Nonlinear Continua Part 6 ppt

4.2.1 Eulerian spatial formulation of the mass-conservation principle Using in Eq.4.17 the expression of Reynolds’ transport theorem given in Eq.4.4 we obtain: G Gw Z w Y wwdY = Z w Y C

Fig 4.2 Discontinuity surface

• ww onwV12

and using the generalized Reynolds’ transport theorem we obtain,

Gw w

Gw

Z

w Y 

w# wdY =

Z

w Y 

C w# Cw

wdY +

Z

w V 

wn · wvw# wdV +

Z

w V 12

wn12 · www# wdV = (4.12a)

Using the generalized Gauss’ theorem (Eq.(4.5)),

Z

w Y  u · (wvw#)wdY =

Z

w V 

wn · wvw#wdV

+ Z

w V 12

wn12 · wv w# wdV (4.12b) hence,

Gw w

Gw

Z

w Y 

w# wdY =

Z

w Y 



Cw#

Cw + u · (wvw#)

¸

wdY (4.12c)

 Z

w V 12

w# wn12 · (wv  ww)wdV = For the region on the positive side ofwn12, in the same way, we get

Gw w

Gw

Z

w Y +

w# wdY =

Z

w Y +



Cw#

Cw + u · (wvw#)

¸

wdY (4.12d) +

Z

w V

w#+ wn12 · (wv+  ww)wdV =

The velocity field of the fictitious particles is coincident with the velocity field of the actual particles everywhere except onwV12 Therefore:

G

Gw

Z

w Y

w# wdY = Gww

Gw

Z

w Y 

w#wdY + Gww

Gw

Z

w Y +

w# wdY = (4.12e) From Eqs.(4.12c) to (4.12e) we get,

G

Gw

Z

w Y

w# wdY =

Z

w Y



Cw#

Cw + u · (wvw#)

¸

wdY +

Z

w V 12

[[w# (wyq  wzq)]]wdV (4.13) where,

[[w# (wyq  wzq)]] = w#+ (wy+

q  wzq)  w# (wy

q  wzq)>

wyq = wn12 · wv>

wzq = wn12 · ww =

In order to obtain a localized version of Reynolds’ transport theorem at the discontinuity surface we consider the arbitrary material volume enclosed

by the dashed line in Fig 4.3

Fig 4.3 Derivation of the jump discontinuity conditionw = w +

 w3

For the enclosed material volume, using Eq.(4.13) and the generalized Gauss’ theorem, we write:

G

Gw

Z

w Y

w#wdY =

Z

w Y

Cw# Cw

wdY +

Z

w V 

wn · wvw# wdV (4.14) +

Z

w V +

wn · wv w# wdV +

Z

w V

[[w# (wyq  wzq)]]wdV>

when wdV+$wV12and wdV $wV12 ; w+$ 0 and w$ 0 we get from Eq.(4.14):

Z

w V 12

¡ [[w# wyq]] + [[w# (wyq  wzq)]]¢ w

Therefore, in order for the above integral equation to be valid for any arbitrary part of the discontinuity surface, we must fulfill

[[w# wyq]] + [[w# (wyq  wzq)]] = 0 (4.16a)

at every point onwV12

Equation (4.16a) is known as the jump discontinuity condition

If we callwU = wzq  wyq the discontinuity’s propagation speed, we can write

[[w# wU]] = [[w# wyq]]= (4.16b) The above equation is known as Kotchine’s theorem (Truesdell & Toupin 1960)

4.2 Mass-conservation principle

In Sect 2.2, Eq.(2.6) introduced the concept of mass of a continuum body B

In the study of continuum media, under the assumptions of Newtonian mechanics, it is postulated that the mass of a continuum is conserved Hence,

G Gw

Z

w Y

wherew = w (w{l> w)

4.2.1 Eulerian (spatial) formulation of the mass-conservation principle

Using in Eq.(4.17) the expression of Reynolds’ transport theorem given in Eq.(4.4) we obtain:

G

Gw

Z

w Y

wwdY =

Z

w Y



Cw

Cw + u · (wwv)

¸

wdY = 0 = (4.18) Since the above equation has to be fulfilled for any control volume inside the continuum, we can write for any point inside the spatial configuration:

Cw

The above partial dierential equation is the localized spatial form of the mass-conservation principle in a Eulerian formulation and it is called the continuity equation

Example 4.3 JJJJJ Using components in a general curvilinear spatial coordinate system, the con-tinuity equation is written as

Cw

Cw +

wyd C

w

Cw{d + wwyd|d = 0=

JJJJJ

For an incompressible material GGww = 0; hence the continuity equation is:

u · wv = 0>

or in components,

wyd|d = 0=

JJJJJ

Let a fluid of densityw = w (w{l> w) have a velocity fieldwv

Let us consider in the spatial configuration a volume

surface(w) that moves with an arbitrary velocity fieldww

Following (Thorpe 1962), we first calculate the fluid mass instantaneously inside the volume

P =

Z

wwdY and using the expression of the generalized Reynolds’ transport theorem in Eq.(4.11), we get

dP

Gw w

Gw

Z

wwdY =

Cw

wdY +

Z

(w)

wn · ww wwdV

wherewnis the external normal of the surface(w)

Using Eq.(4=5) (generalized Gauss’ theorem), we get

Z

u · ¡w

wv¢ w

dY =

Z

(w)

wn · (wwv)wdV and subtracting the above equation from the previous one,

dP

Cw

Cw + u · (wwv)

¸

wdY

+ Z

(w)

wwn · (wwwv)wdV =

Using Eq.(4.19), we see that the first integral on the r.h.s is zero; hence,

dP

Z

(w)

wwn · (wwwv)wdV = The above equation is an integral equation of continuity for a control volume

in motion in the fluid velocity field (Thorpe 1962) JJJJJ

4.2.2 Lagrangian (material) formulation of the mass conservation principle

Equation (4.17) implies that,

Z

 Y

 ({D)dY =

Z

w Y

w (w{d> w)wdY (4.20a) where (> Y ) correspond to the reference configuration and (w> wY ) to the spatial configuration Using Eq.(2.31) in the r.h.s of Eq.(4.20a) and changing variables in the expression of w we obtain,

Z

 Y

 ({D)dY =

Z

 Y

w ({D> w)wM dY > (4.20b)

 Y

Since the above equation has to be fulfilled for any control volume that we define inside the continuum, we can write for any point inside the reference configuration:

and therefore,

G

Gw (

The above equation is the localized material form of the continuity equa-tion

4.3 Balance of momentum principle (Equilibrium)

The principle of balance of momentum is the expression of Newton’s Second Law for continuum bodies Quoting (Malvern 1969):

“The momentum principle for a collection of particles states that the time rate of change of the total momentum of a given set of particles equals the vector sum of all the external forces acting on the particles of the set, provided Newton’s Third Law of action and reaction governs the internal forces The continuum form of this principle is a basic postulate of continuum mechanics”

4.3.1 Eulerian (spatial) formulation of the balance of momentum principle

For a body B in the w-configuration we define its momentum as,

Z

w Y

the resultant of the external forces acting on the elements of mass inside the body are, from Eq.(3.2): Z

w Y

and the resultant of the external forces acting on the elements of the body’s surface are, from Eq.(3.4): Z

w V

Using Eqs.(4.21a-4.21c), we can state Newton’s Second Law for the body

B as,

G

Gw

Z

w Y

wwv wdY =

Z

w Y

wwbwdY +

Z

w V

wtwdV = (4.22)

Using the condition of equivalence between external forces and Cauchy stresses inside a continuum, defined in Eq.(3.7), we get:

G

Gw

Z

w Y

wwvwdY =

Z

w Y

wwbwdY +

Z

w V

wn · w wdV = (4.23) Using in the above the expression of Reynolds’ transport theorem given in Eq.(4.4), we get

Z

w Y



C(wwv)

Cw + u · (wwv wv)

¸

wdY =

Z

w Y

wwbwdY +

Z

w V

wn · wwdV = (4.24) From Example 4.2, we obtain

u · (wwv wv) = wv · £

u (wwv)¤

+ wwv(u · wv) (4.25a) also, from Eq.(2.20b), we get

G (wwv)

C (wwv)

wv · £

u (wwv)¤

> (4.25b) and, from Eq.(4.5) (Generalized Gauss’ Theorem), we get

w V

wn · w wdV =

Z

w Y u · (w)wdY = (4.25c) Using Eqs.(4.25a-4.25c) in Eq.(4.24) we arrive at the integral form of the Eulerian formulation of the balance of momentum principle:

Z

w Y



G

Gw (

wwv) + wwv(u · wv)

¸

wdY =

Z

w Y

£w

wb + u · w¤ w

dY = (4.26) Since the above equation has to be fulfilled for any control volume that

we define inside the continuum, we can write for any point inside the spatial configuration:

G

Gw (

wwv) + wwv(u · wv) = wwb + u · w (4.27a) and using in the above the continuity equation, we have

w G

wv

The above equation is the localized form of the balance of momentum principle in an Eulerian formulation and it is known as the equilibrium equa-tion

Using Eq.(A.62b), in the general Eulerian curvilinear system {w{d}, we get,

u · w = wde|d wge hence, using Eq.(A.55b),

u · w =



Cwde

Cw{d + wve wvdd + wdv wvde

¸

wg

e = From the result in Example A.10, we can easily get

wlop = 1

2

wjpm



C wjlm

C w{o + C wjmo

C w{l  C

wjol

C w{m

¸

= Therefore,

u · w =



Cwde

Cw{d + 1

2

¡w

ve wjdm + wdv wjem¢

µC wj

vm

C w{d + C wjmd

Cw{v  C

wjdv

C w{m

¶¸

wge =

JJJJJ

Example 4.7 JJJJJ

A perfect fluid is defined as a continuum in which, at every point, and for any surface,

wn · w = w wn>

where w is a scalar (no shear stresses)

Since w is a symmetric second order tensor (to be shown in Sect 4.4), its eigenvalues are real and its eigenvectors are orthogonal (Appendix, A.4.1) Referring the problem to the Cartesian system defined by the normalized eigenvectors,wˆewe can write,

wn = wqˆwˆe

w =

3

X

=1

wˆ wˆe wˆe =

Then, for the perfect fluid,

wqˆ wˆ = w wqˆ ( = 1> 2> 3)(qr dgglwlrq rq ) =

The above set of equations is fulfilled only if the three eigenvalues wˆ are equal (hydrostatic stress tensor ) Hence,

w = wswˆe wˆe =

It is easy to show that as the three eigenvalues of w are equal, the above equation is valid in any Cartesian system; hence we can write,

wlm = ws lm >

where ws is the pressure Generalizing the above to any arbitrary coordinate system

w = wswg= Using Eq (A.62b) and the result in Example A.11, the equilibrium equation,

Eq (4.27b), can be written as,

w G

wv

wwb + Cs

C w{l

wjlm wgm = From Eq (A.57) we identify the last term on the r.h.s of the above equation

asuws, hence we can write the equilibrium equation for a perfect fluid as,

w G

wv

wwb + uws = The above equation is known as the Euler equation for perfect fluids Many authors get a minus sign for the second term on the r.h.s because they define

wlm =  ws lm =

JJJJJ

Example 4.8 JJJJJ Following with the topic discussed in Example 4.5 we consider a fluid, moving with a velocity fieldwv, and a moving control volume, moving with a velocity fieldww In this example, following (Thorpe 1962), we are going to analyze the momentum balance inside the moving control volume Using the generalized Reynolds’ transport theorem (Eq.(4.11)) for the fluid momentum,

Gw w

Gw

Z

wwvwdY =

Z C(ww

v) Cw

wdY

+ Z

(w)

wwv(wn · ww)wdV =

From the generalized Gauss’ theorem (Eq.(4.5)),

Z

u · (wwv wv)wdY =

Z

(w)

wn · (wwv wv)wdV =

Subtracting the above equation from the previous one,

Gw w

Gw

Z

wwv wdY =

C(wwv)

Cw + u · (wwvwv)

Ì

wdY

+ Z

(w)

wwv êw

n · (wwwv)ô w

dV =

Using the result in Example 4.2 and Eq.(4.19) (continuity equation),

Gw w

Gw

Z

wwvwdY =

Z

w

ẾCw v

wv · uwv

ả

wdY

+ Z

(w)

wwv êw

n · (wwwv)ô w

dV =

On the r.h.s of the above equation, the term between the brackets in the first integral is the fluid particles material acceleration We can state, using Newton’s second law, that the external force instantaneously acting on the particles inside

wF =

Z

wwawdY =

Hence,

wF = Gw

Gw

Z

wwvwdY +

Z

(w)

ww v êw

n · âw

vwwđô w

dV = JJJJJ

Example 4.9 JJJJJ Let us consider the body B and the particle S on its external surface We define atS a convected coordinate system l with covariant base vectorswegl

in the spatial configuration and egl in the material configuration The con-vected system is defined so as to haveweg1 andweg2 in the plane tangent towV

atwS ; and therefore eg1and eg2 define the plane tangent toV atS

Material and spatial normal vectors (Nanson’s formula)

The external unit normals atS are

wn =

weg1 × weg2

|weg1 × weg2| >

and,

n =

eg1 × eg2

|eg1 × eg2| = Also, the surface-area dierentials are

wdV wn = (weg1 × weg2) d1d2 (D) >

dV n = (eg1 × eg2) d1 d2 (E) =

If we define,

wt1 = d1 weg1 >

wt2 = d2 weg2 >

t1 = d1 eg1>

t2 = d2 eg >

it is obvious from the results in Sect 2.9.1 that

t1 = wT`1

t2 = wT`2 =

We can now define an arbitrary curvilinear system {w{l} in the spatial config-uration and another one {{L} in the material configuration, with covariant base vectorswgl and gL respectively

wt1 = (ww1)n wgn

wt2 = (ww2)n wgn

wT`1 = (ww1)n (w[1)Nn gN

wT`2 = (ww2)n (w[1)Nn gN

wn = wql wgl

n = qL gL

where,

w

[lL = Cw{l

C{L =

We write Eqs.(D) and (E) using the above as,

wdV wql = wlmn (ww1)m (ww2)n

dV qL = LMN (ww1)m (ww2)n (w[1)Mm (w[1)Nn =

Multiplying both sides of the above equation by (w[1)Ll, we get

dV qL (w[1)Ll = LMN (w[1)Ll (w[1)Mm (w[1)Nn (ww1)m (ww2)n but, from Eqs.(A.37e) and (2.34g)

LMN = hLMN

p

|jDE| = Hence, using Eq.(A.37c), we get

dV qL (w[1)Ll = p

| jDE| |w[1| hlmn (ww1)m (ww2)n and again using Eq.(A.37e), we get

dV qL (w[1)Ll = p

|jDE| |w[1|

wlmn

p

|wjde| (

ww1)m (ww2)n =

Therefore,

dV qL (w[1)Ll = wdV wql |w[1|

s

|jDE|

|wjde| and using Eq.(2.34i), we get

wnwdV = wM n · wX1 dV =

The above equation is called Nanson’s formula (Bathe 1996) JJJJJ

For an Eulerian vectorwawe define, using Eq.(2.76a),

wA` = h¡w

[1¢E e

wdei

gE =

Using the generalized Gauss’ theorem (Eq.(4.5)),

Z

w Y u · wa wdY =

Z

w V

wn · wawdV =

In the r.h.s integral we introduce Nanson’s formula derived in Example 4.9; hence,

Z

w Y u · wa wdY =

Z

 V

wM n · wX1 · wadV

= Z

 V

wM n · h¡w

[1¢E e

wdei

g

E

dV

= Z

 V

n · ³

wM wA`´

dV =

Using again the generalized Gauss’ theorem,

Z

w Y u · wawdY =

Z

 Y

wM wA`´

dY = With the notationGLY (·) we indicate a divergence in the reference configu-ration Using in the r.h.s integral Eq.(2.31), we get

Z

 Y

wM ¡

u · wa¢ 

dY =

Z

 Y

wM wA`´

dY = The localized form of the above equation is known as the Piola Identity (Mars-den & Hughes 1983),

wM ¡

u · wa¢

wM wA`´

=

JJJJJ

Example 4.11 JJJJJ

We can write the Piola Identity, derived in the above example, as:

wM wde|e = £w

M (w

[1)Ef wdf¤

|E = After some algebra, we get

wM wde|e = £w

M (w

[1)Ef¤

|E wdf + wM wdf|f =

M (w[1)Ef¤

|E = 0 that is to say

GLY ¡w

M wX1¢

= 0 =

JJJJJ

4.3.2 Lagrangian (material) formulation of the balance of

momentum principle

In the previous section we derived, in the spatial configuration, the integral and localized forms of the balance of momentum principle (equilibrium equa-tions)

In this section we are going to derive, in the material configuration, the integral and localized forms of the equilibrium equations

We are going to refer Eq.(4.23) to volumes and surfaces defined in the material configuration, using Eq.(4.20d) and Nanson’s formula (Example 4.9) G

Gw

Z

 Y

wvdY =

Z

 Y

wbdY +

Z

 V



w

n · wX1 · w dV =

(4.28a)

In the above equation, all magnitudes are written as functions of ({L> w) Using the generalized Gauss’ theorem together with the definition of the first Piola-Kirchho stress tensor, we obtain

Z

 Y

G

Gw (

wv)dY =

Z

 Y

wbdY +

Z

 Y

GLY (wP)dY = (4.28b)

In the above (Malvern 1969),

GLY (wP) = wSDd|D wgd

=



C w

SDd

C{D + wSGd D

GD + wSDg w

[l

D wd lg

¸

wg

d= Equation (4.28b) is an integral form of the equilibrium equations It is im-portant to note that although the integrals are calculated on volumes defined

in the reference configuration, the equilibrium is established in the spatial configuration

The corresponding localized form is,

 G

wv

From Eqs.(4.27b) and (4.29) we get,



w u · w = GLY (w

P)= The above equation is a particular application of the Piola Identity.JJJJJ

In order to write the equilibrium equations in terms of fully material ten-sors we have to pull-back Eq.(4.29)

For the material velocity field:

wV` = êw

!(wyd)ôD 

g

A “physical interpretation” of the pull-back of the material velocity con-travariant components was presented in Example 2.12

In the same way, for the material acceleration,

wa = Gwv

wdd wg

and

h

wA`iD

= êw

! (wdd)ôD

thew˜dDare the components of the material acceleration vector in the convected system {{D} (convected acceleration (Simo & Marsden 1984))

For the external loads per unit mass, we define

wB` = êw

! (wed)ôD 

Since GLY (w

P) is an Eulerian vector,

êw

! âw

SLd|LđôD

= (w[1)DdwSLd|L = (4.34) Therefore, the pull-back of Eq.(4.29) is

wA` = wB` + wX1 · GLY (wP)= (4.35)

In a Lagrangian formulation, the above equation is the localized form of the equilibrium equations