The Modified Cam-Clay Model and the Constitutive Relation Principle of Thermodynamics with Internal Variables

According the theory of thermodynamics with internal variables, the relation between yield function and dissipation function and the condition of associated flow rule in stress space are presented; the elastoplastic matrix of the incremental form of the material constitutive equation is given out, this matrix is determined by the free energy function and the yield function. The Gibbs free energy function of solid phase of saturated soils subjected triaxial compression stress state is presented, and using the constitutive theory of thermodynamics with internal variables, yield function and stress-strain relation of the modified Cam-Clay model is obtained by the free energy function and the dissipation function. These results prove the correctness and feasibility for this constitutive theory to construct elastoplastic constitutive relation of saturated soils.

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Time-Independent Plasticity Related to Critical Point of Free Energy Function and Functional

Journal of Engineering Materials and Technology ◽

10.1115/1.4026232 ◽

2014 ◽

Vol 136 (2) ◽

Cited By ~ 1

Author(s):

Q. Yang ◽

Y. R. Liu ◽

X. Q. Feng ◽

S. W. Yu

Keyword(s):

Free Energy ◽

Critical Point ◽

Thermodynamic Equilibrium ◽

Energy Function ◽

Hessian Matrix ◽

Energy Functional ◽

Internal Variables ◽

Free Energy Function ◽

Free Energy Functional ◽

Appl Math

In this paper, time-independent plasticity is addressed within the thermodynamic framework with internal variables 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). It is shown in this paper that the existence of a free energy function along with thermodynamic equilibrium conditions directly leads to associated flow rules. The time-independent inelastic behaviors can be fully determined by the Hessian matrix at the nondegenerate critical point of the free energy function. The normality rule of Hill and Rice (1973, “Elastic Potentials and the Structure of Inelastic Constitutive Laws,” SIAM J. Appl. Math., 25, pp. 448–461) or the Il'yushin (1961, “On a Postulate of Plasticity,” J. Appl. Math. Mech. 25, pp. 746–750) postulate is just a stability requirement of the thermodynamic equilibrium. The existence of a free energy functional which is not a direct function of the internal variables, along with thermodynamic equilibrium conditions also leads to associated flow rules. The time-independent inelastic behaviors with the free energy functional can be fully determined by the quasi Hessian matrix at the quasi critical point of the free energy functional. With the free energy functional, the thermodynamic forces conjugate to the internal variables are nonconservative and are constructed based on Darboux theorem. Based on the constructed nonconservative forces, it is shown that there may exist several possible thermodynamic equilibrium mechanisms for the thermodynamic system of the material sample. Therefore, the associated flow rules based on free energy functionals may degenerate into nonassociated flow rules. The symmetry of the conjugate forces plays a central role for the characteristics of time-independent plasticity.

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Time-Independent Plasticity Formulated by Inelastic Differential of Free Energy Function

Journal of Non-Equilibrium Thermodynamics ◽

10.1515/jnet-2020-0076 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Qiang Yang ◽

Chaoyi Li ◽

Yaoru Liu

Keyword(s):

Free Energy ◽

Gibbs Free Energy ◽

Stability Condition ◽

Differential Form ◽

Energy Function ◽

Potential Functions ◽

Internal Variables ◽

Free Energy Function ◽

Gibbs Equation ◽

Potential Representation

Abstract The authors presented a time-independent plasticity approach, where a typical plastic-loading process is viewed as an infinitesimal state change of two neighboring equilibrium states, and the yield and consistency conditions are formulated based on the conjugate forces of the internal variables. In this paper, a stability condition is proposed, and the yield, consistency, and stability conditions are reformatted by the inelastic differential form of the Gibbs free energy. The Gibbs equation in thermodynamics with internal variables is a representation to the differential form of the Gibbs free energy by a single Gibbs free energy function. In this paper, we propose the so-called extended Gibbs equation, where the differential form may be represented by multiple potential functions. Various associated and nonassociated plasticity with a single or multiple yield functions can be derived from various representations based on the reformulated approach, where yield and plastic potential functions are in the form of inelastic differentials of the potential functions. The generalized Drucker inequality can only be derived from the one-potential representation as a stability condition. For a multiple-potential representation, the stability condition can be ensured if the multiple potentials are concave functions and possess the same stationary point.

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AN APPLICATION OF HYPERPLASTICITY TO DETERMINE THE BEHAVIOR OF A PLANE STRESS CANTILEVER BEAM

Proceedings of International Structural Engineering and Construction ◽

10.14455/isec.res.2016.146 ◽

2016 ◽

Vol 3 (2) ◽

Author(s):

Hai Than Nguyen ◽

Lam Nguyen Sy

Keyword(s):

Cantilever Beam ◽

Energy Function ◽

Kinematic Hardening ◽

Scalar Potential ◽

Yield Function ◽

Dissipation Function ◽

Surface Model ◽

Free Energy Function ◽

Sophisticated Model ◽

Von Mises

Hyperplasticity is an approach to plasticity theory based on thermodynamic principles. By using this concept, the entire constitutive model behavior can be determined from two scalar potential functions, one is the energy function and the other is the yield function or the dissipation function. In this paper, the hyperplasticity model for homogeneous and isotropic material is derive and implemented in finite element method (FEM) to solve a particular problem in structure analysis, the behavior of a cantilever beam. The Gibbs free energy function and the Von Mises yield function are chosen to derive the elasto-plastic response of structures. Besides, kinematic hardening with single yield surface model is employed to describe the behavior when yield occur. The result shows that the application of hyperplasticity to FEM can be another proper way for non-linear analysis. Moreover, it reveals the ability to develop more sophisticated model using multiple yield surfaces in cases of cyclic loading.

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A canonical ensemble derivation of the McMillan-Mayer solution theory

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19832888 ◽

1983 ◽

Vol 48 (10) ◽

pp. 2888-2892 ◽

Cited By ~ 2

Author(s):

Vilém Kodýtek

Keyword(s):

Free Energy ◽

Partition Function ◽

Energy Function ◽

Special Function ◽

Helmholtz Free Energy ◽

Canonical Ensemble ◽

Free Energy Function ◽

Solution Theory ◽

Pure Solvent ◽

The Common

A special free energy function is defined for a solution in the osmotic equilibrium with pure solvent. The partition function of the solution is derived at the McMillan-Mayer level and it is related to this special function in the same manner as the common partition function of the system to its Helmholtz free energy.

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Run-and-Tumble Motion: The Role of Reversibility

Journal of Statistical Physics ◽

10.1007/s10955-021-02787-1 ◽

2021 ◽

Vol 183 (3) ◽

Author(s):

Bart van Ginkel ◽

Bart van Gisbergen ◽

Frank Redig

Keyword(s):

Free Energy ◽

Diffusion Coefficient ◽

Large Deviations ◽

Energy Function ◽

Internal State ◽

Rate Function ◽

Free Energy Function ◽

Large Deviations Principle ◽

Preferred Direction ◽

State Process

AbstractWe study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

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