Transport theorems, such as that named after Reynolds, are an important tool in the field of continuum physics. Recently, Seguin and Fried used Harrison's theory of differential chains to establish a transport theorem valid for evolving domains that may become irregular. Evolving irregular domains occur in many different physical settings, such as phase transitions or fracture. Here, emphasizing concepts over technicalities, we present Harrison's theory of differential chains and the results of Seguin and Fried in a way meant to be accessible to researchers in continuum physics. We also show how the transport theorem applies to three concrete examples and approximate the resulting terms numerically. Furthermore, we discuss how the transport theorem might be used to weaken certain basic assumptions underlying the description of continua and the challenges associated with doing so.

References

References
1.
Reynolds
,
O.
,
1903
,
The Sub-Mechanics of the Universe
,
Cambridge University Press
,
Cambridge, UK
.
2.
Gurtin
,
M. E.
,
Fried
,
E.
, and
Anand
,
L.
,
2010
,
The Mechanics and Thermodynamics of Continua
,
Cambridge University Press
,
Cambridge, UK
.
3.
Lee
,
J. M.
,
2002
,
Introduction to Smooth Manifolds
,
Springer
,
New York, NY
.
4.
Betounes
,
D. E.
,
1986
, “
Kinematics of Submanifolds and the Mean Curvature Normal
,”
Arch. Rational Mech. Anal.
,
96
(
1
), pp.
1
27
.10.1007/BF00251411
5.
Gurtin
,
M. E.
,
Struthers
,
A.
, and
Williams
,
W. O.
,
1989
, “
A Transport Theorem for Moving Interfaces
,”
Quart. Appl. Math.
,
47
(
4
), pp.
773
777
.
6.
Gurtin
,
M. E.
,
2000
,
Configurational Forces as Basic Concepts of Continuum Physics
,
Springer
,
New York, NY
.
7.
Catalan
,
G.
,
Seidel
,
J.
,
Ramesh
,
R.
, and
Scott
,
J. F.
,
2012
, “
Domain Wall NanoElectronics
,”
Rev. Mod. Phys.
,
84
(
1
), pp.
119
156
.10.1103/RevModPhys.84.119
8.
Stanley
,
H. E.
,
1992
, “
Fractal Landscapes in Physics and Biology
,”
Physica A
,
186
(
1–2
), pp.
1
32
.10.1016/0378-4371(92)90362-T
9.
Bucur
,
D.
, and
Buttazzo
,
G.
,
2005
,
Variational Methods in Shape Optimization Problems
(Progress in Nonlinear Differential Equations and Their Applications 65
),
Birkhäuser, Boston, MA
.
10.
Seguin
,
B.
, and
Fried
,
E.
,
2014
, “
Roughening It—Evolving Irregular Domains and Transport Theorems
,”
Math. Models Methods Appl. Sci.
,
24
(2014),
1729
1779
.10.1142/S0218202514500067
11.
Harrison
,
J.
,
2014
, “
Operator Calculus of Differential Chains and Differential Forms
,”
J. Geom. Anal.
(accepted ms published online).10.1007/s12220-013-9433-6
12.
Marzocchi
,
A.
,
2005
,
Singular Stresses and Nonsmooth Boundaries in Continuum Mechanics
,
Lecture notes from a summer school in Ravello, Italy
.
13.
Rodnay
,
G.
, and
Segev
,
R.
,
2003
, “
Cauchy's Flux Theorem in Light of Geometric Integration Theory
,”
J. Elasticity
,
71
(
1–3
), pp.
183
203
.10.1023/B:ELAS.0000005545.46932.08
14.
Whitney
,
H.
,
1957
,
Geometric Integration Theory
,
Princeton University Press
,
Princeton, NJ
.
15.
Harrison
,
J.
,
1999
, “
Flux Across Nonsmooth Boundaries and Fractal Gauss/Green/Stokes theorems
,”
J. Phys. A: Math. General
,
32
(
28
), pp.
5317
5327
.10.1088/0305-4470/32/28/310
16.
Federer
,
H.
,
1969
,
Geometric Measure Theory
,
Springer
,
New York, NY
.
17.
Flanders
,
H.
,
1973
, “
Differentiating Under the Integral Sign
,”
Am. Math. Monthly
,
80
(
6
), pp.
615
627
.10.2307/2319163
18.
Thomas
,
T. Y.
,
1957
, “
Extended Compatibility Conditions for the Study of Surfaces of Discontinuity in Continuum Mechanics
,”
J. Math. Mech.
,
6
(
3
), pp.
311
322
.
19.
Cermelli
,
P.
,
Fried
,
E.
, and
Gurtin
,
M. E.
,
2005
, “
Transport Relations for Surface Integrals Arising in the Formulation of Balance Laws for Evolving Fluid Interfaces
,”
J. Fluid Mech.
,
544
, pp.
339
351
.10.1017/S0022112005006695
20.
Fosdick
,
R.
, and
Tang
,
H.
,
2009
, “
Surface Transport in Continuum Mechanics
,”
Math. Mech. Solids
,
14
(
6
), pp.
587
598
.10.1177/1081286507087316
21.
Lidström
,
P.
,
2011
, “
Moving Regions in Euclidean Space and Reynolds' Transport Theorem
,”
Math. Mech. Solids
,
16
(
4
), pp.
366
380
.10.1177/1081286510393805
22.
Estrada
,
R.
, and
Kanwal
,
R. P.
,
1991
, “
Non-Classical Derivation of the Transport Theorems for Wave Fronts
,”
J. Math. Anal. Appl.
,
159
(
1
), pp.
290
297
.10.1016/0022-247X(91)90236-S
23.
Angenent
,
S.
, and
Gurtin
,
M. E.
,
1989
, “
Multiphase Thermomechanics With Interfacial Structure 2. Evolution of an Isothermal Interface
,”
Arch. Rational Mech. Anal.
,
108
(
3
), pp.
323
391
.10.1007/BF01041068
24.
Mandelbrot
,
B. B.
,
2002
,
Gaussian Self-Affinity and Fractals
,
Springer-Verlag
,
New York, NY
.
25.
Lamb
,
H.
,
1895
,
Hydrodynamics
,
Cambridge University Press
,
Cambridge, UK
.
26.
Oseen
,
C. W.
,
1911
, “
Über Wirbelbewegung in einer reibenden Flüssigkeit
,”
Arkiv för Matematik, astronomi och Fysik
,
7
(
14
), pp.
1
11
.
27.
Ogden
,
R. W.
,
1984
,
Nonlinear Elastic Deformations
,
Halsted Press/John Wiley & Sons
,
New York, NY
.
28.
Griffith
,
A. A.
,
1921
, “
The Phenomenon of Rupture and Flow in Solids
,”
Philos. Trans. Roy. Soc.
,
221
, pp.
163
198
.10.1098/rsta.1921.0006
29.
Francfort
,
G.
, and
Marigo
,
J. J.
,
1998
, “
Revisiting Brittle Fracture as an Energy Minimization Problem
,”
J. Mech. Phys. Solids
,
46
, pp.
1319
1342
.10.1016/S0022-5096(98)00034-9
30.
Anand
,
L.
,
1996
, “
A Constitutive Model for Compressible Elastomeric Solids
,”
Comput. Mech.
,
18
, pp.
339
355
.10.1007/BF00376130
31.
Gdoutos
,
E. E.
,
Daniel
,
I. M.
, and
Schubel
,
P.
,
2003
, “
Fracture Mechanics of Rubber
,”
Mech., Autom. Control Robot.
,
13
, pp.
497
510
.
32.
Gurtin
,
M. E.
,
Williams
,
W. O.
, and
Ziemer
,
W. P.
,
1986
, “
Geometric Measure Theory and the Axioms of Continuum Thermodynamics
,”
Arch. Rational Mech. Anal.
,
92
(
1
), pp.
1
22
.10.1007/BF00250730
33.
Noll
,
W.
, and
Virga
,
E. G.
,
1988
, “
Fit Regions and Functions of Bounded Variation
,”
Arch. Rational Mech. Anal.
,
102
(
1
), pp.
1
21
.10.1007/BF00250921
34.
Šilhavý
,
M.
,
2006
, “
Fluxes Across Parts of Fractal Boundaries
,”
Milan J. Math.
,
74
(
1
), pp.
1
45
.10.1007/s00032-006-0055-3
35.
Schuricht
,
F.
,
2007
, “
A New Mathematical Foundation for Contact Interactions in Continuum Physics
,”
Arch. Rational Mech. Anal.
,
184
(
3
), pp.
495
551
.10.1007/s00205-006-0032-6
36.
Harrison
,
J.
,
2014
, “
Soap Film Solutions to Plateau's Problem
,”
J. Geom. Anal.
,
24
(1), pp.
271
297
.10.1007/s12220-012-9337-x
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