Human bone fracture behavior and toughness are of interest to both the en- gineering and clinical orthopedic communities as new treatments for bone diseases, such as osteoporosis, are developed. Much of the work on bone fracture has used linear elastic fracture mechanics (LEFM) or the J-integral. These works indicate that bone has a large resistance to fracture that declines with age. However, there is large variability among current studies in results, specimen size, donor tissue demographics, and experimental methods. Here, a quasi-brittle fracture mechanics framework is introduced to evaluate corti- cal bone fracture, supported by in-situ loading and 3D imaging. Quasi-brittle fracture mechanics can provide insights into the behavior of cortical bone be- yond LEFM approaches as we introduce measurements across multiple length scales into the fracture characterization of bone. Quasi-brittle fracture me- chanics allows tools for further development of treatments to address both bone quantity and bone quality. Keywords: Bone, Fracture, Bone Quality, Bone Quantity ∗Corresponding Author at: 585 Purdue Mall, West Lafayette, IN 47906, USA Email address: siegmund@purdue.edu (Thomas Siegmund) Preprint submitted to Journal of Biomechanics October 5, 2023 1. Introduction Recent census reports indicate that the US population is older than ever before, thereby increasing the population at risk for bone fractures [1]. Fragility fractures in those of advanced age are linked to decreased quality of life and increased mortality risk [2]. However, treatments for osteoporosis are among the least effective when compared to other common diseases [3]. The development of effective methods to decrease fracture risk is critical to advancing health outcomes in the current population and reducing burden on healthcare infrastructure. Assessing the need for clinical intervention in bone diseases such as osteoporosis has often relied on measurements of bone mineral density (BMD) and its implied strength through Dual-energy X-ray absorptiometry (DEXA). However, DEXA imaging has been previously re- ported to be imperfect in assessing fracture risk [4, 5]. Fracture mechanics can provide an alternative approach to understanding other factors leading to fragility fracture beyond density and strength [6]. Linear elastic fracture mechanics (LEFM) has typically been used to re- port fracture behavior in bone, either through stress intensity factors, critical distance theory, or via the J-integral. Stress intensity factors are a common tool for characterizing toughness and have been used in several studies with a variety of tissues and experimental methods with a large range in results [7]. The theory of critical distances, derived from LEFM, has been previously applied in fracture mechanics of bone and suggests that the microstructure influences toughness [8]. However, LEFM assumes that no significant dam- age occurs before peak load and the material is linear elastic which is untrue of bone. The J-integral has been commonly used in fracture experiments of bone to demonstrate its strong resistance to fracture through the steep rise in J as a crack grows relative to other materials and bone’s individual ma- terial components [9]. The J-integral does account for plasticity and other non-linear fracture behavior, but does not include a length scale and cannot characterize fracture behavior near the tip. J-integral calculations following ASTM E1820 [10] may also over-predict bone toughness [11]. Fracture toughness of bone has been found to differ with respect to age [12, 13, 14, 15], disease [16], and anatomical direction [7]. Regardless, bone exhibits exceptional toughness and resistance to fracture due to the extrinsic and intrinsic toughening mechanisms present across hierarchical length scales [17]. Of particular interest to this study, are toughening mechanisms derived from the structure and interfaces in bone, such as microcracking, crack deflec- 2 tion at the cement line, crack bridging, and osteon pullout [18, 19, 20]. The accumulation of damage develops a fracture process zone (FPZ) in cortical bone. While much research has employed LEFM to describe bone toughness, the relative size of the FPZ is significant compared to both micro-structural and whole bone length scales and informs the need for an alternative ap- proach. Quasi-brittle fracture mechanics (QBFM) has been developed to ana- lyze fracture in materials that possess a relatively large FPZ and exhibit a size effect [21, 22]. QBFM has been implemented in several materials, such as concrete [21, 23], fiber composites [24], ceramics [25, 22], and rock [26, 27]. QBFM has also been used to characterize bovine bone [28]. Some measurements in the human humerus point to an FPZ size in the longitudi- nal direction of 5-7 mm [29], but no measurements have been reported for human cortical bone in the transverse direction. To determine size effects on strength and/or toughness in engineering materials, multi-specimen ap- proaches are typically used by 1) Evaluating geometrically similar samples of multiple sizes [28, 23], or 2) evaluating samples of the same size with multiple crack lengths [23]. Such approaches become challenging to execute in bone due to the limited material availability of human bone tissue for experiments, high subject specificity, and biological limitations on tissue size. This study aims to apply QBFM to human cortical bone using an alternative approach made possible by in-situ loading with 3D imaging. QBFM further allows interpretation of bone quality and bone quantity contributions to fracture toughness. 2. Materials and Methods 2.1. Materials The diaphysis of a human (92-year-old, male) cadaveric femur was ob- tained through the Indiana University School of Medicine Anatomical Dona- tion Program. A 3D scan of the femur revealed the largest cortical thickness at the distal mid-diaphysis. A section of the femur was extracted at this loca- tion. In this section, the average cortical thickness in the section was ˆtc= 6.9 mm. Considering cortical wall curvature and internal porosities, the largest prismatic bars that could be extracted from the section were of nominal size 4.0 mm ×4.0 mm ×28.0 mm. Specimen were cut using a low-speed saw equipped with a cBN blade (Buehler, Illinois, USA) and ground to square. The sample of interest was dissected from the lateral posterior cortex and 3 was of actual dimensions: B= 4.1 mm, D= 3.9 mm, L= 28.0 mm. An edge notch, of depth a0= 1.7 mm, was introduced into the endosteal face of the SEN(B) specimen using a low-speed saw equipped with a 200 µm thick diamond blade (Buehler, Illinois, USA). The bone volume to total volume ra- tio (BV/TV) of the un-notched beam was determined using a SkyScan 1176 µ-Computed Tomography system (Brucker, Massachusetts, USA; 16.9 µm resolution) and segmented with Simpleware ScanIP (Synopsys, California, USA). BV/TV in the specimen half above the notch was 88%. The elastic modulus of bone matrix was taken to be E0= 19.08 GPa in the longitudinal direction [30]. Osteon diameter, On.Dm, was measured from a polished slice of the beam post-fracture (Fig. 5A) by backscatter SEM as the average minimum Feret diameter of the outlined osteons (Fig. 5B) and found to be On.Dm = 195 .0µm [31]. Figure 1: (A) 3D reconstruction of the femoral shaft (transparent) with section where beam was extracted (opaque). (B) Cross-section shown in (A) of femoral shaft with approximate beam location outlined. (C) Backscatter SEM of beam cross-section. (D) Portion of backscatter SEM indicated in (C) showing outline of osteons used to measure On.Dm (white). The sample presented is a control from a larger study on the pharmaceu- 4 tical treatment of bone. Consequently, the specimen was incubated for 14 days at 37 °C in a solution of phosphate buffered saline supplemented with 1% penicillin-streptomycin and 0.04% vol/vol dimethyl sulfoxide, [32]. After, the sample was kept frozen at -20 °C, and defrosted at 4 °C overnight before fracture experiments. 2.2. Fracture Experiment with 3D Imaging In-situ fracture experiments were conducted in 3-point bending with a Deben CT5000N load cell (Deben, Bury St. Edmunds, UK) in a Zeiss XRA- DIA 510 Versa 3D X-Ray microscope (Carl Zeiss AG, Baden-W¨ urttemberg, Germany). The 3-point bending frame had a span S= 20 mm with rigid X-Ray transparent glassy carbon supports (diameter 5.0 mm) to avoid image artifacts. To maintain hydration, the beam was wrapped in plastic film with a slit around the notch. Displacement was applied at a rate of 0.1 mm/min up to the approximate onset of non-linearity. At that instant, the displace- ment was held constant and the first 3D image was obtained. Subsequently, the displacement was increased at the same rate until a load increase of 10 N was observed and another image was obtained. This sequence was repeated until peak load and resulted in six 3D images. 3D X-ray images were ac- quired with a resolution of 4.5 µm using an exposure time of 5 s for the 801 projections at 120 kV, 10W, 4 ×objective, and a LE2 filter. For the full imaging sequence, the estimated radiation dosage by the manufacturer was 54 Gy, which is on the order of magnitude found not to influence the mechanical or fracture properties of bone [33]. Applied force data was obtained from the load cell of the Deben system. X-Ray projections were processed through XRADIA Scout-and-Scan Recon- structor and then analyzed using Simpleware ScanIP. A recursive Gaussian smoothing filter ( σ= 1 pixel) was applied to reduce image artifacts. Crack mouth opening displacement, CMOD , was obtained from 3D image mea- surement tools in ScanIP. FPZ length, lFPZ, was determined, with respect to pixel size, as the distance from the first image slice orthogonal to the ini- tial crack plane containing a partially separated crack to the last image slice containing a visible crack. The length of the fully separated crack growth, ∆afull, can be similarly measured from the notch root to the last slice of a fully separated crack; then a total effective crack growth can be determined: ∆aeff= ∆afull+lFPZ (1) 5 2.3. Linear Elastic Fracture Mechanics - LEFM Evaluation of the tissue toughness can begin from ASTM guidelines for fracture experiments [10]. The linear elastic fracture toughness, GLEFM is: GLEFM (D) =P2 max E′ρD3g(S/W )(α0) (2) Here, Pmaxis the maximum measured load in the experiment, and B2D= ρD3. The plane strain elastic modulus is E′=E0(BV/TV )/(1−ν2). The normalized initial notch depth is α0=a0/D, where a0is the initial notch depth. The configuration function gdepends on specimen and support geom- etry. Configuration functions are most commonly provided for stress inten- sity factors, K=p (G/E′). Then, the configuration factor for calculations of energy release rates is g=k2. A comprehensive solution of k(S/W )(α) for the SEN(B) specimen in a 3-point bend configuration is given in [34]. For the span-to-width ratio ( S/W = 5) the following expression is provided: k5(α) =√α (1−α)3/2(1 + 3 α)[2.062 + 1 .166α−0.966α2+ 0.388α3] (3) 2.4. Quasi-Brittle Fracture Mechanics - QBFM In materials that are not perfectly brittle, an FPZ of finite length, lFPZ, is present at the crack tip. Within that zone, the material transitions from being fully intact to fully separated. The maximum extent of lFPZis reached at the instance of Pmaxand the elastic equivalent crack extension can be estimated as c= max( lFPZ)/2, [35]. A factor of 1/2 is used to account for a linear decay of traction from the crack tip to the fully separated crack. A normalized effective crack length at Pmaxis defined as: αeff=1 D(a0+c) (4) Then, a corrected toughness measure for the quasi-brittle fracture mechanics approach, GQBFM , can be established by accounting for the fracture process zone. GQBFM =P2 max E′ρD3g(αeff) (5) The two toughness measures can be connected if a linear extrapolation ap- proach is used [35]. Then, the configuration function with respect to the effective crack length in the large limit can be defined as g(αeff) =g(α0) +g′(α0)cf D(6) 6 where g′is (dg/dα ). Combining Eq. 6 with Eqs. 2 and 5 leads to an approx- imation for the quasi-brittle fracture toughness Glin QBFM : Glin QBFM =GLEFM 1 +c Dg′(α0) g(α0) =GLEFM 1 +1 β (7) In Eq. 7 the term β=D cg(α0) g′(α0)plays a central role and characterizes the degree to which a specimen is considered to be either brittle (such that LEFM applies, β >10), quasi-brittle (such that QBFM applied, 0 .1< β < 10) or not crack sensitive (such that neither LEFM or QBFM apply, β < 0.1) [36]. The existence of a finite length FPZ causes GLEFM andGQBFM to be dependent on specimen size. Typically, for engineering materials, a size-independent fracture toughness, Gf, is determined by extrapolation to infinite specimen size ( d→ ∞ ) using Eq. 7, [35]. The value of Gfis then considered a material property. However, for human cortical bone, such an extrapolation is not viable. Due to the biological constraints on the cortical thickness, it is not possible to increase the specimen dimensions much beyond Das would be needed for the determination of Gf. 3. Results Figure 2 depicts the Force - CMOD record from the experiment, and shows a response typical for quasi-brittle solids. After an initial settling pe- riod, the response of the specimen is linear, followed by a nonlinear response initiating around P3with a load maximum, and subsequent load reduction to failure. Figure 3 depicts the reconstruction of the 3D X-ray image at load stage P6= 62.3 N. Movies depicting the reconstructed Haversian Canal and crack structure are provided as Supplementary Material. Reconstructed images provided here are larger than others have reported using smaller samples with cross-sections on the scale of 2 mm or less [37, 7, 38, 14] The crack becomes fully separated from the induced notch at peak load ( P5), at which point the fracture process zone is fully developed and of constant length. Observed in 3D, the crack is shown to be tortuous and interacts with the microstructure at the length scale of On.Dm. Crack bridging (Fig. 3D), crack deflection (Fig. 3B-C), and osteon pullout (Fig. 3A) are observed. The crack extends further internally than at the surface due to plane strain conditions as compared to plane stress conditions at the beam face (see Fig. 7 Figure 2: Force, P, vs. Crack Mouth Opening Displacement, CMOD , for 3-point bending of the SEN(B) specimen. Data points (dots) for each load image pair (#), Pmax=P5= 66.3 N. Fit to data (dashed line). 4). The crack extends in the transverse direction and deflects longitudinally in areas where cement lines are expected based on Haversian canals and osteon diameter. Based on the sequence of reconstructed 3D images at each load step, development of the FPZ can be tracked (Fig. 4). FPZ growth begins around the onset of linearity ( P3, see Fig. 2) and plateaus in size at peak load ( P5). The FPZ begins internally and extends to the surface after increased load. In the sample, max( lFPZ) = 348 µm. As the osteon emerges as the microstructural feature determining crack growth, we express max( lFPZ) =η·On.Dm, with η= 1.8 for the present specimen. Using Eq. 2 with the maximum measured force (Fig. 2, Pmax=P5) and α0= 0.41, the LEFM energy release rate value is GLEFM = 0.46 N/mm. At peak load, from Fig. 4, one determines c= 174 µm which results in αeff= 0.46. Then, the quasi-brittle fracture toughness accounting for the presence of the FPZ (Eq. 5) is GQBFM = 0.60 N/mm. The linearization of Eq. 5 to Eq. 7 provides Glin QBFM = 0.58 N/mm. Furthermore, β= 3.9, is well in the range of quasi- 8 Figure 3: Reconstructed 3D image of the cortical bone specimen under mechanical loading atP6= 62.3 N. Haversian canals (red) and the material separation induced by mechanical loading (blue). (a) A view orthogonal to the plane of the initial notch with examples of osteon pullout (circled). (B,C) Views parallel to the front of the initial notch (outlined) from each side, (D) Perspective view with examples of crack bridging (boxed). brittle fracture conditions. Hypothetically, a specimen size of d >30 mm would be needed to obtain LEFM conditions, a condition that cannot be fulfilled as tc= 6.9 mm. 4. Discussion Cortical bone extracted from the femur of a 92-year-old male emerges as a quasi-brittle solid with GLEFM markedly different from GQBFM . This is confirmed by pronounced FPZ at the crack tip which introduces a size dependence of the fracture characteristics. Prior fracture mechanics stud- ies on transverse fracture toughness of cortical bone have considered smaller 9 Figure 4: Crack measurements with respect to loading state (#). The fracture process zone grows up to peak load and plateaus at lFPZ = 348 µm. As the FPZ stops growing, the fully separated crack begins to grow. Together these contribute to the total crack growth. The crack visible at the beam surface, ∆ asurf, is considerably smaller as compared to that seen internally. specimen dimensions and therefore detected a rather ductile behavior [7]. Such a behavior is the result of the dominant non-linearity of the FPZ. The QBFM approach as applied here indicates that even for the practically largest specimen size possible in the human femur, there is a significant size effect on fracture toughness. Investigation of the fracture surfaces, in conjunction with 3D reconstructions, can lead to insight into the mechanical and bio- logical interactions. Figure 5 shows part of the fracture surface created in the experiment; Distinct height differences emerge at cement lines in the di- rection of crack propagation. Osteons emerge as the fundamental building blocks of the bone tissue, and the failure of the cement line and subsequent osteon fracture determines the FPZ. Discussions on treatment of bone dis- ease often focus on addressing bone quantity or bone quality. Both of these areas encompass a large field of contributing and overlapping variables. The QBFM framework allows these areas to be addressed in terms of the individ- 10 Figure 5: Post-mortem SEM of fracture surface. The crack propagates upward from the machined notch and height differences appear as a result of crack deflection at cement lines and osteon pullout. ual variables’ contribution to fracture toughness. To assess the mechanical resilience of the mid-section of the femur bone, Eq. 7 is rewritten: Gbone=Glin QBFMh 1 +η·On.Dm ˆtc˜g′(˜α0) ˜g(˜α0)i (8) Thereby, the initial crack length is that of a naturally occurring defect, ˜a0∝On.Dm (9) ˜α0= ˜a0/tc (10) The functions ˜ g(˜α0) and ˜ g′(˜α0) are the configuration function and its deriva- tive, respectively, for such defects in the long bone, which can be approx- imated by known configuration functions (e.g. a surface crack in a thick- walled pipe under bending [39, 40]). The factor ηrepresents the length of the FPZ relative to On.Dm. The apparent toughness of the femur, Gbone, can be used together with the LEFM approach, Eq. 2, to determine the maximum load allowable, Pallow, in such bone Pallow=s GboneE′˜ρt3 c ˜g(˜α0)(11) 11 where ˜ ρis a geometry factor for the femur cross-section. We propose to use Eqs. 8 and 11 as a framework for understanding the combined effects of bone quantity and bone quality on bone resilience. The obvious objective is to increase Pallowby increasing Gbonesuch that toughness is increased and frac- ture risk decreases. The equation for Gboneis coupled both mathematically (e.g. ˜ g( ˜α0) is dependent on tc) and biologically (e.g. remodeling balances both tcand On.Dm), but the individual factors can be evaluated for their effect. Aging often reduces bone quantity; reduction of cortical thickness [41] and BV/TV [42] are observed. Reducing tcwill result in a reduced Pallow directly through the t3 cterm, by reducing Gbonedue to the 1 /tcterm, and by increasing ˜ α0and thus also increasing both gandg′/g. A loss in BV/TV due to increased Haversian canal diameter reduces the resilience of the individual osteon to failure and reduces GQBFM andPallow. Measures of bone quality also change with age; On.Dm decreases [31], the size and number of in-vivo microcracks increases [43], and non-enzymatic collagen cross-linking and mineralization also increase [44, 45]. Decreases in On.Dm would reduce the FPZ length and thus reduce GboneandPallow. As micro-crack size and number increases, at least in part due to sluggish remod- eling, in a similar manner to the effects of BV/TV, ˜ α0increases and GQBFM andPalloware reduced. Increased non-enzymatic cross-linking in collagen re- duces the deformation capacity of the individual osteon and the bone matrix as a whole. Such cross-linking reduces bones ability to form crack bridges and extend lFPZbyη, leading to a reduction in GboneandPallow. Increased mineralization moves bone towards a more brittle composition which would alter βand lead to decreased GboneandPallow. Anti-resorptive drugs, such as bisphosphonates and Denosumab are com- mon tools for treating bone loss, and their effects on several of the con- tributing variables in equation 8 have been studied. Cortical thickness in the femoral shaft is reported to not be influenced by bisphosphonate treatment [46, 47] or exercise [48]. However, some research indicates apparent corti- cal thickness increased due to Alendronate and Denosumab treatment [49]. Denosumab has shown some effectiveness in changing cortical thickness at the trochanter [50]. Denosumab is also linked to increased BV/TV at the proximal femoral shaft in women with osteoporosis [51]. In terms of bone quality, bisphosphonate treatment has been shown to not affect the degree of crack deflection [52]. Bisphosphonate treatment has been used to reduce cortical porosity, however, this is often achieved by blocking bone resorption 12 and remodeling which can allow damage to accumulate and drive up the ini- tial defect size [53]. In the treatment of osteoporosis, bisphosphonates have been linked to increased osteon diameter as compared to same-aged peers, yet a cause is unclear [54]. Bisphosphonates can induce hypermineralization which further increases brittleness [44]. Overall, anti-resorptive pharmaceuti- cal treatments may reduce fracture risk by increasing tcand On.Dm, but this effect may be counteracted by increases in microdamage and mineralization. Parametric studies have shown that cement line, interstitial bone tissue, and osteon bone tissue properties have a significant impact on the crack path and degree of osteon deflection [55, 56]. Additionally, the osteon density and material heterogeneity impact fracture properties in experimental and computational studies [57, 58]. The bone ultrastructure, a function of the mineral, collagen, and water content, has been found to affect toughness [59]. Improved toughness and post-yield behavior have been observed in canine and murine bone treated by raloxifene or calcitonin which alters bone water content through non-cell mediated mechanisms [32, 60]. While On.Dm is somewhat dictated by biological sex and age, weight and local strains induced by physical activity can be controlled to change On.Dm [31, 61]. The presented work is limited to a single specimen but motivates the need to incorporate size effect into the evaluation of fracture toughness of human bone. The results show a lower toughness than those typically reported, but are derived from a sample of advanced age, of larger specimen dimensions than typical, and detect crack initiation earlier - all factors which could con- tribute to the lower measured toughness. This sample is exposed to air and drying effects during testing which does not represent anatomic conditions and could alter the true tissue toughness [62]. 5. Conclusions Addressing bone health is key to improving the quality of life among those of advanced age and reducing the burden on healthcare infrastructure. This study shows that a fracture process zone of significant size exists in transverse fracture of the femur. Using quasi-brittle fracture mechanics, a size-accurate toughness can be obtained by accounting for the fracture process zone size in addition to the linear elastic fracture mechanics analysis. The QBFM framework allows for discussion on the relevant structural variables affecting bone quantity and quality. 13 6. Conflict of interest statement The authors declare no conflicts. 7. Acknowledgements This work is supported by NSF (CMMI) grant 1952993. G.G. is sup- ported by NSF-GRF DGE-1842166 and Purdue Doctoral Fellowship. We ac- knowledge the 3D X-Ray Microscope Facility in the Department of Physics, Purdue University, supported by the EVPRP 2017 Major Multi-User Equip- ment Program.