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Geometry of logarithmic strain measures in solid mechanics. (English) Zbl 1348.74039

Summary: We consider the two logarithmic strain measures \[ \begin{aligned} \omega_{\mathrm{iso}}&=\|\mathrm{dev}_n\log U\|=\|\mathrm{dev}_n\log\sqrt{F^TF}\|\text{ and } \\ \omega_{\mathrm{vol}}&=|\mathrm{tr}(\log U)=|\mathrm{tr}(\log\sqrt{F^TF})|=|\log(\det U)|,\end{aligned} \] which are isotropic invariants of the Hencky strain tensor \(\log U\), and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group \(\mathrm{GL}(n)\). Here, \(F\) is the deformation gradient, \(U=\sqrt{F^TF}\) is the right Biot-stretch tensor, \(\log\) denotes the principal matrix logarithm, \(\|\cdot\|\) is the Frobenius matrix norm, \(\mathrm{tr}\) is the trace operator and \(\mathrm{dev}_nX=X-\frac{1}{n}\mathrm{tr}(X)\cdot\mathbf 1\) is the \(n\)-dimensional deviator of \(X\in\mathbb R^{n\times n}\). This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor \(\varepsilon=\mathrm{sym}\nabla u\), which is the symmetric part of the displacement gradient \(\nabla u\), and reveals a close geometric relation between the classical quadratic isotropic energy potential \[ \mu\|\mathrm{dev}_n\mathrm{sym}\nabla u\|^2+\frac{\kappa}{2}[\mathrm{tr}(\mathrm{sym}\nabla u)]^2=\mu\|\mathrm{dev}_n\varepsilon\|^2+\frac{\kappa}{2}[\mathrm{tr}(\varepsilon)]^2 \] in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy \[ \mu\|\mathrm{dev}_n\log U\|^2+\frac{\kappa}{2}[\mathrm{tr}(\log U)]^2=\mu\omega_{\mathrm{iso}}^2+\frac{\kappa}{2}\omega_{\mathrm{vol}}^2, \] where \(\mu\) is the shear modulus and \(\kappa\) denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor \(R\), where \(F=RU\) is the polar decomposition of \(F\). We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity.

MSC:

74B10 Linear elasticity with initial stresses
74B20 Nonlinear elasticity

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