This paper is concerned with the method of memory diagrams developed for solving a problem of frictional elastic contact. Our goal is to establish a link between contact force and displacement for a general plane contact between two arbitrarily-shaped bodies; rolling and torsion are not considered. Description of mechanical interactions between two solids in contact in the presence of friction is a nontrivial task since the desired force-displacement relationships have hysteretic (memory-dependent) character. Arbitrarily changing applied force (or displacement) creates a cumbersome shear stress distribution in the contact zone that has to be adequately parameterized and accounted for. In that regard, it is suggested to consider, instead of complex shear stress distributions, a simpler functional form called memory diagram that contains the same memory information. We have established two integral relationships that link the force and displacement vectors with that internal functional dependency. The integral relationships are supplemented with two other evolution rules for memory diagrams that eventually follow from the Coulomb friction law. The memory diagram is updated with the help of these rules following a given force history. Then the calculated memory diagram is used to update the history of displacement i.e. to produce the desired force-displacement relationship.

The importance of Non-Destructive Testing (NDT) to check the integrity of materials in different fields of industry has increased significantly in recent years. Actually, industry demands NDT methods that allow fast (preferably non-contact) detection and localization of early-stage defects with easy-to-interpret results, so that even a non-expert field worker can carry out the testing. The main challenge is to combine as many of these requirements into one single technique. The concept of acoustic cameras, developed for low frequency NDT, meets most of the above-mentioned requirements. These cameras make use of an array of microphones to visualize noise sources by estimating the Direction Of Arrival (DOA) of the impinging sound waves. Until now, however, because of limitations in the frequency range and the lack of integrated nonlinear post-processing, acoustic camera systems have never been used for the localization of incipient damage. The goal of the current paper is to numerically investigate the capabilities of locating incipient damage by measuring the nonlinear airborne emission of the defect using a non-contact ultrasonic sensor array. We will consider a simple case of a sample with a single near-surface crack and prove that after efficient excitation of the defect sample, the nonlinear defect responses can be detected by a uniform linear sensor array. These responses are then used to determine the location of the defect by means of three different DOA algorithms. The results obtained in this study can be considered as a first step towards the development of a nonlinear ultrasonic camera system, comprising the ultrasonic sensor array as the hardware and nonlinear post-processing and source localization software.