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Continuum Mechanics, part 1

Background, notation, tensors

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The molecular nature of the structure of matter

is well established.

In many cases, however, the individual moleculeis of no concern.

Observed macroscopic behavior is based on

assumption that the material is continuously distributed

throughout its volume and completely fills the space.

The continuum concept of matter is the fundamen-

tal postulate of continuum mechanics.

Adoption of continuum concept means that fieldquantities as stress and displacements are expressed

as piecewise continuous functions of space coordinates

and time.

The Concept of Continuum

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CONTINUUM MECHANICS

generally, it is a non-linear matter,

theoretical foundations are known for more thantwo centuries Cauchy, Euler, St. Venant, ...,

the non-linear mechanics develops quicklyduring the last decades and it is substantially

influenced by the availability of high-performancecomputers and by the progress in numerical andprogramming methods,

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it was the computer and modern mathematical

methods which allowed to solve the difficulttheoretical and engineering problems taking intoaccount the material and geometrical non-linearities and transient phenomena,

still, the most difficult task is the determination ovalidity range of the used mathematical, physicaland computational models.

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The mathematical description of non-linearphenomena is difficult for the efficient

development of formulas it is suitable to use thetensor notation.

The tensor notation can be considered as a

direct hint for algorithmic evaluation of formulas,however, for the practical numerical computationthe matrix notation is preferred.

Note: To a certain extent Maple and Matlab andold Reduce could handle symbolic manipulationin a tensorial notation.

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Notation

Thats why we will talk not only about the tensornotation, which is very efficient for deriving thefundamental formulas, but also about the equivalentmatrix notation, which is preferable for the computer

implementation.

Besides, we will also mention a so called vector

notation, which is currently being used in theengineering theory of strength of material.

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Example

Strain tensor in indicial notation is

ij

Its matrix representation is

[ ]

=

333231

232221

131211

.

Due to the strain tensor symmetrya more compact vector notation is

often being employed in engineering,i.e.

{ } { }T312312332211

= .

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The engineering strain is

{ }

=

=

=

21

21

12

33

22

11

6

5

4

3

2

1

2

2

2

zx

yz

xy

zz

yy

xx

The reason for the appearance of a strangemultiplication factor of 2 will be explained later.

You should carefully distinguish between constants in

klijklij C = and { } [ ] { } C= .

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Continuum mechanics in Solids...scope of the presentation

Tensors and notation

KinematicsFinite deformation and strain tensorsDeformation gradientDisplacement gradientLeft Cauchy deformation gradientGreen-Lagrange strain tensorAlmansi (Euler) strain tensor

Infinitesimal (Cauchy) strain tensorInfinitesimal rotationStretchPolar decomposition

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Continuum mechanics in Solids... cont.

Rigid body motionMotion and flowStress tensorsIncremental quantitiesEnergy principlesTotal and updated Lagrangian approach

Numerical approaches

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Tensors, Notation, Background

Continuum mechanics deals with physical quantities,which are independent of any particular coordinate system.

At the same time these quantities are often specified byreferring to an appropriate system of coordinates.

Such quantities are advantageously represented by

tensors. The physical laws of continuum mechanics areexpressed by tensor equations.

The invariance of tensor quantities under a coordinate

transformation is one of principal reasons for theusefulness of tensor calculus in continuum mechanics.

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Notation being used is not unified.

Deformation and motion of aconsidered body could be observed

from the configuration

at time 0to that at time t,

at time t to that at time t +t.

Notationsymbolic xr,,, cBA

indicial iiijij xcBA ,,,

matrix [ ] [ ] { } { }xcBA ,,,

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Tensors, vectors, scalars

General tensors .. transformation in curvilinear systems

Cartesian tensors .. transformation in Cartesian systems

Tensors are classified by the rank or orderaccording to

the particular form of the transformation law they obey.

In a three-dimensional space (n = 3) the number ofcomponents of a tensor is nN, where Nis the rank

(order) of that tensor.

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Tensors of the order zeroare called scalars.

In any coordinate system a scalar is specified by onecomponent. Scalars are physical quantities uniquelyspecified by magnitude.

Tensors of the order oneare called vectors.In physical space they have three components.

Vectors are physical quantities possessing bothmagnitude and direction.

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Scalars... magnitude only (mass, temperature, energy),will be denoted by Latin or Greek letters in italics as

Ea ,,

Vectors... magnitude and direction (velocity,acceleration), may be represented by directed linesegments and denoted by

{ }x,x

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A vector may be defined with respect to a particularcoordinate system by specifying the components of the

vector in that system.

The choice of coordinate system is arbitrary, but in

certain situations a particular choice may be

advantageous.

The Cartesian rectangular system is represented by

mutually perpendicular axes. Any vector may be

expressed as a linear combination of three, arbitrary,nonzero, noncomplanar vectors, which are called

base vectors.

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Summation rule

When an index appears twice in a term, that index

is understood to take on all values of its range, and theresulting terms summed.

So the repeated indices are often referred to as

dummy indices, since their replacement by any otherletter, not appearing as a free index, does not change

the meaning of the term in which they occur.

===3

1i

iiii babac

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Vectors will be denoted in a following way

Symbolic or Gibbs notation

Indicial notation; a component or all of them

Matrix algebra notation

Note

Tensor indicial notation does not distinguish betweenrow and column vectors

a,,aar

ia

{ }a

{ } { } { } { } { } ( )21T

321

T3

2

1

;...... iin

n

aaaasaaaaa

a

a

a

a

a ===

=

M

M

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Orthogonal transformation xx Direction cosines

)(cos jiij xxa = 9 quantities, 6 of them independent

are stored in 3 by 3 matrixA= aij

Transformation law is

xi = aijxj or {x} = [A]{x} or x = A x

for the first order Cartesian tensors.

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Inverse transformation xx

{ } [ ]{ }xAxxax jjii ==T

Combining forward and inverse transformations for an arbitrary vector

{ } [ ]{ } { } [ ] { }

{ } [ ] [ ] { }

{ } [ ]{ }

{ } { }xxxx

xIxxx

xAAxxaax

xAxxAxxaxxax

ii

kiki

kkjiji

kkjjjiji

==

==

==

====

T

T

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The coefficient[ ] [ ]AAaa kjij

Tor

gives the symbol or variable which is equaleither to one or to zero according to whether

the values iand k are the same or different.

This may be simply expressed by

[ ]Iij or

i.e. by Kronecker delta or unit matrix

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Second order tensor transformation

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Fourth order tensor transformation

rstuluktjsirijkl TaaaaT =' d = 3; DI M T ( d, d, d, d) , T( d, d, d, d) , a( d, d)f or i = 1 t o d

f or j = 1 t o d

f or k = 1 t o df or l = 1 t o d

T ( i , j , k, l ) = 0;f or r = 1 t o d

f or s = 1 t o d

f or t = 1 t o df or u = 1 t o dT(i,j,k,l) = T(i,j,k,l) + a(i,r)*a(j,s)*a(k,t)*a(l,u)*T(r,s,t,u);

next unext t

next s

next rnext l

next knext j

next i

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