Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \(X\left( {x,y,\theta } \right)\) :
-
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
- \(T_{\text{v}}\) :
-
Temperature of the viscoelastic material
- \(\varvec{B}_{k} \left( {x,y} \right)\) :
-
Strain–displacement differential operator matrix, \(k = 1,2,3\)
- \(\varvec{C}_{k}\) :
-
Material properties matrices of the elastic part, \(k = 1,3\)
- \(\varvec{C}_{2}\) :
-
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
- \(D_{\text{s}}\) :
-
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
- \(nkl\) :
-
Number of truncated terms in the Karhunen–Loève expansion
References
Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New York
de Lima AMG, Rade DA, Bouhaddi N (2010) Stochastic modeling of surface viscoelastic treatments combined with model condensation procedures. Shock Vib 17:429–444
de Lima AMG, Da Silva AR, Rade DA, Bouhaddi N (2010) Component mode synthesis combining robust enriched Ritz approach for viscoelastically damped structures. Eng Struct 32:479–1488
Cunha-Filho AG, Briend YPJ, De Lima AMG, Donadon MV (2018) An efficient iterative model reduction method for aeroviscoelastic panel flutter analysis in the supersonic regime. Mech Syst Signal Process 104:575–588
de Lima AMG, Lambert S, Rade DA, Pagnacco E, Khalij L (2014) Fatigue reliability analysis of viscoelastic structures subjected to random loads. Mech Syst Signal Process 43:305–318
Guedri M, de Lima AMG, Bouhaddi N, Rade DA (2008) Robust design of viscoelastic structures based on stochastic finite element models. Mech Syst Signal Process 24:59–77
de Lima AMG, Rade DA, Lépore-Neto FP (2009) An efficient modeling methodology of structural systems containing viscoelastic dampers based on frequency response function substructuring. Mech Syst Signal Process 23:1272–1281
Lambert S, Pagnacco E, Khalij L (2010) A probabilistic model for the fatigue reliability of structures under random loadings with phase shift effects. Int J Fatigue 32:463–474
Pagnacco E, Lambert S, Khalij L, Rade DA (2012) Design optimization of linear structures subjected to dynamic random loads with respect to fatigue life. Int J Fatigue 43:168–177
Sines G (1959) Metal fatigue. McGraw-Hill, New York
Rosa UL, Gonçalves LKS, de Lima AMG (2016) A robust condensation strategy for stochastic systems. In: XXXVII Iberian Latin American congress on computational methods in engineering, Brasília, Brazil
Khatua TP, Cheung YK (1973) Bending and vibration of multilayer sandwich beams and plates. Int J Numer Methods Eng 6:11–24
Drake ML, Soovere JA (1984) A design guide for damping of aerospace structures. Vibration damping workshop, California
Florian A (1992) An efficient sampling scheme: updates Latin Hypercube sampling. Probab Eng Mech 7:123–130
Khalij L, Pagnacco E, Lambert S (2010) A measure of the equivalent shear stress amplitude from a prismatic hull in the principal coordinate system. Int J Fatigue 32:1977–1984
Preumont A (1985) On the peak factor of stationary gaussian process. J Sound Vib 100:15–34
Pitoiset X, Rychlik I, Preumont A (2001) Spectral methods to estimate local multiaxial fatigue failure for structures undergoing random vibrations. Fatigue Fract Eng Mater Struct 24:715–727
Eschenauer H, Koshi J, Osyczka A (1990) Multicriteria design optimization: procedures and applications. Springer, Germany
Deb K (2001) Multiobjetive optimization using evolutionary algorithms. Wiley, London
de Lima AMG, Rade DA, Bouhaddi N (2010) Optimization of viscoelastic systems combining robust condensation and metamodeling. J Braz Soc Mech Sci Eng 32:485–495
Rosenmann MA, Gero JS (1985) Reducing the Pareto optimal set in multicriteria optimization (with application to Pareto optimal dynamic programming). Eng Optim 8:189–206
Srinivas N, Deb K (1993) Multi-objective using nondominated sorting in genetic algorithms. J Evolut Comput 2:221–248
Acknowledgements
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gonçalves, L.K.S., Rosa, U.L. & de Lima, A.M.G. Fatigue damage investigation and optimization of a viscoelastically damped system with uncertainties. J Braz. Soc. Mech. Sci. Eng. 41, 382 (2019). https://doi.org/10.1007/s40430-019-1879-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40430-019-1879-4