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Mixed-Dimensional Electromechanical Coupling of Embedded Beam Networks in Soft Dielectric Composites

L. River Spencer, Manuel K. Rausch, Chad Landis, Jan N. Fuhg

arXiv:2609.35584v1cs.CE

TL;DR

Explicitly resolving dense electroactive fiber networks in three dimensions is costly, although network architecture affects coupled mechanical and electrical response. This paper embeds geometrically exact beams in a dielectric matrix and couples their dielectric response to sampled matrix fields; homogenized examples show architecture-dependent responses.

  • Problem

    Explicit three-dimensional modeling of dense electroactive fiber networks is computationally costly, while studying their coupled responses requires retaining network architecture and matrix interaction.

    Source: §1 · p. 2

  • Method

    Geometrically exact embedded beams are mechanically coupled to a dielectric matrix and electrically driven by its field sampled along beam centerlines, within periodic homogenization.

    Source: §1 · p. 3, §6 · p. 26

  • Results

    Mechanical and electrical responses vary with network architecture; irregular-network normalized sensing changes are approximately 9.6–9.8% versus 10% for the homogeneous ideal matrix under the same fixed-bias isochoric loading.

    Source: §5 · pp. 24, 26

  • Takeaways & Limitations

    The examples support treating fiber-network architecture as a material-design variable for effective electromechanical response.

    Source: §1 · p. 2, §5 · p. 24

    The beam coupling does not resolve finite-radius electric-field variation around individual fibers or an explicit dielectric matrix–fiber interface.

    Source: §5 · p. 26

Abstract

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Soft electroactive composites can exhibit strongly architecture-dependent mechanical and electrical responses, but explicitly resolving dense fiber networks in three dimensions is computationally expensive. We develop a mixed-dimensional finite-element formulation in which electroactive fibers are represented by geometrically exact beams embedded in a deformable dielectric matrix. Unlike existing electroactive beam formulations in which the electric potential is defined directly on the beam, the embedded fibers here are driven by the three-dimensional electric field of the surrounding matrix. The matrix field is sampled along the beam centerlines and enters the beam dielectric enthalpy directly, providing two-way electromechanical coupling without introducing independent electric-potential degrees of freedom on the beams. Beam and matrix mechanics are coupled through projected mortar constraints, and the electromechanical problem is solved monolithically. We use the formulation as the microscale model in a periodic homogenization framework to compute effective stress and electric displacement. Verification studies quantify discretization and field-sampling sensitivity, followed by structured and irregular network examples that demonstrate architecture-dependent mechanical and electrical responses without body-fitted three-dimensional fiber meshes.

1 Introduction

Dense electroactive fiber networks are costly to resolve with body-fitted three-dimensional meshes, yet their architecture affects coupled mechanical and electrical response. This work embeds electroactive beams in a dielectric matrix and couples the matrix field to the beams for periodic homogenization.

1 Introduction

  • Resolving slender fibers, interfaces, and junctions in three-dimensional meshes becomes costly for dense networks with small radii and large aspect ratios.The modeling challenge is to retain explicit network architecture and its coupled interaction with the matrix without resolving full three-dimensional fiber geometry.Source: §1 · p. 2
  • Embedded electroactive beams use the surrounding matrix electric field, sampled along their centerlines, in their dielectric enthalpy, enabling two-way coupling without independent beam potentials.This differs from beam formulations with electric-potential degrees of freedom on the beam.Source: §1 · p. 3
  • The formulation serves as a microscale model in periodic homogenization, where equilibrated RVEs provide homogenized stress and electric displacement.Periodic displacement and potential fluctuations allow local fields to respond to explicit fiber architecture.Source: §1 · p. 3
  • The study assesses single-fiber discretization and field-sampling sensitivity, then examines architecture-dependent responses in structured and irregular networks.The examples demonstrate use with increasingly complex geometries without body-fitted three-dimensional fiber meshes.Source: §1 · p. 3

2 Methods

The methods couple a three-dimensional dielectric matrix to a one-dimensional electroactive beam network and specialize this mixed-dimensional model for periodic homogenization. Matrix and beam fields have distinct kinematics and energy descriptions, with the matrix electric field driving the beam dielectric response.

2 Methods

  • The mixed-dimensional formulation couples a three-dimensional dielectric matrix to a nonmatching one-dimensional beam network using projected mortar mechanics and matrix-field-driven beam dielectric response.The formulation is later specialized to periodic homogenization.Source: §2 · p. 3

2.1 Problem Setting and Continuum Fields

  • Matrix displacement and potential fields are defined on the matrix domain, while beams use centerline displacement and rotation without an independent electric-potential field.A void is an excluded insulating region with traction-free and electrically insulating internal boundary conditions; the matrix referential electric field is the negative potential gradient.Source: §2.1 · p. 3

2.2 Matrix Electromechanical Model

  • The nearly incompressible isotropic electroelastic matrix combines an isochoric response with J-bar volumetric energy and a fixed-potential ideal-dielectric enthalpy.The element-averaged dilatation enters the volumetric energy, while deformation and electric field contribute to the matrix response.Source: §2.2 · p. 4

2.3 Electroactive Beam Kinematics and Section Energy

  • Geometrically exact centerline–rotation beams use incompressible equal-transverse-stretch section kinematics and integrate reduced ideal-dielectric enthalpy over the cross-section.The section reduction does not represent independent cross-section warping or transverse expansion modes; the circular section integral uses a 36-point rule exact for Cartesian polynomial moments through degree ten.Source: §2.3 · pp. 5–6

2.4 Mixed-Dimensional Beam–Matrix Coupling

This section defines an additive embedded beam–matrix formulation in which the matrix electric field drives beam dielectric energy and projected mortar constraints couple the mechanics, and extends the model to periodic RVE calculations. It states a discrete electromechanical Hill–Mandel condition relating macroscopic virtual work to the complete microscopic cell-functional change per reference-cell volume, including coupling and embedded-beam electrical work in the effective response.

2.4 Mixed-Dimensional Beam–Matrix Coupling

  • Beam energy is superposed on the matrix rather than replacing matrix volume, so the formulation represents added reinforcement, not disjoint matrix and fiber phases.Even with identical constitutive coefficients, the additive model does not recover a homogeneous-material response.Source: §2.4 · p. 6

2.4.1 Matrix-to-Beam Electric Coupling

  • Unlike beam-potential formulations [22], the beam dielectric enthalpy uses the matrix electric field sampled at the centerline and transformed into the fixed reference beam basis.The centerline sample is used across each associated cross-section without finite-radius averaging; element-interface gradients are not averaged, making the field a finite-resolution sample whose mesh and placement sensitivity is assessed separately. Its dependence on matrix potential and beam kinematics contributes to both matrix electrical and beam mechanical equilibrium.Source: §2.4.1 · pp. 6–7

2.4.2 Embedded Beam–Matrix Mechanical Coupling

  • Projected mortar operators enforce beam–matrix displacement compatibility weakly without requiring coincident nodes; exact multipliers suit smaller structured cases, while penalties reduce system size for large irregular networks.The penalty form permits a small finite compatibility gap, whereas the exact form enforces the projected constraint.Source: §2.4.2 · pp. 7–8

2.5 Finite-Element Discretization and Nonlinear Solution

  • Quadratic tetrahedral matrix fields and linearly interpolated beam kinematics are assembled with projected coupling terms into a monolithic residual and tangent.Structured center-cross junctions share translations and rotations, while Voronoi branches share translations but retain independent rotations; no rotational beam–matrix mortar constraint is imposed.Source: §2.5 · p. 8

2.6 Periodic Computational Homogenization

  • Periodic RVE calculations prescribe the macroscopic deformation gradient and referential electric field, then solve the microscopic electromechanical problem under periodic constraints.The microscopic fields include matrix fluctuations and beam displacement and rotation; the stationary cell response defines homogenized stress and electric displacement.Source: §2.6 · p. 9

2.6.1 Periodic Boundary Conditions and Load Control

  • Matrix displacement and potential use affine-plus-periodic decompositions, while beams require no separate electrical periodicity because they have no independent electric-potential field.The full-cell average-field identities are consistency checks for unperforated cells and do not apply directly to perforated matrix domains; constant-bias sensing holds the electric field from the initial state during mechanical loading.Source: §2.6.1 · pp. 9–10

2.6.2 Effective RVE Stress and Electric Displacement

  • Effective first Piola–Kirchhoff stress and referential electric displacement are energetic derivatives of the full equilibrium cell functional, including mortar work and matrix-to-beam electric enthalpy.Matrix- and beam-only stress measures are diagnostics, not substitutes for the energetic effective stress.Source: §2.6.2 · p. 11

2.6.3 Macro-Homogeneity

  • Because the microscopic state is stationary, the effective quantities satisfy a discrete macro-homogeneity condition.The state comprises primal unknowns in penalty mode and primal unknowns plus multipliers in exact-mortar mode.Source: §2.6.3 · p. 12
  • The discrete electromechanical Hill–Mandel condition equates macroscopic virtual work with the complete microscopic cell-functional change per reference-cell volume.Source: §2.6.3 · p. 12
  • In exact-mortar mode, the multiplier term contributes to macroscopic mechanical work because its deformation variation need not vanish even when g = 0.Source: §2.6.3 · p. 12
  • The identity includes embedded-beam electrical work through the matrix-to-beam field dependence, accounting for matrix, beam, electrical, and coupling contributions.Source: §2.6.3 · p. 12

2.7 Derived Quantities and Electrical Response Measures

This section defines the derived measures used to assess matrix deformation and report effective mechanical and electrical responses. It distinguishes primary RVE outputs from quantities that require additional postprocessing assumptions.

Matrix deformation measures

  • Matrix deformation is assessed using averages over the enclosing reference-cell volume and the spatial distribution of Jm, including its extrema and deviations from unity.For an unperforated cell with compatible periodic kinematics, Fm = Fmacro; the average Jacobian is a global volume-change diagnostic, not evidence of local incompressibility.Source: §2.7 · p. 12

Mechanical response

  • The primary mechanical output is the effective Cauchy stress σRVE; the effective second Piola–Kirchhoff stress SRVE is available when required.SRVE is obtained from the energetic RVE stress through the stated stress-measure conversion.Source: §2.7 · p. 12

Electrical response

  • The primary electrical output is the homogenized referential electric displacement DRVE, reported by components or magnitude depending on the loading configuration.The magnitude is formed from its three components.Source: §2.7 · p. 12

Fixed-bias response and postprocessing

  • Under fixed referential bias, the electrical measures describe response changes during mechanical deformation; electrode-based charge, capacitance, or voltage estimates are separate postprocessing.That conversion requires assumptions about electrode orientation, fringing, leakage, and mechanical feedback during voltage readjustment.Source: §2.7 · p. 13

3 Numerical Verification and Model Assessment

The verification studies test discretization choices and matrix-to-beam electric-field sampling before comparing network architectures. They assess mechanical and electrical response sensitivity to mesh resolution, interpolation, volumetric treatment, and beam–matrix mesh placement.

3 Numerical Verification and Model Assessment

  • The verification sequence uses a straight fiber for resolution sensitivity, then a curved three-directional network to test interpolation order and volumetric treatment under nonaffine deformation.The curved network's central junction and fiber curvature generate spatially varying matrix deformation under homogeneous macroscopic loading.Source: §3 · p. 13, §3.3 · p. 14

3.1 Material Parameters and Numerical Baseline

  • The baseline uses a beam-to-matrix stiffness ratio of approximately 16.7 and matched dielectric coefficients; quadratic tetrahedra and J-bar are used for the matrix.Exact translational mortar constraints are used for single-fiber and structured-network calculations, while larger irregular networks use penalty coupling.Source: §3.1 · p. 13

3.2 Single-Fiber Discretization Sensitivity

  • Mechanical stress differs from the affine analytical response by at most 4.22 × 10−10; for hm ≥0.16 m, electrical displacement deviates no more than 0.100% from the median.The structured examples use hm = 0.2 m and eight beam elements per straight unit-length fiber as the baseline; curved centerlines are discretized more finely.Source: §3.2 · p. 14

3.3 Matrix Discretization and Volumetric Treatment

  • Quadratic interpolation with J-bar averaging gives 2.48%–2.63% coarse-to-fine stress change, versus 5.36%–5.78% for linear discretization across the tested bulk-modulus range.Without J-bar averaging, the quadratic results range from 2.76% to 3.27%; quadratic interpolation with J-bar averaging is retained for subsequent coupled calculations.Source: §3.3 · p. 15, Figure 4 · p. 16

3.4 Matrix Electric-Field Sampling Sensitivity

  • The electric-field sampling tests separately examine matrix refinement and the position of the beam network relative to the matrix mesh.Both tests assess sensitivity of the homogenized response to finite-resolution sampling in the matrix-to-beam electrical coupling.Source: §3.4 · p. 15

3.4.1 Matrix-Mesh Resolution

  • For h = 0.20 m and 0.15 m, homogenized electric displacement differs from the hf = 0.10 m result by 0.571% and 0.415%, respectively.The curved-cross test uses combined mechanical loading and fixed transverse bias, with only the matrix mesh refined.Source: §3.4.1 · pp. 15–16

3.4.2 Relative Beam–Matrix-Mesh Placement

  • Across tested mesh offsets, the homogenized stress tensor changes by at most 0.069% and the electric-displacement vector by 0.120% relative to the unshifted configuration.The physical network and numerical settings are held fixed while the matrix mesh is translated.Source: §3.4.2 · p. 16

4 Architecture-Dependent Electromechanical Response

The section compares homogenized mechanical and fixed-bias electrical responses across aligned, structured, and irregular fiber architectures. Response patterns vary with loading direction and network organization, with preferred in-plane orientation remaining visible in homogenized response.

4.1 Aligned Fiber Network

  • X-extension gives σ11,RVE = 27.72 kPa versus y-extension σ22,RVE = 4.012 kPa; at F12 = 0.25, shear matches the matrix-only response at reported precision.Fibers carry the imposed stretch under x-loading, whereas transverse loading is primarily matrix dominated.Source: §4.1 · p. 17, Figure 6 · p. 18, Figure 7 · p. 18
  • Under x-extension, E1 bias separates from transverse biases, while y-extension changes response sign and magnitude, so the signal depends on both loading and bias directions.Figure 8 plots relative change in |DRVE| from the initially biased state.Source: Figure 8 · p. 19, §4.1 · p. 19

4.2 Curved and Multidirectional Structured Networks

  • X-extension stresses are 4.803, 4.358, and 13.14 kPa, while the three networks' shear stresses cluster near 5.1 kPa, indicating stronger architecture contrast in extension.The x-extension values correspond to the straight cross, curved cross, and wavy aligned network, respectively.Source: §4.2 · pp. 19–20, Figure 10 · p. 20, Figure 11 · p. 21
  • Curved and straight center crosses have the largest relative electrical changes, respectively, while the wavy aligned network has the smallest despite the largest absolute electric-displacement magnitude.The comparison uses the same mechanical loading and electric bias for all three networks.Source: §4.2 · p. 20, Figure 12 · p. 21

4.3.1 Network Architectures

  • Irregular examples span nominally isotropic, x–y-plane-preferred, and perforated Voronoi networks; the perforated case adds a central insulating void.These comparisons characterize complete network architectures with distributed orientations, branches, and junctions.Source: §4.3 · p. 22, Figure 14 · p. 23

4.3.2 Directional Mechanical Response

  • Loaded normal stresses are 4.699, 4.702, and 3.999 kPa for preferred-network x-, y-, and z-extension; isotropic and perforated realizations vary by less than 0.5% and 1.1%, respectively.The preferred-direction network's out-of-plane response is approximately 15% lower than its in-plane responses.Source: §4.3.2 · p. 23

4.3.3 Fixed-Bias Electrical Response

  • At F11 = 1.1, relative changes in |DRVE| are approximately 9.648%, 9.752%, and 9.611% for nominally isotropic, preferred-direction, and perforated networks, respectively.The preferred-direction network has the largest change and the perforated network the smallest over this loading path.Source: §4.3.3 · p. 24

5 Discussion

Network architecture shapes homogenized mechanical and electrical responses, with the degree of directional variation depending on loading and network type. The discussion also identifies modeling and scope limitations, including unresolved finite-radius electrical fields and the need for device specifications to assess sensing.

5 Discussion

  • Aligned-fiber responses depend strongly on fiber orientation relative to loading, and structured-network differences between architectures are greater under extension than under shear.In irregular networks, the in-plane-biased architecture distinguishes in-plane from out-of-plane loading, while other realizations show comparatively little directional variation.Source: §5 · pp. 24, 26
  • Under fixed-bias isochoric loading, irregular-network responses change by approximately 9.6–9.8%, compared with 10% for the homogeneous ideal matrix.These are architecture-dependent deviations from the kinematic baseline; Figure 17 plots relative homogenized electric-displacement change during x-extension toward F11 = 1.1 at Emacro = 25 e3 MV/m.Source: Figure 17 · p. 25, §5 · p. 26
  • Superposing beam energy on the full matrix domain makes the model an additive reinforcement, not a phase-partitioned composite, and equal beam and matrix properties do not recover a homogeneous material.For straight-fiber loading, the mixed-dimensional and body-fitted models give similar macroscopic stresses despite their different fiber-volume representations.Source: §5 · p. 26
  • Centerline field sampling does not resolve finite-radius electric-field variation or an explicit matrix–fiber interface, although tested mesh-refinement and placement studies show weak sensitivity.A finite-radius or regularized projection would provide a more local description of the field around a fiber.Source: §5 · p. 26
  • The network comparisons vary orientation, curvature, connectivity, and fiber content together, so isolating each descriptor requires matched architecture families; device-level assessment also requires an electrode configuration, measurement circuit, and noise model.Source: §5 · p. 26

6 Conclusion

The paper develops a mixed-dimensional formulation for electroactive fiber networks and uses it in periodic homogenization. Its examples show distinct architecture-dependent responses across aligned and irregular networks.

6 Conclusion

  • Geometrically exact beams are coupled mechanically to the matrix by projected mortar constraints and electrically through matrix fields sampled along beam centerlines, without beam electric-potential degrees of freedom.Source: §6 · p. 26
  • The formulation is used in periodic homogenization to compute effective stress and electric displacement while retaining explicit network geometry without body-fitted three-dimensional fiber meshes.Source: §6 · p. 26
  • Aligned fibers produce strongly directional responses, while curved, multidirectional, and irregular networks retain distinct mechanical and fixed-bias electrical behavior arising from their collective geometry.The framework is intended to study how orientation, curvature, connectivity, and topology influence effective response, with architecture as a longer-term design variable.Source: §6 · p. 26

A Model Settings and Discrete Diagnostics

The appendix collects common material and numerical settings, postprocessing conventions, and discrete diagnostics. It also describes geometric scales relevant to the beam approximation and clarifies how to interpret visualized fields.

A Model Settings and Discrete Diagnostics

  • Table 1 gives the baseline material parameters; the beam section is kinematically incompressible, whereas the matrix uses a finite volumetric penalty.Source: Table 1 · p. 29
  • Exact translational mortar constraints are used for smaller structured problems and verification, while large Voronoi networks use penalty coupling.In exact-mortar calculations, the penalty parameter is used only during continuation and does not enter the final constrained system.Source: A Model Settings and Discrete… · p. 29
  • The local stress plots show matrix Cauchy stress and a reconstructed beam-centerline diagnostic, neither of which is the energetic homogenized stress σRVE.The deformed geometry is displayed with a factor of two that does not scale field values.Source: A Model Settings and Discrete… · p. 29
  • The circular-section quadrature rule recovers area, first moments, cross moment, and quadratic moments to roundoff in the reported moment audit.Source: Table 3 · p. 30, A Model Settings and Discrete… · p. 30
  • Table 4 reports minimum branch-length-to-diameter and sampled curvature-radius-to-fiber-radius ratios, which characterize local geometric scales represented by the beam model.The one-dimensional approximation is best suited to branches long and gently curved relative to fiber diameter; some Voronoi branches violate those assumptions, so the calculations are interpreted at network and homogenized scales.Source: Table 4 · p. 30, A Model Settings and Discrete… · p. 30

B Resolved Three-Dimensional Single-Fiber Comparison

The straight-fiber comparison tests the embedded model against a resolved three-dimensional fiber model under constant-volume loading. The embedded solution is stable over sampled discretizations, while the endpoint stress comparison retains a difference between the models.

B Resolved Three-Dimensional Single-Fiber Comparison

  • The two models use different fiber-volume representations, and the body-fitted calculation is itself a finite-mesh reference, so the observed difference includes any remaining resolved-model discretization error.Source: B Resolved Three-Dimensional… · pp. 30–31
  • The embedded calculations reproduce the affine response to numerical precision over sampled matrix and beam discretizations.The constant-volume loading path uses λ⋆ = 1.10; the resolved calculation uses href = 0.05 m.Source: B Resolved Three-Dimensional… · pp. 30–31
  • At the loading endpoint, the stress-magnitude error is 2.14% and the complete-tensor error is 2.75% between the embedded and body-fitted models.Table 5 summarizes the comparison; Figure 18 shows that the stress-norm difference is essentially unchanged across the embedded discretization sweep.Source: Table 5 · p. 31, B Resolved Three-Dimensional… · p. 31, Figure 18 · p. 31
  • The persistent stress difference is not caused by the sampled embedded beam–matrix discretization, and the affine embedded solution is independent of section quadrature in this test.Source: B Resolved Three-Dimensional… · p. 31

C Verification and Sensitivity Data

This appendix collects numerical data supporting the paper’s verification and sensitivity studies. It reports matrix-discretization, matrix-mesh resolution, beam–mesh placement, and fixed-bias sensing comparisons.

Matrix-discretization study

  • The matrix-discretization study varies interpolation order, mesh resolution, Km/µm, and J-bar treatment, with Table 6 summarizing factorial coverage and full-tensor stress sensitivities.The reported differences are the coarse-to-fine mesh change (dh) and the J-bar on/off difference (dJ) on the coarse P2 mesh.Source: C Verification and Sensitivity Data · p. 31, Table 6 · p. 32

Matrix-mesh resolution study

  • The matrix-mesh resolution study measures coupled shear and fixed-referential-bias response differences against the finest tested mesh, hf = 0.10 m.The differences are relative to hf = 0.10 m, so the zero entries for that mesh are by definition.Source: Table 7 · p. 32, C Verification and Sensitivity Data · p. 32

Beam–matrix-mesh placement study

  • The placement study translates the matrix mesh relative to a fixed physical network and compares homogenized stress and electric-displacement responses with the unshifted configuration.Beam radius, matrix resolution, material parameters, and coupled loading are held fixed.Source: Table 8 · p. 32, C Verification and Sensitivity Data · p. 32

Fixed-bias sensing sensitivity

  • Table 9 reports endpoint normalized fixed-bias sensing signals for nominally isotropic, preferred-direction, and perforated networks under baseline settings and separate quadrature or penalty-parameter variations.Each signal is normalized by the equilibrated initially biased state of the same calculation.Source: C Verification and Sensitivity Data · p. 32, Table 9 · p. 33

D Voronoi Network Construction

The irregular examples use periodic Voronoi-based centerline graphs, with network architecture set by geometric restrictions. Their construction also specifies junction connectivity, branch definitions, and the treatment of the perforated cell.

Voronoi graph construction

  • Periodic Voronoi graphs are built from seeded random sites replicated in a 3 × 3 × 3 tiling; clipped dual edges are filtered, and disconnected components are ranked by periodic-span dimension then vertex count.At junctions, branches share translations but retain independent rotations; a physical branch is a maximal centerline chain between junctions after coincident and opposite-periodic-face endpoints are identified.Source: D Voronoi Network Construction · p. 33

Architecture variants

  • The nominally isotropic network is a finite random realization without prescribed directional bias, while the preferred-direction filter biases retained segments toward the x–y plane.Filtering also changes connectivity, total centerline length, and nominal fiber fraction, so this is not an orientation-only modification at matched fiber content.Source: D Voronoi Network Construction · p. 33

Perforated network

  • The perforated network routes retained centerlines around a central spherical exclusion, while the matrix mesh represents its nominally spherical boundary with a faceted tetrahedral surface.Uniform refinement preserves that existing boundary geometry rather than converging the spherical representation itself.Source: D Voronoi Network Construction · p. 33

Geometric descriptors and examples

  • Table 4 descriptors use the physical-branch definition, and Figure 19 illustrates networks generated with increasing site count at fixed fiber radius and generation parameters.The descriptors include minimum branch slenderness Lmin/d and Rmin/r based on the smallest circumradius away from graph junctions.Source: D Voronoi Network Construction · p. 33, Figure 19 · p. 34

E Homogenized RVE Stress Tensors

The appendix reports homogenized RVE Cauchy-stress tensors for mechanical loading cases at their final loading states. The tensors are component-wise symmetrized for reporting.

Aligned-fiber examples

Lattice and wavy-fiber examples

Voronoi examples

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