Grioli's Theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations
Andreas Fischle Patrizio Neff

The  Chronicles of LITTLE THINGS BCDs III: Gas Clouds in and around Mrk 178, VII Zw 403, and NGC 3738
    Nau Raj Pokhrel
    
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arabic


Abstract



  0.95
    Let F ∈^+(3) and consider the right polar decomposition
     into an orthogonal factor
    (F) ∈(3) and a symmetric, positive definite
    factor U = √(F^TF)∈(3). In 1940 Giuseppe Grioli
    proved that
    
    R ∈ (3)R^TF - 𝕀^2  =  { (F) }  =  R ∈ (3)F - R^2 .

    This variational characterization of the orthogonal
    factor (F) ∈(n) holds in any dimension
    n ≥ 2 (a result due to Martins and Podio-Guidugli).
    In a similar spirit, we characterize the optimal rotations
    
    _μ,μ_c(F)   R ∈ (n) {μ(R^TF - 𝕀)^2
                    + μ_c(R^TF - 𝕀)^2
                   }

    for 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 (F). In mechanics, the weighted energy subject
    to minimization appears as the shear-stretch contribution in
    any geometrically nonlinear, quadratic, and isotropic Cosserat
    theory.






Key words:
Cosserat,
Grioli's theorem,
micropolar,
polar media,
zero Cosserat couple modulus,
euclidean distance to (n)




AMS 2010 subject classification:
  15A24,     22E30,     74A30,     74A35,     74B20,     74G05,     74G55,     74G65,     74N15.   
 




  




§ INTRODUCTION

In 1940 Giuseppe Grioli proved a variational characterization of
the orthogonal factor of the polar decomposition <cit.>.
In order to state this result, let (F) ∈(n) be the
unique rotation characterized as the orthogonal factor of the
right polar decomposition of

    F = (F) U(F),    F ∈^+(n) ,

where U(F) = (F)^TF = √(F^TF)∈(n) denotes
the symmetric positive definite factor (which, in mechanics, is
referred to as the Biot stretch tensor).

Grioli's original result[An exposition of the original
contribution of Grioli in modernized notation has been recently
made available in <cit.>.] is the important special
case of space dimension n = 3 of the following
[Grioli's theorem <cit.>]
  
  Let n ≥ 2 and X^2 X^TX the Frobenius
  norm. Then for any F ∈^+(n), it holds

    R ∈ (n)R^TF - 𝕀^2 = {(F)},
        and thus  min_R ∈ (n)R^TF - 𝕀^2 = U - 𝕀^2 .


The polar factor (F) ∈(n) is the unique energy-minimizing
rotation for any given F ∈^+(n) in any dimension n ≥ 2,
see, e.g., <cit.>. This optimality property has an interesting
geometric interpretation following from the orthogonal invariance of the
Frobenius norm

    R^TF - 𝕀^2 = F - R^2 = ^2_ euclid(F, R)

which reveals a connection to the problem class of matrix distance
(or nearness) problems. In elasticity, a distance of a deformation
gradient (jacobian matrix) F ∇φ∈^+(n) to
a rotation (n) is of interest as a measure for the energy
induced by local changes in length.

In this contribution, we consider a weighted analog of Grioli's
theorem motivated by Cosserat theory and present the energy-minimizing
(optimal) rotations characterized by
[Weighted optimality]
Let n ≥ 2. Compute the set of optimal rotations


    R ∈ (n)(R ;F)R ∈ (n){μ (R^TF - 𝕀)^2
         + μ_c (R^TF - 𝕀)^2}

for given F ∈^+(n) and weights μ > 0,μ_c ≥ 0.
Here, (X) 1/2(X + X^T) and
(X) 1/2(X - X^T) denote the symmetric
and skew-symmetric parts of X ∈^n × n, respectively.

Note that Grioli's theorem stated above is recovered for the
case of equal weights μ = μ_c > 0. In order to express the
connection to the variational characterization of the polar
factor (F), we have introduced the following notation
[Relaxed polar factor(s)]
  Let μ > 0 and μ_c ≥ 0. We denote the set-valued mapping
  that assigns to a given parameter F ∈^+(n) its associated
  set of energy-minimizing rotations by
  
    _μ,μ_c(F)     R ∈ (n)(R ;F) .


In the weighted case, the polar factor (F) is always critical
but not always optimal. In general the global minimizers
_μ,μ_c(F) depend on the parameters μ > 0 and
μ_c ≥ 0 and can substantially deviate from (F).

The optimal rotations in the weighted case _μ,μ_c(F)
have been worked out in two and three space dimensions by the present
authors in a series of papers <cit.>;
cf. also <cit.> and <cit.>
for earlier related work. A visualization of the mechanism of optimal
Cosserat rotations in dimension n = 3 for an idealized nano-indentation
was given in <cit.> and shows that the optimal rotations
can produce interesting non-classical patterns. A final proof of
optimality in any dimension n ≥ 2 has been obtained by Borisov
and the authors in <cit.> and is based on a new
characterization of real square roots of real symmetric matrices.
This contribution presents an overview of these results omitting
the proofs for which we refer to the original contributions.



































Our study of the energy-minimizing rotations _μ,μ_c(F)
is motivated by a particular Cosserat (micropolar)
theory <cit.>, i.e., a continuum theory
with additional degrees of freedom R ∈(n). In this
context, the objective function (R ;F) subject to
minimization in intro:prob:weighted determines the
shear-stretch contribution to the strain energy in any
nonlinear, quadratic, and isotropic Cosserat theory, see also <cit.>.
The arguments to the shear-stretch energy W_μ,μ_c(R ;F) are
the deformation gradient field
F ∇φ: Ω→^+(n)
and the microrotation field R: Ω→(n) evaluated at a
given point of the domain Ω. A full Cosserat continuum model
furthermore contains an additional curvature energy
term <cit.> and a volumetric
energy term, see, e.g., <cit.> or <cit.>.

It is always possible to express the local energy contribution in a
Cosserat model as W = W(), where R^TF
is the first Cosserat deformation tensor. This reduction follows
from objectivity requirements and has already been observed by the
Cosserat brothers <cit.>,
see also <cit.> and <cit.>.
Since  is in general non-symmetric, the most general isotropic
and quadratic local energy contribution which is zero at the reference
state is given by

    μ ( - 𝕀)^2
         + μ_c ( - 𝕀)^2_“shear-stretch energy”  +  λ/2  - 𝕀)^2_“volumetric energy” .

The last term will be discarded in the following, since it couples
the rotational and volumetric response, a feature not present in the
well-known isotropic linear Cosserat models.[The Cosserat brothers never
  proposed any specific expression for the local energy W = W().
  The chosen quadratic ansatz for W = W() is motivated by a direct
  extension of the quadratic energy in the linear theory of Cosserat
  models, see, e.g. <cit.>. We always consider a true volumetric-isochoric
  split in our applications.]

From the perspective of Cosserat theory, the optimal rotations
_μ,μ_c(F) yield insight into the important limit case
of vanishing characteristic length L_ c = 0.[This
  identification requires that the volume term decouples from
  the , e.g.,

    W^ vol() λ/4[( - 1)^2 + (1/ - 1)^2] .

This requirement is quite natural and is satisfied by all linear
Cosserat models <cit.>.] In this context, we can interpret the solutions
of (<ref>) as an energetically optimal mechanical
response of the field R ∈(n) of Cosserat microrotations to
a given deformation gradient F ∇φ∈^+(n).


  The correct choice of the so-called Cosserat couple modulus
  μ_c ≥ 0 for specific materials and boundary value problems is
  an interesting open question. There are indications that a
  non-vanishing μ_c > 0 has never been experimentally observed
  and that such a choice is at least debatable <cit.>.
  The limit case μ_c = 0 is hence of particular interest.


We want to stress that although the term (R ;F) subject to
minimization in (<ref>) is quadratic in the nonsymmetric
microstrain tensor - 𝕀 = R^TF - 𝕀, see,
e.g., <cit.>, the associated minimization
problem with respect to R is nonlinear due to the multiplicative
coupling R^TF and the geometry of (n).


  The energy (R ;F) is a polynomial in the matrix entries,
  hence ∈ C^∞((n),). Further, since the Lie
  group (n) is compact and ∂(n) = ∅, the
  global extrema of  are attained at interior points.


The previous remark hints at a possible solution strategy for
intro:prob:weighted. If all the critical points
R_crit(F) ∈(n) of (R ;F) can be
computed[The smooth manifold (n) has empty boundary.
This implies that a critical point for given F ∈^+(n)
satisfies (R(t) ; F)0 = 0 for
every smooth curve of rotations R(t): (-ϵ, ϵ) →(n) passing through R(0) = R_crit.],
then a direct comparison of the associated critical energy levels
(R_crit ;F) allows to determine the
critical branches which are energy-minimizing.
Clearly, any minimizing critical branch realizes the
reduced Cosserat shear-stretch energy defined as

    W^ red_μ,μ_c: ^+(n) → ,  
      W^ red_μ,μ_c(F) min_R ∈ (n) W_μ,μ_c(R ;F) .


At first, a solution of intro:prob:weighted in three space
dimensions was out of reach (let alone the  n-dimensional problem).
Therefore, we first restrict our attention to the planar case, where
we can base our computations on the standard parametrisation

    R: [-π, π] →(2) ⊂^2× 2,   R() [  cos -sin;  sin  cos ]

by a rotation angle.[Note that π and -π are mapped
to the same rotation. In this text, we implicitly choose π over
-π for the rotation angle whenever uniqueness is an issue.]

It turns out that there are at most two optimal planar rotations
_μ,μ_c^±(F) in the non-classical parameter
range μ > μ_c ≥ 0 and we distinguish these by a sign.
The corresponding optimal rotation angles of
^±_μ,μ_c(F) are denoted by
α^±_μ,μ_c(F). The non-classical minimizers coincide
with the polar factor (F) in the compressive regime
of F ∈^+(2), but deviate otherwise.

The computation of the global minimizers in dependence of F
is not completely obvious even for the planar case.
Hence, the following simplifications of the minimization problem
are helpful.

First, it is useful to introduce
[Parameter rescaling]
  
  
  
  Let μ > μ_c ≥ 0. We define the singular radius  by
  
    > 0 ,
            and further define    λ_μ,μ_c/ρ_1,0 = μ/μ - μ_c ,

  as the induced scaling parameter. Note that ρ_1,0 = 2 and
  λ_1,0 = 1. Further, we define the parameter rescaling
  given by
  
    F_μ,μ_c  λ^-1_μ,μ_c F  = μ - μ_c/μ F   ∈^+(n) .


For μ > 0 and μ_c = 0, we obtain F_μ,0 = F, i.e.,
the rescaling is only effective for μ_c > 0.

Regarding the material parameters, we proved
in <cit.> that for any dimension
n ≥ 2, it is in fact sufficient to restrict
our attention to two parameter pairs:
(μ,μ_c) = (1,1), the classical case, and
(μ,μ_c) = (1,0), the non-classical case.
Hence, somewhat surprisingly, the solutions for arbitrary
μ > 0 and μ_c ≥ 0 can be recovered from these
two limit cases. This is the content of

  
  Let n ≥ 2 and let F ∈^+(n), then
  
    μ_c ≥μ > 0      ⟹  (R ;F)
           ∼ 
          W_1,1(R ;F) ,  and
    μ > μ_c ≥ 0      ⟹  (R ;F)
           ∼ (R ;F_μ,μ_c) .


Here, the equivalence notation means that the energies give rise to
the same global minimizers which we can also state as

  
    _μ,μ_c(F) =
        _1,1(F) = {(F)},    if  μ_c ≥μ > 0
    _1,0(F_μ,μ_c),    if  μ > μ_c ≥ 0



Another important observation can be made introducing the rotation

    R Q^TR^T Q

which acts relative to the polar factor (F) in the
coordinate system given by the columns of Q which span a positively
oriented frame of principal directions of U. This allows us to
transform

    Q^T((R^TF) - 𝕀)Q = Q^T((R^T QDQ^T) - 𝕀))Q
         =(Q^TR^T QDQ^TQ - Q^TQ) = (Q^TR^T Q_ RD - 𝕀) = (RD - 𝕀) .

For fixed choice of Q ∈(n), the inverse transformation allows
to reconstruct the absolute rotation uniquely

    R = (QRQ^T ^T)^T =  QR^TQ^T .

Hence, in the non-classical parameter range represented by the
limit case (μ,μ_c) = (1,0), the minimization problem can
be reduced to the following problem for the optimal relative
rotations.

  
  Let n ≥ 2. Compute the set of energy-minimizing relative
  rotations
  
    _1,0(D)   R ∈ (n)W_1,0(R ;D) = R ∈ (n)(RD - 𝕀)^2 ⊆ (n) .


The decisive point in the solution of prob:opt
in dimensions n ≥ 3 is the characterization of the set
of relative rotations R∈(n) satisfying the
particular symmetric square condition

    (RD - 𝕀)^2 ∈(n)

which is equivalent to the Euler-Lagrange equations.

After having set the stage of the optimization problem on (n),
this overview is now structured as follows: in the next Section 2,
we consider in some detail the planar problem which allows for a
complete solution by elementary techniques and which presents
already the essential geometry which unfolds in dimensions
n ≥ 3. In Section 3, we provide the complete solution for
the three-dimensional case as well as the corresponding reduced
energy expression in terms of singular values of F. We also
provide a geometrical interpretation that allows to view the
minimization problem for μ_c = 0 as a distance problem.
Furtermore, we provide a discussion for which deformation
gradients we can only have the classical response (F).
Finally, in Section 4, we present our results for the general
n-dimensional case.



§ OPTIMAL ROTATIONS IN TWO SPACE DIMENSIONS


In this section, we consider
[The planar minimization problem]
  
  Let F ∈^+(2), μ > 0 and μ_c ≥ 0. The task is to
  compute the set of optimal microrotation angles

    ∈ [-π,π]{μ(R(α)^TF - 𝕀_2)^2
        +
        μ_c (R(α)^TF - 𝕀_2)^2
        } ,

  where
  
    R(α) [  cos -sin;  sin  cos ]∈(2)
          and  [ F_11 F_12; F_21 F_22;      ]∈^+(2) .



In this case we can compute explicit representations of optimal
planar rotations for the Cosserat shear-stretch energy by
elementary means. The parameter reduction strategy described
by lem:parameter_reduction allows us to concentrate
our efforts towards the construction of explicit solutions
to intro:prob:planar on two representative pairs of parameter
values μ and μ_c. The classical
regime is characterized by the limit case (μ,μ_c) = (1,1) and
the unique minimizer is given by the polar factor (F) for
any dimension n ≥ 2.

The non-classical case represented by (μ,μ_c) = (1,0) turns
out to be much more interesting and we compute all global non-classical
minimizers _1,0(F) for n = 2. This is the main contribution
of this section. Furthermore, we derive the associated
reduced energy levels W^ red_1,1(F) and W^ red_1,0(F)
which are realized by the corresponding optimal Cosserat microrotations.
Finally, we reconstruct the minimizing rotation angles for general values
of μ and μ_c from the classical and non-classical limit cases.



 §.§ Explicit solution for the classical parameter range: μ_c ≥μ > 0

The polar factor (F) is uniquely optimal for the classical
parameter range in any dimension n ≥ 2. Let us give an explicit
representation for n = 2 in terms of
∈ (-π,π]. In view of the parameter reduction, distilled
in lem:parameter_reduction, it suffices to compute the set of
optimal rotation angles for the representative limit case
(μ,μ_c) = (1,1).

Thus, to obtain an explicit representation of ∈ (-π,π] which
characterizes the polar factor (F) in dimension n = 2, we
consider

    ∈ [-π,π] W_1,1(R() ;F)
      =  ∈ [-π,π][[  cos -sin;  sin  cos ]^T
    [ F_11 F_12; F_21 F_22;      ]
    -[ 1 0; 0 1 ]]^2 .

Let us introduce the rotation
J [  0 -1;  1  0 ]∈(2).
Its application to a vector v ∈^2 corresponds to multiplication
with the imaginary unit i ∈. In what follows, the quantities
F = F_11 + F_22 and JF = - F_21 + F_12
play a particular role and we note the identity

    F^2 + JF^2 = F^2 + 2 F = U^2 .


The reduced energy
W_1,1^ red(F) min_R∈(n) W_1,1(R ;F)
realized by the polar factor (F) can be shown to be the euclidean
distance of an arbitrary F in ^n× n to (n). For n = 2,
we obtain
[Euclidean distance to planar rotations]
  Let F ∈^+(2), then
  
    W_1,1^ red(F) = ^2(F,(2)) = U - 𝕀^2
        = F^2 - 2 √(F^2 + 2 F) + 2 .

  The unique optimal rotation angle realizing this minimial
  energy level satisfies the equation
  
    [ sin; cos ]
        =1/U[ -JF;   F ] .

  In particular, we have
  (F) = -(JF)·arccos(F/U) ∈ [-π,π].



  
  Let F ∈^+(2), then the polar factor (F) has the
  explicit representation
  
    (F) =
        R() [  cos -sin;  sin  cos ]
        = 1/U[  -F  JF; -JF   F ] .





 §.§ The limit case (μ,μ_c) = (1,0) for μ > μ_c ≥ 0

We now approach the more interesting non-classical limit case
(μ, μ_c) = (1,0) and compute the optimal rotations for
W_μ,μ_c(R ;F). Note that, due to lem:parameter_reduction,
this limit case represents the entire non-classical parameter
range μ > μ_c ≥ 0.

[The formally reduced energy W^ red_1,0(F)]
  
  Let F ∈^+(2). Then, the formally reduced energy
  
    W_1,0^ red(F) min_R  ∈ (2)W_1,0(R ;F)
                           min_R ∈ (2)(R^TF-𝕀)^2

is given by

    W_1,0^ red(F)
        = U - 𝕀^2 = (U - 𝕀)^2 = ^2(F,(2)) ,        if  U < 2
    1/2F^2 - F
                    = 1/2 U^2 - 2 U ,      if  U≥ 2 .



It is well-known that any orthogonally invariant energy density
W(F) admits a representation in terms of the singular values of F, i.e.,
in the eigenvalues of U. Let us give this representation.

  
  Let F ∈^+(2) and denote its singular values by ν_i, i = 1,2.
  The representation of W^ red_1,0(F) in the singular values of F
  is given by
  
    W_1,0^ red(F) = W_1,0^ red(ν_1,ν_2) =
        
          (ν_1 - 1)^2 + (ν_2 - 1)^2 ,     if  ν_1 + ν_2  < 2
    1/2(ν_1 - ν_2)^2 ,     if  ν_1 + ν_2 ≥ 2 .

Note that the previous formulae are independent of the enumeration
of the singular values.



  §.§.§ Optimal relative rotations for mu = 1 and muc = 0


Our next goal is to compute explicit representations of the rotations
^±_1,0(F) which realize the minimal energy level in the
non-classical limit case (μ,μ_c) = (1,0). This is the content
of the next theorem for which we now prepare the stage with the
following

  
  Let D = (σ_1,σ_2) > 0, i.e, a diagonal matrix with
  strictly positive diagonal entries. Then, assuming D≥ 2,
  the equation R(β) D = 2 has the following solutions
    
    β^± = ± arccos(2/D)   ∈ [-π,π] .

    For D < 2, there exists no solution, but we can define
    β = β^± 0 by continuous extension.


Our fig:alpha_rel_plot shows a plot of the optimal
relative rotation angle β(U).
In the classical parameter range 0 < U≤ 2,
(F) is uniquely optimal and β vanishes identically.
In , a classical pitchfork bifurcation occurs. In
particular, due to U(𝕀_2) = 𝕀_2 = 2,
the identity matrix is a bifurcation point of β^±(F).
Further, we note that the branches β^±(U) = ±arccos(2/U)
are not differentiable at U = 2. This has implications
on the interaction of the Cosserat shear-stretch energy with the
Cosserat curvature energy W_ curv.


[Optimal non-classical microrotation angles _1,0^±]
  
  Let F ∈^+(2) and consider (μ,μ_c) = (1,0). The optimal
  rotation angles for  are given by
  
    _1,0^±(F) =
        (F) = arccos(F/U)
              ,  if    U < 2
    (F) ± arccos(2/U)
              ,  if    U≥ 2 .





 §.§ Expressions for general non-classical parameter choices


The reduction for μ and μ_c in lem:parameter_reduction
asserts that the optimal rotations for arbitrary values of
μ > 0 and μ_c ≥ 0
can be reconstructed from the limit cases (μ, μ_c) = (1,1) and
(μ, μ_c) = (1,0). We now detail this procedure which essentially
exploits defi:reduction:rescaling.

Note first that the rescaled deformation gradient F_μ,μ_cλ^-1_μ,μ_c F induces a rescaled stretch tensor

    U_μ,μ_c = √((F_μ,μ_c)^TF_μ,μ_c) = λ^-1_μ,μ_c· U .

The right polar decomposition takes the form
F_μ,μ_c = (F_μ,μ_c) U_μ,μ_c.
From (F_μ,μ_c) = F_μ,μ_cU_μ,μ_c^-1 follows the scaling invariance (F_μ,μ_c) = (F).
For the non-classical parameter range μ > μ_c ≥ 0, the quantity

    U_μ,μ_c = λ_μ,μ_c^-1· U = ρ_1,0/ρ_μ_,μ_cU

plays an essential role. This leads us to

    U_μ,μ_c≥ 2 = ρ_1,0    ρ_1,0/ρ_μ_,μ_c· U≥ρ_1,0    U≥ρ_μ,μ_c .

In particular, this implies that the bifurcation in U
allowing for non-classical optimal planar rotations is
characterized by the singular radius .


  Let F ∈^+(2). For μ_c ≥μ > 0
  the optimal microrotation angle is given by
  
    α_μ,μ_c(F)
        = (F_μ,μ_c)
        = (F) = arccos(F/U) .

  For μ > μ_c ≥ 0, the two optimal rotation angles
  are given by
  
    ^±_μ,μ_c(F) = ^±_1,0(F_μ,μ_c)
      = (F) = arccos(F/U)    ,  if    U < ρ_μ,μ_c
    (F) ∓ arccos(/U)   ,  if    U≥ρ_μ,μ_c .



 

 §.§ Optimal rotations for planar simple shear


We now apply our previous optimality results to simple shear
deformations

    F_γ[ 1 γ; 0 1 ],   γ∈ .

The energy-minimizing rotation angles
_μ,μ_c(γ) _μ,μ_c(F_γ)
for simple shear can be explicitly computed; see also <cit.>
for previous results.

In the classical parameter range μ_c ≥μ > 0 represented
by the limit case (μ,μ_c) = (1,1) the polar rotation
(F_γ) is uniquely optimal.

Let us collect some properties of simple shear F_γ. We have
F_γ^2 = 2 + γ^2 and F_γ = 1, i.e.,
simple shear is volume preserving for any amount γ. This
allows us to compute

    U_γ = √(F_γ^2 + 2 F_γ) = √(4 + γ^2)≥ 2 .

Thus, we have

  Let (μ,μ_c) = (1,0) and let F_γ∈^+(2) be a simple
  shear of amount γ∈. Then,
  
    γ≠ 0
          ⟹  ^±_1,0(F_γ)  ≠ (F_γ) .




  A simple shear F_γ by a non-zero amount γ≠ 0
  automatically generates an optimal microrotational response
  ^±(F_γ) which deviates from the continuum
  rotation (F). This implies that the associated first
  Cosserat deformation tensor ^±_1,0(F_γ) _1,0^±(F_γ)^TF_γ is not symmetric for any
  γ≠ 0.




§ OPTIMAL ROTATIONS IN THREE SPACE DIMENSIONS

In this section, we discuss
[Weighted optimality in dimension n = 3]
  Let μ > 0 and μ_c ≥ 0. Compute the set of optimal rotations
  
  
    R ∈ (3)(R ;F)  R ∈ (3){μ (R^TF - 𝕀)^2
           + μ_c (R^TF - 𝕀)^2}

  for given parameter F ∈^+(3) with distinct singular values
  ν_1 > ν_2 > ν_3 > 0.


The polar factor (F) is the unique minimizer for
W_μ,μ_c(R ;F) in the classical parameter range
μ_c ≥μ > 0, in all dimensions n ≥ 2,
see <cit.>.

Since the classical parameter domain μ_c ≥μ > 0 is very well
understood, we focus entirely on the non-classical parameter
range μ > μ_c ≥ 0. Furthermore, due to the parameter
reduction described by lem:parameter_reduction, which holds
for all dimensions n ≥ 2, it suffices to solve the non-classical
limit case (μ,μ_c) = (1,0), since

    R ∈ (3)W_μ,μ_c(R ;F)  =  R ∈ (3)W_1,0(R ; F_μ,μ_c) .

On the right hand side, we notice a rescaled deformation gradient

    F_μ,μ_cλ^-1_μ,μ_c· F ∈^+(3)

which is obtained from F ∈^+(3) by multiplication with the inverse of
the induced scaling parameter
λ_μ,μ_cμ/μ - μ_c > 0. We note that we use
the previous notation throughout the text and further introduce
the singular radius ρ_μ,μ_c2μ/μ - μ_c.

It follows that the set of optimal Cosserat rotations can be described
by

    _μ,μ_c(F)  = _1,0(F_μ,μ_c)

for the entire non-classical parameter range μ > μ_c ≥ 0.
We are therefore mostly concerned with the case μ_c = 0 in the
present text. Note that for all μ > 0, we have the equality

    ^±_μ,0(F)
       = ^±_1,0(F) .










 §.§ The locally energy-minimizing Cosserat rotations ^±_μ,μ_c(F)

We briefly present the geometric characterization of the optimal
Cosserat rotations ^±_μ,μ_c(F) obtained
in <cit.>. Let R ∈(3) and
let ^2 ⊂^3 denote the unit 2-sphere.
We make use of the well-known angle-axis parametrization of
rotations which we write as [α,  r][The angle-axis parametrization
  is singular, but this is not an issue for our exposition.],
where α∈ (-π,π] denotes the rotation angle
and r ∈^2 specifies the oriented rotation axis.

We recall that it is sufficient to solve for the relative
rotation, i.e., we consider
[Diagonal form of weighted optimality in n = 3]
  
  Let μ > 0 and μ_c ≥ 0 and let
  D = (ν_1,ν_2,ν_3) with
  ν_1 > ν_2 > ν_3 > 0. Compute the set of
  optimal relative rotations
  
    R ∈ (3)(R^T ;D)  R ∈ (3){μ (R D - 𝕀)^2
           + μ_c (R D - 𝕀)^2} .


We stress that the rotation angle of the
relative rotation R is implicitly reversed
due to the correspondence R^T ↔R.

The computation of the solutions to intro:prob_wmm_reduced
by computer algebra together with a statistical verification are
the core results obtained in <cit.> which we
present next.

[Energy-minimizing relative rotations
              for (μ,μ_c) = (1,0)]
  
  Let ν_1 > ν_2 > ν_3 > 0 be the singular values of
  F ∈^+(3). Then the energy-minimizing relative rotations
  solving intro:prob_wmm_reduced are given by
  
    R_1,0^±(F)
            [  cosβ̂^±_1,0 -sinβ̂^±_1,0            0;  sinβ̂^±_1,0  cosβ̂^±_1,0            0;            0            0            1;              ] ,

  where the optimal rotation angles
  β̂^±_1,0∈ (-π,π]
  are given by
    
    β̂^±_1,0(F)
                0 ,         if  ν_1 + ν_2 ≤ 2 ,
     ±arccos(2/ν_1 + ν_2) ,
                   if  ν_1 + ν_2 ≥ 2 .

    Thus, in the non-classical regime ν_1 + ν_2 ≥ 2,
    we obtain the explicit expression
    
    R_1,0^±(F)
              [           2/ν_1 + ν_2 ∓√(1-(2/ν_1 + ν_2)^2)                     0; ±√(1-(2/ν_1 + ν_2)^2)           2/ν_1 + ν_2                     0;                     0                     0                     1 ] .

    In the classical regime ν_1 + ν_2 ≤ 2, we simply
    obtain the relative rotation R_1,0^±(F) = 𝕀,
    and there is no deviation from the polar factor (F)
    at all.


Note that, due to the parameter reduction lem:parameter_reduction,
it is always possible to recover the optimal rotations
^±_μ,μ_c(F)
for general non-classical parameter choices μ > μ_c ≥ 0
from the non-classical limit case
(μ,μ_c) = (1,0); cf. <cit.>
and <cit.> for details.



 §.§ Geometric and mechanical aspects of optimal Cosserat
            rotations


It seems natural to introduce
[Maximal mean planar stretch and strain]
  
  Let F ∈^+(3) with singular values
  ν_1 ≥ν_2 ≥ν_3 > 0. We introduce
  the maximal mean planar stretch  and
  the maximal mean planar strain  as follows:
  
    (F)      ν_1 + ν_2/2 ,  and
    (F)      (ν_1 - 1) + (ν_2 - 1)/2 = (F) - 1 .



In order to describe the bifurcation behavior of _μ,μ_c^±(F)
as a function of the parameter F ∈^+(3), it is helpful to
partition the parameter space ^+(3).
[Classical and non-classical domain]
To any pair of material parameters (μ,μ_c) in the non-classical
range μ > μ_c ≥ 0, we associate a classical domain
_μ,μ_c and a non-classical domain _μ,μ_c.
Here,

    _μ,μ_c   F ∈^+(3)(F_μ,μ_c) ≤ 0 ,
          and  
    _μ,μ_c   F ∈^+(3)(F_μ,μ_c) ≥ 0 ,

respectively.


It is straight-forward to derive the following equivalent characterizations

    _μ,μ_c = F ∈^+(3)(F) ≤λ_μ,μ_c
        = F ∈^+(3)ν_1 + ν_2 ≤2μ/μ - μ_c ,
    _μ,μ_c = F ∈^+(3)(F) ≥λ_μ,μ_c
        = F ∈^+(3)ν_1 + ν_2 ≥2μ/μ - μ_c .


On the intersection
_μ,μ_c∩_μ,μ_c = F ∈^+(3)(F) = 0,
the minimizers _μ,μ_c^±(F) coincide with the polar
factor (F). This can be seen from the form of the optimal
relative rotations in propo:rhat. More explicitly, in
dimension n = 3 and in the non-classical limit case
(μ,μ_c) = (1,0), we have:

    _1,0F ∈^+(3)(F) ≤ 0 ,
        and  _1,0F ∈^+(3)(F) ≥ 0 .

Since the maximal mean planar strain (F) is related to
strain, this indicates a particular (possibly new) type of
tension-compression asymmetry.

Towards a geometric interpretation of the energy-minimizing Cosserat
rotations ^±_1,0(F) in the non-classical limit case
(μ,μ_c) = (1,0), we reconsider the spectral decomposition of
U = QDQ^T from the principal axis transformation in sec:intro.
Let us denote the columns of Q ∈(3) by q_i ∈^2, i = 1,2,3.
Then q_1 and q_2 are orthonormal eigenvectors of U which correspond
to the largest two singular values ν_1 and ν_2 of F ∈^+(3).
More generally, we introduce the following

[Plane of maximal stretch]
  
  The plane of maximal stretch is the linear subspace
  
    P^ ms(F)     q_1,q_2⊂^3

  spanned by the two eigenvectors q_1,q_2 of U associated with the two
  largest singular values ν_1 > ν_2 > ν_3 > 0
  of the deformation gradient F ∈^+(3).


We recall that, due to the parameter
reduction lem:parameter_reduction, it is always possible to
recover the optimal rotations

    _μ,μ_c(F) R ∈ (3)W_μ,μ_c(R ;F)

for a general choice of non-classical parameters μ > μ_c ≥ 0 from the
non-classical limit case (μ,μ_c) = (1,0). However, we defer
the explicit procedure for a bit since it is quite instructive
to interpret this distinguished non-classical limit case first.


  
    at (0,0)
    < g r a p h i c s >
;
    [above] at (-2,0.5) -β̂_1,0;
    [->] (-1.8,0.5) – (-1, -0.25);
    [above] at ( 1.8, 1.3) β̂_1,0;
    [->] (1.8, 1.3) – (-0.1, -0.6);
    [shift=(2,-1.25),scale=0.5]
      [thick,->] (0,0) – (0.95 * -0.975, 0.95 * -0.7);
      [anchor=north] at (0.95 * -0.975, 0.95 * -0.7) q_1;
      [thick,->] (0,0) – (1.2  * 1, 1.2 * -0.275);
      [anchor=north west] at (1.0  * 1, 1.0 * -0.275) q_2;
      [thick,->] (0,0) – (0.025, 1.2);
      [anchor=south] at (0.025, 1.2) q_3;
    
  


figureAction of ^±_1,0(F) in
  axes of principal stretch for a stretch ellipsoid with half-axes
  (ν_1,ν_2,ν_3) = (4,2,1/2). The plane of maximal stretch
  P^ mp(F) is depicted in blue. The cylinder along
  q_3 ⊥P^ mp(F) illustrates that the axis
  of rotation is the eigenvector q_3 of U associated with the
  smallest singular value ν_3 = 1/2 of F. The thin cylinder
  [blue] bisecting the opening represents the relative rotation angle
  β̂ = 0 and corresponds to (F). The outer two
  cylinders [red] correspond to the two non-classical minimizers
  _1,0^±(F). The enclosed angles
  β̂_1,0^± = ±arccos(2/ν_1 + ν_2) are
  the optimal relative rotation angles. This reveals the major
  symmetry of the non-classical minimizers.



  
  For (F) ≤ 0 the maximal mean planar strain is non-expansive.
  By definition, we have F ∈_1,0 in the
  classical domain, for which the energy-minimizing relative rotation
  is given by R_1,0(F) = 𝕀 and there is no deviation
  from the polar factor. In short _1,0^±(F) = (F).


Let us now turn to the more interesting non-classical case F ∈_1,0.

  
  If F ∈_1,0, then by definition (F) > 0
  and the maximal mean planar strain is expansive. The deviation of
  the non-classical energy-minimizing rotations ^±_1,0(F)
  from the polar factor  is measured by a rotation
  in the plane of maximal stretch P^ mp(F) given
  by (F)^T^±_1,0(F) = Q(F)R_1,0^∓(F)Q(F)^T.
  The rotation axis is the eigenvector q_3 associated with the smallest
  singular value ν_3 > 0 of F and the relative rotation angle is
  given by β̂_1,0^∓(F) = ∓arccos(1/(F)).
  The rotation angles increase monotonically towards the asymptotic limits
  
    lim_(F)  → ∞β̂_1,0^±(F)
        =  ±π/2 .

  In axis-angle representation, we obtain
  
    R_1,0^±(F)      ≡  [±arccos(1/(F)),  (0, 0, 1)] ,  and
    ^T^±_1,0(F)      ≡  [∓arccos(1/(F)),  q_3 ] .




  
  For the non-classical limit case (μ,μ_c) = (1,0) we have
  the following formula for the energy-minimizing Cosserat rotations:
  
    ^±_1,0(F)
             (F)    , if   F ∈_1,0 ,
     (F)Q(F)R_1,0^∓(F)Q(F)^T    , if   F ∈_1,0 .

  For general values of the weights in the non-classical range
  μ > μ_c ≥ 0, we obtain
  
    ^±_μ,μ_c(F) ^±_1,0(F_μ,μ_c) ,

  where F_μ,μ_cλ^-1_μ,μ_c F is obtained
  by rescaling the deformation gradient with the inverse of the induced scaling
  parameter λ_μ,μ_cμ/μ - μ_c > 0.

Note that the previous definition is relative to a fixed choice
of the orthonormal factor Q(F) ∈(3) in the spectral
decomposition of U = QDQ^T. Further, right from their variational
characterization, one easily deduces that the energy-minimizing
rotations satisfy
^±_μ, μ_c(Q F) = Q ^±_μ,μ_c(F),
for any Q ∈(3), i.e., they are objective
functions; cf.rem:polar_vs_rpolar.



The domains of the piecewise definition of _1,0^±(F)
in cor:rpolar_formula indicate a certain tension-compression
asymmetry in the material model characterized by the Cosserat
shear-stretch energy W_1,0(R ;F). We can also make a second
important observation. To this end, consider
a smooth curve F(t):(-ϵ,ϵ) →^+(3). If the
eigenvector q_3(t) ∈^2 associated with the smallest singular
value ν_3(t) changes its orientation along this curve, then
the rotation axis of _1,0^±(F) flips as well. Effectively,
the sign of the relative rotation angle β̂_1,0^±(F) is
negated which may lead to jumps.  This can happen, e.g., if F(t)
passes through a deformation gradient with a non-simple singular
value, but it may also depend on details of the specific algorithm
used for the computation of the eigenbasis.

For the classical range μ_c ≥μ > 0, the
polar factor and the relaxed polar factor(s) coincide and trivially
share all properties. This is no longer true for the non-classical parameter
range μ_c ≥μ > 0 and we compare the properties for that range
in our next remark. More precisely, we present a detailed comparison
of the well-known features of the polar factor  which are of
fundamental importance in the context of mechanics.


   Let n ≥ 2 and F ∈^+(n).
  The polar factor (F) ∈(n) obtained from the polar decomposition
  F = (F) U is always unique and satisfies:
  
    [l]
             (Objectivity)      ( Q· F )     =    Q·(F)          (∀ Q ∈(n)) ,
       (Isotropy)      ( F· Q )     =    (F)· Q          (∀ Q ∈(n)) ,
       (Scaling invariance)      ( λ· F )    =    (F)         (∀λ > 0) ,
       (Inversion symmetry)      (F^-1)    =    (F)^-1 .

  The relaxed polar factor(s) _μ,μ_c(F) ⊂(n) is
  in general multi-valued and, due to its variational
  characterization, satisfies:
  
    [l]
             (Objectivity)      _μ,μ_c( Q · F )       =    Q ·_μ,μ_c(F)           (∀ Q ∈(n)) ,
       (Isotropy)      _μ,μ_c( F· Q )        =    _μ,μ_c(F)· Q           (∀ Q ∈(n)) .

  For the particular dimensions k = 2,3, our explicit formulae imply
  that there exist particular instances λ^* > 0
  and F^* ∈^+(k), for which we have
  
    [l]
             (Broken scaling invariance)      ^±_μ,μ_c(λ^* · F^*)    ≠   (F^*)
             ,   and
       (Broken inversion symmetry)      ^±_μ,μ_c(F^*^-1)     ≠   (F^*)^-1   .

  This can be directly inferred from the partitioning of
  ^+(k) = _μ,μ_c ∪ _μ,μ_c
  and the respective piecewise definition of the relaxed
  polar factor(s), see cor:rpolar_formula.

We interpret these broken symmetries as a (generalized) tension-compression
asymmetry.




 §.§ The reduced Cosserat shear-stretch energy

We now introduce the notion of a reduced energy as the energy level
realized by the energy-minimizing rotations _μ,μ_c(F).
[Reduced Cosserat shear-stretch energy]
  The reduced Cosserat shear-stretch energy is defined as
  
    W_μ,μ_c^ red: ^+(n) →,
            
        W_μ, μ_c^ red(F)   min_R ∈ (n) W_μ,μ_c(R ;F) .


Besides the previous definition, we also have the following equivalent
means for the explicit computation of the reduced energy

    W_μ,μ_c^ red(F)     =  W_μ,μ_c(^±_μ,μ_c(F) ;F) ,  and
    
        W_μ,μ_c^ red(F)
            = 
        W_μ, μ_c^ red(D)
         min_R∈(n) W_μ,μ_c(R ; D)
         = 
        W_μ,μ_c(R_μ,μ_c^± ;D) .



  
  Let F ∈^+(3) and ν_1 > ν_2 > ν_3 > 0 the
  ordered singular values of F. Then the reduced Cosserat shear-stretch
  energy (F) admits the following piecewise representation
  
    (F)  =  (ν_1 - 1)^2 + (ν_2 - 1)^2 + (ν_3 - 1)^2 = U - 𝕀^2
             , if  ν_1 + ν_2 ≤ 2, i.e.,  F ∈_1,0 ,
     1/2 (ν_1 - ν_2)^2 + (ν_3 - 1)^2
             , if  ν_1 + ν_2 ≥ 2, i.e., F ∈_1,0 .


Our next step is to reveal the form of the reduced energy for the entire
non-classical parameter range μ > μ_c ≥ 0 which involves the
parameter reduction lemma, but we have to be a bit careful.


  The parameter reduction in lem:parameter_reduction is
  the key step in the computation of the minimizers for general
  non-classical material parameters μ > μ_c ≥ 0.
  It might be tempting, but we have to stress that the general form of
  the reduced energy cannot be obtained by rescaling the singular values
  ν_i ↦λ_μ,μ_c^-1ν_i in the singular value
  representation of W^ red_1,0.


[W^ red_μ,μ_c as a function of the singular values]

Let F ∈^+(n) and ν_1 > ν_2 > ν_3 > 0, the ordered
singular values of F and let μ > μ_c ≥ 0, i.e., a non-classical
parameter set. Then the reduced Cosserat shear-stretch energy
W^ red_μ,μ_c: ^+(3) →
admits the following explicit representation

    W^ red_μ,μ_c(F)  =  μ((ν_1 - 1)^2 + (ν_2 - 1)^2 + (ν_3 -
        1)^2) = μ U - 𝕀^2
           ,  F ∈_μ,μ_c ,
     μ/2(ν_1 - ν_2)^2
        + μ  (ν_3 - 1)^2
        + μ_c/2((ν_1 + ν_2) - )^2
        - μ_c/2·ρ_μ,μ_c^2
           ,  F ∈_μ,μ_c .



Let us consider the contribution of the skew-term to
W^ red_μ,μ_c
given by

    μ_c/2((ν_1 + ν_2) - )^2

as a penalty term for F ∈^+(3) arising for material parameters
in the non-classical parameter range μ > μ_c ≥ 0. This leads to
a simple but interesting observation for strictly positive μ_c > 0.
The minimizers F ∈^+(3) for the penalty term satisfy
the bifurcation criterion

    ν_1 + ν_2 =

for ^±_μ,μ_c(F). In this case
R_μ,μ_c^± = 𝕀 which implies that
R_μ,μ_c^± D - 𝕀∈(3), i.e.,
it is symmetric. Hence, the skew-part vanishes entirely which
minimizes the penalty. In numerical applications, a rotation
field R approximating ^±(F) can be expected to
be unstable in the vicinity of the branching point
ν_1 + ν_2 ≈.
Hence, a penalty which explicitly rewards an approximation to the
bifurcation point seems to be a delicate property. In strong contrast,
for the case when the Cosserat couple modulus is zero, i.e.,
μ_c = 0, the penalty term vanishes entirely. This hints at a
possibly more favorable qualitative behavior of the model in that
case; cf. <cit.>.


We recall that the tangent bundle T(n) is isomorphic to the product
(n)×(n) as a vector bundle. This is commonly referred
to as the left trivialization, see, e.g., <cit.>.
With this we can minimize over the tangent bundle in the following


Let F ∈^n× n. Then

    inf_R ∈ (n)
     A ∈ (n)R^TF - 𝕀 - A^2   =  min_R ∈ (n)(R^TF - 𝕀)^2     min_R ∈ (n) W_1,0(R ;F)  .


In the non-classical limit case (μ,μ_c) = (1,0), the preceding
lemma yields a geometric characterization of the reduced
Cosserat shear-stretch energy as a distance which we find remarkable.

  
  Let n ≥ 2 and consider F ∈^+(n) with singular values
  ν_1 ≥ν_2 ≥…≥ν_n > 0, i.e., not
  necessarily distinct. Then the reduced Cosserat shear-stretch energy
  W_1,0^ red: ^+(n) →
  admits the following characterization as a distance
  
    W_1,0^ red(F)   =  _ euclid^2(F, (n)(𝕀 + (n))) .

  Here, _ euclid denotes the euclidean distance function.

































 §.§ Alternative criteria for the existence of non-classical solutions

For μ > μ_c > 0, i.e., for strictly positive μ_c > 0,
the singular radius satisfies 2μ/μ - μ_c > 2.
We now define a quite similar constant, namely

    ζ_μ,μ_c -  ρ_1,0
    = 2μ_c/μ - μ_c > 0 .

Furthermore, we define the ϵ-neighborhood of a set
𝒳⊆^n× n relative to the euclidean
distance function as

    N_ϵ(𝒳)
      Y ∈^n × n_ euclid(Y, 𝒳) < ϵ .





  Let μ > μ_c > 0, F ∈^+(3) and
  ζ_μ,μ_c2μ_c/μ - μ_c > 0.
  Then we have the following inclusion

    N_1/2ζ_μ,μ_c^2((3))   ⊂  _μ,μ_c .

In other words, for all F ∈^+(3) satisfying
_ euclid(F, (3)) = U - 𝕀^2 < 1/2ζ_μ,μ_c^2,
the polar factor  is the unique minimizer of
W_μ,μ_c(R ;F).




Let F ∈(3), i.e., F = ν_1ν_2ν_3 = 1,
where ν_1 ≥ν_2 ≥ν_3 > 0 are ordered
singular values of F, not necessarily distinct. Then

    (3)   ⊂  _1,0 ,

i.e., F induces a strictly non-classical minimizer. Equivalently,
F = 1 implies the estimate ν_1 + ν_2 ≥ 2.



  
  If we make the stronger assumption ν_1 > ν_2 > ν_3 > 0,
  we obtain a strict inequality ν_1 + ν_2 > 2. In that case,
  F ∈_1,0∖_1,0 is strictly non-classical.



  
  Let μ > 0,
    F ∈^+(3) and assume
  that ν_1 > ν_2 > ν_3 > 0. Then
  
    F   ∈  _μ,0∖_μ,0 ,

  i.e., the minimizers _μ,0^±(F) ≠ are strictly
  non-classical.






 

§ OPTIMAL ROTATIONS IN GENERAL DIMENSION

The key insight for the solution of the minimization problem
in general dimension n ≥ 2 is a new approach to the
analysis of the critical points. The Euler-Lagrange equations for
W_1,0(R ;F) are equivalent to

    (RD - 𝕀)^2  ∈ (n) .

This is a symmetric square condition for the relative rotation
R, since

    (X(R))^2  =  S ∈(n),
          where    
      X(R) RD - 𝕀∈^n × n .

As it is sufficient to compute the optimal relative rotation
R, we simply set R = R for the rest
of this section.


One might suspect that the critical points of W_1,0(R ;D)
are connected to real matrix square roots of real symmetric
matrices. And indeed, the structure of the set of critical
points of W_1,0(R ;D) can be revealed quite elegantly by a specific
characterization of the set of real matrix square roots of real
symmetric matrices. Note that this characterization <cit.>, which is similar in spirit to the standard representation
theorem for orthogonal matrices Ø(n) as block matrices, seems not
to be known in the literature. Due to this representation, the square
roots of interest can always be orthogonally transformed into a
block-diagonal representation which reduces the minimization problem
from arbitrary dimension n > 2 into decoupled one- and two-dimensional
subproblems. These can then be solved independently.
From this point of view, a non-classical minimizer in n=3,
simultaneously solves a one-dimensional and a two-dimensional
subproblem. The one-dimensional problem determines the rotation axis
of the optimal rotations, while the two-dimensional subproblem
determines the optimal rotation angles.

The degenerate cases of optimal Cosserat rotations arising for
recurring parameter values ν_i, i = 1,2,3, in the diagonal
parameter matrix D ∈(n) has not been treated previously
in <cit.>, but is also accessible with the
general approach. Note that this case corresponds to the special
case of two or more equal principal stretches ν_i
which is an important highly symmetric corner case in mechanics.

Combining the results of the two preceding sections, we can now
describe the critical values of the Cosserat shear-stretch energy
W_1,0(R ;D) which are attained at the critical points. The main result
of this section is a procedure (algorithm) which traverses the set of
critical points in a way that reduces the energy at every step of the
procedure and finally terminates in the subset of global minimizers.

Technically, we label the critical points by certain partitions
of the index set {1,…,n} containing only subsets I with one
or two elements. In the last section, we have seen that the subsets
I and a choice of sign for R_I uniquely characterize a critical
point R ∈(n).

Let us give an outline of the energy-decreasing traversal strategy
starting from a given labeling partition (i.e., critical point):


  * Choose the positive sign R_I = +1 for each subset
      of the partition.

  * Disentangle all overlapping blocks for n > 3
     (cf. lem:overlap).

  * Successively shift all 2×2-blocks to the lowest
      possible index, i.e., collect the blocks of size two
      as close to the upper left corner of the matrix R
      as possible (cf. lem:comparison).

  * Introduce as many additional 2× 2-blocks by joining
      adjacent blocks of size 1 as the constraint ν_i + ν_j > 2
      allows (cf. lem:comparison).


The next theorem connects the value of W_1,0(R ;D) realized by
a critical point with its labeling partition and the choice of
determinants R_I which characterize it.
[Characterization of critical points and values]
  
  Let the entries ν_1>ν_2>…>ν_n>0 of D ∈(n).
  Then the critical points R ∈(n) can be classified according
  to partitions of the index set {1,…,n} into subsets of size
  one or two and choices of signs for the determinant R_I for
  each subset I. The subsets of size two I = {i,j} satisfy
  
    |ν_i + ν_j | > 2,          R_I = +1 ,  and
    ν_i - ν_j > 2,      R_I = -1 .

  The corresponding critical values are given by
  
    W_1,0(R ;D) =   ∑_I = {i}
    R_I = 1 (ν_i - 1)^2
      + ∑_I = {i}
    R_I = -1 (ν_i + 1)^2
      + ∑_I = {i,j}
    R_I = 11/2(ν_i - ν_j)^2
      + ∑_I = {i,j}
    R_I = -11/2(ν_i + ν_j)^2 .




  If we allow
  
    ν_1 ≥ν_2 ≥…≥ν_n > 0

  for the entries of D, then the D- and R-invariant
  subspaces V_i are not necessarily coordinate subspaces.
  This produces non-isolated critical points but does not
  change the formula for the critical values.

In order to compute the global minimizers R ∈(n) for the
Cosserat shear-stretch energy W_1,0(R ;D), we have to compare
all the critical values which correspond to the different
partitions and choices of the signs of the determinants
in the statement of theo:critical_values. We may, however,
assume that R_I = +1 for all subsets I,
see <cit.> for further details.

The following lemma shows that blocks of size two are always
favored whenever they exist.

  
  If ν_i + ν_j > 2 then the difference between
  the critical values of W_1,0(R ;D) corresponding to the
  choice of a size two subset I = {i,j} as compared
  to the choice of two size one subsets {i}, {j}
  is given by
  
    -1/2(ν_i+ν_j-2)^2.



Let us rewrite W_1,0(R ;D) in a slightly different form in order to
distill the contributions of the size two blocks in the partition.


  
For the choices of R_I = 1 there holds

    W_1,0(R ;D) = (RD-𝕀)^2 = ∑_i=1^n(ν_i - 1)^2
                      -1/2∑_I = {i,j} (ν_i+ν_j-2)^2.


To study the global minimizers for the Cosserat shear-stretch energy
in arbitrary dimension n ≥ 4, we need to investigate the relative
location of the size two subsets of the partition.


  
  Let R ∈(n) be a global minimizer for W_1,0(R ;D).
  Then R cannot contain overlapping size two subsets, i.e.,
  I = {i_1,i_4}, J = {i_2,i_3}, with i_1 < i_2 < i_3 < i_4.


We are now ready to state the result in the general n-dimensional case.

  
  Let ν_1>ν_2>…ν_n>0 be the entries of D.
  Let us fix the maximum k for which ν_2k-1 + ν_2k > 2.
  Any global minimizer R ∈(n) corresponds to the
  partition of the form
  
    {1,2}⊔{3,4}⊔…⊔{2k-1, 2k}⊔{2k+1}⊔…⊔{n}

  and the global minimum of W_1,0(R ;D) is given by

    W_1,0^ red(D)    min_R ∈(n)(RD - 𝕀)^2
      = ∑_i=1^n(ν_i-1)^2 - 1/2∑_i=1^k (ν_2i-1+ν_2i-2)^2
    
      =   1/2∑_i=1^k (ν_2i-1 - ν_2i)^2 + ∑_i=2k+1^n (ν_i-1)^2 .




The number of global minimizers in the above theorem is
2^k, where k is the number of blocks of size two in
the preceding characterization of a global minimizer as
a block diagonal matrix. All global minimizers are block
diagonal, similar to the previously discussed n = 3 case.



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