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An FE2 model for shear-deformable beams considering periodic lattice-like truss mesostructures

Julian Ochs, Jens Wackerfuß

arXiv:2610.08607v1cs.CE

TL;DR

Prior work had not reported an FE2 framework coupling a beam-based macroscopic model with a truss-based lattice model, despite the need for efficient structural analysis of lattice-like members. This paper develops such a framework with additional constraints and boundary corrections; its studies report RVE-length-independent homogenized quantities and good agreement with fully resolved references.

  • Problem

    An FE2 framework coupling a beam-based macroscopic model with a truss-based lattice model had not been reported, despite the need for efficient analysis of lattice-like load-bearing structures.

    Source: §1 · p. 1, model · p. 3

  • Method

    The framework couples shear-deformable beams with periodic truss lattices, using periodic RVE boundary conditions, additional constraints, and boundary correction factors.

    Source: model · pp. 3–4, §7 · p. 31

  • Results

    The homogenized quantities are independent of RVE length, and FE2 results agree very well with fully resolved reference models across the investigated linear and nonlinear cases.

    Source: §7 · p. 32

  • Takeaways & Limitations

    The homogenized material matrix has the same basic structure as that of a continuum Timoshenko beam.

    Source: §7 · p. 32

    In geometrically nonlinear analysis, the admissible RVE length has an upper bound that depends on the imposed macroscopic strain, introducing an artificial RVE-length dependence.

    Source: §3.5 · p. 10

Abstract

from arXiv · show

The increasing use of lattice-like architectures in load-bearing members, facilitated by advances in additive manufacturing, calls for efficient multiscale approaches to their structural analysis. For slender lattice-like structures, beam models provide a computationally efficient macroscopic representation while retaining the ability to account for the underlying lattice architecture through an homogenization approach. Although finite element squared (FE2) approaches combining beam models at the macroscopic scale with continuum models at the microscopic scale have been established, an FE2 framework combining a beam-based macroscopic model with a truss-based microscopic model of a lattice structure has not yet been reported. To address this gap, an FE2 framework is developed for three-dimensional shear-deformable beams with periodically repeated lattice-like microstructures and periodic displacement boundary conditions. The proposed FE2 framework introduces constraint equations that (a) prevent rigid-body translations and rotations of the representative volume element (RVE) without imposing additional kinematic restrictions and (b) ensure that the homogenized stress resultants and material matrix are independent of the selected RVE length. To mitigate artificial boundary effects arising at the boundaries of the lattice-like RVE, local correction factors are introduced. The framework is assessed through numerical studies of two- and three-dimensional lattice-like structures with different geometries, considering both geometric and material linearity and nonlinearity. The results demonstrate the independence of the homogenized stress resultants and material matrix from the selected RVE length and show good agreement with the corresponding reference models in all investigated cases.

1 Introduction

FE2 couples macroscopic and microscopic models to homogenize complex lattice structures, but beam-to-lattice models remain underdeveloped. This work proposes a category IV beam–truss framework and addresses constraints and boundary effects that can affect homogenized properties.

1 Introduction

  • FE2 homogenization offers an alternative to directly modeling complex lattice geometry, which can be infeasible or prohibitively expensive.The approach replaces lattice geometry with an equivalent homogeneous medium.Source: §1 · p. 1
  • The only earlier category IV study used a two-dimensional micropolar beam and linear displacement boundary conditions, leaving RVE-length effects, nonlinearities, and mesostructural eccentricities insufficiently examined.That study's linear boundary conditions could introduce kinematic restrictions and boundary effects, potentially making homogenized properties depend on RVE length.Source: model · p. 3
  • The proposed FE2 method couples Timoshenko beams with periodic truss-member lattices, investigating two- and three-dimensional structures.Source: model · p. 3
  • Additional constraints suppress parasitic rigid-body rotations of the RVE under periodic boundary conditions.Source: model · p. 3
  • Correction factors address parasitic boundary effects and ensure that homogenized quantities are independent of the selected RVE length.Source: model · p. 4
  • Numerical examples validate the method against reference models, including eccentricities, geometric and material nonlinearities, and solid and open beam cross sections.Source: model · p. 4

2 Lattice-like microstructures

The study considers periodic, slender lattice mesostructures made from straight members, with geometrically stable configurations and smooth ends. Its analysis is restricted to selected two- and three-dimensional lattice types.

2 Lattice-like microstructures

  • The lattices comprise straight members connected at lattice points and repeating unit cells; their geometry, topology, and material data describe each cell.Source: §2 · p. 4
  • The study restricts its two-dimensional lattices to orthorhombic and hexagonal types, and its three-dimensional lattices to orthorhombic types o3I and o3P.Source: §2 · p. 4
  • The mesostructures are slender, with many more unit cells along the beam axis than across its width or height, and smooth planar boundaries at both ends.The h2P(2) cell arrangement is excluded because it does not satisfy the smooth-end condition.Source: §2 · p. 4
  • Members have constant cross sections and material properties, are ideally hinged, and are limited to geometrically stable unit cells h2P(1), o2P(1), o3I, and o3P.The stable configurations are referred to as unit cells 1, 2, 3, and 4, respectively.Source: §2 · p. 4

3 FE2 model

This section presents an FE2 framework coupling a shear-deformable macroscopic beam model with an explicitly resolved periodic truss RVE through localization and homogenization. It derives beam stress resultants from RVE boundary forces independently of RVE length and describes the material matrix for geometrically linear RVEs, including its form under specified symmetry and centering conditions.

3.1 Macro level

  • The FE2 model couples Timoshenko beam elements with finite rotations to a mesoscopic lattice model that determines beam stiffnesses, stress resultants, and material properties.The macroscopic strains follow from the beam kinematics, while the corresponding stress resultants and stiffnesses are obtained by homogenization.Source: §3.1 · pp. 4, 6

3.2 Meso-level

  • Mesoscopic truss RVEs are periodic along the beam axis, and their length is a selectable model parameter subject to scale separation and the heterogeneity present.The RVE width and height are prescribed; its length should be minimized for computational efficiency, and one period suffices for homogeneous axial composition.Source: §3.2 · p. 7
  • Boundary truss members have their axial stiffness halved to enforce periodicity; omitting this modification causes artificial boundary effects, especially for short RVEs.The lattice is represented by pin-jointed truss members, with elastic or J2 elastoplastic behavior considered in the numerical studies.Source: §3.2 · p. 7

3.3 Hill-Mandel condition

  • The macro–meso coupling uses the Hill–Mandel condition, equating macroscopic virtual internal energy with the RVE’s averaged virtual internal energy.In this framework, the volume average reduces to an average over the RVE length.Source: §3.3 · p. 8

3.4 Localization

  • Localization imposes macroscopic strains through periodic displacement conditions on opposing RVE boundaries, satisfying Hill–Mandel for small strains.Applying the conditions separately to the six beam-strain components produces six RVE loading modes.Source: §3.4 · pp. 8–9

3.5 Limitations

  • The localization matrix assumes geometric linearity; under geometric nonlinearity, its use can restrict admissible RVE length and induce artificial length dependence and parasitic axial forces.These effects arise from imposed periodic conditions, including for twist, curvature, and shear; in the reported numerical experiments they became significant only after local instabilities developed, a finding that may depend on lattice geometry, topology, and member stiffnesses.Source: §3.5 · pp. 9–10
  • Parasitic effects vanish as RVE length decreases and are absent for geometrically linear mesoscopic analyses, even with arbitrarily long RVEs.The nonlinear examples also yielded length-independent results in very good agreement with independent reference solutions.Source: §3.5 · p. 10

3.6 Homogenization

  • Homogenization evaluates macroscopic stress resultants and the material matrix only after the constrained RVE reaches equilibrium.The generally nonlinear equilibrium equations are solved with Newton’s method and master–slave elimination.Source: §3.6 · p. 10
  • The homogenized beam stress resultants are computed from RVE boundary forces and are independent of RVE length.Source: §3.6 · p. 11

3.7 Entries of matrix DM

  • For geometrically linear RVE models, DM has 12 independent nonzero entries, with matrix structure independent of lattice unit-cell type and truss-member material properties.Source: §3.7 · pp. 11–12
  • The 12 nonzero components of DM uniquely identify 12 stiffness and geometric parameters corresponding to classical Timoshenko beam stiffness terms.Source: §3.7 · p. 12
  • When the RVE cross section has an axis of symmetry, no eccentricity, and principal y- and z-axes, DM is diagonal; numerical studies verified these conditions and properties.Source: §3.7 · p. 12
  • With geometric and material linearity, the recovered stiffnesses can serve as input for a single-scale model treating the lattice mesostructure as a continuum Timoshenko beam.Source: §3.7 · p. 12

Geometric nonlinearity

Geometric nonlinearity changes the RVE material matrix through the geometric contribution to tangent stiffness, with changes depending on the imposed macroscopic strains. At each macroscopic integration point, FE2 imposes beam strains on the RVE and returns homogenized stress resultants and the material matrix.

Geometric nonlinearity

  • Geometric nonlinearity makes D12, D13, and D14 nonzero, while D15 and D16 also become nonzero even for a centric RVE.The latter changes arise from parasitic axial forces under macroscopic strains γM_z.Source: Geometric nonlinearity · p. 12
  • Under geometric nonlinearity, the matrix remains symmetric and its structure is independent of mesostructural architecture and individual truss-member material behavior.The matrix changes with loading and approaches the geometrically linear structure as the imposed strains decrease.Source: Geometric nonlinearity · p. 12

3.8 FE2 formulation

  • At each macroscopic integration point, FE2 imposes beam strains on the mesoscopic RVE through periodic displacement boundary conditions, solves its equilibrium, and returns homogenized stress resultants and material matrix.The RVE equilibrium is determined with Newton–Raphson for a nonlinear mesostructural model.Source: §3.8 · pp. 12–13

4 Additional constraints

The constraints suppress RVE rigid-body motions without artificial kinematic restrictions and support homogenized quantities independent of RVE length. A discrete constant moment constraint addresses lattice-like RVEs, with additional constraints for eccentric RVEs and the remaining rigid-body motions.

4 Additional constraints

  • The added constraints eliminate possible RVE rigid-body translations and rotations without imposing artificial kinematic restrictions.Together with macroscopic strain constraints, they define the RVE boundary-value problem.Source: §4 · p. 13, §4.3 · p. 14

4.1 Rigid body rotations about y- and z-axis (’discrete constant moment constraint’)

  • For lattice-like RVEs, the new discrete constant moment constraint adapts the continuum constant moment constraint to unit-cell contributions.Its unit-cell contributions are averaged over members using weighted axial stiffness; for linear elasticity, the tangent modulus equals Young’s modulus.Source: §4.1 · p. 13
  • For eccentric RVEs, an additional discrete constant normal force constraint is required to make homogenized shear forces and shear stiffnesses independent of RVE length.This applies when s_y or s_z is nonzero.Source: §4.1 · p. 14

4.2 Remaining rigid body motions

  • The discrete constant moment constraints suppress rotations about the y- and z-axes; additional constraints suppress the three translations and the remaining rotation about the x-axis.The resulting set contains 6 + 3nΓ constraints that depend linearly on the RVE nodal degrees of freedom.Source: §4.2 · p. 14, §4.3 · p. 14

4.4 Symmetry of RVE state variables

  • At equilibrium, the constraints give RVE state variables symmetric or antisymmetric distributions with respect to coordinate directions.The tabulated properties cover two-dimensional RVE types and extend to three dimensions, assuming geometric and material linearity.Source: §4.4 · p. 14

5 Numerical examples I (benchmark problems)

The benchmarks isolate homogenization at a single macroscopic material point and compare effective beam properties with fully resolved reference models, which carry minor uncertainties for some structures. Across homogeneous, heterogeneous, and nonlinear cases, the results test RVE-length independence and compare lattice-like sections with continuum references.

5.1 Homogeneous RVE

  • For homogeneous RVEs, homogenized stiffness components are independent of RVE length; omitting the boundary-member axial-stiffness modification causes up to 7% relative error in EA at small lengths.The tests vary RVE length for two- and three-dimensional structures, including eccentric cases, under geometrically linear, linear-elastic assumptions.Source: §5.1 · pp. 15, 18

5.2 Heterogeneous RVE

  • For the heterogeneous RVE, stiffness components are independent of RVE length, while axial strain and simple shear produce pronounced narrowing and largest member forces around the hole.The tested architecture uses a circular hole in a super unit cell repeated along the RVE length.Source: §5.2 · pp. 18, 20

5.3 Geometric nonlinearity

  • Geometric nonlinearity induces parasitic axial force and additional material-matrix coupling even for a centric RVE; under simple shear, the induced shear force exceeds that axial force by more than one order of magnitude.The axial-force response increases exponentially with loading; the strain load case does not induce parasitic axial force a priori.Source: §5.3 · p. 21
  • With geometric nonlinearity, axial strain and simple shear retain linear, RVE-length-independent responses, but bending and torsion develop weak nonlinearity and artificial length dependence.For the tested load cases, structural instability is the governing limitation.Source: §5.3 · pp. 21, 23

5.4 Material nonlinearity

  • With elastoplastic material behavior, homogenized stress resultants remain independent of RVE length throughout deformation, and loading a single beam-strain component induces only its work-conjugate stress resultant.The tested material has linear isotropic hardening; all other stress resultants are numerically zero, so the material matrix has no coupling terms.Source: §5.4 · p. 23
  • Under material nonlinearity, axial and diagonal members yield in collective transitions, while bending and torsion transition gradually as yielding progresses from boundaries inward, leaving residual stresses after unloading.Simple shear has one sharp elastic-to-inelastic transition when diagonal members yield, producing an almost horizontal shear-force–strain segment despite hardening.Source: §5.4 · pp. 23–24

5.5 C-shaped cross section

  • All 36 homogenized stiffness components for the lattice-like C-section are independent of RVE length and agree very well with reference models.Quantitative agreement with reference and analytical solutions is also reported for lattice-like and continuum C-sections.Source: §5.5 · p. 24
  • For the C-section, lattice-like and continuum models show similar deformation patterns and shear-flow behavior, including vanishing flow at free flange ends and approximately constant flow along the web.The thin-walled continuum distribution is only approximate for the tested section, which is no longer thin-walled.Source: §5.5 · p. 25

6 Numerical examples II

The numerical studies assess the FE2 method on a simply supported lattice-like beam and a spatial frame, comparing modeled responses with fully resolved references. They examine geometric and material behavior across different mesostructures and modeling choices.

6.1 Simply supported beam

  • Under geometric and material linearity, model ’1’ agrees very well with the reference, and the type ’2’ unit-cell beam has substantially higher global stiffness than type ’1’.The comparison is shown in the midspan load–displacement curves.Source: §6.1 · p. 27, Figure 19 · p. 29
  • Model ’3’ reproduces the reference loading and unloading response for both mesostructures with elastoplastic material behavior and geometric linearity at both scales.The response includes linear isotropic hardening at the mesoscopic scale.Source: §6.1 · p. 27, Figure 19 · p. 29
  • For beam models ’2’ and ’4’, geometrically nonlinear mesoscopic modeling agrees very well with references, while geometrically linear variants show small deviations.Model ’2b’ covers materially linear behavior; ’4b’ includes elastoplasticity and nonlinear loading and unloading.Source: §6.1 · pp. 27–28, Figure 19 · p. 29
  • Models ’1’, ’2b’, ’3’, and ’4b’ agree very well with the beam reference, while ’2a’ and ’4a’ show deviations.The deviations occur when the macroscopic model is geometrically nonlinear but the mesoscopic model is geometrically linear.Source: §6.1 · p. 28, Figure 19 · p. 29

6.2 Spatial frame structure

  • For the spatial frame, FE2 and fully resolved models show very good qualitative and load–displacement agreement for both mesostructures, despite nonlinear response most prominently in uy.The type ’4’ cell cantilever is stiffer than the type ’3’ cell cantilever.Source: §6.2 · pp. 29, 31, Figure 20 · p. 30
  • At the frame’s highly loaded section, type ’4’ shows very good local agreement, whereas type ’3’ has approximately 15% deviation associated with restrictive beam kinematics.The kinematic influence propagates particularly from frame corners and clamped boundaries and is more significant for type ’3’.Source: §6.2 · p. 31

7 Conclusions and outlook

The paper develops an FE2 framework coupling shear-deformable beams with periodic truss mesostructures. Numerical studies find RVE-length-independent homogenized quantities and good agreement with fully resolved models, while identifying limits for geometrically nonlinear mesoscopic behavior.

7 Conclusions and outlook

  • The framework couples shear-deformable beams to periodic truss mesostructures using periodic boundary conditions and added constraints that suppress rigid-body modes and yield RVE-length-independent homogenized quantities.Boundary truss axial stiffnesses are adjusted to preserve periodicity and mitigate parasitic boundary effects.Source: §7 · p. 31
  • For linear studies, homogenized quantities are independent of RVE length across centric and eccentric RVEs with compact and open cross sections, and the material matrix has the basic structure of a continuum Timoshenko beam.Very good agreement with fully resolved reference models is obtained.Source: §7 · p. 32
  • With J2 plasticity, homogenized quantities remain RVE-length independent; geometrically nonlinear mesoscopic modeling introduces parasitic axial forces and additional matrix coupling terms with minor influence for the studied mesostructures.The influence of these effects cannot readily be generalized to arbitrary mesostructures.Source: §7 · p. 32
  • More complex two- and three-dimensional beam structures show very good agreement with fully resolved references in linear and nonlinear analyses, with expected quadratic Newton convergence near the solution.Source: §7 · p. 32
  • The conclusions for geometrically nonlinear mesoscopic behavior are limited because parasitic effects cannot readily be generalized to arbitrary mesostructures; the study also excludes lattices whose end boundary nodes are not coplanar.Source: §7 · p. 32
  • Future directions include extending the framework to beam-based mesostructural models and investigating heterogeneity along the beam axis, particularly in three-dimensional mesostructures.Initial results on beam-based mesostructural models are cited as.Source: §7 · p. 32

Statements and Declarations

The statements section reports funding, competing interests, data availability, and author contributions.

Statements and Declarations

  • The authors report that they received no organizational support for the submitted work.Source: Statements and Declarations · p. 32
  • The authors declare no competing interests relevant to the article.Source: Statements and Declarations · p. 32
  • The contribution statement assigns conceptualization and methodology to both authors, with JO also responsible for software, validation, investigation, and visualization.JW is credited with supervision and substantial rewriting and reformulation of the mathematical derivations and scientific presentation.Source: Statements and Declarations · p. 32

A Continuous mesostructures

This appendix derives a simplified form of a constraint for lattice-like mesostructures from the constant moment constraint used for continuum mesostructures. It discusses the strain symmetries and additional conditions relevant to the simplification.

A Continuous mesostructures

  • The lattice-like constraint builds on the constant moment constraint for continuum mesostructures and is simplified for isotropic linear elastic behavior as a basis for extension.Source: A Continuous mesostructures · p. 32
  • The constant moment constraint requires the bending moment distribution My(x) to be symmetric along x, with the same reasoning applying to Mz(x).The bending moment is obtained from the cross-sectional normal stress distribution using the constitutive relation σ = Cε.Source: A Continuous mesostructures · p. 32
  • Preventing pure rigid-body rotation under imposed shear requires omitting a boundary term; the resulting condition depends on strain symmetry in the x-direction.The stated conditions concern εyy and εzz within the RVE.Source: A Continuous mesostructures · p. 33
  • Geometric-linear RVE studies confirmed the strain symmetries required by (A.7), allowing the constraint to simplify; eccentric RVEs require an additional constant normal force constraint to avoid RVE-length-dependent shear stiffnesses.The additional condition is stated for shear stiffnesses GAsy and GAsz.Source: A Continuous mesostructures · p. 33
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