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In a number of engineering situations concerning structures made of quasi-brittle, concrete-like materials, all nonlinearities can be reasonably confined to a locus of possible displacement discontinuities. This locus has a lesser dimensionality (by one) with respect to the problem domain; it encompasses joints, cracks, fracture process zones (described by cohesive crack models) and their possible propagation paths. Linear elasticity is assumed everywhere else for overall analysis purposes. With reference to a very broad class of interface models, i.e. of (holonomic or nonholonomic, inviscid or time-dependent) relationships between displacement jumps and tractions across that locus, the (possibly multiple, if any) solutions of the initial-boundary-value problem of structural analysis are shown herein to be characterized by duality pairs of extremum and min-max properties.