Understanding Viscoelasticity

by Phan-Thien, Nhan

• ISBN13: 9783540433958
• ISBN10: 3540433953
• Format: Hardcover
• Copyright: 2002-08-01
• Publisher: Springer Verlag

Summary

Viscoelasticity (or rheology) is a highly mathematical subject, interfacing several disciplines, from applied mathematics to mechanics and engineering. Other books on rheology are usually comprehensive to such a degree that students find it hard to isolate the important physical principles and develop good analytical skills for solving useful engineering problems. In this book, the necessary background for understanding viscoelasticity is covered; both the continuum and microstructure approaches to modelling viscoelastic materials are discussed, since neither approach alone is sufficient. The book starts with tensor notation, to give students the basic working tools, then addresses kinematics and constitutive equations, and ends with the constitutive modelling of polymer solutions and suspensions. It also includes a series of problems of graded difficulty. It is self-contained and ideally suited for a one-semester course on viscoelasticity.

Table of Contents

Tensor Notation	1	(26)
Cartesian Frame of Reference	1	(2)
Position Vector	3	(1)
Frame Rotation	3	(1)
Orthogonal Matrix	4	(1)
Rotation Matrix	4	(1)
Tensors	4	(5)
Zero- Order Tensors	4	(1)
First-Order Tensor	4	(1)
Outer Products	5	(1)
Second-Order Tensors	5	(1)
Third-Order Tensors	5	(1)
Transpose Operation	6	(1)
Decomposition	6	(1)
Some Common Vectors	7	(1)
Gradient of a Scalar	7	(1)
Some Common Tensors	7	(2)
Tensor and Linear Vector Function	9	(1)
Claim	9	(1)
Dyadic Notation	10	(1)
Tensor Operations	10	(2)
Substitution	10	(1)
Contraction	10	(1)
Transpose	11	(1)
Products of Two Two-Tensors	11	(1)
Invariants	12	(1)
Invariant of a Vector	12	(1)
Invariants of a Tensor	12	(1)
Decompositions	12	(2)
Eigenvalue and Eigenvector	12	(1)
Square Root Theorem	13	(1)
Polar Decomposition Theorem	13	(1)
Cayley-Hamilton Theorem	13	(1)
Derivative Operations	14	(2)
Derivative of det(A)	15	(1)
Derivative of tr(A)	15	(1)
Derivative of tr(A2)	15	(1)
Gradient of a Field	16	(5)
Field	16	(1)
Cartesian Frame	16	(2)
Non-Cartesian Frames	18	(1)
Spherical Coordinates	19	(2)
Integral Theorems	21	(6)
Gauss Divergence Theorem	21	(1)
Stokes Curl Theorem	22	(1)
Leibniz Formula	22	(1)
Problems	22	(5)
Rheological Properties	27	(12)
Viscosity	27	(2)
Shear-Rate Dependent Viscosity	27	(2)
Normal Stress Differences	29	(4)
Weissenberg Rod-Climbing Effect	30	(1)
Die Swell	31	(1)
Flow Down an Inclined Channel	32	(1)
Transient Responses	33	(3)
Small Strain Oscillatory Flow	33	(1)
Stress Overshoot	34	(1)
Stress Relaxation	34	(1)
Relaxation Modulus	35	(1)
Recoil	36	(1)
Elongational Flows	36	(2)
Elongational Viscosity	36	(2)
Viscoelastic Instabilities	38	(1)
Kinematics and Equations of Balance	39	(20)
Kinematics	39	(2)
Reference Configuration	39	(1)
Velocity and Acceleration Fields	40	(1)
Material Derivative	41	(1)
Deformation Gradient and Strain Tensors	41	(4)
Deformation Gradient	41	(1)
Cauchy-Green Strain Tensor	42	(1)
Relative Strain Tensors	43	(1)
Path Lines	43	(1)
Oscillatory Shear Flow	44	(1)
Rivlin-Ericksen Tensors	45	(3)
Small Strain	48	(1)
Equations of Balance	49	(10)
Reynolds Transport Theorem	49	(1)
Conservation of Mass	50	(1)
Conservation of Momentum	51	(4)
Conservation of Energy	55	(1)
Problems	56	(3)
Constitutive Equation	59	(26)
Some Well-Known Constitutive Equations	59	(4)
Perfect Gas	59	(1)
Inviscid Fluid	60	(1)
Fourier's Law	60	(1)
Hookean Solid	60	(2)
Newtonian Fluid	62	(1)
Non-Newtonian Fluid	63	(1)
Weissenberg and Deborah Numbers	63	(1)
Deborah Number	63	(1)
Weissenberg Number	64	(1)
Some Guidelines in Constitutive Modelling	64	(5)
Oldroyd Approach	65	(1)
Principle of Material Objectivity	66	(1)
Objectivity of the Stress	66	(1)
Frame Indifference	67	(2)
Principle of Local Action	69	(1)
Principle of Determinism	69	(1)
Integrity Bases	69	(3)
Isotropic Scalar-Valued Functions	69	(1)
Isotropic Vector-Valued Functions	70	(1)
Isotropic Tensor-Valued Functions	71	(1)
Symmetry Restrictions	72	(1)
Unimodular Matrix	72	(1)
Symmetry Group	73	(1)
Isotropic Materials	73	(1)
Isotropic Elastic Materials	73	(2)
The Simple Material	75	(2)
Simple Fluid	76	(1)
Incompressible Simple Fluid	76	(1)
Fading Memory	76	(1)
Order Fluids	77	(3)
Unsteady Motion	77	(2)
Velocity Field in a Second-Order Fluid	79	(1)
Green-Rivlin Expansion	80	(5)
Problems	80	(5)
Inelastic Models and Linear Viscoelasticity	85	(16)
Inelastic Fluids	85	(3)
Carreau Model	86	(1)
Power-law Model	86	(2)
Linear Viscoelasticity	88	(6)
Simple Shear Flow	90	(2)
Step Strain	92	(1)
Relaxation Spectrum	93	(1)
Correspondence Principle	94	(3)
Quasi-Static Approximation	94	(1)
Circular Couette Flow	95	(2)
Mechanical Analogs	97	(4)
Problems	97	(4)
Steady Viscometric Flows	101	(8)
Kinematics	101	(3)
Steady Parallel Flow	102	(1)
Rectilinear flow	103	(1)
Axial Fanned Flow	103	(1)
Helical Flow	103	(1)
Helicoidal Flow	104	(1)
Stresses in Steady Viscometric Flows	104	(5)
Controllable and Partially Controllable Flows	106	(1)
Problems	106	(3)
Polymer Solutions	109	(22)
Characteristics of a Polymer Chain	109	(3)
Random-Walk Model	109	(3)
Forces on a Chain	112	(3)
Fluctuation-Dissipation Theorem	115	(4)
Langevin Equation	115	(1)
Equi-Partition of Energy	116	(1)
Fluctuation-Dissipation Theorem	116	(1)
Diffusivity Stokes-Einstein Relation	117	(1)
Fokker-Planck Equation	117	(1)
Smoothed-Out Brownian Force	118	(1)
Stress Tensor	119	(3)
Kramers Form	122	(1)
Elastic Dumbbell Model	122	(4)
Langevin Equations	122	(1)
Average Motion	123	(1)
Strong and Weak Flows	124	(1)
Upper-Convected Maxwell Model	125	(1)
Oldroyd-B Model	126	(1)
Main Features of the Oldroyd-B Model	126	(5)
Simple Flows	126	(1)
Multiple Relaxation Time UCM Model	127	(1)
Problems	128	(3)
Suspensions	131	(10)
Bulk Suspension Properties	132	(2)
Dilute Suspension of Spheroids	134	(3)
Epilogue	137	(4)
Problems	138	(3)
References	141	(2)
Index	143