Cumulative Mass Migration An Eulerian Kinematic Foundation for Continuum Solid Mechanics and Its Specification to Small-deformation Isotropic Linear Elasticity

In continuum mechanics, a fundamental dichotomy has long persisted between the descriptive paradigms for fluids and solids: the Eulerian description, tracking physical quantities at fixed spatial points, and the Lagrangian description, following individual material particles. Although well-established for classical problems, the latter faces inherent challenges—namely, mesh distortion and complex interface tracking—in large deformations and fluid-structure interaction. Guided by the rational mechanics pursuit of a mathematically self-consistent and formally unified field theory, this paper introduces the concept of "cumulative mass migration" (CMM) as a fundamental kinematic variable. This variable inherently merges the field characteristics of the Eulerian description with the history-dependent nature of solid deformation, thereby reinterpreting deformation as the transport and redistribution of mass within a fixed space. It is demonstrated that CMM field, defined in a fixed spatial domain, completely encodes the entire deformation history of material points, thereby emancipating the description from reliance on material coordinates. The paper then outlines a descriptive approach parallel to the Lagrangian paradigm yet centered on field variables in a fixed space. For small deformation and isotropic linear elasticity, a complete theoretical system—comprising geometric equations, constitutive relations, and dynamic equations—is constructed, and its physical isomorphism to the Lagrangian framework is established. Finally, the feasibility of introducing internal variables to describe deformation history under finite deformation is discussed, and the potential of this variable system for a unified description of solids and fluids is revealed—demonstrating a mathematical isomorphism between solids and fluids under CMM description.