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Buckling Prediction for Nonlinear Elastic Beams with Soft Inclusions
Thien Tran-Duc, J. E. Bunder, A. J. Roberts
TL;DR
Predicting buckling in heterogeneous beams is challenging because microscale features can affect global instability, and prior patch methods had not established their ability to capture onset and post-buckling. This paper extends sparse patch computations to nonlinear beams with soft inclusions; the computations capture buckling behaviour while reducing simulation cost.
Problem
The onset and nonlinear post-buckling behaviour of heterogeneous structures had not been established for the patch scheme, despite microscale features potentially influencing global instability.
Source: §1 · pp. 2, 5Method
The authors extend the multiscale patch scheme to nonlinear beams with soft inclusions, resolving microscale heterogeneity in sparse patches and coupling them to recover macroscale behaviour.
Source: §1 · p. 5Results
For m ≥6, patch computations accurately capture critical strain and stress, with m=8 agreeing excellently with full-domain results; at r=0.25 they also predict the first three critical points well, particularly the primary unimode.
Source: §4 · p. 36Takeaways & Limitations
Simulations with r=0.5 and r=0.25 preserve essential macroscale dynamics while significantly reducing computation time.
Source: §4 · p. 36
Abstract
from arXiv · showhide
We develop an efficient and accurate multiscale computational framework for predicting the buckling and post-buckling behaviour of elastic beams containing periodically distributed soft inclusions. The framework extends our previous multiscale, patch, computational homogenisation for linear elasticity by incorporating nonlinearity, and thus enables accurate prediction of both the buckling onset and the subsequent post-buckling response. Microscale computations are performed only within a sparse set of small subdomains (patches), while the macroscale behaviour is recovered through a proven patch-coupling algorithm. The scheme is assessed through quarter- and half-domain patch computations for beams with inclusion-to-matrix Young's modulus ratios ranging from 0.001 to 1. The results show that reducing the inclusion stiffness lowers both the critical buckling strain and the critical buckling stress, indicating an increased susceptibility to instability, while producing a milder post-buckling response with smaller transverse deflections and stress drops. Eigenvalue analysis of the Jacobian matrix accurately predicts the onset of instability and the corresponding critical strain and stress. Bifurcation diagrams of the nonlinear buckled configurations under compressive loading, and a quantitative analysis of the effect of the interpolation order on the predicted buckling and post-buckling responses, are also presented. Comparisons with full-domain simulations demonstrate that the proposed framework accurately predicts both the buckling threshold and the post-buckling behaviour while substantially reducing the computational cost. The methodology is readily extendable to heterogeneous beams, plates, shells, and other engineering structures.
1 Introduction
Buckling in heterogeneous structures poses a multiscale challenge because microscale features can influence global instability. This study extends sparse patch computations to nonlinear beams with soft inclusions, targeting buckling onset and post-buckling behaviour.
1 Introduction
- Buckling is a multiscale computational challenge because microscale material and geometric features may influence instability onset and its subsequent macroscale development.Equation-free patch methods address computational cost by simulating selected regions and coupling them to recover system-level behaviour.Source: §1 · p. 2
- Conventional homogenisation reduces cost by replacing microstructure with an equivalent continuum, but averaging may miss localised deformation, stress concentrations, and instability mechanisms.These limitations matter when local microscale features interact with global instability.Source: §1 · p. 4
- Full-domain simulations resolve microstructure throughout the structure, whereas sparse patch and related methods avoid detailed microscale simulation across much of the domain.The introduction also identifies nested macro- and microscale finite-element methods as a related strategy.Source: §1 · pp. 4–5
- Sparse microscale patches retain local heterogeneity and use interpolation and coupling conditions to recover macroscale behaviour, but their ability to predict instability had not been established.Earlier studies reported accurate static and transient predictions for heterogeneous beams using only a small fraction of the domain.Source: §1 · p. 5
- The patch methodology is extended to nonlinear beams with a single layer of soft inclusions and is validated for predicting buckling onset and subsequent nonlinear evolution.The study includes bifurcation points while retaining microscale heterogeneity within simulated patches.Source: §1 · p. 5
- Figure 1 illustrates sparse-patch prediction of snap-through in a randomly heterogeneous beam, including its transition from upward to downward buckling and subsequent oscillations.Only a quarter of the domain is computed, and the figure colors encode axial normal stress.Source: Figure 1 · p. 3, §1 · p. 6
2 Patch algorithm for beam buckling
The beam-buckling scheme combines a nonlinear elastic model with sparse microscale patches and interpolation that couples their responses into a macroscale system. Its discretisation and coupling specify how patch-scale fields and weak dissipation are handled.
2 Patch algorithm for beam buckling
- Green–Lagrange strains drive a Saint–Venant–Kirchhoff stress law, and stress gradients generate internal forces in the motion equation.The framework allows alternative strain-energy functions and corresponding constitutive relations.Source: §2 · p. 7
2.1 Patches
- Each patch contains one centred soft inclusion; sparse patch coverage reduces simulated domain size, while patch spacing controls macroscale accuracy.The illustrated layouts use full-, half-, and quarter-domain coverage; each patch spans one microstructural period.Source: §2.1 · pp. 8–9, Figure 3 · p. 9, §2.3 · p. 14
2.2 Microscale spatial discretisation within patches
- Within patches, centred finite differences on a staggered grid compute strains, using neighbouring-grid interpolation and boundary extrapolation for needed values.Horizontal displacement is linearly extrapolated at the top and bottom boundaries to evaluate boundary stresses.Source: §2.2 · pp. 11–12
2.3 Patch coupling
- Even-order polynomial interpolation uses nearby patches’ edge-adjacent displacement values to set patch edges and feed macroscale coupling back into microscale computations.The coupling preserves the underlying microscale model’s self-adjoint symmetry, which enhances numerical stability.Source: §2.3 · p. 13
2.4 Dissipation
- A Kelvin–Voigt term adds weak viscoelastic dissipation to the motion equation; the specified dimensionless strength is η = 10^-3.The term is incorporated into the microscale discretisation.Source: §2.4 · p. 14
3 Multiscale predictions of buckling
This section assesses sparse-patch multiscale predictions of buckling and post-buckling in beams with soft inclusions, including stress and displacement patterns, bifurcation branches, and interpolation-order effects. It examines buckling-onset accuracy and how patch-system sparsity affects computational time while retaining enough patches to resolve beam-scale structures.
3.1 Displacement and stress of compressed equilibria
- Sparse patch predictions closely match full-domain unbuckled equilibria, while homogeneous and inclusion beams exhibit distinct displacement and stress fields under compression.Homogeneous-beam stresses are uniform beyond short end boundary layers; inclusions induce periodic stress variation, interface-localized shear, nonlinear displacements, and rippled beam edges.Source: §3.1 · pp. 16, 17, 19, Figure 6 · p. 18
3.2 Onset of buckling
- Long-wavelength bending modes destabilize as compression rises, and soft inclusions lower critical strains and yield critical stresses about half those of the homogeneous beam.For the homogeneous beam, the first three full-domain critical strains are 0.093%, 0.191%, and 0.373%; reducing patch sparsity typically overestimates thresholds, especially for higher modes.Source: §3.2 · pp. 20, 22, 23, 25, Table 1 · p. 23, Figure 10 · p. 24
3.3 Buckled beams
- For homogeneous beams, patch sparsities r = 0.5 and 0.25 predict nearly matching buckled shapes and stresses, with slightly different strains for a given deflection.For ζmax = 0.4 and 0.9, predicted strains are 0.103% and 0.128% at r = 0.5, versus 0.104% and 0.134% at r = 0.25.Source: §3.3 · p. 26, Figure 12 · p. 27
3 Multiscale predictions of buckling
- As inclusion stiffness decreases, critical buckling strain and stress fall, with diminishing changes toward the void-like limit; softer inclusions also produce milder post-buckling responses.The inclusions accommodate more compression through local deformation, reducing global transverse deflections and stress relaxation.Source: Heterogeneous soft-inclusion beam · p. 30, Figure 15 · p. 30
Heterogeneous soft-inclusion beam
- Only unimodal buckling branches are stable beyond critical strain; as inclusion stiffness decreases, unimodal and bimodal branches usually occur at lower strains, with smaller differences at the softest values.Each additional buckling mode introduces a pitchfork bifurcation, while bimodal and higher-mode branches remain unstable.Source: Heterogeneous soft-inclusion beam · p. 31, Figure 16 · p. 32
3.4 Order of the interpolation
- Full-domain unimodal buckling predictions are independent of interpolation order, whereas sparse-patch predictions depend on interpolation order.The reported comparison considers orders m = 2, 4, 6, and 8 for homogeneous and soft-inclusion beams across patch sparsities.Source: §3.4 · p. 31, Table 2 · p. 33
- Interpolation orders m = 6 and 8 predict buckling onset and critical strains and stresses in good agreement with full-domain results, while m = 8 is most accurate.For r = 0.25 and 0.5, orders 3 and 4 detect no buckling regardless of inclusion stiffness; m = 8 is used for all r < 1 computations in Section 3.Source: §3.4 · p. 33
3.5 Computing time
- Computational time decreases approximately linearly with sparsity r, while decreasing inclusion modulus from 1 to 10^-3 increases time slightly.The reduction remains semi-linear across investigated inclusion moduli, but patch numbers must remain sufficient for spacing H to resolve emergent macroscale beam structures. The assessment covers patch-system equilibrium and Jacobian eigenmode calculations.Source: §3.5 · pp. 33–34
4 Conclusions
The nonlinear patch framework predicts buckling and post-buckling in soft-inclusion beams using microscale simulations in sparse patches. Higher-order interpolation captures critical buckling measures accurately, while sparse simulations substantially reduce computation time.
4 Conclusions
- The framework extends the patch scheme to nonlinear elasticity to capture large deformations, buckling, and post-buckling while resolving microstructure efficiently.Source: §4 · p. 34
- Increasing deformation localisation within soft inclusions accompanies smaller stress drops and transverse deflections, as inclusions accommodate some imposed compression locally.Source: §4 · p. 36
- Interpolation order m ≥6 captures critical strain and stress, with m = 8 agreeing excellently with full-domain results; quarter-domain patches predict the first three critical points well, especially the primary unimode.For r = 0.25, post-buckling stress is overestimated and beam deflection underestimated.Source: §4 · p. 36
- Sparse microscale simulations at r = 0.5 and r = 0.25 preserve essential macroscale dynamics and significantly reduce computational time for both equilibrium and eigenvalue calculations, with computing-time reduction approximately linear in sparsity r.Source: §4 · p. 36