ContinuumMechanics V2

CONTINUUM MECHANICSEmail: cm.esprit@gmail.com

INTRODUCTION

Continuum mechanics: Continuum assumption: matter is distributed continuously in space

No voids between molecules Density function of the continuum is continuous: Density function of the continuum is continuous:

3limV

mV

V

EXAMPLES OF CONTINUAEXAMPLES OF CONTINUA

Crankshaft

Solids heartBridge

Continua FluidsWaterOilContinuaContinua Fluids OilBlood

Continua

GasAirCO2Steam

WHY CONTINUUM MECHANICS?

Continuum mechanics is used to determine: Stresses & Strains Kinematics and deformation

F il Failure Fluid flow Gas flow Etc…

TENTATIVE OUTLINE OF THE COURSE

scalar product

3. StressVectors

vector algebra

vector calculus

vector product

triple product

Gradient

divergence

Stress vector

Stress tensor

Transformation of stress

Principal stresses

Course

1. Mathematical Preliminairies

4. Constitutive EquationsTensors

Curl

Matrix algebra

dyads

Trensor transformation

Principal stresses

Elastic

Plasticity CourseOutline

5. Elasticity in 2D

Tensor calculus

Index notation

Configuration

Plane stress

Plane strain

Mohr's Circle

Outline

2. KinematicsExample Problems

g

Engineering strain

kinematis of a solid continuum

analysis of deformation

Compatibility equationsCompatibility equations

MATHEMATICAL PRELIMINARIES: VECTORS

A vector quantity is represented by an arrow.

Li f f

Force Vector

Line of force (or pull).

Length represents magnitude.

Arrow head represents

Tail represents point of forceapplicationArrow head represents

direction.

application.

VECTOR REPRESENTATION

A vector quantity is represented by an arrow.

Li f f

Force Vector

Line of force (or pull).

Length represents magnitude.

Arrow head represents

Tail represents point of forceapplicationArrow head represents

direction.

application.

SCALARS

Scalars are quantities which are fully described by a i d l magnitude alone.

Examples are time, energy, speed, distance

VECTORS

Vectors are quantities which are fully described by both a Vectors are quantities which are fully described by both a magnitude and a direction.

Examples include force, momentum, velocity and displacement.

WHICH ARE VECTOR AND WHICH ARESCALARS?

Quantity Example Type

Length 5 m Scalar

Velocity 30 m/sec, East Vector

Speed 30 m/sec Scalar

Temperature 20ºC Scalar

Energy 5 Joules ScalarEnergy 5 Joules Scalar

Displacement 2 m to the right Vector

VECTOR ALGEBRA: ZERO AND UNIT VECTORS

Unit Vector: is a vector with a unit length Unit Vector: is a vector with a unit length

ˆ ˆsuch that 1e e

We can always extract a unit vector from any non zero vector as:

ˆ ˆwhereue u u e

Zero Vector: is a vector with a unit length

where u u e u u eu

Zero Vector: is a vector with a unit length

such that 00 0

VECTOR ALGEBRA: VECTOR ADDITION

t ti it+

commutativity

associativity

u v = v + uu v + w = u v + w

null element

it l t u 0 0 u u

0 opposite element u u 0

VECTOR ALGEBRA: MULTIPLICATION BY ASCALAR

SVECTOR ALGEBRA: SCALAR PRODUCT

lbl tdspacein or plane in the vectorsbe w and,v,uLet

wuvu)wv(u:holdspropertyvedistributiThe2..uvvu :holdsproperty ecommutativ The 1.

scalar. a be clet and

0v0 4.

vcuvuc)vuc( 3.wuvu)wv(u:holdsproperty vedistributi The 2.

2vvv 5.

vectorsnonzeroobetween twangletheisIf

.vuvucos then v and u

vectorsnonzeroobetween tw angle theis If

VECTOR ALGEBRA: VECTOR PRODUCT

VECTOR ALGEBRA: TRIPLE PRODUCT

Scalar triple productp p

. 0 are coplanar A B×C A,B,C p, ,

VECTOR ALGEBRA: VECTOR PRODUCT

Vector triple productp p

VECTOR ALGEBRA: COMPONENTS OF AVECTOR

In a 3-dimensional vector space we can assign an p gortho-normal vector basis:

such that:

ORTHO

NORMAL

VECTOR ALGEBRA: COMPONENTS OF AVECTOR

Any vector can be expressed as a linear combination y pof the basis vectors :

All the expressions we saw previously and be expressed in component form. Vector sum:Vector sum:

Multiplication by scalar: Scalar Product:

Vector Product:

VECTOR ALGEBRA: COMPONENTS OF AVECTOR

Vector Product:

Scalar Triple product:

SUMMATION CONVENTION: FREE AND DUMMY INDICES

Component form:p f

Can be written as:

By convention we write this as:

Examples of expressions using summation convention:

In this expression ‘i’ is a free index and ‘j’ is a dummy index

In this expression ‘k’ is a free index and ‘i’ and ‘j’ are dummy indices.

SUMMATION CONVENTION: KRONECKER DELTA SYMBOL

The Kronecker delta symbol is defined as y f

It i ft i i d t ti i d ti It is often in index notation expressions and equations It is also known as the ‘substitution operator’ because it has

the effect of substituting one index by another.

Orthonormal basis vector are related by:

The scalar product is expressed as:

SUMMATION CONVENTION: PERMUTATION SYMBOL

The permutation symbol is defined as:p y f

Orthonormal basis vector are related by:

The cross product is expressed as:

EPSILON-DELTA RELATIONSHIP

VECTOR TRANSFORMATION LAW

Given two vector bases: ' ' ' '1 2 3 1 2 3ˆ ˆ ˆ ˆ ˆ ˆ, , and , ,B B e e e e e e

The directions of cosines are defined as:

A vector expressed in the two bases as:

1 2 3 1 2 3, , and , ,B Be e e e e e

'ˆ ˆ cosij i j ij e e

' ' ' ' ' ' ' '1 1 2 2 3 3 1 1 2 2 3 3

' ' ' ' ' ' ' '

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ or

ˆ ˆ ˆ ˆ ˆ ˆi i j j

j j i i j j k i i k j j kk ki i ik i

u u u u u u u u

u u u u uu u u

u e e e e e e u e e

e e e e e e Therefore the components of the vector transform

from base B to base B’ as:

j j j j j j

' 1 11 12 13 1' '

2 21 22 23 2'3 31 32 33 3

or k i ik

u uu u u u

u u

3 31 32 33 3

MATRICES

A matrix is an ordered array of scalars:y11 12 13 14

21 22 23 24

31 32 33 34

1,2,3 and 1,2,3,4ij

a a a aa a a a a i ja a a a

A

a matrix with m rows and n columns is said to be an m×nmatrix.

31 32 33 34

an n×n matrix is a square matrix.

MATRIX ADDITION AND MULTIPLICATION BYA SCALAR

TRANSPOSE & INVERSE OF A MATRIX

The transpose of a matrix A is denoted as ATpT

ij jib a B A and

T TT T T A A A B A B

1

T T TAB B A

The inverse of matrix A is denoted as A-1

1 1 AA A A I

The determinant of a matrix is given as:

1 11 1 1 and A I AB B A

ijAA The determinant of a matrix is given as: ijAA 1 2 3det ijk i j kA A AA

EXERCISES

EXERCISES

VECTOR CALCULUS

The del operator is the ‘vector’ 1 2 3ˆ ˆ ˆ ˆix x x x e e e ep

The gradient of a scalar field f(x1,x2,x3) is a vector:1 2 31 2 3 i ix x x x

1 2 31 2 3ˆ ˆ ˆ ˆi

f f f fix x x xf

e e e e The divergence of a vector field is given as:ˆi iAe

31 2

1 2 3div i

i

A AA Ax x x x

A A

ˆA ˆ ˆ ˆe e e The curl of a vector is given as:ˆi iAe

1 2 3

1 2 3

ˆcurl i

j

Aijk kx x x x

A A A

e e eA A e

The Laplacian of a scalar field f(x1,x2,x3) is: 1 2 3A A A

22 ˆ ˆf ff f 2 ˆ ˆj i i i

f fj ix x x xf f

e e

SUMMARY OF VECTOR FORMULAS

TENSOR (EXAMPLE) A vector is defined as an quantity requiring a direction

d it d and a magnitude. A tensor (2nd order) is a quantity that requires TWO

directions and a magnitude. E l Example

If a plane cuts through a continuum subject to externalforces (F1, F2,…) an internal stress vector T will appear.Th i d & di i f T ill d d th The magnitude & direction of T will depend on theorientation of the plane defined by its normal n

Therefore, in order to completely define the stress stateat a point P on the plane two vectors are needed T and nat a point P on the plane two vectors are needed T and n

2ND ORDER TENSORS: DYADS

A quantity that requires TWO directions to be q y qspecified is called a dyad or 2nd Order tensors.

a dyad D is defined as the ‘product’ of two vectors u & d i itt D& v and is written as:

The product uv is known as the dyadic product it is just the vectors written next to each other (it is also known as the

D uv

(indeterminate vector product)

For vectors a and b the following dyadic product ti h ldproperties hold:

2ND ORDER TENSORS: VARIOUS PRODUCTS

A dyadic is defined as: 1 1 2 2 N N D a b a b a b The dot product of a dyad D with a vector v is a vector and is defined as

(this product is NOT commutative): 1 1 2 2 = prefactorN N v D v a b v a b v a b u

The vector product of a dyad D with a vector v is a dyad: 1 1 2 2 = postfactorN N D v v b a v b a v b a w

1 1 2 2 = N N v D v a b v a b v a b E

The dot product of two dyads D and E is a dyad (this product is NOTcommutative):

1 1 2 2 = N N D v a v b a v b a v b F

D E ab cd b c ad F )

The tensor scalar product of two dyads D and E is dyad (this product IScommutative):

The tensor scalar product of two dyads D and E is dyad (this product IS

D E ab cd b c ad F

i i j ja c b d D E ab cd a c b d p y y ( p

commutative): i i j jb c a d D E ab cd b c a d

2ND ORDER TENSORS: TRANSFORM, PRINCIPAL VALUES, INVARIANTS

A dyadic D can be written in component form with respect to an orthonomal basis as:

The identity dyad I is such that:

2ne order tensors (dyads) transform between basis according to:

ˆ ˆ ˆ ˆi j i j i j i ja b D D ab e e e e1 1 2 2 3 3ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i i D I I D I e e e e e e e e

2 order tensors (dyads) transform between basis according to:

The principal values of 2nd order tensor T are defined as the f

' ' ' 'ˆ ˆ ˆ ˆ ˆ ˆi j i j i j ik k jm m ik jm i j k ma b a b a b D ab e e e e e e

roots of :

The determinant results in the Cubic equation known as the characteristic equation:

0 0 det 0ik k i ik ik k ik ikT u u T u T T u u 0

3 21 2 3 0I I I T u u

I1 , I2 and I3 are the first, second and third invariants of tensor T

3 2

1 2 3 0

(t ) d t

I I I

I T I T T T T I T T T T

T u u

1 2 3 1 2 3 (trace); - ; =det ii ii jj ij ij ijk i j j ijI T I T T T T I T T T T

VECTOR CALCULUS: EXERCISES

1. calculate the gradient vector of the following scalar fields:g g

2.

3.

TENSOR CALCULUS: EXERCISES

1.

2.

3.

DEFORMATION: DEFINITIONS

Point: designates a fixed location in space.

Particle: is a small infinitesimal volumetric element of a continuum it is also called a material point

At any time t a continuum with a volume V and bounding surface S will occupy a region R of Spaceregion R of Space.

Configuration: The region occupied by the continuum at a given time t is known as the configuration of the continuum at time t

Deformation: the change in shape of a continuum between an initial(undeformed) configuration and a final deformed configuration.

X: material coordinates vectorx: spatial coordinates vectoru: displacement vector.

DEFORMATION: LAGRANGIAN, EULERIAN

When a continuum deforms its particles move through space this motion is d fi d defined as:

This relation provides the current position of a particle that used to occupy the position at the initial time t=0

This relation is a mapping with one-to-one correspondence, continuous and with pp g p ,continuous first partial derivatives.

The description of the motion of the particles in this way is known as the Lagrangian formulation

Th f l ti i id t b E l i if th t f th ti l f th The formulation is said to be Eulerian if the movement of the particles of the continuum is given as:

the mappings and are unique inverses of one another such that: . This can be true only if the Jacobian of the mappings is not 1( , )tX x xzero:

det 0i iij

j j

x xJX X

DEFORMATION GRADIENT

The gradient of the position coordinate vector xi with respect to material di X l i 2 d d k h i l d f i coordinates Xi results in 2nd order tensor know as the material deformation

gradient given by:

ˆ ˆ ˆ ˆ ˆi ii i j i i j

j j j

x xxX X X

XF x e e e e e

The gradient of the displacement vector ui = xi – Xi produces the material displacement gradient given as:

i ii ix Xu x

Example: calculate the Jacobian, material deformation gradient and the displacement deformation gradient of the following:

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆi ii ii j i j i j ij i j

j j j

x Xu xX X X

XJ u e e e e e e e e F I

1 1 0

1 1 0 ;

0 0 1

det 1 1 1

t

ti

j

t t

ex eX

J F e e

F

1 1 0 0 1 01 0 0

1 1 0 0 1 0 1 0 0 ;0 0 10 0 1 0 0 0

t t

t tiij

j

e ex e eX

J F I

DEFORMATION TENSORS I

î

î

Ca ch deformation tensorCauchy deformation tensor

Green deformation tensor

DEFORMATION TENSORS II The difference is used to measure deformation of a continuum

between two configurations:between two configurations:

Where Lagrangian finite strain tensor

Lagrangian finite strain tensor can be written in terms of displacements as: Lagrangian finite strain tensor can be written in terms of displacements as:

If small deformation theory is assumed the linear lagrangian finite1iu If small deformation theory is assumed the linear lagrangian finite

strain tensor

Interpretation of the linear lagrange strain tensor:

1i

jX

ï ï

Example: find the linear lagrangian strain tensor for a line segment lying originally on the X2 axis.

DEFORMATION TENSORS: OVERVIEWdeformation gradient F i

j

xX

F

displacement gradient Ji i

ijj j

u xX X

J F I

deformation

Cauchy deformation tensor

deformation deformation Tensors

Green deformation tensor

deformation Tensors

Lagrangian finite strain tensor

Non-linear

in terms of material coordinates

in terms of displacementsstrain tensor

Linear

p

DEFORMATION: EXERCISES

1.

2.

3.

4.

5.

6 6.

DEFORMATION: EXERCISES II

1.

2.