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A simple analytical model has been developed to explain a paradoxical situation that has been detected earlier by one of the authors on the basis of the finite element analysis (FEA): deep enough transverse grooves in the pins (bonded components, adherends) resulted, for small size lap shear joints, in an appreciable reduction in, and in a more uniform distribution of, the interfacial stresses. In this analysis we explain, using analytical (ldquomathematicalrdquo) modeling, the physics of the detected situation. The analysis is limited to the evaluation of the shearing stresses, although a similar analysis can be carried out for peeling stresses as well. The numerical example carried out on the basis of the developed model indicates that the ldquoobservedrdquo situation is due to the increase, due to the grooves, in the interfacial compliance of the bonding structure: the grooves transfer (ldquoconvertrdquo) the pin portions located between the grooves and the bonding material into parts of the bonding structure, thereby increasing the thickness and, hence, the compliance of this structure with respect to the shearing deformations. This positive effect overwhelms, as far as the magnitude and the distribution of the interfacial stresses are concerned, the negative effect of the increased, also because of the grooves, axial compliance of the portions of the pins that experience direct action of the external forces. The developed stress model can be used in the analysis and physical design of the lap shear joints and some advanced test specimens in micro-, opto-electronic and photonic packaging. It is concluded that analytical modeling is able not only to come up with simple relationships that clearly indicate ldquowhat affects what and what is responsible for whatrdquo, but, more importantly, can explain the physics of phenomena that neither FEA modeling (ldquosimulationrdquo), nor even actual experimentation are able to.