Geodesics on `39`42`'613A``45`47`'603AGL(3) and nonlinear elasticitymemocs.univaq.it/wp-content/uploads/2014/10/slides_martin_2014.pdf · Geodesics on GL(3) and nonlinear elasticity

Geodesics on GL(3) and nonlinear elasticity

Robert Martin

Chair for Nonlinear Analysis and Modelling,

Faculty of Mathematics,

University of Duisburg-Essen, Germany

joint work with Patrizio Neff, Dumitrel Ghiba, Johannes Lankeit

[email protected]

http://www.uni-due.de/mathematik/ag neff/

October 3, 2014

Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen

Elasticity theory

We consider the deformation of an elastic body:

Ω ⊂ Rn, Ω bounded domain, the reference configuration

ϕ : Ω→ Rn the deformation mapping

ϕ(x) the new position of the material point x ∈ Ω

ϕ(x) = x + u(x), u displacement, ∇u displacement gradient

Ω ϕ(Ω)ϕ


Basic Tensors in linear and nonlinear elasticity

Definitions

F = ∇ϕ (the deformation gradient)

U =√

F T F (the right Biot-stretch tensor)

C = F T F = U2 (the right Cauchy-Green deformation tensor)

V =√

FF T (the left Biot-stretch tensor)

B = FF T = V 2 (the Finger tensor)

ε = sym∇u (infinitesimal strain)

F = R U = V R , U,V ∈ PSym(n) , R ∈ SO(n)


The concept of strain

Strain tensor

A (material) strain tensor is a mapping PSym(n)→ Sym(n) , U 7→ E(U) with

E(QT UQ) = QT E(U)Q ∀ Q ∈ SO(n) ,

E(U) = 0 ⇔ U = 11 .

Note that U = 11 ⇔ F ∈ SO(n).

Strain measure

A strain measure is a mapping PSym(n)→ [0,∞) which measures how much adeformation gradient F deviates from a pure rotation.

The idea of a strain measure is closely related to energy functions in hyperelasticity.


The concept of strain

Strain tensor

A (material) strain tensor is a mapping PSym(n)→ Sym(n) , U 7→ E(U) with

E(QT UQ) = QT E(U)Q ∀ Q ∈ SO(n) ,

E(U) = 0 ⇔ U = 11 .

Note that U = 11 ⇔ F ∈ SO(n).

Strain measure


The idea of a strain measure is closely related to energy functions in hyperelasticity.


Energy functions in isotropic hyperelasticity

Isotropic linear elasticity

The elastic energy for the isotropic linearised model of elasticity is

Wlin(F ) = µ ‖ devn ε‖2 +κ

n[tr ε]2 ,

where

F = ∇ϕ is the deformation gradient,

ε = sym(F − 11) is the infinitesimal strain,

µ > 0 is the shear (distortional) modulus,

κ > 0 is the bulk (compressional) modulus,

‖X‖ = tr X T X denotes the Frobenius matrix norm,

devn ε = ε− 1n

tr(ε) · 11 is the deviatoric (purely distortional) part of ε.



Energy functions

An isotropic energy (density) is a function W : GL+(n)→ [0,∞) with

W (QF ) = W (F ) (objectivity)

W (FQ) = W (F ) (isotropy)

W (F ) = 0 ⇔ F ∈ SO(n).

The corresponding energy I of a deformation ϕ is

I (ϕ) =

∫Ω

W (∇ϕ(x)) dx .

Common additional requirements on W :

smoothness

compatibility with linear elasticity

convexity conditions, coercivity, . . .



Examples

(Compressible) Neo-Hooke energy:

WNH(F ) =µ

2

⟨C − 11, 11

⟩+ κ h(det F ) ;

(Compressible) Mooney-Rivlin energy:

WMR(F ) = C1

⟨C − 11, 11

⟩+ C2

⟨Cof C − 11, 11

⟩+ κ h(F , det F ) ,

C1 =µ1

2, C2 =

µ2

2, µ = µ1 + µ2 ;

Ogden energy:

WOg(λ1, λ2, λ3) =N∑

p=1

µp

αp(λαp

1 + λαp

2 + λαp

3 − 3) ,

2µ =N∑

p=1

µp · αp , µp · αp > 0 ;

. . .


Uniaxial stress response for different energy functions

1 2 3 4 5λ

TBiot

Neo Hooke

Mooney-Rivlin

Ogden

Figure : Uniaxial stretch-stress-curve for different constitutive models


The Hencky energy

Definition (Isotropic Hencky energy [Hencky 1929])

The isotropic Hencky energy is

WH (F ) = µ ‖ devn log U‖2 +κ

n[tr(log U)]2

where

F = ∇ϕ is the deformation gradient,

U =√

F T F is the symmetric right Biot-stretch tensor,

µ > 0 is the shear (distortional) modulus,

κ > 0 is the bulk (compressional) modulus,

log U is the principal matrix logarithm of U and

devn log U = log U − 1n

tr(log U) · 11 is the deviatoric (purely distortional) part oflog U.


The isotropic Hencky strain energy

Advantageous properties of the Hencky strain energy:

3 very good fit to experimental data for moderately large strains [Anand79]

3 fulfils Hill’s inequality: WH is a convex function of log U

3 fulfils the Baker-Ericksen inequality: (σi − σj )(λi − λj ) > 0 if λi 6= λj

3 only 2 Lame-constants, uniquely determined in infinitesimal range, but valid up tomoderate strains [Anand86]

3 describes nonlinear Poynting effect: a circular cylinder lengthens under torsion,with an increase in length proportional to the square of the twist [Bruhns00]


The isotropic Hencky strain energy

Mathematical challenges associated with the Hencky strain energy:

7 WH is not polyconvex and not quasiconvex [Neff2000].

7 WH is not Legendre-Hadamard-elliptic:

D2WH (F ).(ξ ⊗ η, ξ ⊗ η) ≥ c+ · |ξ|2 · |η|2.

However, WH is LH-elliptic in a large neighbourhood of 11 (with admissiblestretches λi ∈ (0.21, 1.4)) [Bruhns2002].

7 WH has subquadratic growth for large deformations.

7 No coercivity: There is no q ≥ 1 such that WH (F ) ≥ c+1 ‖F‖q − c2 .

7 No general existence result is known for elasticity formulation based on WH ,apart from implicit function theorem in the neighbourhood of 11.


Characterisation of energy functions

Energy functions

An isotropic energy is a function W : GL+(n)→ [0,∞) with



W (F ) = 0 ⇔ F ∈ SO(n).

Strain measure


General idea:Characterize energy functions or strain measures as the distance of F to the set ofrotations.


Characterisation of energy functions

Energy functions

An isotropic energy is a function W : GL+(n)→ [0,∞) with



W (F ) = 0 ⇔ F ∈ SO(n).

Strain measure


General idea:Characterize energy functions or strain measures as the distance of F to the set ofrotations.


Energy and strain measures: the linear case

In linearised elasticity, we consider ϕ(x) = x + u(x) with the displacement u : Ω→ Rn.

Infinitesimal rotations

The set of infinitesimal rotations is the set

so(n) = T11 SO(n) = A ∈ Rn×n |AT = −A

which is the set of all skew symmetric matrices in Rn×n.

dist(∇u, so(n)) = infS∈so(n)

dist(∇u, S) = ?


The euclidean distance on matrix spaces

Frobenius norm:

‖X‖2 = 〈X ,X 〉 = tr(X T X ) = ‖ dev X‖2 +1

n[tr X ]2

Weighted Frobenius norm:

‖X‖2µ,κ = µ ‖ dev X‖2 +

κ

n[tr X ]2

Euclidean distance:

disteuclid(X ,Y ) = ‖X − Y ‖µ,κdisteuclid(X ,Y )

X

YRn×n



The euclidean distance of ∇u to the set of (infinitesimal) rotations is

dist2euclid(∇u, so(n)) := min

A∈so(n)‖∇u − A‖2

µ,κ = ‖ sym∇u‖2µ,κ ,

which corresponds to the isotropic elastic energy

W = µ ‖ε‖2 +κ

n[tr ε]2 = µ ‖ dev sym∇u‖2 +

κ

n[tr sym∇u]2 = ‖ sym∇u‖2

µ,κ .



so(n)

Rn×n

∇u

0

sym∇u skew∇u

disteuclid(∇u, so(n))2 = ‖ sym∇u‖2µ,κ = µ ‖ dev sym∇u‖2 + κ

n[tr sym∇u]2



so(n)

Rn×n

∇u

0

sym∇u skew∇u


n[tr sym∇u]2



so(n)

Rn×n

∇u

0

sym∇u skew∇u


n[tr sym∇u]2


Energy and strain measures: the nonlinear case

In nonlinear elasticity, we consider the distance of the deformation gradientF = ∇ϕ ∈ GL+(n) to the set SO(n) of pure rotations.

The euclidean distance of F to the set of rotations is

dist2euclid(∇ϕ,SO(n)) : = min

Q∈SO(n)‖∇ϕ− Q‖2 = min

Q∈SO(n)‖QT∇ϕ− 11‖2

= ‖√∇ϕT∇ϕ− 11‖2 = ‖U − 11‖2

by a well known optimality result characterizing the polar decomposition [GiuseppeGrioli 1940]:

F = RU , R ∈ SO(n) , U ∈ PSym(n) =⇒ minQ∈SO(n)

‖QT F − 11‖2 = ‖U − 11‖2 .





dist2euclid(∇ϕ, SO(n)) : = min


Q∈SO(n)‖QT∇ϕ− 11‖2

= ‖√∇ϕT∇ϕ− 11‖2 = ‖U − 11‖2



‖QT F − 11‖2 = ‖U − 11‖2 .







Q∈SO(n)‖QT∇ϕ− 11‖2

= ‖√∇ϕT∇ϕ− 11‖2 = ‖U − 11‖2



‖QT F − 11‖2 = ‖U − 11‖2 .

G. Grioli. Una proprieta di minimo nella cinematica delle deformazioni finite.

Boll. Un. Math. Ital., 2:252–255, 1940.







Q∈SO(n)‖QT∇ϕ− 11‖2

= ‖√∇ϕT∇ϕ− 11‖2 = ‖U − 11‖2



‖QT F − 11‖2 = ‖U − 11‖2 .

Thus

dist2euclid(∇ϕ, SO(n)) = ‖E1/2‖2

where E1/2 = U − 11 is the Biot strain tensor.

Note the similarity to the Saint Venant-Kirchhoff energy ‖E1‖2µ,κ, where

E1 = 12

(U2 − 11) is the Green-Lagrangian strain.


The euclidean distance on GL+(n): only an extrinsic distance

Reconsider the euclidean distance disteuclid(A,B) = ‖A− B‖ on GL+(n).

Problems:

The Euclidean distance is an arbitrary choice for a distance measure.

disteuclid is not an intrinsic distance measure on GL+(n):Since, in general, A− B /∈ GL+(n), the term ‖A− B‖ depends on the underlyinglinear structure of Rn×n.

A,B ∈ GL+(n) ; A + t(B − A) ∈ GL+(n), thus disteuclid can not becharacterized as the length of a connecting line in GL+(n).

Generally disteuclid(P · A,P · B) 6= disteuclid(A,B), i.e. disteuclid does notrespect the algebraic Lie-group structure of GL+(n).

GL+(n) is not closed in Rn×n under disteuclid and thus GL+(n) is not completein the euclidean metric, e.g. the sequence ( 1

n· 11)n does not converge in GL+(n).

Thus disteuclid is only an extrinsic distance measure on GL+(n).



R = polar(F )

SO(n)

11

GL+(n)

F

dist2euclid(F , SO(n))

= ||U − 11||2 = ||√

F TF − 11||2

Figure : Intuitive sketch of the manifold GL+(n) and SO(n), note that GL+(n) is not compact!



R = polar(F )

SO(n)

11

GL+(n)

F


= ||U − 11||2 = ||√

F TF − 11||2

Figure : Intuitive sketch of the manifold GL+(n) and SO(n), note that GL+(n) is not compact!


GL+(n) as a Riemannian manifold

We view GL+(n) as a Riemannian manifold and consider the geodesic distance onGL+(n):

Let g be a left GL(n)-invariant Riemannian metric g on GL(n). Such a metric isdefined via a transformation of the current tangent vectors to the tangent spaceat the identity:

gA :

TA GL(n)× TA GL(n)→ R

gA(X ,Y ) = 〈A−1X ,A−1Y 〉gl(n), A ∈ GL(n) ,

with a fixed inner product 〈·, ·〉gl(n) on the tangent space T11GL(n) = gl(n) = Rn×n.

The length of a curve γ ∈ C 1([0, 1]; GL+(n)) is

L(γ) =

∫ 1

0

√gγ(t)(γ(t), γ(t)) dt =

∫ 1

0

√〈γ−1γ, γ−1γ〉g dt .

The geodesic distance between P,F ∈ GL+(n) is defined as

distgeod(P,F ) = infL(γ) | γ ∈ C 1([0, 1]; GL+(n)), γ(0) = P, γ(1) = F.


Left GL(n)-invariant, right O(n)-invariant Riemannian metrics

GL+(n)

TAGL+(n) = A · gl(n)

T11GL+(n) = gl(n)

11

A

M

N

A−1M

A−1N

A−1

gA(M,N) = 〈A−1M,A−1N〉gl(n)



GL+(n)

TAGL+(n) = A · gl(n)

T11GL+(n) = gl(n)

11

A

M

N

A−1M

A−1N

A−1

gA(M,N) = 〈A−1M,A−1N〉gl(n)



We consider Riemannian metrics that are left GL(n)-invariant:

gBA(BX ,BY ) = gA(X ,Y ) for all B ∈ GL(n) ,

as well as right O(n)-invariant:

gAQ (XQ,YQ) = gA(X ,Y ) for all Q ∈ O(n) .

right O(n)-invariance ∼= isotropy of the material

left SO(n)-invariance ∼= frame-indifference

left GL(n)-invariance ∼= distgeod(AF ,AP) = distgeod(F ,P) ∀A ∈ GL(n)


Left-invariance in GL+(n): . . . treat similar things similarly

ϕ

A A

Ω ϕ(Ω)

A · Ω A · ϕ(Ω)

dist(Ω, ϕ(Ω))

dist(A(Ω),Aϕ(Ω))

dist(Ω, ϕ(Ω)) ∼ dist(11,∇ϕ) = dist(A,A∇ϕ) ∼ dist(A(Ω),A(ϕ(Ω)))



Definition

The isotropic inner product 〈·, ·〉µ,µc ,κ on the Lie-algebra gl(n) = Rn×n = T11 GL+(n)is

〈X ,Y 〉µ,µc ,κ := µ〈devn sym X , devn sym Y 〉+ µc 〈skew X , skew Y 〉+ κn

tr X tr Y ,

where

〈X ,Y 〉 = tr(X T Y ) is the canonical inner product on gl(n),

devn sym X = sym X − 1n

tr[sym X ] · 11 is the deviatoric part of sym X ,

µ > 0 is the shear modulus,

µc > 0 is the spin modulus and

κ > 0 is the bulk modulus.

Every left GL(n)-invariant, right O(n)-invariant Riemannian metric on GL(n) has theform

gA(X ,Y ) = 〈A−1X ,A−1Y 〉µ,µc ,κ

= µ〈devn sym A−1X , devn sym A−1Y 〉+ µc 〈skew A−1X , skew A−1Y 〉+ κn

tr A−1X tr A−1Y

with constant coefficients µ, µc , κ.


A new perspective: GL+(n) as a Riemannian manifold: intrinsic distance

SO(n)

11

R = polar(F )

GL+(n)

F


= ||U − 11||2 = ||√

F TF − 11||2

dist2geod(F , SO(n)) = ?

Figure : The geodesic distance on GL+(n)


A new perspective: GL+(n) as a Riemannian manifold: intrinsic distance

SO(n)

11

R = polar(F )

GL+(n)

F


= ||U − 11||2 = ||√

F TF − 11||2

dist2geod(F , SO(n)) = ?

Figure : The geodesic distance on GL+(n)


Shortest geodesics on GL+(n)

Every geodesic curve γ : [0, 1]→ GL+(n) connecting F ,P ∈ GL+(n)is of the form [Mielke2002, MartinNeff2014]

γ(t) = F exp(t(sym ξ − µcµ

skew ξ)) exp(t(1 + µcµ

) skew ξ) , (1)

with fixed ξ ∈ gl(n) such that

P = γ(1) = F exp(sym ξ − µcµ

skew ξ) exp((1 + µcµ

) skew ξ) . (2)

Here:

exp : gl(n)→ GL+(n) is the matrix exponential,

sym ξ = 12

(ξ + ξT ) is the symmetric part and

skew ξ = 12

(ξ − ξT ) is the skew symmetric part of ξ

No closed form solution ξ to (2) for given P,F is known, but (1) can be used toobtain a lower bound for distgeod(F , SO(n)) = min

Q∈SO(n)distgeod(F ,Q).


The geodesic distance of F to SO(n)

Lower bound: (can be obtained from the geodesic parameterization)

dist2geod(F , SO(n)) = min

Q∈SO(n)dist2

geod(F ,Q) ≥ minQ∈SO(n)

‖ Log(Q F )‖2µ,µc ,κ

Upper bound: (insert a suitable orthogonal candidate)

dist2geod(F , SO(n)) ≤ dist2

geod(F , polar(F )) ≤ ‖ log(polar(F )T F )‖2µ,µc ,κ

= ‖ log U‖2µ,µc ,κ

= µ‖ dev log U‖2 +κ

n[tr(log U)]2 ,

where

F = R U is the polar decomposition,

R = polar(F ) ∈ SO(n) is the orthogonal polar factor of F and

U =√

F T F ∈ PSym(n).


The geodesic distance of F to SO(n)

Theorem (Log-optimality, Neff et al. 2013)

Let ‖ . ‖ be the Frobenius matrix norm on gl(n), F ∈ GL+(n). Then the minimum

minQ∈SO(n)

‖ Log(QT F )‖2 = ‖ log(polar(F )T F )‖2 = ‖ log(√

F T F )‖2 = ‖ log U‖2 ,

minQ∈SO(n)

µ‖ dev sym Log(QT F )‖2 + µc‖ skew Log(QT F )‖2 +κ

n[tr(Log(QT F ))]2

= µ‖ dev log(U)‖2 +κ

n[tr(log U)]2

is uniquely attained at Q = polar(F ).

The theorem holds for every unitary invariant norm ‖ . ‖ on gl(n,C) as well[Lankeit2013].

Note that the expression Log is used to indicate that the minimum is taken over alllogarithms of QT F (including non-symmetric arguments):

minQ∈SO(n)

‖ Log(QT F )‖2 = min‖X‖ : X ∈ gl(n), exp(X ) = QT F .

Combining this theorem with the upper and lower bound for distgeod(F , SO(n)) yieldsour main result.


Main result

Theorem (Main result)

Let g be any left GL(n)-invariant Riemannian metric on GL(n) that is also rightinvariant under O(n) with constant coefficients µ, µc , κ, and let F ∈ GL+(n). Then:

dist2geod(F , SO(n)) = dist2

geod(F , polar(F )) = µ‖ devn log(U)‖2 +κ

n[tr(log U)]2 .

Thus the geodesic distance of the deformation gradient F to SO(n) is the isotropicHencky strain energy of F . In particular, the result is independent of the spin modulusµc > 0.

For µc = 0, the theorem still holds for the resulting pseudometric.

Furthermore, the result is basically identical for any right GL(n)-invariant, leftO(n)-invariant metric gA(X ,Y ) = 〈XA−1,YA−1〉µ,µc ,κ.


Main result

SO(n)

11

R = polar(F )

GL+(n)

T11GL+(n) = gl(n)

T11SO(n) = so(n)

F

∇u

skew∇u

dist2euclid, gl(∇u, so(n))

= µ||devn sym∇u||2 + κn [tr∇u]2


= ||U − 11||2 = ||√

F TF − 11||2

dist2geod(F , SO(n))

= µ||devn log U||2 + κn [tr(logU)]2

Figure : Main result: The isotropic Hencky energy of F measures the geodesic distance of F toSO(n).


Main result

SO(n)

11

R = polar(F )

GL+(n)

T11GL+(n) = gl(n)

T11SO(n) = so(n)

F

∇u

skew∇u




= ||U − 11||2 = ||√

F TF − 11||2





Main result

SO(n)

11

R = polar(F )

GL+(n)

T11GL+(n) = gl(n)

T11SO(n) = so(n)

F

∇u

skew∇u




= ||U − 11||2 = ||√

F TF − 11||2





Main result

SO(n)

11

R = polar(F )

GL+(n)

T11GL+(n) = gl(n)

T11SO(n) = so(n)

F

∇u

skew∇u




= ||U − 11||2 = ||√

F TF − 11||2





Main result

SO(n)

11

R = polar(F )

GL+(n)

T11GL+(n) = gl(n)

T11SO(n) = so(n)

F

∇u

skew∇u




= ||U − 11||2 = ||√

F TF − 11||2





Main result

SO(n)

11

R = polar(F )

GL+(n)

T11GL+(n) = gl(n)

T11SO(n) = so(n)

F

∇u

skew∇u




= ||U − 11||2 = ||√

F TF − 11||2





Main result - Corollaries

The isochoric and volumetric part can be characterized separately:

Corollaries

Let F ∈ GL+(n). Then

dist2geod,SL(n)

(F

det F 1/n, SO(n)

)= µ ‖ devn log U‖2 ,

dist2geod,R+11

((det F )1/n 11, SO(n)

)=κ

2[ln(det U)]2 ,

where

distgeod,SL is the geodesic distance in SL(n) = F ∈ GL(n) | det F = 1,

distgeod,R+11 is the one-dimensional geodesic distance on R+11,

Fdet F 1/n ∈ SL(n) is the projection on the isochoric part and

(det F )1/n 11 ∈ R+11 is the projection on the volumetric part of F .

Thus the isochoric and volumetric part can be characterized separately.


Materially nonlinear extension of the isotropic Hencky energy

Result

The quantities

‖ devn log U‖ = distgeod,SL(n)

(F

det F 1/n, SO(n)

),

ln(det U) = distgeod,R+11

((det F )1/n 11, SO(n)

)are geometric properties of a deformation gradient F .

Goal

Find a well-behaved energy function

W (F ) = Ψ(‖ devn log U‖, ln(det U))

depending on those quantities alone.

The classical Hencky energy is obtained by letting Ψ(a, b) = µ a2 + κn

b2, hence forsmall strains, the Hencky model can be approximated with an appropriate Ψ.

0.7 1 1.4

σH



Result

The quantities

‖ devn log U‖ = distgeod,SL(n)

(F

det F 1/n, SO(n)

),

ln(det U) = distgeod,R+11

((det F )1/n 11, SO(n)

)are geometric properties of a deformation gradient F .

Goal






0.7 1 1.4

σH



Goal






0.7 1 1.4

σH



Goal






0.7 1 1.4

σH

σH ?


The Hencky energy for large deformations

1 2 3 4 5 6 7λ

TBiot

Treloar 1944

Neo Hooke

Mooney-Rivlin

Ogden

Hencky



Consider the exponentiated Hencky energy with isochoric-volumetric decoupling

WeH(U) =µ

kek‖ devn log U‖2

+κ

2ke k(tr log U)2

, k , k, µ, κ > 0 .




WeH(U) =µ


+κ

2ke k(tr log U)2

, k , k, µ, κ > 0 .

0.8 1 1.2 1.4

k = 2

WeH(x) = 1k e

k ln(x)2

WH(x) = ln(x)2




WeH(U) =µ


+κ

2ke k(tr log U)2

, k , k, µ, κ > 0 .

1 2 3 4 5

k = 1

k = 2

WeH(x) = 1k e

k ln(x)2

WH(x) = ln(x)2


The exponentiated Hencky energy

Uniaxial stress response for the exponentiated Hencky energy, fitted to Treloar’sexperimental data [Treloar1944]:

2 3 4 5 6

strainsoftening

strainhardening

exponentiatedHencky

Hencky

λ

TBiot

The exponentiated Hencky energy describes the effect of strain softening (the Mullinseffect) and strain hardening.




WeH(U) =µ


+κ

2ke k(tr log U)2

, k , k, µ, κ > 0 .

Two-dimensional result (Neff, Lankeit, Ghiba, Martin 2014, work in progress):

Polyconvexity

The two-dimensional exponentiated Hencky energy

WeH(U) =µ

kek‖ dev2 log U‖2

+κ

2ke k(tr log U)2

, µ, κ > 0, k > 13, k > 1

8

is polyconvex (and thus quasiconvex and rank-one convex).

WeH is not polyconvex for n = 3.

Coercivity

The exponentiated Hencky energy is coercive in all Soboloev spaces W 1,q(Ω) with1 ≤ q <∞.




WeH(U) =µ


+κ

2ke k(tr log U)2

, k , k, µ, κ > 0 .

Additional properties satisfied by WeH:

X Baker-Ericksen inequality: (σi − σj )(λi − λj ) > 0 if λi 6= λj

X tension-extension inequality: ∂σi∂λi≥ 0

X pressure-compression inequality: λ ∂σ∂λ≥ 0 for σ = σ 11, F = λ11

X true-stress-stretch invertibility: the mapping U 7→ σ(U) is invertible


The distance between pure rotations

In general, there is no closed form solution to compute distgeod(A,B) forA,B ∈ GL+(n).

Consider the GL(n)-geodesic distance distgeod(P,Q) between P,Q ∈ SO(n).

Can we explicitly compute distgeod(P,Q) ?

Does distgeod(P,Q) only depend on the spin modulus µc ?

Is distgeod(P,Q) equal to the SO(n)-geodesic distance distSO(n)(P,Q) ?
















Geodesics on SO(n)

The Riemannian metric induced on the compact Lie group SO(n)

gQ :

TQ SO(n)× TQ SO(n)→ R

gQ (X ,Y ) = µc 〈Q−1X ,Q−1Y 〉 = µc 〈X ,Y 〉 = µc tr(X T Y ), Q ∈ SO(n)

is bi-invariant (left- and right SO(n)-invariant):

gRQ (RX ,RY ) = gQ (X ,Y ) ,

gQR (XR,YR) = gQ (X ,Y ) for all Q,R ∈ SO(n) .

Geodesics on SO(n) are translated one-parameter groups:

γ(t) = Q · exp(t W ), Q ∈ SO(n), W ∈ so(n) .

The well known SO(n)-geodesic distance between Q1,Q2 ∈ SO(n) is

dist2geod, SO(n)(Q1,Q2) = µc ‖log QT

1 Q2‖2 ,

where

‖M‖ =√

tr MT M =√∑n

i,j=1 M2i j denotes the Frobenius matrix norm and

log denotes the principal logarithm on SO(n).


The geodesic distance on SO(n)

GL+ (n)

P

Q

SO(n)



GL+ (n)

P

Q

SO(n)

distSO(n)(P,Q) = µc ‖logQT1 Q2‖2



GL+ (n)

P

Q

SO(n)

distSO(n)(P,Q) = µc ‖logQT1 Q2‖2

distGL+(n)(P,Q) ?



GL+ (n)

P

Q

SO(n)

distGL+(n)(P,Q) = µc ‖logQT1 Q2‖2 ?



Proposition (Martin, Neff, work in progress)

Let n ∈ 2, 3. Then

distGL+(n)(P,Q) = distSO(n)(P,Q) for all P,Q ∈ SO(n) if µ ≥ µc ,

distGL+(n)(P,Q) 6= distSO(n)(P,Q) for some P,Q ∈ SO(n) if µ < µc .

Proposition

Let n ∈ 2, 3. Then SO(n) is geodesically convex in GL(n) if and only if µ ≥ µc .

Proposition

Let n ≥ 4. Then SO(n) is not geodesically convex in GL(n) if µ < µc .

Conjecture

Let n ≥ 4. Then SO(n) is geodesically convex in GL(n) if µ ≥ µc .







Proposition


Proposition


Conjecture








Proposition


Proposition


Conjecture








Proposition


Proposition


Conjecture




The general case:

Minimize

‖M‖2µ,µc ,κ

= µ ‖ dev sym M‖2 + µc ‖ skew M‖2 +κ

n[tr M]2

over all M ∈ Rn×n with

Q = exp(sym M − µcµ

skew M) exp((1 + µcµ

) skew M) .

Open question (the case n ≥ 4, µ = µc ):

Is

min‖M‖ : exp(MT ) · exp(2 skew M) = Q = ‖ log Q‖

for all Q ∈ SO(n), n ≥ 4?


Open problems

Work in progress:

Find a proof (or a counterexample) for the conjecture on the geodesic convexityof SO(n), n ≥ 4.

Characterize anisotropic Hencky strain energy 〈C. log U, log U〉 as a distance in anappropriate anisotropic Riemannian metric?

Reconsider the well-posedness problem for the quadratic Hencky energy (which isunknown).

Obtain geometric properties of our metric, e.g. the Levi-Civita connectioncoefficients, the Riemannian or Ricci curvature.

Thank You!


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