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Are Microcontinuum Field Theories of Elasticity Amenable to Experiments? A Review of Some Recent Results

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Differential Geometry and Continuum Mechanics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 137))

Abstract

It is well known that the material behavior at the micro- and even more at the nano-scale is size dependent, which is, for example, reflected in a stiffer elastic response. Thus modeling of micro- and nanoelectromechanical systems should be ready to incorporate size dependency as well. However, the classical Boltzmann continuum fails to reproduce the size effect. In this work special attention is paid to higher gradient theories such as the strain gradient theory (of Mindlin’s form-II), the modified strain gradient theory and the couple stress theory for linear elasticity. In particular, the latter will also be investigated in terms of finite elements. A confrontation to the Cosserat- or micropolar theory, the non-local continuum, the fractional calculus and the surface elasticity is carried out.

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Notes

  1. 1.

    Physical linearity denotes a linear dependency of the stress measures on the strain measures.

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Acknowledgments

The present work is supported by DFG MU 1752/33-1.

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Authors and Affiliations

  1. Berlin University of Technology, Einsteinufer 5, 10587, Berlin, Germany

    Christian Liebold & Wolfgang H. Müller

Authors
  1. Christian Liebold
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  2. Wolfgang H. Müller
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Corresponding author

Correspondence to Christian Liebold .

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Editors and Affiliations

  1. Radcliffe Observatory Quarter, Mathematical Institute, Oxford, United Kingdom

    Gui-Qiang G. Chen

  2. Department of Mathematics and Statistics Livingstone Tower, University of Strathclyde, Glasgow, United Kingdom

    Michael Grinfeld

  3. School of Mathem. and Computer Science, Heriot-Watt University, Edinburgh, United Kingdom

    R. J. Knops

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Liebold, C., Müller, W.H. (2015). Are Microcontinuum Field Theories of Elasticity Amenable to Experiments? A Review of Some Recent Results. In: Chen, GQ., Grinfeld, M., Knops, R. (eds) Differential Geometry and Continuum Mechanics. Springer Proceedings in Mathematics & Statistics, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-319-18573-6_9

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