A generalized finite-volume theory is proposed for two-dimensional elasticity problems on rectangular domains. The generalization is based on a higher-order displacement field representation within individual subvolumes of a discretized analysis domain, in contrast with the second-order expansion employed in our standard theory. The higher-order displacement field is expressed in terms of elasticity-based surface-averaged kinematic variables, which are subsequently related to corresponding static variables through a local stiffness matrix derived in closed form. The novel manner of defining the surface-averaged kinematic and static variables is a key feature of the generalized finite-volume theory, which provides opportunities for further exploration. Satisfaction of subvolume equilibrium equations in an integral sense, a defining feature of finite-volume theories, provides the required additional equations for the local stiffness matrix construction. The theory is constructed in a manner which enables systematic specialization through
reductions to lower-order versions. Part I presents the theoretical framework. Comparison of predictions by the generalized theory with its predecessor, analytical and finite-element results in Part II illustrates substantial improvement in the satisfaction of interfacial continuity conditions at adjacent subvolume faces, producing smoother stress distributions and good interfacial conformability.