Geometrical setting of solid mechanics

Arch Rational Mech Anal

Eidel

Martín

2016

We consider the two logarithmic strain measures ω iso = dev n log U = dev n log F T F andwhich 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 GL(n). Here, F is the deformation gradient, U = √ F T F is the right Biot-stretch tensor, log denotes the principal matrix logarithm, . is the Frobenius matrix norm, tr is the trace operator and dev n X = X − 1 n tr(X ) · 1 is the n-dimensional deviator of X ∈ R n×n . This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor ε = sym ∇u, which is the symmetric part of the displacement gradient ∇u, and reveals a close geometric relation between the classical quadratic isotropic energy potentialin linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energywhere μ is the shear modulus and κ denotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonalIn memory of Giuseppe Grioli

Section: )mentioning

confidence: 90%

Geometry of Logarithmic Strain Measures in Solid Mechanics

Arch Rational Mech Anal

Eidel

Martín

2016

“…Rougée [207, pages 131, 302] (see also [82,206] and later extensions by Fiala [73,74,75,76]) identifies Hencky's logarithmic strain measure 2 log U = log C as having (as its Frobenius tensor norm) the length of a geodesic joining two metric states: he endows the set of positive definite matrices PSym(3) (which is not a Lie-group w.r.t. matrix multiplication) with a Riemannian structure (see also [31,157,156]).…”

Section: Previous Work In the Spirit Of Our Investigationmentioning

confidence: 99%

The Exponentiated Hencky-Logarithmic Strain Energy. Part I: Constitutive Issues and Rank-One Convexity

Ghiba

Lankeit

2015

J Elast

126

We investigate a family of isotropic volumetric-isochoric decoupled strain energiesbased on the Hencky-logarithmic (true, natural) strain tensor log U , where µ > 0 is the infinitesimal shear modulus, κ = 2µ+3λ 3 > 0 is the infinitesimal bulk modulus with λ the first Lamé constant, k, k are dimensionless parameters, F = ∇ϕ is the gradient of deformation, U = √ F T F is the right stretch tensor and devn log U = log U − 1 n tr(log U ) • 1 1 is the deviatoric part of the strain tensor log U . For small elastic strains, W eH approximates the classical quadratic Hencky strain energywhich is not everywhere rank-one convex. In plane elastostatics, i.e. n = 2, we prove the everywhere rankone convexity of the proposed family W eH , for k ≥ 1 4 and k ≥ 1 8 . Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for n = 2, 3 and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family W eH is not preserved in dimension n = 3 and that the energiesare not rank-one convex.

“…The convex cone Sym + (n) is frequently also viewed as a Riemannian manifold endowed with the Riemannian metric [6,11,20,21,30]…”

Section: Application To Geodesic Distance On Sym + (N)mentioning

confidence: 99%

The sum of squared logarithms inequality in arbitrary dimensions

Borisov

Proc Appl Math and Mech

Sra

et al. 2016

We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ ℝn whose elementary symmetric polynomials satisfy ek(x) ≤ ek(y) (for 1 ≤ k < n) and en(x) = en(y) , the inequality ∑i(log xi)2 ≤ ∑i(log yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M ⊆ ℂn → ℝ with f(z) = ∑i(log zi)2 has nonnegative partial derivatives with respect to the elementary symmetric polynomials of z. We conclude by providing applications and wider connections of the SSLI. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)