7_elastodynamfluids-handout2

7/28/2019 7_elastodynamfluids-handout2

1/6

Continuum mechanicsVII. Typical problems of elasto-dynamics and

fluid dynamics

Ales Janka

office Math [email protected]

http://perso.unifr.ch/ales.janka/mechanics

Apr 13, 2011, Universite de Fribourg

Ales Janka VII. Problems of elasto-dynamics and fluid dynamics

0. Force equilibria for elasto-dynamicsMass conservation:

t+

yj( vj) =

t+ div(v) = 0 in

Force equilibria (statics) momentum conservation (dynamics)

Analogy for

jFj =

ddt

i

mivi

:

fidy +

ij nj dS =d

dt

vidy

Time derivative of an integral in Euler formulation (cf. III. sect.1)

d

dt

vidy

=

(vi)

t+ vj

(vi)

yj+ vi

vj

yj

dy

=

vi

t+vi

t+vi vj

yj+ vi

vj

yj vi( t+div(v))=0

+ vjvi

yj

dy



2/6

0. Force equilibria for elasto-dynamics

Mass conservation:

t+

yj( vj) =

t+ div(v) = 0 in

Force equilibria (statics) momentum conservation (dynamics)

fidy +

yjij dy =

vi

t+ vj

vi

yj

dy =

Dvi

Dtdy

Hence, for || 0 we get

Dv

Dt div = f

F

in


1. Elasto-dynamics in small deformations, linear materialMomentum conservation:

D2u

Dt2j

ij = Fi ie. 2u

t2 div = F in

Constitutive law: Hookes law, ij = Eijkek, e.g.

= 2e + tr(e) Id where = K2

3Kinematic equation: Cauchy strain tensor e:

ek =1

2[ku +uk] ie. e =

1

2

u + (u)T

in

Boundary conditions: for time t [0,):

u = u(t) on D (Dirichlet-type),

n = g(t) on N = \ D (Neumann-type).Initial condition: for all x :

u(x, 0) = u0(x) andu

t(x, 0) = v0



3/6

2. Fluid dynamics (constitutive law for Newtons fluid)

Elastic solid: material reversibly deformed under applied stressHookes law: hydrostatic pressure p = s:

im = 2 eim

p = s = K e = (3 + 2 ) e =

E

1 2 e

Cauchy stress: ij = ij + sij = ij pij

Fluid: continually deforming (in time) under applied shear stressNewtons fluid: modification of Hookes law for fluids:

ij = 2 eij = 2 eij + e ij

p = s = (3 + 2 ) e = 3 Ke

Dynamic viscosity and second viscosity = 23Note: we have replaced eij and e by their time-derivatives


3. Viscosity and visco-elasticity

Distinction solid vs. fluid is not obvious:

Some materials behave both like a solid and like a fluiddepending on the observation period (asphalt, glass, some plastics)

Example: University of Queensland pitch-drop experiment:(set up in 1927, won the Ig-Nobel Price in October 2005)

Dec 1938 1st drop fellFeb 1947 2nd drop fellApr 1954 3rd drop fell

May 1962 4th drop fellAug 1970 5th drop fellApr 1979 6th drop fell

Jul 1988 7th drop fellNov 28, 2000 8th drop fell



4/6

Digression: General visco-elastic materials

Generalization of constitutive laws for viscoelastic materials:

M

k=1

kks

tk=

N

=1

e

t

Mk=1

kkij

tk=

N=1

eij

t

Maxwell material:(t)

E+

(t)

= e(t)

Kelvin-Voigt material: = E e(t) + e(t)


4. Fluids: Newtons fluidForce equilibrium:

Dvi

Dt

yjij = fi in

Mass conservation / continuity equation:

t+ div(v) = 0 in

Kinematic equation:

eij =1

2

vi

yj+

vj

yi

Constitutive law:

ij = 2 eij + e ij pij



5/6

4. Fluids: incompressible Newtons fluidForce equilibrium:

Dvi

Dt

yjij = fi in

Mass conservation / continuity equation: particle density is

const. in time:

D

Dt=0

+div(v) = 0 ie. div(v) = 0 in

Kinematic equation:

eij = 12

viyj

+ vj

yi

Constitutive law:ij = 2 eij pij


2. Fluids: Navier-Stokes equation

Force equilibria: assembling all to Navier-Stokes equation:

yj vi

yj+

vj

yi pij = fi

Dvi

Dt


vj

yj= 0



6/6



2viyjyj

+ 2vjyjyi

+ p

yi= fi Dvi

Dt


vj

yj= 0




vi

t

+ vi

yjvj

2vi

yjyj +p

yi= fi


vj

yj= 0

7_elastodynamfluids-handout2

Documents