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Highlights • Breathing cracks are modeled as bilinear stiffness which is then incorporated into general FE models to establish for the first time the existence and the salient characteristics of higher-order FRFs; • A very accurate and robust correlation technique is proposed and used to extract very accurately harmonic components present in overall nonlinear vibration responses which are usually orders of magnitudes smaller and are hence difficult to estimate using conventional FFT based signal processing; • A crack detection and identification method is subsequently developed which has been shown to be reliable, accurate and robust in the presence of measurement uncertainties; • Practicality and numerical performances of the proposed methods are demonstrated through extensive case studies based on a very general GARTEUR structure used in an Eurowide test/analysis correlation exercise. Abstract Two types of cracks are often encountered in engineering structural systems and these are the open cracks and the breathing cracks. Existence of open cracks often leads to the loss of physical stiffness, resulting in a mostly linear structure with reduced load bearing capacity and vibration frequencies. The development of breathing cracks not only reduces structural stiffness, but tends to render the otherwise linear structure to become nonlinear, due to their bilinear stiffness characteristics associated with open and closed states. The nonlinear structural vibration responses can then be investigated and potentially employed to detect and identify breathing cracks within a structural system. In the present study, breathing cracks are modeled based on fracture mechanics from which bilinear stiffness values are obtained. These values are then incorporated into finite element models to compute the first- and second-order frequency response functions (FRFs) based on a proposed correlation technique which is both very accurate and resilient against measurement uncertainties. The existence of well-defined second-order FRFs has been firmly established for the bilinear oscillator, as well as a cantilevered beam with breathing cracks. By expressing the bilinear restoring force as a polynomial series, analytical derivation of second-order FRFs of general structural systems such as the GARTEUR AG11 structure with breathing cracks has been established for the first time. Further, a method of identification has been developed to identify the physical parameters of breathing cracks using second-order FRFs. With these new developments that are presented in this paper, a solid foundation has been laid for the potential applications of higher-order FRFs to damage identification and assessment of general real structural systems. Graphical abstract Image, graphical abstract [ABSTRACT FROM AUTHOR]