Mechanical properties exhibited by the materials used in biomedical device components for articulating joints play an important role in determining the implant performance. In the fabrication of complex-shaped parts, the thermomechanical history experienced in different locations of the final part can be substantially dissimilar, which may lead to large differences in the local microstructures and properties. In many instances, it is not feasible to evaluate experimentally the local mechanical properties in the as-manufactured bioimplant prototypes using standardized tests, and use this information in refining the manufacturing cycle to develop implants with improved performance. In order to bridge this critical gap between materials development and manufacturing, we explore here the use of recently developed spherical indentation stress-strain analysis protocols for the mechanical characterization of local properties in the as-manufactured biomedical device prototype. More specifically, this paper presents two main advances: (i) extension of spherical indentation stress-strain analysis protocols needed to extract reliable estimates of elastic modulus and indentation yield strength from polymer matrix composite (PMC) samples and (ii) demonstration of the differences in the properties between samples produced specifically for the standard tension tests and the as-fabricated PMC acetabular socket prototype intended for total hip joint replacement applications. The results of the present study revealed large differences in the mean and variance of the measured moduli and indentation yield strengths in the acetabular socket and the tensile specimen. Based on the extensive micro-computed tomography (micro-CT) analysis, an attempt has been made to rationalize the local property differences on the basis of microstructural attributes.
Hinweise
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Introduction
Standardized tension tests (ASTM Standard E8), compression tests (ASTM Standard E9), and bending tests (ASTM Standard E290) are conventionally used for mechanical characterization and certification of various new materials deployed in advanced technologies, including biomedical applications. These tests represent internationally approved and standardized methods with clear specifications for the sample geometry, test protocols, and analyses of the acquired raw data. Generally, these methods employ a sample geometry that results in a relatively simple macroscopic stress (or strain) field in the sample (often a uniform field) which allows for easy extraction of effective (i.e., homogenized) material properties of interest through fairly simple analyses of the acquired raw data in the tests [1]. However, the geometry of the as-manufactured parts in most advanced technology applications is seldom identical (in fact, often not even similar) to that of the sample geometries employed in the standardized test methods. Often, the defect-free fabrication of engineered parts with different geometries requires significant changes to the thermomechanical process histories employed, which is likely to lead to different local internal material structures (hereafter simply referred to as microstructures) and associated mechanical properties in the differently fabricated parts or even between different locations in a single fabricated part. In order to certify location-specific properties of an as-manufactured part, one must excise samples of desired standard test specimen geometry exhibiting the microstructure of interest, and then conduct the standard tests. Often, the geometry of the manufactured part makes only small volumes of material meeting this criterion available for such tests. In other words, one is forced to explore small-scale testing or testing with small volumes of material extracted from critical locations in the manufactured part [2‐5]. This presents a significant challenge to most material innovation and deployment efforts.
Further complications arise in human healthcare applications, wherein many biomaterials/implants of complex shapes are clinically used. It is often not feasible to produce samples of standard test geometries directly from critical locations of the “as-manufactured” bioimplant. Furthermore, as already mentioned, it is a common practice to optimize the processing history employed for each selected part geometry to ensure defect-free parts (e.g., near full density without cracks). This is particularly acute in manufacturing patient-specific bioimplants. These differences in processing histories should lead to differences in local properties, even when identical material composition and similar processing tools are employed. For example, in compression and injection molding techniques, the mold geometry (especially its thickness) has substantial influence on the local thermophysical conditions (as well as fluid dynamics in the case of injection molding) [6, 7], thereby resulting in different local microstructures and mechanical properties [8].
Anzeige
As mentioned earlier, one option for addressing the challenge outlined above is to employ a miniaturized version of the standard test geometry. For example, it has been demonstrated that micro-scale tensile specimen could be excised from different locations of the heterogeneous bone material to perform the tensile tests and extract local mechanical properties [9, 10]. Similarly, micro-tensile testing has been conducted on specimen excised selectively from dentin (hard, dense material lying under the enamel in teeth) [11‐16]. However, such studies often discuss the following difficulties: (i) miniature test specimen preparation requires excruciating attention to fabrication detail, as the test results are heavily influenced by machining artifacts/defects [2, 9, 11], and (ii) specimen gripping and alignment is particularly arduous in micro-tensile and compressive tests [2, 9, 14‐16]. Due to the high demands on expertise, cost, and effort, the small-scale standard geometry test methods are often employed to obtain only a limited number of measurements in practice.
Indentation testing offers a promising avenue for measuring location-specific properties. The main advantage of indentation is that it can be employed on relatively small material volumes [17, 18] and requires relatively simple sample preparation procedure [19, 20]. Despite the ease of sample preparation, the central difficulty in indentation has historically stemmed from the lack of validated analysis protocols for producing estimates of the intrinsic material properties of interest from the measured raw load-displacement data [21]. This is attributed to the highly heterogeneous stress and strain fields experienced in the indented volume of the sample (underneath the indenter) [22, 23]. In recent years, significant progress has been made in this direction using an indenter of spherical geometry [24]. In particular, new data analysis protocols have been formulated and validated experimentally on a wide variety of metallic samples [24‐27], and further supplemented by numerical simulations employing finite element models of indentation [28, 29]. However, most of the prior work in this field has focused on metallic samples that do not exhibit any rate effects in both elastic and plastic deformation.
The main objective of this study is to explore the viability and utility of extending the recently developed spherical indentation stress-strain protocols to polymer-ceramic composites used in biomedical applications. These protocols are validated in the present work on compression molded acetabular socket prototypes of ceramic-reinforced polymer-matrix composites for potential use in total hip joint replacement (THR) [30]. This biomedical device prototype can be used either as cemented or as non-cemented for THR surgery. Specifically, our focus in this work will be on answering the following central questions: (i) Can the spherical indentation stress-strain protocols be extended to evaluate the mechanical properties of the novel viscoelastic PMCs being developed for bioimplant applications?; and (ii) Can the spherical indentation stress-strain protocols be employed to reliably ascertain the differences in local mechanical properties in the differently shaped final parts made of the same PMC material, but with different processing histories (e.g., standard tensile test specimen and acetabular socket components)?
Background: Spherical Micro-indentation Stress-Strain Protocols for Metals
In this work, we build on the currently employed spherical micro-indentation stress-strain protocols [26, 31] used on metallic samples, which generally exhibit rate-independent elastic-plastic responses. Micro-indenters, unlike nano-indenters, are typically not equipped with continuous stiffness measurement (CSM) capability [32‐34] as controlled high-frequency micro-oscillations are difficult to produce in the larger scale mechanical test protocols. As such, the micro-indentation protocols discussed here are focused on extracting individual points on the indentation stress-strain curves through the use of multiple loading-unloading cycles without the CSM signal. A detailed description of the protocols can be found in prior publications [26, 31]; only a brief summary is provided here. A schematic of the indentation process is provided in Fig. 1, where a number of the variables used in the theory presented next are depicted.
Fig. 1
Schematic of the spherical indentation test showing the variables involved in the analyses of the measured datasets in the present work. The radius of the indenter and the deformed specimen are given by Ri and Rs, respectively. The radius of the area of contact is denoted by a. The total indentation depth is denoted by ht, which is decomposed into elastic and residual components denoted as he and hr, respectively. The depth of the primary indentation zone is estimated to be approximately 2.4a [26, 31]
×
Anzeige
The analysis of the measured load-displacement data in spherical indentation is based largely on Hertz’s theory of elastic contact [22, 35, 36], which provides an analytical solution to the frictionless contact between quadratic surfaces of two isotropic elastic bodies. The following relations are derived from Hertz’s theory [22, 35]:
where P is the indentation load, he is the elastic indentation depth, a is the radius of the contact boundary, and Eeff and Reff are, respectively, the effective elastic modulus and the effective radius of the indenter-specimen system. These effective values are defined by the following relations:
where ν, E, and R denote the Poisson’s ratio, elastic modulus, and radius, respectively [21, 22]. The subscripts s and i refer to the specimen and the indenter, respectively. Eq. (1a) can be algebraically rearranged as (using Eq. (1b))
These algebraic manipulations are motivated by an intuitive definition of the indentation stress as the mean contact pressure (i.e., the total load applied divided by the projected area of contact that is normal to the indenting direction, see Eq. (4a)). Using these definitions, one can express the elastic indentation response as σind = Eeffεind, e. Furthermore, the definition of the indentation elastic strain can be generalized for the case of elastic-plastic indentation as
where h now denotes the total (elastic-plastic) indentation depth [25].
It is important to recognize that the scalar definitions of the indentation stress (Eq. (4a)) and indentation strains (Eqs. (4b) and (4c)) reflect averaged values of the highly heterogeneous stress and strain tensor fields in the primary deformation zone in the sample under the indenter. The definitions presented here have been rationalized and validated in prior work using both experiments [25, 26] and finite element models [28, 29].
The indentation tests are conducted on a relatively flat sample surface, prepared carefully through appropriate metallographic techniques (including mechanical polishing, vibratory polishing, and electropolishing). Mathematically, this implies Rs = ∞, and consequently, Reff = Ri (Eq. (2b)) in the initial elastic regime of indentation. Therefore, Reff is a known quantity for the initial elastic loading regime.
A significant challenge in the analyses of spherical indentation data is the identification of the zero-point, i.e., the point of initial contact. The vendor-supplied software with most indenters often does not produce a sufficiently accurate estimate of the zero-point [25, 37‐39]. Consequently, additional corrections are needed for both the measured load \( \left(\overset{\sim }{P}\right) \) and the measured displacement \( \left(\overset{\sim }{h}\right) \), expressed as
$$ P=\tilde{P}-{P}^{\ast } $$
(5a)
$$ h=\tilde{h}-{h}^{\ast } $$
(5b)
where P∗ and h∗ are the zero-point corrections for the measured load and displacement [25, 26]. Substituting Eqs. (5a) and (5b) into Eq. (1a) and simplifying, one can obtain
Through a least squares regression of the measured indentation load-displacement data in the initial elastic regime to Eq. (6a), one can establish the values of P∗, h∗, and k. Eeff can then be estimated from Eq. (6b). Es can then be estimated from Eq. (2a) provided the indenter’s elastic properties and the sample’s Poisson’s ratio are already known.
After the specimen has undergone plastic deformation, effective radius of the system Reff becomes an unknown quantity (since Rs ≠ ∞). Values of Reff are essential for the computation of the contact radius a, which is needed for the computations of the indentation stress (Eq. (4a)) and the indentation strain (Eq. (4b))—these quantities evolve with increasing indentation depth. Equations (6a) and (6b) allow for estimation of the evolving values of Reff with increasing indentation depths using the Eeff value obtained from the analyses of the initial elastic loading regime. This implies that Eeff is assumed to remain unaltered during the entire elastic-plastic indentation measurement. This is a reasonable assumption as the plastic strains imposed during the indentation tests reported here are typically very small and have not significantly altered the underlying material microstructure. Note that Eqs. (6a) and (6b) can only be employed on the unloading elastic segments (because Hertz’s theory is only applicable to elastic contact). Therefore, in order to recover an indentation stress-strain curve, one needs to implement multiple unloading segments along the loading segment. Note that the superimposed nano-oscillations in a CSM-enabled nano-indenter essentially accomplish this in an automated manner. Since the CSM functionality does not currently exist with micro-indenters, unloading segments are necessary for recovering the indentation stress-strain response.
In the protocols employed in this work, the indenter is unloaded to a pre-determined load value for every load-unload cycle. Each unloading segment is then analyzed with suitably modified versions of Eqs. (6a) and (6b) as
where h denotes the total indentation displacement that can be additively decomposed into an elastic displacement he and a residual displacement hr [26]. Least squares regression of the load-displacement data collected in each unloaded segment to Eq. (7a) allows determination of hr and Reff (using Eq. (6b)), which can then be used to estimate the contact radius a using Eq. (7b). One can then compute the values for the indentation stress and the indentation strain corresponding to the start of each unloading cycle using Eqs. (4a) and (4b). Consequently, analysis of each unloading segment produces one point on the indentation stress-strain curve. The resulting indentation stress-strain curve is used to estimate the indentation yield strength of the sample using a 0.2% plastic indentation strain offset criteria, following the common practice employed in the standardized tensile tests [1].
Extension of Spherical Indentation Stress-Strain Protocols to PMC Samples
As described earlier, a critical step in the analysis of the spherical indentation data is the identification of the elastic segment in the initial loading segment. This is because the value of the indentation modulus estimated for the sample material from this segment is used in the analyses of all subsequent post-yield unloading segments. For samples exhibiting a rate-independent elastic-plastic response, this was achieved through simple load-unload cycles with systematically increasing load levels and identifying the point at which the unloading segment first deviates from the initial loading segment on the load-displacement plots [31]. However, viscoelastic behavior was observed in the bioimplants investigated in this study. In the case of viscoelastic materials, the protocols presented in Sect. 2 are not directly applicable. The material viscoelasticity is expected to most strongly influence the measured load-displacement data during the abrupt path changes from the loading segments to the unloading segments.
The extension of Hertz’s theory for the indentation of a viscoelastic medium has been explored in prior work [22, 40‐44]. The recovery of indentation stress-strain curves from such measurements remains an unsolved problem, with some preliminary progress made in recent years [45]. In the present work, taking advantage of the fact that the PMC samples studied exhibited only small amounts of viscoelasticity, we have explored simple extensions to the protocols developed earlier for metal samples. In these extensions, we have introduced hold segments before and after each unloading segment to allow the material to reach equilibrated conditions. The hold segments are performed in load-controlled mode while the loading and unloading of the indenter is performed in displacement-controlled mode (Fig. 2). It was found that these hold segments allowed reliable extraction of the indentation stress-strain responses in the PMC samples—when the material undergoes a completely viscoelastic response (i.e., no permanent deformation), the end of the load-unload cycle meets the initial loading path of the preceding cycle (Fig. 2). Note that each cycle involves the unloading of the indenter to a small, but non-zero, load level in order to maintain indenter-specimen contact at all times. The displacement rates imposed during loading and unloading segments are set to be equal but opposite in direction.
Fig. 2
An illustrative example of the measured load-displacement cycles during spherical indentation of a viscoelastic solid (a, b) and close-up views of the load-displacement curves pre- and post-yield, showing evidence of accrued permanent deformation (c). For every cycle, the indenter displacement rate is maintained constant, and the duration of holds at load peaks and load valleys are kept constant. The peak load values are incrementally raised every subsequent cycle, while the valley load values are kept at low non-zero values to ensure continuous contact between the indenter and the specimen throughout the entire test (i.e., all of the load-unload cycles). It is seen that the path of the cycle 2 loading is coincident with the path of cycle 1 loading (see the expanded view of point p1 in (c). At a higher load, a residual displacement (hr) can be observed clearly in the load-displacement curve (see the expanded view of point p2 in (c), indicating the onset of viscoplastic deformation. The indenter-specimen schematic shown in (c) is drawn in the unloaded configuration. For this example, it is reasonable to assume that the viscoelastic regime extends to at least 4 N
×
Spherical indentation experiments reported in this paper were performed on the Zwick ZHU2.5 micro-indenter. The load and displacement resolution of this machine is 0.02 N and 0.02 μm, respectively. In a single loading-unloading cycle (see Fig. 2), the specimen was initially indented at a constant displacement rate of 0.05 mm/min in displacement-controlled mode until a pre-specified peak load level was reached. The indenter was then held at that load level in load-controlled mode for 60 s. Then, the indenter was unloaded at a constant displacement rate of 0.05 mm/min in displacement-controlled mode until a load of 0.25 N was achieved. The indenter was then held at 0.25 N in load-controlled mode for 60 s again, thus completing one load-unload cycle (see Fig. 2b for a graphical depiction of these load-unload cycles). It was observed that the material response consistently undergoes viscoelastic creep during the load-controlled hold segments (see Fig. 2a, b). For each indentation test, the initial peak load of the first load-unload cycle was set to 2 N, and thereafter was increased by 0.5 N in each subsequent cycle for a total of 40 load-unload cycles. Note that the indenter was always unloaded to a load value of a relatively low value of 0.25 N (but not zero) in all the load-unload cycles such that the indenter always remained in contact with the sample throughout the entirety of the spherical micro-indentation experiment.
It is important to recognize that the load-unload cycles employed in this work have been designed such that the load-unload segments in subsequent cycles (with systematically increasing peak load levels) are expected to produce overlapping load-displacement paths in the purely viscoelastic regime where there would be no permanent deformation of the specimen. This is mainly because the creep deformations experienced by the viscoelastic material in the loading and the unloading hold segments should cancel each other out as long as the material is responding in the viscoelastic regime, thereby evidencing a complete viscoelastic recovery. This is illustrated in Fig. 2 showing the first three load-unload cycles of a typical indentation raw load-displacement dataset obtained in this study. It is clearly seen that there is full reversibility between the first and second cycles, but not between second and third cycles (see the expanded views of points p1 and p2 in Fig. 2c). This implies that point p1 on the initial loading segment is still in the viscoelastic regime of the indentation test, but not point p2. The protocol presented above has been found to be effective in identifying the pre-yield regime in the initial loading segment of all indentation tests discussed in this study. To the best of our knowledge, this is the first report of a robust protocol for identifying the initial viscoelastic regime in the spherical indentation of a viscoelastic material.
Although the protocol described above helped identify the upper limit of the viscoelastic regime in the spherical indentation test, we encountered new difficulties in extracting a reliable and consistent value of the indentation modulus from the PMC samples studied in this work. Our careful analyses of multiple measurements of the load-displacement curves obtained from the PMC samples made it clear that the data very close to the initial contact needs to be excluded from the analysis. In prior work [46, 47], we had excluded very small early portions of the measured load-displacement curves from the analysis to account for unavoidable surface irregularities on the sample or the indenter (i.e., surfaces not being completely clean, surfaces not ideally conforming to the expected geometry). However, for the heterogeneous PMC samples studied here, we recognized that the deforming region under the indenter (i.e., the indentation zone) expands dramatically in the early portion of the load-displacement curve. This is because the indentation zone starts at zero volume at initial contact and expands quickly with increasing indentation depth. In the PMC samples with large differences in mechanical properties between the matrix and reinforcement (i.e., filler) phases, an important consideration is that the effective modulus in the indentation zone also changes significantly with the indentation depth but attains a stable value only after attaining a significantly large indentation zone size (when the indentation zone captures a representative volume of the heterogeneous material). Furthermore, when heterogeneous materials with relatively large constituents are indented, it is very difficult to obtain a flat sample surface. These problems were found to be particularly acute in our study, because of the large differences in the elastic moduli of the ceramic particles and the polymer matrix (generally referred as high-contrast composites) and the relatively large ceramic fillers present in the material studied here (compared with the indented zone sizes). For all the reasons discussed above, it was found necessary to eliminate the early portion of the measured elastic load-displacement curves in estimating the PMC sample’s indentation modulus (see the selected initial segment shown in green in Fig. 3a, where the data points at lower load levels were not included in the calculation of Eeff). Specifically, in the work presented here, the bottom portion of the elastic load-displacement curve corresponding to less than 0.6 N has been discarded in the analysis (i.e., these data points were exempt from the regression fits described earlier). This specific cutoff value was established by systematically evaluating the accuracy and robustness of the regression fits for different cutoff values in representative initial loading segments conducted on the samples tested in this work. Based on the indentation stress-strain analysis, this lower load limit corresponded to approximately 90–110 μm contact radius. Keeping in mind that the average size of the ceramic fillers in our samples was ~30 μm (Table 1), this lower cutoff level for the load-displacement data is quite reasonable. In other words, we assume that only the measured data above this cutoff reflects the effective response of the PMC.
Fig. 3
a Segments from spherical indentation experiments used in the extraction of the indentation stress-strain responses. b The green segment was used for the estimation of the Young’s modulus. The orange segments (only one is shown for clarity) were used to generate post-elastic points on the indentation stress-strain response. The details of the extraction of the indentation yield strength from the indentation stress-strain response are also shown. The indenter-specimen schematic shown in (a) is drawn in the unloaded configuration
Table 1
Comparison of microstructure features (combined volume fractions and equivalent diameters of HAp and Al2O3 computed through segmentation) and the corresponding spherical indentation results obtained from opposite ends of the acetabular socket rim and tensile specimen. Because of the finite resolution the micro-CT scans (3 μm × 3 μm × 3 μm), it is likely that the difference from the target composition of acetabular socket is due to unresolved fillers that are less than the voxel size
Target filler vol. frac.
Calculated filler vol. frac.
Filler size, deq (μm)
Sample modulus, Es (GPa)
Indentation yield strength, Yind (MPa)
Socket outer rim
0.40
0.28
29.79 ± 38.52
3.97 ± 0.54
60.59 ± 9.71
Socket inner rim
0.40
0.25
24.69 ± 30.34
3.86 ± 0.36
62.15 ± 9.90
Tensile specimen
0.16
0.12
12.47 ± 7.06
3.33 ± 0.07
49.14 ± 1.84
×
Furthermore, small viscoelastic effects near the transition between the peak load hold and the unloading segments were identified as subtle “bulges” near the peak of the curve. It was found that the hold segments reduced this effect significantly, but they did not completely remove them from the measurements. This observation can be reconciled by recognizing that the rate-dependent viscoelastic response of the material is most enunciated at the transition points between hold segments and the loading (and unloading) of the indenter, as these represent points with dramatic changes in strain (or stress) rates. Therefore, data points on the unloading segments located between 85 and 100% of the peak load were also excluded from the linear regression step for extracting the sample modulus (Eqs. (6a) and (6b)). This is reflected in Fig. 3a where the unloading segments selected for analyses (colored green and orange) lie in the middle of the corresponding total unloading segments.
Following the protocols described earlier, we turn our attention to the analyses of the elastic unloading segments after the sample has experienced some amount of permanent deformation (e.g., the orange-colored segment in Fig. 3a). For the same reasons discussed earlier, there is a need to exclude portions of the unloading segments at the very top (a slight bulge caused by viscoelasticity is seen in these portions) and at the very bottom (the size of the deformation zone under the indenter is not large enough to reflect the effective response). In the results presented in this work, only the unloading segment between 85 and 40% of the peak load was employed in the analysis of each unloading cycle after the sample has experienced plastic deformation.
In the indentation of viscoelastic medium, the total indentation depth can be divided into three contributions as h = he + hr + hind. creep. As such, Eqs. (7a) and (7b) can be further modified to accommodate the accrued indenter depth due to indentation creep. However, our interest in this work is not in quantifying hr or hind. creep individually and thus these quantities can be merged into a single term as:
Then, least squares fitting can be employed on the identified unloading segments (e.g., the post-yield unloading segment shown in orange in Fig. 3a) to estimate Reff (using Eqs. (7c) and (6b)) and the contact radius, a (using Eq. (7d)). This then allows computation of the indentation stress (using Eq. (4a)) and the indentation strain (using Eq. (4c)). We therefore extract a single point on the indentation stress-strain curve from each load-unload cycle; the analysis is repeated for every load-unload cycle to draw multiple post-yield indentation stress-strain data points (see orange data points in Fig. 3b).
The pre-yield segment (extraction of indentation modulus using Eqs. (6a) and (6b)) and the post-yield data points (extracted from post-yield unloading segments using Eqs. (7c) and (7d)) can be plotted altogether to recover an indentation stress-strain curve. The resulting indentation stress-strain curve is used to estimate the indentation yield strength using a 0.2% offset indentation plastic strain as already mentioned in Sect. 2. A linear fit is obtained using indentation stress-strain data enclosed within 0.1% and 0.3% plastic indentation strain offsets (lines drawn with the same slope as the indentation modulus corresponding to these plastic indentation strain values), and the intersection between this linear fit and the 0.2% offset line is used to estimate the indentation yield point (Fig. 3b).
Extracting Indentation Stress-Strain Curves from the Viscoelastic HDPE-HA-Al2O3
Fabrication of Standard Tensile Specimen from Polymer-Ceramic Composites
In the present work, the above-described spherical indentation stress-strain protocol has been implemented to extract mechanical properties from a standard tensile test specimen made of HDPE-HAp-Al2O3 composite material. This material was selected because it is being evaluated for use in advanced acetabular cup components [30, 48, 49]. As is typical in most materials development efforts [50], the initial phase of materials development relies on fabricating standard tensile test specimen using an injection molding equipment. Specifically in this study, a twin-screw extrusion was employed at high temperatures to mix the agglomerate of the HDPE-HAp-Al2O3 polymer matrix composite blend to achieve relatively evenly distributed ceramic fillers throughout the entire polymer blend. A defect-free (i.e., fully dense) material made of the desired target composition (HDPE-20 vol.% HA-20 vol.% Al2O3) proved to be unattainable through this approach—the high ceramic filler volume content required a very high applied force inside the twin-screw extruder to effectively mix the HDPE-HAp-Al2O3 blend. Through repeated trials, it was found that a HDPE-9 vol.% HAp-7 vol.% Al2O3 resulted in a fully dense coherent blend of the matrix and fillers (supported by micro-CT images presented later).
The composite constituents were melt mixed in DSM Xplore MC 15 melt extruder at a temperature of 220 °C and twin-screw speed of 60 rpm. A constant force of 8 kN was applied to the twin-screw extruder, and the blend was mixed for 10 min. The extruded composites were then injection molded in Xplore Microinjection Molding Machine at injection pressure of 8 bar with melt temperature at 220 °C and mold temperature at 110 °C to obtain a dog-bone shaped specimen specified by the ASTM D638-03 Type 5 standard. These temperatures were found to be the optimal settings for injection molding of the tensile specimen, thereby achieving relatively evenly distributed ceramic fillers throughout the polymer matrix. The dimensions of the tensile specimen are shown in Fig. 4a.
Fig. 4
a The dimensions of a standard tensile specimen made of HDPE-HAp-Al2O3 used in this study. Spherical indentation is performed across the whole surface of the tensile specimen (indentation direction is normal to the top surface), including both the grip and gage sections (indentations performed at regions highlighted in blue). b Intended application for the PMC (polymer matrix composite) material system in the acetabular socket component for total hip joint replacement (THR). c The dimensions of the acetabular socket prototype made of HDPE-HAp-Al2O3 used in this study. The indenting direction is normal to the exposed surface of the rim. Spherical indentation is performed approximately 1.2 mm away from the inner and outer rim edges (thereby avoiding the free surface throughout the indentation test) to extract local mechanical properties such as elastic modulus and indentation yield strength
×
Fabrication of Polymer-Ceramic Composite-Based Acetabular Socket
In addition to the standard tensile test samples, we have also produced a prototype of an acetabular socket, whose dimensions were selected to be clinically relevant to patient-specific size/shape and are shown in Fig. 4c. Since a higher amount of filler was desired (for improved properties), the sample was produced using a different technique than the injection molding used to produce the tensile test samples. In this work, the acetabular socket was produced by compression molding using a titanate coupling agent. In this process, the desired amount of titanate coupling agent together with HAp and Al2O3 were wet milled at room temperature to form a uniform slurry. The slurry was dried at 50 °C in an oven to yield dried particles, which were then ball-milled with high-density polyethylene polymer for 120 min using horizontal laboratory ball mill. The ball-milled dried powder mix was used for compression molding of the acetabular socket components used in this work.
The compression mold setup used in this work was equipped with a heater, which was heated to 190 °C at the start of the compression mold process cycle. Approximately 30 g of composite material was charged into the cavity of the molding apparatus. The cavity was closed and the composite material was allowed to melt for 5 min, and subsequently compressed at 5 MPa pressure. Then, the heater was switched off and the mold was allowed to cool to 100 °C. After cooling, the acetabular socket was ejected from the mold. The excess material (scraps tend to form between the core and cavity mold) was removed from the edges of the cup. Additional details regarding the processibility of this new PMC composition can be found in prior work [51].
Our previous reports already established improved fretting wear resistance of the compression-molded PMC sample of similar composition samples against alumina, zirconia, and steel counter-body compared with HDPE samples [30, 48, 49, 52]. More importantly, the cytocompatibility as well as short-term and long-term osseointegration in a rabbit model of these novel composite materials are also established [51, 53]. Furthermore, preliminary mechanical characterization was performed on this material (compression molded into a 12-mm-diameter disk coupon) using Vickers indentation test [30, 54]. As noted earlier, we expect the spherical indentation protocols presented in this paper to provide much more reliable estimates of the mechanical properties at different locations in the acetabular sockets produced for this study.
Indeed, the composition and processing conditions are quite different between the standard tensile specimen and the acetabular socket component. However, it is not unusual to see such changes when the manufacturing processes are optimized for different part geometries. Our primary interest here is to characterize reliably the differences (microstructure and mechanical properties) that ensued from the different processing conditions.
3D Microstructure Characterization
3D microstructural characterization of both the tensile test sample and the acetabular socket used in this work was conducted using X-ray micro-computed tomography (micro-CT, Xradia, Zeiss, Jena, Germany). The X-ray source voltage (peak) ranged between 40 and 80 kV, where lower voltage was suitable for the thinner tensile specimen, but a higher voltage was needed for imaging the much thicker acetabular socket prototype. Suitable filters were used to obtain the transmission in the range of 25–35%. For low- and high-resolution scans, 1601 and 3201 projections were acquired, respectively, by rotating the sample by 360°, with pixel binning of 2 (i.e., a 2 × 2 block, which contains a total of 4 neighboring pixels, are merged into one pixel to increase the signal to noise ratio of the orthoslice images) to yield a final reconstructed tomogram of size 1024 × 1024 × 1024 pixels. Low-resolution scans, although they are not shown here, were necessary to precisely locate regions of interest where the high-resolution scans would be taken. The image reconstruction algorithm used in this work is a modified version of the commonly utilized “Filtered back projection” method [55‐57], available in the XMReconstructor software (Xradia, Zeiss, Jena, Germany). This method includes standard beam-hardening correction and a Gaussian smoothing of kernel size 0.5. Manual reconstruction was performed in cases where automatic reconstruction failed, by adjusting the center-shift and beam hardening constant.
Representative high-resolution 3D images are shown in Fig. 5 and differences in filler morphologies were observed between microstructures near the outer and the inner rim of the acetabular socket component, with the microstructures in the tensile test specimen. The fillers at the inner rim showed both platelet-type and equiaxed morphologies, while the fillers at the outer rim showed predominantly equiaxed morphologies. The fillers in the tensile test specimen appeared to be finer and more homogeneously distributed.
Fig. 5
3D reconstructed micro-computed tomograms (micro-CT) showing representative local microstructures near the outer rim (a), inner rim (b), and tensile specimen (c) for visual comparison. Ceramic fillers as large as approximately 350 μm in diameter can be observed in case of the acetabular socket prototype. Differences in filler morphologies can also be observed between (a) and (b); the outer rim microstructure comprised mostly equiaxed fillers (a), whereas the inner rim microstructure comprised a combination of equiaxed fillers as well as platelet-type fillers (b). The tensile specimen microstructure comprised relatively evenly distributed ceramic fillers that showed relatively little variation in size (c)
×
The grayscale contrast in the micro-CT image is a consequence of the difference in densities of individual phases. Significant difference in densities between HDPE matrix and the ceramic fillers provided excellent contrast between these phases. However, both HAp and the Al2O3 phases were resolved in similar grayscales in the micro-CT scans due to similar X-ray attenuation. Although it was not possible to differentiate between these two ceramic fillers, some inferences could be made about their combined size distributions. In this study, the filler size distributions were quantified using the angularly resolved chord length distribution analyses methods reported in recent work [58]. For each micro-CT cubic sub-volume, chord orientations in 3-dimensions (described by two angles, θ and φ, which are selected in 10° intervals within the range 0 ° ≤ θ < 90° and 0 ° ≤ φ < 360°; see Fig. 6a) were used to collect all chord lengths intercepting the filler phases throughout the sub-volume (Fig. 6b). The probability distributions for the chord lengths are shown in the histogram in Fig. 6c. The chord length distributions are fit to a log normal distribution [59] expressed as:
where x is the chord length, y is the probability density, μ is the mean of the logarithmic values of the collected chord lengths, and σ is the standard deviation of the logarithmic values of the collected chord lengths. The resulting log normal distributions and the fitting parameters μ and σ for each microstructure class are also shown in Fig. 6c. The mean value of all the collected chord lengths determined the average filler particle size.
Fig. 6
Demonstration of angularly resolved chord length distribution analysis on an example sub-volume of the micro-CT 3D image is shown in (a). The angularly oriented lines for collecting chord lengths can be described by two angles in θ and φ (note that 0 ≤ θ < 90 and 0 ≤ φ < 360 to avoid repeated computation), and these lines are exercised onto the entirety of the segmented 3D image. A random plane of the segmented 3D image is shown in (b). Chord length distribution analysis was performed on the 3D microstructures shown in Fig. 5, and the resulting histograms are presented in (c). The chord length distributions are in good agreement with a log normal fit. The log normal distribution fitting parameters are provided for each of the distributions shown. The means and standard deviations corresponding to each specimen are presented in the inset table
×
The cubic volumes covered in the micro-CT images analyzed in this work have a 900 μm edge length. The results of the microstructure quantification are summarized in Table 1. Ceramic fillers as large as 350 μm in diameter (although rare) were observed in the microstructure. However, the average size of the ceramic fillers was observed to be approximately 28 μm near the outer rim and 25 μm near the inner rim, and 12 μm for the tensile specimen.
It is suspected that the combined missing volume fraction of the targeted 20% vol. HAp+20% vol. Al2O3 for the socket prototype and 9% vol. HAp+7% vol. Al2O3 for the tensile specimen (see Table 1) may be caused by multiple factors:
a)
Inevitable loss of material during any kind of physical transfer (i.e., ceramic powder transfer into mixing container, transfer of slurry into mold, and ejection of final part from mold).
b)
Non-uniform dispersion of ceramic fillers throughout the HDPE matrix can lead to large deviations from the expected average volume fraction at different locations where images are collected.
c)
The resolution limitations of the imaging technique do not resolve particles less than the voxel size (3 μm × 3 μm × 3 μm).
d)
Segmentation of the tomographic images can be subject to error and bias when visually evaluating the accuracy of the segmented image.
The micro-CT imaging provided excellent contrast between the polymer matrix and the ceramic fillers, which provided confidence in the segmentation results. The error contributions from imaging and segmentation are expected to be minimal. It is expected that the discrepancy between the target and the observed constituent volume fractions are largely due to the nature of the processing technique and the quantified feature sizes, and volume fractions are representative of the actual microstructure.
The inability to differentiate between the HAp and Al2O3 fillers in the micro-CT scans limited our study: (a) we were unable to quantify what fraction of the missing volume fraction was HAp or Al2O3, and (b) we were unable to quantify the variance of the individual phases that were observed in the micro-CT images. Nevertheless, it can reasonably be hypothesized that standard deviations in the filler sizes should be congruent with the variance in mechanical properties for the different samples (i.e., the acetabular socket component, which shows very large variance in ceramic filler sizes, should also have large variance in mechanical properties, and vice-versa for the tensile specimen). Spherical micro-indentation was performed on the acetabular socket prototype and the tensile specimen to extract the mechanical properties of interest (indentation modulus and indentation yield strength) in an effort to validate the spherical stress-strain analysis protocols presented in Sect. 3.
Spherical Micro-indentation of HDPE-HA-Al2O3
The acetabular socket studied in this work was measured to be approximately 7.5 mm in thickness (see Fig. 4c). From microstructure characterization (Sect. 4.3), the largest variations in the local mechanical properties were expected between the inner and outer rims where differences in (non-parent) filler morphologies were observed. In this work, we decided to perform the indents centered approximately near 1.2 mm from the edges. The indentation tests performed in this study used a spherical tip of 6.35 mm in radius. The estimated contact radius (obtained through the indentation stress-strain analysis protocols) at indentation yield (on the extracted indentation stress-strain) was approximately 300 μm. Based on prior FE simulations of the spherical indentation test [25], the depth of the primary indentation zone is estimated to be approximately 2.4 times the contact radius, which is ~720 μm. A primary indentation zone of this size was necessary for collecting effective material response from a polymer-matrix composite with ceramic fillers up to ~350 μm in diameter (see Fig. 5a). In consideration of all of the above, indentations centered at a distance of 1.2 mm from the edge would be adequate for avoiding the free surface of the socket prototype while maintaining a respectable distance between the indentations performed near the inner and the outer edges. The protocols described in Sect. 3 were employed to obtain the desired local mechanical properties of the acetabular socket near the inner and the outer rims. All of the unloading segments used for least squares regression in the spherical indentation analysis yielded R-squared values of 99.5% or higher, which indicates excellent data fit to (Eqs. 6a and 6b).
The results of the spherical indentation tests are tabulated in Table 1 and illustrated in Fig. 7. The data shown in Fig. 7 was fit to a normal distribution [59]:
Fig. 7
Histograms and the fitted probability density normal distributions for the elastic modulus (Es) (a) and indentation yield strength (Yind) (b) obtained from opposite ends of the acetabular socket rim. The normal distribution fitting parameters are provided for each of the distributions shown. There were no dramatic differences between the results obtained from two different places (inner and outer rim)
where x is either the sample modulus (Es) in Fig. 7a, or indentation yield strength (Yind) in Fig. 7b, y is the probability density, μ and σ are the mean and standard deviation. The resulting normal distributions and the fitting parameters are also shown in Fig. 7.
Results showed statistically similar mechanical properties at both the outer and inner rims of the acetabular component. From these results, it is hypothesized that the platelet-type fillers which are exclusively present near the inner rim of the socket are the HAp fillers which do not significantly contribute to mechanical strength. On the other hand, the Al2O3 fillers which are historically known for their role in enhancing mechanical properties are present near both the outer and inner rim of the acetabular socket component. Furthermore, the relatively large standard deviations in mechanical properties can be reasonably explained by the variance in size of ceramic fillers embedded in the microstructure at local indented sites, as demonstrated in Fig. 8. Note that in indentation test 1 (shown in red in Fig. 8), the large ceramic filler approximately 300 μm in diameter is located at the center of the indent site in the test which resulted in high values for sample modulus and indentation yield strength. On the other hand, for the indentation test 2 shown in the same figure (shown in blue in Fig. 8), the lower modulus can be explained by the absence of such large ceramic filler in the indentation deformed zone.
Fig. 8
Plot of the experimentally measured indentation stress-strain response of HDPE-HA-Al2O3 based acetabular socket. Large dissimilarities in mechanical properties as a result of different local microstructures encountered at the indentation sites using optical imaging (ceramic fillers are shown as the darker particles). The scale bar shown in the bottom right corner pertains to both of the indent sites. The large ceramic filler (~300 μm in diameter) located in the center drives up the values of sample modulus and indentation yield strength (Test 1). The lack of large dominant ceramic fillers near the center in the other test shown shows a correspondingly lower sample modulus and indentation yield strength (Test 2). The scale bar as shown pertains to both of the images shown in the figure
×
Spherical indentation was also performed on the tensile test specimen, and the results are summarized in Table 1 and Fig. 9. The data shown in Fig. 9 was also fit to a normal distribution function as previously discussed and the resulting normal distributions and fitting parameters are shown in Fig. 9. Results showed that the averaged values for the sample modulus and the indentation yield strength were significantly lower than the averaged values obtained from the acetabular socket prototype. These findings are consistent with the lower volume fractions of ceramic fillers present in the tensile specimen (refer to Table 1). Additionally, the standard deviations for both sample modulus and indentation yield strength were significantly less than that of the acetabular socket prototype. This is also reasonable because of the narrower size distributions of the ceramic fillers observed in the tensile test specimen (Fig. 5c).
Fig. 9
Comparison of the histograms and the fitted probability density functions for the elastic modulus (Es) (a) and indentation yield strength (Yind) (b) obtained from the tensile specimen and the acetabular socket. The normal distribution fitting parameters are provided for each of the distributions shown. Very different distributions can be observed for the two different specimens
×
The case study presented above has identified many of the benefits of using spherical indentation protocols for evaluating the mechanical properties of as-manufactured engineering parts. One of the main advantages is that it facilitates reliable location-specific measurements of mechanical properties in as-manufactured parts without placing many demands on the part size or geometry. These protocols are aimed mainly at accelerating the iterations that occur naturally in any materials development effort.
Because the spherical indentation protocols are aimed at establishing the local properties, it is quite challenging to directly validate the measurements in all applications. In prior work, a substantial amount of validation was undertaken using finite element models, where one specifies the material’s elastic-plastic constitutive response [60, 61]. A limited amount of experimental validation of these protocols has also been carried out [25, 26]. While most of the prior work was focused on metal samples, the present work has extended the protocols to PMC samples that exhibit viscoelastic-viscoplastic response. Only a limited study of the protocols to polymers was previously explored [45]. Therefore, there is a critical need for future work to rigorously extend the theory to viscoelastic-viscoplastic material, and to conduct suitable validation of the spherical indentation stress-strain protocols. Validated PMC indentation protocols are likely to benefit materials development efforts in many aspects of bioimplant applications, where irregular patient-specific part geometries are to be clinically used.
Conclusion
Spherical indentation protocols presented in this work allow location-specific evaluation of mechanical properties in a complex-shaped component. These protocols exhibit tremendous potential for obtaining critical information related to the component performance and feed that information suitably into the materials manufacturing development cycles. This is especially critical in the deployment of complex shaped heterogeneous materials in advanced technology applications.
In this study, the spherical indentation stress-strain analysis protocols were suitably adapted to extract reliably the sample modulus and the indentation yield strength of the HDPE-HA-Al2O3 molded into two different shapes by different processing methods: (i) an acetabular socket component by compression molding and (ii) a standard tensile test specimen by injection molding. It was seen that the average sample modulus and indentation yield strength measured on the tensile test specimen were significantly lower than the values measured in acetabular socket prototype. These observations are explained by the lower volume fractions of ceramic fillers in the tensile specimen, observed in the corresponding 3D micro-CT datasets from the samples. Furthermore, the lower variance in the measured mechanical properties on the tensile specimen could also be rationalized by a narrower ceramic filler size distribution extracted from the corresponding micro-CT images.
Compliance with Ethical Standards
Conflict of Interest
The authors declare that they have no conflict of interest.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.
