Geometry of Logarithmic Strain Measures in Solid Mechanics

Abstract

Abstract We consider the two logarithmic strain measures $$\begin{array}{ll} {\omega_{\mathrm{iso}}} = ||{{\mathrm dev}_n {\mathrm log} U} || = ||{{\mathrm dev}_n {\mathrm log} \sqrt{F^TF}}|| \quad \text{ and } \quad \\ {\omega_{\mathrm{vol}}} = |{{\mathrm tr}({\mathrm log} U)} = |{{\mathrm tr}({\mathrm log}\sqrt{F^TF})}| = |{\mathrm log}({\mathrm det} U)|\,,\end{array}$$ ω iso = | | dev n log U | | = | | dev n log F T F | | and ω vol = | tr ( log U ) = | tr ( log F T F ) | = | log ( det U ) | , which are isotropic invariants of the Hencky strain tensor $${\log U}$$ log U , 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 = F T F 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 $${{\text dev}_n X = X- \frac{1}{n} \,{\text tr}(X)\cdot {\mathbb{1}}}$$ dev n X = X - 1 n tr ( X ) · 1 is the $${n}$$ n -dimensional deviator of $${X\in{\mathbb {R}}^{n \times n}}$$ X ∈ R n × 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 ‖ dev n sym ∇ u ‖ 2 + κ 2 [ tr ( sym ∇ u ) ] 2 = μ ‖ dev n ε ‖ 2 + κ 2 [ tr ( ε ) ] 2 in 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 ‖ dev n log U ‖ 2 + κ 2 [ tr ( log U ) ] 2 = μ ω iso 2 + κ 2 ω vol 2 , 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 = R U 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.