Upscaling of dislocation walls in finite domains. (English) Zbl 1327.74131
Summary: We wish to understand the macroscopic plastic behaviour of metals by upscaling the micromechanics of dislocations. We consider a highly simplified dislocation network, which allows our discrete model to be a one-dimensional particle system, in which the interactions between the particles (dislocation walls) are singular and non-local.
As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of \(\Gamma\)-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.
As a first step towards treating realistic geometries, we focus on finite-size effects rather than considering an infinite domain as typically discussed in the literature. We derive effective equations for the dislocation density by means of \(\Gamma\)-convergence on the space of probability measures. Our analysis yields a classification of macroscopic models, in which the size of the domain plays a key role.
MSC:
| 74Q05 | Homogenization in equilibrium problems of solid mechanics |
| 74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |
| 82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 82D35 | Statistical mechanics of metals |
Keywords:
plasticity; multiscale; straight edge-dislocations; discrete-to-continuum limit; \(\Gamma\)-convergenceReferences:
| [1] | DOI: 10.1016/S0022-5096(03)00094-2 · Zbl 1106.74331 · doi:10.1016/S0022-5096(03)00094-2 |
| [2] | DOI: 10.1103/PhysRevB.64.224102 · doi:10.1103/PhysRevB.64.224102 |
| [3] | DOI: 10.3934/dcds.2011.31.1427 · Zbl 1239.35015 · doi:10.3934/dcds.2011.31.1427 |
| [4] | DOI: 10.1016/j.jmps.2014.04.014 · doi:10.1016/j.jmps.2014.04.014 |
| [5] | DOI: 10.1016/j.msea.2007.09.074 · doi:10.1016/j.msea.2007.09.074 |
| [6] | DOI: 10.1016/S1359-6454(02)00517-7 · doi:10.1016/S1359-6454(02)00517-7 |
| [7] | DOI: 10.1103/PhysRevB.56.5807 · doi:10.1103/PhysRevB.56.5807 |
| [8] | DOI: 10.1007/s00205-013-0635-7 · Zbl 1282.74067 · doi:10.1007/s00205-013-0635-7 |
| [9] | DOI: 10.1137/S003614100343768X · Zbl 1094.82008 · doi:10.1137/S003614100343768X |
| [10] | ESAIM: Control, Optimisation Calculus Variations 19 pp 1166– (2013) |
| [11] | J. Eur. Math. Soc. 12 pp 1231– (2010) |
| [12] | Discrete Continuous Dyn. Syst. A 23 pp 785– (2009) |
| [13] | DOI: 10.1088/0965-0393/17/7/075010 · doi:10.1088/0965-0393/17/7/075010 |
| [14] | DOI: 10.1007/s10820-008-9090-4 · doi:10.1007/s10820-008-9090-4 |
| [15] | Multiscale Model. Simul. 6 pp 1098– (2007) |
| [16] | SIAM J. Math. Anal. 37 pp 1131– (2006) |
| [17] | DOI: 10.1016/j.ijsolstr.2004.04.021 · Zbl 1075.74020 · doi:10.1016/j.ijsolstr.2004.04.021 |
| [18] | Materials Science and Engineering, An Introduction (2007) |
| [19] | DOI: 10.1016/j.jmps.2005.09.005 · Zbl 1120.74329 · doi:10.1016/j.jmps.2005.09.005 |
| [20] | Interfaces Free Bound 11 pp 291– (2009) |
| [21] | DOI: 10.1137/060657054 · Zbl 1135.74037 · doi:10.1137/060657054 |
| [22] | An Introduction to {\(\Gamma\)}-Convergence (1993) |
| [23] | DOI: 10.1103/PhysRevB.77.184111 · doi:10.1103/PhysRevB.77.184111 |
| [24] | DOI: 10.1088/0965-0393/12/6/003 · doi:10.1088/0965-0393/12/6/003 |
| [25] | Introduction to Dislocations (2001) |
| [26] | Theory of Dislocations (1982) |
| [27] | DOI: 10.1137/090778444 · Zbl 1222.41041 · doi:10.1137/090778444 |
| [28] | DOI: 10.1016/j.msea.2011.09.065 · doi:10.1016/j.msea.2011.09.065 |
| [29] | DOI: 10.1080/14786435.2010.499859 · doi:10.1080/14786435.2010.499859 |
| [30] | DOI: 10.1177/1081286508092616 · Zbl 1257.74036 · doi:10.1177/1081286508092616 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
