Curl bounds Grad on SO(3)

Neff, Patrizio; Münch, Ingo von

doi:10.1051/cocv:2007050

Cited by 71 publications

(25 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The last equality suggest that the parameter values β = γ = α could also be an interesting constitutive choice. Inequality (3.6) 2 admits a (surprising) generalization to exact rotations [28]. Considering the deviator, we observe, moreover…”

Section: The Symmetric Nullspacementioning

confidence: 67%

A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy

Neff¹,

Jeong²

2009

Z Angew Math Mech

101

106

View full text Add to dashboard Cite

We discuss a linear Cosserat model with weakest possible constitutive assumptions on the curvature energy still providing for existence, uniqueness, and stability. The assumed curvature energy is the conformally invariant expression μ L 2 c dev sym ∇ axl A 2 , where axl A is the axial vector of the skewsymmetric microrotation A ∈ so (3), dev is the orthogonal projection on the Lie-algebra sl(3) of trace free matrices and sym is the orthogonal projection onto symmetric matrices. It is observed that unphysical singular stiffening for small samples is avoided in torsion and bending while size effects are still present. The number of Cosserat parameters is reduced from six to four: in addition to the (size-independent) classical linear elastic Lamé moduli μ and λ only one Cosserat coupling constant μc > 0 and one length scale parameter Lc > 0 need to be determined. We investigate those deformations not leading to moment stresses for different curvature assumptions and thereby hypothesize a novel invariance principle of linear, isotropic Cauchy elasticity which is extended to the Cosserat and couple-stress (Koiter-Mindlin) model with conformal curvature.

show abstract

Section: The Symmetric Nullspacementioning

confidence: 67%

A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy

Neff¹,

Jeong²

2009

Z Angew Math Mech

101

106

View full text Add to dashboard Cite

show abstract

“…It is important to realize that the microstructure need not conform to the macroscopic deformation which can be experimentally observed. Hence, it is reasonable to model lattice rotations by a suitable independent degree of freedom, e.g., a microrotation field R. Note further that the elastic rotations R e ∈ SO(3) are usually assumed to be reversible, see, e.g., [25,37], whereas the orthogonal factor of the plastic part R p (F p ) ∈ SO (3) is usually assumed to correspond to irreversible plastic rotations. We further make the assumption that the stored energy content of a copper specimen can be well described by three contributions…”

Section: The Strain Energy Density In Isotropic Multiplicative Plastimentioning

confidence: 99%

The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments

Fischle

Neff

Raabe

2017

Z. Angew. Math. Phys.

Self Cite

View full text Add to dashboard Cite

The rotation polar(F ) ∈ SO(3) arises as the unique orthogonal factor of the right polar decomposition F = polar(F ) U of a given invertible matrix F ∈ GL + (3). In the context of nonlinear elasticity Grioli (1940) discovered a geometric variational characterization of polar(F ) as a unique energy-minimizing rotation. In preceding works, we have analyzed a generalization of Grioli's variational approach with weights (material parameters) µ > 0 and µc ≥ 0 (Grioli: µ = µc). The energy subject to minimization coincides with the Cosserat shear-stretch contribution arising in any geometrically nonlinear, isotropic and quadratic Cosserat continuum model formulated in the deformation gradient field F := ∇ϕ : Ω → GL + (3) and the microrotation field R : Ω → SO (3). The corresponding set of non-classical energy-minimizing rotations rpolar ± µ,µc (F ) := arg min R ∈ SO(3) Wµ,µ c (R ; F ) := µ sym(R T F − 1) 2 + µc skew(R T F − 1) 2represents a new relaxed-polar mechanism. Our goal is to motivate this mechanism by presenting it in a relevant setting. To this end, we explicitly construct a deformation mapping ϕnano which models an idealized nanoindentation and compare the corresponding optimal rotation patterns rpolar ± 1,0 (Fnano) with experimentally obtained 3D-EBSD measurements of the disorientation angle of lattice rotations due to a nanoindentation in solid copper. We observe that the non-classical relaxed-polar mechanism can produce interesting counter-rotations. A possible link between Cosserat theory and finite multiplicative plasticity theory on small scales is also explored.

show abstract

“…The arguments to the shear-stretch energy W µ,µc (R ; F ) are the deformation gradient field F := ∇ϕ : Ω → GL + (n) and the microrotation field R : Ω → SO(n) evaluated at a given point of the domain Ω. A full Cosserat continuum model furthermore contains an additional curvature energy term [26] and a volumetric energy term, see, e.g., [21] or [22].…”

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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Curl bounds Grad on SO(3)

Cited by 71 publications

References 27 publications

A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy

A new paradigm: the linear isotropic Cosserat model with conformally invariant curvature energy

The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments

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

Contact Info

Product

Resources

About