Continuum Mech. Thermodyn. DOI 10.1007/s00161-013-0301-1

O R I G I NA L A RT I C L E

Roger Fosdick · James Serrin

The splitting of intrinsic energy and the origin of mass density in continuum mechanics

Received: 13 March 2013 / Accepted: 2 May 2013 © Springer-Verlag Berlin Heidelberg 2013

Abstract We show that the total intrinsic energy of a body must split into the sum of two terms—an internal energy which depends upon ‘state’ and a kinetic energy which is quadratic in the square of the particle speed. We use the non-relativistic group invariance structure of a generalized form of the balance of energy in continuum thermomechanics, together with a fundamental axiomatic requirement. The fundamental concepts of motion, force, power, heating and intrinsic energy are introduced as primitive, and we derive the notion of mass and its balance. Keywords Continuum mechanics · Thermodynamics · Invariance · Objectivity Mathematics Subject Classification (2000) 74A99 When James Serrin died on August 23, 2012, this work had just been completed. Jim was my close, personal and treasured friend for over 40 years. We collaborated on several works over those years, and we often talked together and socialized on various occasions. I had highest respect for him in all human and professional ways, and there was a definite mutual expression of affection and appreciation. A friendship could not contain more. This paper drew him back to a subject he had worked on years ago, and he was happy to be involved again with a fundamental issue in continuum mechanics. My efforts in this work are dedicated to the memory of James Serrin. He was such a scholar of great breadth and depth—He was wise and witty, and I benefited greatly from his presence. Roger Fosdick 1 Introduction In 1992, Serrin [6] outlined an elementary, but incomplete, strategy for showing that the intrinsic energy of a body, i.e., the volumetric specific energy of a body which depends on its state and particle velocity fields, must split into the sum of the internal energy plus the kinetic energy, as is commonly assumed within the subject of classical continuum thermomechanics. At that time, it was well-known from the works of Noll [5], Green and Rivlin [1,2] and Šilhavý [7], that the classical balance laws of mass, momentum and energy resulted from Communicated by Andreas Öchsner. R. Fosdick (B) Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected] J. Serrin School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected]

R. Fosdick, J. Serrin

a natural invariance condition assumed for either the power or the balance of energy1 . However, Šilhavý [7] had gone further; within a very general setting, he recorded in print the first proof that such a splitting was necessary.2 Our aim here is to prove a ‘splitting theorem’. The existence of mass density automatically will emerge, as a result of the splitting, in the form of a positive scalar ‘inertial field’ that modulates the kinetic energy part of the intrinsic energy. We assume the existence of a fixed frame F and introduce, in addition to the motion of a body and the state of its particles relative to F, the following fundamental concepts of internal action for its arbitrary parts: force, power (as a linear function of the particle velocity), heating and intrinsic energy. Basic to our work, we propose that relative to F, and for arbitrary parts of the body, the time rate of change of the intrinsic energy is in balance with the sum of the power and heating supply, and we require that the ‘form’ of this generalized balance of energy be invariant under a (Euclidean) change of frame F  → F∗ (t). A natural hypothesis which governs the objectivity of traction, heat flux, temperature and radiant heating under a change of frame is posited, and further, it is assumed that relative to the moving frame F∗ (t), there exists a body force field which depends upon both the frame F∗ (t) and the body force that is supplied relative to the fixed frame F. No special transformation property (like Galilean invariance) is required concerning this functional relationship—only that a frame-dependent functional relationship exists. As a third natural hypothesis, we require that the local intrinsic energy of a particle be greater when the particle is in motion than when it is at rest. Finally, we define a process relative to F, and posit as a fundamental axiom, the extent to which a process may be arbitrarily specified by a choice of the body force and radiant heating supply terms. Within this setting, we find that the intrinsic energy necessarily splits into the sum of an internal energy, which depends only on the state of the particles, and the kinetic energy, which is quadratic in the speed of the particles. The existence of a balanced scalar field having the classical properties of mass density, in the sense that it characterizes the inertial intensity of kinetic energy, is a result of the splitting theorem.3 All of the classical balance laws of continuum mechanics relative to F, including those for mass, linear and angular momentum, readily follow, and their forms relative to the moving frame F∗ (t) are derived. The specific form of the body force field relative to F∗ (t) is determined and the effective body force field, i.e., the apparent body force field that enters the balance of linear momentum and contains the Coriolis effect, is uniquely identified. An important conclusion in regard to the Coriolis force field is that it is formally not part of the body force field relative to the moving frame F∗ (t). A characteristic property is that it does not contribute to the power supply and, therefore, does not enter the generalized balance of energy relative to F∗ (t). Nevertheless, it naturally shows up as an essential part of the balance of linear momentum for a moving frame of reference. 2 The setting In the vector space E3 , the concepts of length and angle between elements are meaningful. A frame F in E3 is a basis together with a specified element o ∈ E3 , called the origin. Thus, the elements x ∈ E3 may be 1 Philosophers and other natural scientists, in centuries before, had written about the invariant structure of space-time and interpreted certain physical laws about the evolutionary behavior of a system as statements of ‘causality’, and the ‘mass-force debate’ became an important element of the developments of the day. In the second half of the 20th century, an axiomatic structure began to emerge within the continuum mechanics community that emphasized the balance of energy and the Clausius-Duhem inequality as a basis for the foundation of the equations of continuum thermodynamics, coupled strongly and fundamentally with the invariance condition of frame indifference (often called objectivity). The aims and goals of this development were mainly non-relativistic 2 As a matter of record, J. E. Dunn and R. Fosdick began working on this question of splitting in the late 1960s, and in December, 1969 R.F. presented invited colloquium talks at the University of Illinois and the University of Kentucky titled “A Causality Approach to Continuum Mechanics”. In these talks, a strategy for splitting and the existence of mass and its balance was presented. Serrin was not aware of this work until much later, and the work of Šilhavý [7] was more global than that of Serrin in both direction and emphasis. 3 Debates concerning the concept of mass and its relation to force have captured the interests of the scientific community since the time of Newton in the late 1600s, and they engaged the likes of Leibniz, Euler, Mach and many others for two centuries. At the turn of the 20th century, Wilhelm Ostwald (see the citation by Jammer [3, p. 108] in his masterful historical development of the concept of mass) raised, in his Vorlesungen über Naturphilosophie, a new question in the debate. While the central discussion of the day was whether ‘force’ had priority over ‘mass’ as fundamental, Ostwald was a proponent of the emerging idea that energy was the foremost fundamental concept in physical science. He pursued the question of whether the energy of motion of a physical object was a function of more than its velocity, and he concluded that, apart from the velocity, the special additional property on which the energy of a moving body depends is called ‘mass’. For Ostwald, mass was conceived as merely a capacity for kinetic energy, just as specific heat was considered to be a capacity for thermal energy. Since our derivation of mass density, stemming from (3.25) and continued into (3.29) and what immediately follows (3.29), is intimately connected to the identification of kinetic energy, as is primarily exhibited in (3.21), Ostwald’s interpretation is totally consistent with our conclusion.

Energy splitting and the origin of mass density

characterized in terms of the frame F, which we take as fixed. An isometrically equivalent, time-dependent re-framing of E3 results in a frame F∗ (t) which allows the points of E3 to be represented by x∗ = Q(t)x + c(t), ∀ x ∈ E3 and at each time t, where Q(t) ∈ Orth and c(t) ∈ E3 . For brevity, this transformation is referred to as a change of frame F  → F∗ (t) for Euclidean space-time. Thus, under a change of frame the motion x = χ(X, t) of a body B ⊂ E3 with elements X ∈ B in frame F is represented as x∗ = χ ∗ (X, t) = Q(t)χ (X, t)+c(t) relative to frame F∗ (t). A Galilean change of frame corresponds to the special case Q(t) = Q and c(t) = ta + c, where Q, a and c are constant, independent of time t. It suffices in this work to identify B with the configuration of the body at the time t = 0 in the frame F and to refer to X ∈ B as a material particle. Fields that represent physical quantities generally depend on the frame. For example, under a change of frame F  → F∗ (t), the referential form of the particle velocity vˆ = vˆ (X, t) := ˙ ˙ ˙ t) + Q(t)χ(X, t) + c˙ (t) in χ(X, t) in frame F is represented as vˆ ∗ = vˆ ∗ (X, t) := χ˙ ∗ (X, t) = Q(t)χ(X, frame F∗ (t). A material time derivative of the velocity identifies how the particle acceleration transforms. If a physical space-time-dependent scalar field ϕ, vector field f or tensor field T transforms under a change of frame such that ϕ ∗ = ϕ, f ∗ = Q(t)f or T∗ = QT (t)TQ(t),

(2.1)

then the field is said to be objective. Clearly, the particle velocity field is not objective unless both Q(t) and c(t) are constant in time. It turns out that the particle acceleration field is not objective unless both Q(t) and c˙ (t) are constant in time. Such a change of frame is called a Galilean or inertial change of frame. Suppose a frame F is given. In the following, we let P ⊆ B be an arbitrary part of B with boundary ∂ P , and use the notation Pt := χ(P , t) and ∂ Pt := χ(∂ P , t) to denote the image of these sets at time t under the motion χ(·, t). We introduce the fundamental internal fields: local intrinsic energy ϕˆ = ϕ(X, ˆ t) measured per unit volume of B; traction t = t(x, t; n), defined for all n ∈ Unit and measured per unit area of an oriented surface through x ∈ Bt whose unit normal is n; body force b = b(x, t) dependent on the frame F and measured per unit volume of Bt := χ(B, t); heat flux q = q(x, t) measured per unit area of a surface at x ∈ Bt whose orientation is normal to q; radiant heating r = r (x, t) measured per unit volume of Bt ; Eulerian form of the particle velocity v = v(x, t) at x ∈ Bt , where vˆ (X, t) := v(x, t)|x=χ (X,t) . In these terms, the balance of energy during a time interval I relative to the frame F has the form d (P , t) = P(P , t) + Q(P , t) ∀P ⊂ B, ∀t ∈ I , dt

(2.2)

with the following entries: – (P , t) denotes the total intrinsic energy of P at time t, i.e.,  (P , t) := ϕˆ dv(X).

(2.3a)

P

– P(P , t) denotes the mechanical power supply to P at time t which is defined as a linear function of v with conjugate surface and volume force measures identified as t and b, i.e.,   t · v da(x) + b · v dv(x), (2.3b) P(P , t) := ∂ Pt

Pt

where the unit normal n in t is understood to be the outer unit normal field to ∂ Pt . – Q(P , t) denotes the heating supply to P at time t and is defined according to   Q(P , t) := − q · n da(x) + r dv(x), ∂ Pt

where n is the outer unit normal field to ∂ Pt .

Pt

(2.3c)

R. Fosdick, J. Serrin

Remark 2.1 It is worth emphasizing that the fundamental body force and traction fields b and t which enter the mechanical power supply (2.3b), are, through the requirement of linearity, conjugate to, and independent of, the velocity v of the particle on which the action of power takes place. Thus, it is fundamental that the fields b and t introduced in this work, while acting on particles of the body, are not dependent upon the velocity of the particles, themselves. Because of this, the traction and body force fields may be thought of as ‘work-effective’ fields. ˆ Throughout this work, the deformation gradient F(X, t) := ∇χ(X, t) ∈ Lin+ , with Jacobian J := ˆ t) := det ∇χ > 0, and the absolute temperature θˆ (X, t) > 0 will characterize the state uˆ = u(X, ˆ ˆ ˆ ˆ {F(X, t), θ (X, t)} = {F, θ}(X, t) of a particle X ∈ B at time t in the frame F. We shall write ˆ u = u(x, t) := {F(x, t), θ (x, t)} = {F, θ }(x, t), with u(X, t) := u(x, t)|x=χ (X,t) . We assume that the local intrinsic energy ϕ(X, ˆ t) is determined in the frame F by the state and the particle velocity through a intrinsic energy response function ϕ(X; ¯ ·, ·), so that in (2.3a) we may set   ˆ ϕˆ = ϕ(X, ˆ t) := ϕ¯ X; u(X, t), vˆ (X, t) . (2.4) Under a change of frame F  → F∗ (t) we note that Pt  → Pt∗ := χ ∗ (P , t), ∂ Pt  → ∂ Pt∗ := χ ∗ (∂ P , t), and Bt  → Bt∗ := χ ∗ (B , t). Furthermore, the fundamental fields introduced above will undergo transformations which we shall denote as follows: ϕ → ϕ ∗ = ϕˆ ∗ (X, t); t → t∗ = t∗ (x∗ , t; n∗ ), defined for all n∗ ∈ Unit; b → b∗ = b∗ (x∗ , t); q → q∗ = q∗ (x∗ , t); r → r ∗ = r ∗ (x∗ , t); v → v∗ = v∗ (x∗ , t). Of course, ˙ in this list we know that x∗ = Q(t)x + c(t), which implies that v∗ (x∗ , t) = Q(t)v(x, t) + Q(t)x + c˙ (t). In ∗ ∗ ∗ ∗ ˆ ˆ ˆ ˆ ˆ ˆ addition, we note here that u(X, t) → u = u (X, t) := {F (X, t) = Q(t)F(X, t), θ (X, t)} and vˆ (X, t) → ˙ vˆ ∗ (X, t) = Q(t)ˆv(X, t) + Q(t)χ(X, t) + c˙ (t). We require an invariant structure of the balance of energy under a change of frame as expressed in the following hypothesis: Hypothesis 2.1 The balance of energy recorded in (2.2), (2.3) and (2.4) is form invariant under a change of frame F  → F∗ (t). Accordingly, the balance of energy should appear the same ‘in form’ regardless of the frame in which it is represented. That is, it should express the balance law  d Total intrinsic energy = Mechanical Power + Heating Supply. dt While the definitions of the three entries here are based on the same physical understanding and related mathematical structure regardless of the frame, nevertheless, the six fundamental fields of motion, state, traction, heat flux, body force and radiant heating that appear when this balance is expressed in a particular frame are, in general, frame dependent. Thus, this hypothesis allows us to write the balance of energy during a time interval I in the frame F∗ (t) as d ∗  (P , t) = P ∗ (P , t) + Q ∗ (P , t) ∀P ⊂ B, ∀t ∈ I , dt where ∗



 (P , t) :=

ϕˆ ∗ dv(X),

(2.5)

(2.6a)

P

¯ ·, ·) as with the integrand given by the form invariant function4 ϕ(X;   ∗ ∗ ∗ ∗ ϕˆ = ϕˆ (X, t) := ϕ¯ X; uˆ (X, t), vˆ (X, t)   ∗ ˙ t) + c˙ (t) , = ϕ¯ X; uˆ (X, t), Q(t)ˆv(X, t) + Q(t)χ(X,

(2.6b)

4 Note that because a change of frame F  → F∗ (t) yields F(X, ˆ ˆ t)  → Fˆ ∗ (X, t) = Q(t)F(X, t), where Q(t) ∈ Orth, the function ϕ(X; ¯ ·, ·) is required to be defined on S × E3 , where S denotes the set of all states s = {A, α} with A ∈ Inv := {A ∈ Lin| det A = 0} and α > 0.

Energy splitting and the origin of mass density

and where 

∗

∗

P (P , t) :=

∗



∗

b∗ · v∗ dv(x∗ )

t · v da(x ) + ∂ Pt∗

(2.7a)

Pt∗

is a linear function5 of v∗ , and 

∗

Q (P , t) := −

∗

∗

∗



q · n da(x ) +

∂ Pt∗

r ∗ dv(x∗ ).

(2.7b)

Pt∗

The unit normal n∗ implicit in t∗ in (2.7a) and explicit in (2.7b) is the outer unit normal field to ∂ Pt∗ . Physical considerations suggest that the fields of traction, heat flux, temperature and radiant heating are objective under a change of frame while the body force generally is not. As a minimal transformation requirement, we adopt the following hypothesis: Hypothesis 2.2 i) The fields t, q, θ and r are objective under a change of frame F  → F∗ (t) in the sense that t∗ (x∗ , t; n∗ ) = Q(t)t(x, t; n), q∗ (x∗ , t) = Q(t)q(x, t), θ ∗ (x∗ , t) = θ (x, t) and r ∗ (x∗ , t) = r (x, t), with the understanding that n∗ = Q(t)n for each n ∈ Unit, and x∗ = Q(t)x + c(t). ii) There is a well-determined frame-dependent mapping bF∗ (t) (·) : E3 −→ E3 such that the body force field   satisfies b → b∗ (x∗ , t) = bF∗ (t) b(x, t) , again with the understanding that x∗ = Q(t)x + c(t). In particular, this hypothesis allows us to represent the mechanical power and heating supply terms, (2.7a), (2.7b), for the balance of energy (2.5) as integrals over the domains ∂ Pt and Pt in the forms 

∗

  ˙ t · v + QT (t)Q(t)x + QT (t)˙c(t) da(x)

P (P , t) = ∂ Pt



  ˙ bF∗ (t) (b) · Q(t)v + Q(t)x + c˙ (t) dv(x),

+

(2.8a)

Pt

and 

∗

Q (P , t) := − ∂ Pt

 q · n da(x) +

r dv(x) = Q(P , t).

(2.8b)

Pt

Our final hypothesis ensures the physically suggestive notion that the local intrinsic energy of a particle is greater when it is in motion than when it is at rest: Hypothesis 2.3 For any given a state6 s ∈ S, the intrinsic energy response function ϕ(X; ¯ s, ·) satisfies     ϕ¯ X; s, v > ϕ¯ X; s, 0 ∀v = 0.

(2.9)

  ∂v ϕ¯ X; s, 0 = 0 ∀s ∈ S.

(2.10)

Clearly, this hypothesis implies

5 6

Thus, both t∗ and b∗ are independent of v∗ . See footnote 4.

R. Fosdick, J. Serrin

3 The main theorem For the proof of the main theorem below, it will be convenient to have available the following notion of a process: In words, given a frame F, a process P is an ordered collection of the six fundamental fields of motion, temperature, body force, radiant heating, traction and heat flux such that the balance of energy expressed in (2.2)–(2.4) holds. Concisely, P is a process during a time interval I relative to the fixed frame F if     P := χ (X, t), θ, b, r, t, q (x, t)

(3.1)

satisfies the requirements of (2.2)–(2.4) for all X ∈ B and all t ∈ I , with x = χ(X, t) ∈ Bt . The following axiom expresses a property of a process in a fixed frame that is of fundamental importance to the proof of Theorem 3.1. Axiom 3.1 (Fundamental Axiom) Let 0 := {b0 , r 0 , u0 , u˙ 0 , v0 , v˙ 0 } be an assigned parameter set with u0 = {F0 , θ 0 } and u˙ 0 = {F˙ 0 , θ˙ 0 }, det F0 > 0 and θ 0 > 0. Then, for each choice of {u0 , u˙ 0 , v0 , v˙ 0 } there exists a process P during a time interval I relative to a fixed frame F which takes on the values {b, r }(x0 , t0 ) = ˙ θ˙ }(x0 , t0 ) = u˙ 0 , v(x0 , t0 ) = v0 and v˙ (x0 , t0 ) = {b0 , r 0 }, u(x0 , t0 ) := {F, θ }(x0 , t0 ) = u0 , u˙ (x0 , t0 ) := {F, 0 v˙ at any specified X0 ∈ B and t0 ∈ I with x0 = χ(X0 , t0 ). Remark 3.1 The Fundamental Axiom 3.1 guarantees the existence of a process P during a time interval I ˙ v, v˙ }(x, t) may be assigned equal to 0 at a given particle for which the values of (x, t) := {b, r, u, u, X0 ∈ B and time t0 ∈ I with x0 = χ(X0 , t0 ). The process P may depend on 0 , but, recalling Remark 2.1, the body force field b(x, t) and the traction field t(x, t) for the process P will not depend on the particle velocity v(x, t) that associates with P. Consequently, the limiting values b0 = lim(x,t)→(x0 ,t0 ) b(x, t) and t0 = lim(x,t)→(x0 ,t0 ) t(x, t) are assignable in the parameter set 0 independent of v0 . This observation will be called upon later. Notice that Axiom 3.1 depends upon on the concept of a fixed frame, and accordingly requires that one must assume the existence of a fixed frame. This assumption is essentially that of Newton’s hypothesis concerning absolute space, which contrasted with the Leibnizian notion that there are no preferred frames whatsoever. Whether the main splitting result of Theorem 3.1(II) below can be obtained without the concept of a fixed frame is not known to the authors. On the other hand, for the practical purpose of determining the appropriate dynamical force function f to replace b in the balances of linear and angular momentum noted in Theorem 3.1(III) and (IV) for a given (moving) frame, the notion of a fixed frame, i.e., absolute space, is helpful at the very least.7 Theorem 3.1 Suppose that the balance of energy (2.2)–(2.4) holds during a time interval I subject to the Hypotheses 2.1–2.3 and that Axiom 3.1 applies. Then: (I) There exists a positive referential mass density field ρ0 = ρ0 (X) > 0, ∀ X ∈ B, such that d dt

 ρ dv(x) = 0 with ρˆ := Pt

ρ0 , ∀P ⊂ B, ∀t ∈ I , J

(3.2)

where ρ = ρ(x, t), ρ(X, ˆ t) = ρ(x, t)|x=χ (X,t) . (II) There exists a specific internal energy function8 ε¯ (X; s) := ε¯ (X; A, α) for X ∈ B, defined for all states9 s = {A, α} ∈ S, which is frame indifferent in the sense that ε¯ (X; QA, α) = ε¯ (X; A, α) for all A ∈ Lin+ , α > 0 and Q ∈ Orth+ . Moreover, the field ε = ε(x, t) with εˆ (X, t) = ε(x, t)|x=χ (X,t) such that     ˆ t), θˆ (X, t) , ∀X ∈ B, ∀t ∈ I , εˆ = ε¯ X; uˆ (X, t) := ε¯ X; F(X, 7 We show, within the proof of Theorem 3.1, below, that the total intrinsic energy of a part P splits for a moving frame as it does for a fixed frame. Then, in Sect. 4 we determine the appropriate dynamical force function f to replace b in the dynamical equations analogous to (3.4) and (3.5) that apply for a moving frame. 8 The abridged notation introduced here, though slightly abusive, is suggestive and should cause no confusion. 9 See footnote 4.

Energy splitting and the origin of mass density

satisfies the balance of energy in the classical form d d E(P , t) + K (P , t) = P(P , t) + Q(P , t) ∀P ⊂ B, ∀t ∈ I , dt dt where

 E(P , t) := Pt

1 ρε dv(x), K (P , t) := 2

 ρ|v|2 dv(x).

∂ Pt

(3.4)

Pt

(IV) The balance of angular momentum holds in the form  d ρ(x − o) × v dv(x) dt Pt   = (x − o) × t da(x) + (x − o) × b dv(x) ∀P ⊂ B, ∀t ∈ I , ∂ Pt

(3.3b)

Pt

(III) The balance of linear momentum holds in the form    d ρv dv(x) = t da(x) + b dv(x) ∀P ⊂ B, ∀t ∈ I . dt Pt

(3.3a)

(3.5)

Pt

where o is a fixed point in E3 . Remark 3.2 Theorem 3.1(I) is a statement about the existence of mass and its balance. The referential mass density rightfully could be characterized as an ‘inertial modulus’ because, through the present mass density field ρ defined in (3.2), it fundamentally modulates the influence of acceleration on the internal force structure as seen in the balance of linear momentum (3.4), and it portions the amount of the local intrinsic energy that resides in the body solely due to the speed of its particles, as seen in the structure of the kinetic energy defined in the second of (3.3b). Theorem 3.1(II) states that the intrinsic energy on the left-hand side of the balance of energy (2.2) must split into the sum of the internal energy plus the kinetic energy; this is the standard assumed form of the balance of energy in classical continuum mechanics. In addition, the specific internal energy function is seen to be a frame indifferent function of the state. Theorem 3.1(III)–(IV) records the classical balances of linear and angular momentum, thus completing a summary of the classical balance laws of continuum mechanics relative to a fixed frame F. It is noteworthy that the proof of this theorem does not presuppose the invariance of the body force field under a Galilean change of frame. Proof of Theorem 3.1 Let P is a process during a time interval I relative to a fixed frame F and suppose that Hypotheses 2.1 and 2.2 hold. Consider a special translational Galilean change of frame F  → F∗ (t), for which Q(t) = 1 and c(t) = ta, where a is a constant element of E3 , and subtract the expressions for the balance of energy (2.2) and (2.5) relative to the two frames. Using Hypotheses 2.1 and 2.2, we readily obtain    d  ˆ vˆ + a) − ϕ(X; ˆ vˆ ) dv(X) − a · t da(x) ϕ(X; ¯ u, ¯ u, dt P ∂ Pt    = ba (b) · (v + a) − b · v dv(x) ∀P ⊂ B, ∀t ∈ I , Pt

where, in the present context and for clarity of exhibiting the dependence of bF∗ (t) on the frame F∗ (t) through the single constant vector a, we have set ba (b) := bF∗ (t) (b). Thus, moving the time differentiation inside the first integral and changing its domain of integration from P to Pt , we see that for fixed t ∈ I and a the surface integral over ∂ Pt is bounded by a volume integral over Pt for all P ⊂ B. This presents a condition equivalent

R. Fosdick, J. Serrin

to that used in the proof of the classical Cauchy stress theorem of continuum mechanics which, in the present case, shows that there exists a linear transformation, i.e., the Cauchy stress tensor T = T(x, t), such that   T t da(x) = divTT dv(x). (3.6) t = t(x, t; n) = T (x, t)n and ∂ Pt

Pt

In turn, using (3.6)2 and noting that P ⊂ B is arbitrary, we localize at X ∈ B and t ∈ I with x = χ(X, t) and find that     ¯ u, v + a) − ∂u ϕ(X; ¯ u, v) · u˙ + ∂v ϕ(X; ¯ u, v + a) − ∂v ϕ(X; ¯ u, v) · v˙ ∂u ϕ(X;   (3.7) = J (X, t) a · divTT + ba (b) · (v + a) − b · v , where, we recall, J := det ∇χ is the Jacobian determinant. In (3.7), recalling the notation u = {F, θ }(x, t), and ˙ θ˙ }(x, t), it is natural to introduce and to interpret the notation ∂u ϕ¯ and ∂u ϕ¯ · u˙ through the agreement u˙ = {F, ¯ ∂θ ϕ} ¯ and ∂u ϕ¯ · u˙ := ∂F ϕ¯ · F˙ + ∂θ ϕ¯ θ˙ . ∂u ϕ¯ := {∂F ϕ,

(3.8)

In addition to the local condition (3.7), we observe, by using (3.6)2 , (2.2), (2.3) and (2.4), that the localization of the balance of energy for the process P relative to the fixed frame F at X ∈ B and t ∈ I with x = χ(X, t) has the form ¯ u, v) · u˙ + ∂v ϕ(X; ¯ u, v) · v˙ ∂u ϕ(X;   T = J (divT + b) · v + TT · grad v + divq + r .

(3.9)

Remark 3.3 Before continuing, let us observe, according to (3.6)1 , that the traction field t in the designation (3.1) of a process P may be replaced by the Cauchy stress field T. Thus, with slight abuse of notation, it will be convenient to refer, alternatively, to a process during a time interval I relative to the fixed frame F as the collection     P := χ(X, t), θ, b, r, T, q (x, t) , (3.10) which satisfies (3.9) for all X ∈ B and all t ∈ I , with x = χ(X, t). We now choose a fixed (X0 , t0 ), with X0 ∈ B and t0 ∈ I , and apply the Fundamental Axiom 3.1 to carry out the proof of this theorem. First, emphasizing an earlier key observation concerning Axiom 3.1 on the existence of a process and the definition and admissibility of the corresponding body force and traction fields, we recall from Remark 3.1 that during a time interval I a process P may depend on the parameter set 0 , but, at any time t ∈ I the fields b(x, t), T(x, t) that are associated with P, and consequently the field divTT (x, t), do not depend on the present particle velocity field v(x, t) that is sustained by P. Thus, the limiting values b0 = lim(x,t)→(x0 ,t0 ) b(x, t), T(x0 , t0 ) = lim(x,t)→(x0 ,t0 ) T(x, t) and divTT (x0 , t0 ) = lim(x,t)→(x0 ,t0 ) divTT (x, t) are independent of the choice of v0 in the parameter set 0 . This fact is of crucial importance to the proof below. Step 1 Suppose 0 in the Fundamental Axiom 3.1 is such that u˙ 0 = {F˙ 0 , θ˙ 0 } = 0 and v˙ 0 = 0. Clearly, Axiom 3.1 assures the existence of a process P during a time interval I . Evaluating (3.7) at (X0 , t0 ) with X0 ∈ B, t0 ∈ I and x0 = χ(X0 , t0 ), we see that a · divTT (x0 , t0 ) + ba (b0 ) · (v0 + a) − b0 · v0 = 0. Moreover, because of the identity F˙ = (grad v)F, we see that for this choice (3.9), evaluated at (X0 , t0 ) with X0 ∈ B, t0 ∈ I and x0 = χ(X0 , t0 ), requires   divTT (x0 , t0 ) + b0 · v0 + divq(x0 , t0 ) + r 0 = 0, which is guaranteed to hold because of the existence of P. We now may conclude that ba (b0 ) · (v0 + a) − b0 · v0 is a linear function of a. Consequently, since b0 is fixed independent of v0 , there exists a unique f 0 ∈ E3 , independent of a and affine as a function of v0 ∈ E3 , such that ba (b0 ) · (v0 + a) − b0 · v0 = f 0 · a ∀ a ∈ E3 .

(3.11)

Energy splitting and the origin of mass density

Thus, setting f 0 = m + Mv0 in (3.11), where m ∈ E3 and M is a linear transformation of E3  → E3 , and then choosing v0 = −a, we arrive at Ma · a − (m − b0 ) · a = 0 ∀ a ∈ E3 , which shows that m = b0 and that M ∈ Skew, so f 0 = b0 + Mv0 with M ∈ Skew.

(3.12)

Step 2 Suppose 0 in the Fundamental Axiom 3.1 is such that u˙ 0 = 0. Again, we are assured of the existence of a process P during a time interval I , which may depend on 0 . Evaluating (3.7) at (X0 , t0 ) with X0 ∈ B, t0 ∈ I and x0 = χ(X0 , t0 ), and defining J 0 := J (X0 , t0 ), we get    ¯ 0 ; u0 , v0 + a) − ∂v ϕ(X ¯ 0 ; u0 , v0 ) · v˙ 0 = J 0 a · divTT (x0 , t0 ) + f 0 . ∂v ϕ(X



(3.13)

Also, evaluating (3.9) at (X0 , t0 ) with X0 ∈ B, t0 ∈ I and x0 = χ(X0 , t0 ) we find   ¯ 0 ; u0 , v0 ) · v˙ 0 = J 0 (divTT (x0 , t0 ) + b0 ) · v0 + divq(x0 , t0 ) + r 0 , ∂v ϕ(X which is guaranteed to hold because of the existence of P. Thus, the unit  dividing0by 0|a| 0in   (3.13), introducing a vector k := |a| and passing to the limit |a| → 0, we find that ∂vv ϕ(X ¯ 0 ; u , v )˙v ·k = J 0 k · divTT (x0 , t0 )+  f 0 for all unit vectors k ∈ E3 . Consequently, (3.13) implies   ¯ 0 ; u0 , v0 )˙v0 = J 0 divTT (x0 , t0 ) + f 0 , ∂vv ϕ(X

(3.14)

in which we may substitute (3.12). Then, because divTT (x0 , t0 ) and b0 are independent of v0 , and, in particular, according to Axiom 3.1 the result (3.14) must hold for all v0 , it follows that ∂vvv ϕ(X ¯ 0 ; u0 , v0 )˙v0 = J 0 M. However, recalling that M ∈ Skew and observing that the left-hand side of this last equation is symmetric, we conclude that M = 0 and ∂vvv ϕ(X ¯ 0 ; u0 , v0 )˙v0 = 0. The first of these, together with (3.12), shows that f 0 = b0 and from (3.11) we may conclude that ba (b0 ) = b0 . The second of these, together with the arbitrariness of v˙ 0 due to the Fundamental Axiom 3.1, allows us to conclude that ∂vvv ϕ(X ¯ 0 ; u0 , v0 ) = 0, which shows that there 0 0 exists D(X0 ; u ) ∈ Sym, independent of v , such that ∂vv ϕ(X ¯ 0 ; u0 , v0 ) = D(X0 ; u0 ).

(3.15)

Integrating, we find ∂v ϕ(X ¯ 0 ; u0 , v0 ) = D(X0 ; u0 )v0 + g(X0 ; u0 ), in which we may set g(X0 ; u0 ) ≡ 0 because of Hypothesis 2.3 (in particular, (2.10)). Integrating again, we reach ϕ(X ¯ 0 ; u0 , v0 ) =

 1 ¯ 0 ; u0 , 0), D(X0 ; u0 )v0 · v0 + ϕ¯ 0 (X0 ; u0 ), ϕ¯ 0 (X0 ; u0 ) := ϕ(X 2

(3.16)

which is to hold for all X0 ∈ B and for all u0 and v0 . Also, note that (3.14) and the earlier conclusion that f 0 = b0 yields   D(X0 ; u0 )˙v0 = J 0 divTT (x0 , t0 ) + b0 ,

(3.17)

but this, apparent precursor of the balance of linear momentum, holds only under the assumption in Step 2 that the parameter set 0 satisfies u˙ 0 = 0, and divT(x0 , t0 ) may depend upon this assumption.

R. Fosdick, J. Serrin

Step 3 Now, let 0 be arbitrarily assigned with the assurance from the Fundamental Axiom 3.1 that there exists a corresponding process P during a time interval I , and let us return to (3.7) and (3.9). Evaluating (3.7) at (X0 , t0 ) with X0 ∈ B, t0 ∈ I and x0 = χ(X0 , t0 ), making use of (3.16) and the symmetry of D(X0 ; u0 ), and employing (3.11) along with the earlier conclusion that f 0 = b0 , we readily find      1  ∂u D(X0 ; u0 )u˙ 0 a · a + ∂u D(X0 ; u0 )u˙ 0 v0 + D(X0 ; u0 )˙v0 · a 2   = J 0 divTT (x0 , t0 ) + b0 · a, which must hold for all a. Clearly, the terms linear and quadratic in a must both vanish, which, because of the quadratic term, leads to ∂u D(X0 ; u0 )u˙ 0 = 0 ⇒ D(X0 ; u0 ) = D(X0 ),

(3.18)

the implication as a result of integration after observing that u˙ 0 is arbitrary. Then from (3.18) and the vanishing of the term linear in a we arrive at   D(X0 )˙v0 = J 0 divTT (x0 , t0 ) + b0 , (3.19) wherein 0 is unrestricted. Of course, (3.16) now has the more refined form ϕ(X ¯ 0 ; u0 , v 0 ) =

 1 ¯ 0 ; u0 , 0), D(X0 )v0 · v0 + ϕ¯ 0 (X0 ; u0 ), ϕ¯ 0 (X0 ; u0 ) := ϕ(X 2

(3.20)

which, again, is to hold for all X0 ∈ B and for all u0 and v0 . Noting that (X0 , t0 ) is arbitrary with X0 ∈ B, t0 ∈ I and x0 = χ(X0 , t0 ), we may substitute the conclusion of (3.20) into the balance of energy (2.2), (2.3a) and (2.4), and write the general balance of energy relative to the fixed frame F in the form

  1 d ˆ + D(X)ˆv · vˆ dv(X) = P(P , t) + Q(P , t) ∀P ⊂ B, ∀t ∈ I . (3.21) ϕ¯ 0 (X; u) dt 2 P

Step 4 Suppose P is a process during a time interval I relative to a fixed frame F and that Hypotheses 2.1 and 2.2 hold, and consider a special Galilean change of frame F  → F∗ (t), for which Q(t) = Q and c(t) = 0, where Q ∈ Orth is constant. First, we substitute the conclusion of (3.20) into (2.5) and (2.6a), and use (2.8a) and (2.8b) to rewrite the general balance of energy relative to the special frame F∗ (t). Then, subtracting (3.21), i.e., the general balance of energy relative to the fixed frame F, from this newly constructed balance of energy relative to the special frame F∗ (t), we readily find

  d 1 T ∗ ˆ + (Q D(X)Q − D(X))ˆv · vˆ dv(X) ϕ¯0 (X; uˆ ) − ϕ¯ 0 (X; u) dt 2 P    T = Q bF∗ (t) (b) − b · v dv(x) ∀P ⊂ B, ∀t ∈ I . Pt

Thus, by shifting the time differentiation to the inside of the integration and changing the domain of integration from P to Pt , we may use the arbitrariness of P ⊂ B and t ∈ I to conclude that ∂u ϕ¯ 0 (X; u∗ ) · u˙ ∗ − ∂u ϕ¯ 0 (X; u) · u˙     + (QT D(X)Q − D(X))˙v · v = J QT bF∗ (t) (b) − b · v

(3.22)

for all X ∈ B and t ∈ I with x = χ(X, t). Here, it is convenient to recall the notational agreement associated ˙ θ˙ }, because of the special change of frame we with (3.8) and to observe that while u = {F, θ } and u˙ = {F, have ˙ θ˙ }. u∗ = {QF, θ } and u˙ ∗ = {QF,

Energy splitting and the origin of mass density

Now, let 0 be arbitrarily assigned so that P is a process during a time interval I structured according to the Fundamental Axiom 3.1. Then, evaluating (3.22) at (X0 , t0 ) with X0 ∈ B, t0 ∈ I and x0 = χ(X0 , t0 ), we have ∂u ϕ¯ 0 (X0 ; u∗0 ) · u˙ ∗0 − ∂u ϕ¯0 (X0 ; u0 ) · u˙ 0     + (QT D(X0 )Q − D(X0 ))˙v0 · v0 = J 0 QT bF∗ (t0 ) (b0 ) − b0 · v0 ,

(3.23)

where u∗0 = {QF0 , θ 0 } and u˙ ∗0 = {QF˙ 0 , θ˙ 0 }. Also, according to (3.9), together with (3.19) and (3.20), we have   ∂u ϕ¯ 0 (X0 ; u0 ) · u˙ 0 = J 0 TT (x0 , t0 ) · grad v(x0 , t0 ) + divq(x0 , t0 ) + r 0 .

(3.24)

Note that because of the identity F˙ = (grad v)F, any particular assignment of u0 and u˙ 0 in (3.23) and (3.24) implies an assignment of grad v(x0 , t0 ). Later, we shall see that (3.24) is the basis for the local form of the balance of energy. Returning to (3.23) and choosing 0 such that u˙ 0 = 0, which implies u˙ ∗0 = 0, we readily conclude from the arbitrariness of v0 and v˙ 0 that QT D(X0 )Q = D(X0 ) for all Q ∈ Orth. Thus, we reach the representation formula D(X0 ) = ρ0 (X0 )1 ∀ X0 ∈ B,

(3.25)

where, according to (3.20) and Hypothesis 2.3, ρ0 (·) > 0 is a positive scalar function. For this special change of frame, we also see that bF∗ (t) (b0 ) = Qb0 , but this is not of great interest here. At this point, (3.23) is greatly reduced in form and the left-hand side now contains only the first two terms—the remaining terms canceling out because of the conclusions just reached. To consider its remaining consequences, we find it convenient to use the abridged (and slightly abused) notation, first introduced in the statement of Theorem 3.1(II), i.e., ϕ¯0 (X0 ; A, α) := ϕ¯ 0 (X0 ; s), = {A, α} with A ∈ Inv and α > 0. Thus, we may write ϕ¯0 (X0 and then represent the reduced (3.23) as  T  Q ∂F ϕ¯0 (X0 ; QF0 , θ 0 ) − ∂F ϕ¯ 0 (X0 ; F0 , θ 0 ) · F˙ 0   + ∂θ ϕ¯0 (X0 ; QF0 , θ 0 ) − ∂θ ϕ¯0 (X0 ; F0 , θ 0 ) θ˙ 0 = 0,

where, we recall10 , s

(3.26) ; u∗0 )

= ϕ¯0 (X0

; QF0 , θ 0 ),

(3.27)

which is supposed to hold for all u0 and u˙ 0 , i.e., all F0 ∈ Lin+ , θ 0 > 0, F˙ 0 ∈ Lin and θ˙ 0 ∈ R, and all Q ∈ Orth. Thus, by first choosing F˙ 0 = 0 we see from what remains that the difference ϕ¯ 0 (X0 ; QF0 , θ 0 )− ϕ¯ 0 (X0 ; F0 , θ 0 ) is independent of θ 0 and may depend only on Q and F0 . Then, introducing σ¯ (X0 ; Q, F0 ) := ϕ¯ 0 (X0 ; QF0 , θ 0 ) − ϕ¯ 0 (X0 ; F0 , θ 0 ), and noting the identity ∂F σ¯ (X0 ; Q, F0 ) = QT ∂F ϕ¯ 0 (X0 ; QF0 , θ 0 ) − ∂F ϕ¯ 0 (X0 ; F0 , θ 0 ), which holds for all F0 ∈ Lin+ and θ 0 > 0, we find from (3.27) that σ¯ (X0 ; Q, F0 ) is independent of F0 . Thus, canceling this dependence we conclude that ϕ¯0 (X0 ; QF0 , θ 0 ) − ϕ¯ 0 (X0 ; F0 , θ 0 ) = σ¯ (X0 ; Q), which is to hold for all F0 ∈ Lin+ , θ 0 > 0 and Q ∈ Orth. In particular, this must hold for all Q ∈ Orth+ , and, noting that we have assumed ϕ¯ 0 (X0 ; ·, θ 0 ) to be differentiable, so, in particular, σ¯ (X0 ; ·) is continuous, it follows by an argument of Noll [4, p. 43], which uses the compactness of the proper orthogonal group Orth+ , that σ¯ (X0 ; Q) = 0. This confirms that ϕ¯0 (X0 ; ·, θ 0 ) is frame indifferent in the sense that ϕ¯ 0 (X0 ; F0 , θ 0 ) = ϕ¯ 0 (X0 ; QF0 , θ 0 ) ∀ F0 ∈ Lin+ , Q ∈ Orth+ . 10

See footnote 4.

(3.28)

R. Fosdick, J. Serrin

Step 5 We are now in the position to complete the proof of all but part (IV) of Theorem 3.1. Toward this end, let us first define the scalar field ρ = ρ(x, t) with ρ(X, ˆ t) = ρ(x, t)|x=χ (X,t) such that ρˆ := and observe that d dt

ρ0 > 0, J

(3.29)

 ρ dv(x) = 0 ∀P ⊂ B, ∀t ∈ I . Pt

This confirms (3.2) and proves Theorem 3.1(I). Now, let us recall that in the first four steps of this proof arguments were made and conclusions were drawn at arbitrarily selected X0 ∈ B and t0 ∈ I , with x0 = χ(X0 , t0 ). Thus, it is permissible, and now convenient, to make the replacements X0 → X, t0 → t and x0 = χ(X0 , t0 ) → x = χ(X, t) in all of these prior conclusions. Respecting this replacement, we then conclude from (3.19), (3.25) and (3.29) that ρ v˙ = divTT + b ∀x ∈ Bt , ∀t ∈ I ,

(3.30)

which represents the local form of the balance of linear momentum, and, with (3.6), confirms the global form of the balance of linear momentum (3.4) in Theorem 3.1(III). Again respecting this replacement, we first define the scalar field ε = ε(x, t) with εˆ (X, t) = ε(x, t)|x=χ (X,t) such that ˆ := εˆ = ε¯ (X; u)

1 1 ˆ θˆ ) =: ε¯ (X; F, ˆ θˆ ), ˆ = ϕ¯ 0 (X; F, ϕ¯ 0 (X; u) ρ0 ρ0

(3.31)

wherein we have used the notational agreement (3.26). Then, we see, using (3.29), that (3.24) may be written as ρ ε˙ = TT · grad v + divq + r ∀x ∈ Bt , ∀t ∈ I ,

(3.32)

which is the classical local form of the balance of energy. Moreover, we see, using (3.25), (3.29) and (3.31), that (3.21) may be written as

 d 1 (3.33) ρε + ρv · v dv(x) = P(P , t) + Q(P , t) ∀P ⊂ B, ∀t ∈ I , dt 2 Pt

which confirms the classical global form of the balance of energy (3.3a) of Theorem 3.1(II). In addition, (3.31) and (3.28) show that ε¯ (X; F, θ ) is frame indifferent, as claimed in Theorem 3.1(II), to complete its proof. Step 6 Finally, we turn to a proof of the balance of angular momentum (3.5) of Theorem 3.1(IV). Observe that the conclusions (I), (II) and (III) of Theorem 3.1 apply to processes P during a time interval I relative to the fixed frame F, and that to reach these conclusions we did not use the full objective requirements in Hypothesis 2.2 for arbitrary changes of frame F  → F∗ (t). Observe also that while Hypothesis 2.1 requires the balance of energy to be form invariant under the change of frame F  → F∗ (t), and, therefore, the balance of energy must hold relative to the moving frame F∗ (t), the Fundamental Axiom 3.1 does not apply to a moving frame F∗ (t). Thus, we have given no argument to justify a claim that a balance of linear momentum similar to (3.4) must hold for such a frame. To describe the status of the balance laws for processes P during a time interval I relative to the moving frame F∗ (t), and at the same time establish a proof of the balance of angular momentum stated in Theorem 3.1(IV), we find it convenient to introduce ρ ∗ = ρ ∗ (x∗ , t) with ρˆ ∗ (X, t) = ρ ∗ (x∗ , t)|x∗ =Q(t)x+c(t) such that ρˆ ∗ :=

ρ0 ρ0 ρ0 = = = ρ. ˆ | det ∇χ ∗ | | det Q|J J

Then, because of (3.16), (3.25), (3.28) and (3.31), it follows that the balance of energy during a time interval I relative to the frame F∗ (t), i.e., (2.5), (2.6a), (2.6b), (2.7a) and (2.7b), holds with ∗ (P , t) = E ∗ (P , t) + K ∗ (P , t),

Energy splitting and the origin of mass density

where E ∗ (P , t) :=



ρ ∗ ε∗ dv(x∗ ) =

Pt∗



ρ0 εˆ ∗ dv(X),

P

with ε∗ = ε∗ (x∗ , t), εˆ ∗ (X, t) = ε∗ (x∗ , t)|x∗ =χ ∗ (X,t) , and   1 1 ρ ∗ |v∗ |2 dv(x∗ ) = ρ0 |ˆv∗ |2 dv(X). K ∗ (P , t) := 2 2 Pt∗

P

  Here, we also know that εˆ ∗ = εˆ ∗ (X, t) = ε¯ X; Fˆ ∗ (X, t), θˆ ∗ (X, t) , where Fˆ ∗ = QFˆ and θˆ ∗ = θˆ . Then, because of the frame-indifference conclusion of (3.28) and the definition (3.31) we see that E ∗ (P , t) = E(P , t), so that the energy balance relative to the frame F∗ (t) may be written as d d E(P , t) + K ∗ (P , t) = P ∗ (P , t) + Q(P , t) ∀P ⊂ B, ∀t ∈ I , dt dt

(3.34)

where P ∗ (P , t) is defined in (2.7a) and where we have used (2.8b). Our aim now is to use the change of frame F  → F∗ (t) defined earlier by x∗ = Q(t)x + c(t), the objectivity condition t∗ = Q(t)t from Hypothesis 2.2, and the balance of linear momentum (3.4) of Theorem 3.1(III), to reduce (3.34) to its simplest form. As a first step, it is easy to show that ˙ + Qv + c˙ v¯ ∗ = v¯ ∗ (x, t) := v∗ (x∗ , t)|x∗ =Q(t)x+c(t) = Qx

(3.35a)

and to compute from this that ¨ + 2Qv ˙ + Q˙v + c¨ . v˙¯ ∗ = v˙¯ ∗ (x, t) := v˙ ∗ (x∗ , t)|x∗ =Q(t)x+c(t) = Qx Thus, using the transformation formula b∗ = bF∗ (b) from Hypothesis 2.2, we find   ∗ ∗ ∗ ∗ P (P , t) = t · v da(x ) + b∗ · v∗ dv(x∗ ) ∂ Pt∗

⎛

Pt∗

˙ TQ · ⎜ = P(P , t) + Q ⎝







⎜ +QT c˙ · ⎝

tda(x) + ∂ Pt

⎟ x ⊗ bdv(x)⎠

x ⊗ tda(x) +

∂ Pt

⎛

⎞

 ⎞

Pt

⎟ bdv(x)⎠ +

Pt



 bF∗ (b) − Qb · v¯ ∗ dv(x).



Pt

We also find, using the balance of mass (3.2), that   d ∗ ∗ ∗ ∗ ∗ K (P , t) = ρ v˙ · v dv(x ) = ρ v˙¯ ∗ · v¯ ∗ dv(x) dt Pt∗



=

Pt



 ¨ + 2Qv ˙ + Q˙v + c¨ · v¯ ∗ dv(x) ρ Qx

Pt



=

ρ



   ¨ + 2Qv ˙ + c¨ · v¯ ∗ + Q˙v · Qx ˙ + Qv + c˙ dv(x) Qx

Pt

=

   d ¨ + 2Qv ˙ + c¨ · v¯ ∗ dv(x) K (P , t) + ρ Qx dt Pt   d d ˙ TQ · ρx ⊗ vdv(x) + QT c˙ · ρvdv(x). +Q dt dt Pt

Pt

(3.35b)

R. Fosdick, J. Serrin

˙ T (t)Q(t) ∈ Skew. Together, we have Here, we also have used Q(t) ∈ Orth ⇒ Q d d ∗ K (P , t) − P ∗ (P , t) = K (P , t) − P(P , t) dt dt ⎛ ⎛ ⎞⎞    d ⎜ ⎟⎟ ˙ TQ · ⎜ ρx ⊗ vdv(x) − ⎝ x ⊗ tda(x) + x ⊗ bdv(x)⎠⎠ +Q ⎝ dt ⎛ ⎜d +QT c˙ · ⎝ dt  −

Pt



⎛ ⎜ ρvdv(x) − ⎝

Pt

∂ Pt



Pt

 tda(x) +

∂ Pt

⎞⎞

⎟⎟ bdv(x)⎠⎠

Pt

  ¨ + 2Qv ˙ + c¨ ) · v¯ ∗ dv(x). bF∗ (b) − Qb + ρ(Qx



Pt

However, it readily follows from the balance of linear momentum relative to the frame F, i.e., (3.4) of Theorem 3.1, that the third line, here, vanishes and, with the additional aid of the balance of mass and Cauchy’s stress theorem (3.6)1 , that the second line may be reduced to the expression  T ˙ −Q Q · Tdv(x). Pt

Thus, with the balance of energy relative to the frame F, we find that the balance of energy relative to the frame F∗ (t), expressed in (3.34), is reduced to  ˙ T Q · Tdv(x) Q Pt

    ¨ + 2Qv ˙ + c¨ ) · v¯ ∗ dv(x) = 0 ∀P ∈ B, ∀t ∈ I . + bF∗ (b) − Qb + ρ(Qx Pt

˙ − Qv(x, t) Now, by localizing this result at a fixed time t ∈ I and a fixed x ∈ Bt , and choosing c˙ = −Qx ˙ T Q · T(x, t) = 0 for all Q ∈ Orth and Q ˙ such that in (3.35a) so that v¯ ∗ (x, t) = 0, we may conclude that Q ˙ T Q ∈ Skew. This shows that T(x, t) ∈ Sym for all t ∈ I and x ∈ Bt , which together with the balance Q of mass (3.2), Cauchy’s stress theorem (3.6)1 and the balance of linear momentum (3.4) is easily seen to expresses the balance of angular momentum in the form (3.5) of Theorem 3.1(IV). This completes the proof of Theorem 3.1.   We are now left with the residual localized condition    ¨ + 2Qv ˙ + c¨ ) · v¯ ∗ = 0, bF∗ (b) − Qb + ρ(Qx (3.36) which is supposed to hold at all t ∈ I and x ∈ Bt , and for all processes P during a time interval I and for arbitrary changes of frame F  → F∗ (t). We analyze (3.36) and complete the description of the balance laws for processes P during I relative to the moving frame F∗ (t) in the following section. 4 Observations concerning the moving frame F∗ (t) With the aid of x∗ = Q(t)x + c(t) and (3.35a), and returning to the notation b∗ = bF∗ (b) of Hypothesis 2.2, we may write (3.36) equivalently as      b∗ − Qb − ρ ∗ ω˙ × (x∗ − c) − ω × ω × (x∗ − c) + 2ω × (v∗ − c˙ ) + c¨ · v∗ = 0

Energy splitting and the origin of mass density

for all t ∈ I and all x∗ ∈ Bt∗ , where b = b(x, t)|x=QT (t)(x∗ −c(t)) and ω = ω(t), being the axial vector associated T (t) ∈ Skew, represents the angular velocity of the frame F∗ (t). Because this is supposed to hold ˙ with Q(t)Q for arbitrary v∗ at (x∗ , t), it follows that there is a vector field Ω ∗ = Ω(x∗ , t) such that     b∗ = Qb + ρ ∗ ω˙ × (x∗ − c) − ω × ω × (x∗ − c) + 2ω × (v∗ − c˙ ) + c¨ + Ω ∗ × v∗ in Bt∗ , for all t ∈ I . However, we know that while b∗ = b∗ (x∗ , t) depends on the frame F∗ (t), by the linear function structure of the power P ∗ (P , t) in (2.7a), it is independent of v∗ . Thus, we may conclude that     b∗ = Qb + ρ ∗ ω˙ × (x∗ − c) − ω × ω × (x∗ − c) − 2ω × c˙ + c¨ in Bt∗ ∀t ∈ I , (4.1) and Ω ∗ = Ω ∗ (x∗ , t) = −2ρ ∗ ω(t),

(4.2)

the former of which explicitly characterizes how b∗ = bF∗ (b) depends on b and the frame F∗ (t). It is now easy to develop the form of the balance of linear momentum relative to the moving frame F∗ (t). Toward this end, it is straightforward to show, using the balance of linear momentum relative to the frame F in (3.4), and the balance of mass in (3.2), that ⎛ ⎞    d ⎜ ⎟ ρ ∗ v∗ dv(x∗ ) − ⎝ t∗ da(x∗ ) + b∗ dv(x∗ )⎠ dt Pt∗

=

d dt

d = dt

 Pt



Pt



∂ Pt∗



Pt∗

˙ + Qv + c˙ dv(x) − Q ρ Qx 





 tda(x) −

∂ Pt

˙ + Qv + c˙ dv(x) − Q d ρ Qx dt

bF∗ (b)dv(x) Pt





ρvdv(x) − Pt

    ¨ + 2Qv ˙ + c¨ ) dv(x) bF∗ (b) − Qb + ρ(Qx =−

 bF∗ (b) − Qb dv(x)



Pt

Pt

   =− b∗ − Qb − ρ ∗ ω˙ × (x∗ − c) Pt ∗

   −ω × ω × (x∗ − c) + 2ω × (v∗ − c˙ ) + c¨ dv(x∗ )  Ω ∗ × v∗ dv(x∗ ) =− 

Pt ∗

2ρ ∗ ω × v∗ dv(x∗ ) ∀P ∈ B, ∀t ∈ I .

= Pt ∗

Thus, the balance of linear momentum relative to the frame F∗ (t) has the form    d ρ ∗ v∗ dv(x∗ ) = t∗ da(x∗ ) + f ∗ dv(x∗ ) ∀P ∈ B, ∀t ∈ I , dt Pt∗

∂ Pt∗

(4.3)

Pt∗

where, according to (4.1), f ∗ := b∗ + 2ρ ∗ ω × v∗     = Qb + ρ ∗ ω˙ × (x∗ − c) − ω × ω × (x∗ − c) + 2ω × (v∗ − c˙ ) + c¨ .

(4.4)

R. Fosdick, J. Serrin

Note that f ∗ is not only frame dependent, but also it depends on the particle velocity, and that it is f ∗ , rather than b∗ , that naturally enters the balance of linear momentum (4.3) as an effective body force field relative to a moving frame. This accounts for the so-called Coriolis effect. Finally, using the objectivity requirement t∗ = Qt of Hypothesis 2.2 coupled with t = TT n from (3.6) and n∗ = Qn, we find that t∗ = T∗T n∗ where T∗ = QTQT . Then, T ∈ Sym ⇒ T∗ ∈ Sym so that the balance of angular momentum relative to F∗ (t) also holds. Note that b∗ in the expression for the mechanical power supply P ∗ (P , t) of (2.7a) may be formally replaced by f ∗ defined in (4.4) because the difference, i.e., the so-called Coriolis term, does not contribute to the power supply. 5 Concluding statement The concepts of a fixed frame, motion, force, power, heating, intrinsic energy and a form invariant energy balance law11 under a change of frame have been introduced as fundamental. Based upon Hypotheses 2.1–2.3 and the Fundamental Axiom, we have shown that the local intrinsic energy for a particle of the body must split into the sum of an internal energy depending only upon the particle and the state u of the particle, and a kinetic energy which depends only upon the particle and the particle velocity v, and is proportional to |v|2 . The existence of mass and its balance is shown to be intimately tied to this split. In this work, mass as an inertial property of a body is derived and all classical balance equations of continuum mechanics are obtained from non-relativistic space-time invariance considerations. Herein, we assumed that the radiant heating field r is objective under a change of frame, but allowed the body force to be frame dependent because we wished to emphasize the origin of the basic equations and logical connections within the subject of continuum mechanics governed by a fundamental form of the balance of energy. While we included temperature and the thermodynamic concept of heat in our development, we did not introduce the notion of entropy and investigate its related thermodynamic consequences. We anticipate that a frame-dependent radiant heating can be considered, with similar conclusions as we have reported here. Our thoughts about this matter indicate that a statement of the second law of thermodynamics, embodied in, say, the Clausius-Duhem inequality, and its form invariance under a change of frame, will be an essential element in this development. Acknowledgments We thank Eliot Fried for his helpful comments on an earlier draft of this work.

References 1. 2. 3. 4. 5.

Green, A.E., Rivlin, R.S.: On Cauchy’s equations of motion. Z. Angew. Math. Phys. 15, 290–292 (1964) Green, A.E., Rivlin, R.S.: Multipolar continuum mechanics. Arch. Ration. Mech. Anal. 17, 113–147 (1964) Jammer, M.: Concepts of Mass in Classical and Modern Physics. Harvard University Press, Cambridge (1961) Noll, W.: On the continuity of the solid and fluid states. J. Ration. Mech. Anal. 4, 3–81 (1955) Noll, W.: La mécanique classique, basée sur un axiome d’objectivité. La méthode axiomatique dans les mécaniques classiques et nouvelles. Paris, pp. 47–63 (1963) 6. Serrin, J.: The equations of continuum mechanics as a consequence of group invariance. In: Ferrarese, G. (ed.) Advances in Modern Continuum Mechanics, pp. 217–225. Pitagora Editrice, Bologna (1992) 7. Šilhavý, M.: Mass, internal energy, and Cauchy’s equations in frame-indifferent thermodynamics. Arch. Ration. Mech. Anal. 107, 1–22 (1989)

11 In the early developments of mechanics, this was sometimes referred to as a ‘law of causality’, i.e., a statement about the evolutionary behavior of a system. The notions of ‘cause’ and ‘effect’ were debated as well as was the distinction between force and mass. In the present context, the ‘cause’ is identified as the right-hand side of the balance of energy (2.2)1 , i.e., the power and heating supply. The ‘effect’ is the evolving intrinsic energy of the system on the left-hand side of (2.2)1 .