A 4D geometrical modeling of a material aging

TL;DR

This paper models material aging as the evolution of a 4D Riemannian metric in a material space-time, deriving equations from variational principles and illustrating with examples like creep and chemical degradation.

Contribution

It introduces a novel geometric framework for modeling material aging using a 4D Riemannian metric and derives the governing equations via a variational approach.

Findings

Derived a coupled system of elastostatic and aging equations.

Applied the model to examples like stress relaxation and creep.

Provided a new geometric perspective on material degradation processes.

Abstract

4-dim intrinsic (material) Riemannian metric $G$ of the material 4-D space-time continuum $P$ is utilized as the characteristic of the aging processes developing in the material. Manifested through variation of basic material characteristics such as density, moduli of elasticity, yield stress, strength, and toughness., the aging process is modeled as the evolution of the metric $G$ (most importantly of its time component $G_{00}$) of the material space-time $P$ embedded into 4-D Newtonian space-time with Euclidean metric.\par The evolutional equation for metric $G$ is derived by the classical variational approach. Construction of a Lagrangian for an aging elastic media and the derivation of a system of coupled elastostatic and aging equations constitute the central part of the work. The external and internal balance laws associated with symmetries of material and physical space-time geometries are briefly reviewed from a new viewpoint presented in the paper. Examples of the stress relaxation and creep of a homogeneous rod, cold drawing, and chemical degradation in a tubing are discussed.