An Elementary Proof of the Polar Decomposition Theorem

Martins, L. C.; Podio–Guidugli, Paolo

doi:10.1080/00029890.1980.11995017

Cited by 18 publications

(14 citation statements)

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“…The optimality of the polar factor R p (F ) for n = 3 generalizes to any dimension n ≥ 2, see, e.g., [12], and it is this more general theorem to which we shall refer as Grioli's theorem in this present work. A modern exposition of the original contribution of Grioli has been recently made available in [21].…”

Section: Introductionmentioning

confidence: 93%

“…To state his result, we denote by R p (F ) ∈ SO(n) the unique orthogonal factor of F ∈ GL + (n) in the right polar decomposition F = R p (F ) U (F ) and by U (F ) = R p (F ) T F = √ F T F ∈ PSym(n) the symmetric positive definite Biot stretch tensor. In [6] Grioli proved the special case n = 3 of the following theorem: Theorem 1.1 (Grioli's theorem [6,12,3]). Let n ≥ 2 and X 2 := tr X T X the Frobenius norm.…”

Section: Introductionmentioning

confidence: 99%

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The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

Fischle

Neff

2017

Z Angew Math Mech

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to SO(n). MSC (2010) 15A24, 22L30, 74A30, 74A35, 74B20, 74G05, 74G65, 74N15In any geometrically nonlinear quadratic Cosserat-micropolar extended continuum model formulated in the deformation gradient field F := ∇ϕ : → GL + (n) and the microrotation field R : → SO(n), the shear-stretch energy is necessarily of the formwhere μ > 0 is the Lamé shear modulus and μ c ≥ 0 is the Cosserat couple modulus. In the present contribution, we work towards explicit characterizations of the set of optimal Cosserat microrotations argmin R ∈ SO(n) W μ,μc (R ; F ) as a function of F ∈ GL + (n) and weights μ > 0 and μ c ≥ 0. For n ≥ 2, we prove a parameter reduction lemma which reduces the optimality problem to two limit cases: (μ, μ c ) = (1, 1) and (μ, μ c ) = (1, 0). In contrast to Grioli's theorem, we derive non-classical minimizers for the parameter range μ > μ c ≥ 0 in dimension n = 2. Currently, optimality results for n ≥ 3 are out of reach for us, but we contribute explicit representations for n = 2 which we name rpolar ± μ,μc (F ) ∈ SO(2) and which arise for n = 3 by fixing the rotation axis a priori. Further, we compute the associated reduced energy levels and study the non-classical optimal Cosserat rotations rpolar ± μ,μc (F γ ) for simple planar shear.

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Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

Fischle

Neff

2017

Z Angew Math Mech

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“…Remark 2.2. The minimization property stated in Theorem 2.1 is equivalent to [132] max Q∈SO(n) tr(Q T F ) = max…”

Section: The Euclidean Strain Measure In Nonlinear Isotropic Elasticitymentioning

confidence: 99%

Geometry of Logarithmic Strain Measures in Solid Mechanics

Neff

Eidel

Martín

2016

Arch Rational Mech Anal

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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

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“…The polar factor R p (F ) ∈ SO(n) is the unique energy-minimizing rotation for any given F ∈ GL + (n) in any dimension n ≥ 2, see, e.g., [16]. This optimality property has an interesting geometric interpretation following from the orthogonal invariance of the Frobenius norm…”

Section: Introductionmentioning

confidence: 99%

Grioli’s Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations

Fischle¹,

Neff²

2017

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Let F ∈ GL + (3) and consider the right polar decomposition F = R p (F ) · U into an orthogonal factor R p (F ) ∈ SO(3) and a symmetric, positive definite factor U =This variational characterization of the orthogonal factor R p (F ) ∈ SO(n) holds in any dimension n ≥ 2 (a result due to Martins and Podio-Guidugli). In a similar spirit, we characterize the optimal rotations rpolar µ,µc (F ) := arg minfor given weights µ > 0 and µ c ≥ 0. We identify a classical parameter range µ c ≥ µ > 0 for which Grioli's Theorem is recovered and a non-classical parameter range µ > µ c ≥ 0 giving rise to a new type of globally energy-minimizing rotations which can substantially deviate from R p (F ). In mechanics, the weighted energy subject to minimization appears as the shear-stretch contribution in any geometrically nonlinear, quadratic, and isotropic Cosserat theory.

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An Elementary Proof of the Polar Decomposition Theorem

Cited by 18 publications

References 1 publication

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

The geometrically nonlinear Cosserat micropolar shear–stretch energy. Part I: A general parameter reduction formula and energy‐minimizing microrotations in 2D

Geometry of Logarithmic Strain Measures in Solid Mechanics

Grioli’s Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations

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