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A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology

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Abstract

A curvilinear-coordinate-based finite element methodology is presented as a basis for a straightforward computational implementation of the theory of surface elasticity that mimics the underlying mathematical and geometrical concepts. An efficient formulation is obtained by adopting the same methodology for both the bulk and the surface. The key steps to evaluate the hyperelastic constitutive relations at the level of the quadrature point in a finite element scheme using this unified approach are provided. The methodology is illustrated through selected numerical examples.

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Notes

  1. The documented program can be found at http://www.cerecam.uct.ac.za/code/surface_energy/doc/html/.

  2. Note that we distinguish between the material, spatial and natural configurations. A line element \(\text{ d }{\varvec{X}}\) in the material configuration is mapped to \(\text{ d }{\varvec{x}}\) in the spatial configuration via the linear map \({\varvec{F}}\) and to \(\text{ d }\varvec{\xi }\) in the natural (reference) configuration via \(\varvec{K}\), see Table 3. The material, spatial and natural configurations on the surface are defined in a near-identical fashion to the bulk, see Table 4.

  3. The routine used, rsgene2D, produces a Gaussian height distribution with an exponential auto-covariance. The input parameters were 100 divisions, a surface length of 2, a root mean square height of 0.05, and an (isotropic) correlation length of \(0.25\).

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Authors and Affiliations

  1. Chair of Applied Mechanics, University of Erlangen–Nuremberg, Egerlandstr. 5, 91058 , Erlangen, Germany

    A. Javili & P. Steinmann

  2. Centre for Research in Computational and Applied Mechanics, University of Cape Town, Rondebosch , 7701, South Africa

    A. McBride & B. D. Reddy

Authors
  1. A. Javili
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  2. A. McBride
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  3. P. Steinmann
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  4. B. D. Reddy
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Correspondence to A. Javili.

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Javili, A., McBride, A., Steinmann, P. et al. A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology. Comput Mech 54, 745–762 (2014). https://doi.org/10.1007/s00466-014-1030-4

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