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Travelling wave solutions for a quasilinear model of Field Dislocation Mechanics
| Content Provider | Semantic Scholar |
|---|---|
| Author | Acharya, Amit Matthies, Karsten Zimmer, Johannes |
| Copyright Year | 2010 |
| Abstract | We consider an exact reduction of a model of Field Dislocation Mechanics to a scalar problem in one spatial dimension and investigate the existence of static and slow, rigidly moving single or collections of planar screw dislocation walls in this setting. Two classes of drag coecient functions are considered, namely those with linear growth near the origin and those with constant or more generally sublinear growth there. A mathematical characterisation of all possible equilibria of these screw wall microstructures is given. We also prove the existence of travelling wave solutions for linear drag coecient functions at low wave speeds and rule out the existence of nonconstant bounded travelling wave solutions for sublinear drag coecients functions. It turns out that the appropriate concept of a solution in this scalar case is that of a viscosity solution. The governing equation in the static case is not proper and it is shown that no comparison principle holds. The ndings indicate a short-range nature of the stress eld of the individual dislocation walls, which indicates that the nonlinearity present in the model may have a stabilising eect. We predict idealised dislocation-free cells of almost arbitrary size interspersed with dipolar dislocation wall microstructures as admissible equilibria of our model, a feature in sharp contrast with predictions of the possible non-monotone equilibria of the corresponding Ginzburg-Landau, phase eld type gradient ow model. For walls separating slip states by a full Burgers vector, while dipolar clusters of walls and walls in a dipole can be separated by arbitrarily long dislocation-free cells, we nd that walls in a pile-up cannot be similarly separated. |
| Starting Page | 2043 |
| Ending Page | 2053 |
| Page Count | 11 |
| File Format | PDF HTM / HTML |
| DOI | 10.1016/j.jmps.2010.09.008 |
| Volume Number | 58 |
| Alternate Webpage(s) | https://imechanica.org/files/tw-plastic-preprint.pdf |
| Alternate Webpage(s) | http://imechanica.org/files/tw-plastic_preprint.pdf |
| Alternate Webpage(s) | http://www.maths.bath.ac.uk/~km230/preprints/tw-plastic.pdf |
| Alternate Webpage(s) | http://people.bath.ac.uk/km230/preprints/tw-plastic.pdf |
| Alternate Webpage(s) | http://www.bath.ac.uk/math-sci/bics/preprints/BICS09_12.pdf |
| Alternate Webpage(s) | https://purehost.bath.ac.uk/ws/portalfiles/portal/273593/Matthies_JMPS_2010_58_12_2043.pdf |
| Alternate Webpage(s) | https://doi.org/10.1016/j.jmps.2010.09.008 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
