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arxiv: 2606.17774 · v1 · pith:JAM64UWWnew · submitted 2026-06-16 · 🧮 math-ph · cond-mat.mtrl-sci· math.MP

When Volumetric Growth Selects Surface Growth

Rohn Abeyaratne , Roberto Paroni , Marco Picchi Scardaoni This is my paper

Pith reviewed 2026-06-26 22:49 UTC · model grok-4.3

classification 🧮 math-ph cond-mat.mtrl-scimath.MP
keywords growthelasticityoptimizationsurface growthvolumetric growthvariational methods
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The pith

Optimal growth in elastic solids concentrates on surfaces to minimize external work

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates the relationship between volumetric and surface growth in an optimization-driven framework for linearly elastic solids. Growth is not prescribed by an evolution law but found by solving a constrained optimization problem that minimizes the work done by external loads. In one-dimensional and axisymmetric settings, the optimal solutions are singular and concentrate on boundaries or internal interfaces, even though the model uses a distributed growth strain tensor. This provides a variational mechanism for surface growth to be selected as the optimal form of volumetric growth.

Core claim

Although growth is initially formulated as a volumetric process through a distributed growth strain tensor, the optimal growth distributions that minimize the work performed by external loads are singular and concentrate on boundaries or internal interfaces. This is shown through explicit analytical solutions in one-dimensional and axisymmetric settings.

What carries the argument

Constrained optimization problem that determines the growth strain tensor by minimizing the work of external loads

If this is right

  • Surface growth emerges as the optimal realization of volumetric growth under the minimization principle
  • Optimal growth distributions are singular measures supported on boundaries or interfaces
  • Explicit analytical solutions for growth can be derived in one-dimensional and axisymmetric geometries
  • The framework identifies conditions where volumetric growth reduces to surface growth

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The concentration result may extend to three-dimensional geometries
  • This variational selection could connect to observed surface accretion in biological growth
  • Controlled load experiments might verify whether growth localizes at interfaces
  • Similar optimization could apply to other inelastic processes such as plasticity

Load-bearing premise

Growth is determined as the solution to a static optimization problem minimizing external work rather than by a prescribed evolution law

What would settle it

A calculation or experiment demonstrating that a non-singular volumetric growth distribution achieves lower external work than any surface-concentrated distribution would falsify the claim

read the original abstract

We investigate the relationship between volumetric and surface growth within a recently proposed optimization-driven framework for linearly elastic solids. In this approach, growth is not prescribed through an evolution law; instead, the growth distribution is determined as the solution of a constrained optimization problem. Focusing on processes driven by the minimization of the work performed by external loads in one-dimensional and axisymmetric settings, we derive explicit analytical solutions for the resulting growth distributions. Although growth is initially formulated as a volumetric process through a distributed growth strain tensor, we show that the optimal growth distributions are singular and concentrate on boundaries or internal interfaces. These results provide a variational mechanism through which, under certain conditions, surface growth is selected as the optimal realization of volumetric growth.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates the relationship between volumetric and surface growth in linearly elastic solids within an optimization-driven framework. Growth is formulated as a distributed volumetric strain tensor but is determined by solving a constrained optimization problem that minimizes the work performed by external loads. Explicit analytical solutions are derived in one-dimensional and axisymmetric settings, showing that the optimal growth distributions are singular measures concentrating on boundaries or internal interfaces. This supplies a variational mechanism by which surface growth can emerge as the optimal realization of volumetric growth under the stated conditions.

Significance. If the derivations hold, the result supplies a concrete variational principle explaining the selection of surface growth from an initially volumetric formulation, which is of interest in continuum mechanics of growth. The provision of explicit analytical solutions in elementary geometries is a strength, as it permits direct verification and may serve as a benchmark for more complex models.

minor comments (3)
  1. The abstract states that solutions are 'explicit analytical' but does not indicate the precise functional form of the growth strain or the admissible function space; adding one sentence on the regularity class would improve clarity for readers.
  2. In the axisymmetric case, the transition from distributed to singular growth should be illustrated with a brief remark on how the Euler-Lagrange equation is satisfied in the distributional sense.
  3. A short comparison paragraph with existing surface-growth models (e.g., those based on surface accretion) would help situate the variational selection mechanism.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, the assessment of significance, and the recommendation of minor revision. No major comments appear in the report, so there are no specific points requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper formulates growth via a constrained optimization problem minimizing external work (no evolution law), then derives explicit analytical solutions in 1D and axisymmetric settings showing that minimizers concentrate as singular measures on boundaries or interfaces. This outcome follows directly from the variational problem and its solutions rather than from any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The 'recently proposed framework' is referenced but the central results consist of independent analytical derivations within the stated setting; no quoted step reduces the claimed selection mechanism to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only access prevents exhaustive identification of free parameters or invented entities; the central claim rests on the optimization framework and linear elasticity assumptions whose details are not supplied.

axioms (2)
  • domain assumption The growth distribution is obtained by solving a constrained optimization problem that minimizes work by external loads.
    Stated directly in the abstract as the modeling choice replacing evolution laws.
  • domain assumption Linear elasticity governs the solid under growth.
    The framework is specified for linearly elastic solids.

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discussion (0)

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Reference graph

Works this paper leans on

21 extracted references · 1 linked inside Pith

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