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Review Article

Extending the Transport Theorem to Rough Domains of Integration

[+] Author and Article Information
Brian Seguin

Postdoctoral Fellow
Division of Mathematics,
University of Dundee,
Dundee DD1 4HN, Scotland, UK
e-mail: bseguin@maths.dundee.ac.uk

Denis F. Hinz

Graduate Student
Department of Mechanical Engineering,
McGill University,
Montréal, PQ H3A 0C3, Canada
e-mail: denis.hinz@mail.mcgill.ca

Eliot Fried

Professor
Mathematical Soft Matter Unit,
Okinawa Institute of Science and Technology,
1919-1 Tancha, Onna-son, Kunigami-gun,
Okinawa, Japan 904-0495
e-mail: eliot.fried@oist.jp

For a discussion of these drawbacks, see the introduction of Harrison [11].

Also known as a k-dimensional volume element.

The support of a chain is always a compact set.

The normal time-derivative, under a different name and in a more specific setting, was first introduced by Thomas [18].

This norm is mentioned very briefly, without details, in Sec. 3.

These are more suitable since they are, for example, stable under interactions.

Manuscript received April 4, 2013; final manuscript received December 22, 2013; published online May 29, 2014. Assoc. Editor: Jörg Schumacher.

Appl. Mech. Rev 66(5), 050802 (May 29, 2014) (16 pages) Paper No: AMR-13-1022; doi: 10.1115/1.4026910 History: Received April 04, 2013; Revised December 22, 2013

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.

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References

Figures

Grahic Jump Location
Fig. 1

The curve γ and an approximation of it by a Dirac chain. (a) The curve γ. (b) A depiction of A3, which approximates γ.

Grahic Jump Location
Fig. 2

(a) Depiction of Δu(p;α). The arrow on the right-hand side illustrates –(p; α), the opposite of the original chain (p; α), and the arrow on the left-hand side illustrates (p+u;α), which is a translation of (p; α). (b) Depiction of ΔvΔu(p;α). The arrow on the upper right depicts the original chain (p; α) and the other arrows illustrate chains obtained from this one by translation and inversion.

Grahic Jump Location
Fig. 3

Graph of h˜ at various times. These graphs also appear in Seguin and Fried [10]. (a) Graph of h˜(·,1/4). (b) Graph of h˜(·,3/5).

Grahic Jump Location
Fig. 4

Convergence of the rate of change of area as given by the transport identity in the right-hand side of Eq. (104). (a) Plot of the evolution of A·TT(n) for different values of n, which correspond to the different approximations of the right-hand side of Eq. (104). (b) Convergence of the rate of change of the area A·TT for different snapshots in time. Notice the logarithmic abscissa in (b).

Grahic Jump Location
Fig. 5

Convergence of the rate of change of the area for different snapshots in time using (a) Riemann sums and (b) Simpson's rule. Notice the logarithmic abscissas in (a) and (b).

Grahic Jump Location
Fig. 6

Comparison of A·TT(8),A·Rie,c(8), and A·Sim,c(8) with A·TT(16).

Grahic Jump Location
Fig. 7

(a) Rate of change of the circulation computed with the transport identity (117) for different n with dimensionless integration time step Δτ = 0.001, dimensionless kinematic viscosity ν = 1, and dimensionless circulation Γ = 10. The inset in (a) shows a detailed view of the region around the peak of maximal rate of change of the circulation. (b) Original fractal curve at τ = 0 and deformed fractal curve at τ = 2 for n = 8 and Γ = 10 along with streamlines of the velocity field of the Lamb–Oseen [25, 26] vortex (dashed lines).

Grahic Jump Location
Fig. 8

Snapshots of a cross section of the domain D˜R in the (ξ, ζ)-plane at τ = 0.25. (a) The entire cross section [–1, 1] × [–1, 1]. (b) A detailed view [−0.25, −0.15] × [−0.25, −0.15] of the cross section showing the crack tip.

Grahic Jump Location
Fig. 9

Comparison of ˜·ETT(n) over nondimensional time τ for different resolutions n

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