The contact problem and stress state for indentation by a flat punch with rounded edges is studied. For the contact problem itself analytical solutions are obtained for both surface pressure and interior stress fields. Cases of normal indentation and frictional contact, the latter in both sliding or partial slip conditions, are all treated. The transition from the Hertzian configuration to the contact between a nominally flat pad and contacting flat surface is discussed, and it is found that the strength of the contact decays surprisingly slowly. Regarding the von Mises yield parameter, there is a range of configurations for which the strength is actually higher than the Hertzian one, and the strength decays only when the corner radii are very small. The present solution is therefore a realistic alternative to the classical rigid-flat punch idealization, and has particular application to fretting fatigue tests.

During embryonic development, adult haematopoietic stem cells (HSCs) emerge preferentially in the ventral domain of the aorta in the aorta–gonad–mesonephros (AGM) region. Several signalling pathways such as Notch, Wnt, Shh and RA are implicated in this process, yet how these interact to regulate the emergence of HSCs has not previously been described in mammals. Using a combination of ex vivo and in vivo approaches, we report here that stage-specific reciprocal dorso–ventral inductive interactions and lateral input from the urogenital ridges are required to drive HSC development in the aorta. Our study strongly suggests that these inductive interactions in the AGM region are mediated by the interplay between spatially polarized signalling pathways. Specifically, Shh produced in the dorsal region of the AGM, stem cell factor in the ventral and lateral regions, and BMP inhibitory signals in the ventral tissue are integral parts of the regulatory system involved in the development of HSCs.

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