A Higher-Order Stroh Framework for Weakly Nonlinear Post-Bifurcation Analysis

Abstract

Linear stability analysis predicts the onset of wrinkling, while weakly nonlinear analysis determines whether the bifurcation is supercritical or subcritical through the sign of the cubic amplitude coefficient. When this coefficient vanishes, higher-order terms are needed. In this paper, we develop a higher-order Stroh framework for weakly nonlinear post-bifurcation analysis of finitely deformed elastic solids. For each nonzero harmonic, the same base-state Stroh operator is retained at every order, with nonlinear source terms assembled from lower-order fields to account for geometry, constitutive laws and constraints. This recursive structure, together with Fredholm solvability conditions, enables us to derive cubic, quintic and higher-order amplitude equations. Here, we demonstrate the power of this framework by analysing wrinkling in an incompressible neo-Hookean film bonded to an exponentially graded substrate. The results recover established linear instability benchmarks and the supercritical-subcritical transition in the homogeneous-substrate limit. For graded substrates, the cubic coefficient vanishes on a surface in the parameter space of modulus ratio and substrate gradation. On this surface, the quintic coefficient takes both signs, corresponding to supercritical and subcritical bifurcations. These regions are separated by a curve along which the quintic coefficient also vanishes. Determining the branch direction along this curve requires higher-order terms, which can be calculated within the same framework.

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Cite

BibTeX RIS

BibTeX

@misc{deng2026higherorder,
  title = {{A Higher-Order Stroh Framework for Weakly Nonlinear Post-Bifurcation Analysis}},
  author = {Shiyuan Deng and Han Ye and Yafei Wang and Lizichen Chen and Alain Goriely and Yibin Fu and Weiqiu Chen and Chaofeng Lü and Michel Destrade and Kecheng Li and Yangkun Du},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.14869},
  eprint = {2610.14869},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.14869},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A Higher-Order Stroh Framework for Weakly Nonlinear Post-Bifurcation Analysis
AU  - Shiyuan Deng
AU  - Han Ye
AU  - Yafei Wang
AU  - Lizichen Chen
AU  - Alain Goriely
AU  - Yibin Fu
AU  - Weiqiu Chen
AU  - Chaofeng Lü
AU  - Michel Destrade
AU  - Kecheng Li
AU  - Yangkun Du
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.14869
UR  - https://arxiv.org/abs/2610.14869
AB  - Linear stability analysis predicts the onset of wrinkling, while weakly nonlinear analysis determines whether the bifurcation is supercritical or subcritical through the sign of the cubic amplitude coefficient. When this coefficient vanishes, higher-order terms are needed. In this paper, we develop a higher-order Stroh framework for weakly nonlinear post-bifurcation analysis of finitely deformed elastic solids. For each nonzero harmonic, the same base-state Stroh operator is retained at every order, with nonlinear source terms assembled from lower-order fields to account for geometry, constitutive laws and constraints. This recursive structure, together with Fredholm solvability conditions, enables us to derive cubic, quintic and higher-order amplitude equations. Here, we demonstrate the power of this framework by analysing wrinkling in an incompressible neo-Hookean film bonded to an exponentially graded substrate. The results recover established linear instability benchmarks and the supercritical-subcritical transition in the homogeneous-substrate limit. For graded substrates, the cubic coefficient vanishes on a surface in the parameter space of modulus ratio and substrate gradation. On this surface, the quintic coefficient takes both signs, corresponding to supercritical and subcritical bifurcations. These regions are separated by a curve along which the quintic coefficient also vanishes. Determining the branch direction along this curve requires higher-order terms, which can be calculated within the same framework.
ER  -

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