Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.08623 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929589654126592 |
|---|---|
| author | Dondl, Patrick Heida, Martin Hermann, Simone |
| author_facet | Dondl, Patrick Heida, Martin Hermann, Simone |
| contents | This work examines a discrete elastic energy system with local interactions described by a discrete second-order functional in the symmetric gradient and additional non-local random long-range interactions. We analyze the asymptotic behavior of this model as the grid size tends to zero. Assuming that the occurrence of long-range interactions is Bernoulli distributed and depends only on the distance between the considered grid points, we derive - in an appropriate scaling regime - a fractional p-Laplace-type term as the long-range interactions' homogenized limit. A specific feature of the presented homogenization process is that the random weights of the p-Laplace-type term are non-stationary, thus making the use of standard ergodic theorems impossible. For the entire discrete energy system, we derive a non-local fractional p-Laplace-type term and a local second-order functional in the symmetric gradient. Our model can be used to describe the elastic energy of standard, homogeneous, materials that are reinforced with long-range stiff fibers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_08623 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-local homogenization limits of discrete elastic spring network models with random coefficients Dondl, Patrick Heida, Martin Hermann, Simone Analysis of PDEs 74Q15, 26A33, 74B20 This work examines a discrete elastic energy system with local interactions described by a discrete second-order functional in the symmetric gradient and additional non-local random long-range interactions. We analyze the asymptotic behavior of this model as the grid size tends to zero. Assuming that the occurrence of long-range interactions is Bernoulli distributed and depends only on the distance between the considered grid points, we derive - in an appropriate scaling regime - a fractional p-Laplace-type term as the long-range interactions' homogenized limit. A specific feature of the presented homogenization process is that the random weights of the p-Laplace-type term are non-stationary, thus making the use of standard ergodic theorems impossible. For the entire discrete energy system, we derive a non-local fractional p-Laplace-type term and a local second-order functional in the symmetric gradient. Our model can be used to describe the elastic energy of standard, homogeneous, materials that are reinforced with long-range stiff fibers. |
| title | Non-local homogenization limits of discrete elastic spring network models with random coefficients |
| topic | Analysis of PDEs 74Q15, 26A33, 74B20 |
| url | https://arxiv.org/abs/2411.08623 |