Multi-material topology optimization aims to design structures composed of multiple base materials for optimal performance. However, conventional approaches based on interpolating material parameters implicitly assume that all base materials share an identical constitutive material model, including the same energy density function, inelastic evolution laws, and material parameters. As a consequence, void regions may exhibit nonlinear or inelastic responses, which is physically unacceptable, and the applicability to nonlinear materials with distinct behaviors is severely restricted. To overcome these limitations, we propose an extended discrete material optimization (XDMO) framework [1] that generalizes the original discrete material optimization (DMO) concept [2]. The central idea is to interpolate the governing equations and evolution laws, or equivalently the potential energy, that define the mechanical response of each base material, rather than interpolating material parameters themselves. This theoretical enhancement enables a unified treatment of a broad class of nonlinear and irreversible materials within multi-material topology optimization. The primal, adjoint, and sensitivity equations are derived in a general form, without assuming specific constitutive models, and the formulation is independent of temporal and spatial discretization schemes. Numerical examples, including multi-material topology optimization combining rubber-like and metal-like materials, demonstrate the versatility of the proposed XDMO framework. References [1] J. Han, K. Izui, S. Nishiwaki, Extended discrete material optimization: a generalized framework for multi-material topology optimization of nonlinear material constitutive models, Computer Methods in Applied Mechanics and Engineering (2026) 118682. [2] J. Stegmann, E. Lund, Discrete material optimization of general composite shell structures, International Journal for Numerical Methods in Engineering 62 (2005) 2009-2027.