Notes on Continuum MechanicskPart I. General Theory

Notes on Continuum Mechanics–Part I. General Theory Melissa Morris 3420 Campus Blvd. NE Albuquerque NM 87106 [email protected] September 4, 2018 Abstract The basic mathematical de…nitions and theorems of continuum mechanics are presented in a general setting. Objectivity and invariance under a time-dependent Euclidian transformation are de…ned.

Contents 1 Introduction

1

2 Notation

2

3 The 3.1 3.2 3.3 3.4

Basic De…nitions and Theorems of Continuum Mechanics The Point Mapping, Deformation Gradient, and Jacobian . . . . . The Velocity and Convective Derivative . . . . . . . . . . . . . . . The Velocity Gradient and Other De…nitions . . . . . . . . . . . . Linearization for Small Deformation . . . . . . . . . . . . . . . . .

. . . .

2 2 5 6 7

4 Objectivity and Invariance 4.1 Imposing a Time-Dependent Euclidean Transformation . . . . . . . . . . . . . . . . . . . . . . 4.2 Objectivity and Invariance Under Time-Dependent Euclidean Transformations . . . . . . . . 4.3 Galilean Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 8 10 11

5 A Few Remarks

12

A Tensors A.1 Algebraic Operators . A.2 Di¤erential Operators A.3 Identities . . . . . . . A.4 Theorems . . . . . . . A.5 Computations . . . . .

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12 12 14 15 16 17

Introduction

One of the original intentions I had for developing the M -formulation of ‡uid mechanics in [1] was so that I could include mass di¤usion and still preserve all of the classic results of continuum mechanics. What follows are my notes on the subject as treated generally, that is, for any set of moving continuum points. The issue of what these points might represent in a deformable continuous medium is one that is reserved for later. 1

2

Notation

Throughout these notes, we consider tensors in real three-dimensional space. My tensor notation and de…nitions for the operators are given in [1, §2 and appendix A], and I provide some additional de…nitions below in appendices A.1 and A.2. For a bounded region R, possibly changing in time, I denote its bounding surface as @R, its outwardly pointing normal to the boundary at a point x on the boundary as n@R (x), and the velocity of the boundary at a point x on the boundary as v @R (x).

3 3.1

The Basic De…nitions and Theorems of Continuum Mechanics The Point Mapping, Deformation Gradient, and Jacobian

Let R(i) denote a regular region1 in R3 , called the initial region and corresponding to time t = 0. Every point in R(i) is identi…ed by its initial position, x(i) . Suppose that the initial set of points, R(i) , moves in time. For any time t 0, let R(c) (t) represent the current region, i.e. the set of points comprising the spatial domain at time t. Every point in R(c) (t) may be identi…ed by its current position, x(c) (t). Let us suppose that the initial and current positions have coordinates in a right-handed rectangular Cartesian coordinate system. Assume that for any time t, there exists a point mapping, f (i) x(i) ; t : R(i) ! R(c) (t) ;

(1)

that is twice continuously di¤erentiable with respect to each of its position coordinates and time and which satis…es (2) det rx(i) f (i) 6= 0 for all x(i) 2 R(i) : Then, one may prove that R(c) (t) is a regular region, and by the inverse function theorem provided in appendix A.4, f (i) is spatially invertible, i.e. there exists a unique function, g (c) x(c) (t) ; t : R(c) (t) ! R(i) ; called the inverse point mapping such that h i g (c) f (i) x(i) ; t ; t = x(i) for all x(i) 2 R(i) and

(3)

(4)

i h f (i) g (c) x(c) (t) ; t ; t = x(c) (t) for all x(c) (t) 2 R(c) (t) :

(5)

det rx(c) (t) g (c) 6= 0 for all x(c) (t) 2 R(c) (t) :

(6)

Furthermore, g (c) is twice continuously di¤erentiable and

Next, let us establish a few notational conventions. Functions of the initial position, x(i) , referred to as Lagrangian, will be denoted with "(i)" superscripts, e.g. f (i) ; whereas functions of the current position, x(c) (t), referred to as Eulerian, will have "(c)" superscripts, e.g. g (c) . If (i) represents some function of the initial position, then the corresponding function in terms of the current position at time t is (c)

(i) x(i) =g (c) (x(c) (t);t)

:

(7)

1 Regular regions are de…ned in Kellogg [2, p. 113]. They are bounded by smooth surfaces and are allowed only a …nite number of corners, e.g. parallelopipeds, cylinders, spheres, etc.

2

Note that in view of equation (4), it is also true that (i)

(c)

=

x(c) (t)=f (i) (x(i) ;t)

:

(8)

As further short-hand, I de…ne ( )jx(c) (t) = ( )jx(i) =g(c) (x(c) (t);t)

(9)

( )jx(i) = ( )jx(c) (t)=f (i) (x(i) ;t) ;

(10)

and and I drop the subscripts from the del-operators when it is clear what I am di¤erentiating with respect to, e.g. rf (i) rx(i) f (i) and rg (c) rx(c) (t) g (c) : The displacement, as a function of initial position, is de…ned to be u(i) x(i) ; t = f (i) x(i) ; t

x(i) ;

(11)

and using the above notational conventions and equation (5), the displacement, as a function of the current position, is computed to be u(c) x(c) (t) ; t

u(i)

=

x(c) (t)

= x(c) (t)

g (c) x(c) (t) ; t :

(12)

The deformation gradient and Jacobian are de…ned to be F (i) = rf (i)

(13)

J (i) = det F (i) ;

(14)

and respectively, and the inverse deformation gradient and inverse Jacobian are de…ned as G(c) = rg (c)

(15)

K (c) = det G(c) ;

(16)

and respectively. By the inverse function theorem and the chain rule presented in appendix A.4, F (i)

= G(i) ;

1

(17)

and consequently, 1 ; (18) K (i) where we have used the property (153) of the determinant. For the remainder of this discussion, let (c) represent some arbitrary order tensor function assumed to be su¢ ciently smooth to employ in the operations and theorems below, and let us represent volume integrals over R(i) and R(c) (t) as Z ZZZ J (i) =

(i)

dV (i)

R(i)

(i)

(i)

dx1 dx2 dx3

R(i)

3

(19)

and

Z

ZZZ

dV (c) (t)

R(c) (t)

(c)

(c)

(c)

dx1 (t) dx2 (t) dx3 (t)

(20)

R(c) (t)

and surface integrals over @R(i) and @R(c) (t) as I dA(i) and

I

@R(i)

dA(c) (t) :

@R(c) (t)

Using the de…nitions (14) and (16) and equation (18), the following relations hold: Z Z (c) (i) (i) dV (c) (t) = J dV (i) R(c) (t)

and

Z

(21)

R(i)

(i)

dV

(i)

Z

=

R(i)

(c)

1 dV (c) (t) : J (c)

(22)

R(c) (t)

The above formulas suggest that the Jacobian may be expressed, informally, as the ratio of the current di¤erential volume to the initial di¤erential volume: dV (c) (t) ; dV (i)

J (i) =

(23)

from which we conclude that J (i) > 0

(24)

is always satis…ed and J (i) J (i) J (i)

= 1 for no volume deformation > 1 for expansion < 1 for contraction.

(25)

The motion of an initial region R(i) is said to be incompressible if J (i) x(i) ; t = 1

(26)

is satis…ed for all x(i) 2 R(i) at all times. In addition, as computations (1)-(4) in appendix A.5, we prove the following: (c)

r

= F (c)

1

r

(i)

(27)

x(c) (t)

and r and if the order of

(c)

I

T

(28)

= 0;

is greater than or equal to 1, r

and

J (i) F (i)

@R(c) (t)

n@R(c) (t)

(c) x(i)

(c)

=

1 r J (i)

dA(c) (t) =

I

@R(i)

4

J (i) F (i)

n@R(i)

T

(i)

J (i) F (i)

(29)

T

(i)

dA(i) :

(30)

3.2

The Velocity and Convective Derivative

The velocity, as a function of initial position, is de…ned to be v (i)

@u(i) @t @f (i) using (11). @t

= =

(31) (32)

Integrating de…nition (31) with respect to time, we …nd (i)

u(i) x(i) ; t = u0 where

(i)

u0

x(i) +

Rt

0

v (i) x(i) ; e t de t;

(33)

x(i) = u(i) x(i) ; 0

is the initial condition on the displacement, often chosen to be identically zero. The convective derivative, D=Dt, is de…ned via D

(c)

Dt

(i)

@

=

;

@t

(34)

x(c) (t)

and using this together with de…nition (31), we …nd Du(c) : Dt

v (c) =

(35)

Also, as computation (5) in appendix A.5, we show that D

(c)

Dt

=

(c)

@

+ v (c) r

@t

(c)

:

(36)

The equations, @J (i) = J (i) r v (c) @t and

@ 1=J (c) = @t

1

r

(37)

x(i)

J (c)

v (c) ;

(38)

are proven in computations (6) and (7) of appendix A.5. Note that equation (37) together with condition (26) implies that r v (c) = 0 (39) is satis…ed for incompressible motion. As computation (8) in appendix A.5, we use some of the above properties to obtain d dt

Z

R(c) (t)

(c)

dV

(c)

(t) =

Z

@

(c)

@t

dV (c) (t) +

R(c) (t)

I

n@R(c) (t) v (c)

(c)

dA(c) (t) ;

@R(c) (t)

which is an expression of the Reynolds transport theorem stated in appendix A.4. 5

(40)

3.3

The Velocity Gradient and Other De…nitions

Let us de…ne a few extra quantities that are based on the velocity. The velocity gradient, as a function of current position, is de…ned to be L(c) = rv (c)

(41)

with rate of deformation tensor and vorticity tensor de…ned, respectively, as the symmetric and antisymmetric parts of the velocity gradient: D(c) = L(c)sy

(42)

W (c) = L(c)asy :

(43)

and The angular velocity, as a function of current position, is de…ned to be (c) (t) v (c) ang = x

v (c) :

(44)

The acceleration, as a function of initial position, is de…ned to be a(i)

@v (i) @t @ 2 u(i) by (31) @t2 @ 2 f (i) using (11). @t2

= = =

(45) (46) (47)

Employing (45) with de…nition (34), we …nd a(c) =

Dv (c) : Dt

(48)

The vorticity vector, as a function of current position, is de…ned to be w(c) = r

v (c) :

(49)

Note that by the Helmholtz representation theorem presented in appendix A.4, we can decompose the velocity v (c) into its longitudinal and rotational parts: (c)

(c)

v (c) = v L + v R with

(c)

r and

(50)

vL = 0

(51)

(c)

(52)

r v R = 0. Using this with de…nition (49), we …nd w(c) = r

6

(c)

vR :

(53)

3.4

Linearization for Small Deformation

If the point mapping may be expressed as f (i) x(i) ; t = x(i) + f (i) x(i) ; t ;

(54)

where f (i) x(i) ; t is assumed to be small, then this together with our de…nitions from the previous sections, yields the following approximate quantities: g (c) x(c) (t) ; t = x(c) (t) + g (c) x(c) (t) ; t

with g (c)

f (i)

u(i) = u(i) with u(i)

tr F (i) = r

J (i) = 1 + J (i) with J (i)

(55) (56)

r f (i)

F (i) = 1 + F (i) with F (i)

f (c)

(57) f (i)

(58)

J (c)

1

=1

J (c)

(59)

F (c)

1

=1

F (c)

(60)

@ f (i) @ u(i) = @t @t

(61)

v (i) = v (i) with v (i) (c) (c) v (c) ang = v ang with v ang

a(i) = a(i) with a(i)

x(c) (t)

v (c)

(62)

@ 2 f (i) @ v (i) = @t @t2

(63)

L(c) = L(c) with L(c)

r v (c)

(64)

and w(c) = w(c) with w(c)

v (c) :

r

(65)

Properties (28), (37), and (38), when linearized, become F (i)T

J (i) 1

r

@ J (i) = r @t and @ J (i) @t respectively. Furthermore, if

(c)

(66)

v (c) jx(i) ;

(67)

r

v (c) ;

(68)

+

(c)

(69)

1

=

can be expressed as (c)

where

= 0;

is a constant tensor and

(c)

=

;

is small in comparison, then (i)

=

+

(i)

;

(70)

jx(c) (t) ;

(71)

and (27), (29), and (34)/(36) imply r

(c)

=r

(c)

= r 7

(i)

(c)

r and

D

(c)

=r (c)

Dt

=

@

(c)

@t

(i)

= r =

@

jx(c) (t) ;

(72)

(i)

@t

jx(c) (t) ;

(73)

respectively.

4 4.1

Objectivity and Invariance Imposing a Time-Dependent Euclidean Transformation

Suppose that we have two point mappings, f (i) x(i) ; t : R(i) ! R(c) (t) and

(74)

(i) e (c) (t) ; fe x(i) ; t : R(i) ! R

(75)

each twice continuously di¤erentiable with respect to each of each of its position coordinates and time and satisfying (i) 6= 0 for all x(i) 2 R(i) : (76) det rf (i) 6= 0 and det rfe Then, as we have seen in §3.1, there exist inverse point mappings,

and such that

g (c) x(c) (t) ; t : R(c) (t) ! R(i)

(77)

e (c) (t) ! R(i) ; ge(c) x e(c) (t) ; t : R

(78)

x(c) (t) x(i)

x e(c) (t) x(i)

h i = f (i) g (c) x(c) (t) ; t ; t ; i h = g (c) f (i) x(i) ; t ; t ;

(79) (80)

h i (i) = fe ge(c) x e(c) (t) ; t ; t ; and

(81)

(i) = ge(c) fe x(i) ; t ; t :

(82)

e (c) (t) may be Every point in R(c) (t) may be identi…ed by its current position, x(c) (t), and every point in R (c) identi…ed by its current position, x e (t). (i) (i) (i) x(i) ; t are arbitrary order tensors and, using our earlier notational Suppose that x ; t and e conventions, let us de…ne h i h i (c) (i) (i) (i) x(c) (t) ; t x(i) ; t x ; t (83) x(i) =g (c) (x(c) (t);t) x(c) (t) and e(c) x e(c) (t) ; t

e(i) x(i) ; t

x(i) =e g (c) (x e(c) (t);t)

8

e(i) x(i) ; t

: x e(c) (t)

(84)

As a consequence of (79)-(82), we also have h i (i) x(i) ; t = (c) x(c) (t) ; t

h

x(c) (t)=f (i) (x(i) ;t)

(c)

x(c) (t) ; t

i

(85)

x(i)

and e(i) x(i) ; t = e(c) x e(c) (t) ; t

(c)

x e

(i)

(t)=fe

(

e(c) x e(c) (t) ; t

)

x(i) ;t

:

(86)

x(i)

Next, let us assume that the two point mappings, (74) and (75), are related to one another by a timedependent Euclidean transformation, i.e. (i) x(i) ; t = Q (t) f (i) x(i) ; t + c (t) ; fe

where Q (t) is real orthogonal:

QT (t) Q (t) = Q (t) QT (t) = 1:

(87)

(88)

We assume Q (t) and c (t) to be spatially uniform and twice continuously di¤erentiable and for det Q (t) = +1

(89)

to hold, which means that we do not consider re‡ections. Also, if we wish to enforce (i) x(i) ; 0 = f (i) x(i) ; 0 ; fe

(90)

e (c) (0), then we choose so that R(c) (0) = R

Q (0) = 1 and c (0) = 0:

(91)

Note that Euclidean transformations preserve distances and angles and, therefore, represent rigid-body motion. The …rst term on the right-hand side of (87) represents rigid-body rotation, whereas the second term describes rigid-body translation. Using equation (87) and our de…nitions from §3, we arrive at the following relationships, the details of which are carried out in computations (9)-(21) of appendix A.5: (i) Fe = F (i) QT

Je(i) = J (i)

u e(i) = Q u(i) + Q ve(i) = Q v (i) +

(i) ve(i) ang = x

(i)

e a

(i)

=Q a

dQ +2

dt

v

f (i) +

dt

Q v (i) +

(i)

dQ dt

+

dc dt

f (i) +

d2 Q dt2

e (i) = Q L(i) QT + Q L 9

(93) x(i) + c

1

dQ

(92)

(94) (95)

dc dt

f (i) +

!

(96)

d2 c dt2

(97)

dQT dt

(98)

f W

(i)

e (i) = Q D(i) QT D

=Q W

T

(i)

(99) dQT

Q +Q

(100)

dt

r ve(c) jx(i) = r v (c) jx(i)

w e

and

4.2

(i)

e (i)dev = Q D(i)dev QT D " (i)

= C1;3 C2;4

T

Q +Q

Q L

(101) (102)

dQT dt

! #

r2 ve(c) jx(i) = r2 v (c) jx(i) QT :

(103)

(104)

Objectivity and Invariance Under Time-Dependent Euclidean Transformations

In Jou et al. [3, p.27], the N th -order tensor e(i)

i1 :::iN

is said to be objective if (i) j1 :::jN

= Qi1 j1 (t) : : : QiN jN (t)

(105)

is satis…ed. For a zeroth-order tensor, this means

for a …rst-order tensor,

and for a second-order tensor,

e(i) =

(i)

;

e(i) = Q (t)

e(i) = Q (t)

(106) (i)

(i)

;

(107)

QT (t) :

(108)

If a tensor is objective, this means that it transforms in the expected way if it is measured in an arbitrarily moving coordinate system, or in other words, is not a¤ected by the (time-dependent) rotational or translational rigid-body motion of its material coordinate frame. A tensor is invariant under time-dependent Euclidean transformations if e(i) =

(i)

:

(109)

Note that objective scalars are always invariant and vice versa. Equations (92)-(104), together with the above de…nitions, imply that J, r v, D, and Ddev are objective; J and r v are invariant; and F , u, v, v ang , a, L, W , w, and r2 v are neither objective nor invariant. As computations (22)-(25) appendix A.5, I prove the following properties. 1. If

is an objective tensor of any order N

0, then r

2. If

is an objective tensor of any order N

1, then r

is an objective (non-zero) tensor, then D 3. If tensor.

is objective.

=Dt is objective if and only if

4. If is an objective tensor of any order, then its norm objective scalar. 10

is objective.

is a zeroth order

, as de…ned in appendix A.1, is an

4.3

Galilean Invariance

A tensor is said to be Galilean invariant if it is invariant under the following transformation: (i) fe x(i) ; t = f (i) x(i) ; t + v t + x ;

(110)

where v and x are constant vectors. Comparing this to (87), we see that the above is a special case of a time-dependent Euclidean transformation with Q (t) = 1 and c (t) = v t + x .

(111)

x = 0:

(112)

In order to satisfy (91), we choose If a tensor is Galilean invariant, this means that it remains the same when measured in a coordinate system undergoing translation at a constant velocity. Substituting (111) into relationships (92)-(104), we …nd (i) Fe = F (i)

(113)

Je(i) = J (i)

(114)

u e(i) = u(i) + v t + x (i)

ve

=v

(i)

(115)

+v

(i) (i) ve(i) ang = v ang + x

(116)

v

(117)

e a(i) = a(i) (i)

e L

(118)

= L(i)

(119)

e (i) = D(i) D

f W

(i)

(120)

= W (i)

(121)

r ve(c) jx(i) = r v (c) jx(i)

(122)

w e (i) = w(i)

(124)

e (i)dev = D(i)dev D

and

r2 ve(c) jx(i) = r2 v (c) jx(i) :

(123)

(125)

Therefore, we see that F , J, a, L, D, W , r v, Ddev , w, and r2 v are Galilean invariant quantities, whereas u, v, and v ang , of course, are not. Recall by criterion (105) in section 4.2, the N th -order tensor is objective if e(i)

i1 :::iN

= Qi1 j1 : : : QiN jN

(i) j1 :::jN

is satis…ed. Substituting (111 a) into the above, we …nd that an objective tensor under a Galilean transformation is computed to be e(i)

i1 :::iN

=

i1 j1

=

(i) i1 :::iN :

11

:::

iN jN

(i) j1 :::jN

Therefore by criterion (109), all objective tensors are automatically Galilean invariant. (However, clearly not all Galilean invariant tensors are objective.) As computations (26) and (27) in appendix A.5, I prove the following properties. 1. If

is a Galilean invariant tensor of any order N

0, then r

2. If

is a Galilean invariant tensor of any order N

1, then r

5

is Galilean invariant. is Galilean invariant.

A Few Remarks

Here are few tentative remarks that I will continue to revisit and attempt to clarify in subsequent parts of these notes. There arises a question as to whether or not the continuum mechanical framework detailed above is necessary for modeling ‡uids. Since, as we will soon see, the continuity equation as part of the Navier-Stokes-Fourier theory of ‡uids, allows for an unambiguous interpretation of a material point and its motion, it is easy to take the underlying continuum mechanics for granted. Commonly, the concepts presented here are used for speci…c tasks, such as writing computer codes that incorporate the Lagrangian description, but are they fundamentally necessary for describing ‡uids? Later, when attempting to introduce mass di¤usion into the ‡uid equations of motion, certain issues will arise that make this an essential question. In [1], I made the decision to create a new additional measure of density so that all of the main results of continuum mechanical theory could remain unaltered, but is it also possible to focus purely on an Eulerian description and to abandon the idea of an invertible point mapping and thereby a Lagrangian description? It’s not what I chose to do, but it certainly can be done. However, one should keep in mind that in doing so, one sacri…ces a convenient framework for de…ning, classifying, and relating moving coordinate systems. Among other di¢ culties, this may make it confusing, if not impossible, to extend the theory in order to model relativistic ‡uids.

A

Tensors

Throughout this appendix, indicial notation is used in which sums are taken over repeated indices. Since we are considering tensors in three-dimensional space, the indices run from 1 to 3. Some of the operators de…ned in appendices A.1 and A.2 are generalizations of those de…ned for speci…c order tensors in [1, appendix A].

A.1

Algebraic Operators tensor product: For any M; N 1, the (M + N )-order object, A B , is de…ned component-wise by (A B )i1 :::iM j1 :::jN = Ai1 :::iM Bj1 :::jN : (126) contraction operator: If M 2 and 1 K < L M , then the iK ; iL -contraction of A , CiK ;iL (A ), is an object of order M 2 with components given by [CiK ;iL (A )]i1 :::iK dot product: If M; N

1 iK+1 :::iL

1 iL+1 :::iM

1, then the (M + N A

B

= Ai1 :::iK

1 jiK+1 :::iL

2)-order object, A

1 jiL+1 :::iM

;

(127)

B , is de…ned as

= CiM ;j1 (A B ) ;

(128)

or component-wise, (A

B )i1 :::iM

1 j2 :::jN

12

= Ai1 :::iM

1k

Bkj2 :::jN :

(129)

double dot product: If M; N de…ned as

2, then the (M + N A

B

= CiM

4)-order objects, A

B

and A : B , are

CiM ;j2 (A B )

(130)

CiM ;j1 (A B ) ;

(131)

= Ai1 :::iM

Bklj3 :::jN

(132)

Blkj3 :::jN :

(133)

1 ;j1

and A :B

= CiM

1 ;j2

or component-wise, (A

B )i1 :::iM

2 j3 :::jN

2 kl

and (A : B )i1 :::iM generalized dot operator: If M; N A ( )B

=

CiM

2 j3 :::jN

= Ai1 :::iM

1, then the jM

2 kl

N j-order object, A ( ) B , is de…ned as

CiM N +2 ;j2 : : : CiM jN (A B ) if M Ci1 ;j1 Ci2 ;j2 : : : CiM jM (A B ) if N M

N

N +1 ;j1

;

(134)

or component-wise, [A ( ) B ]i1 :::iM

N

[A ( ) B ]jM +1 :::jN

= Ai1 :::iM

N j1 :::jN

Bj1 :::jN if M

= Ai1 :::iM Bi1 :::iM jM +1 :::jN if N

N

(135)

M;

(136)

e.g. w( )B A ( )w T ( )B A ( )T

= = = =

w B A w T B A T:

(137) (138) (139) (140)

norm: If A is a scalar, then kAk = jAj ; and if A

has order greater than or equal to 1, then p kA k = A ( ) A :

(141)

(142)

determinant:

det A =

(A11 A22 A33 + A12 A23 A31 + A21 A32 A13 ) (A11 A23 A32 + A22 A13 A31 + A33 A12 A21 ) :

(143)

and this may be expressed using the alternating symbol "ijk as det T

= "ijk TiI TjJ TkK = "ijk TIi TJj TKk ;

where (I; J; K) is any even permutation of (1; 2; 3), i.e. (I; J; K) = (1; 2; 3), (2; 3; 1); or (3; 1; 2).

13

(144) (145)

inverse: If det A 6= 0, then A111

=

A121

=

A131

=

A211

=

A221

=

A231

=

A311

=

A321

=

A331

=

A22 A33 A23 A32 det A A13 A32 A12 A33 det A A12 A23 A13 A22 det A A23 A31 A21 A33 det A A11 A33 A13 A31 det A A13 A21 A11 A23 det A A21 A32 A22 A31 det A A12 A31 A11 A32 det A A11 A22 A12 A21 : det A

(146)

A.2

Di¤erential Operators

Let

(x) be a real N th -order tensor function of the Cartesian position vector, x. gradient (Cartesian coordinate de…nition): de…ned component-wise as rx

th

If N

0, then the (N + 1) -order object, rx

ji1 :::iN

=

@

i1 :::iN

@xj

:

, is (147)

th

divergence (Cartesian coordinate de…nition): If N 1, then the (N de…ned as rx = C1;2 rx ;

1) -order object, rx

, is (148)

or component-wise, rx

i2 :::iN

curl (Cartesian coordinate de…nition): If N

=

@

ji2 :::iN

@xj

or component-wise, rx

(149)

1, then the N th -order object, rx

= C1;N +2 CN +1;N +3

rx

:

ii :::iN

1l

=

@

h

ii :::iN

@xj

i

rx 1k

"jkl ;

, is de…ned as (150)

(151)

e.g. rx

i

14

=

@ k "ijk : @xj

(152)

A.3

Identities A is invertible if and only if det A 6= 0. And when this condition holds, 1

det A If T

is a tensor of order

is a tensor of order

is a tensor of order

M T

= AT

is a tensor of order

rT :

(154)

T

(155)

(r T ) + (r ) T :

(156)

=M

rT

+ r M

1, then r ( T )=

If

(153)

1, then r

If T

1 : det A

1, then C1;2 A rT

If T

=

0, then r

w

= (r w)

+w r

:

(157)

If A is a spatially uniform tensor, then r A w = rw AT :

(158)

det A B = det A det B and det AT = det A

(159)

If Q (t) is orthogonal, then 0= and therefore

i d1 d h = Q (t) QT (t) ; dt dt

dQ dt

QT (t) =

dQT Q (t)

dt

:

(160)

r w = tr (rw)

(161)

trA = tr Asy = 1 : A

(162)

A : B C = A B : C = C A : B:

(163)

If B is a spatially uniform tensor, then r

A B = r A

B:

(164)

If C is a spatially uniform tensor, then r

A C

=C r

h

r

(r ) = 0

r (r 15

w) = 0

A

i

:

(165) (166) (167)

A.4

Theorems

Chain Rule. Let us consider the N th -order and …rst-order tensor functions, A and where and

e :X A

e (x; t) =A

x=x b y; t ;

T ! A with X open in R3 , T open in R, and A open in R3

::: 3

T ! X with Y open in R3 .

x b:Y

Their (partial) composition is de…ned to be b =A

A where

e (x; t) y; t = A b :Y A

x=b x(y;t)

;

T ! A.

e is partially If x b is partially di¤erentiable with respect to each of its variables for all points in Y T and if A b di¤erentiable with respect to each of its variables for all points in X T , then A is partially di¤erentiable with respect to each of its variables for all points in Y T . Furthermore, b = ry x b ry A

and

b e @A @A = @t @t

e rx A

+ x=b x(y;t)

@b x @t

x=b x(y;t)

e rx A

x=b x(y;t)

:

Inverse Function Theorem. Let us consider some …rst-order tensor function, y = ye (x; t) ;

where ye : X

and let us assume that ye is R for all points in X T . If

T ! Y with X and Y open in R3 and T open in R, 1 times continuously partially di¤erentiable with respect to x1 , x2 , and x3 det rx ye (x; t) 6= 0 8 (x; t) 2 X

then we can invert ye with respect to x to …nd where

T;

b y; t x=x

x b:Y

T ! X;

x b is R times continuously partially di¤erentiable with respect to y1 , y2 , and y3 for all points in Y h i det ry x b y; t 6= 0 8 y; t 2 Y T :

Furthermore, by the chain rule above,

rx ye

x(y;t) x=b

16

= ry x b

1

T , and

and ry x b

1

= rx ye

y (x;t) y=e

;

and, additionally assuming x b and ye are partially di¤erentiable with respect to t, @e y @t

=

@b x @t

=

@e y @t

x=b x(y;t)

and @b x @t

y=e y (x;t)

1

b ry x

rx ye

1

:

(x) is a tensor function of order greater Divergence Theorem. Assume R is a regular region. If than or equal to 1, which is continuously di¤erentiable with respect to each of its coordinates, then Z I r dV = n@R dA: R

@R

Reynolds Transport Theorem. (See Lin and Segel [4, pp. 347 and 443].) Assume R (t) is a regular region with its boundary moving continuously at a velocity v @R(t) at each of its points. Also, assume that (x; t) is a tensor function of order greater than or equal to 0 that is partially di¤erentiable with respect to time and continuously partially di¤erentiable with respect to each of its spatial coordinates. Then, Z Z I @ d dV = dV + n@R(t) v @R(t) dA: dt @t R(t)

R(t)

@R(t)

Helmholtz Representation Theorem. (See Segel [5, p. 67].) approaches a limit at in…nity, then w can be represented as w =r +r for some scalar

If w (x) is …nite, continuous, and

a

and vector a such that r a = 0:

We de…ne

wL = r and wR = r

a

to be the longitudinal and rotational parts of w, respectively, so that w = wL + wR with r

and

wL = 0 by (166)

r wR = 0 by (167).

A.5

Computations

1. Proof of (27). r

(c)

= rg (c) = G(c) =

F (c)

(i)

r r

x(c) (t) (i) x(c) (t)

1

r 17

by chain rule

by de…nition (15)

(i) x(c) (t)

by (17)

2. Proof of (28). Let us de…ne the second-order tensor, T = J (i) F (i)

T

:

In view of Jacobian de…nition (14) and tensor identity (146), we …nd its components to be (i)

(i)

F23 F32

(i)

(i)

F21 F33

(i)

(i)

F22 F31

(i)

(i)

F12 F33

(i)

(i)

F13 F31

(i)

(i)

F11 F32

(i)

(i)

F13 F22

(i)

(i)

F11 F23

(i)

(i)

F12 F21 :

T11

= F22 F33

T12

= F23 F31

T13

= F21 F32

T21

= F13 F32

T22

= F11 F33

T23

= F12 F31

T31

= F12 F23

T32

= F13 F21

T33

= F11 F22

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

Next, let us de…ne the vector, b = r T: Using the de…nition of the divergence operator (149), we compute its components as b1

=

b2

=

b3

=

@T11 (i) @x1

@T12 (i) @x1

@T13 (i)

@T21

+

(i) @x2

@T22

+

(i)

+

(i)

+

@x2 @T23

+

@x1

+

@x2

@T31 (i)

@x3 @T32

(i)

@x3 @T33

(i)

:

@x3

Next, we substitute the components of T into the above and carry out the computation of b1 to …nd b1

=

@

(i)

(i)

F22 F33

(i) @x1

(i)

=

@F22

(i) F (i) 33 @x1 (i) @F23 (i) F (i) 32 @x1

+ +

(i)

(i)

F23 F32

(i) (i) @F33 F22 (i) @x1 (i) (i) @F32 F23 (i) @x1

+ +

+

@

(i)

(i) @x2 (i) @F13 (i) F (i) 32 @x2 (i) @F12 (i) F (i) 33 @x2

(i)

(i)

F13 F32 + +

Note that

(i)

F12 F33

(i) (i) @F32 F13 (i) @x2 (i) (i) @F33 F12 (i) @x2

+

@ (i)

@x3 (i)

+ +

@F12

(i) F (i) 23 @x3 (i) @F13 (i) F (i) 22 @x3

+ +

(i)

(i)

F12 F23

(i)

(i)

F13 F22

!

(i) (i) @F23 F12 (i) @x3 ! (i) (i) @F22 F13 : (i) @x3

(i)

(i)

Fkj =

@fj

(i)

;

@xk

and since we have assumed f (i) to be twice continuously di¤erentiable with respect to each component of x(i) , it is true that (i) (i) @Fkj @Flj = : (i) (i) @xk @xl

18

Using this property in our previous expression for b1 , it is clear that b1 = 0; and the same type of cancellation occurs when we compute the other two components of b. 3. Proof of (29). r

(c)

jx(i)

=

h

= C1;2

(c)

r

1

= F (i) T r h = r F (i) T 1

r (i)

(i)

by (27)

by (154) i (i) r F (i)

J (i) F (i)

J (i)

by (148)

x(i)

x(i)

= C1;2 F (i)

= r 8 < = :

i

(c)

C1;2 r

1

(i)

T

(i)

T

J (i)

by (155)

J (i) r F (i)

T

r J (i) F (i) T (i) + r J 1(i) J (i) F (i) T J (i) h i (i) 1 r J 1(i) J (i) F (i) r J (i) F (i) T J (i)

(i)

(i)

1

T

(i)

9 = ;

;

where we have used identity (156) in the last step. We see that the second and fourth terms cancel with one another, and (28) implies that the third term vanishes. 4. Proof of (30). I n@R(c) (t)

(c)

dA(c) (t)

Z

=

@R(c) (t)

(c)

r

dV (c) (t) by divergence theorem

R(c) (t)

Z

=

(c)

J (i) r

jx(i) dV (i) by (21)

R(i)

Z

=

r

J (i) F (i)

T

(i)

dV (i) by (29)

T

(i)

R(i)

I

=

J (i) F (i)

n@R(i)

dA(i) by divergence theorem

@Ri

5. Proof of (36). D

(c)

Dt

=

=

= =

@

(i)

@t @

(c)

@t @

+

@f (i) @t

x(c) (t)

r

(c)

by chain rule

(c)

@t @

by (34) x(c) (t)

+ v (i)

x(c) (t)

(c)

@t

+ v (c) r 19

(c)

r

(c)

by (32)

6. Proof of (37). Using de…nition (14) for the Jacobian, formula (144) for the determinant, and de…nition (13)/(147) for the components of the deformation gradient, we can write J (i) = "ljk

(i)

(i)

(i)

(i)

(i)

(i)

@f1 @f2 @f3

:

(168)

@xl @xj @xk

Since we have assumed f (i) to be twice continuously partially di¤erentiable with respect to each component of x(i) and t, di¤erentiation of the above with respect to time yields ! (i) (i) (i) (i) (i) (i) (i) (i) (i) @J (i) @v1 @f2 @f3 @f1 @v2 @f3 @f1 @f2 @v3 = "ljk + + ; (169) (i) (i) (i) (i) (i) (i) (i) (i) (i) @t @x @x @x @x @x @x @x @x @x j

l

k

j

l

k

j

l

k

where we have used equation (32). Next, we note that the chain rule implies @v

(i)

@v

=

(i)

(c)

(i)

@fm

(c)

@x

(i)

@xm (t)

:

(170)

@x

x(i)

Substitution of (170) into equation (169) yields " # (i) (i) (i) (c) (c) (c) @J (i) @f1 @f2 @f3 @v1 @v2 @v3 = "ljk (i) (i) (i) + (c) + (c) (c) @t @xl @xj @xk @x1 (t) @x2 (t) @x3 (t) = J (i) r v (c)

+B

+ B using (168),

x(i)

(171)

x(i)

(172)

where we have de…ned B

0

"ljk @ 0

"ljk @ 0

(c) @v1 (c) @x2 (t) x(i)

(i) @f2 (i) @xl

(c) @v2 (c) @x1 (t) x(i)

(i) @f1 (i) @xj

(c)

@v "ljk @ (c)3 @x1 (t)

+

+

(i)

@v3

+

(i)

xi

(c) @v2 (c) @x3 (t) xi (c)

@f1

@xk

(c)

@x2 (t)

x(i)

Let us compute the …rst term of B:

(c)

"ljk

(c) @v1 jx(i) (c) @x2 (t)

2 6 6 4

(i)

@f2

(i)

@x1

(c)

@x2 (t) (i)

(i)

@x2

(i) @f2 (i) @x2

x(i)

(i)

@f3

(i)

@x3

(i) @f2 (i) @x1

(i) @f3 A (i) @xl

1

(i) @f3 A (i) @xj

(i) @f3 (i) @x3

@f2

(i)

@x2

+

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

@f2

(i)

@x3

(i) @f2 (i) @x3

(i)

@f3

0:

(i)

@x1

(i) @f2 (i) @x2

+

(i) @f3 (i) @x1

+

@xj @xk (i)

(i)

(i)

(i)

@f1 @f3

+

@xl @xk

(i)

@f2

(i)

@x1

@f2

@x3

+

(173)

=

@xl @xj @xk (i)

+

(i)

@f2 @f3

1 (i) (i) (i) @f2 A @f1 @f2 : (i) (i) (i) @xk @xl @xj

@f2 @f2 @f3

@v1

@f2

1

(c) @v1 (c) @x3 (t) x(i)

(i) @f2 (i) @x1

(i)

@f3

(i)

(i)

@x2

(i) @f2 (i) @x3

(i)

(i) @f3 (i) @x2

3

7 7= 5

Since the same type of cancellation occurs in all of the other terms of B, we conclude that B = 0: 20

7. Proof of (38). Evaluating (37) at the current position and using de…nition (34), we …nd DJ (c) = J (c) r v (c) ; Dt which may be rewritten as D 1=J (c) = Dt

1 r v (c) J (c)

or, using (36), @ 1=J (c) = @t

1 J (c)

v (c) r

1 r v (c) : J (c)

Identity (156) gives us our result. 8. Proof of (40): Z d dt

(c)

dV (c) (t)

=

d dt

=

Z

R(c) (t)

Z

@ J (i) @t

Z

J

R(i)

Z

=

R(c) (t)

Z

=

R(c) (t)

Z

=

dV (i) by (21)

R(i)

R(i)

=

(i)

J (i)

R(c) (t)

(i)

" " "

"

(i)

(i)

@

+ (r v )jx(i)

(c)

+ v (c) r (c) + r v (c) (c)

@t

(c)

@

+r

@t

v (c)

(c)

dV (i) by (37)

dV (c) (t) by (22) and (34)

#

dV (c) (t) by (36)

(c)

+ r v (c)

#

#

(c)

Dt @

(i)

c

@t

D

dV (i)

#

dV (c) (t) by (157).

Finally, we may use the divergence theorem to obtain (40). 9. Proof of (92). (i) Fe

(i)

= rfe = rf

(i)

by (13) QT by (87) and (158)

= F (i) QT by (13) 10. Proof of (93). Je(i)

=

(i) det Fe

by (14)

=

det F (i) QT

=

det F (i) det QT (159)

by (92)

= J (i) by (14), (159), (89) 21

11. Proof of (94). (i) = fe

u e(i)

x(i) by (11)

= Q f (i) + c

x(i) by (87)

u(i) + x(i) + c

= Q

= Q u(i) + Q

1

x(i) by (11) x(i) + c

12. Proof of (95). @e u(i) by (31) @t dQ dc u(i) + x(i) + by (94) = Q v (i) + dt dt dQ dc = Q v (i) + f (i) + by (11) dt dt

ve(i)

=

ve(i) ang

=

13. Proof of (96). h

x e(c) (t)

= x(i) = x(i)

ve(i)

ve(c)

i

x(i)

Q v (i) +

by (44)

dQ dt

dc f (i) + dt

!

by (95)

14. Proof of (97). e a(i)

= =

@e v (i) by (45) @t ! dQ dc @ (i) (i) by (95) f + Q v + dt @t dt (i)

= Q a

dQ +2

dt

v

(i)

22

d2 Q +

dt2

f (i) +

d2 c by (45), (32) dt2

15. Proof of (98). e (i) L

=

re v (c) jx(i) by (41) (i) Fe

1

rv (i) by (27) 2 3 r Q v (i) + 6 7 = Q F (i) 1 4 5 by (92) and (95) dQ (i) r dt f 1 0 Q F (i) 1 rv (i) QT + A by (158) and (13) = @ dQT Q F (i) 1 F (i) dt =

= Q

dQT

rv (c) jx(i) QT + Q (i)

= Q L

dt

by (27)

dQT

T

Q +Q

dt

by (41)

16. Proof of (99). e (i) D

e (i)sy by (42) = L 1 0 dQT (i) QT + Q dt + 1@ Q L A by (98) = dQ (i)T T T 2 Q L Q Q + dt

= Q D

(i)

T

Q by (160) and (42)

17. Proof of (100). f W

(i)

e (i)asy by (43) = L 0 1 dQT (i) T Q + Q dt 1@ Q L A by (98) = dQ (i)T T 2 Q L Q QT (t) dt = Q W (i) QT + Q

23

dQT dt

by (160) and (43)

18. Proof of (101). r ve(c) jx(i)

(i)

e = trL

e = trD

(i)

e 1: D

=

by (162) and (42)

(i)

by (162)

Q D(i) QT

1:

=

by (41) and (161)

D(i) QT

= Q:

by (99) by (163)

QT Q : D(i) by (163)

=

= trD(i) by (88) and (162) = tr L(i)

by (162) and (42)

r v (c)

=

x(i)

by (41) and (161)

19. Proof of (102). e (i)dev D

= = =

e (c) D e D

(c)

e (c) D

1 e (c) 1 tr D 3 1 e (c) 1 tr L 3

= Q D(i) QT = Q D(i) QT

by (162) and (42) x(i)

1 r ve(c) 1 3

= Q D(i) QT

x(i)

by (41) and (161) x(i)

1 r v (c) (i) 1 by (99) and (101) 3 x 1 (i) tr L 1 by (41) and (161) 3 1 tr D(i) 1 by (162) and (42) 3

= Q D(i)dev QT 20. Proof of (103). w e(i)

= =

r

ve(c)

C1;3 C2;4

by (49) i re v (c)

x(i)

h

e (i) = C1;3 C2;4 L

= C1;3 C2;4

"

x(i)

by (150)

by (41)

Q L(i) QT + Q

24

dQT dt

! #

by (98)

21. Proof of (104). e (c) jx(i) by (41) r L

r2 ve(c) jx(i) =

=

=

2

1 r 4 J (i) 1

(i) T 1 e (i) by (29) r Je(i) Fe L Je(i) 3 J (i) F (i) T QT 5 by (92), (93), (98) dQT Q L(i) QT + Q dt

"

(i)

(i) T

(i)

dQT

T

!#

Q + by (88) L r J F dt J (i) i 9 8 h < J 1(i) r J (i) F (i) T L(i) QT + = i dQT h by (164) = 1 ; : r J (i) F (i) T dt J (i)

=

= r L(c) jx(i) QT by (29) and (28) = r2 v (c) jx(i) QT by (41)

22. Proof of property (1) in §4.2. Suppose that (i)

= r

is objective and de…ne

(c) x(i)

From these, we compute (i)

(i) and e =

= F (i)

1

r

(i)

(i) = Fe

1

re

(i)

re

(c)

:

(174)

x(i)

by (27)

(175)

and e (i)

= Q F (i)

1

with components written as (i) ji1 :::iN

by (27)

re

(i)

by (92)

(176)

(i) (i) 1 @ i1 :::iN (i) @xk

= Fjk

(177)

and e (i)

ki1 :::iN

From this, we see that if

(i) (i) 1 @ j1 :::jN by (176) (i) @xm (i) lj1 :::jN using (177).

= Qi1 j1 : : : QiN jN Qkl Flm = Qi1 j1 : : : QiN jN Qkl

is an objective tensor, then so is r

23. Proof of property (2) in §4.2. For N (i)

= r

and (105) (178)

according to (105).

1, let us de…ne (c) x(i)

and e(i) =

25

(c) r e

(179) x(i)

from which we compute (i)

=

1 r J (i)

J (i) F (i)

(i)

T

by (29)

(180)

and e(i)

= =

1 r e J (i) 1 r J (i)

(i) Je(i) Fe

J (i) F (i) 1

= QT

J (i)

e(i)

T

by (29)

(i) QT e

T

T e(i)

J (i) F (i)

r

with components given by (i) i2 :::iN

@ 1 (i) J (i) Fjk J (i) @x(i)

=

by (92) and (93)

(181)

by (165)

T

(182)

(i) ki2 :::iN

(183)

j

and (i)

ei2 :::iN

= Qlk

1 @ J (i) @x(i)

(i) T e(i) li2 :::iN

J (i) Fjk

j

1 @ (i) J (i) Fjk J (i) @x(i)

= Qlm Qi2 j2 : : : QiN jN Qlk

T

(i) mj2 :::jN

by (105)

j

= Qi2 j2 : : : QiN jN

mk

= Qi2 j2 : : : QiN jN

1

@

J (i)

@xj

1

@

J (i)

@xj

Hence, from (105), it is clear that if

(i) T

J (i) Fjk

(i)

(i) j2 :::jN

= Qi2 j2 : : : QiN jN

(i) T

(i) mj2 :::jN

J (i) Fjk

(i)

by (88)

(i) kj2 :::jN

by (183).

(184)

is objective, then r

is also.

24. Proof of property (3) in §4.2. Let us de…ne (i)

=

D

(c) (i)

Dt

and e

x(i)

where, by de…nition (34), we have D

(c)

Dt and De

Dt

@e

=

(i)

:

@t

=

26

x(i)

x(c) (t)

(c)

x e

Therefore, using (27) and (176), (i)

(185)

Dt

(186)

@t

(c)

(c)

(i)

@

=

=

De

@

(187)

(t)

(i)

@t

(188)

and (i)

e

=

(i) i1 :::iN

=

or written in component form,

@e

(i)

@

(i) i1 :::iN

;

@t

(189)

(190)

@t

and (i)

ei1 :::iN

@ Qi1 j1 : : : QiN jN @t

=

(i) j1 :::jN

= Qi1 j1 : : : QiN jN

+

by (105)

(i) j1 :::jN

@ (Qi1 j1 : : : QiN jN ) : @t

assumed objective and non-zero, D

From (191) and (105), it is clear that for and only if is a zeroth order tensor. 25. Proof of property (4) in §4.2. objective, then

(i) j1 :::jN

First, in the case for which

(191) =Dt is objective if

is a scalar, it is obvious that if

is

k k=j j is also. If the order of

is N

1, then we compute (i)

q

=

q

= Next, if

is assumed objective, then e(i)

= =

q

e(i)

i1 :::iN

e(i)

(i)

()

(i) i1 :::iN

(i) (i) i1 :::iN :

(192)

i1 :::iN

Qi1 j1 : : : QiN jN q = j1 k1 : : : jN kN q (i) (i) = j1 :::jN j1 :::jN =

and therefore

q

(i)

(i) j1 :::jN Qi1 k1 (i) j1 :::jN

: : : QiN kN

(i) k1 :::kN

(i) k1 :::kN

by (105)

by orthogonality of Q

by (192),

is an objective scalar.

26. Proof of property 1 in §4.3. Suppose that before, we have equation (175): (i)

is Galilean invariant and use de…nitions (174). = F (i)

1

r

(i)

;

and using (113) in (176), we …nd e (i)

= F (i) =

27

(i)

:

1

re

(i)

As

27. Proof of property (2) in §4.3. Suppose that before, we have equation (180): (i)

=

1 J (i)

is Galilean invariant and use de…nitions (179). As

r

J (i) F (i)

(i)

T

and substituting (113) and (114) into (181) yields e(i)

= =

1 r J (i) (i)

:

J (i) F (i)

T

e(i)

References [1] Melissa Morris, A New Continuum Formulation for Materials–Part I. The Equations of Motion for a Single-Component Fluid, arXiv: 1309.4991v2 [physics.‡u-dyn], January 24, 2017. [2] O. Kellogg, Foundations of Potential Theory, Dover Publications, Inc., New York, 1953. [3] D. Jou, J. Casas-Vásquez, and G. Lebon, Extended Irreversible Thermodynamics, 2nd Edition, SpringerVerlag, Berlin, Heidelberg, 1996. [4] C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, Philadelphia, 1988. [5] L. A. Segel, Mathematics Applied to Continuum Mechanics, Dover Publications, Inc, New York, 1987.

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