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Anteprima

ESTRATTO DOCUMENTO

2 ELASTIC AFFINE BODIES

1 Balance principles

1.1 Rigid body

rigid body

A is a body whose placements are such that the deformation between any two of them

is rigid. The space of test velocity fields is made of all the rigid velocity fields.

in any motion at each time t the forces are such that their

Let us assume the following principle:

power in any test velocity field is zero: ( )

out ( ) =

W ∀

0 (1)

v v.

Since for any rigid test velocity field the power takes the form

( )

out ( ) = +

W · · (2)

v f v M W,

p

O O

where is the total force and is the total moment with respect to any position , the principle

f M p

p O

O balance equations

above is equivalent to the following = (3)

f o,

=

skw (4)

M O.

p O

1.2 Affine body and Cauchy stress

affine body

An is a body whose placements are such that the deformation between any two of them

is affine. The space of the test velocity fields is made of all the affine velocity fields.

deformable

The affine body is the simplest model of body. Since for an affine test velocity the

power is ( )

out ( ) = +

W · · (5)

v f v M L,

p

O O

t

if we assumed that at any time the power be zero for any test velocity field, we would get the

following balance equations = (6)

f o,

= (7)

skw O,

M p O =

sym (8)

M O.

p O

The last condition will filter out any force distribution with a symmetric moment tensor. In order

inner power

to allow for such force distributions we can introduce an

( )

in ( ) = + )

W −( · · V , (9)

v z v T L

O R

in any motion at each time t

and assume that ( ) ( )

out in

( ) + ( ) =

W W 0 (10)

v v

balance principle

for any affine test velocity field. Such a new turns out to be equivalent to the

balance equations =

− V (11)

f z o,

R =

− V (12)

M T O.

p R

O

.

DISAT, University of L’Aquila, April 29, 2011 (933) A. Tatone – Mechanics of Solids and Materials.

3

ELASTIC AFFINE BODIES

2 Stress characterization

( )

in

W

inner power

Since the has been introduced for a body which can undergo non rigid defor-

mations, it is reasonable to assume that it vanishes for any rigid test velocity field. Hence

+ =

· · 0, (13)

z v T W

O

for any skew symmetric tensor This condition is equivalent to

W. = (14)

z o,

=

skw (15)

T O.

balance equations

The for an affine body become = (16)

f o,

=

skw (17)

M O,

= V

sym . (18)

M T R

It is worth noting that the moment tensor turns out to be independent of because the total

p O

force is zero.

2.1 Objectivity

In a more general setting we consider two different “observers”. An affine motion is seen by the

first observer as described by such that

φ,

( ) = ( ) + ( )( )

− ∀ ∈ B

t t t

, , (19)

F

p̄ p̄ p̄ p̄ A

φ φ

A O A O

which can be rewritten as ( ) = ( ) + ( )( )

t t t . (20)

F

p p p̄ p̄

A O A O

t

A different observer will see the same body point at time in a different position given by

A

∗ ∗

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

t t t t t , (21)

Q

p q p q

A

A

∗ ( )

t

where are three time functions, and is an orthogonal tensor. Since

Q, Q

q, q ∗ ∗

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

t t t t t , (22)

Q

p q p q

O

O t,

subtracting (22) from (21) we get, using (20) and dropping the argument

∗ ∗ = ( ) = ( )

− − − . (23)

Q QF

p p p p p̄ p̄

A O A O

A O

Hence ∗ ∗ ∗

= + ( )

− (24)

F

p p p̄ p̄

A O

A O

with ∗ = (25)

F QF.

By differentiating (21) with respect to time, we get the relation between the velocities measured

by the two observers ∗ ∗

= + ( ) + ( )

− −

Q̇ Q

ṗ q̇ p q ṗ q̇

A A

A T

∗ ∗ ∗

= + ( ) + ( )

− − (26)

Q̇Q Q

q̇ p q ṗ q̇

A

A T

∗ ∗ ∗

= + ( ) + ( )

− − .

Q Q Q̇Q

ṗ q̇ q̇ p q

A A

DISAT, University of L’Aquila, April 29, 2011 (933) A. Tatone – Mechanics of Solids and Materials.

4 ELASTIC AFFINE BODIES

Q

F QF

Figure 1: Change of observer

For any given test velocity field = + ( )

− , (27)

v v L p p

A O A O

corresponding test velocity field

in a change of observer the is the field

∗ ∗ ∗ ∗ ∗

= + ( )

− (28)

v v L p p

A O A O

such that T

∗ ∗ ∗ ∗

= + ( ) + ( )

− − . (29)

v Qv Q Q̇Q

q̇ q̇ p q

A

A A

Replacing with we get

A O T

∗ ∗ ∗ ∗

= + ( ) + ( )

− − , (30)

v Qv Q Q̇Q

q̇ q̇ p q

O

O O

and, taking the difference, T

∗ ∗ ∗ ∗

= ( ) + ( )

− − −

v v Q v v Q̇Q p p

A O

A O A O

T ∗ ∗

= ( ) + ( )

− −

QL Q̇Q

p p p p

A O A O (31)

T T

∗ ∗ ∗ ∗

= ( ) + ( )

− −

QLQ Q̇Q

p p p p

A O A O

T T ∗ ∗

= ( + )( )

− .

QLQ Q̇Q p p

A O

Thus T T

∗ = + . (32)

L QLQ Q̇Q

material objectivity principle the inner power, for any test velocity field, is

The is stated as follows:

invariant under a change of observer. t

As a consequence we get that at any time and for any change of observer, as defined by the

three time functions ,

Q, q, q ∗ ∗ ∗ ∗

+ = +

· · · · (33)

z v T L z v T L

O

O

for any test velocity field. By using (30) and (32) we obtain

T

∗ ∗ ∗ ∗

+ ( ) + ( )

· − − − ·

z Qv Q Q̇Q z v

q̇ q̇ p q

O O

O

T T

∗ =

+ + − ·

· 0. (34)

T L

T QLQ Q̇Q

DISAT, University of L’Aquila, April 29, 2011 (933) A. Tatone – Mechanics of Solids and Materials.

5

ELASTIC AFFINE BODIES T ∗

= =

Selecting only those changes of observer such that we get

Q̇Q O, Q

q̇ q̇

T T

∗ ∗

+ =

− · − · 0 (35)

Q z z v Q T Q T L

O

which must hold for any test velocity field. Hence

T ∗

= , (36)

z Q z

T ∗

= (37)

T Q T Q.

Replacing these expressions into (34) we obtain

T T

∗ ∗ ∗ ∗ ∗

( ) + ( ) + =

· − − · 0. (38)

z Q Q̇Q T Q̇Q

q̇ q̇ p q

O T =

Selecting those changes of observer such that we get

Q̇Q O

∗ ∗

( ) =

· − 0 (39)

z Q

q̇ q̇

( )

which must hold whatever be the vector . Hence

Q

q̇ q̇

∗ = (40)

z o,

becomes

while (38) T

∗ =

· 0. (41)

T Q̇Q

In order for this condition to hold for any change of observer the stress must be such that

∗ =

skw , (42)

T O

T and (42) implies again (14) and (15).

since is skew symmetric. It is worth noting that (40)

Q̇Q

For from (36) and (40) we get T ∗

= = , (43)

z Q z o

and (42) we get

while from (37)

1 1

T T T T T

∗ ∗ ∗ ∗

= ( ) = ( )

− −

skw T Q

Q T Q Q T Q Q T T

2 2

T ∗ =

= )

( . (44)

skw Q O

T

Q

2.2 Composition with a rigid motion

A different point of view, which turns out to be equivalent to the previous one, consists in com-

t,

with a rigid motion as described, at time by the expression

posing the affine motion (19)

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

t t t t t t t

, , . (45)

Q

p q p q

φ φ

A A

r r

If we set ∗ ( ) = ( ( ) )

t t t

, , (46)

p p

φ A

A r

∗ ( ) = ( ( ) )

t t t

, , (47)

q q

φ r

material objectivity principle the inner power, for any

then (45) turns into (21). Restating the as follows:

test velocity field be invariant for any composition with a rigid motion, we get at the same conclusions

we arrived at by stating the principle in terms of invariance under a change of observer.

DISAT, University of L’Aquila, April 29, 2011 (933) A. Tatone – Mechanics of Solids and Materials.


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DESCRIZIONE DISPENSA

In questo materiale didattico in lingua inglese, relativo a Elastic affine bodies, vengono trattati i seguenti argomenti: balance principles; rigid body; affine body and Cauchy stress; stress characterization; objectivity; composition with a rigid motion; material response; material symmetry group; isotropy; Piola-Kirchhoff stress; constraints and reactive forces.


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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Altri appunti di Mechanics of solids and materials

Spazi Euclidei
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Placements and deformations
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Motions
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Power, forces and moments
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