Normality Structures With Thermodynamic Equilibrium Points

2007 ◽  
Vol 74 (5) ◽  
pp. 965-971 ◽  
Author(s):  
Q. Yang ◽  
R. K. Wang ◽  
L. J. Xue
Keyword(s):  
Thermodynamic Equilibrium ◽  
Equilibrium Points ◽  
Thermodynamic System ◽  
Internal Variables ◽  
System A ◽  
Plastic Dissipation ◽  
Flow Potential ◽  
Yield Surfaces ◽  
Intrinsic Dissipation ◽  
Independent Behavior

Enriched by the nonlinear Onsager reciprocal relations and thermodynamic equilibrium points (Onsager, Phys. Rev., 37, pp. 405–406; 38, pp. 2265–2279), an extended normality structure by Rice (1971, J. Mech. Phys. Solids, 19, pp. 433–455) is established in this paper as a unified nonlinear thermodynamic theory of solids. It is revealed that the normality structure stems from the microscale irrotational thermodynamic fluxes. Within the extended normality structure, this paper focuses on the microscale thermodynamic mechanisms and significance of the convexity of flow potentials and yield surfaces. It is shown that the flow potential is convex if the conjugate force increment cannot not oppose the increment of the rates of local internal variables. For the Rice fluxes, the convexity condition reduces to the local rates being monotonic increasing functions with respect to their conjugate forces. The convexity of the flow potential provides the thermodynamic system a capability against the disturbance of the thermodynamic equilibrium point. It is proposed for time-independent behavior that the set of plastically admissible stresses determined by yield conditions corresponds to the set of thermodynamic equilibrium points. Based on that viewpoint, the intrinsic dissipation inequality is just the thermodynamic counterpart of the principle of maximum plastic dissipation and requires the convexity of the yield surfaces.

Internal Variable Theory Formulated by One Extended Potential Function

2020 ◽  
Vol 45 (3) ◽  
pp. 311-318
Author(s):  
Qiang Yang ◽  
Zhuofu Tao ◽  
Yaoru Liu
Keyword(s):  
Potential Function ◽  
Internal Variable ◽  
The State ◽  
Internal Variables ◽  
State Variables ◽  
Kinetic Rate ◽  
Variable Theory ◽  
Rate Laws ◽  
Flow Potential ◽  
Internal Variable Theory

AbstractIn the kinetic rate laws of internal variables, it is usually assumed that the rates of internal variables depend on the conjugate forces of the internal variables and the state variables. The dependence on the conjugate force has been fully addressed around flow potential functions. The kinetic rate laws can be formulated with two potential functions, the free energy function and the flow potential function. The dependence on the state variables has not been well addressed. Motivated by the previous study on the asymptotic stability of the internal variable theory by J. R. Rice, the thermodynamic significance of the dependence on the state variables is addressed in this paper. It is shown in this paper that the kinetic rate laws can be formulated by one extended potential function defined in an extended state space if the rates of internal variables do not depend explicitly on the internal variables. The extended state space is spanned by the state variables and the rate of internal variables. Furthermore, if the rates of internal variables do not depend explicitly on state variables, an extended Gibbs equation can be established based on the extended potential function, from which all constitutive equations can be recovered. This work may be considered as a certain Lagrangian formulation of the internal variable theory.

EFFECT OF ADDED TIP MASS ON THE NONLINEAR FLAPWISE AND CHORDWISE VIBRATION OF CANTILEVER COMPOSITE BEAM UNDER BASE EXCITATION

2012 ◽  
Vol 12 (02) ◽  
pp. 285-310 ◽  
Author(s):  
M. EFTEKHARI ◽  
M. MAHZOON ◽  
S. ZIAEI-RAD
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Laminated Composite ◽  
Base Excitation ◽  
System A ◽  
Autoparametric Resonance ◽  
The Stability ◽  
Tip Mass ◽  
Made In

In this paper, a comparative study is performed for a symmetrically laminated composite cantilever beam with and without a tip mass under harmonic base excitation. The base is subjected to both flapwise and chordwise excitations tuned to the primary resonances of the two directions and conditions of 2:1 autoparametric resonance. In the literature, the governing nonlinear equations of the same problem without tip mass have been derived using the extended Hamilton's principle. Extension is made in this study to include the effect of a tip mass on the response of the beam. The natural frequencies are obtained numerically using the diversity guided evolutionary algorithm (DGEA). Next, the multiple scales method is applied to determine the nonlinear response and stability of the system. A set of four first-order differential equations describing the modulation of the amplitudes and phases of interacting modes are derived for the perturbation analysis. For verification, the above equations are reduced to the special case of the cantilever beam without tip mass for comparison with existing results. Finally, the effect of the tip mass on the stability of the fixed points and on the amplitude of oscillation about the equilibrium points in both the frequency and force modulation responses is examined.

scholarly journals The dynamics of nutrient, toxic phytoplankton, nontoxic phytoplankton and zooplankton model

2016 ◽  
Vol 5 (1) ◽  
pp. 48
Author(s):  
Hasnaa Fiesal Mohammed Hussien ◽  
Raid Kamel Naji ◽  
Azhar Abbas Majeed
Keyword(s):  
Food Web ◽  
Nonlinear Differential Equations ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Toxic Substance ◽  
Global Dynamics ◽  
Defensive Strategy ◽  
Toxic Phytoplankton ◽  
System A ◽  
Aquatic Food

<p>The objective of this paper is to study the dynamical behavior of an aquatic food web system. A mathematical model that includes nutrients, phytoplankton and zooplankton is proposed and analyzed. It is assumed that, the phytoplankton divided into two compartments namely toxic phytoplankton which produces a toxic substance as a defensive strategy against predation by zooplankton, and a nontoxic phytoplankton. All the feeding processes in this food web are formulating according to the Lotka-Volterra functional response. This model is represented mathematically by the set of nonlinear differential equations. The existence, uniqueness and boundedness of the solution of this model are investigated. The local and global stability conditions of all possible equilibrium points are established. The occurrence of local bifurcation and Hopf bifurcation are investigated. Finally, numerical simulation is used to study the global dynamics of this model.</p>

scholarly journals A Derivation of the Main Relations of Nonequilibrium Thermodynamics

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Vladimir N. Pokrovskii
Keyword(s):  
Nonequilibrium Thermodynamics ◽  
Second Law Of Thermodynamics ◽  
Thermodynamic System ◽  
Internal Variables ◽  
Second Law ◽  
Basic Principles ◽  
First Law Of Thermodynamics ◽  
Sum Of Products ◽  
Thermodynamic Forces ◽  
Production Of Entropy

The principles of nonequilibrium thermodynamics are discussed, using the concept of internal variables that describe deviations of a thermodynamic system from the equilibrium state. While considering the first law of thermodynamics, work of internal variables is taken into account. It is shown that the requirement that the thermodynamic system cannot fulfil any work via internal variables is equivalent to the conventional formulation of the second law of thermodynamics. These statements, in line with the axioms introducing internal variables can be considered as basic principles of nonequilibrium thermodynamics. While considering stationary nonequilibrium situations close to equilibrium, it is shown that known linear parities between thermodynamic forces and fluxes and also the production of entropy, as a sum of products of thermodynamic forces and fluxes, are consequences of fundamental principles of thermodynamics.

Investigation on Modeling Thermal-Hydraulic System of CPR1000 NPP Based on RELAP5

2014 ◽  
Author(s):  
Cong Wang ◽  
Danmei Xie ◽  
Peng Zhang ◽  
Xinggang Yu ◽  
Xiuqun Hou
Keyword(s):  
Hydraulic System ◽  
Nuclear Power ◽  
Power Plants ◽  
Thermodynamic System ◽  
Pressure Variation ◽  
Actual Situation ◽  
Hydraulic Systems ◽  
System A ◽  
Variation Trends ◽  
Steam Parameter

Based on the best-estimate program RELAP5/MOD4.0, a full-scope thermal-hydraulic model with reference to CPR1000 nuclear power plant is established in this paper, which includes the thermal-hydraulic systems of conventional island as well as the primary nuclear island which has already been researched in traditional safety analysis. Therefore, this paper mainly details the numerical model of the turbine and other parts of the conventional island thermodynamic system. A comparison between the calculated results in steady-state and the actual data of reactor demonstrates a fine consistency, thus verifying the accuracy and reliability of the model. In addition, the steam parameter changes are numerically simulated during the steam turbine’s off-design operating condition such as back pressure variation and the variation trends are the same as the actual situation of nuclear power plants.

Two-Temperature Thermodynamics for Metal Viscoplasticity: Continuum Modeling and Numerical Experiments

2016 ◽  
Vol 84 (1) ◽  
Author(s):  
Shubhankar Roy Chowdhury ◽  
Gurudas Kar ◽  
Debasish Roy ◽  
J. N. Reddy
Keyword(s):  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Internal Variable ◽  
Thermodynamic System ◽  
Internal Variables ◽  
Face Centered Cubic ◽  
Dynamic Simulations ◽  
Face Centered ◽  
Atomic Lattices ◽  
Two Temperature

A physics-based model for dislocation mediated thermoviscoplastic deformation in metals is proposed. The modeling is posited in the framework of internal-variables theory of thermodynamics, wherein an effective dislocation density, which assumes the role of the internal variable, tracks permanent changes in the internal structure of metals undergoing plastic deformation. The thermodynamic formulation involves a two-temperature description of viscoplasticity that appears naturally if one considers the thermodynamic system to be composed of two weakly interacting subsystems, namely, a kinetic-vibrational subsystem of the vibrating atomic lattices and a configurational subsystem of the slower degrees-of-freedom (DOFs) of defect motion. Starting with an idealized homogeneous setup, a full-fledged three-dimensional (3D) continuum formulation is set forth. Numerical exercises, specifically in the context of impact dynamic simulations, are carried out and validated against experimental data. The scope of the present work is, however, limited to face-centered cubic (FCC) metals only.

scholarly journals A New 3D Autonomous Continuous System with Two Isolated Chaotic Attractors and Its Topological Horseshoes

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Ping Zhou ◽  
Meihua Ke
Keyword(s):  
Numerical Computation ◽  
Topological Entropy ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Continuous System ◽  
Chaotic Attractors ◽  
System A ◽  
Topological Horseshoes

Based on the 3D autonomous continuous Lü chaotic system, a new 3D autonomous continuous chaotic system is proposed in this paper, and there are coexisting chaotic attractors in the 3D autonomous continuous chaotic system. Moreover, there are no overlaps between the coexisting chaotic attractors; that is, there are two isolated chaotic attractors (in this paper, named “positive attractor” and “negative attractor,” resp.). The “positive attractor” and “negative attractor” depend on the distance between the initial points (initial conditions) and the unstable equilibrium points. Furthermore, by means of topological horseshoes theory and numerical computation, the topological horseshoes in this 3D autonomous continuous system is found, and the topological entropy is obtained. These results indicate that the chaotic attractor emerges in the new 3D autonomous continuous system.

The Explanation of Yield Surface Rotation of Sands by Principle of Thermodynamics with Internal Variables

2014 ◽  
Vol 580-583 ◽  
pp. 308-311
Author(s):  
Jing Yu Chen ◽  
Ying Hai
Keyword(s):  
Stress State ◽  
Yield Surface ◽  
Triaxial Compression ◽  
Constitutive Modelling ◽  
Saturated Soils ◽  
Internal Variables ◽  
Compression Stress ◽  
Surface Rotation ◽  
Saturated Sands ◽  
Yield Surfaces

<p class="p18">Using constitutive modelling principle of thermodynamics with internal variables, the yield surfaces rotation of saturated sands subjected triaxial compression stress state are justified and explained. The explanation for the yield surfaces rotation of saturated sands prove the correctness and feasibility for the principle of thermodynamics with internal variables to construct elastoplastic constitutive relation of saturated soils.</p>

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.

Application of a Robust Minimum-Order Observer to Control the Performance of an Uncertain Multivariable Boiler Unit

2010 ◽  
Author(s):  
Hamed Moradi ◽  
Firooz Bakhtiari-Nejad
Keyword(s):  
Linear Time ◽  
Mechanical Energy ◽  
Internal Variables ◽  
Feed Water ◽  
Boiler Unit ◽  
State Variables ◽  
Minimum Order ◽  
Linear Time Invariant ◽  
System A ◽  
Time Invariant

To obtain economic operation of power plant steam generators, output of mechanical energy must be balanced with the electrical load while maintaining the internal variables within desired ranges. During a boiler unit operation, dynamic variables such as drum pressure, steam temperature and water level of drum must be controlled to achieve an appropriate performance. In this paper, a linear time invariant (LTI) model of a boiler unit is considered where feed-water and fuel mass rates are the control inputs. Due to the inaccessibility of some state variables of boiler system, a robust minimum-order observer is designed to gain an estimate state of the true state. Real dynamic model of boiler unit may associate with parametric uncertainties. In this case, optimum region of poles of observer-based controller are found such that the robust performance of the boiler system against model uncertainties is guaranteed.

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