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Esame di Mechanics of solids and materials docente Prof. A. Tatone

Anteprima

ESTRATTO DOCUMENTO

4 RIGID AND AFFINE MOTIONS

3 Spin axis ( )

t

A skew symmetric tensor , as an endomorphism of a real vector space of dimension three,

W

has a null eigenvalue. In fact, since the characteristic polynomial is of order three there exists at

( )

t

least one real eigenvalue Denoting by a unit eigenvector corresponding to it is

a

λ. λ

o

( ) ( ) = ( )

t t t . (26)

W a a

λ

o o

By (24) = ( ) ( ) · ( ) =

t t t 0. (27)

W a a

λ o o

Hence ( ) ( ) =

t t (28)

W a o.

o ( )

t, t

Let us consider, at a time a line passing through p A

( ) = ( ) + ( )

h, t t h t . (29)

a

p

c o o

A

All body points occupying such positions have the same velocity since by (28),

( ) = ( ) + ( )( ( ) − ( )) = ( ) + ( ) ( ) = ( )

h, t t t h, t t t hW t t t . (30)

W a

ṗ p ṗ ṗ

ċ c

o o o

A A A A

( ) ( )

t t

This property holds for any line parallel to . Hence for each line parallel to there is a

a a

o o

common velocity, possibly different from line to line. Let us consider now any two body points A

Note that the velocity difference (25) is such that, by (28),

e and the difference vector (20).

B T

( ) · ( ) = ( ) ( ) · ( ) = ( ) · ( ) ( ) = − ( ) · ( ) ( ) =

t t t t t t t t t t t 0. (31)

ḋ a W d a d W a d W a

o o o o ( )

( ) t

t and to as well.

Hence the velocity difference is either zero or orthogonal to , by (23), a

d o

Property (31) can be also given a different interpretation. If it is put in the form

( ( ) − ( )) · ( ) =

t t t 0, (32)

a

ṗ ṗ o

B A

it implies that ( ) · ( ) = ( ) · ( )

t t t t , (33)

a a

ṗ ṗ

o o

B A

( )

t

Thus the orthogonal projection of the velocity on turns out to be the same for all body points.

a o ( )

t

Hence the velocity of each body point can be decomposed into the sum of a velocity parallel

v o

( ) ( )

t t

to the axis , which is unique for the whole body, and a velocity orthogonal to .

a a

o o

( ) ( )

t t

Let us consider a straight line passing through and lying on a plane orthogonal to a

p o

A

( ) = ( ) + ( ) ( ) · ( ) =

h, t t hd t t t

, 0. (34)

d a

p

c o

A

Velocities along this line can be expressed as ⊥

( ) = ( ) + ( ) ( ) = ( ) + ( ) + ( ) ( )

h, t t hW t t t t hW t t , (35)

d v v d

ċ o

A A

with ( ) = ( ( ) · ( )) ( )

t t t t

: , (36)

v a a

o o o

A

⊥ ( ) = ( ( ) − ( ))

t t t

: , (37)

v v

ṗ o

A

A

⊥ ( ) ( ) ( )

t t t

where , like , is a vector orthogonal to .

v d a o

A ⊥ ( ) ( ) ( ) ( )

( ) t t t t

t then , which is orthogonal both to

If we choose orthogonal also to W d d

d v A ⊥

( ) ( )

t t h

and by (24) . Then there exists a unique value for

and (31), turns out to be parallel to

a v

o A

such that ⊥ ( ) + ( ) ( ) =

t hW t t (38)

v d o,

A ( ) ( )

t t

a position where the velocity is exactly , parallel to . The straight

selecting, through (34), v a

o o

( )

t spin axis.

line passing through this position and parallel to is called

a o

DISAT, University of L’Aquila, April 23, 2011 (1435) A. Tatone – Mechanics of Solids and Materials.

5

RIGID AND AFFINE MOTIONS

3.1 Axial vector

( ) ( ) ( ) ( )

t t t t

Since transforms any vector into a vector orthogonal both to and , there should

W d d a o

( )

t

be a vector such that

ω ( ) ( ) = ( ) × ( ) ∀ ( ) ∈ V

t t t t t . (39)

W d d d

ω

We can prove that it exists and is unique as follows. Since

( ) ( ) = ( ) × ( ) =

t t t t (40)

W o,

ω ω ω

( ) =

t is an eigenvector corresponding to the eigenvalue 0. Hence it belongs to the same one-

λ

ω ( )

t

dimensional eigenspace as . Setting for any given orthonormal basis

a o ( ) = ( ) + ( ) + ( )

t t t t , (41)

e e e

ω ω ω

ω 2 2 3 3

1 1

the components are obtained through (39)

( ) · = ( ) × · = × · ( ) = · ( ) ⇒ ( ) = ( ) ·

t t t t t t

W e e e e e e e W e e

ω

ω ω ω

2 2 2 3 3 2

1 1 1 1

( ) · = ( ) × · = × · ( ) = · ( ) ⇒ ( ) = ( ) ·

t t t t t t (42)

W e e e e e e e W e e

ω

ω ω ω

2 3 2 3 2 3 2 3

1 1

( ) · = ( ) × · = × · ( ) = · ( ) ⇒ ( ) = ( ) ·

t t t t t t

W e e e e e e e W e e

ω

ω ω ω

3 3 3 2 2 3

1 1 1 1

( ) ( )

t axial vector t defines, through (42), an isomorphism

The vector is called of . Relation (39)

W

ω V

between the space of skew symmetric tensors and the three-dimensional vector space .

3.2 Spin center t, spin center

In a two-dimensional vector space, for a given rigid velocity field at time we call the

( ) =

t

position of a point such that 0. Since

C ṗ C

( ) − ( ) = ( )( ( ) − ( ))

t t t t t , (43)

W

ṗ ṗ p p

A C A C

( ) ( )

t t

if is the spin center then equals the velocity difference and hence it is orthogonal to

p ṗ

C A ( ) ( )

t t

the line joining the placements and . That is why the spin center can be obtained

p p

A C

by the intersection of lines drawn from any two positions and orthogonal to the corresponding

velocities.

4 Rigid motions in coordinate form

Let us consider a two-dimensional Euclidean space and a Cartesian coordinate system defined by

{ } ( )

t t

an origin and an orthonormal basis , . In a rigid motion the rotation at any time

e e R

O 2

1

can be described by ( ) = ( ) + ( )

t t t

cos sin , (44)

R e e e

θ θ 2

1 1

( ) = − ( ) + ( )

t t t

sin cos . (45)

R e e e

θ θ

2 2

1

( )

Then the relation 10 can be transformed into the following one in terms of coordinates

! ! ! !

( ) ( ) ( ) − ( ) −

x t x t t t x̄ x̄

cos sin

θ θ

B A B A

1 1 1 1

= + . (46)

( ) −

( ) ( ) ( )

x t x̄ x̄

x t t t

sin cos

θ θ

B B A

A

2 2 2

2

DISAT, University of L’Aquila, April 23, 2011 (1435) A. Tatone – Mechanics of Solids and Materials.

6 RIGID AND AFFINE MOTIONS

t

Differentiating with respect to we get

!

! ! !

− ( ) − ( )

( )

( ) −

t t

ẋ t

ẋ t x̄ x̄

sin cos

θ θ

A

B B A

1

1 1 1

+ ( )

= t . (47)

θ̇ −

( )

( ) ( ) − ( ) x̄ x̄

ẋ t

ẋ t t t

cos sin

θ θ B A

A

B 2 2

2

2 )

( 46 , the expression

Then we can replace, from ! ! !

− ( ) ( ) ( ) − ( )

x̄ x̄ t t x t x t

cos sin

θ θ

B A B A

1 1 1 1

= (48)

− − ( ) ( ) ( ) − ( )

x̄ x̄ t t x t x t

sin cos

θ θ

B A B A

2 2 2 2

)

( 47 , thus obtaining

into ! !

!

! ( ) − ( )

− ( )

( )

( ) x t x t

t

ẋ t

ẋ t 0 θ̇ B A

A

B 1 1

1

1 +

= . (49)

( ) ( ) − ( )

( )

( ) t x t x t

ẋ t

ẋ t 0

θ̇ B A

A

B 2 2

2

2 ( ) ( )

This is the expression relating the components of the velocities in 17 . Hence the matrix in 49

( )

t

is the matrix of .

W

In general, in a three-dimensional Euclidean space endowed with a Cartesian coordinate sys-

{ } ( )

tem whose orthonormal vector basis is , , , the relation 17 can be transformed into the

e e e

2 3

1

following one in terms of coordinates

 

  

 − ( ) ( ) ( ) − ( )

( )

( ) t t x t x t

ẋ t

ẋ t 0 ω ω B A

A

B 3 2 1 1

1

1  

  

 +

= ( ) − ( ) ( ) − ( )

( )

( ) t t x t x t

ẋ t

ẋ t . (50)

0

ω ω B A

A

B 3 1 2 2

2

2  

  

  

  

 ( ) − ( ) ( ) ( )

( ) −

( ) t t t t

ẋ t x

x

t

ẋ 0

ω ω

2

3 3

3

3 A A

B

B 1

The matrix of is skew symmetric because

W T = − ⇒ · = − · . (51)

W W We e e We

i j i j

5 Affine motion

affine

A motion is said to be if the one-parameter family of deformations (4) is such that for any

t

two body points and and for any

A B ( ) = ( ) + ( )( − )

t t t

, , , (52)

F

p̄ p̄ p̄ p̄

φ φ

B A B A

( ) ( )

t t affine motion

where is a tensor such that det 0. Hence an is defined by the motion of

F F > ( )

deformation gradient t

any body point, say and by the values of the .

F

A,

The velocity field in an affine motion is given by

( ) = ( ) + ( )( − )

t t t

, , . (53)

p̄ p̄ p̄ p̄

φ̇ φ̇

B A B A

( )

Replacing 52 with ( ) = ( ) + ( )( − )

t t t , (54)

F

p p p̄ p̄

B A B A

we get ( ) = ( ) + ( )( ( ) − ( ))

t t t t t . (55)

ṗ ṗ p p

B A B A

0 0

)

( 54

Since from − 1

( ) − ( ) = ( ) ( ( ) − ( ))

t t t t t , (56)

F

p p p p

B A B A

0 0

DISAT, University of L’Aquila, April 23, 2011 (1435) A. Tatone – Mechanics of Solids and Materials.

7

RIGID AND AFFINE MOTIONS

then − 1

( ) = ( ) + ( ) ( ) ( ( ) − ( ))

t t t t t t . (57)

Ḟ F

ṗ ṗ p p

B A B A

Setting − 1

( ) = ( ) ( )

t t t

: , (58)

L Ḟ F

( )

the expression 57 can be written

( ) = ( ) + ( )( ( ) − ( ))

t t t t t . (59)

L

ṗ ṗ p p

B A B A +

t t

If we consider a deformation transforming positions at time into positions at time τ

( ( ) ) = ( ( ) ) + ( )( ( ) − ( ))

t t t t

, , , (60)

F

p p p p

τ τ τ

φ φ t

B A B A

t t t

we get the following expression for the velocities at time

( ( ) ) = ( ( ) ) + ( )( ( ) − ( ))

t t t t

, 0 , 0 0 . (61)

p p p p

φ̇ φ̇ t

B A B A

t t

we obtain

By comparing this expression with (59) − 1

( ) = ( ) ( ) = ( )

t t t 0 . (62)

L Ḟ F Ḟ t

This expression allows us to give a useful characterization of From the polar decomposition of

L.

the deformation gradient ( ) = ( ) ( ) (63)

F R U

τ τ τ

t t t

=

we get, differentiating with respect to time at 0,

τ

( ) = ( ) = ( ) ( ) + ( ) ( ) = ( ) + ( )

t 0 0 0 0 0 0 0 , (64)

L Ḟ Ṙ U R U̇ Ṙ U̇

t t t t t t t

( ) = ( ) = ( ) = ( ) ( )

since 0 implies 0 and 0 Note that 0 is skew symmetric while 0

F I R I U I. Ṙ U̇

t t t t t

is symmetric, since T T T

( ) ( ) = ⇒ ( ) ( ) + ( ) ( ) =

R R I Ṙ R R Ṙ O

τ τ τ τ τ τ

t t t t t t

T

⇒ ( ) + ( ) =

0 0 (65)

Ṙ Ṙ O,

t t

T T

( ) = ( ) ⇒ ( ) = ( )

0 0 . (66)

U̇ U̇ U̇ U̇

τ τ

t t t t

( )

t

If we consider the decomposition of L ( ) = ( ) + ( )

t t t , (67)

L D W

with 1

1 T T

( ( ) + ( ) ) ( ) = ( ( ) − ( ) )

( ) = t t t t t

t , : (68)

: L L W L L

D 2 2

it turns out that ( ) = ( ) ( ) = ( )

t t

0 , 0 . (69)

D U̇ W Ṙ

t t

( ) ( )

t t stretching spin

This is the reason why and are called and respectively.

D W ( ) ( )

affine velocity field t

An is a velocity field whose velocities are given by 59 , where is a

L

tensor.

DISAT, University of L’Aquila, April 23, 2011 (1435) A. Tatone – Mechanics of Solids and Materials.

8 RIGID AND AFFINE MOTIONS

6 Velocity gradient

t

A generic motion at any time can be described by a deformation

(· ) R̄ 7→ E

t

, : . (70)

φ

Let us consider a straight line ( ) = +

h h (71)

c̄ A

R̄ (· ) R

t t

on the shape and for each time the curve , on such that

c

( ) = ( ( ) )

h, t h t

, . (72)

φ

c c̄

( )

t

The tangent vector at 0, is defined as the limit

c 1

( ) − ( )

( ) = h, t t

t 0, . (73)

: lim

d c c

h

h 0 (· )

t t

At each time we can define, for each body point the gradient of the vector field , as the

A, φ

tensor such that ( ) 7→ ( )

t t

, : , (74)

F d̄ d

p̄ A

transforming vectors which are tangent to curves passing through into vectors which are tan-

p̄ A

( ) = ( )

t t

gent to the corresponding curves through , .

p p̄

φ

A A

t

At time the velocity of body point is given by

( ) 7→ ( )

t t

: . (75)

v p ṗ

t A A

t spatial velocity field.

whose domain is the shape of the body at time and which is called The

gradient of this vector field [see A 2] is the tensor such that

v

PPENDIX t 1

1

( ( )) − ( ( )) = ( ) − ( )

∇ ( ) = h, t t h, t t

t 0, lim 0, . (76)

lim v v

v d c c ċ ċ

t t

t h h

→ h

h 0

0

Since from (73) 1

( ) − ( )

( ) = h, t t

t 0, , (77)

lim

ḋ ċ ċ

h

h 0

we get from (76) ∇ ( ) = ( )

t t . (78)

v d ḋ

t

( )

From 74 we get also 1 − 1

( ) = ( + ) − ( ) = ( ) = ( ) ( ) ( )

t t t t t t t

lim , , , . (79)

ḋ d d Ḟ d̄ Ḟ F d

p̄ p̄ p̄

τ A A A

τ

→ 0

τ )

( 78 it turns out, dropping function arguments out,

Replacing this expression into − 1

∇ = . (80)

v ḞF

t +

t t

If we consider the deformation from the shape at a fixed time to any time τ

(· ) ( ) 7→ ( + )

t t

, : , (81)

p p

τ τ

φ A A

t (· )

we can define, for each body point the gradient of , , as the tensor

A, τ

φ t

( ( ) ) ( ) 7 → ( + )

t t t

, : , (82)

F d d

p τ τ

t A

DISAT, University of L’Aquila, April 23, 2011 (1435) A. Tatone – Mechanics of Solids and Materials.


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PUBBLICATO

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DESCRIZIONE DISPENSA

In questo materiale didattico relativo ai moti rigidi e ai moti affini vengono trattati in lingua inglese i seguenti argomenti: i moti; i moti rigidi; asse istantaneo di rotazione; vettore assiale; centro istantaneo di rotazione; moti rigidi in termini di coordinate; moti affini; gradiente della velocità; descrizione di campi di velocità affini.


DETTAGLI
Corso di laurea: Corso di laurea magistrale in ingegneria matematica
SSD:
Università: L'Aquila - Univaq
A.A.: 2011-2012

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Mechanics of solids and materials e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università L'Aquila - Univaq o del prof Tatone Amabile.

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