30 Estimating Frequency-Dependent Mechanical Properties of Materials 315 .t/ DY.t/ P".t/ D t Z 0 Y .t / P". /d!; (30.1) where is an integration variable. Transformation into the frequency domain of Eq. (30.1) yields Q .!/ Di!QY .!/ Q".!/ ; (30.2) where the time derivative and convolution properties of the Fourier transform have been used [18]. Eq. (30.2) represents a complex, frequency-domain expression equivalent to Young’s Law. Thus, the complex elastic modulus QE .!/ is QE .!/ Di!QY .!/ D QE ’ .!/ Ci QE ’’ .!/ ; (30.3) where QE’ .!/ and QE’’ .!/ are the real and imaginary components of QE . The corresponding complex form of Young’s equation is then Q .!/ D QE .!/ Q".!/ : (30.4) 30.2.2 One-Dimensional Wave Propagation in a Bar The One-dimensional equation of motion in a bar is @ @x @2u @t2 D0 (30.5) Taking the Fourier transform of Eq. (30.5), the frequency-domain equation of motion is obtained: @Q @x C !2Qu D0; (30.6) where Q and Qu represent the frequency-domain stress and particle displacement, respectively. Next, the spatial derivative (@/@x) is taken, @2 Q @x2 C !2 @Qu @x D0: (30.7) Finally, it is trivial to transform the definition of strain, ".x; t/ @ @x Œu.x; t/ ; (30.8) into the frequency domain, Q".x;!/ D @ @x ŒQu.x;!/ : (30.9) Substituting Eq. (30.9) into Eq. (30.7) leads to a convenient form of the equation of motion, i.e., QE @2Q" @x2 C !2Q" D0: (30.10) By defining the complex wavenumber as
RkJQdWJsaXNoZXIy MTMzNzEzMQ==