A new constitutive equation that models extensional flow strain hardening based on evolving natural configurations: stability analysis

2004 ◽  
Vol 39 (3) ◽  
pp. 379-387 ◽  
Author(s):  
Liviu-Iulian Palade ◽  
Jay R Walton ◽  
Angiolo Farina
Keyword(s):  
Stability Analysis ◽  
Constitutive Equation ◽  
Strain Hardening ◽  
Extensional Flow

scholarly journals AN EXPLICIT CONSTITUTIVE EQUATION FOR PLANE AND AXISYMMETRIC STEADY FLOWS WITH VISCOELASTIC EFFECTS

2004 ◽  
Vol 3 (2) ◽  
pp. 134
Author(s):  
R. L. Thompson ◽  
P. R. Souza Mendes
Keyword(s):  
Constitutive Equation ◽  
Extensional Flow ◽  
Body Motion ◽  
Simple Shear Flow ◽  
Steady Flows ◽  
Complex Flow ◽  
Dimensionless Numbers ◽  
Dissipative Part ◽  
Viscosity Effects ◽  
Viscoelastic Effects

Non-Newtonian materials respond differently when submitted to shear or extension. A constitutive equation in which the stress is a function of both the rate of deformation and on the type of the flow is proposed and analyzed theoretically. It combines information obtained in shear, extension and rigid body motion in all regions of complex flow. The analysis has shown how to insert some elastic effects in a constitutive equation that depends only on the present time and position. One advantage of the model is that all the steady rheological functions in simple shear flow and in extensional flow are predicted exactly. Another important property that is included is the split of the extensional viscosity in two parts: one dissipative part that is related to the shear viscosity and an elastic part that is related to the first and second normal stress coefficients in shear. A discussion involving the dimensionless numbers that relate elastic and viscosity effects is also given.

Non-Newtonian slender drops in a nonlinear extensional flow

2019 ◽  
Vol 877 ◽  
pp. 561-581 ◽  
Author(s):  
Moshe Favelukis
Keyword(s):  
Stability Analysis ◽  
Power Law ◽  
Analytical Solutions ◽  
Viscosity Ratio ◽  
Extensional Flow ◽  
Shear Thinning ◽  
Newtonian Liquid ◽  
Creeping Flow ◽  
Stationary States ◽  
Power Law Index

In this theoretical report we explore the deformation and stability of a power-law non-Newtonian slender drop embedded in a Newtonian liquid undergoing a nonlinear extensional creeping flow. The dimensionless parameters describing this problem are: the capillary number $(Ca\gg 1)$, the viscosity ratio $(\unicode[STIX]{x1D706}\ll 1)$, the power-law index $(n)$ and the nonlinear intensity of the flow $(|E|\ll 1)$. Asymptotic analytical solutions were obtained near the centre and close to the end of the drop suggesting that only Newtonian and shear thinning drops $(n\leqslant 1)$ with pointed ends are possible. We described the shape of the drop as a series expansion about the centre of the drop, and performed a stability analysis in order to distinguish between stable and unstable stationary states and to establish the breakup point. Our findings suggest: (i) shear thinning drops are less elongated than Newtonian drops, (ii) as non-Newtonian effects increase or as $n$ decreases, breakup becomes more difficult, and (iii) as the flow becomes more nonlinear, breakup is facilitated.

A Proposed Generalized Model for Forming Limit Diagram

2015 ◽  
Author(s):  
Aly El Domiaty ◽  
Abdel-Hamid I. Mourad ◽  
Abdel-Hakim Bouzid
Keyword(s):  
Constitutive Equation ◽  
Strain Rate ◽  
Strain Hardening ◽  
Power Law ◽  
Yield Criterion ◽  
Rate Of Change ◽  
Forming Limit ◽  
Strain Hardening Exponent ◽  
Sensitivity Factor ◽  
Generalized Model

One of the most significant approaches for predicting formability is the use of forming limit diagrams (FLDs). The development of the generalized model integrates other models. The first model is based on Von-Misses yield criterion (traditionally used for isotropic material) and power law constitutive equation considering the strain hardening exponent. The second model is also based on Von-Misses yield criterion but uses a power law constitutive equation that considers the effect of strain rate sensitivity factor. The third model is based on the modified Hill’s yield criterion (for anisotropic materials) and a power law constitutive equation that considers the strain hardening exponent. The current developed model is a generalized model which is formulated on the basis of the modified Hill yield criterion and a power law constitutive equation considering the effect of strain rate. A new controlling parameter (γ) for the limit strains was exploited. This parameter presents the rate of change of strain rate with respect to strain. As γ increases the level of the FLD raises indicating a better formability of the material.

Processing/Structure/Properties Relationships in Polymer Blends for the Development of Functional Polymer Foams

2015 ◽  
Author(s):  
Ali Rizvi ◽  
Chul B. Park
Keyword(s):  
Strain Hardening ◽  
Elastic Properties ◽  
Crystallization Kinetics ◽  
Extensional Flow ◽  
Relaxation Times ◽  
Rheological Behaviour ◽  
Polymer Processing ◽  
Phase Morphology ◽  
Dispersed Phase ◽  
Carbon Dioxide Co2

In this study we present a comprehensive experimental investigation of the effect of polymer blending on the dispersed phase morphology and how the dispersed phase morphology influences the foaming behavior of the semicrystalline polymer matrix using three different material combinations: polyethylene (PE)/polypropylene (PP), PP/polyethylene terephthalate (PET) and PP/polytetrafluoroethylene (PTFE). Samples are prepared such that the dispersed phase domains exhibit either spherical or fibrillated morphologies. Measurements of the uniaxial extensional viscosity, linear viscoelastic properties and crystallization kinetics under ambient pressures and elevated pressures of carbon dioxide (CO2) are performed and the morphological features are identified with the aid of SEM. Batch foaming and lab-scale extrusion foaming experiments are performed, as a screening model for polymer processing, to show the enhancement of the foaming ability as a result of the blend morphology, taking into account the rheological behaviour and the effects of crystallization kinetics. The formation of high aspect ratio fibrils imparts unique characteristics to the semicrystalline matrix such as strain-hardening in uniaxial extensional flow, prolonged relaxation times, pronounced elastic properties and enhanced kinetics of crystallization. In contrast, the regular blends containing spherical dispersed phase domains do not exhibit such properties. Foam processing of the three blends reveals a marked broadening of the foaming window when the dispersed phase domains are fibrillated due to the concurrent increase in crystallization kinetics, improved elastic properties and strain hardening in extensional flow.

A Mechanical Model for TRIP Sheet Steel Forming

2011 ◽  
Vol 221 ◽  
pp. 21-26
Author(s):  
Ti Kun Shan ◽  
Li Liu
Keyword(s):  
Constitutive Equation ◽  
Strain Hardening ◽  
Uniaxial Tension ◽  
Work Hardening ◽  
Trip Steel ◽  
Mechanical Model ◽  
Sheet Steel ◽  
Stress Triaxiality ◽  
Transformation Rate ◽  
Stress Strain Relation

An enhanced elastic-plastic constitutive equation taking into account strain induced transformation and its effect on work hardening of TRIP steel during deformation are investigated. The transformation rate relies on the stress triaxiality. The strain hardening of the TRIP steel takes on parabola shape because of the austenite changed to the martensite during straining. The physical model is verified by comparing with the stress-strain relation of the uniaxial tension experiment. The results showed that the steel keeps a high hardening potential which retards the onset of necking and a good formability thanks to the martensitic strain-induced transformation and the subsequent austenite hardening.

Study on the Strain Hardening Exponent N under the Ultrasonic Action and Establishment of Constitutive Equation

2014 ◽  
Vol 621 ◽  
pp. 82-87
Author(s):  
Lin Chao An ◽  
Xue Li Cheng
Keyword(s):  
Stainless Steel ◽  
Carbon Steel ◽  
Constitutive Equation ◽  
Tensile Test ◽  
Strain Hardening ◽  
Low Carbon Steel ◽  
Manganese Steel ◽  
Strain Hardening Exponent ◽  
Low Carbon ◽  
Ultrasonic Action

The relationship between N and strain hardening exponent curve are obtained by the tensile test of the stainless steel, low carbon steel, medium manganese steel under the ultrasonic action, it is concluded that hardening exponent n of low carbon steel is constant; Hardening exponent N stainless steel has a certain relation with the strain, show the parabolic variation; Strain index N and strain of high manganese steel is linear change. Establish the constitutive equation of low carbon steel under normal and ultrasonic frequency, This equation provides a theoretical basis for the tensile test of low carbon steel, provided with guiding significance and reference value for the application of low carbon steel in engineering practice.

The constitutive equation for a dilute emulsion

1970 ◽  
Vol 44 (1) ◽  
pp. 65-78 ◽  
Author(s):  
N. A. Frankel ◽  
Andreas Acrivos
Keyword(s):  
Constitutive Equation ◽  
Numerical Solutions ◽  
Extensional Flow ◽  
Small Deformation ◽  
Order Theory ◽  
Droplet Surface ◽  
Time Dependent ◽  
Deformation Analysis ◽  
First Order Theory ◽  
Freely Suspended

A constitutive equation for dilute emulsions is developed by considering the deformations, assumed infinitesimal, of a small droplet freely suspended in a time-dependent shearing flow. This equation is non-linear in the kinematic variables and gives rise to ‘fluid memory’ effects attributable to the droplet surface dynamics. Furthermore, it has the same form as the corresponding expression for a dilute suspension of Hookean elastic spheres (Goddard & Miller 1967), and reduces to a relation previously proposed by Schowalter, Chaffey & Brenner (1968) when time-dependent effects become small.Numerical solutions are also presented for the case of a small bubble in a steady extensional flow for the purpose of estimating the range of validity of the small deformation analysis. It is shown that, unlike the drag of a bubble which, in creeping motion, is known to be relatively insensitive to its exact shape, the macroscopic stress field in an emulsion is not well described by the present analysis unless the shapes of the deformed bubbles agree closely with those given by the first-order theory. Thus, the present rheological equation should prove of value in a qualitative rather than a quantitative sense.

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