4.1. Distribution of stress and strain
When models are loaded with the nominal force (force corrected for radular size) in the FEA, we can directly compare effects on stress and strain distribution between species and between tooth types (Figs. 5–8). When models were loaded with 1 N, 0.5 N, and 0.1 N (forces are not corrected for radular size), similar to the situation in the breakings stress experiments, we can, in contrast, directly see the effect of the radular size on the stress/strain distribution.
Overall, as determined in our previous FEA on the teeth of Spekia , the simulated stress and strain under nominal force can be explained by the shape of the teeth and the local mechanical properties that were analysed in previous studies [48, 51]. The simulations of the mechanical behaviour are, in most cases, similar to the observed ones, determined by previous breaking stress experiments under wet condition ([84–85]; see Fig. 3 for the areas of failure in the experiment):
Thick, short, and broader teeth, as the centrals and laterals of Spekia and Lavigeria, are not as prone to deformation and do not show areas of high local stress, which can enable force transmission to the ingesta [see also 25,90]. These teeth are additionally stiffer, leading to a high ability of the material to transmit forces [for the relationship between Young’s modulus and force transmission see e.g. 99–102], supporting puncture mechanics and the resistance to failure [see e.g. 103; review on puncture mechanics see 104]. In our breaking stress experiments these teeth could resist higher forces (mean ± standard deviation; Spekia, laterals: 799.83 ± 313.47 mN, centrals: 979.50 ± 381.14 mN; Lavigeria, laterals: 270.95 ± 87.49 mN, centrals: 700.90 ± 255.52 mN) and were less prone to failure.
The softer, longer, thinner, and slender teeth, in contrast, as the marginals of Spekia and Lavigeria, the laterals and marginals of Bridouxia, and the centrals and laterals of Cleopatra, experience higher strain and stress in the FEA. In our previous breaking stress experiments, these teeth could resist less force, showed a high ability of bending, and usually failed (between 83 and 272 mN, depending on the tooth type and species). Overall, models as well as real teeth (a) can deform more easily during interaction with the ingesta, (b) show areas of high stress, and (c) are more prone to failure.
However, even though central teeth of Bridouxia are relatively thin, they show lesser concentrations of stress and strain in the simulations and additionally could resist higher forces (329.11 ± 128.06 mN) than marginals (96.08 ± 13.33 mN) in previous breaking stress experiments. This could be explained by their mechanical properties, as they have a higher Young’s modulus than the marginals (Fig. 3), and by the morphology of their basis. They are broader thus and possess a large attachment area with the membrane, enabling better stress redistribution [see also 41]. However, the breaking stress experiments also revealed that the laterals can resist to similar force (315.31 ± 104.99 mN) than the centrals, even though they are narrower and have similar mechanical properties. This inconsistency between the FEA and experiments could be explained by the ability of the wet lateral teeth to bend and rely on the lateral teeth from the adjacent rows, gaining support [for the importance of tooth-tooth interaction see also 87–91]. This collective effect, is however difficult to simulate in the FEA.
Within each heterogeneous tooth, the areas of high local stress and strain also correspond to the values of the Young’s modulus. Additionally, the simulated mechanical behaviours of the tooth areas usually reflect real mechanical behaviour observed in breaking stress experiments. Areas that are rather soft, as the styli and bases of the marginals in Bridouxia, Spekia, and Lavigeria and of the laterals in Bridouxia, exhibit a high ability to deform both in simulations and experiments. In docoglossan teeth, this bending behaviour of the stylus has also been previously simulated by the FEA . In our simulations, these areas additionally show high local stress, which corresponds to the areas of failure in the experiment (Fig. 3). In contrast, the stiffer cusps did not deform as much as in the experiment and did not show high local stresses in our FEA. For radular teeth, the importance of the heterogeneous distribution of material properties was previously also determined in docoglossan teeth of Patella and Polyplacophora; here  detected that the tooth’s part, interacting in the ingesta, is harder and stiffer, whereas the underlain parts are softer and more flexible [see also 58 for the flexibility of the stylus]. The co-appearance of harder and softer layers probably leads to a reduction of abrasion in the radular cusps [8, 45] as observed in other structures as well [e.g. 54,105]. The flexibility of both the stylus and basis probably serves as a shock absorber, when interacting with obstacles [see also 58,91], a mechanism also previously reported from other biological structures [e.g. 106–180].
For the homogeneous teeth (with similar Young’s moduli) of Cleopatra we previously detected, that their force-resistance, determined by breaking stress experiments, is the highest in centrals (350.89 ± 49.44 mN), followed by laterals (170.46 ± 32.30 mN), and finally marginals (136.75 ± 16.50 mN). This is contrary to the distributions of stress and strain, obtained from the FEA (here stress and strain are low in the marginals and high in the centrals and laterals). This can be explained by the specific arrangement of teeth: in our models the outer marginal tooth embraces the inner, smaller one, leading to a reduction of high local stress and to a deceased ability of both teeth to deform together. This is the configuration that can be often observed in the SEM (Fig. 1E). In our previous breaking stress experiments, radulae were extracted from the specimens, taped onto glass objects slides, and teeth were stroked into the proposed feeding position. During this latest step, marginal teeth were unfortunately separated in many cases, thus breaking stress experiments were performed with individual and not with naturally interlocking teeth. If and to which extend embracing teeth can resist to higher force should be studied in further experiments. Within Cleopatra’s teeth, we detected high stress and strain at the area between each cusp and stylus, which is in contrast to the experiments, where failure occurred between the stylus and basis (Fig. 3). This can be also explained by the position of teeth in the simulations, as each stylus has a relatively large area of contact with the adjacent stylus from the same tooth type, leading to a distribution of stress across the styli to the underlain radular membrane, but also reducing the stylus’ ability to deform.
When stresses are not corrected for size, we detect, as expected, that smaller radulae show more stress and strain than larger ones, with Bridouxia showing the highest local concentrations of stress/strain, followed by Cleopatra, Spekia, and finally Lavigeria with the lowest ones. Even though the FEA does not allow direct conclusions about failure behaviour or force-resistance of structures, we can suggest the presence of some relationship between the breaking forces, determined in the experiments, with the stress simulations under the defined loads (1 N, 0.5 N, 0.1 N). In the experiment, Cleopatra’s teeth were able to resist to maximal 400 mN (comparable to FEA, loaded with 0.5 N), Bridouxia’s to 460 mN (comparable to FEA, loaded with 0.5 N), Lavigeria’s to 956 mN (comparable to FEA, loaded with 1 N), and Spekia’s to 1360 mN (comparable to FEA, loaded with 1 N). When the stress scales of the FEA models are considered, structural failure should actually occur, when areas of the modelled teeth experience between 1 to 6 MPa. However, as mentioned above, Spekia is able to resist to the highest forces in real experiments, followed by Lavigeria, Bridouxia, and finally Cleopatra. This already indicates that besides the parameter size, other factors seem to be important to reduce stress and strain in living radula. Our here presented models have distinct local Young’s moduli, but are modelled as filled solid bulk materials. Real teeth are, however, composed of fibres and their arrangement, size, and density seem to contribute to the reinforcement of the tooth itself, as it was detected for limpet and chiton teeth [44–46, 109–114]. These parameters, however, await further investigations in the paludomid teeth.
4.2. Functional and trophic specialisations of tooth types
Padilla  summarized previous approaches on radular function and proposed new avenues to gain deeper insight into its functionality and in general to molluscan ecology. She highlighted the importance of the 3D shape, material properties, and interaction of teeth.
By the here presented results of the FEA, that include these parameters, we were able to verify previous hypotheses about tooth functionalities in paludomid gastropods. In soft substrate feeders, all teeth are rather used for collecting particles (monofunctional radula; [51, 85]), whereas in mixed and solid substrate feeders, the centrals and laterals rather loosen food from the substrate, and the marginals collect the particles afterwards (multifunctional radula; [48, 51, 85]. This is supported, as mentioned above, by the simulated mechanical behaviour of teeth under nominal force [see also 49 for Spekia]: the centrals and laterals of Lavigeria and Spekia are rather capable of transferring forces without deformation. The marginals show, in contrast, a high capability of bending at the basis and the stylus, which results in the reduction of failure, but also does not facilitate a direct transfer of forces from the radula to the food. They probably gather the loosened particles in form of an ‘inward raking‘ during retraction of the radula from the ingesta [see 27]. This hypothesis is also supported by another previous approach, involving a physical radular model of Spekia . In that study, we performed dissections of adult specimens, extracted the whole buccal mass with the musculature and radula, documented the anatomy, and mimicked the structures by 3D printing and assembly of fabrics. With this approach, we were able to build the first, relatively simple, but movable radular model. By the manipulation of the radular supporting structures of the model the interaction and ranges of motion of the radular structures could be documented. Hereby we found that the marginal teeth are flexed as consequence from a rotation of the underlain buccal mass musculature and perform a raking motion from the middle of the radula to the outer edges, an ‘inward raking’.
A high ability of bending and the presence of areas of high local stress, leading to a higher risk of breaking, was also simulated by the FEA on the centrals and laterals of Cleopatra. For taenioglossan radulae, it was previously hypothesized that the central teeth are rather used for gathering food [27, 115–116], which seems to be the case for Cleopatra, but, as mentioned above, not for Lavigeria and Spekia. Thus, the functionality of certain tooth types varies greatly between species. We however, also detected by the FEA that the marginals of Cleopatra, when arranged in an embracing position, show less stress and strain. This indicates that in this species marginals are potentially more effective in forcefully loosening algae, which was previously also proposed for the marginal teeth of Littorina .
As mentioned above, we determined by the FEA that with increasing radular size, stress and strain decreases. However, in previous experiments, teeth of Spekia could resist higher forces than those of Lavigeria, and Bridouxia - higher forces than Cleopatra . We therefore propose that the observed size differences between the species studied are the result of adaptations to distinct algae cover types rather than an assistance in foraging on the same algae cover, but with a greater food loosening ability due to larger contact areas between tooth and ingesta. To test this hypothesis, however, the identification, sampling, and finally mechanical testing of the biofilm and algal coverage in Lake Tanganyika is necessary.