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arxiv: 2509.26609 · v2 · pith:5O36F66Inew · submitted 2025-09-30 · 🧮 math-ph · math.MP

Singularities at the vertex of connected angular inhomogeneities under thermal and elastic loading

Yuanpeng Yang , Huiming Yin , Chunlin Wu This is my paper

Pith reviewed 2026-05-21 21:58 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords angular inhomogeneitiesvertex singularitiesequivalent inclusion methodFredholm integral equationeigenvaluesthermal conductivity mismatchelastic stiffness mismatchopening angles
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The pith

Eigenvalues solved from a Fredholm integral equation determine the order of singularities at the vertex where multiple angular inhomogeneities meet under thermal and elastic loads.

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

The paper shows that material mismatches at connected angular regions can be replaced by distributed eigen-temperature-gradients and eigenstrains inside equivalent inclusions that share the matrix properties. These eigen-fields are written in separable form using powers of distance from the vertex and functions of the opening angles. Domain integrals of Green's functions against the eigen-fields then produce a Fredholm integral equation of the second kind whose eigenvalues fix the strength of the resulting singularities in temperature gradient and stress. The formulation automatically includes interactions among any number of regions and reduces to known single-wedge or bimaterial results when two identical regions are placed side by side.

Core claim

By modeling each angular inhomogeneity through Eshelby's equivalent inclusion method with continuously distributed eigen-temperature-gradient for conductivity mismatch and eigenstrain for stiffness mismatch, the boundary-value problem is converted into a Fredholm integral equation of the second kind. Separation of variables in polar coordinates centered at the vertex yields eigen-fields whose radial dependence is a power involving an unknown eigenvalue and the local opening angle. The eigenvalues of this integral equation supply the singularity orders for both the thermal and elastic disturbed fields, and the resulting general formulae capture the combined effects of multiple-region geometry

What carries the argument

Fredholm integral equation of the second kind obtained from domain integrals of Green's functions against the separable eigen-temperature-gradient and eigenstrain fields in the equivalent inclusion model

If this is right

  • Singularity orders depend explicitly on the opening angles of each connected region and on the ratios of thermal conductivity and elastic moduli across interfaces.
  • The same eigenvalue problem recovers the classic bimaterial wedge and infinite-domain wedge solutions when two identical inhomogeneities are placed adjacent to each other.
  • Disturbed temperature, heat flux, displacement, and stress fields are obtained directly from the domain integrals once the eigen-fields are known.
  • The method supplies closed-form dependence of singularity strength on both geometry and material contrast for any number of angular regions meeting at a point.

Where Pith is reading between the lines

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

  • The eigenvalue search could be adapted to time-harmonic or transient loading by allowing complex or frequency-dependent eigenvalues in the same integral operator.
  • The singularity orders obtained here supply the leading terms needed for enriched finite-element or boundary-element meshes at such vertices.
  • Similar integral-equation reductions may apply to other interface problems, such as piezoelectric or poroelastic angular inclusions, by replacing the eigen-fields with the appropriate coupling quantities.

Load-bearing premise

The eigen-fields inside each inclusion admit a separable power-law form in polar coordinates centered at the vertex, with the power set by the eigenvalue and the opening angle of that region.

What would settle it

Numerical solution of the derived Fredholm equation for the classic single-wedge bimaterial configuration produces an eigenvalue whose corresponding singularity order differs from the accepted analytical value in the literature.

Figures

Figures reproduced from arXiv: 2509.26609 by Chunlin Wu, Huiming Yin, Yuanpeng Yang.

Figure 1
Figure 1. Figure 1: Schematic illustration of an infinite domain [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the dominant singularity m of a triangular void versus the opening angle β 1 by Eshelby’s EIM with the classic wedge solution of the re-entrant corner. inhomogeneities 2β 1 = 2β 2 = 3 4 π, and the angle γ between their symmetric lines is 3 4 π, see [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison and verification of heat flux singularity [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison and verification of dominant complex stress singularity [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison and verification of symmetric dominant stress singularity parameter [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison and verification of anti-symmetric dominant stress singularity parameter [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Variation of heat flux singularity parameter [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Variation of heat flux singularity parameter [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Variation of the (a) symmetric (λF ) and (b) anti-symmetric (λS) stress singularity regarding a triangular inhomogeneity of bimaterial problem perpendicular to the interface. The opening angle β 1 ∈ (0, π 2 ) and β 2 = π 2 , Poisson’s ratio ν 0 = ν 2 = 0.3., shear modulus µ 1 = 0, ratio of shear modulus µ 2/µ0 = 0.2, 0.5, 2, 5, 10; (c) and (d) µ 1/µ0 = 1 2 , five ratios of shear modulus µ 2/µ0 = 0.2, 0.5, … view at source ↗
Figure 10
Figure 10. Figure 10: Variation of the (a) first (λF ) and (b) second (λS) stress singularity regarding a triangular inhomogeneity of bimaterial problem oriented parallel to the interface. The opening angle β 1 ∈ (0, π 2 ) and β 2 = π 2 , with Poisson’s ratio ν 0 = ν 1 = ν 2 = 0.3., µ 1/µ0 = 1 2 , five ratios of shear modulus µ 2/µ0 = 0.2, 0.5, 2, 5, 10. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Schematic plot of a triangular subdomain [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
read the original abstract

This paper investigates the singularities at the vertex of multiply connected angular inhomogeneities for heat conduction and elastic deformation. With the aid of Eshelby's equivalent inclusion method (EIM), each inhomogeneity is simulated as an equivalent inclusion, exhibiting the same material properties as the matrix but containing a continuously distributed eigen-field with potential singularities at the vertices and edge lines. Specifically, the eigen-temperature-gradient (ETG) and eigenstrain are utilized to simulate material mismatch of thermal conductivity and stiffness, respectively. Using the separation of variables, the eigen-fields can be formulated in terms of distance to vertices and opening angles, and disturbed thermal/elastic fields are evaluated by domain integrals of Green's function multiplied by eigen-fields, which form Fredholm's integral equation of the second kind. The boundary value problem is reduced to solve for eigenvalues, which are used to determine the order of singularity. The present solution is versatile - by placing two identical inhomogeneities together, it recovers the classic solutions for a single wedge in a bimaterial media or infinite domain. The general and analytical formulae take full consideration of interactions of multiple inhomogeneities and reveal the effects of opening angles and material properties on the thermal and elastic singularities.

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

1 major / 2 minor

Summary. The paper investigates singularities at the vertex of multiply connected angular inhomogeneities for heat conduction and elastic deformation. Using Eshelby's equivalent inclusion method (EIM), each inhomogeneity is modeled as an equivalent inclusion with continuously distributed eigen-temperature-gradient and eigenstrain fields. These eigen-fields are formulated via separation of variables in polar coordinates centered at the vertex, expressed in terms of radial distance raised to an eigenvalue power and functions of the opening angles. Disturbed fields are evaluated through domain integrals of Green's functions, yielding Fredholm integral equations of the second kind. The problem reduces to an eigenvalue problem whose solutions determine the singularity orders. The approach accounts for interactions among multiple inhomogeneities and recovers classic bimaterial wedge solutions when two identical inhomogeneities are combined.

Significance. If the separability assumption for the eigen-fields holds and the resulting eigenvalues accurately capture the singularity orders, the work supplies a general analytical framework for thermal and elastic singularities in complex angular multi-inhomogeneity geometries. It systematically incorporates geometric and material effects as well as inter-inhomogeneity interactions, which may prove useful for analyzing stress and temperature concentrations in composites containing wedge-like or polygonal inclusions. The reduction of the EIM to a Fredholm integral eigenvalue problem is a methodological strength that enables treatment of multiply connected domains without ad-hoc fitting parameters.

major comments (1)
  1. [Eigen-field formulation] Eigen-field formulation (as described in the abstract and the modeling section): the separable ansatz for the eigen-temperature-gradient and eigenstrain fields, written as r^λ f(θ; opening angles), is introduced by direct appeal to separation of variables. In multiply connected angular domains the EIM interface conditions couple the fields across several rays; the manuscript does not demonstrate that a single power-law radial dependence simultaneously satisfies continuity of temperature, heat flux, displacement and traction for all eigen-components. If non-separable terms are required, the extracted eigenvalues would not correspond to the true singularity orders.
minor comments (2)
  1. A brief numerical example or table comparing the recovered eigenvalues for the bimaterial wedge case against known analytic results would strengthen the versatility claim.
  2. Clarification on the discretization scheme used to solve the Fredholm integral equation of the second kind and on the root-finding procedure for the eigenvalues would improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major concern regarding the eigen-field formulation below, offering clarifications on the separability assumption and its consistency with the equivalent inclusion method.

read point-by-point responses

  1. Referee: [Eigen-field formulation] Eigen-field formulation (as described in the abstract and the modeling section): the separable ansatz for the eigen-temperature-gradient and eigenstrain fields, written as r^λ f(θ; opening angles), is introduced by direct appeal to separation of variables. In multiply connected angular domains the EIM interface conditions couple the fields across several rays; the manuscript does not demonstrate that a single power-law radial dependence simultaneously satisfies continuity of temperature, heat flux, displacement and traction for all eigen-components. If non-separable terms are required, the extracted eigenvalues would not correspond to the true singularity orders.

    Authors: We appreciate this observation on the eigen-field ansatz. The power-law form r^λ f(θ; opening angles) is introduced as the leading singular term in the asymptotic expansion near the vertex, following the standard separation-of-variables approach for local singularity analysis in angular domains. Although the overall geometry is multiply connected, the local polar coordinate system centered at the vertex permits this separable representation for the eigen-temperature-gradient and eigenstrain fields. The interface conditions (continuity of temperature, heat flux, displacement, and traction) across all rays are enforced through the Fredholm integral equations of the second kind, which arise from the domain integrals involving the Green's functions and the assumed eigen-fields. The eigenvalue problem is formulated precisely so that non-trivial solutions exist only for specific λ that satisfy these integral equations, thereby ensuring the continuity requirements are met for the chosen eigen-components. This is corroborated by the recovery of classical bimaterial wedge solutions upon combining two identical inhomogeneities. We will revise the manuscript to include an expanded paragraph in the modeling section that explicitly outlines how the integral formulation guarantees satisfaction of the interface conditions for the separable ansatz. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses standard EIM and separation ansatz to obtain independent eigenvalue problem

full rationale

The paper models inhomogeneities via EIM with eigen-temperature-gradient and eigenstrain fields, assumes the standard separable power-law form r^λ f(θ) via separation of variables, evaluates disturbed fields through Green's function integrals to obtain a Fredholm integral equation of the second kind, and solves the resulting eigenvalue problem for singularity orders. This chain is self-contained against external benchmarks: it recovers known wedge and bimaterial solutions by construction of the setup rather than by redefining inputs, relies on classical Green's functions and Eshelby equivalence without load-bearing self-citations, and does not fit parameters to the target singularity orders. No step reduces the claimed result to its own inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the applicability of separation of variables to the eigen-fields and on the validity of representing material mismatch through continuously distributed eigen-temperature-gradient and eigenstrain fields inside equivalent inclusions.

axioms (1)
  • domain assumption Separation of variables can be applied to express the eigen-fields in terms of radial distance to the vertex and the opening angles of the angular regions.
    Invoked to formulate the eigen-temperature-gradient and eigenstrain fields prior to constructing the integral equation.

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