[Paper Review] Evolution of Continuum from Elastic Deformation to Flow

Jianhua Xiao|ArXiv.org|Nov 19, 2005

Elasticity and Material Modeling7 references5 citations

TL;DR

This paper proposes a unified formulation linking elastic deformation to flow in continuum mechanics by integrating Chen's additive decomposition of the deformation gradient with Eringen and Truesdell polar decompositions. It derives explicit expressions for strain from both additive and multiplicative decompositions, reveals plastic deformation as irreversible local rotation, and shows that dynamic deformation requires non-symmetric stress while static deformation requires symmetric stress—resolving the paradox of cracking and buckling. The framework enables a continuous evolution theory from elasticity to flow.

ABSTRACT

Traditionally, the deformation of continuum is divided into elastic, plastic, and flow. For a large deformation with cracking, they are combined together. So, for complicated deformation, a formulation to express the evolution of deformation from elastic to flow will help to understand the intrinsic relation among the related parameters which relate the deformation with a stress field. To this purpose, Eringen polar decomposition and Trusedell polar decomposition are formulated by explicit formulation of displacement field, based on Chen additive decomposition of deformation gradient. Then the strain introduced by the multiplicative decomposition and the strain introduced by the additive decomposition are formulated explicitly with displacement gradient. This formulation clears the intrinsic contents of strains defined by taking the Eringen polar decomposition and Trusedell polar decomposition. After that, it shows that the plastic deformation can be expressed as the irreversible local average rotation. For initial isotropic simple elastic material, the path-dependent feature of classical plasticity theory is naturally expressed in Chen strain definition. It is founded that for initially isotropic material the motion equations require a non-symmetric stress for dynamic deformation and a symmetric stress for static deformation. This controversy between dynamic deformation and static deformation can be used to explain the cracking or buckling of solid continuum. Finally, the research shows that the flow motion of continuum can be expressed by the same formulation system. So, it forms an evolution theory from elastic deformation to flow of continuum.

Motivation & Objective

• To establish a continuous theoretical framework describing the transition from elastic deformation to flow in solids.
• To clarify the intrinsic physical meaning of strains defined via Eringen and Truesdell polar decompositions.
• To express plastic deformation as irreversible local average rotation within a consistent kinematic framework.
• To resolve the contradiction between symmetric and non-symmetric stress requirements in static and dynamic deformations.
• To unify the description of elastic, plastic, and flow behaviors under a single formulation system.

Proposed method

• Utilizes Chen's additive decomposition of the deformation gradient to separate elastic and plastic parts.
• Applies Eringen and Truesdell polar decompositions to the deformation gradient with explicit dependence on displacement fields.
• Derives explicit strain expressions from both additive and multiplicative decompositions using displacement gradients.
• Introduces the concept of irreversible local average rotation to represent plastic deformation.
• Analyzes motion equations under static and dynamic loading conditions to identify stress symmetry requirements.
• Extends the formulation to describe flow motion, completing the evolution from elasticity to flow.

Experimental results

Research questions

• RQ1 How can the transition from elastic deformation to flow be described using a unified kinematic framework?
• RQ2 What is the physical interpretation of strain in Eringen and Truesdell polar decompositions when expressed via displacement gradients?
• RQ3 How is plastic deformation mathematically represented as irreversible local average rotation in this formulation?
• RQ4 Why does dynamic deformation require a non-symmetric stress tensor while static deformation requires a symmetric one?
• RQ5 Can the same mathematical framework describe both elastic and flow regimes in continuum mechanics?

Key findings

• The strain defined via Chen's additive decomposition naturally captures the path-dependent behavior of classical plasticity in initially isotropic materials.
• Plastic deformation is rigorously expressed as irreversible local average rotation, providing a kinematic basis for plasticity.
• Dynamic deformation necessitates a non-symmetric stress tensor due to rotational inertia effects, while static deformation requires a symmetric stress tensor.
• The discrepancy between symmetric and non-symmetric stress requirements explains the onset of cracking or buckling in solid continua.
• The same formulation system successfully describes flow motion, completing the evolution theory from elasticity to flow.
• Explicit expressions for strain from both additive and multiplicative decompositions clarify their intrinsic physical content.