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

Anteprima

ESTRATTO DOCUMENTO

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.

9

RIGID AND AFFINE MOTIONS ( )

t

transforming vectors which are tangent to curves through into vectors tangent to the corre-

p A

( + )

t we get

sponding curves through . Instead of (79)

p τ

A 1

( ) = ( + ) − ( ) = ( ( ) ) ( )

t t t t t t

lim , (83)

ḋ d d Ḟ d

p

τ t A

τ

→ 0

τ

and finally ∇ = . (84)

v Ḟ

t t

7 Affine velocity fields

The meaning of the velocity gradient can be illustrated in the following way. In a two-dimensional

t.

space let us consider a body in the shape of a square at a time Let us consider also an orthonor-

mal basis whose vectors are parallel to the sides of the square and denote the matrix of the velocity

gradient by

g g

11 12 (85)

g g 22

21

The shape the body takes in a sufficiently short time

The velocity field is described by (59).

interval can be described by the expression

τ ( + ) = ( ) + ( ) + ( )

t t t o

p p ṗ

τ τ τ

B B B (86)

= ( ) + ( ) + ( )( ( ) − ( )) + ( )

t t t t t

L o

p ṗ p p τ τ

B O B O

( ) =

t

If we assume that the center is at rest ( 0) we get

ṗ O

( + ) = ( ) + ( )( ( ) − ( )) + ( )

t t t t t . (87)

L o

p p p p

τ τ τ

B B B O ( )

t

1, 2, 3, show the shapes the body takes according to the values of given by the

Figures L

matrices in the tables below arranged in the same order as the shapes

1 0 0 1

0 0 0 0

Fig. 1

0 0 0 0

1 0 0 1

1

1 0 0 2

1

0 0 0

2

Fig. 2 1

0 0

0 2

1 0 1

0

2 1

0 0 0 2

1

0 0 0

2

Fig. 3 1

0 0 0

2

1 0 0

0

2

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

10 RIGID AND AFFINE MOTIONS

Figure 1: Illustration of the velocity gradient

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

11

RIGID AND AFFINE MOTIONS

Figure 2: Illustration of the velocity gradient (symmetric part)

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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