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Research Papers

Biaxial Normal Strength Behavior in the Axial-Transverse Plane for Human Trabecular Bone—Effects of Bone Volume Fraction, Microarchitecture, and Anisotropy

[+] Author and Article Information
Arnav Sanyal

Orthopaedic Biomechanics Laboratory,
Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720
e-mail: arnavsanyal@berkeley.edu

Tony M. Keaveny

Orthopaedic Biomechanics Laboratory,
Department of Mechanical Engineering,
University of California,
Berkeley, CA 94720
Department of Bioengineering,
University of California, Berkeley, CA 94720
e-mail: tmk@me.berkeley.edu

1Corresponding author.

2Please address all reprint requests to Tony M. Keaveny.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received July 9, 2013; final manuscript received September 23, 2013; accepted manuscript posted October 14, 2013; published online November 6, 2013. Assoc. Editor: Kristen Billiar.

J Biomech Eng 135(12), 121010 (Nov 06, 2013) (9 pages) Paper No: BIO-13-1309; doi: 10.1115/1.4025679 History: Received July 09, 2013; Revised September 23, 2013; Accepted October 14, 2013

The biaxial failure behavior of the human trabecular bone, which has potential relevance both for fall and gait loading conditions, is not well understood, particularly for low-density bone, which can display considerable mechanical anisotropy. Addressing this issue, we investigated the biaxial normal strength behavior and the underlying failure mechanisms for human trabecular bone displaying a wide range of bone volume fraction (0.06–0.34) and elastic anisotropy. Micro-computed tomography (CT)-based nonlinear finite element analysis was used to simulate biaxial failure in 15 specimens (5 mm cubes), spanning the complete biaxial normal stress failure space in the axial-transverse plane. The specimens, treated as approximately transversely isotropic, were loaded in the principal material orientation. We found that the biaxial stress yield surface was well characterized by the superposition of two ellipses—one each for yield failure in the longitudinal and transverse loading directions—and the size, shape, and orientation of which depended on bone volume fraction and elastic anisotropy. However, when normalized by the uniaxial tensile and compressive strengths in the longitudinal and transverse directions, all of which depended on bone volume fraction, microarchitecture, and mechanical anisotropy, the resulting normalized biaxial strength behavior was well described by a single pair of (longitudinal and transverse) ellipses, with little interspecimen variation. Taken together, these results indicate that the role of bone volume fraction, microarchitecture, and mechanical anisotropy is mostly accounted for in determining the uniaxial strength behavior and the effect of these parameters on the axial-transverse biaxial normal strength behavior per se is minor.

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Figures

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Fig. 1

Definition of the chronological yield point. This graph depicts the stress-normalized strain responses in the longitudinal (solid line) and transverse (dotted line) directions for a single specimen loaded biaxially in the longitudinal and transverse directions. The normalized strains occur at the same instant in time for both responses. For this loading, the ratio of maximum applied strain in the longitudinal to transverse direction was 0.73, and yielding first occurred along the transverse direction, which defined the chronological yield point.

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Fig. 2

The longitudinal (solid) and transverse (open) yield points and the respective yield ellipses for one specimen. Each ellipse is represented by five parameters, the major and minor diameters (a and b, respectively), the coordinates of the center (h,k), and the angle of tilt of the major axis with respect to the horizontal (ϕ). The shaded region bounded by the two intersecting ellipses defines the elastic region.

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Fig. 3

A dual-ellipse fit to the normalized longitudinal and transverse yield strength data pooled from all specimens. The yield strength in each quadrant was normalized by the respective specimen-specific uniaxial strengths.

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Fig. 4

(a) Dual-ellipse fit, (b) single-ellipse fit, and (c) quartic super-ellipse fit to the (same) pooled normalized chronological yield strength data

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Fig. 7

Distribution of yielded tissue at the biaxial yield point in thin (∼0.45 mm) longitudinal slices taken from three 5 mm cube specimens subjected to two biaxial loading cases (top row: longitudinal compression and transverse tension in a ratio of about 5:1; bottom row: longitudinal tension and transverse compression, also in a ratio of about 5:1). The percentage value denotes the proportion (percentage) of total tissue yielded in the overall cube specimen at the biaxial yield point. Red regions denote tissue yielded in tension and blue regions denote tissue yielded in compression. BV/TV = bone volume fraction; EA = elastic anisotropy. (The reader is referred to the online version of this article (doi:10.1115/1.4025679) for interpretation of the references to color.)

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Fig. 6

Variation of biaxial strength with elastic anisotropy at a constant bone volume fraction (BV/TV) of 0.09, 0.18, and 0.20 for two biaxial loading cases: longitudinal compression and transverse tension in a ratio of 5:1 (dotted line); and longitudinal tension and transverse compression, also in a ratio of 5:1 (solid line). For longitudinal compression and transverse tension, biaxial strength was always defined by the transverse direction. However, for longitudinal tension and transverse compression, the biaxial strength was defined by the longitudinal direction up to an elastic anisotropy of ∼6, beyond which the biaxial strength was defined by the transverse direction, leading to a change in the relation between biaxial strength and elastic anisotropy.

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Fig. 5

Comparison of dual-ellipse fit (solid) and a quadratic fit (dotted) to the pooled normalized “equivalent” yield strength data

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