Constitutive Relations for the Nonelastic Deformation of Metals

1976 ◽  
Vol 98 (3) ◽  
pp. 193-202 ◽  
Author(s):  
E. W. Hart
Keyword(s):  
Strain Rate ◽  
Real Time ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
State Variables ◽  
Simple Manner ◽  
Loading Direction ◽  
Deformation State ◽  
Experimental Background

Constitutive relations for metal nonelastic deformation are proposed. The descriptiion is entirely in terms of real time strain rate. The equations are fully three-dimensional, and the description is incremental. Changes of loading direction are treated in a simple manner. The current deformation state of the metal is characterized by a set of state variables that themselves evolve by well defined laws. The experimental background for the theory is also described.

scholarly journals Phase Transition at the Metric Elastic Universe

1996 ◽  
Vol 168 ◽  
pp. 569-570
Author(s):  
Alexander Gusev
Keyword(s):  
Phase Transition ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Vacuum State ◽  
Space Time ◽  
Deformation Tensor ◽  
De Sitter ◽  
Initial State ◽  
Deformation State ◽  
The Universe

At the last time the concept of the curved space-time as the some medium with stress tensor σαβon the right part of Einstein equation is extensively studied in the frame of the Sakharov - Wheeler metric elasticity(Sakharov (1967), Wheeler (1970)). The physical cosmology pre- dicts a different phase transitions (Linde (1990), Guth (1991)). In the frame of Relativistic Theory of Finite Deformations (RTFD) (Gusev (1986)) the transition from the initial stateof the Universe (Minkowskian's vacuum, quasi-vacuum(Gliner (1965), Zel'dovich (1968)) to the final stateof the Universe(Friedmann space, de Sitter space) has the form of phase transition(Gusev (1989) which is connected with different space-time symmetry of the initial and final states of Universe(from the point of view of isometric groupGnof space). In the RTFD (Gusev (1983), Gusev (1989)) the space-time is described by deformation tensorof the three-dimensional surfaces, and the Einstein's equations are viewed as the constitutive relations between the deformations ∊αβand stresses σαβ. The vacuum state of Universe have the visible zero physical characteristics and one is unsteady relatively quantum and topological deformations (Gunzig & Nardone (1989), Guth (1991)). Deformations of vacuum state, identifying with empty Mikowskian's space are described the deformations tensor ∊αβ, wherethe metrical tensor of deformation state of 3-geometry on the hypersurface, which is ortogonaled to the four-velocityis the 3 -geometry of initial state,is a projection tensor.

Constitutive Modeling of a Thermoplastic Olefin Over a Broad Range of Strain Rates

2006 ◽  
Vol 128 (4) ◽  
pp. 551-558 ◽  
Author(s):  
Yan Wang ◽  
Ellen M. Arruda
Keyword(s):  
Strain Rate ◽  
Strain Hardening ◽  
Three Dimensional ◽  
Elastic Response ◽  
Strain Rates ◽  
Strain Behavior ◽  
Rate Dependent ◽  
State Dependent ◽  
Deformation State ◽  
Ann Arbor

A microstructually motivated, three-dimensional, large deformation, strain rate dependent constitutive model has been developed for a semi-crystalline, blended, thermoplastic olefin (TPO) (Wang, Y., 2002, Ph.D. thesis, The University of Michigan, Ann Arbor, MI). Various experiments have been conducted to characterize the TPO and to verify the modeling approach (Wang, Y., 2002, Ph.D. thesis, The University of Michigan, Ann Arbor, MI). The model includes a quantitative rate-dependent Young’s modulus, a nonlinear viscoelastic response between initial linear elastic response and yield due to inherent microstructural irregularity, rate and temperature dependent yield with two distinctive yield mechanisms for low and high strain rates, temperature-dependent strain hardening, plastic deformation of crystalline regions, and adiabatic heating. It has been shown to accurately capture the observed TPO stress-strain behavior including the rate-dependent initial linear elastic response; temperature, strain rate, and deformation state-dependent yield; temperature and deformation state-dependent strain hardening; and pronounced thermal softening effects at high (impact) strain rates. The model has also been examined for its ability to predict the response in plane strain compression based on material parameters chosen to capture the uniaxial compression response. The model is predictive of the initial strain rate dependent stiffness, yield, and strain hardening responses in plane strain. Such predictive capability demonstrates the versatility with which this model captures the three-dimensional anisotropic nature of TPO stress-strain behavior.

A Simplified Method to Account for Plastic Rate Sensitivity With Large Deformations

1979 ◽  
Vol 46 (4) ◽  
pp. 811-816 ◽  
Author(s):  
N. Perrone ◽  
P. Bhadra
Keyword(s):  
Strain Rate ◽  
Large Deformations ◽  
Three Dimensional ◽  
Rate Sensitivity ◽  
Approximate Solutions ◽  
Dimensional Structure ◽  
Initial Kinetic Energy ◽  
Maximum Strain ◽  
Membrane Action ◽  
Deformation State

A string supported impulsively loaded mass is used to study large deformation rate sensitivity effects where membrane action is dominant. It is found that an overall correction factor can be devised using physical properties associated with the average strain rate. Maximum strain rate occurs with a velocity field corresponding to the deformation state wherein half the initial kinetic energy has been dissipated. (If V0 is initial velocity, V0/2 is associated with maximum strain rate.) Exact and approximate solutions for a broad range of parameters serve to illustrate and verify the procedure. A discussion is presented to show how the same methodology could also be applied via a modal approach to an arbitrary three-dimensional structure undergoing large deformations, if the primary mechanism for energy absorption is from membrane action.

scholarly journals Soft catenaries

2011 ◽  
Vol 691 ◽  
pp. 165-177 ◽  
Author(s):  
Ken Kamrin ◽  
L. Mahadevan
Keyword(s):  
Three Dimensional ◽  
Maxwell Fluid ◽  
State Variables ◽  
Minimal Models ◽  
Dimensional Solution ◽  
Slender Body Theory ◽  
Deformation State ◽  
Dimensional Finite Element ◽  
Complete Deformation ◽  
Three Dimensional Finite Element

AbstractUsing the classical catenary as a motivating example, we use slender-body theory to derive a general theory for thin filaments of arbitrary rheology undergoing large combined stretching and bending, which correctly accounts for the nonlinear geometry of deformation and uses integrated state variables to properly represent the complete deformation state. We test the theory for soft catenaries made of a Maxwell fluid and an elastic yield-stress fluid using a combination of asymptotic and numerical analyses to analyse the dynamics of transient sagging and arrest. We validate our results against three-dimensional finite element simulations of drooping catenaries, and show that our minimal models are easier and faster to solve, can capture all the salient behaviours of the full three-dimensional solution, and provide physical insights into the basic mechanisms involved.

scholarly journals Simultaneous Invariants of Strain and Rotation Rate Tensors and Their Admitted Region

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Igor Vigdorovich ◽  
Holger Foysi
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Turbulent Flows ◽  
Reynolds Stress ◽  
Building Block ◽  
Rotation Rate ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Strain Rate Tensor ◽  
Reynolds Stress Tensor

The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensorSand rotation rate tensorΩand calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensorsSandΩare studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensorsSandΩ, in which the scalar coefficients depend on simultaneous invariants of these tensors.

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