Size effects and stochastic behavior of nanoindentation pop in.

A statistical model for pop in initiated at preexisting dislocations during nanoindentation is developed to explain size-dependent pop-in stresses. To verify theoretical predictions of this model, experiments were performed on single-crystal Mo, utilizing indenter radii that vary by over 3 orders of magnitude. The stress where plastic deformation begins ranges from the theoretical strength in small volumes, to 1 order of magnitude lower in larger volumes. An intermediate regime shows wide variability in the stress to initiate plastic deformation. Our theory accurately reproduces the experimental cumulative probability distributions, and predicts a scaling behavior that matches experimental behavior.

Influence of indenter tip geometry on elastic deformation during nanoindentation.

Nanoindentation with a Berkovich indenter is commonly used to investigate the mechanical behavior of small volumes of materials. To date, most investigators have made the simplifying assumption that the tip is spherical. In reality, indenter tips are much more complex. Here, we develop a new method to describe the tip shape using the experimentally determined area function of the indenter at small depths (0-100 nm). Our analysis accurately predicts the elastic load-displacement curve and allows the theoretical strength of a material to be determined from pop-in data. Application of our new method to single crystal Cr3Si shows that the predicted theoretical strengths are within 12% of the ideal strength G/2pi, where G is the shear modulus.

Anisotropic properties of human tibial cortical bone as measured by nanoindentation.

The purpose of this study was to investigate the effects of elastic anisotropy on nanoindentation measurements in human tibial cortical bone. Nanoindentation was conducted in 12 different directions in three principal planes for both osteonic and interstitial lamellae. The experimental indentation modulus was found to vary with indentation direction and showed obvious anisotropy (one-way analysis of variance test, P < 0.0001). Because experimental indentation modulus in a specific direction is determined by all of the elastic constants of cortical bone, a complex theoretical model is required to analyze the experimental results. A recently developed analysis of indentation for the properties of anisotropic materials was used to quantitatively predict indentation modulus by using the stiffness matrix of human tibial cortical bone, which was obtained from previous ultrasound studies. After allowing for the effects of specimen preparation (dehydrated specimens in nanoindentation tests vs. moist specimens in ultrasound tests) and the structural properties of bone (different microcomponents with different mechanical properties), there were no statistically significant differences between the corrected experimental indentation modulus (Mexp) values and corresponding predicted indentation modulus (Mpre) values (two-tailed unpaired t-test, P > 0.5). The variation of Mpre values was found to exhibit the same trends as the corrected Mexp data. These results show that the effects of anisotropy on nanoindentation measurements can be quantitatively evaluated.

Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nano-indentation.

The elastic modulus and hardness of secondary osteonal and interstitial bone was examined through the thickness of the cortex of human femora of various ages by nano-indentation. There was a clear difference between the stiffness and hardness of secondary osteonal and interstitial bone, the latter being stiffer (F(1,48)=56.0, P<0.001). There were some differences between the bones of different subjects; however, there were no differences that could be reliably associated with the chronological age of the subject, or with differences in location through the thickness of the cortex (F(2,48)=0.21, P=0.810). Previous studies have been equivocal in relating changes in the macroscopic 'composite' material stiffness of bone to the age of the individual. By combining the results of the nano-tests with histological measures, we were able to produce a good relationship of the microstructural properties at the matrix level with the bending modulus of whole bone (R(2)=0.88, P<0.001) and this improved further by taking into account the age of the individual (R(2)=0.94, P<0.001). Our results suggest that using differences in the volumetric proportions of secondary osteons versus interstitial bone, and the properties of these elements/structures in isolation may be a more accurate method of determining differences in elastic modulus of whole bone between individuals of various ages.

Effects of anisotropy on elastic moduli measured by nanoindentation in human tibial cortical bone.

Many biological materials are known to be anisotropic. In particular, microstructural components of biological materials may grow in a preferred direction, giving rise to anisotropy in the microstructure. Nanoindentation has been shown to be an effective technique for determining the mechanical properties of microstructures as small as a few microns. However, the effects of anisotropy on the properties measured by nanoindentation have not been fully addressed. This study presents a method to account for the effects of anisotropy on elastic properties measured by nanoindentation. This method is used to correlate elastic properties determined from earlier nanoindentation experiments and from earlier ultrasonic velocity measurements in human tibial cortical bone. Also presented is a procedure to determine anisotropic elastic moduli from indentation measurements in multiple directions.

Correlations between osteocalcin content, degree of mineralization, and mechanical properties of C. carpio rib bone.

Osteocalcin is one of the most abundant noncollagenous proteins in bone. It is strongly associated with the mineral phase of bone, and has long been associated as a marker of bone turnover. However, its relationship to bone composition, strength, and structure is unclear. Carp rib bone is an excellent model for the study, because osteocalcin represents almost 60% of the total extractable noncollagenous proteins found in it. Because of the abundance of osteocalcin relative to other extractable proteins, any changes in the properties of carp rib bone would be more likely influenced by the osteocalcin concentration. To test the hypotheses that the concentration of osteocalcin is reflected in other properties of bone, the correlations between the osteocalcin concentration and the mineral content, microstructural properties, and physical characteristics of the bone mineral crystals were determined utilizing radioimmunoassay (RIA), spectrophotometry, nanoindentation, and small-angle X-ray scattering (SAXS) techniques, respectively. Osteocalcin concentration was found to be correlated to the molar Ca/P ratio and inversely correlated to the elastic modulus and hardness in the longitudinal plane. This study provides evidence for a putative relationship between the concentration of osteocalcin and the microstructural mechanical properties of bone. Correlations were also found between the mechanical properties in the longitudinal plane and both the phosphate content and the molar Ca/P ratio. However, no relationships could be identified between osteocalcin concentration and several parameters of bone crystals, as determined by SAXS.

The anisotropic Young's modulus of equine secondary osteones and interstitial bone determined by nanoindentation.

The equine radius is a useful subject for examining the adaptation of bone histology to loading because in life the anterior cortex is loaded almost entirely in tension, the posterior cortex in compression. The histology of the two cortices is correspondingly different, the osteones and the interstitial lamellae in the posterior cortex having a more transversely oriented fibre arrangement than those in the anterior cortex. Presumably as a result of this histological difference, the posterior cortex is stronger in compression than the anterior cortex; the anterior cortex is stronger in tension than the posterior cortex. We here use nanoindentation to examine how the Young's modulus of elasticity of secondary osteones and interstitial lamellae in the anterior and posterior cortices varied as a function of angle. The anterior osteones were stiffer than the posterior osteones when tested in the direction parallel to the bone's long axis, but became progressively relatively less stiff as the angle increased; at 90 degrees, they were less stiff than the posterior osteones. Although the interstitial lamellae were stiffer than their neighbouring osteones, the same relationship between anterior and posterior interstitial lamellae as a function of angle was found as for the osteones. The anisotropy of these Young's moduli determined by nanoindentation shows a close relationship with what was to be expected from the histological findings.

The distribution of osteocalcin, degree of mineralization, and mechanical properties along the length of Cyprinus carpio rib bone.

This study examined the spatial distribution of selected biochemical and mechanical properties along the length of carp rib bone. Carp rib bone was chosen because of its unusually high osteocalcin content relative to other extractable proteins. The amount of osteocalcin was significantly lower (p<0.01) at the most distal section, relative to all other sections. The amount of phosphate (p<0.05) and the elastic modulus in the longitudinal plane (p<0.0001) were found to be significantly higher in the most distal section, relative to the most proximal section. There was no significant difference in the calcium distribution, molar Ca/P ratio, or elastic modulus in the transverse plane. It was speculated that the distal section contains less mature bone. The methods illustrate the potential usefulness of nanoindentation to characterize the mechanical properties of bone, relative to its biochemical composition.

Relationship between ultrastructure and the nanoindentation properties of intramuscular herring bones.

The nanoindentation technique was used to characterize the indentation modulus of intramuscular herring bone (Clupea harengus) along its length, from early to fully mineralized regions. Both the indentation modulus and modulus anisotropy ratio, indentation modulus longitudinally/indentation modulus transversely, increased nonmonotonically with mineralization, from 1.1 in the early stages of mineralization to 2.1 in the fully mineralized region. The indentation modulus in the fully mineralized region decreased from 19.9+/-2.0 GPa in the longitudinal direction to 11.4+/-1.5 GPa in the transverse directions. In the earliest stages of mineralization, the indentation modulus was 3.8+/-0.7 GPa in the longitudinal direction and 3.5+/-0.3 GPa in the transverse direction. High-intensity synchrotron x rays were used to examine two parameters of crystal texture, the coherence length and the angular spread. Low angular spread was seen along the c axis of bone crystallite. However, the large angular spread observed along the (300) plane indicated an imperfection of crystallite periodicity in the direction perpendicular to the longitudinal axis in the bone. The relatively high coherence length was related to the low angular spread. In general, the indentation modulus increased with increasing crystallite size, but the indentation modulus decreased with increasing crystallite size along the (300) plane.

Variations in the individual thick lamellar properties within osteons by nanoindentation.

The nanoindentation method was used to examine variations in the individual thick lamellar properties within completed secondary osteons as a function of distance from the osteonal center (haversian canal). In general, there is a decline in both elastic modulus and hardness from the center of the osteon outward. Because some of the osteons may have a different general trend than others, an analysis of covariance was also carried out. The overall analysis was highly significant for both elastic modulus and hardness. Also, osteon number was significant as a factor, indicating that there was some difference in the overall thick lamellar properties of the different osteons. An unpaired t-test showed statistically significant differences (p = 0.0005 and 0.0004, respectively) between thick lamellar properties obtained from most of the inner two osteonal lamellae (E = 20.8 +/- 1.3 GPa and H = 0.65 +/- 0.06 GPa) and those from outermost two osteonal lamellae (E = 18.8 +/- 1.0 GPa and H = 0.55 +/- 0.05 GPa). In general, lamellar properties from near to the center of the osteon were greater than those from the outermost osteonal lamella. The mechanical properties of osteons are also significantly lower than those of the interstitial bone (p < 0.0001). The ratio (E1/E2) of the elastic moduli of the outermost osteonal lamella (E1) (considered to be the soft part of the osteons) and that of interstitial bone (E2) was approximately 0.7. These results may have important implications for the mechanical contribution of individual osteons to bone biomechanics.

Mechanical and morphological variation of the human lumbar vertebral cortical and trabecular bone.

The nanoindentation technique was used to characterize the variation in the elastic modulus and hardness of human lumbar vertebral cortical and trabecular bone. The elastic modulus (and in most cases, the hardness as well) of axially aligned trabeculae cut in the transverse direction was significantly greater than in other orientations of vertebral cortical and trabecular bone. In all cases, the elastic modulus and hardness of bone in the load-bearing direction was greater than in corresponding bone types cut in the other directions. Scanning electron micrographs of cortical shell revealed the Haversian-like canal systems expected in secondary cortical bone, but it was difficult to differentiate by morphology cortical from trabecular bone in the human lumbar vertebrae.

Elastic properties of microstructural components of human bone tissue as measured by nanoindentation.

The elastic properties of several microstructural components of dry human vertebrae (T-12 and L-1) and tibiae have been investigated in the longitudinal and transverse directions using nanoindentation. The largest Young's modulus was that for the interstitial lamellae in the longitudinal direction (25.7 +/- 1.7 GPa). This was followed in decreasing order by osteons in the longitudinal direction (22.4 +/- 1.2 GPa), trabeculae in the longitudinal direction (19.4 +/- 2.3 GPa), an average over osteons and interstitial lamellae in the transverse direction [16.6 +/- 1.1 GPa (it was difficult to microstructurally distinguish osteons from interstitial lamellae in the transverse direction)], and trabeculae in the transverse direction (15.0 +/- 2.5 GPa). An ANOVA statistical analysis revealed that the values all are significantly different (p < 0.05). Since the elastic moduli in the longitudinal direction are all greater than in the transverse, measurable elastic anisotropies exist in the components. The hardnesses also varied among the microstructural components in the range 0.52-0.74 GPa.

The elastic properties of trabecular and cortical bone tissues are similar: results from two microscopic measurement techniques.

Acoustic microscopy (30-60 microm resolution) and nanoindentation (1-5 microm resolution) are techniques that can be used to evaluate the elastic properties of human bone at a microstructural level. The goals of the current study were (1) to measure and compare the Young's moduli of trabecular and cortical bone tissues from a common human donor, and (2) to compare the Young's moduli of bone tissue measured using acoustic microscopy to those measured using nanoindentation. The Young's modulus of cortical bone in the longitudinal direction was about 40% greater than (p<0.01) the Young's modulus in the transverse direction. The Young's modulus of trabecular bone tissue was slightly higher than the transverse Young's modulus of cortical bone, but substantially lower than the longitudinal Young's modulus of cortical bone. These findings were consistent for both measurement methods and suggest that elasticity of trabecular tissue is within the range of that of cortical bone tissue. The calculation of Young's modulus using nanoindentation assumes that the material is elastically isotropic. The current results, i.e., the average anisotropy ratio (E(L)/E(T)) for cortical bone determined by nanoindentation was similar to that determined by the acoustic microscope, suggest that this assumption does not limit nanoindentation as a technique for measurement of Young's modulus in anisotropic bone.

Effects of drying on the mechanical properties of bovine femur measured by nanoindentation.

Effects of drying on the measurement of mechanical properties of bone by nanoindentation methods have been examined. Tests were conducted to measure the elastic modulus and hardness of two cross-sectional cortical specimens obtained from adjacent areas of bovine femur. One specimen was thoroughly dried in air prior to testing while the other was stored in deionized water. The properties of osteons and interstitial lamellae showed statistically significant differences (p<0.0001) and were therefore investigated separately. Drying was found to increase the elastic modulus by 9.7% for interstitial lamellae and 15.4% for osteons. The hardness was also found to increase by 12.2% for interstitial lamellae and 17.6% for osteons.

Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation.

An experimental investigation was undertaken to measure the intrinsic elastic properties of several of the microstructural components of human vertebral trabecular bone and tibial cortical bone by the nanoindentation method. Specimens from two thoracic vertebrae (T-12) and two tibiae were obtained from frozen, unembalmed human male cadavers aged 57 and 61 years. After drying and mounting in epoxy resin nanoindentation tests were conducted to measure Young's modulus and the hardness of individual trabeculae in the vertebrae and single osteons, and interstitial lamellae in the tibiae. Measurements on the vertebral trabeculae were made in the transverse direction, and the average Young's modulus was found to be 13.5 +/- 2.0 GPa. The tibial specimens were tested in the longitudinal direction, yielding moduli of 22.5 +/- 1.3 GPa for the osteons and 25.8 +/- 0.7 GPa for the interstitial lamellae. Analysis of variance showed that the differences in the measured moduli are statistically significant. Hardness differences among the various microstructural components were also observed.