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Research Papers

# Modeling of Dynamic Fracture and Damage in Two-Dimensional Trabecular Bone Microstructures Using the Cohesive Finite Element Method

[+] Author and Article Information
Vikas Tomar1

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556vikas.tomar@nd.edu

1

Corresponding author.

J Biomech Eng 130(2), 021021 (Apr 08, 2008) (10 pages) doi:10.1115/1.2903434 History: Received January 06, 2007; Revised September 19, 2007; Published April 08, 2008

## Abstract

Trabecular bone fracture is closely related to the trabecular architecture, microdamage accumulation, and bone tissue properties. Micro-finite-element models have been used to investigate the elastic and yield properties of trabecular bone but have only seen limited application in modeling the microstructure dependent fracture of trabecular bone. In this research, dynamic fracture in two-dimensional (2D) micrographs of ovine (sheep) trabecular bone is modeled using the cohesive finite element method. For this purpose, the bone tissue is modeled as an orthotropic material with the cohesive parameters calculated from the experimental fracture properties of the human cortical bone. Crack propagation analyses are carried out in two different 2D orthogonal sections cut from a three-dimensional $8mm$ diameter cylindrical trabecular bone sample. The two sections differ in microstructural features such as area fraction (ratio of the 2D space occupied by bone tissue to the total 2D space), mean trabecula thickness, and connectivity. Analyses focus on understanding the effect of the rate of loading as well as on how the rate variation interacts with the microstructural features to cause anisotropy in microdamage accumulation and in the fracture resistance. Results are analyzed in terms of the dependence of fracture energy dissipation on the microstructural features as well as in terms of the changes in damage and stresses associated with the bone architecture variation. Besides the obvious dependence of the fracture behavior on the rate of loading, it is found that the microstructure strongly influences the fracture properties. The orthogonal section with lesser area fraction, low connectivity, and higher mean trabecula thickness is more resistant to fracture than the section with high area fraction, high connectivity, and lower mean trabecula thickness. In addition, it is found that the trabecular architecture leads to inhomogeneous distribution of damage, irrespective of the symmetry in the applied loading with the fracture of the entire bone section rapidly progressing to bone fragmentation once the accumulated damage in any trabeculae reaches a critical limit.

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## Figures

Figure 1

(a) An 8mm diameter cylinder of trabecular bone from an ovine (sheep) femur (obtained from the Tissue Mechanics Laboratory at the University of Notre Dame du Lac directed by G. L. Niebur (1)) and (b) a digital 2D scan of a cross section of the same sample (in both figures, the dotted lines show principal material coordinate systems used for describing the continuum orthotropic properties in the current research)

Figure 2

Irreversible bilinear cohesive law

Figure 3

FE mesh of (a) Microstructure 1 and (b) Microstructure 2 analyzed in the presented research (blue (in color)∕dark shade (in black and white) in the microstructure window is bone and red (color)∕dark shade (black and white) surrounding microstructure window are homogeneous elements with effective apparent cortical bone properties)

Figure 4

Dynamic fracture analysis setup for carrying out CFEM calculations (load is applied along the y-axis direction)

Figure 5

(a) Damage distribution in both microstructures at loading velocity V0=0.5m∕s and time t=25μs and (b) the maximum principal stress distribution in both microstructures at loading velocity V0=0.5m∕s and time t=25μs

Figure 6

A comparison of damage density as a function of simulation time and loading velocity in (a) Microstructure 1 and in (b) Microstructure 2 (damage density is normalized with respect to the available bone surface area for fracture)

Figure 7

A comparison of fracture energy dissipation as a function of crack density and loading velocity in (a) Microstructure 1 and in (b) Microstructure 2 (crack density is normalized with respect to the available bone surface area for fracture)

Figure 8

A comparison of fracture energy release rate as a function of crack density and loading velocity in (a) Microstructure 1 and in (b) Microstructure 2

Figure 9

A comparison of (a) fracture energy dissipation as a function of crack density and (b) fracture energy release rate as a function of crack density in both microstructures at V0=0.1m∕s

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