The shear-lag composite model has been utilized to predict the stiffness of bone, dentin, and enamel.
These materials are comprised of collagen and hydroxyapatite. The apatite crystals are assumed to be
behave like short fibers that are aligned with the axis of loading, and the modulus equation is a function
of the geometry of the apatite crystals, resulting in the following equation:
Ecomposite = 0.60(VapatiteEapatite) + VcollagenEcollagen
a) Using the Voigt, Reuss, and shear-lag models, calculate the elastic modulus of a collagenhydroxyapatite
composite for volume fractions of HA ranging from 0 to 1. Generate a plot with
the elastic moduli plotted vs the volume fraction of HA. How does the shear-lag model compare
with the Voigt and Reuss models? Assume Ecollagen = 1 GPa, EHA = 70 GPa.
b) On the same graph, denote the elastic moduli for cortical bone, dentin, and enamel (use values
from the course textbook). Based on your results, what would be the approximate volume
fraction of HA for bone and dentin if the shear-lag model is assumed to be valid?
c) What do you notice about the shear-lag model prediction for enamel based on the previous
assumptions? If the form of hydroxyapatite comprising enamel has a higher elastic modulus
(130 GPa), what is the predicted volume fraction HA for enamel?
The shear-lag composite model has been utilized to predict the stiffness of bone, dentin, and enamel.
These materials are comprised of collagen and hydroxyapatite. The apatite crystals are assumed to be
behave like short fibers that are aligned with the axis of loading, and the modulus equation is a function
of the geometry of the apatite crystals, resulting in the following equation:
Ecomposite = 0.60(VapatiteEapatite) + VcollagenEcollagen
a) Using the Voigt, Reuss, and shear-lag models, calculate the elastic modulus of a collagenhydroxyapatite
composite for volume fractions of HA ranging from 0 to 1. Generate a plot with
the elastic moduli plotted vs the volume fraction of HA. How does the shear-lag model compare
with the Voigt and Reuss models? Assume Ecollagen = 1 GPa, EHA = 70 GPa.
b) On the same graph, denote the elastic moduli for cortical bone, dentin, and enamel (use values
from the course textbook). Based on your results, what would be the approximate volume
fraction of HA for bone and dentin if the shear-lag model is assumed to be valid?
c) What do you notice about the shear-lag model prediction for enamel based on the previous
assumptions? If the form of hydroxyapatite comprising enamel has a higher elastic modulus
(130 GPa), what is the predicted volume fraction HA for enamel?
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