International audienceStrong incompatibility stresses may develop at grain or twin boundaries because of elastic and plastic anisotropies. Their knowledge attwin boundaries may be of interest for a better understanding of the mechanical behaviour of fcc materials that can display lamellar twin structures, such as twinning-induced plasticity (TWIP) steels or general nanotwinned materials. In this paper, incompatibility stresses arising at general twin boundaries are explicitly derived for a given twin volume fraction. They are deduced from the solutions of the general infinite bicrystal, which is equivalent to a periodic layered structure. In the case of pure elasticity andtwin boundaries, the result is of remarkable simplicity. The incompatibility stress field reduces to a shear stress acting upon a plane orthogonal to twin plane. Simple analytical expressions of the resolved shear stresses are also determined according to the twin-boundary orientation, the twin volume fraction and the elastic anisotropy factor. Such expressions allow performing a comprehensive study of slip initiation. In particular, there exists a large physical domain, depending on the three above parameters, where simultaneous slip parallel to twin plane in the parent and in the twin is greatly promoted. There is also a restricted domain where simultaneous single slip parallel to twin plane is promoted. The conditions for these promotions are realistic considering the literature data on TWIP steels. The present results, hence, support the high ductility and strong contribution of kinematic hardening observed in TWIP steels and agree with composite hardening models with single- and multi-slip-deforming grains