A thermomechanical framework for rate-independent dissipative materials with internal functions A.M. Puzrin a, *, G.T. Houlsby b a Technion, Israel Institute of Technology, Faculty of Civil Engineering, 32000 Haifa, Israel b Oxford University, Oxford, UK Recieved in ®nal revised form 8 May 2000 Abstract This paper builds on previous work by Houlsby and Puzrin (Int. J. Plasticity 16 (2000) 1017) in which a framework was set out for the derivation of rate-independent plasticity the- ory from thermodynamic considerations. A key feature of the formalism is that the entire constitutive response is determined by knowledge of two scalar functions. The loading history is eectively captured through the use of internal variables. In this paper, we extend the con- cept of internal variables to that of internal functions, which represent in®nite numbers of internal variables. In this case the thermodynamic functions are replaced by functionals. We set out the formalism necessary to derive constitutive behaviour within this approach. The principal advantages of this development is that it can provide realistic modelling of kinematic hardening eects and smooth transitions between elastic and elastic±plastic behaviour. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Thermodynamics; B. Constitutive behaviour; B. Elastic±plastic material; C. Energy methods; 1. Introduction Houlsby and Puzrin (2000) set out a general framework for the derivation of the constitutive behaviour of rate-independent elastic±plastic materials within the set- ting of thermodynamics. A key element of the approach that they developed was that the constitutive behaviour is entirely speci®ed by the knowledge of twofunctions: an energyfunction and either the dissipationfunction or the yieldfunction. They set out how the incremental constitutive behaviour could be derived from these functions. They also explored the relationships between alternative forms of the energy func- International Journal of Plasticity 17 (2001) 1147±1165 www.elsevier.com/locate/ijplas 0749-6419/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0749-6419(00)00083-8 * Corresponding author. Fax:+972-4-823-7149. E-mail address: puzrin@tx.technion.ac.il (A.M. Puzrin).