From Instability and Time Dependence on the Microscale to Stability and Time Independence on the Macroscale

1995 ◽  
Vol 117 (4) ◽  
pp. 368-372 ◽  
Author(s):  
Daniel C. Drucker
Keyword(s):  
Time Dependence ◽  
Inelastic Deformation ◽  
Factor Of Safety ◽  
Constitutive Relations ◽  
Plasticity Theory ◽  
Time Dependent ◽  
Dynamic Activity ◽  
Micro Scale ◽  
Simplifying Assumptions ◽  
Structural Metals

The quasistatic inelastic deformation of ductile structural metals observed on the macroscale reflects a diversity of dynamic inelastic effects on the microscale. The generation, motion, and immobilization of dislocations are primary among them, but a host of other activities such as the opening and growth of cracks and voids, also may contribute. Dynamic activity on the microscale is strongly time-dependent on the time scales of importance to the microscopic processes. Also, the atomic configurations of single dislocations and groups of dislocations are highly unstable over a significant portion of each path of rapid motion. Nevertheless, engineers continue to design structures and machines with a reasonable factor of safety against failure on the basis of conventional plasticity theory with its assumption of both time-independence and stability (normality and convexity). This discussion of the validity of these simplifying assumptions for macroscopic constitutive relations despite instability and time-dependence on the atomic- and micro-scale expands upon a recent paper with Ming Li.

Constitutive Relationships for the Time-Dependent Deformation of Metals

1976 ◽  
Vol 98 (1) ◽  
pp. 47-51 ◽  
Author(s):  
A. R. S. Ponter ◽  
F. A. Leckie
Keyword(s):  
High Temperature ◽  
Strain Hardening ◽  
Potential Function ◽  
Inelastic Deformation ◽  
Kinematic Hardening ◽  
Constitutive Relations ◽  
Thermal Softening ◽  
Time Dependent ◽  
Polycrystalline Metal ◽  
State Potential

The paper discusses the constitutive relations for the inelastic deformation of a polycrystalline metal at high temperature. Commencing from a description of a dislocation structure in terms of strain hardening and thermal softening, the general form of the constitutive relation is developed in terms of a potential function. The existence of a stationary state potential is established and generalizations of isotropic and kinematic hardening are described.

scholarly journals Renormalization Group Approach to Pandemics as a Time-Dependent SIR Model

2021 ◽  
Vol 8 ◽  
Author(s):  
Michele Della Morte ◽  
Francesco Sannino
Keyword(s):  
Renormalization Group ◽  
Time Dependence ◽  
Recovery Rate ◽  
Sir Model ◽  
Time Dependent ◽  
Cumulative Number ◽  
Group Approach ◽  
Renormalization Group Approach

We generalise the epidemic Renormalization Group framework while connecting it to a SIR model with time-dependent coefficients. We then confront the model with COVID-19 in Denmark, Germany, Italy and France and show that the approach works rather well in reproducing the data. We also show that a better understanding of the time dependence of the recovery rate would require extending the model to take into account the number of deaths whenever these are over 15% of the cumulative number of infected cases.

scholarly journals The Diffraction of Transient Electro-Magnetic Waves by a Wedge

1958 ◽  
Vol 11 (2) ◽  
pp. 95-103 ◽  
Author(s):  
A. C. Butcher ◽  
J. S. Lowndes
Keyword(s):  
Wave Equation ◽  
Time Dependence ◽  
Half Plane ◽  
Line Source ◽  
Time Dependent ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Electro Magnetic ◽  
Infinite Wedge ◽  
Magnetic Waves

Much of the work on the theory of diffraction by an infinite wedge has been for cases of harmonic time-dependence. Oberhettinger (1) obtained an expression for the Green's function of the wave equation in the two dimensional case of a line source of oscillating current parallel to the edge of a wedge with perfectly conducting walls. Solutions of the time-dependent wave equation have been obtained by Keller and Blank (2), Kay (3) and more recently by Turner (4) who considered the diffraction of a cylindrical pulse by a half plane.

scholarly journals A Computational Model of Viscoelastic Composite Materials for Ligament or Tendon Prostheses

2000 ◽  
Vol 9 (3) ◽  
pp. 096369350000900
Author(s):  
P. Vena
Keyword(s):  
Composite Materials ◽  
Constitutive Relations ◽  
Strain Energy Function ◽  
Time Dependent ◽  
Element Formulation ◽  
Viscoelastic Composite ◽  
Uniaxial Test ◽  
The Matrix ◽  
Tissue Reconstruction ◽  
Constitutive Formulation

A constitutive model and a finite element formulation for viscoelastic anisotropic materials subject to finite strains is expounded in this paper. The composite material is conceived as a matrix reinforced with stiff fibres. The constitutive relations are obtained by defining a strain energy function and a relaxation function for each constituent. By means of this approach, the viscoelastic properties of the material constituents can be taken into account and therefore different time dependent behaviour can be assigned to the matrix and to the reinforcing fibres. The response provided by this kind of constitutive formulation allows for the description of mechanical behaviour for either natural anisotropic tissues (such as tendons and ligaments) and for the composite materials which are currently adopted for tissue reconstruction. The main features of those mechanical properties observed in an ideal uniaxial test are: a non linear stress-strain response and a time dependent response which is observed in relaxation of stresses for a prescribed constant stretch and in a moderate strain rate dependence of the measured response.

scholarly journals Constitutive relations for compressible granular flow in the inertial regime

2019 ◽  
Vol 874 ◽  
pp. 926-951 ◽  
Author(s):  
D. G. Schaeffer ◽  
T. Barker ◽  
D. Tsuji ◽  
P. Gremaud ◽  
M. Shearer ◽  
...  
Keyword(s):  
Steady State ◽  
Granular Flow ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Constitutive Relations ◽  
Practical Interest ◽  
Time Dependent ◽  
Direct Result ◽  
Wide Range ◽  
Inertial Regime

Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology that does not suffer from such defects is proposed. In the framework of compressible $I$-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

scholarly journals DARBOUX CLASS OF COSMOLOGICAL FLUIDS WITH TIME-DEPENDENT ADIABATIC INDICES

2000 ◽  
Vol 15 (15) ◽  
pp. 979-990 ◽  
Author(s):  
H. C. ROSU
Keyword(s):  
Mathematical Method ◽  
Cosmological Constant ◽  
Time Dependence ◽  
Adiabatic Index ◽  
Time Dependent ◽  
Accelerating Universe ◽  
Constant Parameter ◽  
Darboux Transformations ◽  
Scale Factors ◽  
Experimental Findings

A one-parameter family of time-dependent adiabatic indices is introduced for any given type of cosmological fluid of constant adiabatic index by a mathematical method belonging to the class of Darboux transformations. The procedure works for zero cosmological constant at the price of introducing a new constant parameter related to the time dependence of the adiabatic index. These fluids can be the real cosmological fluids that are encountered at cosmological scales and they could be used as a simple and efficient explanation for the recent experimental findings regarding the present day accelerating universe. In addition, new types of cosmological scale factors, corresponding to these fluids, are presented.

scholarly journals Time dependence of recruitment and derecruitment in the lung: a theoretical model

2002 ◽  
Vol 93 (2) ◽  
pp. 705-713 ◽  
Author(s):  
Jason H. T. Bates ◽  
Charles G. Irvin
Keyword(s):  
Mechanical Ventilation ◽  
Theoretical Model ◽  
Lung Injury ◽  
Time Dependence ◽  
Computer Model ◽  
Lung Volume ◽  
Critical Pressure ◽  
Time Dependent ◽  
Lung Elastance ◽  
Parallel Arrangement

Recruitment and derecruitment (R/D) of air spaces within the lung is greatly enhanced in lung injury and is thought to be responsible for exacerbating injury during mechanical ventilation. There is evidence to suggest that R/D is a time-dependent phenomenon. We have developed a computer model of the lung consisting of a parallel arrangement of airways and alveolar units. Each airway has a critical pressure (Pcrit) above which it tends to open and below which it tends to close but at a rate determined by how far pressure is from Pcrit. With an appropriate distribution of Pcrit and R/D velocity characteristics, the model able to produce realistic first and second pressure-volume curves of a lung inflated from an initially degassed state. The model also predicts that lung elastance will increase transiently after a deep inflation to a degree that increases as lung volume decreases and as the lung becomes injured. We conclude that our model captures the time-dependent mechanical behavior of the lung due to gradual R/D of lung units.

The Time Dependent Behavior of Linear Viscoelastic Polymers

2011 ◽  
Vol 675-677 ◽  
pp. 435-438
Author(s):  
Wei Xiang Zhang ◽  
Xing Shao ◽  
Zhao Ran Xiao
Keyword(s):  
Mechanical Behavior ◽  
Mechanical Characteristics ◽  
Mechanical Response ◽  
Constitutive Relations ◽  
Time Dependent ◽  
Numerical Examples ◽  
General Methodology ◽  
Inverse Transform ◽  
Dependent Behavior ◽  
Linear Viscoelastic

Polymers have been proved to have attractive mechanical characteristics, which made it desirable to choose these materials over traditional materials for numerous types of applications. As the uses of polymers increase, a thorough understanding of the mechanical behavior of these materials becomes vital in order to perform innovative and economical designs of various components. The main objective of this paper is to develop an effective method with the use of the Laplace inverse transform to describe the time dependent mechanical response of viscoelastic polymers. This general methodology is based on differential constitutive relations for viscoelastic polymers, avoiding the use of relaxation integral functions. As its application, the creep and relaxation properties of the materials are exhibited in the numerical examples.

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