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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2508.16333 |
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| _version_ | 1866916934210027520 |
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| author | Gudoshnikov, Ivan |
| author_facet | Gudoshnikov, Ivan |
| contents | Plasticity with softening and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete degradation of the set of admissible stresses. We present a state-dependent sweeping process which solves the evolution of elasto-plastic Lattice Spring Models with arbitrary placement of softening, hardening and perfectly plastic springs. Using numerical simulations of regular grid lattices with softening we demonstrate the emergence of non-symmetric shear bands with strain localization. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions. These solutions correspond to fixed points in the implicit catch-up algorithm and we observe a discontinuous bifurcation with the exchange of stability of those fixed points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16333 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs Gudoshnikov, Ivan Optimization and Control Soft Condensed Matter Mathematical Physics 74C05, 74N30, 47J20, 47J26 Plasticity with softening and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete degradation of the set of admissible stresses. We present a state-dependent sweeping process which solves the evolution of elasto-plastic Lattice Spring Models with arbitrary placement of softening, hardening and perfectly plastic springs. Using numerical simulations of regular grid lattices with softening we demonstrate the emergence of non-symmetric shear bands with strain localization. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions. These solutions correspond to fixed points in the implicit catch-up algorithm and we observe a discontinuous bifurcation with the exchange of stability of those fixed points. |
| title | Elastoplasticity with softening as a state-dependent sweeping process: non-uniqueness of solutions and emergence of shear bands in lattices of springs |
| topic | Optimization and Control Soft Condensed Matter Mathematical Physics 74C05, 74N30, 47J20, 47J26 |
| url | https://arxiv.org/abs/2508.16333 |