7.3 Mechanical properties of trabecular bone

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7.3.1 Trabecular bone behavior and large variability

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Stress-strain behavior for compressive and tensile loading of bovine (left) and human vertebral (right) trabecular bone, showing the wide range in strength that is typical for trabecular bone.

The bovine bone here is 10 times stronger and stiffer than the human vertebral bone. X indicates tensile fracture. For compression, the curves have been truncated. Left: Keaveny et al. (1994) J Biomechanics, 27: 1137-1146: and right: Kopperdahl and Keaveny (1998) J Biomechanics, 31:601-608.

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7.3.2 Trabecular bone apparent density

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Dependence of compressive on-axis strength on apparent density for trabecular bone for two different sites bovine tibial (BPT) and human vertebral (HVB) trabecular bone. The difference in slopes in the linear relationship for each is due to the different architectures, being mainly plate-like in the bovine bone and rodlike in the human vertebral bone. When all the data are pooled, there is a strong squared power law relationship, with r 0.94. (Bone Mechanics Handbook. Editor SC Cowin. CRC Press Boca Raton, 2001)

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7.3.3 Trabecular bone crush strength and age

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Age-related reductions in compressive strength of human femoral and vertebral trabecular bone.

Note that for any given age, there remains much scatter in the data; that is, age is not a good determinant of behavior.

Data from Mosekilde et al. (1987) Bone 8:79-85 and McCalden et al. (1997) J Bone Jt Surg 79A:421-427.

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7.3.4 Trabecular bone yield asymmetry

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Compressive and tensile yield strains vs. human anatomic site for trabecular bone specimens tested on-axis. Compressive yield strains were always greater than tensile yield strains.

Bars denote ±1 SD. The numbers within the bars show the significant (p < 0.05) correlation coefficients against apparent density, confirming that in most sites there was no dependence on density, and at most only a weak correlation.

From Morgan and Keaveny (2001) J Biomechanics 34: 569-57.

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7.3.5 Trabecular bone yield anisotropy

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7.3.5.1 Yield stress anisotropic – yield strain isotropic

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Dependence of yield strain (left) and Young’s modulus (right) on specimen orientation in tension and compression, for dense bovine trabecular bone. For the off-axis orientation, the specimen axes were offset 30-40 degrees from the principal trabecular (on-axis) direction. Error bars show ±1 SD. In contrast to the yield strains, which were isotropic, but asymmetric, elastic modulus and yield stress (not shown) were clearly anisotropic. From Chang et al. (1999) J Orthop Res 124582-585

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7.3.6 Fatigue of trabecular bone

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S-N fatigue curve for human vertebral trabecular bone

, showing two run-out specimens (marked with X). Applied stress has been divided by Young’s modulus to minimize scatter.

From Haddock et al. (2004) J Biomechanics 37:181-187.

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7.3.7 Post-yield damage of trabecular bone

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1. Typical compressive post-yield behavior of trabecular bone for loading, unloading, and reloading along path 3-b-e as shown. The initial modulus upon reloading (EINT REL) is statistically equal to the Young’s modulus E. The reloading curve shows a sharp decrease in modulus, denoted by ERESIDUAL- This modulus is similar to the “perfect-damage” modulus (Epp, dashed line), which would occur if the material behaved in a perfectly damaging manner with cracks occurring at yield.¹⁹

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2. Dependence of percent modulus reduction (between E and ERESIDUAL) and strength reduction on the plastic strain that develops with overloading.²⁰

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7.3.8 Failure prediction

• Von Mises stress criterion is not good for bone (particularly when shear stresses are high)

• Tsai-Wu is a much better metric of strength, however, requires significantly more experiments to establish the criterion \[\begin{split} F_1 \sigma_1 + F_2 \sigma_2 + F_3 \sigma_3 &\\ + F_{11} \sigma_1^2 + F_{22} \sigma_2^2 + F_{33} \sigma_3^2 &\\ + 2 F_{12} \sigma_1 \sigma_2 + 2 F_{13} \sigma_1 \sigma_3 + 2 F_{23} \sigma_2 \sigma_3 &\\ + F_{44} \sigma_4^2 + F_{55} \sigma_5^2 + F_{66} \sigma_6^2 &=1 \end{split}\] (Note vector representation of stress 1-6)

• Requires tension, compression, and torsion tests in longitudinal and transverse specimens

• This is a big challenge in practical prediction of bone failure

  • Most studies only compare stresses without predicting failure

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Due to overlapping material and to provide continuity, the lectures slides for Lecture 1 are included in the materials for Lecture 0.

Due to overlapping material and to provide continuity, the lectures slides for Lecture 1 are included in the materials for Lecture -1.

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19. @Bartel2006 ↩︎

20. Keaveny et al. (1999) J Orthop Res, 17:346-353 ↩︎