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A coordinate-free approach to surface kinematics

Published online by Cambridge University Press:  18 May 2009

A. I. Murdoch
A. I. Murdoch
Affiliation:
Department of MathematicsUniversity of StrathclydeGlasgow G1 1XH Scotland
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A thin three-dimensional material system, such as a thin shell or fluid-fluid interface, is often modelled as a bidimensional continuous body which at any instant “occupies” some geometrical surface. The time evolution of such surfaces is usually described in terms of curvilinear coordinates [2], [4], [6], a procedure which can mask the geometry involved. An alternative, coordinate-free, approach has been employed [1], [3] which patently exhibits the fundamental geometric (and algebraic) aspects of the kinematics of deforming surfaces. The foundations of this approach are presented in Section 2, following introductory remarks on notation and calculus in Euclidean point spaces, and hitherto unpublished results are developed in Section 3. Account is taken both of material and non-material surfaces: in the former case (surface) mass is conserved (this will be true for thin solid shells) while in the latter context mass exchange with contiguous phases is possible (as is to be expected in the case of fluid-fluid interfaces). The results are also pertinent to singular surfaces [2], [6], [7] (such as shock waves) which are not endowed with intrinsic material attributes but rather with discontinuities of bulk quantities.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 32 , Issue 3 , September 1990 , pp. 299 - 307
Copyright
Copyright © Glasgow Mathematical Journal Trust 1990

References

REFERENCES

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