Fatigue damage investigation and optimization of a viscoelastically damped system with uncertainties

  • L. K. S. GonçalvesEmail author
  • U. L. Rosa
  • A. M. G. de Lima
Technical Paper


The need for larger and more efficient structures has led to the development of numerical models to approach real-world complex behaviors. Normally, engineering systems consider deterministic parameters, while in real-world situations, they are subjected to parametric uncertainties, resulting in differences between the calculated responses. This work is devoted to performing numerical and computational studies of fatigue damage analyses of viscoelastically damped systems in the frequency domain subjected to parametric uncertainties. Here, the discretization of the random field is performed using the finite element method combined with the Karhunen–Loève expansion and the fatigue damage index is estimated by applying the so-called Sines’ global criterion. Optimization techniques were applied in order to obtain the optimal design of surface viscoelastic treatments in terms of fatigue. Therefore, the main contribution intended for the present study is to show the importance of considering optimal–robust parameters to estimate the failure probability of engineering structures with constrained viscoelastic layers. After presenting the underlying foundations, the numerical results are presented and the main features of the proposed methodology are highlighted.


Viscoelastic damping Uncertainties Fatigue Sines’ criterion Robust multiobjective optimization 

List of symbols

\(X\left( {x,y,\theta } \right)\)

Bidimensional random field

\(X\left( {x,y} \right)\)

Mean value of the bidimensional random field

\(\left( {\lambda_{r} ,f_{r} } \right)\)

r-Eigensolutions of the KL expansion

\(\xi_{r} \left( \theta \right)\)

Gaussian random variables

\(\varvec{M}^{{\left( {\text{e}} \right)}} \left( \theta \right)\)

Stochastic elementary mass matrix

\(\varvec{K}_{k}^{{\left( {\text{e}} \right)}} \left( \theta \right)\)

Stochastic elementary stiffness matrix of the elastic part, \(k = 1,3\)

\(\varvec{K}_{2}^{{\left( {\text{e}} \right)}} \left( {\omega ,T_{\text{v}} ,\theta } \right)\)

Stochastic elementary stiffness matrix of the viscoelastic part

\(\bar{\varvec{M}}_{r}^{{\left( {\text{e}} \right)}}\)

Random elementary mass matrix

\(\bar{\varvec{K}}_{kr}^{{\left( {\text{e}} \right)}}\)

Random elementary stiffness matrix of the elastic part, \(k = 1,3\)

\(\bar{\varvec{K}}_{2r}^{{\left( {\text{e}} \right)}}\)

Random elementary frequency and temperature-independent stiffness matrix of the viscoelastic part

\(G\left( {\omega ,T_{\text{v}} } \right)\)

Complex modulus function

\(G\left( {\omega ,T_{\text{v}} ,\theta } \right)\)

Random complex modulus function with uncertain temperature


Temperature of the viscoelastic material

\(\varvec{B}_{k} \left( {x,y} \right)\)

Strain–displacement differential operator matrix, \(k = 1,2,3\)


Material properties matrices of the elastic part, \(k = 1,3\)


Frequency- and temperature-independent material properties matrix of the viscoelastic part

\(\varvec{U}\left( {\omega ,T_{\text{v}} ,\theta } \right)\)

Vector of the random displacements

\(\varvec{F}\left( \omega \right)\)

Vector of the applied external forces

\(\varvec{H}\left( {\omega ,T_{\text{v}} ,\theta } \right)\)

Frequency response function matrix of the stochastic system

\(\varvec{H}\left( {\omega ,T_{\text{v}} } \right)\)

Frequency response function matrix of the deterministic system

\(\varvec{\varPhi}_{s} \left( {\omega ,T_{\text{v}} ,\theta } \right)\)

Power spectral density (PSD) matrix of the random stress response

\(\varvec{\varPhi}_{f} \left( \omega \right)\)

Power spectral density (PSD) matrix of the random loadings

\(\varvec{s}\left( {t,T_{\text{v}} ,\theta } \right)\)

Time-domain random stress vector of the viscoelastic system


Sines’ fatigue index

\(R_{i} \left( \theta \right)\)

Maximum measurements of the stress amplitudes

\(J_{2a} \left( \theta \right)\)

Second invariant of the random deviatoric stress tensor


Number of truncated terms in the Karhunen–Loève expansion



The authors are grateful to CNPq for the continued support to their research activities, especially through the research Grant 302026/2016-9 (A.M.G. de Lima). It is also important to express the acknowledgements to FAPEMIG, especially to research projects PPM-0058-18 (A.M.G. de Lima) and APQ-01865-18 (Manzanares-Filho).


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • L. K. S. Gonçalves
    • 1
    Email author
  • U. L. Rosa
    • 1
  • A. M. G. de Lima
    • 1
  1. 1.School of Mechanical EngineeringFederal University of UberlândiaUberlândiaBrazil

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