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Anteprima

ESTRATTO DOCUMENTO

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.

6 ELASTIC AFFINE BODIES

3 Material response

The characterization of the relationship between stress and motion is based on these principles:

the stress is determined by the past history of the motion.

Principle of determinism:

• the stress in a body point does not depend on the deformation in any other

Principle of local action:

• point at a finite distance. At any time the deformation

Let us consider an affine motion as described by the expression (19).

is defined by its value in and by the deformation gradient Since by the principle of local

F.

p O

action the stress cannot depend on the value of the deformation in a particular position, then it

response function t

will depend on the gradient. Hence the at time gets the following form

t

b

= ( ) , (48)

T T F

t

where denotes the history of the motion through the deformation gradient. A very important

F elastic materials, the

class of materials is made up of the characterized by the following property:

stress depends only on the current value of the deformation gradient. The response function becomes

b

= ( ) . (49)

T T F

Note that the response function contains the description of a placement on which the deforma-

p̄,

tion applies, characterized by a zero stress.

φ

Let us consider for a body made up of an elastic material the affine motion (20) and the corre-

as seen by a different observer through (21). The response from the point

sponding motion (24)

of view of the second observer will be, by (25),

∗ ∗

b b

= ( ) = ( ) . (50)

T T F T QF

principle of material objectivity

Since the leads to (37), by replacing there (49) and (50) we get

T b

b ( )

( ) = (51)

T QF Q

T F Q T

=

This condition has to be fulfilled by the response function for any By choosing , where

Q. Q R

is the rotation in

R = (52)

F RU,

we get as a necessary condition to be fulfilled T

b b

( ) = ( ) , (53)

T F R T U R

or equivalently T b b

( ) = ( ) . (54)

R T F R T U

Viceversa, if a response function has the property (53), then it satisfies (37) for any For, replac-

Q.

in (50) we get

ing (53) T T T T T

∗ b b b b

= ( ) = ( ) ( )( ) = ( ( ) ) = ( ) = (55)

T T QF QR T U QR Q R T U R Q Q T F Q QTQ

which is equivalent to (37). Hence the property (53) characterizes all the elastic materials and is

reduced form of the response function for elastic materials.

called

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

7

ELASTIC AFFINE BODIES

F F

Q b b

( ) = ( )

Figure 2: A rotation belongs to the material symmetry group if .

Q T FQ T F

4 Material symmetry group after

Note that if the affine deformation is applied a rigid deformation such that

φ φ r

= ( ) = ( ) + ( )

− (56)

Q

p̄ p̌ p̌ p̌ p̌

φ φ

A A O A O

r r

=

⋆ ◦

then the composition : is again an affine deformation such that

φ φ φ r

⋆ ( ) = ( ( )) = ( ( )) + ( )

− . (57)

FQ

p̌ p̌ p̌ p̌ p̌

φ φ φ φ φ

A A O A O

r r

=

Since the deformation gradient turns out to be the corresponding stress is

F FQ,

b b

⋆ ⋆

= ( ) = ( ) . (58)

T T F T FQ

The group made up of the rotations such that, when followed by the same deformation, leave

Q

the response unchanged, i.e. such that b b

( ) = ( ) ∀ , (59)

T FQ T F F

symmetry group of the material.

is called

As an example, if the rotation in Fig 2 and in Fig. 3 belongs to the symmetry group, any

Q F

gives rise to the same stress. It is worth noting that the application of after leads to a different

F Q

configuration than the one reached by applying just even though the shapes are the same.

F,

4.1 Isotropy V isotropic.

Those materials whose symmetry group is the whole rotation group of are called Since

T

=

for isotropic materials (59) holds for any rotation it should hold in particular for , thus

Q Q R

becoming T

b b

( ) = ( ) , (60)

T RUR T F

becomes

which on turn, by (53), T T

b b

( ) = ( ) . (61)

T RUR R T U R turns out to be isotropic. For, be-

Viceversa, any material whose response function satisfies (61)

cause the polar decomposition of is

FQ T

= = ( )( ) , (62)

FQ RUQ RQ Q UQ

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

8 ELASTIC AFFINE BODIES

F F F

Q Q

b b

( ) = ( )

Figure 3: A material is isotropic if for any rotation

T FQ T F Q.

first, then (61) and again (53), we get

by applying (53) T T T

b b b b

( ) = ( ) ( )( ) = ( ) = ( ) (63)

T FQ RQ T Q UQ RQ R T U R T F

5 Piola-Kirchhoff stress

The expression for the inner power can be written replacing the current volume with the volume

before deformation =

· ·

V V

det . (64)

T L T L F

R R̄

This allows us to define a stress depending on in the following way. In an affine test velocity

S, T,

field the difference between the velocities at and è given by

p p

A O

( )

− . (65)

L p p

A O

Since ( ) = ( )

− − , (66)

L LF

p p p̄ p̄

A O A O

the same difference can be obtained by applying to the difference between the corresponding

LF

R̄.

positions in We can define a new stress as the tensor such that

S

=

· · ∀

V V , (67)

T L S LF L.

R R̄

we get

s By (64) =

· ·

V V

det , (68)

T L F S LF

R̄ R̄

and then T

=

· · ∀

det (69)

T L F SF L, L.

Hence it turns out that T

=

det , (70)

T F SF

from which we get the sought expression T − 1

= ( ) det (71)

S T F F.

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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