Analyzing the effect of hydration on the wedge indentation fracture behavior of cortical bone

Author’s Accepted Manuscript Analyzing the effect of hydration on the wedge indentation fracture behavior of cortical bone Kevin Hoffseth, Connor Randall, Srinivasan Chandrasekar, Paul Hansma, Henry T.Y. Yang www.elsevier.com/locate/jmbbm

PII: DOI: Reference:

S1751-6161(17)30001-2 http://dx.doi.org/10.1016/j.jmbbm.2017.01.001 JMBBM2172

To appear in: Journal of the Mechanical Behavior of Biomedical Materials Received date: 5 April 2016 Revised date: 24 December 2016 Accepted date: 2 January 2017 Cite this article as: Kevin Hoffseth, Connor Randall, Srinivasan Chandrasekar, Paul Hansma and Henry T.Y. Yang, Analyzing the effect of hydration on the wedge indentation fracture behavior of cortical bone, Journal of the Mechanical Behavior of Biomedical Materials, http://dx.doi.org/10.1016/j.jmbbm.2017.01.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Analyzing the effect of hydration on the wedge indentation fracture behavior of cortical bone Kevin Hoffseth1*, Connor Randall1,2, Srinivasan Chandrasekar3, Paul Hansma2, Henry T. Y. Yang1 1

Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA 93106, USA 2 Department of Physics, University of California Santa Barbara, Santa Barbara, CA 93106, USA 3 Center for Materials Processing and Tribology, School of Industrial Engineering, Purdue University, West Lafayette, IN 47907, USA *Corresponding author

*Corresponding author. Kevin Hoffseth, Department of Mechanical Engineering, Engineering Building II, Room 2355, University of California, Santa Barbara, Santa Barbara, CA 93106-5070. (805) 893 5087. [email protected]

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Abstract: Hydration directly affects the mechanical properties of bone. An initial and basic procedure shows both wedge indentation fracture experiments under plane strain conditions in cortical bone and numerical simulation with finite elements agree that dry bone fractures much more easily than fully hydrated bone submerged in an aqueous environment, such as in the body of an animal. The wedge indentation experiments were performed with high speed video microscopy, under dry and fully hydrated (submerged) conditions. The numerical simulation, specifically finite element analysis using cohesive elements to simulate fracture, was utilized to capture plasticity, fracture initiation and propagation, and to study the applicability of brittle material based indentation fracture theory. Experiment and theory give similar results for the dependence of depth of fracture initiation, and size of plastic zone, on hydration state. Comparison of fracture propagation characteristics between wet and dry bone are examined and discussed. This research demonstrates the ability to quantitatively assess the effect of hydration on the fracture initiation, propagation, and plastic zone size of cortical bone, through an approach using simple wedge indentation, with important implications for efforts in developing methods to understand clinical diagnostic testing and general fracture behavior of living bone in the ultimate interest of health care purposes.

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1. Introduction: In recent years there has been a growing interest in investigating and understanding mechanical properties and performance of bone. A basic step to reach this goal has been to study the response of bone under indentation. The resistance to indentation in bone can reveal much about the health and structure of the host body, including fracture resistance (Diez-Perez et al., 2010; Guerri-Fernandez et al., 2013; Randall et al., 2013; Farr et al., 2014; Malgo et al., 2015; Duarte Sosa et al., 2015; Mellibovsky et al., 2015; Rudang et al., 2015). Understanding mechanical properties such as plasticity and fracture in bone, how to analyze their impact under in-vivo or clinical loading conditions using theory, experiment and computation, and how they are affected by hydration and microstructure holds promising importance in treatments ranging from novel diagnostics to orthopaedic surgery. Bone is the main load bearing organ of vertebrate animals, making it a specific point of interest in science and medicine since before the well-known experiments of Galileo. Bone can be classified into two main types, cortical (or compact) and trabecular (or spongy), where cortical bone will be the focus of study here, at the characteristic Haversian, or osteonal level. Cortical bone is a biological hierarchical composite material, with different structural configurations corresponding to different scales of length. On the centimeter scale the boundary between cortical and trabecular bone becomes distinguishable. On an even smaller scale, at the sub-millimeter length, the characteristic Haversian, or osteonal system of circular subunits of lamellae enclosing canals filled with blood and nutrients are visible (Rho, 1998; Currey, 2002; Olszta, 2007). As a biological material, situated inside the body, bone has a natural dependence on hydration state. In the past, many insightful bone studies were performed with dry bone, but the importance of hydration has increasingly been more commonly recognized and incorporated (Riley et al., 1975; Hensberger et al., 2002; Gupta et al., 2006; Guidoni et al., 2015) and explored further, for example in terms of microstructural influence on strength and fracture toughness (Nyman et al., 2006). Indentation of cortical bone for mechanical property investigation has been explored in various research. Early investigations attempted conventional hardness testing methods, such Rockwell, Vickers or Knoop (Zysset, 2009), while others have utilized nanoindenters with various tip shapes (Hensberger et al., 2002; Fan et al., 2003; Guidoni et al., 2015), microindentation of different types (Yin et al., 2012; Zhang et al., 2008) or atomic force microscope (Thurner, 2009). Recently, methods of Reference Point Indentation have been utilized in bone investigations, holding promise for handheld in-vivo measurements of bone in living patients. Work has shown correlations with fracture risk and indentation response (Diez-Perez et al., 2010; Gallant et al., 2013; Farr et al., 2014) with a type of cycled Reference Point Indentation instrument, while lately efforts have examined bone response with a new instrument, a hand-held Reference Point Indentation instrument named the Osteoprobe, utilizing a single rapid, dynamic, indentation, with maximum indentation distance being the primary variable (Bridges et al., 2012; Randall et al., 2013; Hoffseth et al., 2015). Fracture is a key topic in research of cortical bone, and work to understand damage and fracture in bone under various loading continues through the present (Guo et al., 1998; O'Brien et al., 2005; Nalla et al., 3

2005; Yan et al., 2007; Ritchie et al., 2009; Zimmermann et al., 2009; Zimmermann et al., 2011; Li et al., 2013). Bone response can reveal critical information about bone health as a whole, with important implications for fracture risk (Zhang et al., 2010; Rho et al., 1998; Zysset, 2009). Finite element analysis has been increasingly used as an investigative tool to examine the behavior of fracture in cortical bone, with work including general fracture toughness (Hogan, 1992; Abdel-Whab et al., 2012; Ural et al., 2013; Donaldson et al., 2014), the difference in aging with virtual compact tension fracture test specimens (Ural et al., 2005) or on multiple length scales (Budyn et al., 2008), response with models built from quantitative computed tomography scanning (Crawford et al., 2003), and the influence of cement lines and microstructure (Raeisi Najafi et al., 2007; Mischinski et al., 2009). Cohesive elements in particular have been previously used to investigate fracture in cortical bone under various conditions (Yang et al., 2006; Ural et al., 2006, 2007, 2013; Mischinski et al., 2009). Lately, work with cohesive elements was used to compare Mode I fracture results in cortical bone with results from standard double cantilever beam testing, including those on the influence of hydration (Morais et al., 2010). Additionally, they have been used to study brittle fracture (Xu et al., 1994), indentation fracture (Lee et al., 2012), and impact damage (Camacho et al., 1996). Recently, cutting type wedge indentation experiments have been performed (Li et al., 2014; Sugita et al., 2009; Kasiri et al., 2010; Li et al., 2014) with implications for surgical knowledge and fundamental mechanics, particularly the fracture response in bone under a cutting type indentation. The similarity to this growing body of wedge indentation work in bone to that of research in the field of brittle materials must be noted, with beneficial implications. Starting with fundamental work on brittle fracture (Barrenblatt, 1962) and moving through to work on brittle indentation fracture, where theory was used to develop an elastic-plastic or rigid-plastic framework to describe stresses and deformation under the indentation load (Lawn et al., 1975; Lawn et al., 1980), and the use of cohesive elements in modeling brittle fracture (Tvergaard et al., 1992; Xu et al., 1994; Ortiz et al., 1996), indentation fracture in brittle materials has been explored and refined, with continued development. A schematic illustrating the application of brittle indentation fracture theory to bone is shown in Figure 1. Studying the influence of hydration on the behavior of cortical bone underneath indentation by both experiment and finite element simulation, especially in regards to any in-elastic deformation or plasticity present and the subsequent fracture, will help in understanding the applicability of existing brittle indentation fracture theories to cortical bone and understanding of cortical bone behavior under indentation in-vivo, completely hydrated, where experimental studies have been lacking. The purpose of the current study was to develop an approach to analyze the effect of hydration on the indentation fracture behavior of cortical bone underneath a wedge shaped indenter, with longitudinal osteonal orientation in plane strain condition. The study used two different hydration states: dry and completely hydrated under total submergence in solution. The difference in wedge indentation fracture behavior under the specified conditions was examined through a combined approach of both experiments and finite element analysis with cohesive elements, with focus on phenomena including the apparent in-elastic, or plastic behavior prior to fracture, plastic zone size, and the fracture behavior itself, characterized by crack initiation and propagation, with basic analysis to characterize toughness.

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2. Materials & Methods 2.1 Cortical Bone preparation Whole bovine femurs were obtained freshly harvested from a mature animal. The bone was sectioned roughly into 6 mm cubes of cortical bone by vertical bandsaw (Gryphon) and then ground to finish using a polishing wheel (SBT Model 900) with copious amount of fluid keeping the specimens hydrated while processing. Final dimensions were approximately 5mm cubed. Cube faces planned for viewing through video capture were finished to a suitable surface texture ideal for video capture using 1200 grit. Orientation of the osteonal microstructure was aligned normal and parallel to the cube edges during sectioning, with slight deviation in microstructure alignment among samples due to material preparation and natural variation. Finished sample cubes were then frozen until test, wrapped gauze soaked with Hanks Buffered Salt Solution (HBSS, Thermofisher Scientific). Samples for dry testing were set aside a day prior to actual tests, allowed to rise to room temperature while wrapped in soaked gauze and submerged in solution, then placed in a desiccation chamber for 24 hours. For samples set aside for wet testing, on the day of testing they were allowed to rise to room temperature while wrapped in soaked gauze and submerged in solution.

2.2 High Speed Video Capture of Indentation Fracture Indentation fracture events were captured by high speed video camera (PCO.dimax) focused through a glass plate (common borosilicate glass) and a custom fixture apparatus, capable of retaining an amount of solution needed to completely submerge the tests samples. A framerate of 200 frames/second was used, in order for the best match between image size and frequency, and to attempt to effectively capture initiation and propagation. An optical zoom of 10x was obtained through an attached objective. Fiber optic lighting was arranged around the fixture in order for sample illumination with multiple lighting units required due to the framerate and optical properties resulting from sample submersion. Digital data of each experimental run was collected in a series of raw images (1 image per frame), and transferred to hard-drive. The experimental setup is shown in Figure 2. A total of nine samples were tested, five dry and four submerged, respectively, as detailed in the following. The fixture itself was composed of multiple parts, machined out of aluminum, primarily a frame with a large cavity and threaded bolt holes, a large plate with viewing opening and bolt holes, a glass plate for the viewing, and rubber gland to prevent solution loss/leakage. For experiment and video capture, the fixture was setup inside a MTS universal test machine (QTest/50LP) with a compression platen driving the wedge indenter tip into the bone at a displacement controlled rate of 20 microns per second. Target displacement was 500 microns. Loading was kept quasi-static to eliminate inertial effects. The indenter tip was a 90° wedge ground out of a high speed steel tool-bit, with tip radius of <15 microns. The fixture was designed with a channel to accommodate the insertion of the wedge tool, with two small set screws used to press the wedge tool against the fixture and glass and minimize out of plane movement of the wedge as it is indented. This was necessary to prevent bubbles or bone debris obscuring the wedge tip and indentation site.

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Sample bone cubes were oriented in the fixture with the osteonal microstructure running vertically, as specified before, with the prepared surface facing the camera through the glass plate, and tool tip indenting downward. Flexible shims were used to orient the bone samples correctly under glass plate. For dry sample testing, the fixture was dried and cleaned prior to test, then samples were inserted and fixture tightened, while dry. For wet, fully submerged sample testing, solution was first added to the fixture to about half capacity, then bone samples were taken from submersion in room temperature solution and oriented in the fixture. After tightening of the face plate, solution was added through the tool channel to completely immerse the sample. An example of the optical view of the bone through the camera can be seen in Figure 3, while a progression of stills showing emergence of cracking and fracture can be seen in Figure 4.

2.3 Finite Element Analysis For finite element simulations of the indentation, the vertically oriented cortical bone was modeled as a homogenous, isotropic, elastic-plastic solid continuum body with an embedded interface, representing a cement line boundary between osteons. The majority of the bone was modeled with four node, eight degree-of-freedom, quadrilateral, plane strain elements. The cement line interface was modeled with four node, eight degree-of-freedom cohesive elements, chosen for their ability to capture material separation and surface creation suitable for investigation of fracture. The interface of cohesive elements was embedded in a single vertical, central alignment, in order to best simulate the expected fracture. The overall finite element mesh can be seen in Figure 5. The mesh was generated through a combination of both elements formed by ordered node position and elements formed through automatic generation, using an advancing front method (ABAQUS 6.14, Dassault Systemes). A convergence study was performed numerically, with the results of fracture initiation depth versus number of nodes shown in Figure 6. The effect of mesh refinement was studied from coarse to fine, specifically focusing on the finer mesh of a majority of nodes located in the zone under the indenter. Node number was varied from 980 to 53,706, with resulting fracture initiation depth showing monotonic convergence after 10,000 to 16,000 nodes. There were a total number of 16,205 elements in the final mesh, with greater refinement under the area of initial indentation contact and along the interface composed of cohesive elements. ABAQUS/Standard solver was chosen to solve the finite element stiffness equations. The behavior of cohesive elements are dictated by a traction-separation relationship which describes the tractions needed to separate the element surfaces at a specific magnitude, until failure by complete separation and creation of new surface. A schematic of the relationship used is shown in Figure 7. The parameters used to define the traction law are Gc (work per unit area to separate material, equivalent to the energy release rate, and GIc in pure Mode I loading),

σ (traction sustained by the element), σc

(traction at initial failure, i.e. maximum cohesive strength), δ (separation between the element faces, i.e. displacement), δi (separation at initial failure), and δc (maximum separation at complete failure and zero traction). The critical traction at which damage starts and proceeds until total failure is specified by 6

a Maximum Stress Criterion (ABAQUS 6.14, Dassault Systemes), in this case tensile stress, most relevant for the predominant Mode I fracture behavior studied here. The traction needed to maintain the cohesive displacement then declines linearly until the point of total failure, at maximum displacement, noted by δc in Figure 7, after which the element does not transmit any tensile stress, only possible compressive stresses to resist interpenetration of elements, with visualization as a new crack surface. While various traction separation relationships have been explored (Tvergaard, 1992; Needleman, 1994), a linear triangle shape was assumed and used in this work for simplicity, as used in previous literature (Ural, 2012). The numerical model was developed by varying parameter values taken from literature to reach agreement with experimental results. Parameter values were chosen from the range seen in literature (Rho et al., 1998; Currey, 2002; Ural et al., 2006; Zhang et al., 2008; Zysset, 2009; Ural et al., 2013), and iteratively fit to match simulation to experiment, ranging between upper and lower bound literature values to find the best comparative fit, with values shown in Table 1. It is of interest to discuss the convergence criterion for the iterative method. First, a convergence study of the depth of fracture initiation as the finite element mesh is refined is shown in Figure 6. It is seen that the depth of fracture initiation begins to converge monotonically at the level of 10,000 nodes. Although the criteria for the convergence of the depths of fracture initiation and the subsequent crack lengths as related to the various values of cohesive strength and energy release rate are hard to define uniquely, using these results to match the data obtained by experiment was adopted as a method to choose the various parameter values. Cohesive strength values in particular were varied, starting from cement line strengths seen in literature (O'Brien et al., 2005; Mischinski et al., 2009). Maximum separation displacement, the max separation of the cohesive element, was directly related to the work of separation, or energy release rate, as shown in Figure 7, and adjusted in conjunction to those parameters. The cement line is weaker than the other cortical bone material surrounding it, and plays a critical role in preferred crack propagation, and deflecting and blunting of cracking. The plane strain elements were modeled with elastic-perfectly plastic behavior, in order to simulate the amount of inelastic deformation observed. The deformation in bone is known to be primarily sustained as microcracking forming under tension or shear, and as such plasticity is used here as a way to accommodate the deformation in a continuum sense without necessitating the creation and propagation of many discrete micro-cracks in the finite element analysis. The wedge shaped indenter was modeled as a rigid body, with full angle of 90 degrees and tip radius no greater than 15-20 microns. The bone was in the shape of a block 5 mm square, identical to the dimensions of the sample blocks used in experiment. The indentation was modeled in plane strain conditions, matching that of experiment. The model was simply supported on the bottom, and horizontally constrained at the bottom left corner to avoid rigid body motion during simulation. The contact between wedge and top block surface was modeled as rough contact, where no slip is allowed between the master and slave surfaces after initial contact, formed by the rigid wedge and block surface nodes. Loading was displacement controlled, under static conditions. A two part progression of a hydrated bone simulation can be seen in Figures 8b and 8c, which can be compared with experimental photos in Figure 4, showing a similar progression. Figure 8a displays an example of the developing 7

inelastic, or plastic zone underneath wedge indentation prior to fracture initiation, noting characteristics about the process, including the elastic and plastic regions, the elastic-plastic boundary, and the location of the embedded cohesive element layer.

3. Results As detailed, indentation fracture under wedge indentation in cortical bone was investigated using experiments with high speed video capture of plane strain wedge indentation and with finite element simulations using cohesive elements to model fracture phenomenon. Experiments showed a large difference between submerged and dry samples in the depth of major fracture initiation. At comparable depths of indenter penetration (roughly 110 to 120 µm), video analysis shows wet, submerged cortical bone with average of depth of crack initiation deeper than dry bone. See Table 1. Crack lengths appeared to show a difference as well, with dry samples showing longer cracks in the lens view, when compared to hydrated bone at the same wedge depth, although comparison was difficult due to the rapid fashion in which cracks propagated post initiation. Indentation of a submerged sample can be seen in Figs. 3, 4 and 9a. The submerged, hydrated samples showed more predictable and stable deformation and fracture, as compared to the dry samples. It is noted that observations from experiment show fracture along the softer thin cement line layers dominates after an initial amount of in-elastic deformation, or plasticity, which is in agreement with the assumptions and observations in the literature. Figure 4 shows a progression set of video stills with slight annotation showing crack initiation and propagation under indentation in a wet bone sample. All wet samples displayed fracture and crack propagation at and along cement lines, with less crack crossover to adjacent cement lines as compared to the dry sample. Dry samples showed slightly more inclination to cross through or over osteons to another cement line, but a general median vent shaped crack was still constantly observed. Microstructural features on the bone samples were more easily seen on the samples under full submergence, where the solution appeared to assist the optical setup and clarity. Visible in the photos are osteonal microstructural units running vertically, bounded by cement lines, with Volkmann’s canals and osteocytes showing as dark elongated globular shapes and spots, respectively. Finite element simulations effectively captured the inelastic, or equivalent plastic zone development and fracture initiation depth and propagation under indentation using cohesive elements, with fitting of cohesive element properties along the interface (virtual cement line) to match wet and dry cortical bone. A comparison is shown in Figure 9, a though c, with overlay, to illustrate the similarity in behavior, especially the depth of crack initiation. It is noted the simulation shows a slight degree of pileup, where the experiments do not appear to exhibit any significant pile-up or sink-in on the length scale observed. A typical progression of the wedge indentation process along with display of tensile stress contours is displayed in Figure 10. The effect of cohesive strength and energy release rate on depth of visible crack initiation can be seen in Figure 11. The wet and dry models showed differences in behavior, especially in regards to indentation force versus distance, Figure 12, and to the depth of crack initiation and growth of the resulting crack, as seen in Figure 13. The crack initiation depths showed agreement with those measured in experimental data, seen in Table 2, and Figs. 11, 13, though the 8

following crack progression showed some deviation. The depth at which the large median vent crack initiated was taken as the depth of the inelastic zone underneath the indenter. The models showed a visible crack length of wet bone was approximately 1/5 the length of dry bone at comparable depths of indentation, seen in Figure 13. One observation related to this would be that a larger process zone was seen ahead of the crack tip in the wet simulations as opposed to that in the dry simulations, as expected from a tougher material, and noted previously in other works (Morais et al., 2010). Values for the process zone lengths in the simulations at visible crack lengths of approximately 1 mm may be seen in Table 1. The slight differences in overall results when compared to experiment may be influenced by the choices in idealization of cracking, and post-fracture cohesive behavior parameters. An interesting display of the difference in wet vs. dry behavior can be seen in force vs. wedge distance relations obtained from the simulations, illustrating the greater amount of damage tolerance wet bone shows versus dry, as noted in Figure 12. The dry bone opens a large crack earlier, and completely fails earlier as well. With the observance of fracture in vertically oriented cortical bone in both experimental observation and finite element simulation apparently opening up at roughly the bottom edge of the inelastic, nonreversible deformation, or equivalent elastic-plastic boundary in bone, shown in Figure 8a, efforts were made to calculate the critical energy release rate, noted here as TIC, applying brittle indentation fracture methods. By using equations for an edge crack from early work on indentation fracture (Lawn et al., 1975), which studied indentation fracture in brittle materials assuming the opening of a median vent at the elastic plastic boundary, basic toughness values were calculated. Values can be seen in Table 3, where values are defined at the force, F; plastic zone depth, z; crack length C; total depth c = z + C, and Poisson’s ratio, ν. As expected from observed behavior, wet bone has a larger TIC value than dry, roughly 40% greater, 4.73 versus 2.35 (kJ/m2). Additionally, they are within order of magnitude agreement with values seen in the literature (Currey, 2002). However, given the existence of a process zone ahead of the visible crack, especially in hydrated bone, methods based on linear elastic fracture are inexact and different approaches should be considered (Yang et al., 2006; Morais et al., 2010). Only when the traction free crack length is much larger than the process zone will fracture occur as defined by linear elastic fracture mechanics (Yang et al., 2006). Since the process zones as recorded in hydrated simulations in Table 1 are on the order of 1/2 to 4/5 of the total visible crack length at the studied indentation depths, it is clear these conditions are not met for hydrated cortical bone.

4. Discussion The results reported show the approach of using experimental and finite element methods of investigations holds value in capturing the initial fracture behavior of cortical bone under indentation, and the good agreements between the two methods, as discussed above, show promise for greater understanding of the significance hydration plays in behavior of cortical bone. The comparison with both to elastic plastic fracture calculation methods developed to investigate brittle indentation fracture show possible benefits, but also sharp drawbacks to the applicability. Further, the response of cortical bone under plane strain wedge indentation has important implications for understanding general fracture behavior in cortical bone, both in terms of mechanical parameters, and for efforts in developing 9

bone health diagnostics and orthopaedic surgery. Research on recent developments such as Reference Point Indentation (Randall et al., 2013) motivates the search for deeper understanding of the deformation and fracture in bone in order to link clinical results with physical behavior. The spreading application of Reference Point Indentation in medical and research areas to study bone, along with efforts to understand the corresponding measurements in regards to bone health or response, have developed rapidly in the last few years and shown promising initial results. Correspondingly, there is a need to fundamentally understand the mechanical behavior that the indentation measures in bone, a complex task. This motivates efforts to find basic methods of evaluating bone’s mechanical and material properties amenable to engineering approaches, such as proposed here, with plane strain wedge indentation examined though finite element and experimental methods under the framework of elasticplastic brittle indentation and fracture a logical first approach. Correspondingly, this work has used novel experimental and computational methods to directly evaluate and compare fracture behavior in dry and wet cortical bone under plane strain indentation, showing the important effect of hydration has on bone properties, improving its ability to deform and resist loading and fracture, and the influence on apparent inelastic deformation, viewed as plasticity. Additionally, viewing and modeling the fracture process in cortical bone under indentation as a general system similar to that developed for classical brittle materials using conventional elastic-plastic theory demonstrates a simple and practical approach in obtaining first order values on bone toughness. However these single values cannot account for the process zone and rising R-curve behavior seen in cortical bone. It must be noted however, that while conventional brittle indentation fracture theory uses a single yield strength, bone displays different yield strengths under tension and compression. In order for model simplicity, and for more direct comparison to brittle indentation fracture theory, a single yield strength for bone was used in the finite element simulations. Other work shows that use of a Drucker-Prager yield criterion, which utilizes differing yield strengths, has no resulting pileup in some indentation simulations (Mullins et. al, 2009; Hoffseth et. al, 2015). While the experimental work was excellent in capturing the difference in wet and dry behavior with a novel fluid filled setup, the fixture as used showed some drawbacks. Efforts were made to collect experimental force vs. wedge distance data, but difficulties with driving the wedge in the sealed fixture prevented collection of accurate data, and restricted select tests. Extensive crack propagation behavior was difficult to record as well, due to the restriction on field of view. The restricted field of view allowed observation of the area where fracture initiated and of the inelastic zone size, but hindered capture of large developed cracks which could be compared with crack length size versus wedge indentation depth data from finite element simulations, seen in Table 4. The displacement oriented data still proved itself valuable in understanding and analyzing the behavior equivalent to elastic-plastic indentation fracture. Indeed, images captured from this study of wedge indentation in cortical under full liquid submersion show insights into fracture initiation and propagation in three specific cases of both dry and fully hydrated cortical bone. Research efforts may also find future value in pursuing greater investigation into crack progression at large crack lengths, and consideration of bridging and shielding factors in the fracture process would also prove helpful in guiding future modeling directions and experimental efforts. 10

One intriguing result of the research was that cortical bone behavior under wedge indentation prior to large fracture nucleation and propagation appears to be analogous to behavior predicted by elastic plastic brittle indentation theory. The behavior of the material inside the inelastic, or pseudo plastic zone is of great interest, especially the presence and interaction of micro-cracks in the region prior to larger cracking. Though the present work shows order of magnitude but slightly higher values of critical energy release rate when compared to those appearing in the literature, the process zone results from simulation with the basic elastic-plastic indentation model show the application of the model may not capture the total fracture behavior. A logical next step for finite element efforts would be for more detailed models with additional cohesive zones, or multiscale efforts, requiring more computational efforts with further mesh refinement. Additionally, a focus solely on hydrated cortical bone behavior would show the respective roles of the elastic, plastic and cohesive material parameters in the overall cracking response. Due to the focus on comparing hydrated and dry and the applicability of brittle indentation fracture theory to cortical bone, efforts were made to make fits of parameters for simulations to match experiment. Other matching parameter sets of finite element mechanical properties may be possible when considering additional mechanical behavior of cortical bone, but this initial work shows the impact of the cohesive element traction law in particular for bone indentation fracture work. The similar grouping of the three wet finite element parameter sets, with values in Table 1, and their difference when compared to the dry parameter set in Figure 13, displays the effect of the cohesive element energy release rate, GC. It should be noted that work was restricted to the cortical bone orientation chosen, with osteons oriented parallel to the direction of indentation. This is an ideal material orientation to study influence on bone fracture characteristics due to the clear and predictable cleavage along cement lines, but it would be expected that other orientations will deviate from this ideal behavior. Given the variability of the microstructure and the stochastic nature of crack initiation and propagation, the current study is not intended to draw statistical conclusions. Rather, focus was on the general behavior of these illustrative cases. The work may show value as an expected lower bound of indentation fracture toughness for bone, as the vertical osteonal orientation is arguably the least tough (easier to open in Mode I fracture). This approach may be an initial foundation upon which to build, using brittle material analytical expressions as a base to start developing expressions for cortical bone indentation fracture, and showing the existing drawbacks. Other previous work has studied toughness of bone under indentation with analysis of surface cracking using Vickers or cube corner indenters (Mullins et al., 2009), similar to some research on brittle indentation, however, this depends on the ability to image the surface crack length, which is currently not possible under in-vivo conditions. The logical next step is to connect fracture properties with other measured quantities, such as depth of indentation, load, or a combination thereof.

5. Concluding Remarks The research carried out here examines the effect of hydration on indentation fracture of cortical bone, and demonstrates promise using experimental and numerical procedures and methods to further analyze the indentation fracture behavior of cortical bone under full hydration mimicking in-vivo conditions. The conclusions of this research based on the selected case studied may be summarized as: 11

1) Fracture behavior of cortical bone is contingent upon hydration state. Submerged (wet) cortical bone is able to withstand indentation deeper and longer longitudinally along the osteonal microstructure prior to large median vent fracture, and better minimize the progression of the crack. The illustrative sample studies provided some quantified data numbers; 2) Simulating the indentation fracture of longitudinally oriented cortical bone with finite elements and in particular cohesive elements to mimic the cement line at osteonal boundaries shows promise and good agreement with experimental results, allowing investigation with greater depth into the indentation phenomena and the influence of constituent properties within the bulk cortical bone; 3) Calculating fracture toughness of cortical bone in an approach developed for elastic-plastic indentation of brittle material shows simplicity and usefulness, but does not account for non-linear fracture behavior seen, and as such is appropriate for only long, developed indentation cracking. This research may lead to simpler basic analysis of indentation fracture toughness in cortical bone, useful to understanding of bone fracture in-vivo, with potential applications to clinical diagnostic testing and health monitoring and caring of bones.

Acknowledgements The authors would like to gratefully acknowledge funding by NSF through grant CMMI-1031244. Additionally, the authors would like to acknowledge the support of the UC Santa Barbara Mechanical Engineering Department, the UC Santa Barbara Physics Department, and the Department of Industrial Engineering, Purdue University.

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Figures and Tables

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Fig. 1. Progression of wedge indentation process, with 1) showing the setup before indentation, 2) displaying the region of plasticity appearing after initial indentation, and 3) showing the fracture initiating upon further loading at the elastic plastic boundary

Fig. 2. Experimental setup showing high speed microscopy equipment and liquid cell fixture, containing the sample.

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Fig. 3. Example of raw video still from high speed microscopy experiments

Fig. 4. Sequential view of fracture caused by indentation of wedge, from (a) to (b), with (c) identical to (b), with crack highlighted with arrows under enhanced contrast (color online version).

Fig. 5. Finite element mesh used for simulating indentation fracture, with 16,321 nodes and 16,205 elements. Local region highlighted shown at right.

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Fig. 6. Finite element convergence study with depth of initiation point of fracture versus mesh refinement, shown for dry bone parameter set, converging to a fracture initiation depth of 0.066 mm.

Fig. 7. The traction-separation relation of cohesive elements used in the finite element simulations.

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Fig. 8. Finite element simulation of the indentation and fracture with (a) a schematic showing the plastic zone prior to fracture, with the light grey showing extent of plastic field, and from (b) to (c) a sequence of images from 90 degree wedge simulation, with crack highlighted by arrows, similar to that of the experimental sequence seen in Fig. 4.

Fig. 9. A comparison of experimental (a) and finite element (b) results, at same scale and same indentation depth with 90° wedge, showing good agreement with regions of fracture initiation (arrow 1) and progressing tip (arrow 2), as well as plastic zone (arrow 3), and also showing a direct overlay of the two results (c).

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Fig. 10. Typical finite element simulation for wet bone showing initiation and propagation of median vent crack as a result of max tensile stress at the interface, with color showing stresses in the horizontal direction, or S11 (red being maximum tensile value and blue being maximum compressive value). From left to right, the depths of wedge indentation are 8 µm, 67 µm, 100 µm, 108 µm, 117 µm.”

Fig. 11. Finite element results showing effect of cohesive energy release rate Gc vs. crack initiation depth, for a range of cohesive strength values.

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Fig. 12. Finite element results for wedge force vs. wedge indentation distance, showing differing behavior for wet and dry cortical bone.

Fig. 13. Finite element results for visible crack length vs. wedge indentation distance, with wet and dry parameter fits, see Table 3 for parameter details.

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Table 1: Finite element modeling parameters and results for dry and wet fits.

Wedge depth at initiation (µm) 66

Max σc depth (µm) 74

Process zone length (µm)

200

Visible Crack Initiation depth (µm) 315

20

200

482

108

74

684

0.375

20

150

553

116

63

542

0.563

20

200

456

108

33

884

σc

Gc

(MPa)

Dry

(kJ/m )

E (GPa)

Y (MPa)

100

0.2

22

Wet

100

0.5

Wet 2

75

Wet 3

75

2

218

Table 2: Results of experiment for dry (N=3) and wet (N=3) cortical bone at similar depth, with target of around 120 µm, for average wedge indentation depth, depth/size of the inelastic, or plastic deformation zone, z, and crack length, C, with standard deviation in parentheses.

Wedge Depth (µm) Dry Wet

115.98 (14.64) 126.83 (12.90)

Depth of deformation zone, z (µm) 251.76 (26.59) 515.71 (112.68)

Crack length, C (µm) 742.54 (243.92) 689.42 (99.99)

Table 3: Calculated critical energy release rate parameters from finite element results at ~116 µm indentation depth using established methods (Lawn et al., 1975)

Dry Wet

F (N) 109.6 133

z (µm) 315 482

C (µm) 2,670 454

c (z+C) (µm) 2,985 986

ν 0.3 0.3

TIC (kJ/m2) 2.35 4.73

Table 4: Crack length properties from finite element modeling, wedge depth (µm) versus crack length (mm). Visible Crack length C (mm) Dry Wet

Wedge depth of 100 µm 2.08 0

Wedge depth of 117 µm 2.67 0.454

Wedge depth of 125 µm 3.02 0.816

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Highlights:     

Effect of hydration on indentation fracture of cortical investigated Wedge indentation and resulting fracture in cortical bone examined Phenomena simulated with finite elements Simulations validated by experiment with high speed microscopy Hydration affects bone plasticity and fracture initiation

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