Lie group symmetry analysis for granular media stress equations
Johnpillai, I. Kenneth, McCue, Scott W., & Hill, James M. (2005) Lie group symmetry analysis for granular media stress equations. Journal of Mathematical Analysis and Applications, 301(1), pp. 135157.

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Abstract
The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasistatic deformations of a cohesionless Coulomb–Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of onedimensional Lie algebras, a complete set of groupinvariant solutions is derived. By this it is meant that any groupinvariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.
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ID Code:  40053 

Item Type:  Journal Article 
Refereed:  Yes 
Keywords:  Airy stress function, Granular flow, CoulombMohr, plane strain, Lie group symmetries 
DOI:  10.1016/j.jmaa.2004.07.010 
ISSN:  0022247X 
Subjects:  Australian and New Zealand Standard Research Classification > MATHEMATICAL SCIENCES (010000) > APPLIED MATHEMATICS (010200) > Theoretical and Applied Mechanics (010207) 
Divisions:  Past > QUT Faculties & Divisions > Faculty of Science and Technology Past > Schools > Mathematical Sciences 
Copyright Owner:  Copyright 2005 Elsevier 
Deposited On:  14 Feb 2011 23:27 
Last Modified:  10 Aug 2011 14:58 
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