Issue 47

D. Benasciutti et alii, Frattura ed Integrità Strutturale, 47 (2019) 348-366; DOI: 10.3221/IGF-ESIS.47.26 349 criterion on their own. Users, in implementing the criterion, may feel not sure to have correctly understood the basic principles of the method and have doubts that their own programs are really free from errors and/or misinterpretations, and that they return a output truly correct. The possibility to have access to detailed results for a number of simple case studies, to be used as a reference, may thus help to solve possible doubts. The “Projection-by-Projection” (PbP) method is fatigue criterion for multiaxial loadings. It has first been proposed about ten years ago [1,2] as a conventional time-domain criterion, in which stress time-histories are processed directly. It was then reformulated in the frequency-domain to address multiaxial random loadings [3,4]. Indeed, like for other multiaxial criteria, a time-domain definition may become computationally time-consuming (and thus impractical), for example when considering medium-to-large finite element models, in which it is required to process long digitalized multiaxial time- histories in hundreds or thousands of nodes [5]. This is perhaps the main limitation (along with others theoretical issues here not mentioned) that motivated the researchers’ effort of reformulating multiaxial criteria (PbP included) from time- to frequency-domain. Unlike their time-domain counterparts, frequency-domain multiaxial criteria (also called multiaxial spectral methods) are indeed able to reduce the overall computational time drastically, while keeping high levels of accuracy. In multiaxial spectral methods, the fatigue damage and life are estimated directly from Power Spectral Density (PSD) data (i.e. spectra and cross-spectra), which characterize a multiaxial random stress in the frequency-domain [5]. This approach proves to be computationally efficient especially if it can exploit results of FE structural dynamic analyses. It has furthermore been demonstrated that the PbP method, unlike other frequency-domain criteria, is sensitive to the local state of stress in a structure as it correctly takes into account the degree of correlation between stress components. At the same time, the method can also account for different material behaviors depending on different types of multiaxial stress [1-4]. This is of great relevance in engineering applications, in which several materials need to be scrutinized and compared for the same component geometry, or it is required to perform a geometric optimization to make the best use of a given material. Though the PbP spectral method has been presented in a number of scientific articles published so far [3,4], it is our belief that this work – entirely devoted to its practical application – would be helpful to anyone who wishes to implement the method on his own. After summarizing some theoretical formulas to characterize deviatoric and hydrostatic stress components in frequency-domain, the work discusses all the steps necessary to implement the PbP method. A first numerical case study is considered, in which various biaxial stress states are applied to three different materials. Each analysis step is commented on in detail to give an exhaustive description of the procedure. Finally, the case of a simple 2D structure subjected to random excitations is considered to show how the PbP method can be easily embedded into a FE durability analysis. F REQUENCY - DOMAIN DESCRIPTION OF A MULTIAXIAL RANDOM STRESS efore discussing the PbP criterion and the numerical examples, it is useful to summarize the main quantities used in the following (a nomenclature is also given at the end of the article). The next paragraphs will assume that the reader is somehow familiar with the basic concepts and terminology of the frequency-domain approach to random processes. Otherwise, a short review is provided in Appendix A. Equations will be developed for a biaxial stress state; extension to a three-dimensional stress state is straightforward. Description of multiaxial stress state in physical space Assume that σ ( t ) represents the time-varying stress tensor in a point of a mechanical component ( t is the time variable). If the point is located on the surface, where fatigue cracks usually nucleate, the tensor σ ( t ) will have three non-redundant stress components σ x ( t ), σ y ( t ), τ xy ( t ) (i.e. biaxial or plane stress), where σ and τ are normal and shear stress, respectively. Each stress σ x ( t ), σ y ( t ), τ xy ( t ) is assumed to be a zero-mean and covariant stationary random stress (the term “stationary” means, roughly speaking, that the random stress has statistical properties almost constant over time). The stress components in σ ( t ) are usually reordered into a stress vector x ( t ) = (σ x ( t ), σ y ( t ), τ xy ( t )), also written as x ( t ) = ( x 1 ( t ), x 2 ( t ), x 3 ( t )). This vector is characterized in the frequency-domain by the two-sided PSD matrix:            ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( * , * , , * , , , f S f S f S f S f S f S f S f S f S f xy xy yy xy xx xy yy yy yy xx xy xx yy xx xx S (1) B