Solid Mechanics

Faculty of Engineering, LTH

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Multi-scale modeling

Multi-scale modeling

Phenomena occurring over a range of length and time scales can play critical roles in the deformation and fracture of a wide variety of materials. In particular, the deformation and fracture of ductile metals typically involves complex interactions between large numbers of dislocations, grain and phase boundaries and mechanical defects such as micro-cracks. These interactions are strongly influenced by the chemical environment. As a consequence, developing a predictive analysis capability requires modeling phenomena over a range spanning atomic length and time scales (Angstroms and pico-seconds) to crystal or polycrystal length scales (microns, millimeters and larger) and service loading time scales (seconds to years).

Whereas the validity of conventional plasticity models at a macroscopic level is not questioned, this is not the case when smaller length scales are approached. When turning to microlength scales, say 1-10 microns, experimental investigations have shown a strong influence of length scale. This dependence of length scale is not captured in standard plasticity models. A method for solving problems where plastic flow is represented in terms of the collective motion of discrete dislocations has been presented by Van der Giessen and Needleman (1995). In this approach a number of natural length scales exsits, e.g. burgers vector, obstacle distance, nuclation site distance etc. The range of boundary value problems that can be solved directly using discrete dislocations is, however, limited due to the large computational demands of discrete dislocation plasticity, even for two-dimensional plane strain problems. In three dimensions the computational demands are even greater.


Hierarchical multi-scale modeling

A methodology for coupling a region of discrete dislocations to a region modeled using conventional continuum crystal plasticity have been developed. When a region modeled by continuum crystal plasticity is combined with a discrete dislocation plasticity region, displacement and traction matching conditions must be imposed at the interface. Also, since dislocations can pass from the discrete dislocation region to the crystal plasticity region and vice versa, a strategy for handling the passage o dislocation through the interface must be developed. Another issue is the appropriate matching of the mechanical properties between the discrete dislocation and continuum plasticity regions. In the present project a general approach for the coupling of discrete dislocation plasticity and continuum plasticity regions have been developed. The method for coupling of tractions and displacements across the interface is valid for three-dimensional analyses. However, there are significant issues in three dimensions associated with the passing of dislocation loops across the interface, such as the treatment of a portion of a loop crossing the interface, which are not addressed here. Below, a the result of a two-dimensional numerical examples is shown.

Fatigue crack growth- An application of the 'Hierarchical multi-scale approach'

Fatigue crack growth is the growth of a crack under cyclic loading conditions at a driving force that is smaller than is required for the same crack to grow under monotonic loading conditions. Fatigue crack growth is often analyzed using a fatigue crack growth law that is specified a priori. In contrast, the the crack growth can follow naturally as a consequence of the solution to an initial/boundary value problem.




  1. Hybrid discrete dislocation models for fatigue crack growth
    W. A. Curtin, V. S. Deshpande, A. Needleman, E. Van der Giessen and Mathias Wallin
    International Journal Of Fatigue, 32, 1511-1520, 2010
  2. Multi-scale plasticity modeling: Coupled discrete dislocation and continuum crystal plasticity
    Mathias Wallin, William Curtin, Matti Ristinmaa and Alan NeedlemanJournal of the Mechanics and Physics of Solids, 56, 3167-3180, 2008
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