The gradient displacement field of a micro-structured strong inter-face of a three-dimensional multi-material is regarded as a gradient-Young measure so that the stored strain energy of the material is defined as a bifunctional of displacement/Young measure state variables. We propose a new model by computing a suitable variational limit of this bifunctional when the thickness and the stiffness of the strong material are of order ε and 1/ε respectively. The stored strain energy functional associated with the model in pure displacements living in a Sobolev space is obtained as the marginal map of the limit bifunctional. We also obtain a new asymptotic formulation in terms of Young measure state variable when considering the other marginal map.