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Equivalent Spectral Method to Estimate the Fatigue Life of Composite Laminates Under Random Vibration Loadings

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Mechanics of Composite Materials Aims and scope

The power spectral density (PSD) of an equivalent stress is proposed to analyze the fatigue life of composite laminates under random vibration loadings using the Tsai–Hill criterion. The vibration fatigue life can be estimated based on the rainflow amplitude probability density function p(S) of the equivalent stress, which is obtained from this PSD, combined with the S-N curve of the materials. The fatigue life analysis of a random vibration fatigue experiment of composite laminates was carried out, and the calculation results were in accordance with experimental data, which indicates that the fatigue life analysis model proposed gives a satisfactory prediction accuracy.

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    D. Y. Gao, W. X. Yao & J. Huang

  2. Aero-engine Thermal Environment and Structure Key Laboratory of Ministry of Industry and Information Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    D. Y. Gao & W. D. Wen

  3. Key Laboratory of Fundamental Science for National Defense-Advanced Design Technology of Flight Vehicle, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    W. X. Yao & J. Huang

Authors
  1. D. Y. Gao
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  2. W. X. Yao
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  3. W. D. Wen
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  4. J. Huang
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Corresponding author

Correspondence to D. Y. Gao.

Additional information

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 57, No. 1, pp. 139-160, January-February, 2021.

Appendix

Appendix

A. Power Spectral Density (PSD) and spectral moments

For a stationary ergodic random Gaussian process X (t) with a zero mean, its stochastic properties can be described by its autocorrelation function RX (τ) in the time domain, which can be calculated by the equation:

$$ {R}_X\left(\tau \right)=E\left[X(t)X\left(t+\tau \right)\right]. $$
(A.1)

It can also be described by the two-side power spectral density SX (ω) in the frequency domain, which forms a Fourier transform pair together with RX (τ):

$$ {\displaystyle \begin{array}{c}{S}_X\left(\omega \right)={\int}_{-\infty}^{+\infty }{R}_X\left(\tau \right){e}^{- j\omega \tau} d\tau, \\ {}{R}_X\left(\tau \right)=\frac{1}{2\pi }{\int}_{-\infty}^{+\infty }{S}_X\left(\omega \right){e}^{j\omega \tau} d\omega .\end{array}} $$
(A.2)

Since the negative frequency components in SX (ω) has little sense in engineering, the one-side power spectral density GX (ω) can be defined as

$$ {G}_X\left(\omega \right)=\Big\{{\displaystyle \begin{array}{c}2{S}_X\left(\omega \right),\omega \ge 0,\\ {}\kern1.24em 0\kern2em \omega <0.\end{array}} $$
(A.3)

Then, the spectral moments can be defined as

$$ {m}_i={\int}_0^{\infty }{\omega}^i{G}_X\left(\omega \right) d\omega, $$
(A.4)

where usually i = 0, 2, 4, … .

In this method, the number ν + of cycles in a unit time is described by the expected rate of zero-crossing with a positive slope:

$$ {\nu}^{+}=\frac{1}{2\pi}\sqrt{\frac{m_2}{m_0}}. $$
(A.5)

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Gao, D.Y., Yao, W.X., Wen, W.D. et al. Equivalent Spectral Method to Estimate the Fatigue Life of Composite Laminates Under Random Vibration Loadings. Mech Compos Mater 57, 101–114 (2021). https://doi.org/10.1007/s11029-021-09937-2

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  • DOI: https://doi.org/10.1007/s11029-021-09937-2

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