A decomposition theorem of motion

Int. J. Engng Sci. Vol. 34, No. 4, pp. 417-423, 1996

Pergamon

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0020-7225/96 $15.00+ 0.00

0020-7225(95)00093-3

A DECOMPOSITION

THEOREM

OF MOTION

LIQIU WANG School of Mechanical and Production Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798 (Communicated by C. G. SPEZIALE) Abstract--A general motion is shown to be composed of an isochoric simple extension, a simple shear, a rotation and a pure dilatation. This helps to visualize a general motion in stages and to study the effect of the motion on the heat flux vector field. The result may also be regarded as a general decomposition theorem for tensors and matrices.

1. I N T R O D U C T I O N

Heat transfer research aims to predict and control heat quantities exchanged. Associated with the heat quantities, there exists a heat flux vector field (a typical example of Cauchy fluxes) whose scalar product with the normal to the surface yields the surface density of the flux [1]. Heat flux vector is proportionally related to the temperature gradient through the classical Fourier law, which is only valid for heat transfer in rigid bodies. For heat transfer processes accompanied by the macroscopic relative motion between two sides exchanging heat, the heat flux vector is related to the temperature gradient through a generalized Fourier law [2] which contains three heat flux coefficients 4~i (i = 0, 1, 2) to be determined through experiments. The heat flux coefficients 4~i (i = 0, 1, 2) are functions of thermophysical properties of material and six invariants of strain rate tensor D and temperature gradient V0. Because of varieties of possible motions, it is very complicated and difficult to carry on such experiments without the guidance of some theories. One way to simplify the experiments is to decompose a complex motion into several simple motions, if possible. A review of the literature reveals that a motion may be decomposed in various ways to serve different purposes. In the kinematics of a rigid body, the motion is decomposed into a translation and a rotation. The kinematical analysis of a fluid medium led to a theorem that decomposed the general motion into a translation, a rotation and a deformation at any point of a fluid body. To investigate hydrodynamic instabilities and coherent structures in turbulence, the velocity fields in fluid flow were decomposed into a spatial and temporal model of orthogonal eigenfunctions [3, 4]. In atmospheric sciences, the atmospheric flow was decomposed by three methods to formally partition the observed atmospheric flow and to verify atmospheric general circulation models [5]. The coherent nature of fluids results in a system of motion equations which exhibits different characters in different flow domains. In particular, it may be parabolic in some flow domains, elliptic in some other domains and hyperbolic in still other domains. The velocity gradient also appears to be steep in some flow domains but mild in some other domains. These characters provide a possibility to decompose flow domain for effectively making theoretical, numerical or experimental observations on fluid flows. This may be regarded as one kind of motion decomposition biased on flow domains. The boundary layer theory and domain decomposition spectral collocation method [6] belong to this kind of motion decomposition. The motivation for the present work comes from the desire to establish a new decomposition theorem of motion aiming to decompose a general motion into several simple motions. An arbitrary homogenous motion is shown to be composed of a pure dilatation and an isochoric motion (Theorem 3). The latter is composed of an isochoric simple extension, a simple shear 417

418

L. WANG

and a rotation (Theorem 4). H e r e a motion is called homogeneous if its deformation gradient is independent of material points. This helps to visualize a general motion in stages. The result is also valid at any one point in a non-homogeneous motion (note that we can only decompose a non-homogeneous motion locally). The new decomposition theorem differs from the previous ones mainly in two aspects. The first of these is that the four motions into which a general motion is to be decomposed are very basic and simple. The second aspect is that it is valid for any motion and may be regarded as a general decomposition theorem for tensors and matrices. The kinematics of basic motions into which a general motion is to be decomposed is well known and can be found in the books of continuum mechanics. We will quote some results about their deformation gradient and the strain rate directly in proving the decomposition theorem. The proof of the theorem is divided into four separate theorems with Theorems 1 and 2 as the basis for proving Theorems 3 and 4. A synthesis of these theorems is presented after the proof of the four separate theorems.

2. D E C O M P O S I T I O N T H E O R E M S OF M O T I O N THEOREM 1. A motion from x to r may be viewed as a motion from x to a configuration r*, followed by a motion from r* to r. If F1 is the deformation gradient of the first motion, and F2 and D2 are the deformation gradient and the strain rate of the second motion with r* as reference, then the deformation gradient F and the strain rate D of x--* r are: F=F2F1

(1)

D = D2 + I)1

(2)

and where 1 DI = 2 (El "[- LT) L1 = F2LIF2-1 with Lt as the velocity gradient of the first motion. PROOF. Since dr = F2 dr* and dr* = F1 dx then dr = F2F1 dx. Also dr = F dx. The definition of tensor equivalence concludes that F = F2Ft.

The velocity gradient L of x---, r can, therefore, be written as L = ~F-1 = L2 + El where L2 is the velocity gradient of the second motion, and L 1 = F2LIF2 |.

Decomposition t h e o r e m of motion

419

Hence, 1 O = ~. (L + L T) = DE

+ Dl

in which 1 DE = ~ (1,2 + I ~ ) 6, =

1

(£, +

Regarding F as a general tensor, this theorem then implies that any decomposition theorem of tensor and matrix such as polar decomposition theorem and singular-value decomposition theorem [7] may be used to decompose the motion. Conversely, every element tensor or matrix in any decomposition theorem of tensor or matrix would be associated with some kind of flow. Similar associated flows have been used in finding the singular-value decomposition of a matrix [7] and in the classical Q R algorithm [8, 9]. THEOREM 2. Deformation gradient F for any isochoric plane strain can be decomposed into a rotation Q and a simple shear S, i.e. F = QS.

(3)

PROOF. Let F be the gradient of an arbitrary isochoric plane strain and let S = 1 + y(t)m ® n

(4)

m . u3 = n . u3 = 0

(5)

Iml = Inl = 1

(6)

m . n = 0.

(7)

for some y, m and n such that

and

Then S is a simple shear in a plane with u3 as the normal [see equation (13) for the definition of U3] •

Furthermore, if we can find such y, m and n which also satisfy srs = C = FrF

(8)

then Q

(9)

= FS -I

will be a rotation because det F det Q - det---S- 1

(10)

and QQr

= FS-~S-rFr

=

F(SrS)-IF r

= FF-,F-rFr

=1

(11)

and equation (9) will imply equation (3). Hence, we need to find y, m, and n satisfying equations (5)-(8). Now

srs = 1+ y(m®n +n®m) + y2n®n

(12)

and for any isochoric plane strain in a plane urn-u2 we have'that C = A2ul ®ul + A-2u2®u2 + ua®ua

(13)

where A is one (associated with ul) of the eigenvalues of right stretch tensor U (U = X/-C), and

420

L. W A N G

Ul, U2, U3 are eigenvectors of U or C. The problem reduces to finding y, m, n for given A, ul, uz such that the right-hand sides of equations (12) and (13) agree. Note that [10] det(C - A21) = 0

(14)

substituting equations (12) (as C) into equation (14) gives

or

• =

(16,

Since the motion is a plane strain in the plane u~-u2, it follows that m, n must be in the plane with u3 as the normal. This, with the requirements of equations (6) and (7), allows us to write m = cos O/u~ - sin 0/112"[

sin

n

!

(17)

O/U1 + COS O/U2J

or

ul = cos O/m + sin O/n "[ [

u:

- s i n O/m + cos O/nJ

(18)

where O/is the angle between U1 and m. Note that Cul=A2ut

(19)

then m'Cul:A2ul-m or

u l . C m = A2ul-m. Substituting equations (12) and (18) into this gives cos O/+ y sin O/= A2 cos O/

i.e. A2 - 1 tg O/=

Y

(20)

Thus we have shown that, given F (and C = F r F ) , we can find a simple shear S [equation (4)] satisfying equations (5)-(8). Finally, equation (9) shows that F can be written as the product of a rotation and a simple shear

F = QS which means that the motion can be decomposed into a rotation and a simple shear by T h e o r e m 1. THEOREM 3. An arbitrary homogeneous motion can be d e c o m p o s e d into a pure dilatation and an isochoric motion. PROOF. Note that for an arbitrary deformation gradient F, J = det F > 0 then we can represent F as F = F2FI = FtF2

(21)

Decomposition theorem of motion

421

where F2 = fl/31 is a pure dilatation, and F~ = J-U3F is an isochoric motion because 1 det FI = det(J-1/3F) = ~ det F = 1 for homogeneous; motion [J = J(t)]. This, with Theorem 1, establishes the truth of Theorem 3. THEOREM 4. An arbitrary isochoric motion can be decomposed into an isochoric simple extension along a3, a simple shear in a plane normal to u3 and a rotation, where u3 is one eigenvector of the right stretch tensor U. PROOF. Let F be the deformation gradient of arbitrary isochoric motion. Since detF= 1 >0 the polar decomposition theorem gives 3

F = RU = R ~ XiUi @ Ui

(22)

i=1

where R is a rotation ( = FU-~), ui are eigenvalues of U (U = V~), and Ai are eigenvalues of U. Note that det F = det R det U = det U = A~A2A3= 1 for isochoric motion, equation (22) then can be written as F = R U = RF2F1

(23)

where FI = ~ / ~ 2

el @ Ul "t- V~- 1~2 u2 @ U2 "t- ,~.383@ U 3

= V'&A2 1 + (A3 - V~IA2)u3 ® u3 F2 =

(24)

~ l V ~ 3 Ul @ U 1 -'l-/~2V~3 U2(~) U2 -t'- U 3 @ U 3.

(25)

Note that det F1 = ~

~

A 3 = AIAEA 3 =

1

i.e. VAIA2 = 3~A~. Hence F~ is an isochoric simple extension along u3. Since det F2 = AIV~3 AEV~3 = AIA2A3 ~- 1 then AlVa3 =

1 A2V~3 "

Therefore, F2 is an isochoric plane strain with AlVa3, JX2V~3 and 1 as the principle stretches (eigenvalues). Applying Theorem 2 to F2 gives F2 = QS

(26)

where Q is a rotation and S is a simple shear in the plane normal to u3. Substituting equation (26) into equation (23) we get F = RQSFI = (~SFI £S $4/4-D

(27)

L. W A N G

422

Table 1. Expressions of Fl, S, Q and F4

S

FI F I = AV~-~I~Ul ®Ul

S=l+Tm~n

+ V~-~A2 u 2 ® n 2

T = I ( A - 1/A)I

-I- ,~3U3~ n 3

m = cos ~ u~ - sin ~ u 2

Q

F4

Q = j- I/3FU-IF*S-I

F4 = j1/31

F* = A l V a 3 e l ~ U l + ~2V~3 U2 ~ U2

n = sin OfU1 -[- COS OfU2

-~- U 3 ~ U 3

A = AlVa3 tg a = (A 2 - 1 ) / y

in which 0 = R Q is a rotation because det 0 = det (RQ) = det R det Q = 1 and ¢~r(~ = Q T R r R Q = Q r Q = 1. By considering F as a general tensor, the theorems derived above may be used as a general decomposition theory for tensors and matrices. Collecting all results from Theorems 1-4 shows that an arbitrary homogeneous motion can be decomposed into an isochoric simple extension, a simple shear, a rotation and a pure dilatation in order, or a pure dilatation, an isochoric simple extension, a simple shear and a rotation in order. Let F be the deformation gradient for an arbitrary homogeneous motion, J be the determinate of F, and Ai, ui (i = 1, 2, 3) be the eigenvalues and eigenvectors of the right stretch tensor U of J-1/3F. Then the decomposition theorem can then be written as F = F4QSF1

(28)

F = QSF~F4

(29)

or

where F1 is an isochoric simple extension along u3; S is a simple shear in the plane u~-u2; Q is a rotation; and F4 is a pure dilatation. The expressions of the F1, S, Q and F4 are summarized in Table 1. This allows us to write the strain rate D for an arbitrary homogeneous motion as D = --)1 + QDsQ -I + Q I ) , Q -t 3J

(30)

D, = -~(t) - ~ (m ® n + n ® m)

(31)

where

1

i'~1 = SL1S -1 = SF'lFi-lS -I.

3.

(32) (33)

CONCLUSION

The decomposition theorem derived in the present work not only shows the possibility to decompose a general homogeneous motion into four simple and basic motions, but also provides a method to construct each component. The theorem is also valid at each point in a non-homogeneous motion. Furthermore, the result may be used as a general decomposition theorem for tensors and matrices as well.

Decomposition theorem of motion

423

Acknowledgement--The author is indebted to the referee for his/her critical review on the original manuscript and constructive, valuable comments and suggestions.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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