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Research Papers

# Hamilton’s Principle of Entropy Production for Creep and Relaxation Processes

[+] Author and Article Information
Q. Yang1

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, P. R. Chinayangq@tsinghua.edu.cn

Y. R. Liu, J. Q. Bao

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, P. R. China

An overall discussion on thermodynamic state space has been provided by Muschik et al. (7).

The introduction of Lagrange formalism into the irreversible thermodynamics in full state space can be found in Refs. 9-10, etc.

The conjugate forces given by Eq. 5 are conservative. It should be noted that the conjugate forces can be nonconservative. Rice (2,11) and Hill et al. (12) addressed this possibility. They showed that the normality structures established in Sec. 3 still hold for nonconservative conjugate forces.

Creep or relaxation usually refers to fixed uniaxial stressing or straining on materials in rheology. The creep or relaxation defined in this paper as fixed stress tensor $σ$ or strain tensor $ε$ and temperature $ϑ$ on materials can be viewed as a triaxial generalization.

1

Corresponding author.

J. Eng. Mater. Technol 132(1), 011018 (Dec 09, 2009) (5 pages) doi:10.1115/1.4000302 History: Received August 12, 2008; Revised March 19, 2009; Published December 09, 2009; Online December 09, 2009

## Abstract

In this paper, two subspaces of the state space of constrained equilibrium states for solids are proposed and addressed. One subspace, constrained affinity space, is conjugate-force space with fixed temperature and internal variable. It is revealed in this paper that the remarkable properties of the kinetic rate laws of scalar internal variables, established by Rice (1971, “Inelastic Constitutive Relations for Solids: An Internal Variable Theory and Its Application to Metal Plasticity,” J. Mech. Phys. Solids, 19, pp. 433–455) and elaborated by Yang (2005, “Normality Structures With Homogeneous Kinetic Rate Laws,” ASME J. Appl. Mech., 72, pp. 322–329; 2007, “Normality Structures With Thermodynamic Equilibrium Points,” ASME J. Appl. Mech., 74, pp. 965–971), are all located in constrained affinity space. Furthermore, the flow potential function monotonically increases along any ray from the origin in constrained affinity space. Another subspace, constrained configuration space, is the state space with fixed external variables. It is shown that the specific free and complementary energies monotonically decrease and increase, respectively, along the path of motion of the thermodynamic system of the material sample in constrained configuration space. For conservative conjugate forces, Hamilton’s action principle is established in constrained configuration space, and the action is the entropy production of the thermodynamic system in a time interval. The thermodynamic processes in constrained configuration space are just creep or relaxation processes of materials. The Hamilton principle can be considered as a fundamental principle of rheology.

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