The linear elastic properties of a soft tissue exhibiting a unidirectional arrangement of reinforcing fibers may be described in terms of the five independent elastic stiffness coefficients C11, C13, C33, C44, and C66. In previous studies, ultrasonic measurements of these coefficients for formalin fixed specimens of bovine Achilles tendon and normal human myocardium were reported. In the present study these results are used to analyze the anisotropy of Young's modulus of these tissues. For formalin fixed tendon a value of 1.37 GPa is obtained for Young's modulus along the fiber axis of the tissue, and a value of 0.0706 GPa is obtained perpendicular to the fibers. For formalin fixed myocardium, values of 0.101 and 0.0311 GPa parallel and perpendicular to the fibers, respectively, are obtained. Based on the results for the angular dependence of Young's modulus from unidirectional specimens of myocardium, a model is introduced to estimate these features for the more complicated fiber architecture of the left ventricular wall.
The anisotropy of Young's modulus in human cortical bone was determined for all spatial directions by performing coordinate rotations of a 6 by 6 elastic stiffness matrix. The elastic stiffness coefficients were determined experimentally from ultrasonic velocity measurements on 96 samples of normal cortical bone removed from the right tibia of eight human cadavers. The following measured values were used for our analysis: c11 = 19.5 GPa, c22 = 20.1 GPa, c33 = 30.9 GPa, c44 = 5.72 GPa, c55 = 5.17 GPa, c66 = 4.05 GPa, c23 = 12.5 GPa. The remaining coefficients were determined by assuming that the specimens possessed at least an orthorhombic elastic symmetry, and further assuming that c13 = c23 c12 = c11 - 2c66. Our analysis revealed a substantial anisotropy in Young's modulus in the plane containing the long axis of the tibia, with maxima of 20.9 GPa parallel to the long axis, and minima of 11.8 GPa perpendicular to this axis. A less pronounced anisotropy was observed in the plane perpendicular to the long axis of the tibia. To display our results for the full three-dimensional anisotropy of cortical bone, a closed surface was used to represent Young's modulus in all spatial directions.
Recent studies from our laboratory have detailed the anisotropy of velocity of quasilongitudinal-mode ultrasonic waves through formalin fixed samples of normal human myocardium and bovine Achilles tendon. Results of these studies were used to determine the elastic stiffness coefficients c33, corresponding to the propagation of longitudinal-mode waves parallel to the fiber axis of the tissue, and c11, corresponding to the propagation of longitudinal-mode waves perpendicular to the fiber axis. For a tissue possessing a unidirectional arrangement of fibers with a random transverse distribution, three additional coefficients, c13, c44, and c12, are needed to describe its linear mechanical properties completely. Direct ultrasonic measurements of these coefficients in solids typically require the propagation of transverse-mode waves through the sample. Such measurements are difficult to perform in soft tissues because transverse-mode ultrasonic waves are very highly attenuated by the tissue. This study therefore employs a technique to estimate c13 based on measurements of the velocity of quasilongitudinal-mode ultrasonic waves for numerous angles of propagation relative to the fiber axis of the tissue. Analysis of data obtained from formalin fixed bovine Achilles tendon and human myocardium yield estimated values for c13 of 3.17 and 2.46 GPa, respectively.
Experimental results of the anisotropy of the frequency dependence of ultrasonic attenuation in unidirectional graphite/epoxy composite are presented. This material is made of graphite fibers aligned along a single axis and embedded in epoxy resin with a random transverse distribution. A sample with flat and parallel faces was interrogated in reflection mode with 5 MHz longitudinal and transverse wave contact transducers. A log-spectral subtraction method was used to obtain the frequency-dependent signal loss. The results show that for the 9 waves generated (3 longitudinal and 6 transverse) the signal loss was approximately linearly dependent on frequency over the useful bandwidth of the measurement system. Thus, a least-squares line fit to the data divided by the total propagation length in the sample provided the slope of attenuation. The slope of attenuation for longitudinal waves propagating parallel to the fibers was substantially less than those with propagation perpendicular to the fibers. In contrast, for transverse waves, the slopes of attenuation associated with either propagation or polarization parallel to the fibers were larger than those corresponding to both polarization and propagation perpendicular to the fibers.
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