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Surface plasticity: theory and computation

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Abstract

Surfaces of solids behave differently from the bulk due to different atomic rearrangements and processes such as oxidation or aging. Such behavior can become markedly dominant at the nanoscale due to the large ratio of surface area to bulk volume. The surface elasticity theory (Gurtin and Murdoch in Arch Ration Mech Anal 57(4):291–323, 1975) has proven to be a powerful strategy to capture the size-dependent response of nano-materials. While the surface elasticity theory is well-established to date, surface plasticity still remains elusive and poorly understood. The objective of this contribution is to establish a thermodynamically consistent surface elastoplasticity theory for finite deformations. A phenomenological isotropic plasticity model for the surface is developed based on the postulated elastoplastic multiplicative decomposition of the surface superficial deformation gradient. The non-linear governing equations and the weak forms thereof are derived. The numerical implementation is carried out using the finite element method and the consistent elastoplastic tangent of the surface contribution is derived. Finally, a series of numerical examples provide further insight into the problem and elucidate the key features of the proposed theory.

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Notes

  1. The assumption of isotropy in this work is for the sake of simplicity.

  2. The Lie derivative of a spatial surface tensor field \(\hat{\varvec{f}}(\hat{\varvec{x}}, t)\), relative to a vector field \(\hat{\varvec{v}}\) is obtained by \(\pounds _{\hat{\varvec{v}}} \hat{\varvec{f}} = \hat{\varvec{\varphi }}_* \left( \frac{\mathrm {D}}{\mathrm {D}t} \hat{\varvec{\varphi }}{}^{-1}_* (\hat{\varvec{f}})\right) \), where \(\hat{\varvec{\varphi }}_*^{-1} \) is the surface pull-back and \(\hat{\varvec{\varphi }}_* \) the surface push-forward operators.

  3. Note that the most general set of arguments for the surface free energy contains \(\hat{\varvec{F}}_\mathrm{p}\) and \(\hat{\varvec{F}}\) in Euclidean space. Imposing the invariance under superposed rigid body motion onto the intermediate configuration results in \(\hat{\varvec{C}}_\mathrm{e}\), where \(\hat{\varvec{C}}_\mathrm{e}\) is the elastic right Cauchy–Green strain in the intermediate configuration. Imposing further invariance under superposed rigid body motion on the spatial configuration finally results in \(\hat{\varvec{b}}_\mathrm{e}\), see [59] for further details.

  4. The spatial symmetry operator is , where , for spatial second-order surface tensors. Its Material counterpart is defined as \(\{\hat{\bullet }\}^\mathrm {Sym} = \mathbb {I}^\mathrm {Sym}:\{\hat{\bullet }\}\), where \(\mathbb {I}^\mathrm {Sym}= \frac{1}{2}[\hat{\varvec{I}}\, {\overline{\otimes }}\,\hat{\varvec{I}} + \hat{\varvec{I}}\, {\underline{\otimes }}\,\hat{\varvec{I}}]\).

  5. Note that in Eq. (19) it is assumed that plastic spin \(\hat{\varvec{W}}_\mathrm{p}= \varvec{0}\), thus \(\hat{\varvec{L}}_\mathrm{p}\) is symmetric.

  6. The surface trace operator for spatial second order tensor is defined as \(\widehat{\mathrm {trace}}\{\hat{\bullet }\} = \{\hat{\bullet }\}:\hat{\varvec{i}}\). In the material configuration the surface trace operator is defined correspondingly as \(\widehat{\mathrm {Trace}}\{\hat{\bullet }\} = \{\hat{\bullet }\}:\hat{\varvec{I}}\).

  7. Note that henceforth only the classical example for metal plasticity, i.e. the von Mises-type yield criterion is considered. Thus, we only take into account the simplest plasticity model, i.e. J\(_2\) type flow theory with isotropic hardening to be developed on the surface. This is to motivate a surface elastoplasticity model and examine its computational aspects.

  8. The term volumetric has a different meaning on the surface. A surface volumetric deformation describes a deformation that changes the area. A volumetric deformation in the bulk however changes the volume. Nonetheless, we use the same term for both the bulk and the surface for the sake of simplicity.

  9. We mention the assumptions made for the numerical part of the current manuscript: first, the surface stress response is isotropic. Second, the plastic spin on the surface is assumed to vanish. Third, the main focus here is on metal plasticity meaning that plastic yielding is isochoric, i.e. \(\hat{J}_\mathrm{p}= 1\), which justifies the decoupling of the surface strain energy. Note that the same assumptions are also made for the bulk elastoplasticity.

  10. Alternatively, a formulation adapting the flow rule in [62] to surface plasticity is possible.

  11. This is the case when \(\hat{K}([\hat{F}_\mathrm{p}]_{\tau +1})\) is a non-linear function of \([\hat{F}_\mathrm{p}]_{\tau +1}\) and consequently a non-linear function of \(\Delta \gamma \) since \([\hat{F}_\mathrm{p}]_{\tau +1} = [\hat{F}_\mathrm{p}]_\tau + \sqrt{2/3}\Delta \gamma \).

  12. In the following derivations we drop the super- and sub-index \(\mathrm{trial}\) and \({\tau +1}\) for the sake of brevity.

  13. Due to the negligible energetic contributions of the side surfaces only the energetics of x-y surfaces are taken into account.

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Acknowledgements

The first author gratefully acknowledges the support by the Cluster of Excellence “Engineering of Advanced Materials”.

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Authors and Affiliations

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

    A. Esmaeili & P. Steinmann

  2. Department of Mechanical Engineering, Bilkent University, 06800, Ankara, Turkey

    A. Javili

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  1. A. Esmaeili
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Correspondence to A. Javili.

Appendix A: Convergence behavior

Appendix A: Convergence behavior

In this section we present some data on the convergence behavior of the computational problem at hand. The \(L_2\) norms of the residual of few increments for some of the cases discussed in Sect. 5 are given in Table 3.

Table 3 \(L_2\) norm of the residual for case 4, case 3 and case 6
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Esmaeili, A., Steinmann, P. & Javili, A. Surface plasticity: theory and computation. Comput Mech 62, 617–634 (2018). https://doi.org/10.1007/s00466-017-1517-x

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