Topological–Historical Continuum Mechanics: A Three-Axiom Framework for Finite Deformation, Plasticity, Defects, Memory, Fracture, Contact, Waves, Structural Reduction, and Certified Computation

Abstract

We construct a unified framework for continuum mechanics from three physical axioms: microscopic ontic definiteness and finite localization, physical reality of the spacetime–vacuum substrate, and persistent causal source–response coupling. Continuum fields are defined through controlled coarse graining of finite localized carriers and their causal histories rather than introduced as primitive physical entities. The resulting theory distinguishes fundamental assumptions, derived balance laws, constitutive representation hypotheses, material realizations, numerical approximations, experimental inputs, and unresolved proof obligations. The framework develops geometric kinematics, mass and momentum balance, thermodynamic admissibility, finite elasticity, multiplicative elastoplasticity, gradient and measure-valued defects, causal memory, damage, fracture, contact, friction, adhesion, wear, solid waves, and dimensionally reduced structural models. Classical hyperelasticity, \(J_2\) plasticity, phase-field fracture, viscoelasticity, beam and plate theories, and local continuum mechanics arise only as declared material realizations or controlled limits. Weak formulations, incremental variational principles, energetic and balanced-viscosity solutions, compactness requirements, stability conditions, and failure domains are stated explicitly. Results that cannot yet be rigorously closed are retained as concrete and verifiable proof obligations. A computational certification layer accompanies the theory. Reproducible benchmarks address radial cavitation, measure-valued dislocation reactions, thermofrictional contact, causal wave scattering, structural reduction, rate-independent fracture, dynamic cohesive closure, energy–momentum integration, and two-dimensional plasticity–defect–memory coupling. Each benchmark reports governing equations, geometry, parameters, discretization, solver tolerances, conservation and dissipation residuals, refinement behavior, and machine-readable artifacts. Parameter ownership, sensitivity, uncertainty propagation, separated calibration and validation data, and noncompensatory model-rejection rules connect the mathematical theory to experiments. The resulting construction is a certificate-oriented continuum theory rather than a claim of universal numerical or empirical completion. It identifies precisely which conclusions are proved, conditionally derived, computationally verified, experimentally constrained, or still unresolved, while preserving classical continuum mechanics on its validated domain. Keywords **continuum mechanics; finite deformation; elastoplasticity; defect mechanics; causal memory; gradient plasticity; damage mechanics; fracture; contact and friction; configurational forces; energetic solutions; balanced-viscosity solutions; weak solutions; thermodynamic consistency; structure-preserving algorithms; uncertainty quantification; model validation; topological mechanics; historical disturbance field; certified computation**