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A third-medium approach for electro-thermo-mechanical contact considering Joule heating

Maximilian Vorwerk

arXiv:2609.08031v1math.NAcs.NA

TL;DR

Coupled electric current transfer, Joule heating, and transient heat conduction across closing interfaces have received little attention in third-medium contact. The paper presents a monolithic third-medium formulation with deformation-dependent transport switching and independent-field regularization, and its numerical examples reproduce contact-activated current and thermal response, including localized transport at rough interfaces.

  • Problem

    Coupled electric current transfer, Joule heating, and transient heat conduction across closing interfaces have received little attention within third-medium contact.

    Source: §1 · p. 2

  • Method

    A monolithic finite element formulation couples mechanical contact, electrical and thermal transport switching, Joule heating, and an independently interpolated deformation-gradient-like field for regularization.

    Source: §5 · pp. 21–22

  • Results

    Current is suppressed before contact and activates with a conducting path; Joule heating raises temperature transiently, while temperature-dependent conductivity reduces current density and rough-interface transport localizes at contact spots.

    Source: §5 · p. 22

  • Takeaways & Limitations

    The formulation captures rough-interface evolution from isolated asperity contact to an almost continuous conducting interface without explicitly tracking the evolving contact zone.

    Source: §5 · p. 22

    The transport laws are deformation-dependent third-medium constitutive models rather than direct physical-gap representations, and their activation criteria require configuration-specific calibration.

    Source: §5 · p. 22

Abstract

from arXiv · show

Connectors are ubiquitous in technical systems and frequently combine mechanical contact with electric current transfer and heat generation. To capture these interacting processes, an electro-thermo-mechanical contact formulation based on the third-medium concept is proposed. Mechanical contact, electric current flow, and heat conduction are represented within a unified finite element framework without explicit contact-surface tracking. Deformation-dependent switching functions govern the onset of electrical and thermal transport during interface closure. Electrical conduction is coupled to the transient thermal problem through Joule heating, while the temperature dependence of the electrical conductivity provides the corresponding feedback on the current flow. The coupled displacement, electric potential, and temperature fields are solved monolithically and combined with an independently interpolated deformation-gradient-like field for robust third-medium regularization under severe compression. Numerical examples demonstrate contact-induced current transfer, subsequent Joule heating and transient temperature evolution, as well as localized current paths during progressive closure of rough interfaces.

1 Introduction

Coupled electric current transfer, Joule heating, and transient heat conduction across closing interfaces have received little attention in third-medium contact. This work extends the third-medium approach to electrically conducting contact and assesses its coupled thermal response numerically.

1 Introduction

  • Conventional contact formulations rely on explicitly defined interfaces, with contact search and active-set treatment becoming involved for large sliding, self-contact, and unknown boundaries.Prior electro-thermo-mechanical approaches also rely on explicit contact interfaces.Source: §1 · p. 1
  • The work extends third-medium contact to electrical current transfer, Joule heating, and transient heat conduction across closing interfaces.Electrical activation depends on deformation, and the formulation includes temperature-dependent electrical conductivity.Source: §1 · p. 2
  • The formulation builds on a mixed low-order third-medium framework whose independently interpolated deformation-gradient-like field supports stabilization under severe compression.This approach avoids direct evaluation of second displacement derivatives [29].Source: §1 · p. 2

2 Continuum formulation of deformable conductors

The formulation couples mechanical equilibrium, electric current conservation, and transient heat transport in deformable conductors and a fictitious third-medium. Compression-dependent electrical and thermal transport, Joule heating, and mixed stabilization represent evolving contact and support the third-medium under severe compression.

2.1 Boundary value problem, kinematics and weak form of a conductor

  • The continuum problem combines quasi-static mechanical equilibrium, stationary current balance, and transient heat transport, with electrical dissipation entering as a volumetric Joule heat source.The fields are displacement, electric potential, and temperature, and the weak formulation combines their contributions.Source: §2.1 · p. 3
  • Thermal expansion is modeled through a multiplicative deformation split, using a linear isotropic thermal stretch restricted to in-plane directions for the plane-strain problems.The linear relation is intended for moderate temperature changes around the reference state.Source: §2.1 · p. 4

2.2 Behavior of the conducting contact domains

  • Conducting solids use a hyperelastic mechanical model, Fourier heat conduction, and electrical dissipation, while deformation-induced changes in scalar electrical conductivity are neglected.Deformation still affects the referential conductivity tensor through the geometric transformation.Source: §2.2 · pp. 4–6

2.3 Behavior of the current and temperature conducting third-medium

  • The fictitious third-medium acts as a contact barrier and enables electrical and thermal transport as compression evolves, using separate deformation-dependent switching functions.Its electrical conductivity inherits the solid's temperature dependence, while thermal conductivity is activated separately.Source: §2.3 · pp. 6–8
  • The third-medium has negligible mechanical stiffness before contact, while its rapidly increasing energy prevents complete collapse of the intermediate layer.The present two-dimensional plane-strain specialization retains only the isochoric energy contribution, whose energy becomes unbounded as J approaches zero.Source: §2.3 · p. 6
  • An independently interpolated deformation-gradient-like field Θ and gradient regularization improve robustness under severe compression without requiring second derivatives of displacement.The mixed formulation is intended to preserve low-order finite-element efficiency while improving robustness under extreme compression.Source: §2.3 · pp. 6–7

2.4 Switching functions within the third-medium

  • Electrical transport switches on discontinuously when J < Jcrit, whereas thermal transport increases continuously with compression through a smooth, non-negative activation law.Jcrit = 0.01 is a problem-dependent model parameter, not a universal physical gap or material parameter.Source: §2.4 · pp. 8–9, Figure 1 · p. 9

2.5 Temperature-dependent electrical conductivity

  • Solid electrical conductivity decreases with increasing temperature, and the third-medium inherits this temperature dependence while also applying deformation-dependent electrical activation.For θ ≥ θ0, the denominator lower bound is inactive and the relation reduces to the standard linear resistivity approximation.Source: §2.5 · p. 9, Figure 2 · p. 10

3 The multi-field third-medium finite element implementation

The implementation discretizes displacement, electric potential, temperature, and an independently interpolated deformation-gradient-like field, solving the coupled primary fields monolithically for a fixed electrical active set. Its transient thermal discretization and consistent tangent account for the coupled transport and Joule-heating terms.

3 The multi-field third-medium finite element implementation

  • For a fixed electrical active set, displacement, electric potential, temperature, and the independently interpolated deformation-gradient-like field Θ are solved monolithically within Newton-Raphson iterations.Electrical-state changes use an active-set-type update.Source: §3 · p. 10
  • Temperature evolution is discretized using an implicit Euler scheme, approximating the material time derivative from temperatures at successive time steps.The increment is denoted Δt.Source: §3 · p. 10
  • The consistent tangent is generally non-symmetric, with blocks representing deformation-dependent transport, temperature-dependent conductivity, and Joule-heating contributions.Direct electro-mechanical feedback on the mechanical balance is neglected, although the electric problem depends on deformation.Source: §3 · p. 11
  • Displacement, electric-potential, and temperature fields use linear T1 interpolation; Θ is independently interpolated with the same linear shape functions in the third-medium.Element contributions use six-point Gaussian quadrature.Source: §3 · pp. 10–11

4 Numerical examples

The numerical examples range from analytical verification to application-oriented contact problems, evaluating the coupled response of deformation-dependent electrical and thermal conductivity in the third-medium formulation.

4.1 Analytical verification of the electro-thermal material response

4.2 Contact between two conducting blocks

  • Current starts only when λ = 1 establishes conduction; under insulated Case 1, retained Joule heat raises temperature monotonically.Source: §4.2 · p. 15, Figure 7 · p. 16
  • Without heat removal, conductivity reduction lowers current; with heat removed through isothermal boundaries, warming remains moderate and current nearly constant.Source: Figure 8 · p. 16, §4.2 · p. 17

4.3 Contact between two conducting blocks and non-symmetric loading

4.4 Contact between blocks with microstructural surface roughness

5 Conclusion and outlook

The third-medium formulation couples mechanical contact, electrical and thermal transport, and Joule heating without explicit contact tracking. Numerical examples demonstrate the coupled response, while the transport laws require problem-specific calibration.

5 Conclusion and outlook

  • The monolithic formulation represents mechanical contact through a deformable intermediate medium and electrical and thermal transport through deformation-dependent switching functions.Joule heating and temperature-dependent electrical conductivity provide electro-thermal coupling without explicit contact detection, gap evaluation, or mechanical contact active sets.Source: §5 · p. 21
  • An independently interpolated deformation-gradient-like field, penalty coupling, and gradient regularization provide robustness under severe compression while retaining compatibility with low-order finite elements.Source: §5 · p. 22
  • The numerical examples capture current activation after contact, Joule heating, temperature-dependent feedback on current density, and progressive current spreading across rough interfaces.The predicted temperature evolution also depends decisively on thermal boundary conditions.Source: §5 · p. 22
  • The transport laws are third-medium constitutive models, not direct representations of a physical interface gap, and their activation criteria require problem-specific calibration.Future work should calibrate them against experimental electrical and thermal contact-resistance data and investigate geometrically objective interface measures.Source: §5 · p. 22
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