Cyclic Viscoplastic Constitutive Equations, Part I: A Thermodynamically Consistent Formulation

1993 ◽  
Vol 60 (4) ◽  
pp. 813-821 ◽  
Author(s):  
J.-L. Chaboche
Keyword(s):  
Constitutive Equations ◽  
Evolution Equations ◽  
Internal Variables ◽  
Isotropic Hardening ◽  
State Variables ◽  
Inelastic Analysis ◽  
Hardening Process ◽  
Classical Models ◽  
Energy Dissipated ◽  
Consistent Formulation

Cyclic viscoplastic constitutive equations are increasingly used for the inelastic analysis of structures under severe thermomechanical conditions. The purpose of the paper is to show how the classical models can be modified in order to follow the general principles of thermodynamics with internal variables. Using the restrictive framework of standard generalized materials, the state variables associated to various kinds of kinematic and isotropic hardening are selected. The evolution equations for these internal variables are then formulated in a slightly less restrictive form. For each hardening process, the separation of the total plastic work into energy dissipated as heat and energy stored in the material is discussed in detail.

Parameter Evaluation for a Unified Constitutive Model

1993 ◽  
Vol 115 (2) ◽  
pp. 157-162 ◽  
Author(s):  
P. E. Senseny ◽  
N. S. Brodsky ◽  
K. L. DeVries
Keyword(s):  
Constitutive Model ◽  
Strain Rate ◽  
Least Squares ◽  
Constitutive Equations ◽  
Laboratory Data ◽  
Constant Strain Rate ◽  
Nonlinear Least Squares ◽  
Evolutionary Equation ◽  
Internal Variables ◽  
Isotropic Hardening

Parameters for the unified constitutive model MATMOD [1] were evaluated for rock salt (NaCl) by using nonlinear least squares to fit the model to isothermal laboratory data. MATMOD incorporates two internal variables that represent the effects of both kinematic and isotropic hardening. The constitutive equations contain nine parameters that must be evaluated to model isothermal deformation. Laboratory data from stress relaxation, constant strain rate, and long-term creep tests were used. The latter two test types included staged tests in which the strain rate or stress was changed step-wise during the test. The test conditions were precisely controlled by a computer and the constitutive equations were integrated to simulate the laboratory conditions closely. The MATMOD parameters were then evaluated by fitting the integrated equations to the laboratory data using nonlinear least squares. The model fits the data well, but the fit may be improved by changing the evolutionary equation for the internal variable that accounts for isotropic hardening.

Phenomenological Modeling of Hardening and Thermal Recovery in Metals

1988 ◽  
Vol 110 (1) ◽  
pp. 1-8 ◽  
Author(s):  
K. S. Chan ◽  
S. R. Bodner ◽  
U. S. Lindholm
Keyword(s):  
Evolution Equations ◽  
Thermal Recovery ◽  
Internal Variables ◽  
Isotropic Hardening ◽  
Nickel Base Superalloy ◽  
Cyclic Tests ◽  
Phenomenological Modeling ◽  
Nickel Base ◽  
Base Superalloy ◽  
Good Agreement

Modeling of hardening and thermal recovery in metals is considered within the context of unified elastic-viscoplastic theories. Specifically, the choices of internal variables and hardening measures, and the resulting hardening response obtained by incorporating saturation-type evolution equations into two general forms of the flow law are examined. Based on the analytical considerations, a procedure for delineating directional and isotropic hardening from uniaxial hardening data has been developed for the Bodner-Partom model and applied to a nickel-base superalloy, B1900 + Hf. Predictions based on the directional hardening properties deduced from the monotonic loading data are shown to be a good agreement with results of cyclic tests.

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.

scholarly journals An extension of Gronwall inequality in the theory of bodies with voids

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1161-1167
Author(s):  
Marin Marin ◽  
Praveen Ailawalia ◽  
Ioan Tuns
Keyword(s):  
Boundary Value Problem ◽  
Internal State ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniqueness Result ◽  
Internal Variables ◽  
State Variables ◽  
Gronwall’S Inequality ◽  
Initial Boundary Value

Abstract In this paper, we obtain a generalization of the Gronwall’s inequality to cover the study of porous elastic media considering their internal state variables. Based on some estimations obtained in three auxiliary results, we use this form of the Gronwall’s inequality to prove the uniqueness of solution for the mixed initial-boundary value problem considered in this context. Thus, we can conclude that even if we take into account the internal variables, this fact does not affect the uniqueness result regarding the solution of the mixed initial-boundary value problem in this context.

A Dislocation Density Based Constitutive Model for Cyclic Deformation

1996 ◽  
Vol 118 (4) ◽  
pp. 441-447 ◽  
Author(s):  
Y. Estrin ◽  
H. Braasch ◽  
Y. Brechet
Keyword(s):  
Cyclic Loading ◽  
Dislocation Density ◽  
Constitutive Model ◽  
Constitutive Equations ◽  
Cyclic Deformation ◽  
Internal Variables ◽  
Density Evolution ◽  
Alloy 800H ◽  
Material Response ◽  
Dislocation Densities

A new constitutive model describing material response to cyclic loading is presented. The model includes dislocation densities as internal variables characterizing the microstructural state of the material. In the formulation of the constitutive equations, the dislocation density evolution resulting from interactions between dislocations in channel-like dislocation patterns is considered. The capabilities of the model are demonstrated for INCONEL 738 LC and Alloy 800H.

TUTORIAL ON NEUROBIOLOGY: FROM SINGLE NEURONS TO BRAIN CHAOS

1992 ◽  
Vol 02 (03) ◽  
pp. 451-482 ◽  
Author(s):  
WALTER J. FREEMAN
Keyword(s):  
Dynamic Range ◽  
Basins Of Attraction ◽  
State Transitions ◽  
State Variables ◽  
Dynamic Range Compression ◽  
The Core ◽  
Neuron Populations ◽  
Classical Models ◽  
Sigmoid Curve ◽  
Trigger Zone

Those classical models are reviewed that are most widely used by neurobiologists to explain the dynamics of neurons and neuron populations, and by modelers to implement artificial neural networks. Each neuron has input fibers called dendrites that integrate and an axon that transmits the output. The differing fiber architectures reflect these dissimilar dynamic operations. The basic tools to describe them are the RC model of the membrane, the core conductor model of the fibers, the Hodgkin–Huxley model of the trigger zone, and the modifiable synapse. Populations additionally require description of macroscopic state variables, the types of nonlinearity (most importantly the sigmoid curve and the dynamic range compression at the input to the cortex), and the types and strengths of connections. The properties of these neural masses can be characterized with the tools of nonlinear dynamics. These include description of point, limit cycle, and chaotic attractors for the cerebal cortex, as well as the types and mechanisms for the state transitions between basins of attraction during learning and perception.

On the Modeling of Evolving Elastic and Plastic Anisotropy in the Finite Strain Regime – Application to Deep Drawing Processes

2012 ◽  
Vol 504-506 ◽  
pp. 679-684 ◽  
Author(s):  
Ivaylo N. Vladimirov ◽  
Michael P. Pietryga ◽  
Vivian Tini ◽  
Stefanie Reese
Keyword(s):  
Deep Drawing ◽  
Evolution Equations ◽  
Finite Strain ◽  
Kinematic Hardening ◽  
Plastic Anisotropy ◽  
Material Model ◽  
Internal Variables ◽  
Time Step ◽  
Finite Element Software ◽  
Structure Tensors

In this work, we discuss a finite strain material model for evolving elastic and plastic anisotropy combining nonlinear isotropic and kinematic hardening. The evolution of elastic anisotropy is described by representing the Helmholtz free energy as a function of a family of evolving structure tensors. In addition, plastic anisotropy is modelled via the dependence of the yield surface on the same family of structure tensors. Exploiting the dissipation inequality leads to the interesting result that all tensor-valued internal variables are symmetric. Thus, the integration of the evolution equations can be efficiently performed by means of an algorithm that automatically retains the symmetry of the internal variables in every time step. The material model has been implemented as a user material subroutine UMAT into the commercial finite element software ABAQUS/Standard and has been used for the simulation of the phenomenon of earing during cylindrical deep drawing.

Constructing Poisson and Dissipative Brackets of Mixtures by using Lagrangian-to-Eulerian Transformation

2010 ◽  
Vol 26 (2) ◽  
pp. 219-228
Author(s):  
K.-C. Chen
Keyword(s):  
Evolution Equations ◽  
Poisson Brackets ◽  
State Variables ◽  
Component Mixture ◽  
Dissipative Part ◽  
Canonical Variables ◽  
Lagrangian Variable ◽  
Two Component ◽  
Lagrangian Form

AbstractThis paper aims to construct the bracket formalism of mixture continua by using the method of Lagrangian- to-Eulerian (LE) transformation. The LE approach first builds up the transformation relations between the Eulerian state variables and the Lagrangian canonical variables, and then transforms the bracket in Lagrangian form to the bracket in Eulerian form. For the conservative part of the bracket formalism, this study systematically generates the noncanonical Poisson brackets of a two-component mixture. For the dissipative part, we deduce the Eulerian-variable-based dissipative brackets for viscous and diffusive mechanisms from their Lagrangian-variable-based counterparts. Finally, the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, are derived by constructing its Poisson and dissipative brackets.

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