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120

4 Kinematics

Theorem 4.1. The motion of S is (globally) rigid if and only if

D D 0:

(4.20)

Proof. To prove that (4.20) follows as a necessary condition, we observe that a rigid

motion implies vi .x; t / D vi .x0 ; t / C ij l !j .t /.xl x0l /: The velocity gradient

@vi

D

@xk

ij k !j

is then skew-symmetric, so that

2Dik D

@vi

@vk

C

D

@xk

@xi

ij k

C

kj i

!j D 0:

To prove that D D 0 is a sufficient condition, by (4.15) and (4.16) we obtain

@vi

D Wij ;

@xj

Wij D

Wj i :

(4.21)

The system (4.21) of 9 differential equations with the three unknown functions vi .x/

can be written in the equivalent form

d vi D Wij dxj ;

(4.22)

so that (4.21) has a solution if and only if the differential forms (4.22) can be

integrated. If the region C of the kinetic field is simply connected, then a necessary

and sufficient condition for the integrability of (4.22) is

@Wij

@Wih

D

:

@xh

@xj

By cyclic permutation of the indices, two additional conditions follow:

@Wj i

@Wj h

D

;

@xi

@xh

@Whj

@Whi

D

:

@xj

@xi

Summing up the first two, subtracting the third one and taking into account (4.21)2 ,

we derive the condition

@Wij

D 0;

@xh

4.3 Rigid, Irrotational, and Isochoric Motions

121

which shows that the skew-symmetric tensor Wij does not depend on the spatial

variables and eventually depends on time. Then, integration of (4.22) gives

vi .x; t / D v0i .t / C Wij .t /.xj

x0j /;

t

u

and the motion is rigid.

The motion of S is defined to be irrotational if

1

r

2

!D

v D 0:

(4.23)

Again suppose that the region C is simply connected; then it follows that the

motion is irrotational if and only if

v D r';

(4.24)

where ' is a potential for the velocity, also defined as the kinetic potential.

Let c C be a region which is the mapping of c

C under the equations of

the motion. This region is a material volume because it is always occupied by the

same particles. If during the motion of S the volume of any arbitrary material region

does not change, then the motion is called isochoric or isovolumic, i.e.,

Z

d

dc D 0:

dt c

By changing the variables .xi /

written as

d

dt

! .XL /, the previous requirement can be

Z

Jdc D 0;

c

and, since the volume c is fixed, differentiation and integration can be exchanged,

and because of (4.19), we get

Z

Z

J r v dc D

c

r v dc D 0

8c

C:

c

Next, we conclude that a motion is isochoric if and only if

r v D 0:

(4.25)

Finally, an irrotational motion is isochoric if and only if the velocity potential

satisfies Laplace’s equation

' D r r' D 0;

whose solutions are known as harmonic functions.

(4.26)

122

4 Kinematics

4.4 Transformation Rules for a Change of Frame

As we will discuss in Chap. 7, constitutive equations are required to be invariant

under change of frame of reference. Intuitively, this requirement means that two

observers in relative motion with respect to each other must observe the same stress

in a given continuous body. For this and other reasons it is of interest to investigate

P rv, D, and W transform under a change of frame of reference.

how the tensors F,

Let us briefly recall that a frame of reference can be considered as an observer

measuring distances with a ruler and time intervals with a clock. In general, two

observers moving relatively to each other will record different values of position

and time of the same event. But within the framework of Newtonian mechanics, it

is postulated that: distance and time intervals between events have the same values

in two frames of reference whose relative motion is rigid.

Suppose that the first observer is characterized by the reference system Oxi and

time t and the second one by O 0 xi0 and t 0 . The above requirement can be expressed

in analytical terms as

xi0 D Qij .t /xj C ci .t /;

t 0 D t C a;

(4.27)

where Q D .Qij /, c D .ci /, and a are a proper orthogonal tensor, an arbitrary

vector, and an arbitrary scalar quantity, respectively.

The change of frame of reference expressed by (4.27) is a time-dependent rigid

transformation known as a Euclidean transformation (see Chap. 1).

A scalar field g, a vector field q, and a tensor field T are defined to be objective

if, under a change of frame of reference, they transform according to the rules

g 0 .x0 ; t 0 / D g.x; t /;

q0 .x0 ; t 0 / D Q.t /q.x; t /;

T0 .x0 ; t 0 / D Q.t /T.x; t /QT .t /:

(4.28)

In this case we say that two observers are considering the same event from two

different points of view.

It was already observed in Sect. 3.6 that the deformation gradient F and the left

and right Cauchy–Green tensors transform as follows:

F0 D QF;

C0 D C;

B0 D QBQT :

(4.29)

Moreover, the following additional transformation rules hold under a change of

frame (4.27):

P

FP 0 D QFP C QF;

4.5 Singular Moving Surfaces

123

P T;

r 0 v0 D QrvQT C QQ

D0 D QDQT ;

P

:

W0 D QWQT C QQ

T

(4.30)

Equation (4.30)1 is obtained by differentiating (4.29)1 with respect to time.

Furthermore, differentiating the relation xi0 D Qij .t /xj C ci .t / with respect to

time and xh0 , we get

@vj @xk

@v0i

P @xj

0 D Qij

0 C Qij

@xh

@xk @xh

@xh0

and (4.30)2 is proved. Moreover, the orthogonality condition QQT D I allows us to

ÁT

P T C QQ

P T D 0, i.e., QQ

P T D

P T , so that the skew-symmetry of

write QQ

QQ

P T is derived. By recalling the definitions (4.12) and (4.13) of D and W, (4.30)3;4

QQ

can easily be proved.

It follows that only the rate of deformation tensor D can be considered to be

objective.

4.5 Singular Moving Surfaces

In continuum mechanics it is quite common to deal with a surface that is singular

with respect to some scalar, vector, or tensor field and is moving independently of

the particles of the system. Typical phenomena include acceleration waves, shock

waves, phase transitions, and many others. Due to the relevance of these topics, this

section and the next one are devoted to the kinematics of singular surfaces.

Let

f .r; t / D 0

(4.31)

be a moving surface †.t /. Given a point r 2 †.t /, consider the straight line a of

equation r C s n, where n is the unit vector normal to †.t / at r. If f .r; t C / D 0 is

the equation of †.t C /, let the intersection point of a with the surface †.t C / be

denoted by y. The distance between †.t / and †.t C / measured along the normal

at r is given by s. /, and this parameter allows us to define the normal speed of

†.t / as the limit

cn D lim

!0

s. /

D s 0 .0/:

(4.32)

124

4 Kinematics

The limit (4.32) can also be expressed in terms of (4.31) if it is observed that s. / is

implicitly defined by equation

f .r C s n; t C / Á '.s; / D 0;

so that, from Dini’s theorem on implicit functions, it follows that

Â

s 0 .0/ D

@'=@

@'=@s

Ã

D

.0;0/

@f =@t

:

rf n

Since rf is parallel to n and has the same orientation, it can be written as

cn D

@f =@t

:

jr f j

(4.33)

The velocity of the surface with respect to the material particles instantaneously

lying upon it is called the local speed of propagation and is given by .cn v n/, if

the current configuration is regarded as a reference configuration.

Note that we obtain (4.33) by using the implicit representation of the surface

†.t /. If we adopt a parametric form of †.t /, i.e., r D g.u˛ ; t /, where u˛ , ˛ D 1; 2;

are the parameters on the surface, then f .g.u˛ ; t /; t / D 0 and (4.33) gives

cn D

1

rf

jr f j

@r

@g

Dn

:

@t

@t

(4.34)

It can be proved that the velocity of the surface is independent of the parametric

representation of †.t /.

In fact, if U˛ with u˛ D u˛ .Uˇ ; t / are new parameters, then the parametric

equations of the surface will be R.U˛ ; t / D r.u˛ .Uˇ ; t // and (4.34) will be

expressed in terms of @R=@t instead of @r=@t . Since

@R

@r @u˛

@r

D

C ;

@t

@u˛ @t

@t

and @r=@u˛ are tangent to the surface †.t /, it follows that

n

@R

@r

Dn

;

@t

@t

and both parametric representations lead to the same value of cn .

Let r D g.u˛ ; t / be a surface †.t / and let .t / be a moving curve on it,

with parametric equations r D G.s; t / D g.u˛ .s; t /; t /, where s is the curvilinear

abscissa on .t /. The orientation of the tangent unit vector to the curve .t / is

fixed according to the usual rule that is moving counterclockwise on .t / for an

4.5 Singular Moving Surfaces

125

observer oriented along n. Furthermore, the unit vector normal to .t / on the plane

tangent to †.t / is expressed by † D

n.

According to these definitions, the velocity of the curve .t / is given by the

scalar quantity

w D

@G

@t

†

Áw

†;

(4.35)

and it can be proved that w † is independent of the parametric representation of

†.t / and .t / (see Exercise 7).

A moving surface †.t / is defined to be singular of order k 0 with respect

to the field .x; t / if the same definition applies to the fixed surface S of <4 of

Eq. (4.31) (see Sect. 2.5).

It is convenient to write the relationships found in Sect. 2.5 in terms of variables

.xi ; t /: To this end, let <4 be a four-dimensional space in which the coordinates

.xi ; t / are introduced, and let the fixed surface S of <4 be represented by (4.31) (see

Fig. 4.1).

The unit vector N normal to S has components

ND

Â

Ã

1

@f

gradf

D

rf;

;

jgradf j

jgradf j

@t

(4.36)

where gradf D .@f =@xi ; @f =@t /. Taking into account (4.33) and observing that

the unit normal n in <3 to the surface f .r; t / D 0, with t fixed, has components

.@f =@xi /= jrf j, we find that (4.36) can be written as

ND

jrf j

.n; cn / Á ˇ.n; cn /:

jgradf j

Fig. 4.1 Singular surface in the space-time

(4.37)

126

4 Kinematics

With this notation, if †.t / is a surface of order 1 with respect to the tensor field

T.x; t /, then (2.48) and (4.37) give the following jump conditions:

ŒŒrT D n ˝ A.x; t /;

ÄÄ

@T

D cn A.x; t /;

@t

(4.38)

(4.39)

where A.x; t / D a jr f j = jgradf j.

If the surface is of order 2, then (2.50) leads to the following conditions:

ŒŒrrT D n ˝ n ˝ A.x; t /;

ÄÄ

@T

D cn n ˝ A.x; t /;

r

@t

ÄÄ 2

@T

D cn2 A.x; t /:

@t 2

(4.40)

(4.41)

(4.42)

4.6 Time Derivative of a Moving Volume

Let V .t / be a moving volume with a smooth boundary surface @V .t / having

equation g.x; t / D 0 and unit outward normal N. Suppose that its velocity

component along the normal is given by

1 @g

:

jrgj @t

VN D

When dealing with balance equations, it is useful to express the time derivative of a

given quantity, defined over a volume which is not fixed but is changing with time.

To do this, we need to use the following relation:

d

dt

Z

Z

f.x; t / dc D

V .t/

V .t/

@f

dc C

@t

Z

fVN d :

(4.43)

@V .t/

To prove (4.43), we observe that

d

dt

Z

f .x; t / dc

V .t/

1

D lim

t !0 t

ÂZ

Ã

Z

f.x; t C t / dc

V .tCt/

f.x; t / dc :

V .t/

4.6 Time Derivative of a Moving Volume

127

The right-hand term can also be written in the form

Z

Z

V .tCt/ f.x; t

C t / dc

f.x; t C t / dc

V .t/

Z

C

Œf.x; t C t /

f.x; t / dc;

V .t/

and we see that the difference of the first two integrals gives the integral of f.x; t C

t / over the change of V .t / due to the moving boundary @V .t /. By neglecting terms

of higher order, we have

Z

Z

f.x; t C t / dc

f.x; t C t / dc

V .tCt/

V .t/

Z

D

f.x; t C t /VN t d

@V .t/

so that in the limit t ! 0, (4.43) is proved.

In the same context, let †.t / be a singular surface of zero order with respect

to the field f.x; t / and .t / D †.t / \ V .t /. If f .x; t / D 0, n, and cn denote the

equation of †.t /, its unit normal, and its advancing velocity, respectively, we have

cn D

1 @f

jrf j @t

and the following formula can be proved:

d

dt

Z

Z

f.x; t / dc D

V .t/

V .t/

@f

dc C

@t

Z

Z

fVN d

@V .t/

ŒŒfcn d :

(4.44)

.t/

To this end, suppose the surface †.t / divides the volume V .t / into two regions,

V and V C . In order to apply (4.43), we note that @V has velocity VN , while .t /

is moving with velocity cn if .t / is considered to belong to V .t /, or with velocity

cn if it is part of V C . Furthermore, the field f on .t / will be f or fC , depending

on whether .t / belongs to V or V C . We derive (4.44) by applying (4.43) to V C

and V and subtracting the corresponding results.

As a particular case, if V .t / is a fixed volume (VN D 0), then from (4.44) it

follows that

Z

Z

Z

@f

d

dc

f.x; t / dc D

ŒŒfcn d :

(4.45)

dt V

V @t

.t/

As a further application of (4.44), consider a material volume c.t / C.t / that is

the mapping of the initial volume c

C by the equations of motion x D x.X; t /.

128

4 Kinematics

Since this moving volume is a collection of the same particles of S.t /, its boundary

@c.t / is moving at normal speed v N, and (4.44) can be written as

d

dt

Z

Z

@f

dc C

@t

f.x; t / dc D

c.t/

c.t/

Z

Z

f ˝ v Nd

ŒŒfcn d :

@c.t/

(4.46)

.t/

By applying the generalized Gauss’s theorem (2.42) to the second integral of the

right-hand side of (4.46), we derive

Ä

Z

Z

d

dt

f.x; t / dc D

c.t/

c.t/

Z

@f

C r .v ˝ f/ dc

@t

ŒŒf.cn

vn / d :

(4.47)

.t/

In applications the need often arises to consider a material open moving surface,

i.e., a surface which is the mapping, by the equations of motion x D x.X; t /, of a

surface S represented in the reference configuration by the equation r D r.u˛ /: In

this case the following formula holds:

d

dt

Ä

Z

Z

u.x; t / N d D

S.t/

S.t/

@u

Cr

@t

.u

v/ C vr u

Nd ;

(4.48)

where N is the unit vector normal to S.t /.

To prove (4.48), we first observe that due to (3.19) we can write

d

dt

Z

u.x; t / N d D

S.t/

d

dt

d

D

dt

Z

ui d

Z

i

S.t/

S

Z

LD

@XL

ui J

d

@xi

S

d

dt

Â

Ã

@XL

ui J

d

@xi

L:

(4.49)

In addition, from (4.19) it follows that

d

dt

Â

J

@XL

@xi

Ã

D Jr v

@XL

d @XL

CJ

I

@xi

dt @xi

(4.50)

and to evaluate the derivative on the right-hand side it is convenient to recall that

@XL @xh

D ıih :

@xi @XL

Therefore,

d @XL

D

dt @xi

@xP h @XL

;

@xi @xh

4.6 Time Derivative of a Moving Volume

129

and (4.50) becomes

d

dt

Â

Ã

@XL

@XL

J

D Jr v

@xi

@xi

J

@xP h @XL

:

@xi @xh

By using this expression in (4.50) we find that

d

dt

Z

Z

.uP C ur v

u.x; t / N d D

S.t/

u rv/ N d

S.t/

and (4.48) is proved by applying known vector identities.

Finally, it is of interest to investigate how (4.48) can be generalized to the case

in which the material surface S.t / intersects the moving surface †.t /, which is

supposed to be singular with respect to the field u. Let .t / D S.t / \ †.t / be the

intersection curve between S.t / and †.t /, and let S .t / and S C .t / be the regions

into which S.t / is subdivided by .t /. If denotes the unit vector tangent to .t /

and s the unit vector tangent to S.t / such that s ; ; N define a positive basis (see

Fig. 4.2), then we can easily prove the following generalization of (4.48):

d

dt

Ä

Z

Z

u.x; t / N d D

S.t/

S.t/

Z

@u

Cr

@t

n

ŒŒu

.w

.u

v/ C vr u

v/

.t/

S

ds;

Nd

(4.51)

where w is the velocity of .t /.

Remark 4.1. When dealing with continuous systems characterized by interfaces

(examples include shock waves, Weiss domains in crystals, phase transitions, and

many other phenomena), it is useful to evaluate the following derivative:

d

dt

Z

ˆS d ;

S.t/

Fig. 4.2 Singular curve on a surface

130

4 Kinematics

where ˆ S .x; t / is a field transported by the nonmaterial surface †.t / and S.t / D

c.t / \ †.t /, where c.t / is a material or fixed volume. The formula (4.51) does not

allow us to consider this case, and as a result many authors have paid attention to

this problem (see for example Moeckel [37], Gurtin [16], Dell’Isola and Romano

[9]). Recently, Marasco and Romano [34] have presented a simple and general

formulation that will be further discussed in Volume II.

4.7 Exercises

1. Given the motion

Ã

Â

t 2

x1 D 1 C

X1 ;

T

x 2 D X2 ;

x 3 D X3 ;

find the Lagrangian and Eulerian representation of the velocity and acceleration.

The displacement components in the Lagrangian form are given by

u1 D x1

Ã

Â

t

t2

X1 D 2 C 2 X1 ;

T

T

u2 D u3 D 0;

and in Eulerian form they are

u1 D

2.t =T / C .t =T /2

x1 ;

Ã

Â

t 2

1C

T

u2 D u3 D 0:

The velocity components in the Lagrangian representation are

v1 D

2

@u1 .X; t /

D

@t

T

Â

Ã

t

1C

X1 ;

T

v2 D v3 D 0;

and in Eulerian form they are

v1 D

1

2

Ã x1 ;

Â

t

T

1C

T

v2 D v3 D 0:

Finally, the acceleration components in Lagrangian form are

a1 D

2

X1 ;

T2

a2 D a3 D 0;

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