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The exponentiated Hencky-logarithmic strain energy: part III—coupling with idealized multiplicative isotropic finite strain plasticity

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Abstract

We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric–isochoric decoupled strain energies

$$F \mapsto W_{\rm eH}(F):= \widehat{W}_{\rm eH}(U) := \left\{ \begin{array}{lll} \frac{\mu}{k}\,e^{k\, \| {\rm dev}_n \log {U}\|^2}+\frac{\kappa}{2\, {\widehat{k}}}\,e^{\widehat{k}\,[ {\rm tr}(\log U)]^2}&\quad \text{if}& \det\, F > 0,\\ + \infty & \quad \text{if} & \det F \leq 0,\end{array} \right.$$

based on the Hencky-logarithmic (true, natural) strain tensor \({\log U}\) . Here, \({\mu > 0}\) is the infinitesimal shear modulus, \({\kappa=\frac{2 \mu+3\lambda}{3} > 0}\) is the infinitesimal bulk modulus with λ the first Lamé constant, \({k,\widehat{k}}\) are additional dimensionless material parameters, \({F=\nabla \varphi}\) is the gradient of deformation, \({U=\sqrt{F^T F}}\) is the right stretch tensor, and dev n \({\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot{\mathbb{1}}}\) is the deviatoric part of the strain tensor \({\log U}\) . Based on the multiplicative decomposition \({F=F_e\, F_p}\) , we couple these energies with some isotropic elasto-plastic flow rules \({F_p\,\frac{\rm d}{{\rm d t}}[F_p^{-1}]\in-\partial \chi({\rm dev}_3 \Sigma_{e})}\) defined in the plastic distortion F p , where \({\partial \chi}\) is the subdifferential of the indicator function \({\chi}\) of the convex elastic domain \({\mathcal{E}_{\rm e}({\Sigma_{e}},\frac{1}{3}{\boldsymbol{\sigma}}_{\mathbf{y}}^2)}\) in the mixed-variant \({\Sigma_{e}}\) -stress space, \({\Sigma_{e}=F_e^T D_{F_e}W_{\rm iso}(F_e)}\) , and \({W_{\rm iso}(F_e)}\) represents the isochoric part of the energy. While \({W_{\rm eH}}\) may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of \({W_{\rm eH}}\) on the one hand and the elastic domain in \({\Sigma_{e}}\) -stress space on the other hand are closely related. Thus, the new formulation remains elliptic in elastic unloading at any given plastic predeformation. In addition, in this domain, the true stress–true strain relation remains monotone, as observed in experiments.

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  1. Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, 45127, Essen, Germany

    Patrizio Neff & Ionel-Dumitrel Ghiba

  2. Department of Mathematics, Alexandru Ioan Cuza University of Iaşi, Blvd. Carol I, no. 11, 700506, Iaşi, Romania

    Ionel-Dumitrel Ghiba

  3. Octav Mayer Institute of Mathematics of the Romanian Academy, Blvd. Carol I, no. 8, Iaşi Branch, 700505, Iaşi, Romania

    Ionel-Dumitrel Ghiba

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  1. Patrizio Neff
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Communicated by Victor Eremeyev, Peter Schiavone and Francesco dell'Isola.

Dedicated to David J. Steigmann, a great scientist and good friend.

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Neff, P., Ghiba, ID. The exponentiated Hencky-logarithmic strain energy: part III—coupling with idealized multiplicative isotropic finite strain plasticity. Continuum Mech. Thermodyn. 28, 477–487 (2016). https://doi.org/10.1007/s00161-015-0449-y

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