Applied Mechmatreianics of Solids (a.F

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9/13/13 Applied Mechanics of Solids (A.F. Bower) Appendix A: Vectors and Matrices

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1. Objectives and Applications >1.1 Defining a Problem >

1.1.1 Deciding what to calculate

1.1.2 Defining geometry

1.1.3 Defining loading

1.1.4 Choosing physics

1.1.5 Defining material behavior

1.1.6 A representative problem

1.1.7 Choosing a method of

analysis

2. Governing Equations >

2.1 Deformation measures

>

2.1.1 Displacement and Velocity

2.1.2 Deformation gradient

2.1.3 Deformation gradient from two

deformations

2.1.4 Jacobian of deformation

gradient

2.1.5 Lagrange strain

2.1.6 Eulerian strain

2.1.7 Infinitesimal Strain

2.1.8 Engineering Shear Strain

2.1.9 Volumetric and Deviatoric strain2.1.10 Infinitesimal rotation

2.1.11 Principal strains

2.1.12 Cauchy-Green deformation

tensors

2.1.13 Rotation tensor, Stretch

tensors

2.1.14 Principal stretches

2.1.15 Generalized strain measures

2.1.16 Velocity gradient

2.1.17 Stretch rate and spin

2.1.18 Infinitesimal strain/rotation rate

2.1.19 Other deformation rates

2.1.20 Strain equations of

compatibility

2.2 Internal forces >

2.2.1 Surface traction/body force

2.2.2 Internal tractions

2.2.3 Cauchy stress

2.2.4 Kirchhoff, Nominal, Material

stress

2.2.5 Stress for infinitesimal motions

2.2.6 Principal stresses2.2.7 Hydrostatic, Deviatoric, Von

Mises stress

2.2.8 Stresses at a boundary

2.3 Equations of motion >

2.3.1 Linear momentum balance
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2.3.2 Angular momentum

balance

2.3.3 Equations using other

stresses

2.4 Work and Virtual

Work >

2.4.1 Work done by Cauchy

stress

2.4.2 Work done by other

stresses2.4.3 Work for infinitesimal

motions

2.4.4 Principle of virtual work

2.4.5 Virtual work with other

stresses

2.4.6 Virtual work for small

strains

3. Constitutive Equations >

3.1 General requirements

3.2 Linear elasticity >

3.2.1 Isotropic elastic behavior3.2.2 Isotropic stress-strain laws

3.2.3 Plane stress & strain

3.2.4 Isotropic material data

3.2.5 Lame, Shear, & Bulk modulus

3.2.6 Interpreting elastic constants

3.2.7 Strain energy density (isotropic)

3.2.8 Anisotropic stress-strain laws

3.2.9 Interpreting anisotropic

constants

3.2.10 Anisotropic strain energy

density

3.2.11 Basis change formulas

3.2.12 Effect of material symmetry

3.2.13 Orthotropic materials

3.2.14 Transversely isotropic

materials

3.2.15 Transversely isotropic data

3.2.16 Cubic materials

3.2.17 Cubic material data

3.3 Hypoelasticity

3.4 Elasticity w/ large rotations

3.5 Hyperelasticity >

3.5.1 Deformation measures3.5.2 Stress measures

3.5.3 Strain energy density

3.5.4 Incompressible materials

3.5.5 Energy density functions

3.5.6 Calibrating material

models

3.5.7 Representative

properties

3.6 Viscoelasticity >

3.6.1 Polymer behavior

3.6.2 General constitutive

equations

3.6.3 Spring-damper

approximations

3.6.4 Prony series

3.6.5 Calibrating constitutive
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aws

3.6.6 Calibrating material models

3.6.7 Representative properties

3.7 Rate independent plasticity

>

3.7.1 Plastic metal behavior

3.7.2 Elastic/plastic strain

decomposition

3.7.3 Yield criteria

3.7.4 Graphical yield surfaces3.7.5 Hardening laws

3.7.6 Plastic flow law

3.7.7 Unloading condition

3.7.8 Summary of stress-strain

relations


3.7.10 Principle of max. plastic

resistance

3.7.11 Drucker's postulate

3.7.12 Microscopic perspectives

3.8 Viscoplasticity >

3.8.1 Creep behavior

3.8.2 High strain rate behavior

3.8.3 Constitutive equations

3.8.4 Representative creep properties

3.8.5 Representative high rate

properties

3.9 Large strain plasticity >

3.9.1 Deformation measures

3.9.2 Stress measures

3.9.3 Elastic stress-strain

relations

3.5.4 Plastic stress-strainrelations

3.10 Large strain viscoelasticity

>



3.10.3 Stress-strain energy

relations

3.10.4 Strain relaxation


3.11 Critical state soils >

3.11.1 Soil behavior3.11.2 Constitutive laws (Cam-

clay)

3.11.3 Response to 2D loading


3.12 Crystal plasticity >

3.12.1 Basic crystallography

3.12.2 Features of crystal

plasticity



3.12.5 Elastic stress-strain

relations

3.12.6 Plastic stress-strain

relations


3.13 Surfaces and interfaces >
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. .

3.13.2 Contact and friction

4. Solutions to simple problems >

4.1 Axial/Spherical linear elasticity

>

4.1.1 Elastic governing equations

4.1.2 Spherically symmetric

equations

4.1.3 General spherical solution

4.1.4 Pressurized sphere4.1.5 Gravitating sphere

4.1.6 Heated spherical shell

4.1.7 Axially symmetric equations

4.1.8 General axisymmetric solution

4.1.9 Pressurized cylinder

4.1.10 Spinning circular disk

4.1.11 Interference fit

4.2 Axial/Spherical elastoplasticity

>

4.2.1 Plastic governing equations


equations

4.2.3 Pressurized sphere

4.2.4 Cyclically pressurized sphere

4.2.5 Axisymmetric equations

4.2.6 Pressurized cylinder

4.3 Spherical hyperelasticity >

4.3.1 Governing equations


equations

4.3.3 Pressurized sphere

4.4 1D elastodynamics >

4.4.1 Surface subjected to pressure4.4.2 Surface under tangential

loading

4.4.3 1-D bar

4.4.4 Plane waves

4.4.5 Wave speeds in isotropic

solid

4.4.6 Reflection at a surface

4.4.7 Reflection at an interface

4.4.8 Plate impact experiment

5. Solutions for elastic solids >

5.1 General Principles >


5.1.2 Navier equation

5.1.3 Superposition &

linearity

5.1.4 Uniqueness & existence

5.1.5 Saint-Venants principle

5.2 2D Airy function solutions >

5.2.1 Airy solution in rectangular

coords

5.2.2 Demonstration of Airy solution

5.2.3 Airy solution in polar coords

5.2.4 End loaded cantilever5.2.5 Line load perpendicular to

surface

5.2.6 Line load parallel to surface

4.4.7 Pressure on a surface

4.4.8 Uniform ressure on a stri
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4.4.8 Stress near a crack tip

5.3 2D Complex variable

solutions >

5.3.1 Complex variable solution

5.3.2 Demonstration of CV

solution

5.3.3 Line force

5.3.4 Edge dislocation

5.3.5 Circular hole in infinite

solid

5.3.6 Slit crack

5.3.7 Bimaterial interface crack

5.3.8 Rigid flat punch on a

surface

5.3.9 Parabolic punch on a

surface

5.3.10 General line contact

4.3.11 Frictional sliding contact

4.3.12 Dislocation near a surface

5.4 3D static problems >

5.4.1 Papkovich-Neuber potentials5.4.2 Demonstration of PN

potentials

5.4.3 Point force in infinite solid

5.4.4 Point force normal to surface

5.4.5 Point force tangent to surface

5.4.6 Eshelby inclusion problem

5.4.7 Inclusion in an elastic solid

5.4.8 Spherical cavity in infinite

solid

5.4.9 Flat cylindrical punch on

surface5.4.10 Contact between spheres

4.4.11 Relations for general

contacts

4.4.12 P-d relations for

axisymmetric contact

5.5 2D Anisotropic elasticity >


5.5.2 Stroh solution

5.5.3 Demonstration of Stroh solution

5.5.4 Stroh matrices for cubic

materials

5.5.5 Degenerate materials

5.5.6 Fundamental elasticity matrix

5.5.7 Orthogonality of Stroh matrices

5.5.8 Barnett/Lothe & Impedance

tensors

5.5.9 Properties of matrices

5.5.10 Basis change formulas

5.5.11 Barnett-Lothe integrals

5.5.12 Uniform stress state

5.5.13 Line load/dislocation in infinite

solid

5.5.14 Line load/dislocation near asurface

5.6 Dynamic problems >

5.6.1 Love potentials

5.6.2 Pressurized spherical

cavity
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5.6.3 Rayleigh waves

5.6.4 Love waves

5.6.5 Elastic waves in

waveguides

5.7 Energy methods >

5.7.1 Definition of potential energy

5.7.2 Minimum energy theorem

5.7.3 Simple example of energy

minimization

5.7.4 Variational approach to beam

theory

5.7.5 Estimating stiffness

5.8 Reciprocal theorem >

5.8.1 Statement and proof of

theorem

5.8.2 Simple example

5.8.3 Boundary-internal value

relations

5.8.4 3D dislocation loops

5.9 Energetics of dislocations >

5.9.1 Potential energy of isolatedloop

5.9.2 Nonsingular dislocation

theory

5.9.3 Dislocation in bounded solid

5.9.4 Energy of interacting loops

5.9.5 Peach-Koehler formula

5.10 Rayleigh Ritz method >

5.10.1 Mode shapes, nat. frequencies,

Rayleigh's principle

5.10.2 Natural frequency of a beam

6. Solutions for plastic solids >

6.1 Slip-line fields >

6.1.1 Interpreting slip-line fields

6.1.2 Derivation of slip-line fields

6.1.3 Examples of solutions

6.2 Bounding theorems

>

6.2.1 Definition of plastic dissipation

6.2.2 Principle of min plastic

dissipation

6.2.3 Upper bound collapse theorem

6.2.4 Lower bound collapse theorem

6.2.5 Examples of bounding theorems6.2.6 Lower bound shakedown

theorem

6.2.7 Examples of lower bound

shakedown theorem

6.2.8 Upper bound shakedown

theorem

6.2.9 Examples of upper bound

shakedown theorem

7. Introduction to FEA >

7.1 Guide to FEA >

7.1.1 FE mesh7.1.2 Nodes and elements

7.1.3 Special elements

7.1.4 Material behavior

7.1.5 Boundary conditions

7.1.6 Constraints
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9/13/13 Applied Mechanics ofSolids (A.F. Bower) Appendix A: Vectors and Matrices


7.1.7 Contacting surface/interfaces

7.1.8 Initial conditions/external fields

7.1.9 Soln procedures / time

increments

7.1.10 Output

7.1.11 Units in FEA calculations

7.1.12 Using dimensional analysis

7.1.13 Scaling governing equations

7.1.14 Remarks on dimensional

analysis7.2 Simple FEA program

>

7.2.1 FE mesh and connectivity

7.2.2 Global displacement vector

7.2.3 Interpolation functions

7.2.4 Element strains & energy density

7.2.5 Element stiffness matrix

7.2.6 Global stiffness matrix

7.2.7 Boundary loading

7.2.8 Global force vector

7.2.9 Minimizing potential energy7.2.10 Eliminating prescribed

displacements

7.2.11 Solution

7.2.12 Post processing

7.2.13 Example code

8. Theory & Implementation of

FEA >

8.1 Static linear elasticity >

8.1.1 Review of virtual work

8.1.2 Weak form of governing

equns

8.1.3 Interpolating displacements

8.1.4 Finite element equations

8.1.5 Simple 1D implementation

8.1.6 Summary of 1D procedure

8.1.7 Example 1D code

8.1.8 Extension to 2D/3D

8.1.9 2D interpolation functions

8.1.10 3D interpolation functions

8.1.11 Volume integrals

8.1.12 2D/3D integration schemes

8.1.13 Summary of element

matrices8.1.14 Sample 2D/3D code

8.2 Dynamic elasticity >


8.2.2 Weak form of governing eqns


8.2.4 Newmark time integration

8.2.5 Simple 1D implementation

8.2.6 Example 1D code

8.2.7 Lumped mass matrices

8.2.8 Example 2D/3D code

8.2.9 Modal time integration

8.2.10 Natural frequencies/mode

shapes

8.2.11 Example 1D modal dynamic

code

8.2.12 Example 2D/3D modal
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ynamc co e

8.3 Hypoelasticity >




8.3.4 Newton-Raphson iteration

8.3.5 Tangent moduli for hypoelastic

solid

8.3.6 Summary of Newton-Raphson

method8.3.7 Convergence problems

8.3.8 Variations on Newton-Raphson

8.3.9 Example code

8.4 Hyperelasticity >


8.4.2 Weak form of governing

eqns


8.4.4 Newton-Raphson iteration

8.4.5 Neo-Hookean tangent

moduli

8.4.6 Evaluating boundary

integrals

8.4.7 Convergence problems

8.4.8 Example code

8.5 Viscoplasticity >




8.5.4 Integrating the stress-strain

law

8.5.5 Material tangent

8.5.6 Newton-Raphson solution8.5.7 Example code

8.6 Advanced elements >

8.6.1 Shear locking/incompatible

modes

8.6.2 Volumetric locking/Reduced

integration

8.6.3 Incompressible materials/Hybrid

elements

9. Modeling Material Failure >

9.1 Mechanisms of failure >

9.1.1 Monotonic loading9.1.2 Cyclic loading

9.2 Stress/strain based criteria >

9.2.1 Stress based criteria

9.2.2 Probabilistic methods

9.2.3 Static fatigue criterion

9.2.4 Models of crushing

failure

9.2.5 Ductile failure criteria

9.2.6 Strain localization

9.2.7 High cycle fatigue

9.2.8 Low cycle fatigue

9.2.9 Variable amplitude

loading

9.3 Elastic fracture mechanics >

9.3.1 Crack tip fields

9.3.2 Linear elastic fracture
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9.3.3 Calculating stress intensities

9.3.4 Using FEA

9.3.5 Measuring toughness

9.3.6 Values of fracture toughness

9.3.7 Stable tearing

9.3.8 Mixed mode fracture

9.3.9 Static fatigue

9.3.10 Cyclic fatigue

9.3.11 Finding cracks9.4 Energy methods in fracture >

9.4.1 Definition of energy release

rate

9.4.2 Energy based fracture

criterion

9.4.3 G-K relations

9.4.4 G-compliance relation

9.4.5 Calculating K with

compliance

9.4.6 Integral expression for G

9.4.7 The J integral

9.4.8 Calculating K using J

9.5 Plastic fracture mechanics >

9.5.1 Dugdale-Barenblatt model

9.5.2 HRR crack tip fields

9.5.3 J based fracture mechanics

9.6 Interface fracture mechanics

>

9.6.1 Interface crack tip fields

9.6.2 Interface fracture

mechanics

9.6.3 Stress intensity factors

9.6.4 Crack path selection10. Rods, Beams, Plates & Shells

>

10.1 Dyadic notation

10.2 Deformable rods - general

>

10.2.1 Characterizing the x-section

10.2.2 Coordinate systems

10.2.3 Kinematic relations

10.2.4 Displacement, velocity and

acceleration


10.2.6 Strain measures

10.2.7 Kinematics of bent rods

10.2.8 Internal forces and moments

10.2.9 Equations of motion

10.2.10 Constitutive equations

10.2.11 Strain energy density

10.3 String / beam theory >

10.3.1 Stretched string

10.3.2 Straight beam (small

deflections)

10.3.3 Axially loaded beam

10.4 Solutions for rods >10.4.1 Vibration of a straight beam

10.4.2 Buckling under gravitational

loading

10.4.3 Post buckled shape of a rod

10.4.4 Rod bent into a helix
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10.4.5 Helical spring

10.5 Shells - general >

10.5.1 Coordinate systems

10.5.2 Using non-orthogonal

bases


10.5.4 Displacement and

velocity


10.5.6 Other strain measures

10.5.7 Internal forces and

moments

10.5.8 Equations of motion

10.5.9 Constitutive relations

10.5.10 Strain energy

10.6 Plates and membranes >

10.6.1 Flat plates (small strain)

10.6.2 Flat plates with in-plane

loading

10.6.3 Plates with large

displacements10.6.4 Membranes

10.6.5 Membranes in polar

coordinates

10.7 Solutions for shells >

10.7.1 Circular plate bent by pressure

10.7.2 Vibrating circular membrane

10.7.3 Natural frequency of

rectangular plate

10.7.4 Thin film on a substrate

(Stoney eqs)

10.7.5 Buckling of heated plate10.7.6 Cylindrical shell under axial

load

10.7.7 Twisted open walled cylinder

10.7.8 Gravity loaded spherical shell

A: Vectors & Matrices

B: Intro to tensors

C: Index Notation

D: Using polar coordinates

E: Misc derivations

Problems

1. Objectives and Applications >1.1 Defining a Problem

2. Governing Equations >

2.1 Deformation measures

2.2 Internal forces

2.3 Equations of motion

2.4 Work and Virtual

Work

3. Constitutive Equations >

3.1 General requirements

3.2 Linear elasticity

3.3 Hypoelasticity3.4 Elasticity w/ large rotations

3.5 Hyperelasticity

3.6 Viscoelasticity

3.7 Rate independent plasticity

3.8 Viscoplasticity
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3.9 Large strain plasticity

3.10 Large strain viscoelasticity

3.11 Critical state soils

3.12 Crystal plasticity

3.13 Surfaces and interfaces

4. Solutions to simple problems >

4.1 Axial/Spherical linear elasticity

4.2 Axial/Spherical elastoplasticity

4.3 Spherical hyperelasticity

4.4 1D elastodynamics

5. Solutions for elastic solids >

5.1 General Principles

5.2 2D Airy function solutions

5.3 2D Complex variable

solutions

5.4 3D static problems

5.5 2D Anisotropic elasticity

5.6 Dynamic problems

5.7 Energy methods

5.8 Reciprocal theorem

5.9 Energetics of dislocations5.10 Rayleigh Ritz method

6. Solutions for plastic solids >

6.1 Slip-line fields

6.2 Bounding theorems

7. Introduction to FEA >

7.1 Guide to FEA

7.2 Simple FEA program

8. Theory & Implementation of

FEA >

8.1 Static linear elasticity

8.2 Dynamic elasticity

8.3 Hypoelasticity

8.4 Hyperelasticity

8.5 Viscoplasticity

8.6 Advanced elements

9. Modeling Material Failure >

9.1 Mechanisms of failure

9.2 Stress/strain based criteria

9.3 Elastic fracture mechanics

9.4 Energy methods in fracture

9.5 Plastic fracture mechanics

9.6 Interface fracture mechanics

10. Rods, Beams, Plates & Shells>

10.1 Dyadic notation

10.2 Deformable rods - general

10.3 String / beam theory

10.4 Solutions for rods

10.5 Shells - general

10.6 Plates and membranes

10.7 Solutions for shells

A: Vectors & Matrices

B: Intro to tensors

C: Index NotationD: Using polar coordinates

E: Misc derivations

FEA codes

Maple

Matlab
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Report an error

Appendix A

Review of Vectors and Matrices

A.1. VECTORS

A.1.1 Definition

For the purposes of this text, a vector is an object which has magnitude and direction. Examples include forces,

electric fields, and the normal to a surface. A vector is often represented pictorially as an arrow and

symbolically by an underlined letter or using bold type . Its magnitude is denoted or . There are two

special cases of vectors: the unit vector has ; and the null vector has .

A.1.2 Vector Operations

Addition

Let and be vectors. Then is also a vector. The vector may be shown

diagramatically by placing arrows representing and head to tail, as shown in the

figure.

Multiplication

1. Multiplication by a scalar. Let be a vector, and a scalar. Then is a vector. The

direction of is parallel to and its magnitude is given by .

Note that you can form a unit vectorn which is parallel to a by setting .

2. Dot Product (also called the scalar product). Let a and b be two vectors. The

dot product ofa and b is a scalar denoted by , and is defined by

,

where is the angle subtended by a and b.Note that , and

. If and then if and only if ; i.e. a and b are

perpendicular.

3. Cross Product (also called the vector product). Let a and b be two

vectors. The cross product ofa and b is a vector denoted by .

The direction ofc is perpendicular to a and b, and is chosen so that (a,b,c)

form a right handed triad, Fig. 3. The magnitude ofc is given by

Note that and .

Some useful vector identities

A.1.3Cartesian components of vectors

Let be three mutually perpendicular unit vectors which form a right handed triad, Fig. 4. Then

are said to form and orthonormal basis. The vectors satisfy

We ma ex ress an vectora as a suitable combination of the unit vectors , and . For exam le, we ma
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write

where are scalars, called the components ofa in the basis . The components ofa have

a simple physical interpretation. For example, if we evaluate the dot product we find that

in view of the properties of the three vectors , and . Recall that

Then, noting that , we have

Thus, represents the projected length of the vectora in the direction of , as

illustrated in the figure. Similarly, and may be shown to represent the

projection of in the directions and , respectively.

The advantage of representing vectors in a Cartesian basis is that vector addition

and multiplication can be expressed as simple operations on the components of

the vectors. For example, let a, b and c be vectors, with components , and ,

respectively. Then, it is straightforward to show that

A.1.4 Change of basis

Let a be a vector, and let be a Cartesian basis. Suppose that the components of a in the basis

are known to be . Now, suppose that we wish to compute the components of a in a

second Cartesian basis, . This means we wish to find components , such that

To do so, note that

This transformation is conveniently written as a matrix operation

,

where is a matrix consisting of the components ofa in the basis , is a matrix consisting ofthe components ofa in the basis , and is a `rotation matrix as follows

Note that the elements of have a simple physical interpretation. For example, ,

where is the angle between the and axes. Similarly where

is the angle between the and axes. In practice, we usually know the angles between the axes

that make up the two bases, so it is simplest to assemble the elements of by putting the cosines of the known

angles in the appropriate places.

Index notation provides another convenient way to write this transformation:

You dont need to know index notation in detail to understand this all you need to know is that


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The same approach may be used to find an expression for in terms of . If you work through the details, you

will find that

Comparing this result with the formula for in terms of , we see that

where the superscript Tdenotes the transpose (rows and columns interchanged). The transformation matrix

is therefore orthogonal, and satisfies

where [I] is the identity matrix.

A.1.5Use ful vector operations

Calculating areas

The area of a triangle bounded by vectors a, band b-a is

The area of the parallelogram shown in the picture is 2A.

Calculating angles

The angle between two vectors a and b is

Calculating the normal to a surface .

If two vectors a and b can be found which are known to lie in the surface, then the unit normal to the

surface is

If the surface is specified by a parametric equation of the form , where s and t are two

parameters and r is the position vector of a point on the surface, then two vectors which lie in the plane

may be computed from

Calculating Volumes

The volume of the parallelopiped defined by three vectors a, b, c is

The volume of the tetrahedron shown outlined in red is V/6.

A.2. VECTOR FIELDS AND VECTOR CALCULUS

A.2.1. Scalar field.

Let be a Cartesian basis with origin O in three dimensional space. Let

denote the position vector of a point in space. A scalar field is a scalar valued function of position in space. A

scalar field is a function of the components of the position vector, and so may be expressed as . The

value of at a particular point in space must be independent of the choice of basis vectors. A scalar field may


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.

A.2.2.Vector field


denote the position vector of a point in space. A vector field is a vector valued function of position in space. A

vector field is a function of the components of the position vector, and so may be expressed as . The

vector may also be expressed as components in the basis

The magnitude and direction of at a particular point in space is independent of the choice of basis vectors. A

vector field may be a function of time (and possibly other parameters) as well as position in space.

A.2.3.Change of basis for scalar fields.

Let be a Cartesian basis with origin O in three

dimensional space. Express the position vector of a point relative to O

in as

and let be a scalar field.

Let be a second Cartesian basis, with origin P. Let

denote the position vector of P relative to O. Express the

position vector of a point relative to P in as

To find , use the following procedure. First, express p as components in the basis , using

the procedure outlined in Section 1.4:

where

or, using index notation

where the transformation matrix is defined in Sect 1.4.

Now, express c as components in , and note that

so that

A.2.4. Change of basis for vector fields .

Let be a Cartesian basis with origin O in three dimensional

space. Express the position vector of a point relative to O inas

and let be a vector field, with components


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, .

denote the position vector of P relative to O. Express the

position vector of a point relative to P in as

To express the vector field as components in and as a

function of the components ofp, use the following procedure. First,

express in terms of using the procedure outlined

for scalar fields in the preceding section

fork=1,2,3. Now, find the components of v in using the procedure outlined in Section 1.4.

Using index notation, the result is

A.2.5. Time derivatives of vectors

Let a(t)be a vector whose magnitude and direction vary with time, t. Suppose that is afixedbasis, i.e.

independent of time. We may express a(t) in terms of components in the basis as

.

The time derivativeofa is defined using the usual rules of calculus

,

or in component form as

The definition of the time derivative of a vector may be used to show the following rules

A.2.6. Using a rotating basis

It is often convenient to express position vectors as components in a basis which rotates with time. To writeequations of motion one must evaluate time derivatives of rotating vectors.

Let be a basis which rotates with instantaneous angular velocity . Then,

A.2.7. Gradient of a scalar field.

Let be a scalar field in three dimensional space. The gradient of is a vector field denoted by or

, and is defined so that

for every position r in space and for every vectora.


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denote the position vector of a point in space. Express as a function of the components ofr .

The gradient of in this basis is then given by

A.2.8. Gradient of a vector field

Let v be a vector field in three dimensional space. The gradient ofv is a tensor field denoted by or

, and is defined so that

for every position r in space and for every vectora.


denote the position vector of a point in space. Express v as a function of the components of r, so that

. The gradient of v in this basis is then given by

Alternatively, in index notation

A.2.9. Divergence of a vector field

Let v be a vector field in three dimensional space. The divergence ofv is a scalar field denoted by or

. Formally, it is defined as (the trace of a tensor is the sum of its diagonal terms).


denote the position vector of a point in space. Express v as a function of the components ofr: .

The divergence ofv is then

A.2.10. Curl of a vector field.

Let v be a vector field in three dimensional space. The curl of v is a vector field denoted by or .

It is best defined in terms of its components in a given basis, although its magnitude and direction are not

dependent on the choice of basis.


denote the position vector of a point in space. Express v as a function of the components ofr .

The curl of v in this basis is then given by


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Using index notation, this may be expressed as

A.2.11 The Divergence Theorem.

Let V be a closed region in three dimensional space, bounded by an

orientable surface S. Let n denote the unit vector normal to S, taken so

that n points out ofV. Let u be a vector field which is continuous and has

continuous first partial derivatives in some domain containing T. Then

alternatively, expressed in index notation

For a proof of this extremely useful theorem consult e.g. Kreyzig,Advanced Engineering Mathematics,Wiley,

New York, (1998).

A.3. MATRICES

A.3.1 Definition

An matrix is a set of numbers, arranged in m rows and n columns

A square matrix has equal numbers of rows and columns

A diagonal matrix is a square matrix with elements such that for

The identity matrix is a diagonal matrix for which all diagonal elements

A symmetric matrix is a square matrix with elements such that

A skew symmetric matrix is a square matrix with elements such that

A.3.2 Matrix operations

Addition Let and be two matrices of order with elements and . Then

Multiplication by a scalar. Let be a matrix with elements , and let kbe a scalar. Then

Multiplication by a matrix. Let be a matrix of order with elements , and let be a matrix

of order with elements . The product is defined only ifn=p, and is an matrix

such that

Note that multiplication is distributive and associative, but not commutative, i.e.


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The multiplication of a vector by a matrix is a particularly important operation. Let b and c be two vectors with

n components, which we think of as matrices. Let be an matrix. Thus

Now,

i.e.

Transpose. Let be a matrix of order with elements . The transpose of is denoted .

If is an matrix such that , then , i.e.

Note that

Determinant The determinant is defined only for a square matrix. Let be a matrix with

components . The determinant of is denoted by or and is given by

Now, let be an matrix. Define the minors of as the determinant formed by omitting the ith

row and jth column of . For example, the minors and for a matrix are computed as

follows. Let

Then

Define the cofactors of as

Then, the determinant of the matrix is computed as follows

The result is the same whichever row i is chosen for the expansion. For the particular case of a matrix

The

determinant may also be evaluated by summing over rows, i.e.


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and as before the result is the same for each choice of column j. Finally, note that

Inversion. Let be an matrix. The inverse of is denoted by and is defined such that

The inverse of exists if and only if . A matrix which has no inverse is said to besingular. Theinverse of a matrix may be computed explicitly, by forming the cofactor matrix with components as

defined in the preceding section. Then

In practice, it is faster to compute the inverse of a matrix using methods such as Gaussian elimination.

Note that

For a diagonal matrix, the inverse is

For a matrix, the inverse is

Eigenvalues and eigenvectors. Let be an matrix, with coefficients . Consider the vector

equation

(1)

where x is a vector with n components, and is a scalar (which may be complex). The n nonzero vectors x

and corresponding scalars which satisfy this equation are the eigenvectors and eigenvalues of .

Formally, eighenvalues and eigenvectors may be computed as follows. Rearrange the preceding equation to

(2)

This has nontrivial solutions forx only if the determinant of the matrix vanishes. The equation

is an nth order polynomial which may be solved for . In general the polynomial will have n roots, which may

be complex. The eigenvectors may then be computed using equation (2). For example, a matrixgenerally has two eigenvectors, which satisfy

Solve the quadratic equation to see that

The two corresponding eigenvectors may be computed from (2), which shows that

so that, multiplying out the first row of the matrix (you can use the second row too, if you wish since we chose

to make the determinant of the matrix vanish, the two equations have the same solutions. In fact, if ,

you will need to do this, because the first equation will simply give 0=0 when trying to solve for one of the

ei envectors


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which are satisfied by any vector of the form

wherep and q are arbitrary real numbers.

It is often convenient to normalize eigenvectors so that they have unit length. For this purpose, choosep and q

so that . (For vectors of dimension n, the generalized dot product is defined such that

)

One may calculate explicit expressions for eigenvalues and eigenvectors for any matrix up to order , but

the results are so cumbersome that, except for the results, they are virtually useless. In practice,

numerical values may be computed using several iterative techniques. Packages like Mathematica, Maple or

Matlab make calculations like this easy.

The eigenvalues of a real symmetric matrix are always real, and its eigenvectors are orthogonal, i.e. the ith and

jth eigenvectors (with ) satisfy .

The eigenvalues of a skew symmetric matrix are pure imaginary.

Spectral and singular value decomposition. Let be a real symmetric matrix. Denote the n

(real) eigenvalues of by , and let be the corresponding normalized eigenvectors, such that

. Then, for any arbitrary vectorb,

Let be a diagonal matrix which contains the n eigenvalues of as elements of the diagonal, and let be

a matrix consisting of the n eigenvectors as columns, i.e.

Then

Note that this gives another (generally quite useless) way to invert

where is easy to compute since is diagonal.

Square root of a matrix. Let be a real symmetric matrix. Denote the singular value

decomposition of by as defined above. Suppose that denotes the square

root of , defined so that

One way to compute is through the singular value decomposition of


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(c) A.F. Bower, 2008

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Applied Mechmatreianics of Solids (a.F