Influence of Structure and Geometry on the Compressive Deformation Behavior of Macadamia Integrifolia and Bertholletia Excelsa Shells: A Validated Finite Element Simulation Study

Macadamia (Macadamia integrifolia) and Brazil nuts (Bertholletia excelsa) have an impressive mechanical resistance. To better understand the key structural features and foam‐like deformation behavior, mechanical compression tests coupled with numerical finite element simulations are performed. Models with different degrees of structural complexity highlight the effects of geometry where even small changes strongly influence the stress distribution and deformation behavior. Circumferential tensile stresses play a vital role in the failure process. Voids or vascular bundles in the shell wall redistribute stresses and bolster the overall strength of structures by preventing early crack propagation. The void arrangement has the potential to inspire the design of lightweight materials with strategically incorporated porosity, leading to improved mechanical performance. Linear elastic models, despite their oversimplification, are valuable for gaining crucial insights about the influence of geometry. The mechanical behavior is further influenced to different extents by isotropy and orthotropy. The distinctive architecture of nutshells offers valuable insights into the design of composite structures with controlled failure, capable of withstanding high compression loads combining foam‐like behavior and fibrous layers.


Introduction
In nature, a wide range of astonishingly efficient structural and functional solutions has emerged through evolution.The interdisciplinary field of bioinspiration characterizes promising natural structures to deduce innovative solutions to technological problems. [1]Prominent examples are various nutshells and seed coats, some of which are very tough and extremely strong.Thereby, they protect the seeds against environmental influences, and especially against mechanical damage, whether it is due to falling from the plant when they are ripe or from being crushed by animals. [2]The coats of macadamia seeds (Macadamia integrifolia, colloquially addressed as "nuts") and the mesocarp of Brazil nuts (Bertholletia excelsa) are two outstanding examples when comparing fracture forces normalized by shell thickness.They outperform other nuts such as almond, hazelnut, walnut, pecan, and peanuts in terms of their resistance to fracture. [3,4]Macadamia seed coats even outperform ceramics and glasses regarding specific strength and Young's modulus. [5]Common nutcrackers are usually inadequate to open them, requiring fracture forces between 1.6 and 2.7 kN. [6,7] Brazil nuts tolerate falls from trees as high as 50 m, and compression fracture forces above 10 kN are needed to crack their mesocarp. [8]herefore, these seed coats and nut shells have come to the fore as inspiration to develop lightweight impact and puncture-resistant materials. [4,9]hile fracture forces of nutshells and seed coats can be measured by compression testing, stress distributions and identification of regions critical for failure need further calculations.Analytical solutions, however, exist only for compression of thin-walled shells with R/t ≃ 30 (R radius, t wall thickness) between rigid plates, for example, performed for elastic spherical shells with R/t = 10 and 50 in contact with a rigid plate. [10,11]here is no theory to calculate the stress distribution of the relatively thick-walled macadamia seed coats (R/t % 2) and Brazil nutshells (R/t % 5.5).Numerical methods such as the finite-element method (FEM) offer an alternative to calculate stresses and deformations and to deduce fracture strength values. [12]Occasionally, FEM has been used to model the mechanical properties of nutshells.Table 1 gives an overview.
Modeling macadamia seed coats as a simple hollow sphere with a constant wall thickness and linear elastic material properties showed a maximum tensile stress of about 188 MPa in the inner layer of the shell for a compressive force of 1.5 kN, suggesting that failure is initiated at the inner surface. [5]Most works addressed the question of how to crack nuts without damaging the kernel, as this is of economic importance.Based on FEM simulations on a walnut shell idealized as a combination of a cylinder and two cones closing the ends and assuming isotropic, linear elastic material properties, a static uniform line load perpendicular to the shell's suture line was predicted as the best way to extract the kernel as this loading situation led to the most advantageous crack path. [13] more detailed model of a walnut shell was obtained from 3D-imaging data, applying an orthotropic linear elastic, ideally brittle material model. [14]The load was applied by a handheld nutcracker (stainless steel) at a dynamic loading rate of 2 m s À1 .The findings revealed successful kernel extraction; however, the authors suggested that the common nutcracker design should be changed to reduce plastic deformation.The same geometry extraction and material model was used for the simulation of the static compression of hazelnuts between two rigid plates. [15]quivalent stresses of about 10 MPa on the plate were calculated.For a pecan nutshell, the model geometry was also deduced from 3D imaging data. [16]A high-speed impact (hit and run) loading scenario revealed that stresses of about 7 MPa are necessary to induce a crack, for an impact force of 996 kN.
Nutshells and seed coats have a hierarchical microstructure with a complex 3D distribution of pores in different sizes. [3,4]o the best of our knowledge, there are no FEM studies on the stress and strain distribution in seed coats or nutshells considering the hierarchical structure, specifically porosity.
To better understand the deformation behavior and the reasons for the high compressive strength of macadamia seed coats and Brazil nut mesocarp, we performed a series of FE simulations.We deliberately modified the structure, yielding models with different complexity.We examined stress distributions and the diameter widening of the structures that helped us identify the structural features on the length scales of the vascular bundles and the macrogeometry which are critical for failure initiation.

Materials
Macadamia (Macadamia integrifolia) seeds were provided by MAC nuts, WA, Australia.Brazil nut fruits (Bertholletia excelsa) were purchased from Mercadolivre, Brazil.Figure 1 shows the general features of both biological shell structures.Being aware that, botanically, macadamia is no nut and that the mesocarp is only one layer of the Brazil nut, for simplicity, in the following we will address both, the macadamia seed coat and the mesocarp of the Brazil nut as (nut)shells.
The macadamia seed coat is round and has a smooth brown outer surface.A hemispherical suture connects the hilum and the micropyle (Figure 1a).A dense network of vascular bundles pervades the shell, with the highest density near the hilum and a preferred orientation of the longitudinal axis of the vascular bundles in the direction from the hilum toward the micropyle (Figure 1b,c).The shell of the Brazil nut consists of three layers: the exocarp (not shown), which rots quickly, the mesocarp, and the endocarp.Mechanically, the mesocarp is the strongest layer. [4]It has an uneven surface and possesses structural ridges on the inside meeting near the peduncle (Figure 1d,e).On the opposite side of the peduncle is the opercular opening.The shell contains multiple seeds, each of which is protected with a hard tegument.The mesocarp contains a high number of voids, comprising vascular bundles and microcracks (Figure 1f ).The dimensions of the macadamia and Brazil nuts used for the mechanical tests are summarized in Table 2.
The moisture content was determined gravimetrically with a standard-level analytical balance (0.0001 g resolution, HR-200, A&D Company Ltd., Tokyo, Japan) on fractured pieces of each of the macadamia seed coats and Brazil nutshells tested in compression.The pieces were kept at 60 °C for 21 h when no further weight change was observed.The macadamia coats had an average humidity of 7 wt% (standard deviation: 0.6 wt%).The mean value for the Brazil nutshells was 6 wt% (standard deviation: 1 wt%).
To further investigate the influence of material inhomogeneities, artificial shells were printed based on the microcomputed tomography (μCT) data of one macadamia and one Brazil nutshell (see Section 2.2) from polylactic acid (PLA; 3D-printer Anet A8; Shenzhen Anite Technology Co., Ltd., Shenzhen, China).The printing direction was parallel to the direction of the suture for the macadamia shell, and parallel to the axis connecting the peduncle and opercular opening, that is parallel to the ridges, for the Brazil nutshell.Due to size and time restrictions during printing, the printed Brazil nut was scaled down to half the original size.We are confident that this downscaling does not induce a size effect in the elastic range because the structural features influencing the elastic properties of the polymer are still small as compared to the size of the part.This will of course influence the visibility of structural features.However, the features of interest for this investigation, that is the vascular bundles and crack-like structures, were larger than 60 μm and were therefore resolved sufficiently well.

Mesh Generation from μCT
To evaluate the influence of geometry and porosity in the size range of vascular bundles on the elastic deformation and the stress distribution in the shell material, we developed FE models from the μCT data.As previously shown, the tegument of the macadamia seeds and the exo-and endocarp of the Brazil nut hardly contribute to the strength and toughness of the seeds/nuts during compression. [3,17]Therefore, only the macadamia seed coat and the mesocarp layer of the Brazil nut were taken into account for the simulations.An adaptive linear tetramesh (C3D4 elements) was generated from the μCT data with the open-source software "3D Slicer". [18]he first step was segmentation, to discriminate between the shell material and the vascular bundles.Two mesh sets were created by the "segment mesher" extension of the software: one for the shell material and one for the vascular bundles. [19]he latter were idealized as virtually empty tubes, consisting of a very low-stiffness material (see Section 2.5 below), and the tube walls were assumed to be part of the shell material.Figure 2a shows the resulting mesh with 2 106 241 elements.
Compared to the macadamia shell, the Brazil nutshell contains many more and intertwined vascular bundles (Figure 2c).Therefore, to reduce calculation time, we first used 3D Slicer to generate one set of mesh elements, comprising both the voids and the shell material.Then the open-source software "Bonemat" was used to differentiate the shell material from vascular bundles and other porosities in the same size range, based on the two different grayscale values of the two materials in the μCT scan.Darker areas, indicating lower material density, were considered as voids.Subsequently, Bonemat was used to partition the mesh into two element sets. [20]The software classifies mesh elements according to the grayscale value of the μCT data.The threshold was found by visual comparison of the grayscale values in the μCT dataset with the resulting submeshes comprising 409 780 elements when combined.

Geometry Variations
To understand which geometrical features have the most impact on the stress and strain distributions, simplified, nonporous rotational (biaxial) ellipsoid geometries were constructed (Table 3).
The dimensions of the simplified geometries were based on the μCT scans: therefore, for the ellipsoid representing macadamia, a height of 22 mm and a width of 24 mm with a wall thickness of 6 mm were chosen; for the Brazil nut, a height of 110 mm and a width of 100 mm with a wall thickness of 10 mm fit the scanned nut best (Figure 2b,d).M-A and B-A represent the highest degree of abstraction: they are ellipsoids with constant wall thicknesses.With each geometry variation, another structural feature was added according to the structural features of the biological shells.For the macadamia nutshell, these were the hilum, the suture, and the micropyle (Figure 1a).The Brazil nutshell contained the peduncle on one side and the opercular opening on the opposite side (Figure 1d).Inside the shell, the structural ridges joined near the peduncle, resembling a spike (Figure 1f ).M-vb, M, and M-P were models generated from the μCT scans with material properties of the macadamia shell (M-vb, M) and PLA (M-P).Likewise, B, B-vb, and B-P were generated with the properties of the Brazil nutshell and PLA, respectively.Note that specimen B-P was half the original size of the scanned Brazil nutshell.Consequently, to achieve a comparable stress response, only half the displacement was used for loading B-P as compared to the other models.

Simulation Setup
Each geometry was loaded in two directions between rigid plates using the FEM Software ABAQUS. [21]The simulation was static and nonlinear.In Figure 3, the boundary conditions are shown for parallel and perpendicular loading.For macadamia, "parallel" and "perpendicular" mean that the loading direction was parallel and orthogonal, respectively, to the suture.For the Brazil nutshell, the "parallel" and "perpendicular" loading directions were along and orthogonal to the structural ridges, respectively.The bottom plate was locked in translation and rotation, and the upper rigid plate was loaded by given displacements.The maximum displacement u in the simulation was 0.35 mm for the macadamia nut and 2 mm for the Brazil nut.These displacements were chosen based on laboratory experiments, [3,4] At the contact surface, friction coefficients of 0.5 and 0.492 were set for the biological and polymer models, respectively, which corresponded to values reported for friction between wood or PLA and steel. [22,23]][27] In contrast, similar tests performed on Brazil nutshell material revealed orthotropic elastic behavior. [17,24]As no values for the Poisson's ratio of the nutshell materials are known, we chose a value of 0.35 based on reports for wood as the closest alternative material. [28]For PLA, the value was set to 0.36. [24]he vascular bundles were modelled as quasi empty tubes with a value for the Young's modulus close to 0 MPa.To analyze the influence of the Young's modulus on the stress and strain distribution, we performed further calculations on the macadamia (M-vb) and Brazil nutshell (B-vb) models, assigning Young's modulus values between 1000 and 6000 MPa to the shell material.
A mesh sensitivity study was conducted, in which the number of elements was varied until the resulting stresses and displacements changed by less than 2%, ensuring that the mesh density was sufficient for accurate simulation of the mechanical behavior.

Compression Tests
Compression tests were carried out on macadamia (n = 18) and Brazil (n = 8) nuts at room temperature using a universal testing machine (0.001 N and 1 μm force and displacement resolution, respectively; UPM-inspect retrofit 100 kN, Hegewald & Peschke Meß-und Prüftechnik GmbH, Nossen, Germany) with a crosshead displacement rate of 1 mm min À1 .For comparison, 3D-printed polymer nuts were also tested.A maximum preload of 5 N was applied to hold the specimens in place.Two loading directions were applied, as described for the FEM simulations (compare Figure 4).For both species, half of the specimens were tested in each, the parallel and the transverse loading direction, until a crack was visible and a force drop was observed.
For calculation of the displacement fields by digital image correlation (DIC), photos of the nuts were captured every second during the experiments using the timelapse feature of a digital camera (Sony DSC-RX10 II, Sony Europe B.V., Berlin, Germany).Since DIC needs a surface pattern with clearly discernible contrast, the rather smooth macadamia nuts were primed with white color on which black paint was sprayed, yielding a black-on-white speckle pattern.The Brazil nuts did not need such coloring, as the surface itself provided sufficient contrast.The images were evaluated with the DIC software GOM correlate (GOM GmbH, Braunschweig, Germany) using 19 Â 16 pixel facets.The diameter widening of the specimens perpendicular to the loading direction was measured by calculating the elongation for at least four lines between opposite edge points near the longitudinal center of each tested nutshell, with the digital extensometer application of the software.Average values of the measurements were used to represent the diameter widening.For the macadamia nutshell in parallel loading, this widening was measured along the equator between the western and eastern points (d pa in Figure 3a).For perpendicular loading, it was measured also in the widest region of the spherical shell structure, from the suture line to the opposite side (d pe in Figure 3a).For the Brazil nutshell in the parallel loading case, the widening was also measured in the widest region, between the eastern and western points (d in Figure 3b), and in the perpendicular orientation, the widening was measured along the same line, that is orthogonal to a line connecting the peduncle and the opercular opening.The forces were normalized by the average diameters of the specimens or of the simulation geometries.This normalization was chosen because not all nutshells broke completely and the thickness varies along the fracture line.With increasing displacement, the force increases in an approximately linear relationship.As expected, due to the lower Young's modulus, the polymer specimen behaves less stiff as compared to the biological shells.Accordingly, variation of Young's modulus of the macadamia shell material in model M-vb also revealed higher structural stiffness with increasing modulus (data not shown).In both loading directions and for both, the nutshell material and the polymer, the experimental results match the simulations M-vb and M-P very well.Neglection of the vascular bundles (model M) still leads to a very good match, with only slightly higher forces as compared to the model with vascular bundles (M-vb).

Macadamia Nutshell
In contrast, simplifying the shape to an ellipsoid has a pronounced stiffening effect for the parallel loading case: for the same displacements, model M-A predicts higher forces than models M and M-vb.Adding the hilum-mimicking notch (model M-B) leads to even higher forces, which, interestingly, decreases almost to the level of M-A when a depression, mimicking the suture, is added (model M-C).For the perpendicular loading case, all ellipsoid models predict stiffening to the same amount.
Figure 4c,d shows the widening of the diameter normal to the loading direction versus the compressive displacement, as simulated in comparison with the experimental results.For clarity, only one typical result is shown representing the lab experiments (for all results, see Figure S1, Supporting Information).The diameters of both, the "real" specimen and the printed counterpart, widen to a similar amount when compressed to the same percentage in the parallel direction.All models predict an almost linear relationship, with lower-diameter widening values for the models M-vb, M-P, M and M-A, and higher values for the models M-B and M-C.
In contrast, loading in the perpendicular direction results in less widening of the polymer specimen: especially in the beginning, the slope of the graph is higher compared to the simulation M-P, reaching a plateau at a compression of %0.6%.The biological shell is reproduced with the simulations of M-vb.In comparison, the discrepancies between the polymer specimen and M-P are slightly higher.The increases in diameter in M-A and M-B are identical, but higher than those of the real geometries.M-C results in a somewhat higher widening level.Interestingly, for the biological shell and the simulated structures alike, the relative widening of the diameter is much lower than the relative decrease in height.
Figure 5 shows contour plots of the principal stress in sections parallel to the loading direction (for comparison with the results of others, who often show the Mises stress state, see Figure S2, Supporting Information).Here, the maximum or minimum principal stresses are shown, depending on the maximum absolute value.Both principal and Mises stresses show similar distributions; however, principal stresses also reveal whether tension or compression prevails.
Generally, the highest compressive stresses are concentrated on the top and the bottom where the load is introduced, that is, where the boundary conditions are applied.The inner surfaces of the shells show that the values decrease toward the center.The models based on the biological shell geometry (Figure 5a,b) show high tensile stresses at the suture line (arrow 1) as well as under the hilum and above the micropyle for the parallel loading scenario.In the perpendicular case, these areas experience compressive stresses.In the outer shell walls, tensile stresses are observed (arrow 2) that expand to the inside in the middle of the shell.For perpendicular loading, tensile stresses are overall more dominant compared to compressive stresses, especially at the inner shell wall.The vascular bundles within M-vb lead to a reduction of the compressive stresses (arrow 3).The Mises stresses (Figure S2, Supporting Information) further highlight the stress raising effect of the voids and vascular bundles.The elliptical models (Figure 5c,d) generally show a more homogeneous stress distribution with compressive stresses prevailing on the inside and tensile stresses toward the outside of the walls.Addition of the notch in M-B results in a decrease in compressive stresses below the notch (arrow 4) in the parallel loading case.Concurrently, tensile stress regions expand, accompanied by elevated stresses on the opposing side.Further, the compressive stress levels are overall higher.Adding the slight depression ("suture") leads to higher tensile stresses on the inside near the load introduction area (arrow 5).Furthermore, the central region of the shell experiences a greater influence from tension along the suture line, a phenomenon not mirrored in M (arrow 6).In M, higher stresses align with the suture line.
For the perpendicular loading case (Figure 5d), the stress distributions vary only slightly for the different models.The inner shell wall also shows areas of tensile stresses in the middle (arrow 7), which become thinner and follow the depression in M-C (arrow 8).In contrast, in M, the suture experiences compressive stresses, while in the inner part of the shell wall tensile stresses dominate.

Brazil Nutshell
Figure 6a,b shows load-displacement curves normalized by the average diameter of the shells, as simulated in comparison with one typical experimental result (for all results, see Figure S3, Supporting Information).The forces rise almost linearly with increasing displacement.Please note that the experimental results obtained from the polymer specimen and B-P are based on a smaller specimen (half the size of the real nutshell).This is the reason why there is a force drop in the polymer specimen in the perpendicular direction due to larger deformation.
In the parallel loading direction, the forces predicted by the simulations B-vb and B-P are close to those measured in the experiments.For the perpendicular loading direction, the specimen shows lower force levels compared to B-vb while the printed specimen shows higher levels than B-P.Neglection of the vascular bundles and pores in simulation B leads to slightly higher forces.Accordingly, variation of the Young's modulus of the Brazil shell material in model B-vb also revealed higher structural stiffness with increasing modulus (data not shown).
The simplified geometries B-A to B-E show an even stiffer behavior than B. In the parallel loading case, adding a notch (B-B) and a foot (B-C) leads to an increase in stiffness as compared to the simple ellipsoid (B-A).Adding the hole in the foot (B-D) slightly reduces the stiffness, while the inner ridges (modeled by a spike-like structure) (B-E) increase the stiffness again.In the perpendicular loading case, all idealized, elliptical models show the same force answer to the displacement-controlled loading in the simulation.
In Figure 6c,d the diameter widening is plotted versus the vertical compression.For the tested specimen, a typical curve is shown (for all results, see Figure S3, Supporting Information).Compared to the simulation B-vb, the biological shell either shows a considerably lower (parallel) or a considerably higher (perpendicular) diameter widening, for the same compressive displacement.In contrast, the diameter widening of B-P is closer to its experimental counterpart.However, the curve progressions differ.In the parallel loading case, the diameter of the polymer specimen widens more in the beginning, reaching nearly a plateau.The simulation on the other hand shows low values for the widening in the beginning followed by a linear increase.The perpendicular case of the polymer specimen leads to a response similar to the simulation of B-P.Neglection of the vascular bundles in model B leads to a slightly higher widening than in B-vb only in the parallel loading direction; otherwise, the widening is lower.The curve of the pure ellipsoid B-A is close to B with only slightly higher values.Adding the notch (B-B) again leads to slightly higher widening values in the parallel loading scenario.Adding the foot (B-C), the foot with a central hole (B-D) and the inner spikes (B-E) leads to even higher diameter changes as compared to the other models; the differences between these three geometries are very small.For the perpendicular loading scenario, all the simplified models B-A to B-E show the same levels of diameter widening.
Figure 7 and 8 show the distributions of the principal stresses for the Brazil nutshells.Contour plots of the Mises stresses are summarized in Figure S4 and S5, Supporting Information, for better comparison with the works of others.The two evaluations show similar stress distributions and regions of high stresses, critical for failure, are revealed.However, the Mises stresses do not indicate whether tension or compression prevails.
In Figure 7, the principal stress distributions in longitudinal sections, parallel to the loading direction, and on the inner, exposed surface of the shell are shown for the simulations B-vb and B. As also seen in the macadamia models, maxima of the compressive stresses are observed in the contact areas, and the stress levels decrease toward the center.Contrary to macadamia, overall higher compressive stresses are seen in the perpendicular loading case (Figure 7b).The shell walls experience tensile stresses in most regions, besides the regions near the load introduction points.Model B, where the voids and vascular bundles are closed, shows a more homogeneous stress distribution.The voids in model B-vb reduce the compressive stresses, as can be seen in the area marked with arrow 1.The Mises stresses (Figure S4, Supporting Information) further highlight the stress-raising effect of the voids.
Figure 8 displays the principal stress distributions of the simplified elliptic geometries in comparison to the simulation B. For parallel (Figure 8a) and perpendicular (Figure 8b) loading, the maximum compressive stress values occur near the contact areas, decreasing inwards.Tensile stresses are dominant in the outer shell wall, whereas, in the center, tensile stresses become more dominant toward the inside.Compared to B-A, adding the notch (B-B) leads to a reduction of compressive stresses near the load introduction points in the parallel loading condition (arrow 1).Adding the foot (B-C) results in higher tensile stresses below the notch, and reduction of the compressive stress near the load introduction points (arrow 2).The hole in the foot in B-D has a negligible effect on the stress distribution.Adding the spike (B-E) leads to a further compressive stress reduction near the load introduction points (arrow 3) and higher tensile stresses in the center of the shell wall (arrow 4).High tensile stresses are also seen in the upper part of the spike (arrow 5).The inner shell wall of the biological shell model B shows larger areas affected by tensile stresses.In contrast to the parallel loading case, hardly any differences are observed between the geometries for the perpendicular loading case (Figure 8b).

Fracture Patterns
The cracking patterns of the nuts vary depending on the direction of the applied force.In the case of macadamia nuts, when compressed parallel to the suture line, fracture and direction of crack propagation are consistently observed along that line (Figure 9a, left).When subjected to perpendicular loading, crack growth consistently occurred perpendicular to the suture line.Often, the crack passed through the micropyle (Figure 9a, middle).In certain instances, the primary crack randomly diverted from a straight course, resulting in a zigzag line, following the suture over a certain distance at the perimeter of the nut (Figure 9a, right).The fracture patterns of the printed shells mirrored the observations made for the majority of the biological shells.
In the case of parallel loading of the Brazil nuts, the area surrounding the opercular opening undergoes initial destruction before any visible horizontal deformation occurs in the central region of the nutshell, as shown in Figure 9b.The primary crack runs parallel to the loading direction, approximately in the middle between two structural ridges and extending from the peduncle to the opercular opening (Figure 9b).When subjected to perpendicular loading, the Brazil nutshell fractures parallel to the loading direction, in the middle between the peduncle and the opercular opening (Figure 9c).Notably, the polymer specimen also exhibited the described fracture behavior.The fracture surface of the biological Brazil nutshells is irregular and contains many voids (Figure 9b,c, right).

Discussion
Macadamia and Brazil nutshells resist impressively high compressive forces prior to fracture. [4,9]To understand the structural features leading to this high strength, knowledge of the inner stresses and strains is needed.Our FE models make it possible to assess main relationships between the structures and the loads sustained.The nutshell geometries can be likened to ellipsoids.However, they do not fulfil the conditions of a thinwalled container and the geometry deviates from a sphere.Therefore, currently there is no analytical solution able to describe the states of stress encountered.Moreover, analytical approaches usually assume isotropic material properties and neglect the influence of porosity and fiber orientation which have important implications on the stress and strain state in the nutshells.We therefore applied FEM to assess the stress and strain states using models with different degrees of geometric and structural complexity.In this manner, we could show an important influence of the geometry in the region of load introduction affected by the presence of vascular bundles and microcracks.In the following, we discuss these influences in detail and propose a failure model.

Contact Radius Evaluation: Comparative Analysis with Hertz Theory
Macadamia and Brazil nutshells are ellipsoidal in shape with a circular cross section and a relatively small difference between the long and short axes.Therefore, as a first approximation, Hertz theory for plate loading of spheres in compression can be used to estimate the contact radii.For a sphere with radius R loaded by a rigid plate to a compressive displacement of u, the Hertz contact radius, a, is given by [29] a The resulting depth of the elastic deformation, t, is then given by [29] t Using spheres with radii corresponding to the longer and smaller axes of the ellipsoid, models M-A and B-A provide upper and lower bounds.For the simulations, average values for both loading cases are given.For both the macadamia and Brazil nut models, the analytical solution results in higher contact radii than those predicted by the simulations (Table 5).The depth, t, of the elastic deformation is much smaller than the wall thickness, highlighting the pronounced stress concentration in the load introduction regions.The differences between the analytical and the simulation results span from %25% to %50%, with much higher values predicted by the Hertz theory.
The differences between the analytical and the simulation results amount to %30% to 50% for the contact radii and the elastic deformation depth of both shells, highlighting the importance of considering the deviation of the shape of the nutshells from a sphere.

Macadamia Nutshell
Fracture of the macadamia nutshell nearly always begins at the suture, parallel to the loading direction.This finding coincides with our previous findings on a much higher number of nuts. [9]gure 9.Typical fracture patterns of the shells: a) macadamia: left, parallel loading case: crack along the suture; middle, perpendicular loading case: straight crack through the micropyle, orthogonal to the suture; right, perpendicular loading case: "zigzag" crack propagation, crack follows the suture over a certain length but avoids the micropyle.b) Brazil nut, parallel loading sequence: left, preloading state; middle, right: with increasing compression loading, the "foot" around the opercular opening is increasingly destroyed; right, cracks (solid black line) propagate parallel to the structural ridges and the loading direction; the inset shows a structural ridge inside a fractured piece.c) Brazil nut, perpendicular loading case: left, preloading state; middle, loading state before the fracture occurs; right, fractured state.
Interestingly, also the printed polymer sample showed this behavior.As the suture constitutes an area of smaller wall thickness, stresses are considerably higher along this line, especially the tensile stresses in the parallel loading case.Thus, the suture acts as a predetermined breaking point, especially close to the micropyle where the wall is the thinnest (compare Figure 1c).
The simulations further indicate a higher diametrical widening for parallel loading.This is due to the larger contact area when loading is introduced to the hilum region.Here, the stiffness is relatively lower, due to the high density of vascular bundles, which is not the case for the perpendicular loading case.For most nut specimens tested, the model predictions apply.However, there are some exceptions where very low widening was observed in the parallel loading case (see Figure S1, Supporting Information).The deviations observed are likely due to the presence of additional defects such as voids or microcracks within the structure that are too small to be seen in the μCT scan or that were not present in the scanned nut.Depending on their orientation, these defects may induce an even more foam-like behavior to the nutshells.Therefore, the resulting forces observed for these nuts are relatively low.
Under perpendicular loading conditions, we observed two different fracture patterns.The nutshells exhibiting a straight fracture path parallel to the loading direction and through the micropyle show more pronounced widening.In contrast, nutshells with a more tortuous crack path exhibited less widening.This suggests that nutshells with a more brittle fracture behavior (straight crack path) behave more like hard, elastic shells, while those with a tortuous crack path show a more foam-like behavior.
In our simulations, we included some, but not all, structural characteristics of the macadamia nutshell.We replicated the network of vascular bundles, but macadamia nutshells possess a hierarchical sandwich structure, consisting of multiple layers with different cell geometries. [3]The sandwich structure, and, especially, the preferred orientation of the sclerenchyma fibers and their central pores in one of the layers, are important for the stress distribution and the deformation behavior.Considering more structural features of the complex nut microstructure in the simulations is therefore expected to yield more realistic results.This conclusion is supported by the good match of the experimental and simulation widening results of the printed polymer nutshells (compare Figure 5c,d).A comparison between the experimental and simulation results, specifically in the linear-elastic range, suggests that the printing direction of the components has a relatively minor impact before reaching the point of fracture.

Brazil Nutshell
There are some distinct differences between macadamia and Brazil nutshells, though both benefit from a hierarchical structure.The Brazil nutshell has no (weakening) suture line; rather, structural ridges lead to a more homogeneous stress distribution in the parallel loading direction. [17]The structural ridges and their effect on the stress distribution explain the higher fracture forces observed in the parallel as compared to the perpendicular loading case.Further, the Brazil nutshell has an overall irregular surface resulting in a more pointwise load introduction in both loading directions, and stepwise flattening of the surface leads to a continuously increasing contact area with increased load.Depending on the curvature of each individual nut near the contact surface, the size of the contact area will be smaller or larger, as the stress distribution in B-A indicates.Here, in the parallel direction, the ellipsoid shape shows a greater curvature; thus, the contact areas are smaller as compared to the perpendicular direction.
The compressive failure in the case of parallel loading occurs along a line through the thinner wall region between the structural ridges and extends through the peduncle and the opercular opening.In the case of perpendicular loading, the load was introduced to a face between the structural ridges.In this case, cracks primarily tear a structural ridge apart, propagating in the loading direction between the peduncle and the opercular opening.In contrast to the findings reported in ref. [17], we did not observe crack paths through the peduncle for this case.Instead, our simulations indicate that the area surrounding the peduncle is stiffer, due to the convergence of structural ridges.According to ref. [4], the fibers of the primary layer of the mesocarp are aligned parallel to the loading direction in the case of perpendicular loading.The individual fibers become separated from each other due to circumferential tension stresses, and cracks, therefore, can easily propagate between the fibers.During parallel loading, the crack must traverse through the fibers, which leads to a higher resistance against crack propagation.
The printing direction of the polymer geometries has a similar effect as the fibers in the nutshell on the failure of the specimen.While the anisotropy of the printed geometry does not influence the elastic behavior, fracture is strongly influenced by the printing direction, and the individual layers begin to separate from each other when exposed to tensile stresses.
For parallel compression, we observe differences between the experimentally measured diameter widening and the predictions from the simulations (Figure 6c).Notably, the simulations B-vb and B suggest pronounced initial deformation in the region surrounding the opercular opening, preceding any notable widening along the central axis.The same region experiences early failure in the experiment, yet the diameter increase of the specimen is comparably low.The 3D-printed specimen exhibits greater initial diameter widening compared to B-P, suggesting that the lower portion around the opercular opening possesses sufficient strength to deform the surrounding material before undergoing catastrophic failure.This observation suggests that a linear elastic material model is inadequate for accurately describing the behavior of the shell and the printed geometry, particularly in the contact area.The higher diameter widening predicted by the models B-vb, B, and B-P for the perpendicular loading direction as compared to the parallel loading case can be explained as follows.First, the presence of the opercular opening leads to reduced resistance, allowing for easier widening of the specimen.Second, the larger and more symmetrical contact area resulting from the curvature of the specimen also contributes to an increased diameter widening because the stresses are more evenly introduced into the shell walls, while stress concentration near the region of load introduction appears to protect the walls from higher stresses.
This explanation is supported by comparing the experimental and simulation results of the polymer specimen where the simulation of B-P reproduced the experimentally observed deformation very well.

Failure Model
Putting together the knowledge about the distribution of structural features and different loading scenarios makes it possible to predict the behavior of the nuts.Figure 10 illustrates the stress state in the nut simulations, based on absolute principal stresses taken from the FEM results.
In the contact compressive stresses oriented along the loading direction are more dominant in the inner shell region, while the outer part of the shell wall experiences mainly circumferential tensile stresses.In the equator of the shells, tensile stresses dominate over the entire thickness of the wall, as has also previously been reported for macadamia nut models. [5]hese tensile stresses are the main reasons for crack initiation in nutshells subjected to a compressive load.If loading is applied parallel to the suture of macadamia nutshells, the tensile stresses pull the notch-like structures (suture, micropyle, vascular bundles) open, leading to the lower resistance to failure observed in compression tests.We hypothesize that the lower density of vascular bundles in the outer shell in the equatorial region is a design feature of macadamia nuts, protecting them against premature failure.In the perpendicular loading case (Figure 10a, right sketch), the micropyle experiences compression from the inside, while the surrounding shell area is subjected to circumferential tensile stresses, making this region more prone to fracture. [9]Our own and the experimental findings of others show that fracture always follows the suture, connecting the hilum and the micropyle. [9]ompression loading of Brazil nuts parallel to their inner ridges (Figure 10b, left sketch) leads to bending loading of the inner shell near the opercular opening and the region where the structural ridges converge.Thus, these structures are "pulled open" and they act as notches, with high stress concentrations (Figure 7a).Accordingly, the fracture follows a line from the peduncle to the opercular opening.In the case of perpendicular loading (Figure 10b, right sketch), circumferential tensile stresses dominate, leading to fracture along a line from one contact point to the other (Figure 9b).Microcracks, voids, and vascular bundles act as notches (Figure 7), leading to a rough fracture surface (Figure 9b).
These findings are useful in creating bioinspired structures from relatively brittle based materials, introducing controlled deformation and adding design features that lead to benign failure under overload conditions.Of particular interest are voids, where both shape and orientation may be tailored in two ways.The long axes of the voids in the Brazil nut are aligned with the long axes of the fibers. [4]Therefore, upon application of external load, closing of these voids not only prevents significant diameter widening due to a more foam-like behavior, it also results in increased stiffness of the entire structure.On the other hand, the voids act as notches.While this might be potentially detrimental, it also leads to localized noncritical failure into microcracks, thereby dissipating energy, resulting in a more benign failure behavior.Similarly, in the macadamia shell, the arrangement, orientation, and density of vascular bundles in regions which experience less detrimental stress states can be used to design functional materials with graded porosity allowing transport of liquid while affecting failure resistance less than more homogeneous arrangements of microstructures.

Conclusion
By combining the results of compression loading experiments of naturally grown shells and of FEM simulations from models with different degrees of structural complexity, we provide valuable insights into the stress distribution in different nutshells, representing thick-walled shell structures.Together with existing knowledge about the microstructure, our work explains reasons for the deformation and fracture behavior of two very different, but similarly strong, nutshells: macadamia and Brazil nuts.
Major insights for the development of impact and punctureresistant lightweight spherical containers include the following.1) Models with simplified geometries highlight that even small structural changes have a major impact on the stress distribution and the deformation behavior.They further emphasize the critical role of the contact area with the compression plates.Circumferential tensile stresses play a vital role in the failure process.2) The idealized linear elastic material model, despite its oversimplification, has proven to be valuable for gaining crucial insights into the influence of geometry.3) The distinctive architecture of nutshells offers valuable insights into the design of composite structures capable of withstanding high compression loads and combining foam-like behavior and fibrous layers.4) Voids or vascular bundles in the shell wall redistribute stresses and bolster the overall strength of structures by preventing early crack propagation.The void arrangement inspires the design of lightweight materials with strategically incorporated porosity, leading to improved mechanical performance.5) The mechanical behavior of nutshell materials is influenced to different extents by isotropy and orthotropy.This focuses attention on considering direction-dependent properties in which the fibers are not placed perpendicular to the circumferential tension stresses.

Figure 1 .
Figure 1.a) Macadamia seed coat shown from the hilum and the micropyle side, with the connecting outer suture visible as fine line; b) μCT: reconstructed longitudinal section through a macadamia seed coat with the edible seedling inside; vascular bundles are visible as black entities, with a higher density near the hilum (top); c) μCT: 3D-reconstructed volume of the macadamia seed coat, highlighting the network of the vascular bundles; d) Brazil nut showing the outer geometry of the mesocarp and the peduncle and opercular opening; e) μCT: reconstructed longitudinal section of Brazil nut with mesocarp, endocarp, tegument, and edible seeds; the mesocarp is the thick white layer positioned outside the endocarp and containing (black) vascular bundles and crack-like structures; f ) μCT: reconstructed 3D data showing structural ridges and the network of vascular bundles and crack-like structures.

Figure 2 .
Figure 2. a) Meshes of the macadamia shell material (brown) and the vascular bundles (blue); the inset shows the mesh of the vascular bundles only, highlighted in blue; b) longitudinal sections of the idealized, ellipsoid "macadamia" shell geometries: M-A: yellow lines; M-B = M-A plus v-notch, representing the micropyle (blue); M-C = M-B plus hemispherical depression, representing the suture (pink); c) meshes of the Brazil nutshell material (brown) and the vascular bundles, voids and microcracks (blue); the inset shows the inner voids highlighted in blue; d) longitudinal sections of the idealized ellipsoid "Brazil nut" shell geometries: B-A: yellow lines, B-B = B-A plus sharp notch, representing the peduncle (dark blue); B-C = B-B plus "foot" opposite the peduncle (black); B-D = B-C plus "hole," representing the opercular opening (pink); M-E = M-D plus "spike," representing the part of the inner ridges near the peduncle (light blue).

Figure 3 .
Figure 3. FEM boundary conditions: a) macadamia, b) Brazil nutshell, both shown for the "parallel" loading case on the left-hand side and the "perpendicular" loading case on the right-hand side.The top plate was allowed to move in the orthogonal direction for defined displacement values (u), the bottom plate was blocked in translation and rotation.d pa , d pe , and d mark the lines along which the horizontal diameter widening (orthogonal to the loading direction) was measured.For the Brazil nutshell, d is used for both loading directions.

Figure
Figure 4a,b shows load-displacement curves for macadamia as calculated by FEM in comparison with one representative, typical curve measured in the laboratory experiments (for all experimental results, see Figure S1, Supporting Information).The forces were normalized by the average diameters of the specimens or of the simulation geometries.This normalization was chosen because not all nutshells broke completely and the thickness varies along the fracture line.With increasing displacement, the force increases in an approximately linear relationship.As expected, due to the lower Young's modulus, the polymer

Figure 4 .
Figure 4. Response of the macadamia nutshell, the printed geometry, and the simulation models to compressive loading: a,b) force-displacement curves with force normalized by the diameter of the specimen c,d) diameter widening relative to original diameter plotted versus the compressive displacement relative to the original height of the specimens in loading direction ("compression").
vb vb v vb v vb vb v vb b b vb vb vb b b b b v v v v vb v v vb b vb vb b b b v vb v v v v v v vb v vb b b b b b vb v v vb vb b b b v v v v v vb b b b b v vb b b b b v vb v vb b b b v v vb v v v v vb vb b b b b b b b v v vb b b b v v v v vb b vb b b v v v vb b b b b

Figure 5 .
Figure 5. Distribution of the principal stresses shown for half spheres sectioned parallel to the loading direction through the hilum and micropyle: a,b) comparison between the models M-vb and M (left and right contour plot, respectively): a) parallel and b) perpendicular loading direction; c,d) comparison between the nonporous macadamia nutshell model M and the elliptical idealizations: c) parallel and d) perpendicular compression.Note the different stress scales for panels (a,b) and (c,d).Please note that for the "principal stress" minimum or maximum values are shown, depending on the maximum absolute value.

Figure 6 .
Figure 6.Results of in silico and in vitro compressive, displacement-controlled loading experiments of the Brazil nutshell, the printed geometry, and the simulation models: a,b) force-displacement curves with force normalized by the average diameter of the specimen; c,d) diameter widening relative to the original diameter plotted versus the compressive displacement relative to the original height of the specimens in loading direction ("compression").

Figure 7 .
Figure 7. Distribution of the principal stresses in longitudinal sections (parallel to the loading direction) of the Brazil nut shell: a) parallel and b) perpendicular compression.The left and right contour plots display the results for the models with and without vascular bundles, B-vb and B, respectively.Please note that for the "principal stress" minimum or maximum values are shown, depending on the maximum absolute value.

Figure 8 .
Figure 8.Comparison of the maximum principal stress states between the nonporous Brazil nutshell model and equivalent ellipsoids, shown in sections parallel to the loading directions: a) parallel and b) perpendicular loading.Please note that for the "principal stress" minimum or maximum values are shown, depending on the maximum absolute value.

Figure 10 .
Figure10.Failure model for parallel (left) and perpendicular (right) loading case for a) macadamia, b) Brazil nut; the pale red and blue shading denote regions primarily under tension and compression, respectively.The yellow channels in the macadamia sketches mark vascular bundles (sized up to show their long axis).For clarity, and to highlight the preferred orientation of the bundles, they are only shown in the region below the hilum, where their density is especially high, and in the equator region, where high tensile stresses act orthogonal to their long axis.The dots in the Brazil nutshell represent voids, cracks, and vascular bundles.

Table 1 .
Summary of published FEM studies on "nutshells".The voxel size for the macadamia data was 16 μm, and for the Brazil nut data, it was 60 μm.Accordingly, the resolution of the Brazil nut data was %4 times smaller.

Table 2 .
Height and width (average AE standard deviation) of the macadamia and Brazil nut shells used for mechanical testing."Height" denotes the diameter parallel to the suture of the macadamia shells and parallel to the ridges for the Brazil nut shells.

Table 3 .
Geometries used for the simulation.

Table 4 .
Material properties applied in the FE models for parallel (||) and perpendicular (⊥Þ loading.