization of friction drag, which classifies them as nekton (Ritterbush & Bottjer, 2012)

Fig. 10 Westermann’s morphospace, describing the shell of various ammonoids, is associated with a specific life mode (shown in the bottom left of figure) (Adapted from Ritterbush, 2016).

Shell Ribs

An overview of shell growth

The progressive formation of the shell is subdivided into two distinct growth processes: surface growth and volume growth. Surface growth is described as the formation of new layers within an environment that has no supporting matter (Rudraraju et al., 2019). However, volume growth is characterized as the local deposition of matter upon the pre-existing substrate. The latter does not lead to an increase in the number of surfaces during this phenomenon (Rudraraju et al., 2019).

The design of the mechanical model in relation to the formation of the ribs

With the aim of shedding light on the mechanical matrix of shell formation, models are needed. The one developed by Rihito Morita offers an explanation for some of the morphological characteristics of shells by considering its structure as a double membrane elastic tube (Chirat et al., 2013). This is a simplified model because it does not account for the accretion, or surface growth component, i.e. the enlargement of an organism in relation to an agglomeration of matter. On the other hand, another mechanical model that illustrates the formation of the ammonite shell, a mollusc subclass, is based on the research conducted by Derek E. Moulton and Régis Chirat (Chirat et al., 2013).  This model highlights a strong link between time and the evolution of shell growth. Unlike the previous model, accretion growth is strongly considered in this analysis. In addition, the notable element of this system is the mechanical deformation of the mantle. By considering the edge of the mantle as a circular band, this circular band gives it a one-dimensional elastic structure that varies the angle of the shell opening (Fig. 11) (Erlich et al., 2018). The contact between this ring and the generative zone induces mechanical forces. The generative zone is the region that connects the previously calcified part with the mantle’s edge. It is important to consider the presence of periostracum in this model – a thin flap that envelopes the outer rim of the shell.  This model also induces that the rate of growth of this band is fixed. This assumes that its growth occurs incrementally. There is, therefore, a gradual and uniform deposition of matter during this process (Chirat et al., 2013). Thus, it makes it possible to increase the opening with each iteration.

Fig. 11 The design of the model that highlights commarginal ribs. The shell which is presented in the form of an elastic ring has a radius r(z) and an opening angle φ with respect to the growth axis z (Adapted from Moulton et al., 2015).

Commarginals and anti-marginals are forms of the ribs present on the surface of the shell (Erlich et al., 2018). The notable difference between these two categories is the direction of the pattern. The ornamental structure for the commarginals is illustrated as parallel to the growth lines. Conversely, the pattern is perpendicular to the growth lines for antimarginals (Fig. 12). The main factors influencing shell formation are growth rate, shell geometry and mechanical properties of tension and compression. (Chirat et al., 2013).

 Fig. 12 Commarginal ornamentations that are parallel to the growth lines (Promicroceras planicosta) are presented in (a). While antimarginal ornamentations that are perpendicular to growth lines (Tridacna squamos) are presented in (b) (Adapted from Erlich et al., 2018).

It is therefore possible to introduce many variables within this model, such as the radius from the mantle (Rz), the radius of the calcified shell (rz), the direction of growth (z) and the angle of opening with respect to the growth axis ϕ (z) (Moulton et al., 2012). Through the balance between the elastic tensile force of the mantle and the compressive force generated by the generative zone, it is possible to determine the value of the radius and that of the angle of the orientation following a new iteration. Thus, the strength of the generative region is reflected in this formula:

The variable KGZ represents the resistance due to the change of direction.  Since the framework of this geometry is circular, all forces have a radical direction (Chirat et al., 2013). The differential equation of the opening angle (ϕ) is illustrated in this form:

This first-order linear differential equation has the variable k (k=Es⁄(KGZ)) which is the relative stiffness. It is generated by the mantle on the generative zone.  This variable is constant during the development of the shell.  The radius r(Z) is represented as follows: s(z)-β. Since β (a parameter that is related to the thickness and radius of the mantle/periostracum) is negligible, then r ≈ s.  The balance of exclusively radical forces translates in this way:

The variable n is closely related to this equation:

The variable Es is the axial value. Model analysis deepens thanks to this geometric relationship, which is a first-order linear differential equation: 

It is important to introduce conditions to this equation as the location denoted z = 0 (Hammer, 2000).  Through these equations, the model highlights the mechanical characteristics of the mantle/periostracum, the rigidity of the generative region as well as the growth of the mantle/periostracum. 

These equations form a system of equations. Thus, its resolution makes it possible to find the value of the radius of the shell r(z) and the angle of opening ϕ(z)  after a new iteration.  Nevertheless, the function of R(z) is not a linear equation (Chirat et al., 2013).  It will be possible to make an approximation of R thanks to this linear function:           

The variable γ represents the growth rate of the periostracum that is subject to no physical stress.  When z is zero (the initial position), R0 is the initial radius.  Thus, it is important to carry out linearization to solve this system of equations. Then, it is necessary to make several changes of variable. This is achieved through these relationships:

The variable L represents the length of the shell. Thus, the two parameters that do not have a dimension of this system are

It is necessary to apply an initial condition:

In addition, the rate of expansion must correspond to the initial orientation

These conditions therefore make it possible to find the solution to this system of equation: 

Of which, the values of the variables (ω, A et υ) are

,

and

.  This solution determines the value of the radius and the angle of the orientation following a new iteration.

It is possible to draw several conclusions thanks to these equations concerning the formation of ribs. Thus, the turning of the periostraccum outwards generates a shift between r(z) and R(Z). In addition, in the event that the radius of the mantle is greater than the radius of the aperture, this generates tension within the shell (Erlich et al., 2018). It is important to note that this phenomenon does not produce thorns, as the degree of instability is insufficient. This force that is generated by the mantle is moving outwards (ϕ>0). The radius of the shell in the next iteration will be larger because the generative zone is pushed outwards. Thus, a tension is exerted by the mantle/periostraccum (point A) (Moulton et al., 2015). A force is exerted by the calcifying generative zone that has properties similar to a torsion spring (Moulton et al., 2018). In terms of direction, this force opposes the force of the mantle, because it generates a compressive force of the shell towards the interior.   Due to an overextension, the orientation of the generative zone therefore changes inwards (ϕ<0).  As a result, there is a narrowing of the radius (point B). The compression exerted by the generative zone begins to accumulate in the mantle/periostraccum (point C). When the state of compression subsides, the orientation of the generative zone again tends outwards.  This process therefore corresponds to a cycle (point D) (Moulton et al., 2015). Thus, this periodicity between the states of compression and tension is the main cause of the regularity of the ribs as demonstrated by this equation:

Through this equation, it is important to note that the shell tends to oscillate around a reference radius R(z) (the green line) due to the states of compression and tension (Fig. 13) (Moulton et al., 2012).

Fig. 13 The comparison of the numerical solution with the analytical solution. The oscillation of the shell around a reference radius denoted s in this graph (the green line) due to the states of compression and tension. Point A represents the state of tension during the flaring of the periostraccum. Point B represents a decrease in voltage. Point C represents the beginning of the compression state. Then, point D represents the end of this compression state  (Adapted from Moulton et al., 2015).

The principle of mechanical equilibrium is essential in this phenomenon because a continuous oscillatory system of tension and compression is exerted on the circular band of the mantle (Moulton et al., 2018). Through the value of the α present in the differential equation of the opening angle (φ), it is possible to identify the state of compression and tension. Thus, it is presented in this way s(z)-β)R(z) < 1(compression) and s(z)-β)R(z) >1(tensile) (Moulton et al., 2012).  In addition, in modeling the mantle-shell system, it is important to emphasize the relationship between the growth rate of the mollusc and the rate of the diameter of the shell opening. The more the growth rate increases, the more the rate of the diameter of the opening of the shell also increases. This relationship leads to less pronounced ribs and a smoother surface (Moulton et al., 2018). These decreasing oscillations can be explained by the model of the elastic band of Molton et al (Moulton et al., 2015).  Indeed, a rapid extension of this ring which represents the growth rate of the shellfish decreases the propagation of the recall force. Oscillations in this mantle-shell system are closely related to this force (Erlich et al., 2016). It is important to note that the amplitude of the veins also establishes a direct correlation with the size of the shell opening. This is therefore based on Buckman’s first law of covariation (Fig. 14). The lateral and ventral ribs are equal for Figure A. The B-structure is more compressed than the A structure along the lateral axis (the opening of the shell is smaller). The amplitude of the lateral ribs is proportionally smaller. Structure C is enlarged along a lateral axis (the opening of the shell is larger). Thus, the amplitude of the lateral veins is proportionally larger. It is important to note the amplitude of the ventral ribs is constant (Hammer & Bucher, 2005).

Fig. 14 Illustration of Buckman’s law of covariation through numerical modeling of hypothetical ammonoid structures. The first horizontal row represents an apertural view, while the horizontal second row represents a side view  (Adapted from Hammer & Bucher, 2005).

The linear model of Molton and Chirat also establishes a relationship with the number of ribs.  The argument of the cosine function is present in equation 14 to determine the number of oscillations along  the shell (Moulton et al., 2018).

Figure 15 shows that due to the term ln z’, the number of ribs decreases according to a logarithmic tendency. The variable z must be between

(Erlich et al., 2016).  It is important to identify

. Thus, a function is highlighted,

. The formula that determines the total number of ribs:

.

Fig. 15 The number of ribs as a function of the rate of expansion of the shell. The blue curve represents the number of ventral ribs, while the green curve represents the number of dorsal ribs (Adapted from Moulton et al., 2015).

Spine formation

Molluscan shell spine description

Some species of mollusks grow shells with long cone-shaped ornaments protruding from the surface, called spines. These spines can take many shapes and forms, varying in length, width, and geometry as pictured in Fig. 16.

Fig. 16 Examples of mollusks with shell spines (Chirat et al., 2013).

Spines on mollusk shells have been present in various families of mollusks over the millennia. They have brought many advantages to improve mollusk survival (Ponder et al., 2019). Whether it’s resistance to some species of predators (Beaty & Rollins, 2002), or resistance to some fish that break seashells (Palmer, 1979), or even resistance to predation from other mollusks (Stone, 1998), spines have evolved multiple times to protect the mollusks that wear them.

Indeed, “spines in mollusk seashells are classically interpreted as having repeatedly evolved as a defense” (Chirat et al., 2013) and various studies have observed this repeated evolution in multiple families of (Vermeij & Carlson, 2000) (West et al., 1991). 

Molluscan shell formation 

The mantle forms a layer separating the mollusk’s interior from its shell. This mantle is responsible for shell formation. It does so by secreting molecules that calcify to form a solid shell. Mollusk shells grow in two distinct manners: surface growth and volume growth (Rudraraju et al., 2019). When shells undergo surface growth, a new shell material is secreted at the shell edge, extending its surface area. When shells undergo volume growth however, a new shell layer is being formed on top of a previous one, thereby only adding volume but no new surface area. 

Molluscan shell spine origin

Mollusk shell spines are secreted from the mantle edge, just like any other area of the shell. The mantle edge, a living tissue attached to the interior of the shell, is responsible for the secretion of the shell material; therefore its orientation in space upon secretion will determine the shape of the shell secreted. Consequently, if the mantle edge undergoes an initial deformation, the next shell layer’s geometry will be deformed accordingly (Chirat et al, 2013) (Checa & Crampton, 2002). If the mantle and shell edge were of the same length, then the former would perfectly adhere to the latter and no deformation would occur. That is why the initial deformation of the mantle edge is suspected to be caused by growth in the mantle tissue, affecting its ability to adhere properly (Chirat et al., 2013) (Checa & Crampton, 2002). Due to its elasticity, the mantle edge will then deform by bending upwards at the centre of the deformation and downwards at the ends of the deformation, increasing portions of its curvature to a point higher than that of the rest of the shell edge. But this deformation will not remain limited to a single shell layer. As shown in Fig. 17, since the mantle edge adheres to the shell edge, the newly deformed shell increases the deformation in the mantle edge, sharpening the shape of the spine with each new layer until “the middle section pinches, causing the spine to fold on itself” (Chirat et al., 2013).

Fig. 17 Graphic depiction of the mantle edge adhesion to the shell edge adhesion (Adapted from Chirat et al., 2013).

Rod elasticity

To understand the way in which the mantle edge deforms to know the shape it will take and what form the spines will be, it is necessary to understand the stresses the mantle edge undergoes as it grows. That is why Moulton et al. (2013) developed a model of morphoelastic rods, such as mollusk mantle edges, to model the stress and stress response materials of the same type as the mollusk mantle would undergo.  

Of particular relevance to this paper is the model of Moulton et al. on morphoelastic rods attached to support structures, which very closely resembles the scenario of a mantle edge attached to a shell edge. Their model sets up a reference frame relative to the orientation of the mantle by setting up direction vectors d1 and d2 parallel to the rod cross section. s is defined as the arc length of the mantle edge, its distance along the mantle edge, while r(s) is defined as the “centreline”, the line at the central axis perpendicular to the mantle cross section and parallel to the cross product of the direction vectors. is the function defining the curve of the supporting shell edge. The function A is used to describe the space between the elastic mantle and the shell edge. Therefore, the mantle and shell edges are attached together from point r(A) to point ρ(A). Furthermore, the vector pointing from the point of attachment on the mantle to the point of attachment on the shell edge, denoted as the distance E, would then be: 

(Adapted from Moulton et al., 2013)

Fig. 18 below provides a 3-D view depiction of the mantle shape and centerline, while Fig. 19 shows all vectors d1, d2, r(A), ρ(A), and E.

Fig. 18 3-D illustration of the rod-shaped mantle edge and its coordinate system, with its centerline represented as a dotted line (Adapted from Moulton et al., 2013).

Fig. 19 Depiction of the mantle centerline (r) the coordinate system, the shell edge (), the points of attachment and the displacement vector between the two points (Moulton et al., 2013).

To model the reaction acting on the mantle equation for the force and moment on the mantle were derived. f(s) is the spring force acting on the mantle edge, stretching it outwards. Its value is calculated as follows:

with |Eo| being the length between the mantle and shell edge at rest, making |E|–|Eo| the distance of extension or compression travelled by the mantle edge, as seen in Fig. 20. So, the force will be oriented in the same direction as the unit vector parallel to E, and the larger the current distance between the mantle and shell edge is compared to the rest distance, the greater |E|–|Eo| will be and the greater the force will be. If there is no distance separating the mantle and shell edge, the force is null.

Fig. 20 Extension of the mantle edge distance from the shell edge (Adapted from Moulton et al., 2013).

The moment l caused by this spring force is calculated by computing the cross product between the vector pointing from the centerline to the point of attachment on the mantle edge and the vector of the spring force, as pictured in Fig. 21. 

(Adapted from Moulton et al., 2013)

Fig. 21 Moment l calculated from the force f(s) and the distance r(A)– r (Adapted from Moulton et al., 2013).

It is this moment that in turn alters the orientation of the mantle. After being twisted, the mantle edge will rotate to reach an equilibrium state, but as pictured in Fig. 22, this equilibrium orientation differs from its original position.

Fig. 22 Depiction of the moment caused by the spring force on the mantle from a point of view parallel to the mantle cross section. After the mantle is twisted by the spring force, its new rest position is no longer the same as its original position (Moulton et al., 2013).

Internal factors impacting molluscan spines 

When the mantle grows, the forces applied on it changes, and it deforms to adjust to the new equilibrium state for its dimensions and new forces applied on it. In other words, the mantle edge deforms in the most stable manner possible to minimize the potential mechanical energy on it. But mantle edges don’t deform in the same manner in different species of mollusk. Two main factors determine how a mantle edge will deform (and consequently which shape the spine will take): mantle growth rate and mantle stiffness. 

In the context of modelling shell spines, the growth rate is defined as the excess rate relative to the shell edge (how much more the mantle edge has grown than the shell edge over a period of time.) Changes in the growth rate affect the intensity of the mantle edge’s curved deformation, limiting the height the formed spine can take.

If the growth rate is large, the mantle edge grows a large distance than the shell edge for a period of time. As seen on the right of Fig. 23, larger growth rates have been shown to make the spines of a mollusk shell take shorter and more curved forms (Chirat et al., 2013). If a growth rate is large, the additional tissue has less space to fit when adhering to the shell edge, rendering the deformation more pronounced. The more pronounced deformation curves make the new shell layers more likely to go around the spine faster, closing the shape of the spine before it can grow taller.

The opposite process occurs if the growth rate is small (Chirat et al., 2013). The mantle edge grows a small distance more than the shell edge for a period of time, leading to taller and less curved forms due to less pronounced deformations as seen on the left end of Fig. 23. 

Fig. 23 Impact of increasing growth rate on spine geometry.

The mantle stiffness, on the other hand, is the tendency of the mantle to bend and deform when it cannot adhere properly. The mantle tissue has been observed to be non-uniform (Jackson et al., 2006), leading Chirat et al. to construct a model of shell spine formation based on variations in mantle stiffness where “bending stiffness of the elastic mantle is considered to be a function of position”. Their model showed that mantle stiffness could account for the width of the spines formed. In their model, they derived Equation 18:

in which Eb is the stiffness of the mantle edge and s0 is the location along the surface of the mantle edge. Therefore, the constants b1 and b2 control the intensity and scope of the stiffness, respectively. As can be seen in Fig. 24, Chirat et al. (2013) found that an increase in b1 led to weaker stiffness of the mantle edge and consequently a higher curvature at that point. Increases in b2 led to a wider range of decreased stiffness.

Fig. 24 Effects of constants b1 and b2 on mantle stiffness and mantle curvature (Chirat et al., 2013).

Fig. 25 below shows how variation in each factor is what contributes to the diverse array of shell spine geometry:

Fig. 25 Effects of mantle growth rate and mantle stiffness on spine geometry (Chirat et al., 2013).


Mechanical properties of molluscan shells

While the shell’s growth is guided by external pressure, the mollusc, like all living organisms, exists in the context of an ecosystem, where survival is also determined by stresses: environmental and predatory ones. Among the molluscan shells’ many functions, its principal one is to resist mechanical stresses it experiences to avoid catastrophic failure. The question of interest, therefore, is how exactly the mollusc shell succeeds in protecting its host: which macroscopic and microscopic features are relevant, and how do they dissipate energy and contribute to the magnitude and types of mechanical stresses the shell can withstand?

Contributions of macrostructures 

To begin, we will consider the macroscopic feature, dentition, which concerns the mollusk’s tooth shape and placement. Molluscs have 2 major dentition sites, the hinge and commissure sites, which both play a significant role in the shell’s ability to withstand shear stresses (L.D. Coen in 1985) (Fig. 26).

Fig. 26 Torsional force applied to Noetia ponderosa and Mercenaria mercenaria at failure. Shells were altered to remove hinge dentition, commissure dentition, and both at once (Coen, 1985, p. 483).

Both Noetia ponderosa and Mercenaria mercenaria demonstrate relatively high shear stress capacity with both sets of dentitions intact, but upon manual removal of dentition at each location, the shear strength decreases significantly (Coen, 1985, p. 483). The macroscopic features of these shells, such as dentition, are not arbitrary, but  rather serve an essential, protective role (Coen, 1985, p. 483). 

Overview of molluscan shell material structures

It is not only the macroscopic properties that bestow upon mollusc shells their impressive strength. The shells of all molluscs share a basic hierarchical microarchitecture which is responsible for a large portion of this strength. Generally, biomineralized calcium carbonate crystals are arranged into specific configurations and encased in an organic phase which acts as a type of glue (Bernard et al., 2006).

For example, consider one of the most famous of these structures, nacre. Nacre consists of a composite of calcium carbonate, mineralized in the form of aragonite, and organized into vertically stacked and laterally interlocking hexagonal pallets (Wegst et al., 2015). This tessellated lamellar structure (Yadav et al., 2018) is surrounded and held together by an organic protein and polysaccharide matrix (Jackson et al., 1988, p. 416). What is so striking about this organization is how the combination of these individual components yields a substance whose toughness is much greater, not only than that of its individual components, but also of a mixture of its components (Fig. 27). Nacre, in fact, has a work fracture around 3000 times higher than that of pure aragonite, despite having a 95% aragonite composition (Jackson et al., 1988, p. 416). The same is true for other common hierarchical biostructures, like bone.

Fig. 27 Plot of the Young’s modulus against Fracture toughness of various natural composites. For comparison, each composite’s constituent substances, as well as homogenous mixtures of these constituents, are plotted as well (Wegst et al., 2015).

The drawback is that toughness can lend itself to fragility on large scales. However, the molluscan shell is much more complex and possesses multiple other mechanisms to dissipate energy. Figure 28 summarizes the main design features of nacre, which dissipate energy when mechanical stress is applied. One of the most important ones is the organic biopolymeric mortar which separates laterally and vertically adjacent plates and relieves localized stress by allowing for controlled minor inelastic deformations. 

Fig. 28 The various micro and nano architectural features of nacre which function to relieve stress via the dissipation of energy away from the brittle mineralized aragonite plates (Wegst et al., 2015).

Other features, like mineral bridges, keep pallets aligned and resist stresses such as torsion, while nano-asperities on the palette surfaces resist inelastic shearing during the fracture of larger mineral bridges (Wegst et al., 2015).

This general hierarchical structure provides the base for a wide variety of molluscan shells, and much diversity has been observed within this framework. For example, the shell of Strombus gigas (Fig. 29) takes on an almost identical composite structure, only it consists of tessellated and lamellar aragonite rods, or ‘lamellae’ (instead of hexagonal pallets), organized in orthogonally cross sectioned layers and separated by an organic proteinaceous matrix (Kamat et al., 2000). 

Fig. 29 Strombus gigas, or the queen conch (Wikiwand, 2022).

Contributions of micro and nano structures

Now that the relevant features of the nano-scale framework have been summarized, the mechanical behavior of molluscan shells can be rationalized at a structural level. To begin, fracture mechanics (or cracking) will be discussed in both Haliotis rufescens (red abalone) (Fig. 30) and Strombus gigas (Fig. 29). Fracture cracks can be introduced using specialized machinery and resulting damage can be observed with the aid of micro-imaging tools.

Fig. 30 Haliotis rufescens, or red abalone (Abalone Shells variety., 2022).

Fracture analysis of the red abalone shell provides a fantastic example of a nano-scale architectural response to applied mechanical stress. The red abalone shell has two main shell regions: the more brittle, ‘prismatic’ outer region, and the nacreous interior region. Figure 31 (Li et al., 2004, p. 617) depicts simplified diagrams of atomic force microscope (AFM) and scanning electron microscope (SEM) images of the fracture patterns observed in the prismatic and nacreous shell layers, respectively, post-crack indentation (Li et al., 2004, p. 617). 

i)

 ii)

Fig. 31 SEM (figures labelled “a”) and AFM (figures labelled “c” or “d”) images of prismatic and nacreous layers reacting to induced fractures. Figures labelled “b” are schematics depicting crack deflection mechanisms. i) and ii) distinguish between the prismatic nacreous shell layers, respectively (Li et al., 2004, pp. 616-617).

Note the differences in crack propagation, as well as how the nano-scale structure responds to stress input. In the prismatic layer, radial cracks spread outwards interrupted for long distances. There are no mechanisms for stress redistribution. However, radial cracks are much shorter in the nacreous region as stress is distributed amongst the layers of nacre (Li et al., 2004, p. 616). Adjacently interlocked hexagonal tablets separate from each other to form ‘slip bands’, thus redistributing the stress applied. This ‘crack deflection’ constrains damage to stay around the crack site, while minimizing the stress concentration (Li et al., 2004, p. 617). Additionally, the group who performed this fracture analysis detected slight elastic deformation of the aragonite pallets right at the crack sites, which increases energy dissipation but was surely not expected of such a brittle substance (Li et al., 2004, p. 617). 

Fascinatingly, similar crack deflection behavior was demonstrated in the shell of Strombus gigas. This is particularly interesting as Strombus gigas builds a tessellated and lamellar array of orthogonal layered aragonite rods, not hexagonal pallets (Kamat et al., 2000). During bending deformation, cracks form between parallel lamellae and arrest at lamellar cross junctions. This dissipates low-load stress, but as stress increases the cracks pierce the orthogonally layered lamellae. However, even this process is dampened as some of the energy is dissipated via bridging, which occurs between perpendicularly oriented lamellae at cross junctions (Barthelat et al., 2009, p. 22). Despite slight structural differences, the result is nearly identical.

The organic phase separating mineralized aragonite plays a key role by allowing for crack deflection and energy dissipation, but the other nanostructures in nacre are equally important. The nanopillars, small aragonite mineral bridges that connect vertically adjacent tablets, are essential for resisting the shear stress of torsion, which has been measured to be 41.5 ± 14.7 MPa at 2.0 ± 0.8% strain (Alghamdi et al., 2017). Comparatively, simple aragonite crystals subjected to equivalent tests displayed torsional resistances of 14.5 ± 2.2 MPa at 1.4 ± 0.2% strain (Alghamdi et al., 2017). These values were calculated using experimental measurements and equations 19 and 20, which can be visualized and more easily interpreted in Figure 32. 

Fig. 32 A visual representation of the significance of each variable in Equations 19 and 20, which are used to evaluate the torsional resistance of nacre (Alghamdi et al., 2017).

Torsional stress at failure is calculated using the basic formula

where T is torque, r is the cross-section radius, 𝜌 distance between an arbitrary point on the cross-section to the center (0 ≤ 𝜌 ≤ r) and J = 0.5πr4 is the polar moment of inertia (Alghamdi et al., 2017). To calculate the maximum sliding distance (smax), we use a simple application of the definition of the radian:

Where γmax is the shear strain at the external edge, Δθ twisting angle, and l is the height of two vertically adjacent tablets (Alghamdi et al., 2017).

Surprisingly, mathematical modeling has shown that the thin nanopillars, made of brittle aragonite mineral, account for more than 95% of nacre’s shear torsional resistance (Fig. 33).

Fig. 33 Contribution of various nanostructures of nacre to torsional stress resistance. Note that polygonal vertices represent separate conditions and trials, as well as the type of stress applied to the nacreous tablets (Alghamdi et al., 2017).

Conclusion

Using advantageous mechanical processes, mollusks have developed shells that solved their problem of insufficient protection. Most notable among these mechanical processes is the minimization of mechanical energy across the shell, which plays a major role in determining the geometry of shells’ macroscopic and microscopic structures and allowing efficient growth side by side with the mollusk. Indeed, shells’ macroscopic coils, ribs, and spines form to improve the shells’ structural stability, as they minimize the mechanical energy across the shell surface following the imbalance created by external and internal stresses. More specifically, coiling and ribs increase the resistance to external stresses, whereas the formation of ribs and spines appease the internal stresses on the mantle and defend from shell-crushing predators.

Other stresses of concern to the molluscan shell include those applied by environmental and predatorial entities. Macroscopic shell features and the hierarchical nanoarchitecture of the shell’s material protect the mollusc by dissipating applied mechanical energy using a variety of surprisingly simple mechanisms.  

The relevance of such an in-depth analysis of the aragonite-protein structure that lies at the heart of the shell’s energy-dissipation mechanism is its application in design. Understanding the mechanisms outlined in this paper informs how engineers can replicate and leverage these technologies and properties. For example, work by Yadav et al. (2018) looks to advance materials engineering via the “biomimicking of hierarchical molluscan shell structure” using layer-by-layer 3D printing techniques. The possible applications of this armour of nature are both vast and exciting.

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