Topologically interlocking material (TIM) systems are composed of convex polyhedral units placed such that building blocks restrict each other's move- ment. Here, TIM tubes are considered as rolled monolayers of such assem- blies. The deformation response of these assembled tubes under diametrical loading is considered. This investigation employs experiments on additively manufactured physical realizations and nite element analysis with contact interactions. The internal load transfer in topologically interlocking tubes is rationalized through inspection of the distribution of minimum principal stress. A thrust-line (TL) model for the deformation of topologically in- terlocking tubes is established. The model approximates the deformation response of the assembled tubes as a the response of a collection of Mises trusses aligned with paths of maximum load transfer in the system. The pre- dictions obtained with the TL-model are in good agreement with results of nite element models. Accounting for sliding between building blocks in the TL-model yields a predicted response more similar to experimental results with additively manufactured tubes. Keywords: Architectured Material Systems, Topological Interlocking, Tube Preprint submitted to International Journal of Engineering Science August 23, 2021 Structures 1. Introduction Architectured material systems comprise of periodic geometric material features larger than the microstructure of the underlying material (e.g. a grain size) but smaller than the overall component of interest. Material architecture is of interest as a design approach as it can be used to tailor the response of structures to speci c applications [1, 2]. Synthetic architectures exist as lattice materials [3], kirigami or origami [4], and as assemblies of individual building blocks [5]. Many of these material architectures draw inspiration from natural occurrences of periodic geometry [6]. Plants and animals have evolved to possess skins, scales, armors, and other extremities that give enhanced functionality with structurally unique constitution [7, 8, 9]. Bamboo for example is composed of stacked tubular segments that each possess multi-cell micro-architecture [10], with the architecture seen as responsible for high compression and bending strength. Seed and nut shells were recently shown to possess a micro-architecture that results in high durability [11]. Seed shells composed of topologically interlocking 3D puzzle-type components were demonstrated to possess higher the strength that seed shells built from a ber-type material. Epithelial cells in embryos have been found to take a scutoid shape during their growth, which allows the collection of cells to take on a tubular shape [12]. This collection of scutoid-shaped cells is much softer than the puzzle-type component assembly in nut shells, but still provides structure to a developing embryo. The driving mechanism in these examples of naturally resilient materials is a segmented 2 and interlocking material architecture. It can thus be hypothesized that using this bioinspired principle, traditionally monolithic structures could be replaced with assemblies of interlocking building blocks to achieve a strong and resilient engineered structural response. Topologically interlocking material (TIM) systems are one example of an architectured material system that utilizes the principle of material architec- ture through interlocking building blocks. Historically, interlocking building block systems have been used to construct oors and walls, see discussion in [13, 14]. Recent e orts have revived these concepts in the context of a modern stone architecture for the 21st Century [15, 16, 17, 18, 19, 20]. In the context of engineering material systems, the terminology Topologically Interlocking Material Systems has been introduced [5, 21, 22, 23, 24]. When considering an underlying brittle solid, these studies demonstrate an increase in toughness, or energy absorption capability, for the segmented system rel- ative to the corresponding monolithic system. Even more signi cant, [25] demonstrated the simultaneous increase of strength and toughness for the segmented system over the corresponding monolithic one. Recent advances have focused on the altering of the mechanical properties of the TIM systems by manipulating the segmentation architecture. In [26] it was shown that heterogeneously segmented system do possess higher strength and toughness than those made of all identical building blocks. Furthermore, in [27] it was shown that the response of segmented systems depends strongly on the geometry of individual building blocks. The unique load response of TIM systems is a result of the load transfer within the system, which di ers from that in corresponding monolithic sys- 3 tems. The periodic arrangement of building blocks and their interaction via contact allows load transfer to be described by the concept of force networks, and the deformation of TIM systems is then obtained by applying fundamen- tal mechanics of material concepts to the deformation of the material in the force networks. While there exists a description of the mechanical response of planar TIM systems utilizing this force network concept [22, 28, 29], the behavior of non-planar TIM systems under external loading is not well de- ned. With the realization of a wide array of non-planar TIM geometries being plausible [15, 16, 30, 31, 32], it is relevant that the mechanics of these structures be fully understood. Tunnels constructed of interlocking building blocks have been shown to have higher energy absorption capacity than stan- dard construction methods [33]. Thermoacoustic instabilities in combustion chambers have also been mitigated with the energy absorption properties of TIM construction [34]. Further motivation is derived from the already mentioned study of interlocking seed wall structures [11]. The present study considers the mechanical behavior of TIM tubes un- der diametrical loading and investigates how tube aspect ratio and building block contact conditions in uence the overall system response. Both an ide- alized system and a system conforming to experimental conditions is con- sidered. Thereby, the mechanical response as predicted from FE-models can be compared to that of a closed form thrust line model and to a measured experimental response. 4 2. Methods 2.1. Material System Architecture TIM tubes are constructed from truncated quasi-tetrahedral building blocks. The plurality of building blocks is arranged to form a tubular struc- ture. The discrete character of the TIM systems requires the presence of an external frame, or geometric constraint. These frames conform to the TIM tube at one of their axial surfaces. For the tubes, such frames constrain the collection of building blocks at each tube end. Each of the constraining frames is equivalent to one ring of fused-together building blocks. Figure 1 shows an example of a TIM tube con guration. (a) (b) Figure 1: (a) Example of a TIM tube assembled by 150 individual building blocks and constrained by two rigid frames at the tube ends. (b) Detail of the axial cross-section of TIM tube. 5 The building block geometry is de ned by square prototiles of edge length, a0, placed tangentially to a cylinder radius R. The square prototile edge length is related to the number, nf, of building blocks (i.e. the number of facets) along the tube perimeter as a0= 2Rtan(=n f). Each edge of the square prototiles lies in its own plane oriented at the interlocking angle int relative to a cylindrical coordinate system with its origin at the tubes center. Angles are de ned from the rzplane for vertical edges and from the rplane for horizontal edges. Alternating intbetween its positive and its negative value when traversing the square prototile perimeter results in the intersections of the bounding planes being the vertices of quasi-tetrahedral building blocks. The resulting quasi-tetrahedra are of two distinct shapes as a result of the two possible orientations of planes relative to the reference cylinder surface. All tetrahedral building blocks are truncated to a desired tube wall thick- ness,t. The number of building blocks along the z-axis,nh, sets the tube height ash= (nh+2)a0. The tube aspect ratio is de ned as '=h=(R+t). In the present study, three values of tube aspect ratios are considered, '= 1.6, 2.3, and 3.0. TIM tubes are constructed with an interlock angle of int= 17. Tube structures built as an arrangement of non-interlocking building blocks, int= 0, are considered for comparison. 2.2. Experimental Methods Physical realizations of TIM tubes were created by the Polyjet additive manufacturing (AM) processes using an Objet350 Connex (Stratasys Ltd.) 3D printer. As-manufactured TIM tubes undergo diametral loading by a at punch, Fig. 2. The at punch is positioned at the mid-height of the TIM 6 (a) (b) (c) Figure 2: (a) Experimental setup for diametrical loading experiments of TIM tubes by a at punch. A ber light is inserted into the tube such that the back illumination enables the visualization of the TIM system architecture. Detailed view of (b) interlocking building blocks ( int=17) and (c) non-interlocking building blocks ( int=0) surrounded by support material. tube and initially sits ush with the outer face. The load response of TIM tubes under quasi-static applied displacement conditions captured using a BOSE ElectroForce 3200 mechanical test instrument. The TIM tubes are printed as integrated assemblies of all building blocks and end frames in a PolyJet (tm) print process. Thereby, building blocks (VeroWhite (tm), Stratasys Ltd.) were separated by a layer (0.3 mm thick- ness) of of the PolyJet support material FullCure705 (Stratasys Ltd.). Build- ing blocks were separated by a layer (0.3 mm thickness) of of the Polyjet support material FullCure705 (Stratasys Ltd.). The support material was removed from the TIM tube external surface but left in place between the building blocks. 7 In order to realize a system where the support material (mortar) pro- vides minimal mechanical strength, the support material is embrittled. The FullCure support material in an as-printed state is already signi cantly more compliant and weaker than the VeroWhite material used for the building blocks. In addition, aging through exposure to UV light degrades the strength of the support material [35]. Specimens were aged for seven days before the mechanical loading experiments. The resulting TIM tube systems then are dominated by the geometry of the building blocks, their mechanical behavior, and interactions rather than by the support material lling the gap. The end frames of the TIM tubes were connected to one another by use of a threaded steel rod. Nuts and washers are threaded onto each end of the rod and tightened to near zero pressure by monitoring the pressure through thin lm pressure sensors. Thereby, the TIM tubes is axially constrained with zero pre-load. The sti ness of the steel rod connecting frames is approximately 20E6 times that of the elastic connectors considered in the nite element models. Thus, the resulting conditions are such that the ends of the tubes are essentially xed in space. For visualization purposes a glass ber light tube is inserted through the end plates to provide back-light illumination. 2.3. Finite Element Models TIM tubes are constructed from linear elastic solids with elastic modulus Eand Poisson's ratio . Building blocks interact with each other through contact and friction. TIM tubes are loaded diametrically by a at punch. Thereby, one of the building block located at the mid-height position of the TIM tube is loaded with the end face of a cylinder ( d < a 0) whose axis is aligned with the r-coordinate, Figs 3a and 3b. 8 (a) (b) Figure 3: TIM tubes and loading conditions: (a) Loading via a at punch and xed frame boundary condition (FP) and (b) loading via a at punch and frames connected via an elastic element (shown in red), the system rests on a at surface (FP-E). TIM building blocks were assigned the following mechanical properties, to replicate the polymer used in experiments: E=1827 MPa and =0.35. Both the frames and the load-inducing punch were considered as rigid bodies. A density value of = 950 kg/m3was used. A mass scaling factor of 100 was employed to decrease computational cost and suppress the kinetic energy. Two sets of contact conditions are considered in the FE simulations. In an idealized system condition, the contact sti ness is high and approaches a hard contact. In that case, contact between all building blocks in the system was characterized by a linear pressure-overclosure formulation with sti ness 50 times that of the nite elements, Kc=63.95 kN/mm. Friction is described by a Coulomb friction model. The coecient of friction considered was =0.6, as this is the upper bound of for dry materials. In FE-models seeking to approximate the experimental system, the contact and friction interaction 9 can depart signi cantly from the idealized conditions. The actual value of the contact sti ness is then determined through a t to the experimental data. Furthermore, an extended Coulomb friction model with a shear stress cut-o 0is employed. The actual parameter value of the contact sti ness and 0are determined through a t to the experimental data. Two types of constraint con gurations are considered. First, the con- straining frames are considered as rigid and spatially xed, Fig. 3a. Such a boundary condition (FP) resembles a set-up where the TIM tube serves as a connector between two sti , and strongly connected components. Alter- natively, rigid frames are constrained to one another by an elastic element with its ends positioned at the centroids of the two opposing frames. In such a con guration (FP-E), the TIM tube is self-constrained and can be handled as an independent part. During loading, the tube is considered to be resting on a at, rigid surface, Fig. 3b. The sti ness Ksof the elastic connection between the two frames is an additional system parameter. Here, it was considered as equal to the axial sti ness of the equivalent monolithic tubeKM=EAM=h, whereAM= (R+t=2)2(Rt=2)2. Consequently, Ks=59.7 kN/mm for '= 1:6,Ks=41.5 kN/mm for '= 2:3, andKs=31.8 kN/mm for '= 3:0. FE-models of the TIM tube systems are solved for displacement-controlled loading by a at punch. Computations are executed by use of an explicit FE algorithm (ABAQUS Explicit) but only the quasi-static response is of con- cern. In addition to the force-displacement response of the punch, computed values of external work Wex, strain energy Uel, frictional dissipation Ufand contact penalty work Ucwere recorded. Other system energy components 10 (viscous dissipation, arti cial energy, and kinetic energy) were negligible be- fore total system failure occurred. In idealize model con gurations, only Wex, UelandUfare of concern and Ucis negligible. In model con gurations rep- resenting the experiments, only only Wex,UcandUf. In that case Uel, the strain energy, is insigni cant. TIM building blocks and monolithic tubes were discretized with linear, hexahedral, reduced integration elements. Enhanced hourglass control was used with the reduced integration elements. The mesh convergence of the simulations was investigated by monitoring maximum reaction force ( Fmax) values with mesh densities ranging from 5 to 20 seeds per building block unit edge. At 15 seeds per polyhedra edge the di erence in Fmaxbetween seed sizes was less than 1%, so building blocks in all computations were meshed with 15 seeds per edge. With this value, individual TIM building blocks were comprised of 3600 elements (type C3D8RH of the ABAQUS/Explicit code). For details of the convergence study, see Appendix A. 3. Experimental Results Figure 4 depicts measured force-displacement data for both the TIM tubes and the non-interlocking tubes. The response of the TIM tubes ( int=17) features an initial elastic regime followed by a sharp drop in measured force, Fig. 4a. The sti ness of TIM tubes decreases with tube aspect ratio. One observes a rather abrupt load drop past a peak load value. Final failure occurs at displacements approximately equal to the tube wall thickness. Non-interlocking tubes ( int=0) exhibit sti ness and strength that are in- dependent of tube aspect ratio, Fig. 4b. A sharp drop in measured force 11 is still observed past the peak load, however. Using elementary mechanics of materials concepts, the strength of support material is computed to be 0=Fmax=(4a0t)=0.08 MPa. (a) (b) Figure 4: Normalized data of the measured force-displacement response for: (a) TIM tubes (int= 17) - FP and (b) non-interlocking tubes ( int=0) - FP. Details of the deformation and failure mechanism can be deduced from Fig. 5. In the experiments, one observes that the entire TIM tube face is de ected initially, Fig. 5a. The loaded building block is pushed in by the punch after support material between blocks begins to fail, Fig. 5b. If push- out was the only active deformation mechanism, then only this building block would incur signi cant displacement. This is however, not the case in the topologically interlocked con guration. Figures 5b and 5c demonstrate that the deformation of the TIM tube systems a ects not only the building block in contact with the punch. In Fig. 5b adjacent building blocks are found to be tilted from their initial position, and in Fig. 5c building blocks even further removed from the loaded block are not only tilted but also shifted radially outward from their initial position. After the push-out of the center 12 (a) (b) (c) (d) (e) (f) (g) (h) Figure 5: Snapshots of deformed TIM and non-interlocking ( int= 0) tubes ('= 1:6) during loading experiment. (a-d) TIM tube ( int= 17) - FP: (a)u=a 0<0.2, (b)u=a 0=0.5, (c)u=a 0=1.0 and (d) removed center block. (e-h) Non-interlocking tube ( int= 17) - FP: (e)u=a 0<0.2, (f)u=a 0=0.5, (g)u=a 0=1.0 and (h) removed center block. block at nal stages of deformation, the TIM tube structure remains intact minus this building block. The deformation and failure of non-interlocking tubes on the other hand is entirely due to the push-out of the loaded building block. Before support material failure the entire tube face is de ected like in TIM tubes, Fig. 5e. Afterwards, the loaded building block is pushed in while the positioning of surrounding blocks remains unchanged, Fig. 5f and 5g. The non-interlocking tube remains intact minus the pushed-out building block after failure, Fig. 5h. 13 4. Analysis 4.1. Idealized Models Figure 6a depicts the computed force-displacement response of TIM tubes with xed frame boundary conditions and at punch loading. The response of the TIM tube is skew-parabolic and displays gradual failure beyond the maximum load. TIM tubes are found to fail at an applied displacement 1.3 to 1.7 times the tube thickness. Figure 6b documents the computed force- displacement response of TIM tubes with elastically constrained frames and at punch loading. The shape of the response and at punch loading is preserved in case of the model with elastically connected frames; force levels are lower but the applied displacement to failure is slightly higher. (a) (b) Figure 6: Normalized force-displacement response for: (a) TIM tubes ( int= 17) - FP and (b) TIM tubes ( int= 17) - FP-E. TIM tube internal load transfer is visualized as contour plots of the min- imum principal stress, p;min, in Fig. 7. Load transfer within TIM systems is dominated by compressive stresses, with tensile loads restricted to the frames, 14 and the connection between frames (if present). This is distinct from the load transfer in a monolithic structure, where material is loaded by bending under transverse indentation and a distribution of tensile and compressive stresses exists. Distributions of p;min on axial sections of the TIM tubes were created at the instance of the force maximum, and at force levels of half the force maximum pre- and post-peak. These paths of load transfer can be considered thrust-lines forming a force network within the system. The thrust-lines are initially oriented diag- onally along the exposed cross-section such that they originate underneath the punch and terminate at the binding frames, Fig. 7a. The evolution of thrust-lines is a result of the displacement of the center building block as well as the rotation of neighboring building blocks. When the system strength is reached the thrust-lines in this cross-section are oriented with the tube axis, Fig. 7b. These thrust-lines are observed to remain present even at displace- ments larger than the wall thickness of the TIM tube, Fig. 7c. Due to the slip of the center block relative to its neighbors, the shape of the thrust-line is modi ed and now straddles the inner tube surface rather than extending from the tube exterior to the interior. The load transfer is not limited to only a few building blocks. The domain of the tube under stress spreads from the loaded building block to many points along the tube perimeter. Other thrust-lines composing the over-arching force network are evident from the visible stress contours outside of the cut cross-section. These thrust-lines also originate underneath the punch, but terminate at di erent points along the frames' perimeter. Force networks remain intact up until some point at which the center building block is punched out from the tube. This observation was 15 consistent for all aspect ratios considered. (a) (b) (c) Figure 7: Contours of the computed minimum principal stress, p;min , from FE-models representing the idealized con guration of a TIM tube ( int= 17;'= 1:6) - FP: (a) u=a 0= 0:25, (b)u=a 0= 0:75, and (c) u=a 0= 1:25. Contour scale: [0 (grey), -50 (red)] MPa. The computed system energies for int=17are depicted in Fig. 8. At initial stages of deformation, the elastic strain energy dominates over the frictional dissipation. As deformation progresses, strain energy stored is re- duced in magnitude and frictional dissipation becomes dominant. The tube aspect ratio determines if slip or elastic deformation dominates the system deformation. For the smallest aspect ratio, deformation to about the peak load is dominated by elastic energy. For the two cases with higher aspect ratio, deformation far into the softening regime remain dominated by elastic energy. In the highest aspect ratio case, '= 3:0, essentially the entire defor- mation is dominated by elastic deformation and slip only becomes relevant during the nal stages of loading. The thrust-line (TL) model approach of [22, 28, 29] is followed to establish an analytical model for the calculation of the force-displacement response. Thrust-lines are domains within the system of high values of compressive 16 Figure 8: Ratios of computed model energies Uel=WexandUf=Wexfor the idealized model con guration of TIM tubes ( int=17) - FP. principal stress, Fig. 7. In the model, thrust-lines are represented as trusses. Pairs of such trusses form a collection of Mises trusses. The mechanical analysis of Mises trusses is then used to establish equations to predict the load response. Each truss is assigned a local coordinate system such that all truss coordinate systems share a y-axis. The reactions Fxi(thrust force) and Fyi(shear force) at the abutments (the rigid end frames) of each of the Mises trusses (i) are computed as: Fxi(i) =EAi 1cos(0i) cos(0ii) cos(0ii) (1) Fyi(i) =EAi 1cos(0i) cos(0ii) sin(0ii) (2) Here, the change in angle of inclination of the truss iisi=0itan1(2hi=Li) withhi=h0iu. The load response of each Mises truss relates to the initial 17 truss height ( h0i), the truss span ( Li), constitution ( E), and an equivalent cross-section area ( Ai). The elastic modulus of the trusses is assumed to be that of the VeroWhite (tm) polymer used in realizations of TIM tubes. Then, the truss section area value was estimated by tting the response calculated with the TL-model to that from the FE-model. The net reaction force for each Mises truss is: Fi= 2Fyi. (a) (b) (c) Figure 9: Deformed TIM tube geometry ( '= 1:6) andp;min contours on (a) an axial section with Mises truss F1overlaid and (b) an oblique section with an o -axis Mises truss overlaid. In the depicted instance i= 4. The circumferential distance between abutments ofF1andFiis (i1)a0. Contour scale: [0 (grey), -100 (red)] MPa. (c) All Mises trusses (red) for the present model geometry are overlaid on translucent TIM tube geometry. Mises trusses are arranged to align with thrust-lines observed in minimum principal stress contours from FE-models, Fig. 9. All truss apexes intersect on the face of the loaded building block at the point of punch contact. Truss members span from this point to discrete abutments along the rigid end 18 frames. Truss 1 is aligned with the TIM tube axis, Fig. 9a. The abutment of a truss ( i >1) is placed at the tube end such that the circumferential distance from the abutment of truss 1 is ( i1)a0. The oblique trusses (i>1) then span the tube diagonally, Fig. 9b, and are symmetric about the tube mid-plane. This leads to a pyramidal arrangement of Mises trusses with the loaded building block at the apex of the pyramid, Fig. 9c. A truss ( i) must be entirely within the tube wall thickness, thus satisfying the condition: 1cos(i1) n <t R; i= 1;n: (3) The number of Mises trusses needed to be considered for a given nis (2p1), wherepis the largest integer isatisfying the condition in eqn. 3. The total load response of the TIM tube is then equivalent to the sum of the reactions of Mises trusses, eqn. 4. F=F1+ 2pX i=2Fi: (4) There are two types of support con gurations for the Mises trusses. One group of trusses has a termination at the end frames at the tube's inner surface, Fig. 9a. The second group of trusses has a termination at the end frames at the tube's outer surface, Fig. 9b. A truss iis in the rst group if its span is entirely within the tube wall thickness (in this con guration) by satisfying the condition in eqn. 5. Otherwise the truss iis in the second group. For the tubes considered here, trusses 1 and 2 are in the rst group, and trusses 3, 4, 5, and 6 are in the second group. 19 1 4s [4R2t2]2[cos(2i )1] 4R2cos(i )t2cos(i )4R2t2Rt 2(5) Consequently, there are two equations for the calculation of the initial angle of inclination of the Mises trusses with respect to the local truss span, 0i. The number of Mises trusses in the rst group for a given nis (2q1), whereqis the largest integer isatisfying the condition in eqn. 5. The number of Mises trusses in the second group is then (2 p1)(2q1). This is described in terms of axial building block count N, unit sizea0, tube radius R, and facet angle = 2=nin eqns. 6 and 7. tan(0i) =2Rf1cos[(i1) ]g+tf1 + cos[(i1) ]gp a2 0N2+ 4R2sin2[(i1) ]; i<q (6) tan(0i) =(2R+t)f1cos[(i1) ]gp a2 0N2+ 4R2sin2[(i1) ]; iq (7) Figure 10 shows the force-displacement responses of TIM tubes as com- puted with the TL-model. For comparison, results obtained with the FE- models of identically sized TIM tubes are shown ( xed frames, at punch loading). The TL-model is tted to the results from the FE-model by matching the initial sti ness of the TIM tube with '= 1:6. We consider Ai=const: , and a value of EAi=29,597 N in the TL-model is found to best replicate the results of the respective FE-model. The TL-model then used is used for the analysis of TIM tubes with '= 2:3 and'= 3:0. For all aspect ratios, the TLM overall predicts the initial sti ness and strength well. The load decay past the peak force is more pronounced in the data from FE-models, and the TL-model consistently predicts signi cantly larger dis- 20 placements to failure. This nding is attributed to sliding between building blocks in FE-models which is not accounted for in the TL-model. Figure 10: The force-displacement response as computed with the FE-model and the thrust-line model (TLM) for TIM tubes with '=1.6, 2.3, and 3.0. 4.2. Realistic Models A comparison of the predicted TIM tube response for the idealized system and the respective physical realization reveals signi cant di erence. Both the FE-model and the TL-model thus require modi cations from their original form. It is found that a decrease in the contact sti ness and the activation of a friction limit in FE-models leads to a response approximating the experimen- tal results of the TIM tubes, Fig. 11. First, the contact sti ness is lowered to approximate the initial sti ness of the TIM tubes. A contact sti ness 21 value ofKc=1250 was found to provide a good match to all experimental data. Subsequently, the frictional shear stress limit is set to the strength of support material determined in experiments, 0=0.08 MPa. With these model parameters, the FE-model predicts sti ness and strength to again de- cline with increasing tube aspect ratio, Fig. 11. The failure of the TIM tubes is now predicted to occur at much reduced magnitudes of applied displace- ments. The FE-model predicts the presence of signi cant load drops past the peak load, which is also in agreement with the experimental observation. However, the FE-models predict the load drops to occur a later stages of loading than observed in the experiments. Figure 11: Normalized force-displacement response as obtained from FE-analysis with parameter estimates for experimentally measured response: TIM tubes ( int= 17) - FP. Computed distributions of p;min on axial sections of the TIM tubes again 22 were are depicted at the instance of the force maximum, and at force levels of half the force maximum pre- and post-peak. The shape of the force network as obtained from the the FE-model for the experimental system is initially similar to the one in the FE-model of the idealized system, Fig. 12a vs. Fig. 7a. Deviations from the idealized model occur at larger displacements due to the increased amount sliding between building blocks. In addition, while the center building block being displaced radially inward, other building blocks in the TIM tube are displaced radially outward, Fig. 12b and 12c. This leads to an earlier interruption of thust-lines within the TIM tube and a smaller displacement to failure. (a) (b) (c) Figure 12: Contours of the computed minimum principal stress, p;min , in from FE-models representing the experiments. (a-c) TIM tube ( int= 17;'= 1:6) - FP: (a) u=a 0= 0:1, (b)u=a 0= 0:3, and (c)u=a 0= 0:7. Contour scale: [0 (grey), -1 (red)] MPa. The computed system energies for int=17are depicted in Fig. 13. With the realistic contact conditions, frictional dissipation becomes dominant in the system energy much earlier compared to idealized con gurations. Now, slip dominates the system deformation before the peak load is reached for all aspect ratios. The ratio of frictional dissipation to stored elastic energy is also higher than in idealized con gurations. Again, the dominance of sliding 23 occurs at later stages of loading for higher aspect ratio TIM tubes. Figure 13: Ratios of computed model energies Uc=WexandUf=Wexfor the model con g- uration representing the experimental system: TIM tubes ( int=17) - FP. The TL-model for TIM tube behavior must be modi ed to account for sliding between building blocks. This is accomplished by allowing some frac- tion of applied displacement increments to account for sliding. This con g- uration is implemented as a modi cation of the reference con guration for the Mises truss. Thereby, the initial angle of a truss, 0i, is decreased by a fraction of the angle change idue to the applied displacement. The fraction of angle change to be assumed as slip is referred to as the slip factor, Sf. The onset of slip is determined with a individual truss force criterion. Slip will not be introduced to an individual truss unless the shear component of the truss reaction exceeds a critical force, Fc. As suggested in [36], the critical force is related to interface frictional strength such that Fc= 0Acwhere 0is the joint shear strength and Acis the contact area between building blocks. 24 Deformation due to slip in TIM tubes is then described within the TL-model by eqn. 8. 0i=8 >< >:0i; ifFi<F c 0iSfi;ifFiFc(8) The predictions obtained with the slip-inclusive TL-model are presented in Fig. 14. The parameter values of TL-model are again estimated to match the sti ness of the TIM tube with '= 1:6, i.e.EAi=const: = 180 N. From a comparison to experimental data and FE-results, it can be seen that the TL-model approximates the initial sti ness of the TIM tubes well. A critical force value of Fc=1.0 N results in the model-predicted strength matching experiment results. The decay of indentation force with additively manufactured TIM tubes is best replicated with a slip factor of Sf=0.8. All of the above values are used in the TL-model with dimensions of TIM tubes with'= 2:3 and 3.0. The additively manufactured TIM tube sti ness and strength are reasonably approximated for each aspect ratio. 5. Discussion We de ne the geometry of TIM tubes as consisting of two types of quasi- tetrahedral building blocks. This is in contrast to the interlocking con gura- tion emerging from a planar grid of prototiles. In that case only one uniquely shaped building block is needed, yet de ning interlocking geometry from a quasi-cylindrical grid of prototiles introduces two complementary polyhedra. TIM tubes were realized by a PolyJet (tm) additive manufacturing process which creates a material architecture that is functionally similar to those of 25 (a) (b) (c) Figure 14: Comparison of measured TIM tube force-displacement response (EXP) to FE- model (FEA) and TL-model (TLM) predictions. (a) '=1.6, (b)'=2.3, and (c) '=3.0. [37]. This entails an arrangement of interlocking building blocks separated by weak interfaces. The TIM tubes require the presence of binding frames. In a planar TIM system, the frame may be a single piece running the perimeter of the assembly. TIM tubes on the other hand require two separate frame parts to constrain the collection of building blocks, which can be xed to one another such that the system can be handled independently. Experiments performed on additively manufactured TIM tubes demon- strated that interlocking building blocks result in a sti er, stronger, and 26 tougher con guration than non-interlocking blocks. The failure of non- interlocking tubes ( int=0) in experiments was solely due to the push-out of the loaded block after the shear failure of surrounding material. On the other hand, the response and failure of TIM tubes ( int=17) is more com- plex due to interactions between interlocking building blocks. The sharp decline in load capacity noticed in AM TIM tubes is a result of the failure of the support matrix between building blocks. Before support matrix failure, the TIM building blocks are essentially bonded. This results in the sti er response before the drop in measured load. After support matrix failure sliding becomes very relevant in the TIM tube response. The soft contacts between building blocks enables the sliding of building blocks relative to one another more easily. As a result the TIM building block located under the at punch is displaced further relative to neighboring blocks. Interactions in the system caused neighboring blocks to be pushed radially outward, result- ing in a less stringent con nement of building blocks. The failure of AM TIM tubes is then a result of cumulative interactions between building blocks that ultimately results in the removal of the loaded block. This is in contrast to the shear-failure push-out observed in non-interlocking tubes, and as such, a theory for the deformation response of TIM tubes is needed. Numerical simulations of the loading of idealized TIM tubes predict a skew-parabolic load response with no sharp drops in load capacity. The force- displacement responses of all TIM tubes displayed softening to some extent. This leads to a more gradual failure than in monolithic brittle materials, and is consistent with previous ndings in planar TIM systems [22, 26, 29]. Considering an elastic connection between frames leads to lower sti ness and 27 strength predictions, but preserves the shape of the response. Ultimately, this shows that TIM tubes can be realized by xing frames together with some connection and that the structure's response would be characteristic of TIM systems. The deformation of TIM tubes is a combination of the elastic deformation of building blocks and sliding between building blocks. During the loading of TIM tubes, deformation shifts from mostly elastic deformation to mostly sliding deformation. FE-model energies show that higher aspect ratio TIM tubes are more resistant to sliding. The ultimate failure of each aspect ratio TIM tube occurs after the removal of the center building block. By modifying the contact conditions in FE-models a TIM tube response like in experiments can be replicated. Scaling down contact sti ness and im- posing a frictional shear stress limit results in the lower sti ness and dramatic failure observed in experiments. This is due to the contact modi cations al- lowing increased slip between building blocks. The locomotion of building blocks during deformation also re ects experimental observations. Building blocks other than the loaded one are displaced from their initial positions dur- ing the deformation of the TIM tube, which was not observed in the idealized simulations. There is also little energy stored as strain in building blocks, meaning the TIM tube behaves like an collection of rigid bodies under these conditions. This shows that the interaction between blocks is an important factor in the deformation and response of TIM tubes. Weaker interactions between blocks result in a more compliant and lower strength system. This is similar to the ndings of [38] where the addition of soft, rubber interfaces between concrete TIM building blocks lowered the TIM system strength. Analysis of the TIM tube systems through equivalent truss structures 28 enabled the force-displacement response to be recreated, like in planar TIM systems. While a network of identical, orthogonal trusses with local displace- ment criteria was representative of planar TIM systems, crossed trusses of varying span and height were able to capture the sti er, stronger response of TIM tubes. Previously the sti ness of planar TIM systems were approx- imated based on material properties, TIM building block size, and building block count. Since trusses were not restricted to spanning the distance be- tween frames while remaining orthogonal to one another, mechanical prop- erties were dependant on a larger set of TIM tube parameters. For the Mises truss, F1, the response sti ness is proportional to the height of the tube. The dimensions of diagonal trusses in the TIM tube, however, are described in terms of tube radius and unit count around the perimeter. Because of this the sti ness of a diagonal truss Fiis proportional to a function of material properties, tube radius, building block size, and building block count. Thus, the total sti ness of TIM tubes also includes tube radius in its formulation. The TL-model provides a good approximation of the sti ness and strength of TIM tubes with larger values of 'based on a selection of a parameter value forEAi. However, the basic TL-model does not approximate the experimen- tal results well. In both the numerical models and experiments, interfaces between TIM building blocks are not perfectly bonded. The TL-model does not account for sliding, so there exist discrepancies in the failure mechanisms between each. Idealized contact conditions are initially used in numerical models so that sliding between blocks is minimal. As a result the TL-model strength predictions are relatively similar to idealized numerical model re- sults. In experiments on additively manufactured TIM tubes, more sliding 29 between building blocks was allowed due to the compliant support matrix. The result is that the TL-model predicts much higher strengths and displace- ments to failure than that of additively manufactured TIM tubes. Reformulating the TL-model to account for system deformation in the form of slip yields a load response similar to experimental results. Imposing a critical force cuto for individual trusses in the model controls the onset of slip and predicted strength of TIM tubes. This is similar to the frictional shear stress cuto used in the FE-models for the experimental con guration, and may be compared to the failure of support material between blocks in experiments. The global failure mode predicted by the TL-model depends on the fraction of applied displacement allocated to slipping by the slip factor. With a small slip factor the TL-model predicted response displays a gradual failure. This is because only a small amount of applied displacement is accounted for as slip, and the TL-model functions the same otherwise. With larger slip factors more most of the applied displacement is accounted for as slip and noticeable drops in force can be observed. Large slip factors best replicate the large amounts of sliding in additively manufactured TIM tubes after the failure of the support matrix between building blocks. Either formulation of the TL-model does not work well to predict the response of TIM tubes with int=0. Whenint=0there is a considerable amount of slip in assemblies, even at the onset of loading. This is unlike experimental results, where slip is initiated after the strength of the support material is overcome. As a result the thrust-line evolution in TIM tubes with int=0is unique compared to TIM tubes with int=17. To accurately pre- dict the response of TIM tubes with int=0, the TL-model must be modi ed 30 further to account for this earlier slip between building blocks. The discrepancies in predicted sti ness between experimental, FE-model, and TL-model results may be a result of the formulation of the TL-model. It is possible that the e ective truss area Aivaries with tube ', between trusses Fi, or with de ection u. The area Airepresents the contact area between building blocks within a single force network component. In [39] a de ection- dependent contact area is used in a model for Voussier beams. The rotation of beam segments is considered to update the contact area used in analysis. In the case of TIM tubes a similar method could be applied to account for the rotation and sliding of building blocks. The present thrust-line TIM tube model does not account changes in contact area. Aiwas assumed constant for all trusses. Also, truss abutments were restricted to have xed abutments located on the TIM tube's bounding frames. Each TIM unit within the frame is also limited to having one truss abutment. No more diagonal frame-to- frame trusses can be incorporated into the model without thrust-lines leaving the tube volume, so TIM tube response formulation may include trusses with smaller spans and dynamic abutments somewhere within the plurality of TIM building blocks. The implication of this is that load transfer in a TIM tube is more complex than previously seen planar TIM systems. In addition to equivalent trusses now being non-orthogonal, it seems that xed abutment trusses spanning frame-to-frame yield an incomplete system response. Only a handful of possible TIM tube con gurations were considered in this study - all of which were crafted from square tessellations. It is pos- sible to create TIM tubes with other tessellation patterns, and the result of doing so is unknown. Additionally, only tube height was altered in this 31 study, with tube radius, tessellation size, and interlocking angle being held constant. Modifying these other parameters would certainly in uence TIM tube response, and this should be kept in consideration. 6. Conclusion TIM tubes composed of repeated quasi-tetrahedral units were consid- ered as a novel topologically interlocking material system. The present work demonstrated the signi cant increase in strength and toughness that can be achieved by interlocking geometries relative to non-interlocking geome- tries. Mechanisms in the realizations of strong and tough biologically-inspired structures were explained with fundamental mechanics of materials concepts. The deformation and load response of TIM tubes with multiple boundary conditions were closely monitored in FE computations. TIM tube response was found to exhibit a gradual failure in computations, like previously demon- strated in planar TIM systems. The strength and failure of TIM tubes was found to depend on the the dimensions of the tube as well as the contact conditions between building blocks. The response of TIM tubes was found to be a product of the formation of force networks. Force network geometry was used to describe equivalent truss structures in a thrust-line model of the sys- tem. The formulated thrust-line model was able to approximate the sti ness of the computed and experimental TIM tube load response. Modifying the model to include slip between building blocks resulted in a better approxima- tion of TIM tube strength and failure under experimental conditions. This model contains a new parameter which is not present in planar TIM models (tube radius), so the mechanical response can be further modulated with this 32 parameter. Acknowledgment: This work is supported by the National Science Foun- dation under Grant No. 1662177. 33 Appendix A. Mesh Convergence To determine an appropriate mesh density for TIM building blocks, re- sults using a range of mesh seeds per side is considered. Mesh densities of 5 to 20 seeds per building block edge are considered. The geometry used in this convergence study was that of the TIM tube with '=3.0. The TIM tube considered was subjected to loading by at punch indentation and a xed end boundary condition. The computed value of the computed maximum reaction force, Fmax, is used to determine an investigate mesh density. Val- ues ofFmaxin dependence of mesh density are shown in Fig. A.15a. Fmaxis normalized with by the material elastic modulus multiplied by the building block cross-section area. A mesh density of 15 seeds per side was selected asFmaxvalues do not vary by more than 1% with higher mesh density. It was also found that certain mesh densities cause a large viscous dissipation contribution in the computed solution. The reason for the presence of vis- cous dissipation is to resolve contact issues between edges and corners of some elements. Figure A.15b shows the ratio of viscous dissipation to total applied work in models with various mesh densities. It was decided that the viscous dissipation contribution is acceptable at the selected mesh density of 15 seeds per side. 34 (a) (b) Figure A.15: (a) Maximum computed force relative to mesh density and (b) ratio of viscous dissipation energy (VD) to total applied work (WK) relative to mesh density.