Historical review of internal state variable theory for inelasticity Mark F. Horstemeyer * , Douglas J. Bammann Mechanical Engineering Department, Mail Stop 9552, 210 Carpenter Building, Mississippi State University, Mississippi State, MS 39762, USA article info Article history: Received 24 September 2009 Received in final revised form 2 June 2010 Available online 7 July 2010 Keywords: Inelastic internal state variable Constitutive model Thermodynamics Creep-plasticity Continuum damage mechanics abstract A review of the development and the usages of internal state variable (ISV) theory are pre- sented in this paper. The history of different developments leading up the formulation of the watershed paper by Coleman and Gurtin is discussed. Following the Coleman and Gur- tin thermodynamics, different researchers have employed the ISV theory for dislocations, creep, continuum damage mechanics (CDM), unified-creep-plasticity (UCP), polymers, composites, biomaterials, particulate materials, multiphase and multiphysics materials, materials processing, multiscale modeling, integrating materials science (structure–prop- erty relations) into applied mechanics formulations, and design optimization under uncer- tainty for use in practical engineering applications. Ó 2010 Elsevier Ltd. All rights reserved. 1. Historical setting Internal state variable (ISV) theory has been growing in its influence over the past 20–30 years. The confluence of in- creased computing power, finite element method developments, and experimental validation methods have positioned ISV theory to have great impact in the design of thermomechanical structural components and failure analysis. Fig. 1 shows a summary roadmap of the birth of ISV theory and how different phenomena and mechanisms have been brought into the ISV theory over the years. What is ISV theory? Essentially, it is a material model (or constitutive model) that can capture the mechanical history of a material and be used to predict mechanical properties such as strength and failure of a material. It can be used for elastic behavior but its strength is for inelasticity, such as plasticity, viscoelasticity, viscoplasticity, creep, and damage. It really can be used for any material (polymer-based, ceramic-based, and metal alloys). The convergence of thermodynamics, kine- matics, kinetics, and mechanics provided the impetus for the mathematical formality of ISV theory as presented by Coleman and Gurtin (1967). From that time, ISV theory was used to address different material systems over time. 2. Thermodynamics The inelastic internal state variable (ISV) theory owes much of its development to the early thermodynamic works of Car- not (1824), Joule (1843), and Clausius (1850), which later lead to the development of state variable thermodynamics con- structed by Helmholtz (1847), Maxwell (1875), Kelvin Thomson (1851), and Gibbs (1873a,b) in the late 19th century and early 20th century. The notion of ISV was morphed into thermodynamics by Onsager (1931) and was applied to continuum mechanics by Eckart (1940, 1948). The basic idea behind the theory of an ISV is that, in order to uniquely define the Helmholtz free energy of a system undergoing an irreversible process, one has to expand the dimensions of the state space of deformation and temperature 0749-6419/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2010.06.005 * Corresponding author. Tel.: +1 662 325 7308; fax: +1 662 325 7223. E-mail address: mfhorst@me.msstate.edu (M.F. Horstemeyer). International Journal of Plasticity 26 (2010) 1310–1334 Contents lists available at ScienceDirect International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas