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2 cauchy continuum

1 Balance principle

Cauchy continuum is a body model whose placements are such that any deformation is a smooth

map of the Euclidean space without any of the restrictions like those characterizing affine or

E,

rigid bodies. Let us consider a body and any placement

B : (1)

B → E.

p

In order to describe a mechanical interaction between the body and the exterior world we define

(out)

a linear function , called outer power which, for any placement transforms any velocity

W p,

field into a scalar.

The linear space of the test velocity fields is the collection of all the smooth velocity fields on

If in a test velocity field we denote the velocities at and by

R. p p

A B (2)

,

v v

A B

the power has the representation (out) (v ) = + (3)

, .

W · ·

v f v f v

A B A A B B

The vectors and are called external forces applied to body points and respectively.

f f A B

A B

The case we are interested most is where and are such that the set im (the body shape)

B p p

is a subset that is the closure of an open set, whose boundary is piecewise smooth. A

∂R

R ⊂ E

test velocity field is a function : (4)

,

7→

v v

p A A

with domain The outer power has in general the following representation

R. Z Z

(out) (v) = + (5)

dV dA.

W · ·

b v t v

∂R

R

The vector fields and on and respectively, are called bulk force distribution and surface

∂R

R

b t,

force distribution (or tractions). The inner power is assumed in general to have the following

representation Z

(in) (v) = (z + (6)

dV,

W − · · ∇v)

v T

R

where and are descriptors of the stress (T is called Cauchy stress) and is the gradient of

∇v

z T

test velocity field.

We assume, as the balance principle, that in any shape the total power is zero for any test

velocity field : (out) (in)

(v) + (v) = 0 (7)

W W ∀v.

2 Balance equations

By substituting the expression for the divergence of a tensor field (see Appendix 2)

T

= div div (8)

· ∇v − ·

T T v T v

we get

in the expression for the inner power (6)

Z Z Z Z T

(z + = + div div (9)

dV dV dV dV

− · · ∇v) − · · −

v T z v T v T v

R R R R

DISAT, University of L’Aquila, April 26, 2011 (1409) A. Tatone – Mechanics of Solids and Materials.


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DETTAGLI
Corso di laurea: Corso di laurea magistrale in ingegneria informatica e automatica
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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