6 Evaluating Stresses from Measured Strains in Viscoelastic Body Using Numerical Laplace Transformation 57 where F(ω) is the Laplace transformation of the function f (t). It is observed that the Laplace transformation and the inverse transformation can be represented in the form of the Fourier transformation and the inverse transformation. When the infinite sum of integrals is truncated by the N−1 term and the number of data is N, Eqs. (6.4) and (6.5) can be expressed as F(n ω) = t N−1 k=0 f(k t) · e−γk t /e−2πikn/N (6.6) f(k t) = eγk t 2π ω N/2−1 k=0 F(n ω) · e 2πikn/N (6.7) where Δω, Δt, n, and ω are the angular frequency interval, the time interval, the data number, and the angular frequency, respectively. It is possible to utilize fast Fourier transformation (FFT) for the above equation, and the computation can be performed by setting Nto a power of 2. When computing using FFT, the input function is regarded as one cycle. Assuming that this period is NΔt, the appropriate value of γ is γ =4/(NΔt)~5/(NΔt) [9]. However, the appropriate values of the numerical Laplace transformation cannot be obtained because the results are affected by the computational nature of FFT. 6.5 Numerical Laplace Transformation Considering the Nature of FFT This section shows a computation method that is not affected by the nature of FFT. The results of the numerical Laplace transformation by FFT are effective in the range of 0 < ω < ωs/2 where ωs is the sampling angular frequency. This is caused by the FFT nature where the discrete data is used. The results are obtained in the range of 0<ω<ωs by doubling the number of the input strain data. Since the number of the input data is doubled, ωs is also doubled but ωis constant. Therefore, the results in the range of 0<ω<ωs is obtained appropriate values. When the Laplace inverse transformation is computed using the effective values in the range of 0 < ω < ωs/2, appropriate values cannot be obtained. It is considered that transformation solutions of the input used for the Laplace inverse transformation using FFT need the complex conjugate. Therefore, transformation solutions of conjugate complex data are obtained by doubling the number of data. Then, the additional data are plotted symmetrically with the value at ωs/2 as the boundary. Since the number of the input data is doubled, ωs/2 is also doubled but ωis constant. 6.6 Computing Stresses Using Proposed Method In order to show the effectiveness of the numerical Laplace transformation and the inverse transformation methods proposed in previous section, the through-thickness strains are calculated from in-plane strains. Also, the stresses are calculated from obtained values and in-plane strains using constitutive equation of the viscoelastic body. K(t), G(t), and υ(t) are expressed as by using a Prony series as follows expressed as [10] K(t) =K0 + N i=1 Kie−t/ρi (6.9) G(t) =G0 + N i=1 Gie−t/ρi (6.10) υ(t) =υ0 − N i=1 υie−t/ρi (6.11)
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