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The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(1):67-88; doi:10.1093/qjmam/hbn026
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© The author 2009. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Inverse membrane problems in elasticity

P. Pathmanathan{dagger}

( University of Oxford, Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD )

S. J. Chapman{ddagger}

( University of Oxford, Mathematical Institute, 24–29 St Giles, Oxford OX1 3LB )

D. J. Gavaghan§

( University of Oxford, Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD )

{dagger} Corresponding author <pras{at}comlab.ox.ac.uk>

{ddagger} <chapman{at}maths.ox.ac.uk>

§ <gavaghan{at}comlab.ox.ac.uk>

Received 7 August 2008. Revise 18 November 2008. Accepted 12 December 2008.


  Abstract

The inverse elasticity problem of determining the undeformed, deflated, configuration of a nonlinear elastic membrane, given the deformed configuration enclosing an incompressible fluid under known pressure, is considered. It is shown that, in practical cases, it is enough to determine only the undeformed metric tensor, and it is also shown how the two- and three-dimensional cases are fundamentally different. For the three-dimensional case, we set up and classify the partial differential equations to be solved, prove existence of an undeformed state given an undeformed metric and study the axisymmetric case in detail.


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