دانلود کتاب Magneto-Active Polymers: Fabrication, characterisation, modelling and simulation at the micro- and macro-scale

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Cover
Half Title
Also of Interest
Magneto-Active Polymers: Fabrication, characterisation, modelling and simulation at the micro- and macro-scale
Copyright
Dedication
Preface
Acknowledgment
Contents
Acronyms
Notation
Blackboard
Calligraphic
Fraktur
Greek
Italics
Latin
Miscellaneous
Operators
Scripts
Tensor notation
Index notation
Table of Units
1. Introduction
2. Fabrication of magneto-active polymer composites
2.1 A general discussion on MAP fabrication
Composition
Fabrication apparatus and procedure
2.2 Characterisable MAP: From lab to fab
Composition
Preparation and curing methodology
3. Experimental apparatus and testing procedure
3.1 Parallel-plate rotational rheometer
3.1.1 Stress-controlled deformation
Small amplitude oscillatory shear (SAOS)
Large amplitude oscillatory shear (LAOS)
3.1.2 Magneto-rheological device
3.1.3 Influence of rotor geometry on magnetic field
Surface finish of parallel rotor
Smooth parallel versus cone rotor
3.1.4 Caveats to consider when performing experiments
Elastomeric polymers
Magneto-active polymer composites
3.2 Experimental methodology
4. Magneto-mechanical characterisation of magneto-active polymer composite
4.1 Unfilled matrix (PDMS)
4.2 Microstructure developed in the cured filled MAP
4.3 Strain amplitude dependence of the composite MAP (PDMS with a CIP filling)
4.3.1 Response to mechanical loading
Microstructural interpretation of the material response
4.3.2 Response to magneto-mechanical loading
Microstructural interpretation of the magneto-mechanical response
4.3.3 Influence of rotor geometries on the response of isotropic MAPs
4.3.4 Modelling of isotropic MAPs at a fixed frequency
4.4 Frequency dependence of the composite MAP (PDMS with CIP filling)
4.4.1 Response to mechanical loading
4.4.2 Response to magneto-mechanical loading
5. Introduction to continuum magneto-mechanics
5.1 Continuum setting
5.1.1 Continuum domain
5.1.2 Kinematics
5.1.3 Line, area and volume transformations
5.1.4 Time derivatives
5.1.5 Fundamentals of electromagnetism
Electrostatics: Coulomb’s law for charged particles
Magnetostatics: Ampére’s law for electric circuits
Lorentz force
5.2 Continuum theorems for materials with discontinuities
5.2.1 Control volumes with surface discontinuities
Kinematics of moving interfaces
Gauss’ (divergence) theorem
Reynolds transport theorem
Global balance law and localisation
5.2.2 Control surfaces with line discontinuities
Kelvin–Stokes theorem
Reynolds transport theorem
Global balance law and localisation
Summary of localised balance laws
5.3 Governing equations
5.3.1 Balance laws
Electromagnetics
Magneto-mechanics
5.3.2 Spatial relationships
5.3.3 Summary of balance laws
5.3.4 Ponderomotive force and moment
5.3.5 Formulations for magnetic potentials
5.3.5.1 Magnetic vector potential
5.3.5.2 Magnetic scalar potential
5.3.6 Weak formulation of conservation laws
5.3.6.1 Mechanical contribution
5.3.6.2 Magnetic contribution with parameterisation in terms of B
5.3.6.3 Magnetic contribution with parameterisation in terms of H
5.3.7 Variational formulations
5.3.7.1 Parameterisation in terms of B
First variation
Linearisation
5.3.7.2 Parameterisation in terms of H
First variation
Linearisation
5.4 Thermodynamics
5.4.1 First law of thermodynamics: Energy balance
5.4.2 Second law of thermodynamics: Entropy balance
5.4.3 Parameterisation of isothermal energy functions
6. General aspects of computational simulation of coupled problems
6.1 Finite element discretisation
6.1.1 Displacement field
Choice of ansatz for the displacement field
6.1.2 Magnetic vector potential field
Choice of ansatz for the magnetic vector potential field
6.1.3 Magnetic scalar potential field
Choice of ansatz for the magnetic scalar potential field
6.1.4 Summary of finite element implementation
6.1.5 Tools automating the computation of finite element linearisations and constitutive model tangent moduli
6.2 Evaluation of definite integrals
6.3 Solution of a time/load increment
6.3.1 Solution to the time-independent non-linear problem using the Newton-Raphson method
6.3.2 Formation and solution of the discrete system of linear equations
Block Gaussian elimination
6.3.3 Special considerations when using the magnetic scalar potential formulation
7. Constitutive modelling
7.1 Preliminaries to magneto-mechanical energy functions
7.1.1 Volumetric-isochoric split
7.1.2 Invariants for isotropic media
7.1.3 Transverse isotropy
7.2 Viscomagneto-viscoelasticity
Numerical examples: Viscomagneto-viscoelasticity
Stepwise magnetic induction with no deformation
Time-dependent magnetic induction with no deformation
Numerical example: Magneto-viscoelasticity
7.3 Transverse isotropy and particle chain dispersion
Numerical examples: Dispersed chain-like particle structures
Uniaxial deformation
Inflation and extension of a magnetoelastic tube
8. Phenomenological modelling of the curing process
8.1 A continuum framework for the curing of polymers
8.1.1 Curing in viscomagneto-viscoelastic materials
8.1.2 Curing with field-sensitive shrinkage effects
Numerical examples: Curing in the presence of a magnetic field
Curing without shrinkage and anisotropy
Curing with shrinkage and anisotropy
9. Homogenisation
9.1 First-order homogenisation of magneto-coupled materials
9.1.1 Hill’s condition on the equality of micro- and macro-scale virtual powers
9.1.2 Consistent boundary conditions arising from the equality of virtual power
9.1.2.1 Mechanical contribution to virtual power
Fluctuation in potential field
Spatially piecewise constant applied traction
9.1.2.2 Magnetic contribution to virtual power: Magnetic vector potent
Fluctuation in potential field
Spatially piecewise constant applied “traction”
9.1.2.3 Magnetic contribution to virtual power: Magnetic scalar potent
Fluctuation in potential field
Spatially piecewise constant applied “traction”
9.1.2.4 Upper and lower bounds on the averaged material response
9.1.2.5 Implementation of constraints for periodically repeating Rves
Numerical example: Boundary conditions and the homogenised material response
9.1.3 Computation of consistent tangent moduli for the macro-scale problem
9.1.3.1 Perturbation method
9.1.3.2 Algorithmically consistent tangent moduli
9.1.4 Homogenisation of curing using the Mori–Tanaka method
Numerical examples: Micromechanical model of curing
9.2 The stochastic finite element method
9.2.1 Extension into stochastic dimensions
9.2.2 Defining material discontinuities using level set functions
9.2.3 Basis function selection
Global stochastic basis
Local stochastic basis
Extended FEM framework
Numerical examples: SFEM with uni- and multi-variate uncertainties
Elasticity
Magneto-mechanics
10. Modelling and computational simulation at the micro-scale
10.1 Single particle representative volume element problem
10.1.1 Solution accuracy and finite element discretisation
10.1.1.1 Analytical solution
10.1.1.2 Error analysis
Numerical example: Error analysis of a magnetostatic RVE
10.1.2 Computation of magnetic forces and torques
10.1.2.1 Computations derived from the strong form
10.1.2.2 Computations derived from the weak form
Numerical examples: Force and torque measurements on magnetostatic RVEs
10.2 Micro-structural studies of MAP compositions
10.2.1 Full resolution simulation of a prototype magnetostatic microstrucural model
10.2.2 Influence of microstructural organisation in a representative volume element on the response characteristics of a prototype magnetoelastic material
10.2.2.1 Effective stiffness moduli in isotropic, orthotropic and transversely isotropic microstructures
10.2.2.2 Magnetic forces generated in dispersed chain-like structures
11. Modelling and computational simulation at the macro-scale
11.1 Macro- to micro-scale transition using the FE2 approach
11.2 Immersion of magnetic bodies in free space
11.2.1 Mesh motion in the free space
Numerical example: MAP with dispersed chain-like particle structures immersed in free space
11.3 Mixed variational approach for quasi-incompressible media
Variational formulation
First variation
Finite element discretisation
Solution of linear iteration step
Numerical example: Quasi-incompressible magneto-active valve
12. Further reading
Transient behaviour
Dissipative material response
Micromagnetism
Surface effects
Electroelasticity
A. Identities
A.1 Operation identities
Cross product
Cofactor (of a non-singular tensor)
Permutation operators
Cross product with a scalar factor
Cross product with a common operator
Scalar triple product
Scalar triple product with common operator
Vector triple product
Jacobi identity
Jumps and averages
A.2 Generic differential identities
Divergence
Curl
Curl of a gradient
Divergence of a curl
Curl of a curl
Gradient of a scalar
Divergence of a scaled vector
Gradient of a scaled vector
Curl of a scaled vector
Divergence of a vector cross product
Gradient of a vector cross product
Divergence of the tensor product of two vectors
Divergence of a scaled tensor
Divergence of a vector-tensor inner product
Divergence of a tensor-tensor inner product
Divergence of a vector-tensor outer product
Divergence of a vector-tensor cross product
Curl of a cross product
Cross product of a curl
Gradient of a vector-vector inner product
A.3 Differential and rate identities: Continuum mechanics
Derivatives with respect to the deformation gradient tensor [559]
Derivatives with respect to the right Cauchy–Green deformation tensor [559]
Material time derivatives [51, 559]
B. Calculus
B.1 Microscopic theorems
Dirac delta function [133]
Gradient operator [414]
Divergence operator [414, 133, 230]
Laplace operator [133, 230]
Curl operator [414, 133]
B.2 Continuum theorems
Volume transformation
Area transformation by Nanson’s formula
Integration by parts
Kelvin–Stokes theorem
Divergence theorem
Gradient theorem
Piola identity [204]
Leibniz integral rule
B.2.1 Materials without discontinuities
Reynolds transport theorem: Control surface
Global balance law: Control volume
Global balance law: Control surface
B.2.2 Materials with discontinuities
Reynolds transport theorem: Control volume
Reynolds transport theorem: Control area
B.3 Continuum identities
Volume integral of a curl
Volume integral of the divergence of a contravariant quantity
Area integral of the curl of a covariant vector
C. Derivations and proofs
C.1 Fundamentals of electromagnetism
C.1.1 Electrostatics Divergence of electric field vector
Negative gradient of electric potential field
C.1.2 Magnetostatics
Perfect path integral over a closed circuit [230]
Force acting on a circuit
Divergence of magnetic induction vector
Curl of magnetic vector potential field
C.2 Continuum mechanics for magnetoelasticity
Push forward of referential magnetic field, magnetisation and electric current
Push forward of referential magnetic induction
Pull back of magnetic fundamental constitutive relation
Referential surface electric charge density
Referential total surface current vector
Jump of spatial magnetic field and induction vectors for decoupled magnetic fields
Cross product of the spatial surface normal and the jump of the spatial magnetic field and induction vectors for decoupled magnetic fields
Cross product of the reference surface normal and the jump of the reference magnetic field and induction vectors for decoupled magnetic fields
C.3 Stress tensors
C.3.1 Definitions
Spatial ponderomotive stress tensor
Two-point ponderomotive stress tensor
Spatial Maxwell stress tensor
Two-point Maxwell stress tensor
Spatial magnetisation stress tensor
Two-point magnetisation stress tensor
C.3.2 Divergences
Ponderomotive stress tensor
Magnetisation stress tensor
Maxwell stress tensor
C.3.3 Jumps
Magnetisation stress tensor
Maxwell stress tensor
Ponderomotive stress tensor
C.4 Ponderomotive forces and tractions
C.4.1 Definitions of the Lorentz forces
Spatial total ponderomotive force vector
Referential ponderomotive body force density vector
Referential ponderomotive traction vector
C.5 Thermomechanical and electromagnetic balance laws
C.5.1 Derivation of spatial statement of Maxwell’s equations in a non-relativistic Eulerian reference frame
Eulerian reference frame
A note on the setting for Maxwell’s equations
Maxwell’s equations: Gauss’ law
Maxwell’s equations: Gauss’ magnetism law
Maxwell’s equations: Faraday’s law
Maxwell’s equations: Ampére’s law
Conservation of electric charge
C.5.2 Derivation of spatial statement of mechanical conservation laws (for magnetostatic systems)
Conservation of mass
Traction continuity
Balance of linear momentum for polar media [523, 133, 204]
Balance of angular momentum for polar media [523, 133, 204]
C.5.3 Derivation of additional balance and jump identities for potent
Magnetic vector potential
Magnetic scalar potential
C.5.4 Transformation of conservation laws to their referential description
C.5.4.1 Time derivatives
Material nominal time derivative: Scalar densities
Material nominal time derivative: Vector densities
Material nominal time derivative: Flux vectors [406, 133]
C.5.4.2 Balance laws and fundamental relations in a volume Pull back
Pull back of Maxwell’s equations: Gauss’ law
Pull back of Maxwell’s equations: Gauss’ magnetism law
Pull back of transient Maxwell’s equations: Faraday’s law
Pull back of transient Maxwell’s equations: Ampére’s law
Referential balance of angular momentum
C.5.4.3 Jump conditions and fundamental relations on a surface
Jump condition associated with Gauss’ law
Jump condition associated with Ampére’s law
Jump condition associated with Gauss’ magnetism law
Cauchy stress theorem
Pull back of traction continuity
C.5.5 Weak formulation of quasi-static balance of linear momentum
C.6 Legendre transformations
Total free energy
Free field energy
Two-point magnetisation stress tensor
Spatial magnetisation stress tensor
Two-point Maxwell stress tensor
Spatial Maxwell stress tensor
Two-point ponderomotive stress tensor
Spatial ponderomotive stress tensor
C.7 Thermodynamics
C.7.1 Work performed by the Lorentz forces
Total ponderomotive power: Magnetic contribution
Total ponderomotive power: Electric contribution
Total ponderomotive power
C.7.2 Boundary contributions to external power
Mechanical contribution
Thermal contribution
Electromechanical contribution
C.7.3 Combined contribution of external mechanical and electromagnetic
C.8 Homogenisation
C.8.1 Relationship between macroscopic and microscopic field quantities
Deformation gradient tensor
Piola stress tensor
Magnetic induction vector
Magnetic field vector
C.8.2 The Hill–Mandel condition Quasi-static mechanical problem
Quasi-static magnetic problem: MSP formulation
Quasi-static magnetic problem: MVP formulation
C.8.3 Algorithmically consistent tangent moduli
C.9 Constitutive modelling
C.9.1 Free energy function derivatives
C.9.2 Invariants
Cayley–Hamilton theorem [80]
Reducibility of magnetoelastic pseudo-invariant
C.9.3 Invariant derivatives [504]
First derivatives
Second derivatives
C.9.4 Free space stored energy function derivatives
First partial derivatives
Second partial derivatives
C.9.5 Volumetric / deviatoric split of free energy function
Deviatoric projection tensor
Component of the deviatoric part of material elasticity tensor
C.10 Time integrators for rate-dependent materials described by internal variables
Summary of calculations for incremental update of internal variable
Bibliography
Image reproduction
Index