Bone-inspired microarchitectures achieve enhanced fatigue life

Significance

Microarchitectured materials, such as foams and lattice structures, can achieve high stiffness and strength while remaining extremely lightweight. Applying high-porosity microarchitectured materials to durable devices, such as vehicles, however, will require the materials to also resist failure during cyclic loading. Here, we identify an aspect of microstructure in cancellous bone that greatly influences failure under cyclic loading and show that the effect is generalizable to synthetic microarchitectured materials. Our findings demonstrate that a common design strategy to improve stiffness and strength of microarchitectured materials comes at the cost of impaired service life. Our findings are useful for the design and application of microarchitectured materials and additionally provide insight into human health in situations of osteoporosis.

Abstract

Microarchitectured materials achieve superior mechanical properties through geometry rather than composition. Although ultralightweight microarchitectured materials can have high stiffness and strength, application to durable devices will require sufficient service life under cyclic loading. Naturally occurring materials provide useful models for high-performance materials. Here, we show that in cancellous bone, a naturally occurring lightweight microarchitectured material, resistance to fatigue failure is sensitive to a microarchitectural trait that has negligible effects on stiffness and strength—the proportion of material oriented transverse to applied loads. Using models generated with additive manufacturing, we show that small increases in the thickness of elements oriented transverse to loading can increase fatigue life by 10 to 100 times, far exceeding what is expected from the associated change in density. Transversely oriented struts enhance resistance to fatigue by acting as sacrificial elements. We show that this mechanism is also present in synthetic microlattice structures, where fatigue life can be altered by 5 to 9 times with only negligible changes in density and stiffness. The effects of microstructure on fatigue life in cancellous bone and lattice structures are described empirically by normalizing stress in traditional stress vs. life (S-N) curves by √ψ, where ψ is the proportion of material oriented transverse to load. The mechanical performance of cancellous bone and microarchitectured materials is enhanced by aligning structural elements with expected loading; our findings demonstrate that this strategy comes at the cost of reduced fatigue life, with consequences to the use of microarchitectured materials in durable devices and to human health in the context of osteoporosis.

Microarchitectured materials can achieve high stiffness and strength per unit mass through underlying geometry rather than material composition (1–4). Recent developments in additive manufacturing and lattice design software allow for rapid optimization of lattice density and architecture to meet stiffness, strength, and/or energy absorption demands with low-density microarchitectures (5). Advancements in micro- and nanofabrication have allowed for the design of microarchitectured materials from a variety of different substrates with high stiffness and strength (6–11). Resistance to fatigue failure is not as often considered in the design of lattice microstructures. However, microarchitectured materials can be susceptible to fatigue failure, because their complex geometry results in stress concentrations that can be an order of magnitude greater than stresses applied to the bulk material, thereby promoting the initiation and propagation of fatigue damage (12–14). Balancing the needs for fatigue life with stiffness, strength, and other desired material properties is a major challenge for the use of microarchitectured materials in durable devices.

Naturally occurring materials can display exceptional mechanical performance and are useful models for the design of microarchitectured materials (13, 15). Bone is a biological material with high stiffness and strength relative to density. Whole bones consist of an outer shell made of dense tissue known as cortical bone that surrounds a foam-like tissue known as cancellous bone. Cancellous bone consists of a network of interconnected plate-like and rod-like struts called trabeculae (∼50 to 300 μm in thickness). Trabeculae in cancellous bone are preferentially aligned in the direction of stresses generated by habitual physical activity, resulting in a transversely isotropic microstructure. Although microarchitecture is widely recognized as a contributor to the mechanical performance of cancellous bone, to date, only density/porosity and fabric tensor (a measure of anisotropy) have been shown to be major contributors to cancellous bone stiffness and strength; all other aspects of microarchitecture provide only negligible contributions (16). The effect of microarchitecture on the fatigue properties of cancellous bone is not as well studied.

The stiffness and strength of cancellous bone and other cellular solids have been studied for some time and are related to density through power law relationships (4, 17). Although there are analytical methods of relating the foam density to fatigue life (number of cycles to failure, N_f) (17), the fatigue life of foams is better explained by normalized stress vs. life (S-N) relationships of the form (18, 19)

$$\frac{\sigma }{E_0}=A{N}_f^B,$$ [1]

where σ is the maximum compressive stress, E₀ is the initial Young’s modulus (alternatively, yield stress or plateau stress is used), N_f is the number of cycles to failure, and A and B are empirical constants (in cancellous bone, A ranges from 0.0091 to 0.013, and B ranges from −0.121 to −0.094 [20]). Although the normalized S-N relationship improves prediction of fatigue life, the empirical constants differ among microstructures (cancellous bone from different animals, distinct lattice microstructures, etc.) in ways that are not yet well understood but are commonly attributed to local deformation mechanisms (bending vs. stretching) or in synthetic microarchitectured materials, flaws generated during manufacturing (19).

Results and Discussion

To better understand the effects of microarchitecture on fatigue life, we determined the relationship between microstructure and fatigue damage processes in high-porosity (>90%) cancellous bone from the vertebral bodies of deceased human donors (n = 44 specimens from 18 donors) (Materials and Methods). Cyclic compressive loading (0 to compressive stress) was applied in the direction of habitual loading in vivo. Fatigue loading was suspended at a specified amount of cyclic loading (determined by accumulated cyclic strain), and the resulting amount and location of microscopic damage within the microstructure (microscopic cracks and accumulation of submicroscopic cracks; referred to as “microdamage” in the bone literature) were detected using contrast agents (Fig. 1 A and B and SI Appendix, Supplementary Methods and Materials and Fig. S1) (21, 22). Microarchitecture was assessed using 3-dimensional (3D) images and analyzed using a morphological decomposition approach that isolates each individual strut within the structure and classifies the strut as plate like or rod like as well as determining its orientation relative to loading (Materials and Methods and Fig. 1 C and D) (23). The amount of tissue damage caused by fatigue loading was correlated with maximum applied apparent strain (SI Appendix, Table S1) but was not correlated with specimen density, other specimen-average measures of microstructure, or measures of plate-like trabeculae (the primarily load-carrying elements). Surprisingly, the amount of tissue damage was reduced in specimens with thicker rod-like struts (Fig. 1E and SI Appendix, Table S2) (R² = 0.76, P < 0.01). This finding was unexpected, since rod-like struts in cancellous bone are primarily oriented transverse to the applied load, constitute on average only 20% of the solid volume of high-porosity cancellous bone (SI Appendix, Table S3), carry only a small proportion of longitudinally oriented loads, and have negligible effects on stiffness and strength in the longitudinal direction (24) (transversely oriented elements have similarly small effects in static properties of nanolattice structures [25]).

Fig. 1.

Microarchitecture influences fatigue damage accumulation in cancellous bone. (A) The creep fatigue curve of cancellous bone is shown with the three phases of fatigue loading indicated. Cyclic compressive loading of cancellous bone was stopped at different points along the creep fatigue curve (data points) to determine patterns of damage accumulation (fatigue life was estimated as in ref. 21). (Inset) Cyclic loading waveform is shown. The 3D images of cancellous bone with (B) green indicating damage, (C) plate-like and rod-like struts, and (D) strut orientation relative to anatomical position (longitudinal, oblique, and transverse). (E) The amount of damage in cancellous bone (damage volume fraction, DV/BV) was correlated with maximum applied strain, but specimens with thicker rod-like trabeculae experienced less damage accumulation (R² = 0.76, P < 0.01). Error bars indicate the SDs as determined from the linear mixed effects model. (F) Early in fatigue life, strut failure occurs primarily in transversely oriented rod-like struts; final mechanical failure is characterized by widespread failure of longitudinally oriented plate-like struts.

To better understand the effect of rod-like struts on fatigue failure, we examined the distribution of tissue damage at different points during the fatigue loading process. The failure of individual trabeculae during fatigue loading occurs nonlinearly with cycle number and differs by trabecular type/orientation (Fig. 1F). Early in fatigue, failure of struts occurs primarily in rod-like trabeculae; substantial damage accumulation in plate-like trabeculae does not occur until overt failure (Fig. 1F). The pattern of strut failure is also related to orientation: failed rod-like trabeculae are predominately transversely oriented, while failed plate-like trabeculae are predominately oriented longitudinally (SI Appendix, Fig. S2). We attribute the pattern in failure of individual trabeculae to the distribution of tensile stresses generated by loading: finite element models indicated that apparent compressive loading results in tensile stresses in rod-like trabeculae (primarily transversely oriented) and compressive stresses in plate-like trabeculae (primarily longitudinally oriented) (SI Appendix, Fig. S3). These findings suggest that, in cancellous bone, transversely oriented trabeculae act as sacrificial elements during cyclic loading by accumulating tissue damage and thereby, protecting the load-carrying, longitudinally oriented, plate-like trabeculae, the failure of which is indicative of final fatigue failure.

Tissue heterogeneity is also a major contributor to damage accumulation in cancellous bone (22, 26) and is, therefore, a potential explanation for our findings in human bone tissue. To isolate the effects of microstructure from those associated with material heterogeneity, we generated 3D models of cancellous bone microstructures using a high-resolution projection stereolithography printer (Fig. 2 A and B) (M1; Carbon) (27). Cancellous bone microstructures (Fig. 2B) were modified by adding material to the surface of transverse trabeculae in 1 of 3 increments: no modification (original geometry), +20 µm on the surface (an average increase in rod thickness of 20 ± 5%; mean ± SD), or +60 µm on each surface (an average increase in rod thickness of 45 ± 14%). Because transverse rod-like trabeculae constitute only a small portion of the solid volume and carry only a small portion of longitudinal loads, thickening of rod-like struts had only a small effect on density (Fig. 2C) (increase of 11 ± 8%; mean ± SD) and apparent stiffness (22 ± 19% increase in longitudinal Young’s modulus) (Fig. 2D). When applied uniformly across the microstructure, such small increases in density and stiffness cause only small changes in fatigue life that are well described by the normalized S-N relationship (19). However, when applied only to rod-like trabeculae, fatigue life was increased by as much as 2 orders of magnitude (Fig. 2E) and followed entirely new normalized S-N relationships (SI Appendix, Fig. S4).

Fig. 2.

Models of cancellous bone generated using additive manufacturing show that fatigue life is sensitive to small changes in microarchitecture. (A) Digital images of human vertebral cancellous bone were edited and printed into (B) high-resolution 3D models. Increases in the thickness rod-like struts had small effects on (C) density and (D) stiffness (Young’s modulus in first cycle of loading) yet resulted in (E) increases in fatigue life by as much as 2 orders of magnitude. (Lines connect samples derived from the same bone specimen; 2 magnitudes of normalized cyclic stress are shown. Scatter is a result of variations in microstructure among the 5 bone samples.) (F) A microcomputed tomography image of a 3D-printed sample of cancellous bone after fatigue loading to failure. A radioopaque dye penetrant indicated regions of accumulated damage. (Inset) Magnified view. (G) The amount of damage (damage volume fraction, DV/BV) generated by fatigue loading to failure was reduced in 3D-printed specimens with greater thickness of rod-like struts.

To confirm that damage accumulation in the 3D models did not differ considerably from that of cancellous bone, we examined damage (microscopic cracking, constrained microcracks, etc.) in the additively manufactured specimens after loading using a radioopaque dye penetrant. Locations of damage accumulation identified with the dye penetrant were distributed throughout the structure in a manner qualitatively similar to that seen in cancellous bone (Fig. 2F). Furthermore, printed specimens with thicker rod-like struts showed reduced damage accumulation (Fig. 2G). Hence, damage accumulation during fatigue loading is modified by the thickness of rod-like trabeculae in both materials. Furthermore, finite element models of the specimens indicated that the average tensile stresses in rod-like trabeculae (predominately transversely oriented) were greater than those in plate-like trabeculae (predominately longitudinally oriented) (SI Appendix, Fig. S3), demonstrating that the localization of damage follows stress distributions within the microarchitecture as seen in true cancellous bone. Together, these findings indicate that small increases in mass applied to transversely oriented structural components of the microstructure can reduce tensile stresses, leading to disproportionately large beneficial effects on fatigue life.

To determine if our findings are generalizable to other cellular solids and other deformation mechanisms (bending vs. stretching), we created printed models of an octet truss (1) as well as an octet truss modified to have plate-like and rod-like elements mimicking the microstructure and anisotropy of cancellous bone (Fig. 3A). Cancellous bone microstructure shows bending-dominated behavior, the octet truss shows a stretching-dominated deformation behavior, and the bone-like microarchitecture displayed a combination of both stretching and bending deformation behaviors (SI Appendix, Fig. S5). In the bone-like microarchitectures, increases in transverse strut thickness resulted in an increase in the fatigue life by a factor of 8 (Fig. 3B), with only a small change in density (+4%) or longitudinal stiffness (+20%). In the octet truss, increases in transverse strut thickness resulted in an increase in fatigue life by a factor of 5 (Fig. 3B), with only minor changes in density (+10%) or longitudinal stiffness (+14%) (Fig. 3B and SI Appendix, Table S4). In contrast, when the modified octet truss model was rotated 90° so that the thickened elements were vertically oriented and oblique to the applied loads, the fatigue life was reduced by a factor of 9 compared with the model without thickened struts (Fig. 3B), demonstrating that the effect of transverse elements on fatigue life is related to the proportion of material oriented transverse to loading rather than the thickness of the transverse struts per se. To understand the extent to which the transversely oriented material influenced fatigue damage accumulation, we performed nonlinear finite element models of cyclic loading. Fatigue damage involves a local irreversible energy-dissipating process resulting in increases in inelastic dissipation energy. Finite element models of the first 5 to 25 cycles of loading indicate that the fatigue life of the octet and bone-like microarchitectures with and without thickened struts is closely related to the inelastic dissipation energy per unit work (Fig. 3C). Although these models are limited to the first few cycles of loading, inelastic dissipation energy and stress triaxiality stabilize by this point (SI Appendix, Fig. S5G). Hence, increases in the transverse volume fraction (ψ; the proportion of the solid volume oriented transverse to loading) in these microarchitectured materials reduce the amount of inelastic energy dissipation and damage accumulation during cyclic loading, just as thicker rod-like trabeculae (predominately transversely oriented) experienced less damage accumulation in cancellous bone (Fig. 1E). Following a single overload (50% strain), both bone (28) and microarchitectured materials (25, 29) can recover a large proportion of the applied strain, an effect attributed to elastic deformations in transversely oriented struts (however, in our work, the energy loss coefficient during fatigue loading did not vary much with alterations in strut thickness [SI Appendix, Fig. S6]). Our finding here suggests that transversely oriented struts are also important for resisting fatigue failure under intermediate and high cycle fatigue, such as that commonly applied to durable devices. Together, these findings show that the effects of transversely oriented material on fatigue life are not unique to cancellous bone but extend to synthetic microarchitectured materials. That our findings regarding damage accumulation are consistent in human bone tissue (a biological ceramic polymer composite) as well as a polymer used in additive manufacturing further supports the idea that the effect is due to geometry and may not be limited to one class of constituent material.

Fig. 3.

Transverse volume influences fatigue life in repeating cellular solids. (A) Images of the bone-inspired microstructure and an octet truss are shown. (Scale bar: 5 mm.) (B) The fatigue life of microarchitectured materials printed as designed or with rod-like struts thickened (colored) is shown. Thickening transverse struts increases fatigue life, while thickening vertically oriented struts reduces fatigue life (specific stiffness, E₀/ρ is also shown). (C) Fatigue life of the lattice structures is related to the inelastic dissipation energy per unit work determined from finite element models. (D) Fatigue life for 3D-printed specimens at different applied cyclic loading (σ/E₀, noted in microstrains) is shown with lines indicating regression model fits (Eq. 2) (R² = 0.82). Symbols indicate bone (●), bone-like microstructure (▲), and octets (♦).

We developed an empirical model to characterize the relationship between fatigue life (N_f), applied cyclic normalized stress (noted as σ/E₀), and transverse volume fraction (ψ) for bone, bone-like microarchitectures, and octet trusses generated with additive manufacturing. Surprisingly, the regression models identified a predictive equation only slightly different from Eq. 1:

$$\frac{\sigma }{E_0}\frac{1}{\sqrt{\psi }}=A{N}_f^B,$$ [2]

in which A and B are empirical constants (A = 51,903, B = −0.14, R² = 0.82) (regression coefficients are in SI Appendix, Table S5). Although there remains unexplained variance in Eq. 2 (likely due to differences in tensile stresses among the bone, bone-like, and octet microstructures), this modification to the normalized S-N relationship accounts for the differences in normalized S-N relationships among the microstructures and provides a simple means of considering fatigue life during the design/selection of microarchitectured materials.

In ultralightweight microarchitectured materials, lattice structures in which struts are aligned with expected loading and underloaded struts are removed can achieve more efficiency in terms of specific stiffness, strength, and energy absorption (25, 27). However, our findings demonstrate that such a strategy can come at the cost of large reductions in fatigue life due to reductions in the proportion of material oriented transverse to loading (as a result of reductions in the thickness and/or number of transversely oriented struts) (SI Appendix, Fig. S7). Future applications of high-porosity microarchitectured materials to durable products, such as vehicles, will require balancing performance in terms of stiffness, strength, and energy absorption with costs associated with replacement and repair due to fatigue failure.

Our findings indicate an effect of microstructure on fatigue; however, material properties can also influence the accumulation of microscopic damage due to fatigue loading. In particular, material heterogeneity (designed or naturally occurring) has the potential to influence fatigue failure. Furthermore, as strut size approaches that of the critical size for flaw insensitivity, there can be a large increase in the strength of a microarchitectured material (the so-called “smaller is stronger” concept) (8, 9, 11, 30). It is possible that such approaches could also be used to improve fatigue life, although little is known about the fatigue properties of such materials.

Our findings are also relevant to osteoporosis-related fractures in the elderly. Osteoporosis is characterized by degradations in cancellous bone microstructure illustrated by drastic reductions in the number and robustness of transversely oriented trabeculae (31). However, the stiffness, strength, and energy absorption of cancellous bone are overwhelmingly determined by longitudinally oriented trabeculae, and the mechanical importance of transversely oriented trabeculae is traditionally viewed as minor and limited to resisting occasional off-axis loading (32, 33) or providing simple support for the more mechanically relevant longitudinally oriented trabeculae (31). Our findings challenge this view by showing that transversely oriented trabeculae are key determinants of fatigue performance under longitudinal loading. While many osteoporosis-related fractures in humans are caused by a single overload sensitive to strength (fall from standing height, heavy lifting, etc.), the most common type of osteoporosis-related fracture, vertebral fractures in the spine, frequently occurs in the absence of a discrete loading event, implicating a contribution of fatigue damage (34). Preferential loss of transversely oriented trabeculae during aging, therefore, causes reductions in the fatigue life of cancellous bone that are disproportionately larger than the reductions in stiffness and strength, potentially explaining the long-held but poorly described relationship between cancellous bone microstructure and clinical fractures.

Materials and Methods

Mechanical Characterization and Damage Assessment in Cancellous Bone.

This study examined human vertebral cancellous bone (n = 44 cancellous bone specimens from 10 male and 8 female donors 62 to 92 y of age) as reported in 2 prior studies (21, 22). Human tissue was received from a nonprofit human tissue bank (National Disease Research Interchange) and is therefore exempt from institutional review board review. Cylindrical specimens 8 mm in diameter and 27 mm in length were oriented in the cranial–caudal direction and were press fit into end caps to avoid artifacts during mechanical testing (SI Appendix, Supplemental Materials and Methods).

Fatigue loading was applied using a 4-Hz haversine waveform cyclically between 0 N and a compressive load corresponding to σ = E₀ × 0.0035 mm/mm, where σ is stress and E₀ is the initial Young’s modulus of the specimen. Fatigue loading of the remaining specimens was applied at room temperature and stopped before failure (defined as 4% apparent strain [22]) at a predetermined magnitude of cyclic strain. After loading, specimens were carefully removed from the testing device and bulk stained with contrast agents that identify microscopic tissue damage (microdamage in the bone literature), and 3D images were collected. Microscopic damage detected with this methodology includes damage 5 to 10 μm in size, including both individual microscopic cracks as well as aggregation of submicroscopic cracks. Hence, individual submicroscopic cracks generated at lower levels of structural hierarchy (for example, those caused by individual lamellae, individual mineral crystals, etc.) are only detected in aggregate. More details on microdamage detection are in SI Appendix, Supplementary Methods and Materials and refs. 21 and 22.

Cancellous Bone Microarchitecture.

Microarchitecture was assessed using specimen average measurements [BoneJ; http://bonej.org/ (35); 21-μm isotropic voxel microcomputed tomography images collected prior to fatigue loading] (SI Appendix, Table S6) as well as morphological analysis of each individual trabecula (SI Appendix, Table S7). Morphological analysis involved classifying each trabecula as rod like or plate like based on Digital Topological Analysis (Fig. 1C) (ITS software; Columbia University). Failure of cancellous bone has been referred to as accumulation of failed trabeculae (28, 36). Here, we consider failure of a trabecula to occur when at least 10% of the volume of the trabecula includes microdamage (37). Variation in this definition of failure did not influence conclusions.

Additive Manufacturing of Modified Cancellous Bone Microstructures.

Microcomputed tomography images of cancellous bone samples (21, 22) that had been collected before mechanical loading were modified digitally by adding material to surfaces. A high-resolution stereolithography system (M1; Carbon) was used to generate 3D models of the modified and unmodified microstructures from a urethane methacrylate polymer resin (UMA 90; Carbon; E = 2 GPa) (27) at 1.5 times isotropic magnification (12-mm diameter, ∼30-mm length). The accuracy of the printed geometries was confirmed using microcomputed tomography images (SI Appendix, Tables S7 and S8). Models generated through additive manufacturing were submitted to cyclic fatigue loading from 0 to a normalized initial compressive stress magnitude σ/E₀ corresponding to 9,500, 6,500, or 4,500 µε until failure (4% applied strain). A total of 45 models of cancellous bone microstructure were used for this experiment (5 distinct microstructures × 3 distinct rod thicknesses × 3 different normalized stress magnitudes). Damage in polymer specimens was identified using a radio opaque dye penetrant.

Bone-like microarchitectures derived from the octet truss were designed by elongating the longitudinal axis (the longitudinal axis) and adding plates to achieve porosity, transverse volume fraction, longitudinal volume fraction, and plate volume fraction similar to the cancellous bone specimens (SI Appendix, Fig. S5).

Finite Element Modeling.

Finite element modeling was used to characterize the stress distributions in bone and lattice structures generated using additive manufacturing. Models were meshed with elastic perfectly plastic brick elements (1.5 million per model), and quasistatic analyses were carried out for cyclic compressive loading. Inelastic dissipation energy was defined as that portion of the internal strain energy that is dissipated by rate-independent and rate-dependent plastic deformation calculated as

$$E_p={\displaystyle \underset{0}{\overset{t}{\int }}\left({\displaystyle {\int }_V{\sigma }^c:{\dot{\epsilon }}^{pl}dV}\right)d\tau },$$

where E_p is the inelastic dissipation energy, σ^c is the stress derived from the constitutive equation, and $\dot{\epsilon }$^pl is the plastic strain rate. These properties are integrated in time and over the volume of the specimen.

To characterize the primary deformation mechanism (bending dominated or stretching dominated) within each microstructure, the stress triaxiality was determined at each point within the model. Stress triaxiality is defined as

$$Triaxiality=-\frac{p}{q},$$

where p is the hydrostatic stress defined as positive when stress is compressive and q is the von Mises stress. By this definition, points under uniaxial tension would have a triaxiality of +0.33, and points under pure uniaxial compression would have a triaxiality value of −0.33 (SI Appendix, Fig. S5).

Data Sharing.

All data, documentation, and custom code used in this work are available from the corresponding author on reasonable request.