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2 PLACEMENTS AND DEFORMATIONS

1 Placement and deformation

B { } body points, placement

Denoting by a set . . . made up of a is a map

A, B, B→E

: (1)

p points)

which assigns a position to each point, in such a way that different body points (or simply

B

shape

take different positions. The of the body is the set

R =

: im (2)

p.

B

configuration

We call of the collection of the couples

( ) ∀ ∈ B , (3)

A, p A A R̄ R

placement.

where is a For any two placements and corresponding to shapes and there

p p̄ p,

deformation,

exists a bijective map, called R̄ → R

: , (4)

φ

− 1

= ◦ ( ) ∈ B ( )

defined as : , which moves every point from position to position

p p̄ A p̄ A

φ ( ) = ( ( )) . (5)

p A p̄ A

φ

displacement field,

We can define also the ( ) 7→ ( ( ) − ( )) ∀ ∈ B

: . (6)

u p̄ A p A p̄ A A

2 Rigid deformations and affine deformations rigid

If the deformation is an isometry which leaves the orientation unchanged we call it a

φ

deformation. In a rigid deformation, however we choose a body point the position of any other

A

body point in the placement is given by the following expression

B p ( ) = ( ) + ( − ) , (7)

R

p̄ p̄ p̄ p̄

φ φ

B A B A

V

where is a rotation of .

R affine deformation

If is an affine map which leaves the orientation unchanged then we call it an

φ

homogenous deformation.

or a However we choose a body point the position of any other body

A

point in the placement is given by the following expression

B p ( ) = ( ) + ( − ) , (8)

F

p̄ p̄ p̄ p̄

φ φ

B A B A

V

where is an endomorphism of , such that det 0. Note that the line

F F >

( ) = +

h h (9)

c̄ A

is transformed, through (8), into the curve

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

h h h h

: (10)

F Fā.

p̄ p̄ p̄

φ φ φ

c c̄ c̄

A A A

This curve is nothing but the line ( ) = ( ) +

h h (11)

a

φ

c A

whose tangent vector is = (12)

a Fā.

Hence an affine deformation transforms lines into lines.

DISAT, University of L’Aquila, April 23, 2011 (1432) A. Tatone – Mechanics of Solids and Materials.

3

PLACEMENTS AND DEFORMATIONS

3 Composition of deformations

Let us consider the following composition of two affine deformations

=

: . (13)

φ φ φ

[ ] [ ]

2 1

For any two body points and the first deformation is such that

A B, ( ) = ( ) + ( − ) . (14)

F

p̄ p̄ p̄ p̄

φ φ

B A B A

[ ]

1

[ ] [ ]

1 1

Applying the second deformation we get

( ) + ( ( ) − ( ))

( ) = (15)

F

p̄ p̄ p̄

p̄ φ φ φ

φ φ

φ A B A

B [ ]

2

[ ] [ ] [ ]

[ ] [ ]

[ ] 1 1 1

1 2

2

and, by substituting (14),

( ) + ( − )

( ) = . (16)

F F

p̄ p̄ p̄

p̄ φ

φ φ

φ A B A

B [ ] [ ]

2 1

[ ]

[ ] [ ]

[ ] 1

1 2

2 is described by

Hence for the composition (13) ( ) = ( ) + ( − ) (17)

F

p̄ p̄ p̄ p̄

φ φ

B A B A

where = . (18)

F F F

[ ] [ ]

2 1

nor composition (18) are commutative.

Note that in general neither composition (13)

By using the polar decomposition = (19)

F RU,

every affine deformation can be expressed, after choosing any body point as

A,

φ ( ) = ( ) + ( − ) (20)

RU

p̄ p̄ p̄ p̄

φ φ

B A B A

and then decomposed into a translation

( ) = ( ) + ( − ) , (21)

p̄ p̄ p̄ p̄

φ φ

B A B A

[ ] [ ]

0 0

( ) = ( ) ( )

such that , followed by a stretch, while holding fixed,

p̄ p̄ p̄

φ φ φ

A A A

[ ]

0

( ( )) = ( ( )) + ( ( ) − ( )) = ( ) + ( − ) , (22)

U U

p̄ p̄ p̄ p̄ p̄ p̄ p̄

φ φ φ φ φ φ φ

B A B A A B A

[ ] [ ] [ ] [ ] [ ] [ ]

1 0 1 0 0 0

( )

followed in turn by a rotation, with center ,

φ A

( ( ( ))) = ( ) + ( ( ( )) − ( )) . (23)

R

p̄ p̄ p̄ p̄

φ φ φ φ φ φ φ

B A B A

[ ] [ ] [ ] [ ] [ ]

2 1 0 1 0

( ) principal di-

The orthogonal lines through generated by the eigenvectors of called the

U,

φ A

rections of the stretch, are invariant under the deformation . The distance between any two

φ [ ]

1

positions along the principal directions changes by a factor equal to the corresponding eigen-

( )

principal stretches.

value of called the The line through generated by the eigenvector of

U, p̄

φ A

rotation axis,

called is invariant under the deformation .

R, φ [ ]

2

DISAT, University of L’Aquila, April 23, 2011 (1432) A. Tatone – Mechanics of Solids and Materials.

4 PLACEMENTS AND DEFORMATIONS

4 Affine deformations in coordinate form

E

Given a Cartesian coordinate system in a Euclidean space of dimension two, the positions of

∈ B

any two body points in the placement can be described by the expressions

A, B p̄

= + +

x̄ x̄ ,

e e

p̄ o

A A A 2

1

1 2 (24)

= + +

x̄ x̄ .

e e

p̄ o

B B B 2

1

1 2

The positions of the same two body points in the placement can be described by the expressions

p

( ) = + +

x x ,

e e

φ o

A A A 2

1

1 2 (25)

( ) = + +

x x .

e e

φ o

B B B 2

1

1 2

For a rigid deformation, by substituting (24) and (25) into (7), we get

+ + = + + + ( − ) + ( − )

x x x x x̄ x̄ x̄ x̄ , (26)

e e e e R e e

o o

B B A A B A B A

2 2 2

1 1 1

1 2 1 2 1 1 2 2

which implies + = + + ( − ) + ( − )

x x x x x̄ x̄ x̄ x̄ . (27)

e e e e Re Re

B B A A B A B A

2 2 2

1 1 1

1 2 1 2 1 1 2 2

θ

We can parameterize the rotation tensor through an angle in the following way

= +

θe θe

cos sin , (28)

Re 2

1 1

= − +

θe θe

sin cos . (29)

Re 2 2

1 ( )

Then, by equating components in the vector equation 27 we obtain the coordinate description

of the rigid deformation ! !

!

! −

− x̄ x̄

θ θ

x

x cos sin B A

A

B 1 1

1

1 +

= . (30)

θ θ x̄ x̄

x

x sin cos B A

A

B 2 2

2

2

For an affine deformation, by substituting (24) and (25) into (8) and setting

= +

f f , (31)

Fe e e 2

1 11 1 21

= +

f f , (32)

Fe e e

2 22 2

12 1

we obtain the following coordinate description

! ! ! !

x x f f x̄ x̄

B A B A

11 12

1 1 1 1

= + . (33)

x x f f x̄ x̄

B A B A

22

21

2 2 2 2

5 Parameterizations E

Given a Cartesian coordinate system in a Euclidean space of dimension two, the position of a

) = = ( )

( s x̄ s x̄

24 . If we set : , : , then 24

body point in the placement is described by

A p̄ A A A A

1 1 2 2

becomes = + +

s s . (34)

e e

p̄ o

A A A 2

1

1 2

Hence, for each position there exists a couple of coordinates

7→ ( )

s s

, (35)

p̄ A A A

1 2

DISAT, University of L’Aquila, April 23, 2011 (1432) A. Tatone – Mechanics of Solids and Materials.

5

PLACEMENTS AND DEFORMATIONS

and viceversa any couple of coordinates defines a position

( ) 7→

s s

, . (36)

A A A

1 2 R̄ =

parameterization

Denoting by such a function, called of the body shape : im we get

p̄,

κ = ( )

s s

, . (37)

p̄ κ

A A A

1 2

In general the parameterization of a body shape is independent of the coordinate system. If the

dimension of the Euclidean space is 2, we assume that the domain of the parameterization is the

2

closure of an open set of . Since the deformation gradient is never singular the dimension of

F

R

the shape does not change. φ

A coordinate description of a deformation can be given through two scalar functions and

φ 1

φ such that

2 ( ( )) = + ( ) + ( )

s s φ s s φ s s

, , , . (38)

e e

φ κ o

2 2 2 2 2

1 1 1 1 1

If the dimension of the Euclidean space is 3, we assume that the domain of the parameterization

3 φ φ φ

is the closure of an open set of . In this case we need three scalar functions , and in

R 2 3

1

order to give a coordinate description of the deformation.

6 Deformation gradient = ( )

s s

Let us consider a placement and the line through ,

p̄ p̄ κ

O 2

1

( ) = +

h h

: . (39)

e

c̄ O

1 1

and the parameterization (37) we obtain the following

By using a coordinate system as in (34)

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

h h s s h s h s s h, s

: , . (40)

e e e e

p̄ κ κ

c̄ o

O 2 2 2 2

1 1 1 1 1 1 1

This line is transformed by into the curve

φ ( ) = ( ( )) = ( ( + ))

h h s h, s

: . (41)

φ φ κ

c c̄ 2

1 1 1

If the deformation is not affine then this curve is not a straight line in general. By using the

φ

coordinate expression (38) for we get the following coordinate expression for the curve (41)

φ,

( ) = + ( + ) + ( + )

h φ s h, s φ s h, s , (42)

e e

c o 2 2 2 2

1 1 1 1 1

( ) = + ( ) + ( )

φ s s φ s s

0 , , . (43)

e e

c o 2 2 2 2

1 1 1 1 1

( )

Hence the tangent vector at is given by

φ O 1

′ ( ) − ( )) = +

= ( h ∂ φ ∂ φ

0 , (44)

lim e e

c c

c 2 2

1 1 1 1 1 1

1 h

h 0

where is the derivative with respect to the first argument. The line

1 ( ) = + = ( ) + = + + ( + ) = ( + )

h h s s h s s h s s h

: , , (45)

e e e e

p̄ κ κ

c̄ o

O

2 2 2 2 2 2 2

1 1 1 1

is transformed into the curve ( ) = ( ( )) = ( ( + ))

h h s s h

: , . (46)

φ φ κ

c c̄

2 2 2

1

DISAT, University of L’Aquila, April 23, 2011 (1432) A. Tatone – Mechanics of Solids and Materials.


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DESCRIZIONE DISPENSA

In questo materiale didattico relativo a posizionamenti e deformazioni vengono trattati in lingua inglese i seguenti argomenti: posizionamento e deformazione; deformazioni rigide e deformazioni affini; composizione di deformazioni affini; deformazioni affini in termini di coordinate; parametrizzazioni; gradiente della deformazione; Gradiente dello spostamento; deformazione locale; dilatazioni e scorrimenti.


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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