Balance Equations
5.1 General Formulation of a Balance Equation
The fundamental laws of continuum mechanics are integral relations expressing conservation or balance of physical quantities: mass conservation, momentum balance, angular momentum balance, energy balance, and so on. These balance laws can refer to amaterial volume, intended as a collection of the same particles, or to a fixed volume.
In rather general terms, a balance law has the following structure:
d dt
Z
c.t /
f.x; t / dcD Z
@c.t /
ˆNdC Z
c.t /
rdc; (5.1)
wherec.t /is an arbitrarymaterial volumeof the configurationC.t /of the system.
The physical meaning of (5.1) is the following: the change that the quantityf exhibits is partially due to the fluxˆacross the boundary ofc.t /and partially due to the source termr:
© Springer Science+Business Media New York 2014
A. Romano, A. Marasco,Continuum Mechanics using MathematicaR, Modeling and Simulation in Science, Engineering and Technology, DOI 10.1007/978-1-4939-1604-7__5
137
138 5 Balance Equations If we refer to afixed volumeV, then in addition to the fluxˆwe need to consider the transport offthroughoutV with velocityv, so that (5.1) assumes the form
d
When the configurationC.t /is subdivided in two regions by a moving singular surface†.t /, then the previous balance law in the integral form (5.1) or (5.2), the derivation rules (4.45) and (4.46), as well as the generalized Gauss theorem (2.46) allow us to derive the following local form of the balance equation and jump condition
@f
@t C r .f˝vCˆ/rD0 inC.t /†.t /;
ŒŒf.vncn/CˆnD0 on †.t /: (5.3) In many circumstances, on the basis of reasonable physical assumptions, we are led to the following integral balance equation:
d
In this case, Gauss’s theorem cannot be applied to the surface integral on the right-hand side; consequently, (5.3) cannot be derived. Anyway, if sis supposed to depend on .x; t / as well as on the unit vector N normal to @c.t / (Cauchy’s hypothesis), thenCauchy’s theoremcan be proved1:
Theorem 5.1. If in the integral momentum balance law the tensor of orderr has the structuresDs.x; t;N/, then there is a tensorˆ.x; t /of order.rC1/such that
sDˆN: (5.5)
Proof. For sake of simplicity but without loss of generality,sis supposed to be a vector. If a material volumec.t /C.t /is considered such thatc.t /\†.t / D ;, then from (5.3) and (4.46) it follows that
Z
1In [43], W. Noll proves that Cauchy’s hypothesis follows from the balance of linear momentum under very general assumptions concerning the form of the function describing the surface sources.
IfAis the area of the surface @cand the functions under the integral are regular, then (5.6) gives
1 A
Z
c
'idcD 1 A
Z
@c
si.N/ d:
By applying the mean-value theorem to the volume integral,2we have 1
Avol.c/'i.i; t /D 1 A
Z
@c
si.N/ d:
wherei,iD1; 2; 3, are the coordinates of a suitable point internal toc. In the limit A!0(or, which is the same forcmerging into its internal pointx), the following result is obtained:
Alim!0
1 A
Z
@c
si.N/ d D0: (5.7)
Now, if'andsare regular functions, it must hold that
s.N/D s.N/; (5.8) as we can prove by applying (5.7) to a small cylinderchaving height2, bases with radiusand by considering the surfacethrough the internal pointx(see Fig.5.1).
Let1,2, and3be three points located on the surface of the cylinder, on the base whose normal isN, and on the other base of normalN, respectively; moreover, let N1the unit normal to the lateral surface ofc.
By applying the mean-value theorem, it follows that Z
@c
si.x;N/ d D23si.1;N1/C2si.2;N/C2si.3;N/;
where the argumentt has been omitted for sake of simplicity.
N
N t
Fig. 5.1 Small cylinder to evaluates(N)
2The mean value theorem applies to each component of the vector function, and not to the vector function itself.
140 5 Balance Equations
Fig. 5.2 Cauchy’s tetrahedron
Since the term23C22represents the areaAof the cylinderc, the previous relation gives (5.7) whenA!0.
Now to prove (5.4), we use Cauchy’s tetrahedron argument. At a pointx 2 C draw a set of rectangular coordinate axes, and for each direction N choose a tetrahedron such that it is bounded by the three coordinate planes throughx and by a fourth plane, at a distancefromx, whose unit outward normal vector isN (see Fig.5.2).
Let0be the area of the surface whose normal isNand leti .i D1; 2; 3/be the area of each of the three right triangles whose inward normal is given by the basis unit vectorei. Sincei D0jNijand the volumecofis equal to0=3, if in (5.7) cdenotes, it follows that
lim!0
1 0.1CP3
iD1jNij/
"
si.1;N/0C X3 iD1
si.j;ej/jNjj0
# D0;
and in addition, ifNj > 0 .j D1; 2; 3/, then from (5.8) it must be that
s.x;N/D X3 jD1
s.x;ej/Nj; (5.9)
which is valid even if some Nj D 0, because of the continuity assumption on s.x;N/.
Conversely, ifN1is negative, then1D 0N1and in (5.9) the terms.x;e1/N1 will replaces.x;e1/N1; assuming the reference.e1;e2;e3/in place of.e1;e2;e3/ still gives (5.9).
If
ˆij Deis.x;ej/, (5.10)
then (5.9) can be written as
si Dsei DˆijNj: (5.11)
SinceNandsare vectors andNis arbitrary, (5.11) requiresˆijto be the components
of a second-order tensor so that (5.5) is proved. ut
In the presence of electromagnetic fields additional balance laws are needed.
They have the following structure:
d dt
Z
S.t /
uNd D Z
@S.t /
adsC Z
S.t /
gNdC Z
.t /
kds; (5.12) whereu;a;g, andkare vector fields,S.t /is any arbitrary material surface, and .t / is the intersection curve ofS.t /with the singular moving surface†.t /:
To obtain the local form in the regionC as well as the jump conditions on†.t /, the rule (4.48) and the Stokes theorem (2.48) have to be applied to the first integral on the right-hand side, so that
@u
@t Cvr uD r .aCvu/Cg inC.t /;
.n ŒŒu.wv/CaCk/†D0 on†.t /: (5.13) Moreover,wNDvN,wnDcn, since .t /belongs to both the material surface S.t /and the singular surface†.t /; therefore, (5.13)2can be written
.ŒŒ.cnvn/uun.wv/CnaCk/†D0: (5.14) In the basis.;n;N/it holds that
wvDŒwvC cnvn
sin2˛ n.cnvn/cos˛ sin2˛ N;
where cos˛DnN, so that (5.13)2assumes the form
.ŒŒ.cnvn/uCnaCk/†CŒŒnu.cnvn/cot˛D0: (5.15)
142 5 Balance Equations Since the material surfaceS.t /is arbitrary, the relation (5.15) must hold for any value of˛. If, as an example, we first suppose that ˛ D =2and that it has an arbitrary value, we derive the following:
ŒŒ.cnvn/uCnaCkD0;
ŒŒun.cnvn/DŒŒuncnŒŒunvnD0: (5.16) In literature, the following relation is usually proposed:
nŒŒu.cnvn/nCaCkD0 on†.t /: (5.17) We note that it is equivalent to (5.16)1if and only if (5.16)2is satisfied.
Some consequences of (5.16)2are the following:
1. ifŒŒunD0, thenŒŒvnD0;
2. if the surface†.t / is material, then (5.16)2 is identically satisfied and (5.16)1
reduces to
nŒŒaCkD0: (5.18)
Remark 5.1. The balance equations presented here are sufficiently general to allow the formulation of all the physical principles considered in this volume. However, they do not work for describing phenomena such as phase transitions (see [34]) or shock waves, with transport of momentum and energy on the wavefront. For these phenomena the reader is referred to Volume II.