A very simple frictional plasticity model for a granular material is presented, including the effects of dilation. The novelty lies in the fact that this is described within the hyperplasticity framework, expressed using the terminology of convex analysis. This allows a consistent mathematical treatment of the dilation constraint. The Fenchel Dual is used to link the force and flow potentials. The resulting model accommodates non-associated flow within a rigorous mathematical framework that ensures compliance with the Laws of Thermodynamics.

Accepted: 2019-08-16

Published online: 2019-09-12

Keywords: Convex analysis, friction, dilation, plasticity

@article{OGEO_2019__1__A3_0, author = {Houlsby, Guy T.}, title = {Frictional Plasticity in a Convex Analytical Setting}, journal = {Open Geomechanics}, eid = {3}, publisher = {Alert Geomaterials}, volume = {1}, year = {2019}, doi = {10.5802/ogeo.2}, language = {en}, url = {opengeomechanics.centre-mersenne.org/item/OGEO_2019__1__A3_0/} }

Houlsby, Guy T. Frictional Plasticity in a Convex Analytical Setting. Open Geomechanics, Volume 1 (2019) , article no. 3, 10 p. doi : 10.5802/ogeo.2. https://opengeomechanics.centre-mersenne.org/item/OGEO_2019__1__A3_0/

[1] Application of thermomechanical principles to the modelling of geotechnical materials, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Volume 453 (1997), pp. 1975-2001 | Article | Zbl 0933.74045

[2] A thermomechanical analysis of a family of soil models, Géotechnique, Volume 52 (2002) no. 7, pp. 507-518 | Article

[3] A more fundamental approach to plastic stress-strain relations, Proc. of 1st US National Congress of Applied Mechanics, 1951 (1951), pp. 487-491

[4] A definition of a stable inelastic material, Journal of Applied Mechanics (1959), pp. 101-106 | Zbl 0088.17103

[5] Elastic moduli of soils dependent on pressure: a hyperelastic formulation, Géotechnique, Volume 55 (2005) no. 5, pp. 383-392 | Article

[6] Study of plasticity theories and their applicability to soils (1981) (Ph. D. Thesis)

[7] Interpretation of dilation as a kinematic constraint, Modern approaches to plasticity, Elsevier: New York, 1992, pp. 19-38

[8] A thermomechanical framework for constitutive models for rate–independent dissipative materials, International journal of Plasticity, Volume 16 (2000) no. 9, pp. 1017-1047 | Article | Zbl 0958.74011

[9] Principles of hyperplasticity: an approach to plasticity theory based on thermodynamic principles, Springer Verlag, London, 2006, 375 pages

[10] Plasticity: mathematical theory and numerical analysis, Springer, 1999 | Zbl 0926.74001

[11] The thermodynamics of plasticity and fracture, Cambridge University Press, 1992

[12] On unilateral constraints, friction and plasticity, New variational techniques in mathematical physics, CIME, 1974, pp. 173-322

[13] Stress-strain relations for clays: an energy theory, Géotechnique, Volume 17 (1967) no. 4, pp. 348-358 | Article

[14] On the generalized behaviour of ‘wet’ clay, Engineering Plasticity, Cambridge University Press, 1968, pp. 535-610 | Zbl 0233.73047

[15] Convex Analysis, Princeton University Press, NJ, 1970 | Article | Zbl 0229.90020

[16] New results in subdifferential calculus with applications to convex optimization, Applied Mathematics and Optimization, Volume 32 (1995) no. 3, pp. 213-234 | Article | MR 1340042 | Zbl 0828.49015

[17] Critical state soil mechanics, McGraw–Hill London, 1968

[18] Fundamentals of Soil Mechanics, Wiley, New York, 1948

[19] An Introduction to Thermomechanics, North Holland, Amsterdam, 1977 ((2nd edition 1983)) | Zbl 0358.73001