Elsevier

Journal of Biomechanics

Volume 44, Issue 7, 29 April 2011, Pages 1285-1290
Journal of Biomechanics

Residual stress distribution in rabbit limb bones

https://doi.org/10.1016/j.jbiomech.2011.01.038Get rights and content

Abstract

The presence of the residual stresses in bone tissue has been noted and the authors have reported that there are residual stresses in bone tissue. The aim of our study is to measure the residual stress distribution in the cortical bone of the extremities of vertebrates and to describe the relationships with the osteon population density. The study used the rabbit limb bones (femur, tibia/fibula, humerus, and radius/ulna) and measured the residual stresses in the bone axial direction at anterior and posterior positions on the cortical surface. The osteons at the sections at the measurement positions were observed by microscopy. As a result, the average stresses at the hindlimb bones and the forelimb bones were 210 and 149 MPa, respectively. In the femur, humerus, and radius/ulna, the residual stresses at the anterior position were larger than those at the posterior position, while in the tibia, the stress at the posterior position was larger than that at the anterior position. Further, in the femur and humerus, the osteon population densities in the anterior positions were larger than those in the posterior positions. In the tibia, the osteon population density in the posterior position was larger than that in the anterior position. Therefore, tensile residual stresses were observed at every measurement position in the rabbit limb bones and the value of residual stress correlated with the osteon population density (r=0.55, P<0.01).

Introduction

Living tissue, like blood vessels, is subject to residual stresses (Fung, 1990). The presence of residual stresses in bone has also been noted (Tadano and Okoshi, 2006). A sin2 ψ method of X-ray diffraction as the measurement method of the residual stress in bone tissue has been proposed and residual stresses in the bone tissue of bovine femurs have been reported (Yamada and Tadano, 2010). The residual stress is defined as the stress that remains in bone tissue without any external forces. This previous study of bovine femoral diaphyses showed that the residual stresses in the bone axial direction were tensile, and the residual stresses in the bone tissue were discussed as a factor in the tissue strength.

Cortical bone has a hierarchal and composite structure formed by hydroxyapatite (HAp) like mineral particles and collagen matrix. The HAp in bone tissue has a hexagonal crystalline structure, and X-ray diffraction can be used to measure the interplanar spacings of HAp crystals (Fujisaki et al., 2006, Gupta et al., 2006, Almer and Stock, 2007, Fujisaki and Tadano, 2007, Tadano et al., 2008, Giri et al., 2009). When bone tissue deforms, the displacement of the lattice planes of the HAp crystals change almost proportionally. It has been shown that the distance between the lattice planes of the HAp crystals change proportionally to the deformation of the bone tissue (Fujisaki and Tadano, 2007). The HAp crystal strain can be calculated by the deformation of the interplanar spacing compared with a reference state (Fujisaki et al., 2006, Tadano et al., 2008). Based on this, the residual stresses in bone tissue can be measured using the sin2 ψ method of X-ray diffraction.

In general, residual stress is generated in a material by the indeterminate structure. It is well known that bone is usually replaced by new bone tissue with constructing osteon structures (Currey, 2002, Fung, 1990). Since the new tissue develops under in vivo loadings as the non-deformed state, an indeterminate structure may be generated by the difference of the deformation between the old and new phases. Further, the mechanical properties (e.g. elastic modulus) are also different in these phases (Gibson et al., 2006, Rho et al., 1999). Due to the nonuniform structures in bone tissue, residual stress may remain around the replaced region without any external forces.

The aim of this study is to measure the residual stress distribution in the cortical bone of the extremities of vertebrates and to describe the relationships with the osteon structures and mechanical loadings in vivo. In the experiments, the study used the bones of rabbit extremities and measured the residual stresses in the bone axial direction at anterior and posterior positions on the cortical surface; the osteon population densities at the sections at the measurement positions were observed by microscopy.

Section snippets

Methods

Bragg's law, the fundamental equation of X-ray diffraction, is expressed as the following equation:2dsinθ=nλ

Using characteristic X-rays with a unique wavelength λ, Eq. (1) relates the Bragg angle θ to the interplanar spacing d at a specific lattice plane (h k l).

When bone tissue deforms, the interplanar spacing d of the HAp crystals in the tissue changes. The angle of inclination ψ is defined as the angle between the normal direction of the specimen surface and the diffracted lattice plane. As

Specimens

The study used bones of the extremities of adult rabbits (Japanese White Rabbits, female, average age 16.5 weeks, average weight 3.0 kg), and Fig. 3 shows the bones of the extremities. In this study, three femur, three tibia/fibula, three humerus, and three radius/ulna specimens were used. The specimens were 60 mm long in the bone axial direction and cut using a slow speed diamond wheel saw (SBT650: South Bay Technology Co., USA). Although the preparation may release some amount of residual

Residual stresses distribution

Fig. 6 shows the distribution of residual stress in the rabbit limb bones. Each bar indicates the average of the three specimens in each position, and errors are the corresponding standard deviations. Tensile residual stresses were observed at every measurement position. The average stresses at the hindlimb bones and the forelimb bones were 210 and 149 MPa, respectively, showing that the hindlimb bones were subject to tensile residual stress 1.4 times higher than that in the forelimb bones. In

Discussion

The values of the measured residual stresses depend on the stress constant Kx, and the stress constant was calculated from the product of the elastic modulus Ex and the coefficient k*x, as Eq. (9). Here, the k*x stands for the ratio of the tissue strain to the HAp strain as in the following equation:kx=1[(1+ν)360πtanθ0]εxεxHεxεxH

Fujisaki and Tadano (2007) showed that the elastic modulus was proportional to εH/ε as in the following equation:EεHε1k

Therefore, it can be said that the stress

Conflict of interest statement

No actual or potential conflicts of interest exist.

Acknowledgement

This work was supported by Grant-in-Aid for Scientific Research (A), MEXT (No. 19200035) and Grant-in-Aid for JSPS Fellows (No. 09J00736).

References (22)

Cited by (20)

  • Hydration and radiation effects on the residual stress state of cortical bone

    2013, Acta Biomaterialia
    Citation Excerpt :

    It has been noted that the macroscale residual stress is released when the bone is cut into smaller pieces [7]. Numerous studies have employed synchrotron radiation to elucidate the residual stress in both hydroxyapatite crystals and collagen fibrils within the cortical bone material [7–12]. The sin2ψ method was used to determine the residual stress of a bulk bovine femoral diaphysis [13,14] and found hydroxyapatite particles to be under tensile stress at the surface, transitioning to compressive stress a few millimetres below the surface [15].

  • Effects of growth on residual stress distribution along the radial depth of cortical cylinders from bovine femurs

    2013, Journal of Biomechanics
    Citation Excerpt :

    It suggests that the impurity effects on the value of residual stresses can be negligible. Further, the same value of Kd was used to calculate the residual stresses in both the young and mature bones in this study because the Kd value was not different between these groups according to the measurement of a young bone specimen (3×28×0.6 mm) using the previous procedure (Yamada et al., 2011a). The limitation of this study was the use of the air-dried specimens cut out from the whole femurs.

  • Influence of osteon area fraction and degree of orientation of HAp crystals on mechanical properties in bovine femur

    2013, Journal of Biomechanics
    Citation Excerpt :

    The X-ray diffraction profiles were measured by a scintillation counter between 2θ=11.0° and 12.5°, which includes the diffraction angle of the (002) plane of HAp crystals. The Bragg angle of the plane in Eq. (2) was defined as half of the angle at the peak position of the X-ray diffraction profile, and the peak position was determined by applying the full width at two-thirds maximum (FWTTM) method (Yamada et al., 2011). The HAp crystal strain εH was measured three times at each tissue strain condition (ε=0, 500 μ, 1000 μ, 1500 μ, and 2000 μ) with the small tensile test device attached to the X-ray diffractometer.

View all citing articles on Scopus
View full text