Volumetric Growth in Linear Elasticity Driven by an Optimality Criterion

TL;DR

This paper presents a novel framework for modeling volumetric growth in linear elasticity as an optimization process, where the growth tensor is implicitly determined by constrained minimization rather than phenomenological laws.

Contribution

It introduces an optimization-driven formulation of volumetric growth within linear elasticity, incorporating equilibrium, mass-balance, and irreversibility constraints, with finite element discretization for numerical implementation.

Findings

Finite element discretization results in a finite-dimensional constrained minimization problem.

The evolution of growth can be interpreted as a projected gradient flow.

Numerical examples demonstrate the effectiveness of the proposed framework.

Abstract

Using linearized elasticity as a convenient mechanical framework, we show that volumetric growth can be formulated as an optimization-driven process in which the growth tensor is determined implicitly by constrained optimization rather than prescribed through phenomenological evolution laws. At each incremental step, the displacement and growth fields satisfy equilibrium, mass-balance constraints, and an irreversibility condition enforcing accretive growth, while an objective functional encodes the driving mechanism of the process. Finite element discretization leads to a finite-dimensional constrained minimization problem in the growth variables alone and makes explicit the interpretation of the evolution as a projected gradient flow. Numerical examples illustrate the proposed framework.