continuum mechanics || continuum mechanics in curvilinear coordinates

Chapter 13Continuum Mechanicsin Curvilinear Coordinates

13.1 Introduction

In this chapter the basic equations of continuum mechanics are presented in generalcurvilinear coordinates. In Sect. 13.3 on deformation analysis the use of materialcoordinates are applied. Convective time derivatives of tensors were briefly intro-duced in Sect. 11.4, but is now properly defined and presented in Sect. 13.4. Gen-eral equations of motion are given in Sect. 13.5. The basic equations in Elasticityare the subject of Sect. 13.6, and the equations of Fluid Mechanics are presented inSect. 13.7.

13.2 Kinematics

In order to describe the motion and deformation of a continuum a reference con-figuration Ko of the material body is introduced. This configuration will usually bea real configuration of the body, representing the body at a reference time to, butnot necessarily. In Ko a general coordinate system Y is introduced. The coordinatesYK are used to identify particles of the body, and we use the expression particle Y .To identify places in space we introduce another general coordinate system y. Inthe present configuration K of the body, at the present time t, the particle Y is atthe place y, where yi are the coordinates of the particle in the coordinate system y.In special cases it is convenient to consider the Y -system as a material coordinatesystem that moves and deforms with the body. YK are then material coordinates.

A third coordinate system is the Cartesian coordinate system x with origin O.The particle Y has the Cartesian coordinates Xi, while the place y has the Cartesiancoordinates xi. All three coordinate systems Y , at the reference time to, y, and x arefixed in the same reference Rf, such that one-to one correspondence between thecoordinate sets Y and X , and between y and x exist:

Xi = Xi(Y ) ⇔ YK = YK(X), xi = xi(y) ⇔ yi = yi(x) (13.2.1)

599

600 13 Continuum Mechanics in Curvilinear Coordinates

Note that lower case Latin (or Greek) letters are used as indices in the x- and they-systems, while upper case Latin (or Greek) letters are used as indices in theY -system. We may identify one and the same particle by either Y or X , and oneand the same place by either y or x.

The positions of a particle Y in the configurations Ko and K are given by theplace vectors ro and r respectively, as shown in Fig. 13.2.1. The motion of the bodyat the present time t may be given by the place vector r(Y,t), the coordinate functionsyi(Y, t), or by the displacement vector:

u(Y,t) = r(Y,t)− ro(Y ) (13.2.2)

The velocity v and the acceleration a of a particle are:

v = r =∂r(Y,t)∂ t

, vi =∂yi

∂ t, vK = vi gK

i (13.2.3)

a = v =∂v(Y,t)∂ t

= r (13.2.4)

The symbols gKi are Euclidean shifters as defined by the formulas (12.7.4).

In the reference description we use the Lagrangian coordinates (YK ,t) as inde-pendent variables. The material derivative of a tensor N(Y,t) of order n is a newtensor of the same order and defined by:

N =∂N(Y,t)∂ t

⇔ NK ..L ≡ N[gK , ..,gL]= ∂

∂ t

(NK ..L

)(13.2.5)

In the spacial description we use the Eulerian coordinates (yi, t) as independentvariables. The material derivative of a tensor N(y, t) of order n is a new tensor ofthe same order and defined by:

N =∂∂ t

N(y(Y,t),t) = ∂tN(y,t)+ grad N(y, t) · v(y,t) ⇔

x1

x3

x2

r

u

ro

Ko, to

Y1

y1

v = vKgKY3 Y2

y3

y2

K, t

v= vigi

ORf

Fig. 13.2.1 General coordinate systems Y and y, and Cartesian coordinate system x

13.3 Deformation Analysis 601

Ni..j ≡ N

[gi, ..,g j]= ∂tNi..

j + Ni..j∣∣k vk (13.2.6)

Note that Ni..j are components of the tensor N and not the material derivatives of

the tensor components Ni..j. The latter are given by:

∂∂ t

[Ni..

j(y(Y,t),t)]=∂∂ t

[Ni..

j(y,t)]+

∂∂yk

[Ni..

j(y, t]

vk (13.2.7)

and do not represent a tensor in general coordinates Y , only when the coordinatesystem Y is Cartesian.

Material differentiation follows the standard differentiation rules. For example,the material derivative of the vector a = Bc has the components:

ai = Bki ck + Bk

i ck (13.2.8)

This result is easy to prove by use of (13.2.6).Because the unit tensor 1 and the permutation tensor P are constant tensors it

follows that:gi j = gi j = 0, εi jk = ε i jk = 0 (13.2.9)

The definition (13.2.6) may be used to find the following expression for the par-ticle acceleration a(y,t):

a = v = ∂tv + gradv · v = ∂tv +(v · ∇) v ⇔ ai = ∂t vi + vi

∣∣k vk (13.2.10)

If N is a steady tensor field: N = N(y), the material derivative may be expressedby the absolute derivative with respect to time:

N =dNdt⇔ Ni..

j =δNi..

j

δ t(13.2.11)

13.3 Deformation Analysis

13.3.1 Strain Measures

The section is based on the presentation in Chap. 5 Deformation Analysis and issubdivided into five parts: 1. Small strains and small deformations, 2. Deformationkinematics, 3. General deformation analysis, 4. Deformation analysis in convectivecoordinates, and 5. Deformation of material surfaces.

It has been shown in Sect. 5.2 that the deformation in the neighborhood of aparticle Y , or X , is determined by the deformation gradient F defined by:

dr = F · dro, F = Grad r =∂r∂ro

(13.3.1)


dro and dr represent a material line element in the reference configuration Ko and inthe present configuration K. In the x-system the relations (13.3.1) are represented by:

dxi = Fik dXk, Fik =∂xi

∂Xk(13.3.2)

In a general coordinate system y the motion is given by y(Y, t) and the relations(13.3.1) have the component representations:

dyi = FiK dY K , Fi

K =∂yi

∂YK(13.3.3)

FiK are two-point-components of the deformation gradient F.

The lengths dso and ds of the material line element in Ko and K are found from:

(dso)2 = dro · dro = gKL dY K dY L, (ds)2 = dr · dr = gi j dyi dy j (13.3.4)

Using (13.3.1) and (13.3.3) we may write:

(ds)2 = dr · dr = (F · dro) · (F · dro) ⇒(ds)2 = dro · C · dro = CKL dY K dY L (13.3.5)

The Green deformation tensor C is defined by:

C = FT F ⇔ CKL = FiK FiL (13.3.6)

If we imagine that the Y -system is imbedded in the continuum and moves and de-forms with the body, it follows from (13.3.5) that the components CKL also representthe fundamental quantitites of this convective Y -system. This interpretation of CKL

will be utilized below.The Green strain tensor E is defined by:

E =12

(C−1) ⇔ EKL =12

(CKL−gKL) (13.3.7)

The strain measures at a particle Y : the longitudinal strain ε in the directione, the volumetric strain εv, and the shear strain γ with respect to two orthogonaldirections e and e, were defined in Sect. 5.1. The expressions for these in generalcoordinates are:

ε =ds−dso

dso=√

e · C · e−1 =√

eK CKL eL−1

=√

1 + 2e · E · e−1 =√

1 + 2eK EKL eL−1 (13.3.8)

εv =dV −dVo

dVo= detF−1 =

√detC−1 =

√det(1 + 2E)−1

=√

det(CL

K

)−1 =√

det(δL

K + 2ELK

)−1 (13.3.9)


sinγ =e · C · e√

(e · C · e) (e · C · e) =eK CKL eL√

(eM CMN eN)(ePCPQ eQ)

=2e · E · e√

(1 + 2e · E · e) (1 + 2e · E · e) =2 eK EKL eL√(1 + ε)(1 + ε)

(13.3.10)

ε is the longitudinal strain in the direction e.When the motion is expressed by the displacement vector: u(ro, t) = r(ro, t)−ro,

the deformation will be defined in terms of the displacement gradient H:

H =∂u∂ro

= F−1, HKL ≡ uK |L = FKL−gKL (13.3.11)

It now follows that:

C = 1 + H + HT + HT H ⇔ CKL = gKL + uK |L + uL|K + uN∣∣K uN |L (13.3.12)

E =12

(H + HT + HT H

) ⇔ EKL =12

(uK |L + uL|K + uN

∣∣K uN |L

)(13.3.13)

13.3.2 Small Strains and Small Deformations

The special but very important case of small strains has been discussed in Sect. 5.3.Small strains may be defined by the inequality: norm E << 1. The expressions(13.3.8–13.3.10) for the primary strain measures are reduced to:

ε = e · E · e = eK EKL eL, γ = 2e · C · e = 2 eK EKL eL, εv = trE = EKK (13.3.14)

Small deformations imply small strains and small rotations. The condition ofsmall deformations is defined by:

norm H =√

tr(HHT

)=√

uK |L uL|K � 1 (13.3.15)

Note that the components uK |L are dimensionless. The expressions (13.3.12, 13.3.13)for the Green deformation tensor and the Green strain tensor may be approximatedby:

C = 1 + H + HT ⇔ CKL = gKL + uK |L + uL|K (13.3.16)

E =12

(H + HT) ⇔ EKL =

12

(uK |L + uL|K) (13.3.17)

If we further can assume that the displacements are small, such that the Ko and K areconfigurations close to one another, the place coordinates yi may be used as particle


coordinates. The expressions (13.3.17) are now written as:

E =12

(H + HT) ⇔ Ei j =

12

(ui| j + u j

∣∣i

)(13.3.18)

The three characteristic strain measures at a particle Y : the longitudinal strain ε inthe direction e, the shear strain γ with respect to two orthogonal directions e and e,and the volumetric strain εv are now expressed by:

ε = e · E · e = ei ui| j e j, γ = 2e · E · e = ei(

ui| j + u j∣∣i

)e j, εv = trE = ui

∣∣i

(13.3.19)

The rotation tensor for small deformations R is defined by:

R =12

(H−HT) ⇔ R i j =

12

(ui| j− u j

∣∣i

)(13.3.20)

The principal directions of the strain tensor E rotates a small angle determined bythe rotation vector:

z =12∇×u ⇔ zi =

12ε i jk uk| j =

12ε i jk uk, j =

12ε i jk Rk j (13.3.21)

The small strain tensor field E(y,t) has to satisfy the compatibility equations,which in Cartesian coordinates are presented in Sect. 5.3.9 as (5.3.40). Since eachterm in these equations are components of tensors, the equations may be directlytransformed to general coordinates y:

Ei j∣∣kl + Ekl |i j− Eil| jk− E jk

∣∣il = 0 (13.3.22)

Written out in detail these equations become very long and quite complex. Sincethey will not be used directly in the following, we shall refrain from presentingthem. Malvern [28] presents a complete listing of the compatibility equations inorthogonal coordinates and in cylindrical coordinates.

The equations of compatibility for small deformations were presented and dis-cussed in the Sect. 5.3.9 and 5.3.10. An interesting but mathematically rathercomplex derivation of the compatibility equations, relevant also for large deforma-tions, is based on the following arguments. First we introduce a convected coor-dinate system Y , formally to be presented in Sect. 13.3.5. This coordinate systemis imbedded in the continuum, which implies that the system moves and deformswith the material. The compatibility equations result from the requirement that theconvected Y -system shall at all times represent a Euclidean space. This implies thata one-to-one correspondence exists between any coordinate set x in a fixed Carte-sian system and a set Y in the convected material coordinate system at all times.However, the compatibility equations for large deformations seem to be of littleimportance in applications and will not be presented in the present exposition.


13.3.3 Rates of Deformation, Strain, and Rotation

The velocity gradient tensor L, the rate of deformation tensor D, and the rate ofrotation tensor W at the time t are defined by:

L =∂v∂r≡ gradv ⇔ Lik = vi|k (13.3.23)

D =12

(L+ LT ) ⇔ Dik =

12

(vi|k + vk|i) (13.3.24)

W =12

(L−LT ) ⇔ Wik =

12

(vi|k− vk|i) =12

(vi,k−vk,i) (13.3.25)

The last equality follows from the fact that the rate of rotation tensor isantisymmetric.

The longitudinal strain rate ε in the direction e, the shear strain rate γ withrespect to two orthogonal directions e and e, and the volumetric strain rate εv havebeen defined in Sect. 5.4. The expressions in general coordinates are:

ε = e · D · e = ei Dik ek = ei vi|k ek (13.3.26)

γ = 2e · D · e = 2ei Dik ek = ei (vi|k + vk|i) ek (13.3.27)

εv = trD = div v = Dii = vi

∣∣i (13.3.28)

The principal directions of D rotate with the angular velocity:

w =12∇×v ⇔ wi =

12ε i jk vk, j =

12ε i jk Wk j (13.3.29)

13.3.4 Orthogonal Coordinates

This section will present coordinate strains for small deformations, coordinate strainrates, and rates of rotation in orthogonal coordinates. First we present the physicalcomponents of the displacement vector u and the velocity vector v:

u = u(i) eyi ⇔ u(i) = u · ey

i , v = v(i) eyi ⇔ v(i) = v · ey

i (13.3.30)

where eyi are the unit tangent vectors to coordinate lines in the orthogonal system y:

eyi =

1hi

gi = hi gi (13.3.31)


The physical components of the small strain tensor E and the rate of deformationtensor D are defined by:

εii = E(ii) = eyi · E · ey

i =1

hi2 Eii =

1

hi2 ui|i no summation

γi j = 2E(i j) = 2eyi · E · ey

j =2

hi h jEi j =

1hi h j

(ui| j + u j

∣∣i

)i �= j (13.3.32)

εii = D(ii) = eyi · D · ey

i =1

hi2 Dii =

1

hi2 vi|i no summation

γi j = 2D(i j) = 2eyi · D · ey

j =2

hi h jDi j =

1hi h j

(vi| j + v j

∣∣i

)i �= j (13.3.33)

Using the formulas (12.5.67, 12.5.68) we obtain from the formulas (13.3.32):

εii = E(ii) =1

h2i

Eii =

[∂∂yi

(u(i)hi

)+∑

j

hi, j

hi h ju( j)

]no summation w. r. to i

γi j = 2E(i j) =2

hi h jEi j =

hi

h j

∂∂y j

(u(i)hi

)+

h j

hi

∂∂yi

(u( j)h j

)i �= j

εv = trE = E(ii) =1h ∑i

∂∂yi

(hu(i)

hi

)(13.3.34)

Similarly, using the formulas (12.5.67, 12.5.68) we obtain from the formulas(13.3.33):

εii = D(ii) =1

h2i

Dii =

[∂∂yi

(v(i)hi

)+∑

j

hi, j

hi h jv( j)

]no summation w. r. to i

γi j = 2D(i j) =2

hi h jDi j =

hi

h j

∂∂y j

(v(i)hi

)+

h j

hi

∂∂yi

(v( j)h j

)i �= j

εv = trD = D(ii) =1h ∑i

∂∂yi

(hv(i)

hi

)(13.3.35)

The physical components of the vorticity vector c = rot v are according to formula(12.5.63):

c(i) =hi

h ∑j,kei jk [hkv(k)] , j (13.3.36)

In Cartesian coordinate systems the vorticity vector c and the rate of rotation tensorW are according to (5.4.12), (5.4.15), and (4.3.10) related through:

ci = ei jk Wk j ⇔ Wi j =−12

ei jk ck in Cartesian coordinate systems (13.3.37)


In orthogonal coordinate systems we define the physical components of W throughthe similar relation:

c(i) = ei jkW (k j) ⇔ W (i j) =−12

ei jk c(k) (13.3.38)

From (13.3.36) and (13.3.38) we obtain:

W (i j) =1

2hi h j[(hi v(i)) , j−(h j v( j)) ,i] (13.3.39)

The results of applying (13.3.32–13.3.39) in cylindrical coordinates and sphericalcoordinates are presented in the formulas (5.3.13–5.3.16) and (5.4.18 –5.4.23).

13.3.5 General Analysis of Large Deformations

The displacement and deformation of a differential material line element from thereference configuration Ko to the present configuration K may be decomposed intoa deformation of pure strain and a rigid-body motion as discussed in Sect. 5.5 andillustrated in Fig. 5.5.3. The decomposition, which does not necessarily representthe actual displacement and deformation of the material, may be considered in twoalternative ways. First, let the material be subjected to pure strain through the rightstretch tensor U, transforming the line element dro emanating from the particle Y tothe line element dr:

dr = U · dro ⇔ dY L = ULK dY K (13.3.40)

Then the line element dr is rotated to give the element dr:

dr = R · d r ⇔ dY N = RNL dY L (13.3.41)

Finally, the line element dr is given the displacement u to its final position at place y:

dr = d r ⇔ dyi = giN dY N ⇒ (13.3.42)

dr = d r = RU · dro ⇔ dyi = giN RN

L ULKdY K (13.3.43)

The alternative displacement and deformation of the material element starts with adisplacement u, followed by a rotation R, and then a stretch given by the left stretchtensor V:

dr = VR · dro ⇔ dyi = V ij R j

k gkK dY K = V i

j R jK dY K (13.3.44)

When the present configuration K is used as reference configuration it is conve-nient to introduce the inverse deformation gradient:

F−1 ≡ ∂ro

∂r⇔ F−1K

i =∂YK

∂yi(13.3.45)


such that:dro = F−1 · dr (13.3.46)

The inverse deformation tensor, also called the Cauchy deformation tensor, is de-fined by:

B−1 = F−T F−1 ⇔ B−1i j = F−1K

i F−1K j (13.3.47)

It follows that:

(dso)2 = dro · dro = dr · B−1 · dr = dyi B−1

i j dy j (13.3.48)

The Euler strain tensor, also called the Almansi strain tensor is defined by the ex-pression:

(ds)2− (dso)2 = 2 dr · E · dr (13.3.49)

under the condition that the tensor is symmetric. From (13.3.4)2 and (13.3.48,13.3.49) we obtain:

E =12

(1−B−1) , Ei j =

12

(δi j−B−1

i j)

(13.3.50)

The following formulas are to be derived in Problem 13.1:

Ei j =12

(gi · u, j +g j · u,i−u,i · u, j) =12

(ui| j + u j

∣∣i− uk|i uk

∣∣∣j

)(13.3.51)

E = FT E F ⇔ EKL = Ei j FiK F j

L (13.3.52)

13.3.6 Convected Coordinates

The coordinate system Y introduced in the reference configuration Ko is now as-sumed to be imbedded in the continuum, which implies that the coordinate systemmoves and deforms with the material. The system is called a convected coordinatesystem. The base vectors and the fundamental parameters for the Y -system are placeand time functions. The base vectors are denoted:

cK(Y,t) =∂r(Y,t)∂YK

⇒ cK(Y,to) = gK(Y ) (13.3.53)

The reciprocal base vectors cK(Y,t) are defined by:

cK · cL = δKL (13.3.54)

The fundamental parameters for the Y -system are:

CKL = CKL(y,t) = cK · cL, CKL = CKL(Y, t) = cK · cL (13.3.55)


The length dso and ds of a material line element in the two configurations Ko and Kare respectively given by:

(dso)2 = gKL dY K dY L, (ds)2 = CKL dY K dY L (13.3.56)

The components CKL(Y,t) now represent two tensors in the Y -system: In (13.3.5,13.3.6) CKL(Y, t) are components in Ko of the deformation tensor C. In (13.3.56)2

CKL(Y, t) are fundamental parameters in the Y -system and components of the unittensor 1 in K. In particular:

CKL(Y,to) = gKL(Y ) (13.3.57)

The components CKL(Y,t) also represent two tensors in the Y -system: contravari-ant components in Ko of the deformation tensor C and fundamental parameters inthe Y -system and components of the unit tensor 1 in K. In particular:

CKL(Y,to) = gKL(Y ) (13.3.58)

It follows from the expressions (13.3.7) that in convected coordinates the com-ponents EKL of the strain tensor represent half of the change in the fundamentalparameters from gKL in Ko to CKL in K for the convected Y -system.

If the y-system in K is chosen such that the coordinate system coincides with theconvective Y -system at time t then:

FiK =

∂yi

∂YK= δ i

K (13.3.59)

and (13.3.59) shows that the matrix in of the Green strain tensor and the matrix inthe Euler strain tensor become identical.

If the motion from Ko to K is given by the displacement vector u(Y, t) in(13.2.2), then:

cK =∂r∂YK

=∂ro

∂YK+∂u∂YK

= gK + u,K (13.3.60)

CKL = cK · cL = gK · gL + gK · u,L + u,K · gL + u,K · u,L ⇒CKL = gKL + uK |L + uL|K + uN

∣∣K uN |L (13.3.61)

Note that the displacement components uK and uK are related to the Y -system in Ko:

uK = u · gK , uK = u · gK (13.3.62)

and that covariant differentiation is to be performed in Ko with Christoffel symbolsbased on the fundamental quantities gKL.

Alternatively we may use the displacement components uK and uK related to theY -system in K:

uK = u · cK , uK = u · cK (13.3.63)


For simplicity we use the same symbols for the displacement components here.Using the result (13.3.60), we obtain:

gKL = gK · gL = (cK−u,K) · (cL−u,L) = cK · cL− cK · u,L−u,K · cL + u,K · u,L

⇒ CKL = gKL + uK‖L + uL‖K− uN∥∥

K uN‖L (13.3.64)

Covariant differentiation is now marked by a double vertical line to indicate that it isto be performed in K and with Christoffel symbols computed from the fundamentalquantities CKL. The strain components EKL in the formulas (13.3.13) will have todifferent forms:

EKL =12

(uK |L + uL|K + uN

∣∣K uN |L

)(13.3.65)

EKL =12

(uK‖L + uL‖K− uN

∥∥K uN‖L

)(13.3.66)

Note the two sets of the strain components EKL in (13.3.65) and (13.3.66) are identi-cal, while the displacement components uK and the covariant differentiations are notthe same. Comparing the formulas (13.3.66) for the components of the Green straintensor with formulas (13.3.51) for the components of the Euler strain tensor, we seethat the two components set are identical if we choose a y-system that coincideswith the Y -system at the present time t.

Example 13.1. Simple ShearThe deformation simple shear may be described by:

x1(Y1,Y2,t) = Y1 +β (t)Y2, x2(Y2) = Y2 , x3(Y3) = Y3, β (to) = 0 (13.3.67)

xi are Cartesian coordinates and YK are convected coordinates chosen such that YK =xiδ i

K at the reference time to. Figure 13.3.1 shows the deformation of a block of thematerial. The Green strain tensor and the coordinate strains will be determined.

It follows from (13.3.67) that:

(∂xi

∂YK

)=

⎛⎝1 β 0

0 1 00 0 1

⎞⎠ ,

(∂YK

∂xi

)=

⎛⎝1 −β 0

0 1 00 0 1

⎞⎠

Fig. 13.3.1 A material blockin Ko at time to and in Kat time t. Convected coordi-nates Y

Y2x2

x1

Y1

Y2x2

x1

Y1

K KY Y

b)a)

13.4 Convected Derivatives of Tensors 611

(CKL) =(∂xi

∂YK

∂xi

∂YL

)=

⎛⎝1 β 0β 1 +β 2 00 0 1

⎞⎠ ,

(B−1

KL)

=(∂YK

∂xi

∂YL

∂xi

)=

⎛⎝1 +β 2 −β 0−β 1 00 0 1

⎞⎠

(EKL) =(

12

(CKL−gKL))

=12

⎛⎝0 β 0β β 2 00 0 0

⎞⎠

The non-zero coordinate strains are obtained from (5.2.17) and (5.2.20):

ε22 =√

1 + 2e2 · E · e2−1 =√

1 + 2g2√g22· E · g2√

g22−1

=

√1 + 2

E22

g22−1 =

√1 +β 2−1

sinγ12 =2e1 · E · e2√

1 + 2e1 · E · e1√

1 + 2e2 · E · e2

=2

g1√g11· E · g2√

g22√1 + 2

g1√g11· E · g1√

g11

√1 + 2

g2√g22· E · g2√

g22

=2

E12√g11 g22√

1 + 2E11

g11

√1 + 2

E22

g22

=2β2√

1 + 212β 2

⇒ sinγ12 =β√

1 +β 2

For small deformations: β � 1⇒ β 2� β , the strain matrix and the non-zero coor-dinate strain are:

(EKL)≈ (Ei j) =12

⎛⎝0 β 0β 0 00 0 0

⎞⎠ , γ12 = β

13.4 Convected Derivatives of Tensors

A tensor quantity that is independent of the choice of reference, is called a referenceinvariant tensor or objective tensor. Tensor quantities dependent of the choice ofreference, are called reference related tensors. These concepts were introduced anddiscussed in Sect. 11.2. It was pointed out that the material derivative of an objectivetensor not necessarily is objective. One form of time derivative of an objective tensor


leading to a new objective tensor is the corotational derivative of the objective tensor,defined in Sect. 11.3. Other objective time derivatives are the lower-convected andupper-convected derivatives of objective tensors, briefly presented in Sect. 11.4. Weare now in the position to give a proper definition of these convected derivatives inthe setting of convected coordinates.

We apply convected coordinates Y with base vectors cK(y, t). The materialderivatives of these base vectors are:

cK =∂∂ t

∂r∂YK

=∂∂YK

∂r∂ t

=∂v∂YK≡ v,K (13.4.1)

v is the particle velocity. The components of the material derivatives of the basevectors are covariant derivatives of the velocity components:

cK = vL∥∥

K cL (13.4.2)

It is straight forward to show that, see Problem 13.2.:

cK =− vK∥∥

L cL (13.4.3)

Let a(Y, t) be a convected vector field, i.e. a vector field associated with the par-ticles in the material we are considering:

a = aK cK = aK cK (13.4.4)

The material derivative of a is:

a = aK cK =[∂aK

∂ t+ aL vK

∥∥L

]cK , a = aK cK =

[∂aK

∂ t−aL vL

∥∥K

]cK (13.4.5)

a is a reference related vector field, while the two vectors defined by the compo-nents:

∂caK ≡ ∂aK

∂ t, ∂caK ≡ ∂aK

∂ t(13.4.6)

are objective. The expressions (13.4.6) are called convected differentiated vectorcomponents. The vectors defined in (13.4.5) are two different vector fields. In orderto determine the components of these vectors in a reference fixed coordinate systemy, we rearrange (13.4.5):

∂caK = aK−aL vK∥∥

L , ∂caK = aK + aL vL∥∥

K (13.4.7)

Because these equations are tensor equations, they may directly be transformed tothe fixed y-system:

∂cai = ai−ak vi∣∣k =

∂ai

∂ t+ ai∣∣k vk−ak vi

∣∣k (13.4.8)

∂cai = ai + ak vk∣∣∣i=∂ai

∂ t+ ai|k vk + ak vk

∣∣∣i

(13.4.9)

13.4 Convected Derivatives of Tensors 613

Note that the convective derivatives of the contravariant and the covariant vectorcomponents do not result in one and the same vector. The vector defined by (13.4.8)is called the upper-convected derivative of the vector a and the vector defined by(13.4.9) is called the lower-convected derivative of a. The two derivatives are alsopresented as:

a∇ = a−La ⇔ a∇i = ai− vi∣∣k ak upper-convected derivate of a (13.4.10)

aΔ = a+ LT a ⇔ aΔi = ai + vk∣∣∣iak lower-convected derivate of a (13.4.11)

Confer the definitions (11.4.1–11.4.2) in Cartesian coordinates.Convected differentiated tensor components are defined by their representation

in a convected Y-system in which they are given directly by the material derivativesof the tensor components. As an example we consider a 2. order tensor B. Theconvected derivatives of the components BK

L in the Y -system are defined by:

∂cBKL ≡ ∂

∂ tBK

L (13.4.12)

If B is an objective tensor then so is the tenor defined by the components (13.4.12).One set of components of this tensor in the y-system are denoted ∂cBi

j and arefound as follows. Let a and b be two objective vectors. The scalar α = B [a, b] mayalternatively be computed from:

α = BKL aK bL, α = Bi

j ai b j (13.4.13)

Then we may write:

α = ∂cBKL aKbL + BK

L ∂caK bL + BKL aK ∂cbL (13.4.14)

α = Bij ai b

j + Bij ai b j + Bi

j ai b j (13.4.15)

The equations (13.4.14) are transformed to the y-system:

α = ∂c Bij ai b

j + Bij ∂c ai b j + Bi

j ai ∂cb j (13.4.16)

The formulas (13.4.8, 13.4.9) are used for ∂cai and ∂cb j in (13.4.16), and then(13.4.15) is subtracted from (13.4.16). The result is:

[∂cBi

j − Bij + Bk

j vi∣∣k−Bi

k vk∣∣∣

j

]ai b

j = 0

Because the vectors a and b may be chosen arbitrarily, the expression in the bracketsmust be zero. Thus we have the result:

∂cBij = Bi

j−Bkj vi∣∣k + Bi

k vk∣∣∣

j(13.4.17)


When the material derivative and the covariant derivatives in (13.4.17) are writtenout in detail, we shall see that the terms with Christoffel symbols are eliminated,and that the result is, see Problem 13.3:

∂cBij =

∂∂ t

Bij + Bi

j,k vk−Bkj vi

,k + Bik vk

, j (13.4.18)

Oldroyd [37] derived the formula (13.4.18) by evaluating the material derivativeof the transformation equation:

BKL =

∂YK

∂yi

∂y j

∂YLBi

j (13.4.19)

Followed by a substitution of the result into the transformation equation:

∂c Bij =

∂yi

∂YK

∂YL

∂y j∂c BK

L (13.4.20)

See Problem 13.4.Now we define two tensors by their components in the fixed y-system and the

corresponding components in the convected Y -system:

∂c Bi j in they-system ⇔ ∂∂ t

BKL in the Y -system (13.4.21)

∂c Bi j in they-system ⇔ ∂∂ t

BKL in the Y -system (13.4.22)

Then the following formulas may be derived, see Problem 13.5:

∂cBi j = Bi j + Bk j vk∣∣∣i+ Bik vk

∣∣∣j

(13.4.23)

∂cBi j = Bi j−Bk j vi∣∣k−Bik v j

∣∣k (13.4.24)

The results (13.4.8, 13.4.9) and (13.4.17, 13.4.23, 13.4.24) show a pattern forconstructing the convected derivatives of objective tensors of any order. Note thatthe convective derivatives of the tensor components of different types do not re-sult in one and the same tensor. For a 2. order tensor B it is customary to let thecomponents (13.4.23) defined the lower-convected derivative of the tensor B, whilethe components (13.4.24) defined the upper-convected derivative of the tensor B.Special symbols are introduced for these two tensors:

BΔ = B+ LT B+ BL ⇔BΔi j = Bi j + vk

∣∣i Bk j + Bik vk

∣∣j

lower-convected derivative of B (13.4.25)

B∇ = B−LB−BLT ⇔B∇i j = Bi j− vi

∣∣k Bk j−Bik v j

∣∣k

upper-convected derivative of B (13.4.26)

Confer the definitions (11.4.3, 11.4.4) in Cartesian coordinates.

13.5 Stress Tensors. Equations of Motion 615

13.5 Stress Tensors. Equations of Motion

13.5.1 Physical Stress Components

The stress vector t on a surface through a particle with unit normal n is determinedby the Cauchy stress theorem:

t = T · n ⇔ ti = T ik nk (13.5.1)

T is the Cauchy stress tensor. The normal stress σ and the shear stress τ on thesurface are given by:

σ = n · t = n · T · n = ni T ik nk, τ =

√t · t−σ2 =

√ti ti−σ2 (13.5.2)

The stress vector ti, normal stress σi, and shear stress τi on a coordinate surfaceyi = constant are:

ti = T · gi√gii

=T ki√

giigk (13.5.3)

σi =gi√gii· T · gi√

gii=

T ii

gii , τi =

√T ki T i

k

gii − (σi)2 (13.5.4)

The shear stress τi has two components on the surface yi = constant:

(τi)k =gk√gkk· ti =

gk√gkk· T · gi√

gii=

T ik√

gkk giik �= i (13.5.5)

The physical components of the stress vector ti are according to the general defini-tion (12.3.5):

τki = T ki√

gkk

gii (13.5.6)

Figure 13.5.1 shows the physical components τ12 and τ22 of the stress vector t2.Green and Zerna [19] call the components (13.5.6) physical stress components. Asseen from Fig. 13.5.1, τ12 (and τ32) are shear stresses, while τ22 represents in generalboth a shear stress and a normal stress.

Fig. 13.5.1 Stresses on asurface and on a coordinatesurface y2 = constant

y2 y1

g2

t2

t

n

τ τ22τ12σ


Truesdell [50] defines physical stress components differently from Green andZerna. First the stress tensor t and the unit normal surface vector n are expressed inphysical components:

t(k) = tk√gkk , n(i) = ni√gii (13.5.7)

Then the Cauchy stress theorem (13.5.1) is expressed as:

t(k) = T (ki)n(i) , T (ki) = T ki

√gkk

gii(13.5.8)

Since the physical components n(i) are dimensionless and the physical componentst(k) have dimension force per unit area, the dimension of the components T (ki) isalso force per unit area. T (ki) are called physical stress components because theyare the proper coefficients in the linear relations (13.5.8) between the physical com-ponents of the unit normal vector and the physical components of the stress vector.Note that the components T (ki) are not in general symmetric. It may be shown,Problem 13.6, that the two sets of physical components defined by (13.5.6) and(13.5.8) are related through:

τki =∑j

T (k j)gi j√

g j j

gii (13.5.9)

In orthogonal coordinate systems:

gi j = gi j = 0 for i �= j , gii =1gii = h2

i

Hence:

τki = T (ki) = hkhiTki =

hk

hiT k

i (13.5.10)

T (ki) for k = i is the normal stress σi on the coordinate surface yi = constant, whileT (ki) for k �= i are orthogonal components of the shear stress τi on the same sur-face. In the following the physical stress components are only used in orthogonalcoordinates, and for practical reasons we shall use the notation T (ki). Note that:

T (ki) = eyk · T · ey

i (13.5.11)

eyi are unit tangent vectors to the coordinate lines.

Example 13.2. Physical Stress Components in Cylindrical CoordinatesThe physical components of the stress tensor in cylindrical coordinates are ex-

pressed with alternative symbols as shown below. Expressed in terms of the mixedtensor components and the contravariant tensor components we find:

(T (i j))≡

⎛⎜⎝σR τRθ τRz

τθR σθ τθz

τzR τzθ σz

⎞⎟⎠≡

⎛⎜⎝

TRR TRθ TRz

TθR Tθθ Tθz

TzR Tzθ Tzz

⎞⎟⎠as shown in Fig. 13.5.2

13.5 Stress Tensors. Equations of Motion 617

Fig. 13.5.2 Physical stresscomponents in cylindricalcoordinates

θR

TRR

TZR

TθR

TZθ

TθZ

TRθ

Tθθ

TRZ

TZZ

=

⎛⎜⎝

T 11 T 1

2 /R T 13

R T 21 T 2

2 RT 23

T 31 T 3

2 /R T 33

⎞⎟⎠=

⎛⎜⎝

T 11 RT 12 T 13

R T 21 R2 T 22 RT 23

T 31 RT 31 T 33

⎞⎟⎠ (13.5.12)

13.5.2 Cauchy Equations of Motion

The Cauchy equations of motion (3.2.35) have the following representation in ageneral curvilinear coordinates:

T ik∣∣∣k+ρ bi = ρ ai ⇔ T ik,k +T lkΓi

lk + Til Γklk +ρ bi = ρ ai (13.5.13)

By using the result (12.5.6), we obtain the alternative form:

1√g∂∂yk

(√gT ik)

+ T lkΓilk +ρ bi = ρ ai (13.5.14)

In orthogonal coordinates we use the formulas (12.5.59), (13.5.10), and (12.5.64)and can rewrite (13.5.14) to:

∑k

1h∂∂yk

(hhk

T (ik))

+hi,k

hi hkT (ik)− hk,i

hi hkT (kk)+ρ b(i) = ρ a(i) (13.5.15)

The Cauchy equations in cylindrical coordinates and in spherical coordinates arepresented in (3.2.39, 3.2.40, 3.2.41) and (3.2.42, 3.2.43, 3.2.44).


13.6 Basic Equations in Elasticity

The classical theory of elasticity has been presented in Chap. 7. The most importantequations of the theory will now be transformed to general coordinates.

In the Cauchy equations of motion represented now by (13.5.13, 13.5.14, 13.5.15)the particle acceleration will be expressed through the displacement vector u(y,t):

a = u ⇔ ai = ui (13.6.1)

The constitutive equations for isotropic, linearly elastic material are represented byHooke’s law, which in tensorial form, i.e. invariant form, and in Cartesian form, isrepresented by (7.2.8). The invariant form and the form in general coordinates are:

T =η

1 +ν

[E+

ν1−2ν

(trE)1]⇔ T i

j =η

1 +ν

[Ei

j +ν

1−2νEk

kδij

](13.6.2)

The strain tensor may be expressed by the displacement gradients:

Ei j =12

(ui| j + u j

∣∣i

)(13.6.3)

The Navier equations, i.e. the equations of motion expressed through the displace-ment u(y,t), are in invariant form and Cartesian form given by (7.6.28). The invariantform and the form in general coordinates are:

∇2u+1

1−2ν∇(∇ · u)+

ρμ

(b−a) = 0 ⇔ ui∣∣kk +

11−2ν

uk∣∣∣ik+ρμ(bi−ai)= 0

(13.6.4)

In any orthogonal coordinate system the Navier equations (13.6.4) may be expandedin details by application of formula (12.5.98)2 for ∇2u and formula (12.5.97) for∇(∇ ·u).

As an example of the Navier equations in special coordinate system, the follow-ing equations apply in cylindrical coordinates:

∇2uR− uR

R2 −2

R2

∂uθ∂θ

+1

1−2ν∂∂R

(∇ · u)+ρμ

(bR− uR) = 0

∇2uθ +2

R2

∂uR

∂θ− uθ

R2 +1

1−2ν1R∂∂θ

(∇ · u)+ρμ

(bθ − uθ ) = 0

∇2uz +1

1−2ν∂∂ z

(∇ · u)+ρμ

(bz− uz) = 0 (13.6.5)

where:

∇2 =1R∂∂R

(R∂∂R

)+

1R2

∂ 2

∂θ 2 +∂ 2

∂ z2 , ∇ · u =1R∂∂R

(RuR)+1R∂uθ∂θ

+∂uz

∂ z(13.6.6)

13.7 Basic Equations in Fluid Mechanics 619

13.7 Basic Equations in Fluid Mechanics

This section will only present the most important equations pertaining to classicalfluid mechanics. The equations presented in invariant form and in terms of Carte-sian components in Chap. 8 will now be transformed to general coordinates and inparticular to orthogonal coordinates.

The continuity equation for a control volume V with control surface A is:

∫

V

∂tρ dV +∫

A

ρ vini dA = 0 (13.7.1)

The continuity equation for a place becomes:

ρ+ρ vi∣∣i = 0 ⇔ ∂tρ+

(ρ vi)∣∣

i = 0 (13.7.2)

The Reynolds transport theorem takes the form:

B =∫

V

β ρ dV =∫

V

∂t (β ρ) dV +∫

A

β ρ vi ni dA (13.7.3)

The equation of continuity:

∂ρ∂ t

+∇ · (ρv) = 0 ⇔ ∂ρ∂ t

+(ρ vi)∣∣

i = 0 (13.7.4)

By application of formula (12.5.58) we may rewrite the component form to:

∂ρ∂ t

+1√g

(√gρ vi) , i = 0 (13.7.5)

Irrotational flow, defined by the condition:

c≡ ∇×v = 0 ⇔ vk|i = vi|k ⇔ vk,i = vi,k (13.7.6)

implies potential flow:

v = ∇φ ⇔ vi = φ , i ⇔ vi = φ |i (13.7.7)

For potential flow of an incompressible fluid the velocity potential φ must satisfythe Laplace equation:

∇2φ = 0 ⇔ φ |ii = 0 (13.7.8)

The thermal energy balance equation (6.3.14) contains the heat flux vector:

h =−κ ∇θ ⇔ hi =−κ θ , i (13.7.9)


κ is the thermal conductivity and θ is the temperature. The heat flux vector givesthe heat flux q(n,r, t) per unit area through a material surface at the place r withunit normal n:

q =−h · n =−hi ni (13.7.10)

The stress power per unit volume is:

ω ≡ T : D = T : L = T ik vi|k (13.7.11)

The thermal energy balance equation (6.3.14) may now be presented:

ρ ε =−∇ · h+ T : D ⇔ ρ ε =− hi∣∣i + T ik vi|k ⇔

ρ ε =− 1√g

(√ghi) , i +T ik

(vi,k− v jΓ j

ik

)⇔

ρ ε =1√g

(κ√

ggikθ ,k

), i +T ik

(vi,k− v jΓ j

ik

)(13.7.12)

For an incompressible fluid we may introduce cin(θ ) as the specific heat, suchthat:

ε = cin θ (13.7.13)

13.7.1 Perfect Fluids ≡ Eulerian Fluids

The Eulerian fluid is defined by the constitutive equation:

T =−p1 , p = p(ρ ,θ ) ⇔ T i j =−pgi j (13.7.14)

With this expression for the stress tensor the Cauchy equations of motion (13.5.13)give the Euler equations:

∂v∂ t

+(v · ∇)v =− 1ρ∇p + b ⇔ ∂vi

∂ t+ vk vi

∣∣k =− 1

ρp|i + bi ⇔

∂vi

∂ t+ vk(

vi,k +vl Γilk

)=− 1

ρgik p,k +bi (13.7.15)

13.7.2 Linearly Viscous Fluids ≡ Newtonian Fluids

The Newtonian fluid is defined by the constitutive equation (8.4.6), which we nowpresent as:

13.7 Basic Equations in Fluid Mechanics 621

T =−p(ρ ,θ )1 + 2μD+(κ− 2μ

3

)(trD)1 ⇔

T ij =−p(ρ ,θ )δ i

j + 2μDij +(κ− 2μ

3

)Dk

k δij ⇔

T ij =−p(ρ ,θ )δ i

j + μ(

vi∣∣

j + v j∣∣i

)+(κ− 2μ

3

)vk∣∣∣kδ i

j (13.7.16)

When the constitutive equation is substituted into the Cauchy equation of motion(13.5.13) we get the Navier-Stokes equations:

∂v∂ t

+(v · ∇)v =− 1ρ∇p +

μρ∇2 v +

1ρ

(μ3

+κ)∇(∇ · v)+ b ⇔

∂vi

∂ t+ vk vi

∣∣k =− 1

ρp|i + μ

ρvi∣∣kk +

1ρ

(μ3

+κ)

vk∣∣∣ik+ bi ⇔

∂vi

∂ t+ vk(

vi,k + vl Γi

lk

)=− 1

ρgik p,k + bi

+μρ

glk [vi,lk + vr (Γi

rl,k +ΓsrlΓ

isk−Γi

rsΓslk

)+ vr

,kΓirl + vr

,l Γirk− vi

,r Γrlk

]

+1ρ

(μ3

+κ)

gil[

1√g

(√gvk)

,k

],l

(13.7.17)

13.7.3 Orthogonal Coordinates

The basic formulas will now be presented in general orthogonal coordinates. Weshall apply formulas developed in Sect. 12.5.

The equation of continuity is:

∂ρ∂ t

+1h∑i

(hhiρ v(i)

), i = 0 (13.7.18)

v(i) are the physical velocity components v(i) = vi hi.We use the expressions (12.5.65–12.5.68) for the covariant derivatives of vector

components and obtain the formulas for the physical components of the particleacceleration in the Euler equations and in the Navier-Stokes equations:

a(i) =Dv(i)

Dt+∑

k

v(k)hi hk

[hi,kv(i)−hk,iv(k)

](13.7.19)

where we have introduced the operator:


DDt

=∂∂ t

+∑k

v(k)hk

∂∂yk

(13.7.20)

In cylindrical coordinates the physical acceleration components become:

aR =∂vR

∂ t+ vR

∂vR

∂R+

vθR∂vR

∂θ+ vz

∂vR

∂ z− v2

θR

aθ =∂vθ∂ t

+ vR∂vθ∂R

+vθR∂vθ∂θ

+ vz∂vθ∂ z

+vRvθ

R

az =∂vz

∂ t+ vR

∂vz

∂R+

vθR∂vz

∂θ+ vz

∂vz

∂ z(13.7.21)

In spherical coordinates the physical acceleration components become:

ar =∂vr

∂ t+ vr

∂vr

∂ r+

vθr∂vr

∂θ+

vφr sinθ

∂vr

∂φ− v2

θ + v2φ

r

aθ =∂vθ∂ t

+ vr∂vθ∂ r

+vθr∂vθ∂θ

+vφ

r sinθ∂vθ∂φ

+vrvθ

r− cotθ

rv2φ

aφ =∂vφ∂ t

+ vr∂vφ∂ r

+vθr

∂vφ∂θ

+vφ

r sinθ∂vφ∂φ

+vrvφ

r+

vθvφr

cotθ (13.7.22)

The constitutive equations of the Newton fluid in terms of physical compo-nents are:

T (i j) =−p(ρ ,θ )δi j + 2μD(i j)+(κ− 2μ

3

)D(kk)δi j (13.7.23)

The physical components T (ij) and D(ij) are given by the formulas (13.5.10) and(13.3.33) respectively.

In order to obtain the physical components of the Navier-Stokes equations weneed the expressions for ∇2v and ∇(∇ ·v) presented respectively by the formulas(12.5.98)2 and (12.5.97). Then the Navier-Stokes equations become:

a(i) =− 1ρhi

p, i +b(i)+1ρ

(μ3

+κ) 1

hi

{1h∑k

[hhk

v(k)],k

}, i

+μρ

{∇2v(i)+∑

k

[hi

h

(hhi,k

h2i h2

k

),k v(i)

+(

1hi

(1h

(hhk

),k

), i− hi

h

(hhi,k

h2i h2

k

),k

)v(k)]

(13.7.24)

+∑k

[2

h2i h2

k

hi (hkhi,k v(k),i−hihk, i v(k),k)]}

Problems 623

where:

∇2() =1h∑k

[h

h2k

(),k],k (13.7.25)

The Euler equations for inviscid fluids are incorporated in these equations.The thermal energy balance equation (6.3.14) becomes:

ρDεDt

=−1h∑k

[hhk

h(k)],k + T (i j) D(i j) (13.7.26)

If Fourier’s law applies, the thermal energy balance equation becomes:

ρDεDt

=1h∑k

[κ

h

h2k

θ ,k

],k + T (i j) D(i j) (13.7.27)

Problems

Problem 13.1. Derive the formulas (13.3.51, 13.3.52).

Problem 13.2. Derive (13.4.3).

Problem 13.3. Show that (13.4.18) follows from (13.4.17).

Problem 13.4. Derive the expression (13.4.18) for the convective derivatives of thecomponents of an objective tensor of 2. order tensor B by applying the Oldroydmethod described in connection with (13.4.19, 13.4.20).

Problem 13.5. Derive (13.4.23, 13.4.24).

Problem 13.6. Show that the two sets of physical components defined by (13.5.6)and (13.5.8) are related through (13.5.9).

Problem 13.7. Use (13.5.15) to develop the Cauchy equations in cylindrical coor-dinates and in spherical coordinates. The results are listed as (3.2.39, 3.2.40, 3.2.41)and (3.2.42, 3.2.43, 3.2.44).

continuum mechanics || continuum mechanics in curvilinear coordinates

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