Free-Material Optimization in a Peridynamic World

We study free-material optimization for nonlinear bond-based nonlocal diffusion and its localization as the interaction horizon $\delta$ vanishes. Bond conductivities are subject to pointwise bounds and a global material budget constraint. After normalization, the asymptotic behavior of these problems is governed by the fraction ${\theta }_{\delta }$ of potentially active bonds that can be assigned the maximal conductivity. We identify three distinct limiting regimes. If ${\theta }_{\delta }\to 0$, the normalized optimal values converge to those of the classical vanishing-material optimal design problem, equivalently formulated as a Lipschitz-constrained state problem or a measure-valued flux problem. For ${\theta }_{\delta }\to \theta \in (0,1)$, we obtain a directional local free-material problem governed by the upper tail of the directional $p$-strain over position-direction space. Thus the state variable localizes, while the limiting constitutive law retains an internal directional variable inherited from the bonds. If ${\theta }_{\delta }\to \theta \ge 1$, the limit is the isotropic local $p$-Dirichlet problem. A convex minimax argument eliminates the normalized conductivity variable exactly, after which the analysis combines convex duality, nonlocal compactness, and $\mathrm{\Gamma }$-convergence for convex integrands. We prove a unified $\mathrm{\Gamma }$-convergence result for the reduced state energies whenever ${\theta }_{\delta }\to \theta \in [0,+\mathrm{\infty })$, yielding convergence of normalized optimal values and compactness and identification of optimal states throughout the full parameter range. At the vanishing-material endpoint, we additionally recover optimal measure-valued local fluxes from optimal nonlocal two-point fluxes.