Review of multiaxial vibration fatigue spectral methods—with open-source support

Šonc, Jaša (author), Slavič, Janko (author)

In vibration fatigue, dynamic excitation such as base acceleration or force can produce a multiaxial stress response. In the frequency domain, this stress response is characterised by a cross-power-spectral-density (cross-PSD) matrix that contains the spectral content of each stress component and the phase relations between them. For fatigue-life estimation under stationary excitation, the multiaxial stress cross-PSD is reduced to an equivalent uniaxial stress PSD compatible with established spectral damage models. Several criteria have been proposed for this reduction, each differing in the assumptions made about phase relations, the required material parameters, and computational cost. This paper provides an implementation-oriented review of ten such criteria: three critical-plane variants (maximum normal, maximum shear, and combined normal-and-shear stress), the equivalent von Mises stress and its adaptation for out-ofphase stress components, the Carpinteri–Spagnoli frequency-domain reformulation, the frequency-based multiaxial rainflow, a thermoelasticity-based criterion, the Niesłony hydrostatic–deviatoric combination, and the equivalent Lemaitre stress with multiaxial S–N interpolation. As a result of this research, all ten methods are also implemented in FLife (version 2.2.0), an open-source Python package that extends the uniaxial spectral framework to multiaxial inputs. Previously these implementations were distributed across separate source papers; here they share a single input format and a single workflow, so that any subset can be compared on the same dataset without reimplementation. A comparative overview highlights trade-offs between simplicity, sensitivity to non-proportional loading, material-parameter requirements, and computational cost, supporting method selection in multiaxial vibration-fatigue practice. The criteria are also applied to a common dataset, where the equivalent stress varies by up to about a factor of 1.7 across the ten criteria.

• Language: English
• Keywords: multiaxial vibration fatigue, spectral methods, equivalent stress, power spectral density, open source, reproducibility, FLife
• Work type: Article
• Typology: 1.02 - Review Article
• Organization: FS - Faculty of Mechanical Engineering
• Publication status: Published
• Publication version: Version of Record
• Year: 2026
• Number of pages: Str. 1-12
• Numbering: Vol. 1, iss. 1, art. 2606004129
• PID: 20.500.12556/RUL-183972
• UDC: 531:620.178.3:534
• ISSN on article: Y510-3088
• COBISS.SI-ID: 282507011
• Publication date in RUL: 23.06.2026

Record is a part of a journal

• Title: Applied nonlinear dynamics and vibrations
• Publisher: Scilight Press
• COBISS.SI-ID: 281090563

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: večosno vibracijsko utrujanje, spektralne metode, ekvivalentna napetost, spektralna gostota moči, odprta koda

Projects

• Funder: ARIS - Slovenian Research and Innovation Agency
• Project number: P2-0263-2019
• Name: Mehanika v tehniki