Geometry of Logarithmic Strain Measures in Solid Mechanics

By Patrizio Neff, Bernhard Eidel, Robert J. Martin. 2015

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AbstractWe consider the two logarithmic strain measures $$\begin{array}{ll} {\omega_{\mathrm{iso}}} = ||{{\mathrm dev}_n {\mathrm log} U} || = ||{{\mathrm dev}_n {\mathrm log} \sqrt{F^TF}}|| \quad \text{ and } \quad \\ {\omega_{\mathrm{vol}}} = |{{\mathrm tr}({\mathrm log} U)} = |{{\mathrm tr}({\mathrm log}\sqrt{F^TF})}| = |{\mathrm log}({\mathrm det} U)|\,,\end{array}$$ωiso=||devnlogU||=||devnlogFTF||andωvol=|tr(logU)=|tr(logFTF)|=|log(detU)|, which are isotropic invariants of the Hencky strain tensor $${\log U}$$logU, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group $${{\rm GL}(n)}$$GL(n). Here, $${F}$$F is the deformation gradient, $${U=\sqrt{F^TF}}$$U=FTF is the right Biot-stretch tensor, $${\log}$$log denotes the principal matrix logarithm, $${\| \cdot \|}$$‖·‖ is the Frobenius matrix norm, $${\rm tr}$$tr is the trace operator and $${{\text dev}_n X = X- \frac{1}{n} \,{\text tr}(X)\cdot {\mathbb{1}}}$$devnX=X-1ntr(X)·1 is the $${n}$$n-dimensional deviator of $${X\in{\mathbb {R}}^{n \times n}}$$X∈Rn×n. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor $${\varepsilon={\text sym}\nabla u}$$ε=sym∇u, which is the symmetric part of the displacement gradient $${\nabla u}$$∇u, and reveals a close geometric relation between the classical quadratic isotropic energy potential $$\mu {\| {\text dev}_n {\text sym} \nabla u \|}^2 + \frac{\kappa}{2}{[{\text tr}({\text sym} \nabla u)]}^2 = \mu {\| {\text dev}_n \varepsilon \|}^2 + \frac{\kappa}{2} {[{\text tr} (\varepsilon)]}^2$$μ‖devnsym∇u‖2+κ2[tr(sym∇u)]2=μ‖devnε‖2+κ2[tr(ε)]2in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy $$\mu {\| {\text dev}_n log U \|}^2 + \frac{\kappa}{2}{[{\text tr}(log U)]}^2 = \mu {\omega_{{\text iso}}^2} + \frac{\kappa}{2}{\omega_{{\text vol}}^2},$$μ‖devnlogU‖2+κ2[tr(logU)]2=μωiso2+κ2ωvol2,where $${\mu}$$μ is the shear modulus and $${\kappa}$$κ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor $${R}$$R, where $${F=RU}$$F=RU is the polar decomposition of $${F}$$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.

Subfields: Biomedical Engineering, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/adbd3d419b422e2987fed881e7bd19a4d0727533