Geometry of Logarithmic Strain Measures in Solid Mechanics

Abstract

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,$${F}$$Fis the deformation gradient,$${U=\sqrt{F^TF}}$$U=FTFis the right Biot-stretch tensor,$${\log}$$logdenotes the principal matrix logarithm,$${\| \cdot \|}$$‖·‖is the Frobenius matrix norm,$${\rm tr}$$tris the trace operator and$${{\text dev}_n X = X- \frac{1}{n} \,{\text tr}(X)\cdot {\mathbb{1}}}$$devnX=X-1ntr(X)·1is 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$${\mu}$$μis the shear modulus and$${\kappa}$$κdenotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor$${R}$$R, where$${F=RU}$$F=RUis 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.