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Remarks on instabilities and ellipticity for a fiber-reinforced compressible nonlinearly elastic solid under plane deformation

Author(s): J. Merodio; R. W. Ogden
Journal: Quart. Appl. Math. 63 (2005), 325-333.
MSC (2000): Primary 74B20
Posted: February 18, 2005
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Abstract: In this paper we examine loss of ellipticity and associated failure for fiber-reinforced compressible nonlinearly elastic solids under plane deformation. The analysis concerns a material model that consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. We examine reinforcement that introduces additional stiffness under shear deformations. It is shown that loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. More particularly, under uniaxial loading in the fiber direction loss of ellipticity may be associated with different failure mechanisms. Under compression these include fiber kinking and fiber splitting, while under extension the relevant mechanism is matrix failure.

References:

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J. Merodio, R.W. Ogden. Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation. Arch. Mech. 54, 525-552 (2002). MR 1964402 (2004b:74012)

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J. Merodio, R.W. Ogden. Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation. Int. J. Solids Structures 40, 4707-4727 (2003).MR 1994903 (2004f:74013)

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J. Merodio, R.W. Ogden. A note on strong ellipticity for transversely isotropic linearly elastic solids. Q. J. Mech. Appl. Math. 56, 589-591 (2003).MR 2026873

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J. Merodio, R.W. Ogden. Tensile instabilities and ellipticity in fiber-reinforced compressible nonlinearly elastic solids. Int. J. Engng Sci., submitted (2004).

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Additional Information:

J. Merodio
Affiliation: Department of Structural and Mechanical Engineering, E.T.S. de Ingenieros Industriales y de Telecomunicación, University of Cantabria, 39005, Santander, Spain
Email: merodioj@unican.es

R. W. Ogden
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom
Email: rwo@maths.gla.ac.uk

PII: S0033-569X-05-00954-1
Keywords: Nonlinear elasticity, loss of ellipticity, fiber reinforcement, reinforcing models, transverse isotropy
Received by editor(s): June 18, 2004
Posted: February 18, 2005
Copyright of article: Copyright 2005, Brown University


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