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

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.

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

2013 ◽  
Vol 353-356 ◽  
pp. 837-841 ◽  
Author(s):  
Jing Yu Chen ◽  
Ying Hai
Keyword(s):  
Free Energy ◽  
Constitutive Relation ◽  
Energy Function ◽  
Yield Function ◽  
Dissipation Function ◽  
Saturated Soils ◽  
Internal Variables ◽  
Free Energy Function ◽  
Modified Cam Clay ◽  
Cam Clay

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.

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

2014 ◽  
Vol 136 (2) ◽  
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.

A correlation between thermodynamic properties, thermal expansion and electrical resistivity of Ag–28% Cu nanopowders processed by the mechanical alloying route

2015 ◽  
Vol 17 (42) ◽  
pp. 28322-28330 ◽  
Author(s):  
Speranta Tanasescu ◽  
Alexandru Milea ◽  
Oana Gingu ◽  
Florentina Maxim ◽  
Cristian Hornoiu ◽  
...  
Keyword(s):  
Electrical Resistivity ◽  
Free Energy ◽  
Thermal Expansion ◽  
Thermodynamic Properties ◽  
Mechanical Alloying ◽  
Crystallite Size ◽  
Gibbs Free Energy ◽  
Energy Function ◽  
Free Energy Function

The relative electrical resistivity, Gibbs free energy function and crystallite size of the Ag–28% Cu nanopowders as a function of temperature.

Simulation of a Biological System On an Analog Computer

SIMULATION ◽  
1964 ◽  
Vol 2 (4) ◽  
pp. R-9-R-18
Author(s):  
E.C. DeLand
Keyword(s):  
Mathematical Model ◽  
Free Energy ◽  
Gibbs Free Energy ◽  
Energy Function ◽  
Biological System ◽  
Chemical Species ◽  
Analog Computer ◽  
Free Energy Function ◽  
The Mathematical Model ◽  
Method Of Steepest Descent

The purpose of this paper is to discuss a method for the construction of a mathematical model of a large biological system. This method, based on Gibbs' free energy hypothesis, uses the format of mathematical programming, while the actual computation is ac complished by the method of steepest descent. The biological system chosen to exemplify the mathe matical method was the respiratory function of the blood in the human lung. This method is based on the postulate that chemical mixtures tend toward a reaction equilibrium which minimizes the potential, or free energy, of the system. We may thus write down the classical Gibbs free energy function for each chemical species, and require that total free energy relative to some standard state be minimized under the conditions of the experiment. The solution of the equilibrium problem consists of a set of mole numbers which minimizes the free energy function, subject to equations for conservation of mass and nonnegativity. The analog computer solution of the respiration model was undertaken not only to give fast, sensitive tests of the mathematical model and its assumptions, but also to obtain a simulation of the time depend ent system. Examples of the mechanization of the equations are presented in this paper, and also results are computed for the static equilibrium of a canoni cal model.

scholarly journals Multiscale Approach for the Modeling of Chemo-Magneto-Thermo-Mechanical Couplings – Reversible Framework

2018 ◽  
Vol 941 ◽  
pp. 2290-2295 ◽  
Author(s):  
Olivier Hubert ◽  
Karine Labernhe-Taillard ◽  
Mame-Daro Fall ◽  
Xu Yang Chang ◽  
Maxime Savary ◽  
...  
Keyword(s):  
Free Energy ◽  
Gibbs Free Energy ◽  
Multiscale Modeling ◽  
Energy Function ◽  
Multiscale Approach ◽  
Free Energy Function ◽  
Unified Approach ◽  
Modeling Approach ◽  
Scale Change

We focus in this paper on a multiscale modeling approach of the materials’ reversible behavior involving couplings of the chemo-magneto-thermo-mechanical type. It is shown that it is possible to take into account a large variety of these coupled environments by a unified approach using the springs of the scale change and the build of an appropriate Gibbs free energy function. The approach is well suited to situations where some fields can be considered homogeneous at a relevant scale and where free deformation can be defined.

scholarly journals Thermodynamic properties prediction of Mg-Al-Zn melts based on the atom and molecule coexistence theory

2019 ◽  
Vol 55 (2) ◽  
pp. 135-145
Author(s):  
Man-Cang Zhang ◽  
Sheng-Chao Duan ◽  
Rong-Huan Xu ◽  
Ming Zou ◽  
Shi-Wen Dong ◽  
...  
Keyword(s):  
Free Energy ◽  
Thermodynamic Properties ◽  
Gibbs Free Energy ◽  
Energy Function ◽  
Composition Range ◽  
Mass Action ◽  
Free Energy Function ◽  
Structural Units ◽  
Coexistence Theory ◽  
Calculated Mass

A developed and verified thermodynamic model based on the atom and molecule coexistence theory (AMCT) is employed to predict activities relative to pure liquids in standard state in Mg-Al, Mg-Zn, Al-Zn and Mg-Al-Zn melts through the calculated mass action concentrations of structural units, i.e., Ni. According to AMCT, Ni can be extrapolated and calculated by the chemical equilibrium constant of a structural molecule, i.e., Ki, in the Mg-Al-Zn ternary system and binary subsystems. In this paper, the standard Gibbs free energy function, for reported activities and mixing thermodynamic properties in Mg-Al, Mg-Zn and Al-Zn melts, was regressed and optimized. The results showed that Ki and Ni were deduced by Gibbs free energy function at the studied temperature. The results of calculating thermodynamic properties in the full composition range for liquid Mg-Al-Zn from 880 to 1100 K, as well as Mg-Al from 923 to 1073 K, Mg-Zn from 880 to 973 K and Al-Zn from 1000 to 1073 K, are presented in the paper by coupling with Ni and AMCT. An excellent agreement is noticed between the calculated values of this study and measured thermodynamic data from the references, suggesting that the AMCT can be well applied to describe and predict the activities of the Mg-Al-Zn system and its subsystems.

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