# Including friction in the mathematics of classical plasticity

H W Chandler, C M Sands

Research output: Contribution to journalArticle

9 Citations (Scopus)

### Abstract

In classical plasticity there are clear mathematical links between the dissipation function and the consequent yield function and flow rule. These links help to construct constitutive equations with the minimum of adjustable parameters. Modelling granular materials, however, requires that the dissipation function depends on the current stress state (frictional plasticity) and this changes the mathematical structure—altering the links and invalidating the associated flow rule. In this paper we show, for a large family of dissipation functions, how much of the structure remains intact when frictional dissipation is included. The surviving links are examined using straightforward physically based graphical insight and well-established mathematical techniques leading to a central result, which provides a mathematical justification for the procedural features of hyperplasticity. This should allow hyperplasticity to be used more widely and certainly with increased confidence.

As an example of the effectiveness of the general method, two specific dissipation functions are constructed from the simple physical concepts of sliding friction and granule damage. One is based on a Drucker–Prager cone and the other a Matsuoka–Nakai cone, both incorporate kinematic hardening and a compactive cap. In each Case a single smooth yield function with consistent flow rules is produced. The computational usefulness of an inequality derived in the paper is demonstrated in the generation of the figures showing yield surfaces and flow directions by means of a simple maximization procedure.
Original language English 53-72 20 International Journal for Numerical and Analytical Methods in Geomechanics 34 1 26 May 2009 https://doi.org/10.1002/nag.806 Published - Jan 2010

### Fingerprint

mathematics
Plasticity
plasticity
dissipation
friction
Friction
Cones
constitutive equation
hardening
Granular materials
sliding
Constitutive equations
kinematics
Hardening
Kinematics
damage
modeling

### Keywords

• Legendre–Fenchel
• constitutive behaviour
• friction
• granular material
• optimization
• yield condition

### Cite this

In: International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 34, No. 1, 01.2010, p. 53-72.

Research output: Contribution to journalArticle

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AB - In classical plasticity there are clear mathematical links between the dissipation function and the consequent yield function and flow rule. These links help to construct constitutive equations with the minimum of adjustable parameters. Modelling granular materials, however, requires that the dissipation function depends on the current stress state (frictional plasticity) and this changes the mathematical structure—altering the links and invalidating the associated flow rule. In this paper we show, for a large family of dissipation functions, how much of the structure remains intact when frictional dissipation is included. The surviving links are examined using straightforward physically based graphical insight and well-established mathematical techniques leading to a central result, which provides a mathematical justification for the procedural features of hyperplasticity. This should allow hyperplasticity to be used more widely and certainly with increased confidence.As an example of the effectiveness of the general method, two specific dissipation functions are constructed from the simple physical concepts of sliding friction and granule damage. One is based on a Drucker–Prager cone and the other a Matsuoka–Nakai cone, both incorporate kinematic hardening and a compactive cap. In each Case a single smooth yield function with consistent flow rules is produced. The computational usefulness of an inequality derived in the paper is demonstrated in the generation of the figures showing yield surfaces and flow directions by means of a simple maximization procedure.

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