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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
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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)
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Energy functions in isotropic hyperelasticity
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, . . .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Energy functions in isotropic hyperelasticity
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 ;
. . .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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]
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Characterisation of energy functions
Energy functions
An isotropic energy is a function W : GL+(n)→ [0,∞) with
W (QF ) = W (F ) (objectivity)
W (FQ) = W (F ) (isotropy)
W (F ) = 0 ⇔ 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.
General idea:Characterize energy functions or strain measures as the distance of F to the set ofrotations.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Characterisation of energy functions
Energy functions
An isotropic energy is a function W : GL+(n)→ [0,∞) with
W (QF ) = W (F ) (objectivity)
W (FQ) = W (F ) (isotropy)
W (F ) = 0 ⇔ 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.
General idea:Characterize energy functions or strain measures as the distance of F to the set ofrotations.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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) = ?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Energy and strain measures: the linear case
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Energy and strain measures: the linear case
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Energy and strain measures: the linear case
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Energy and strain measures: the linear case
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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 .
G. Grioli. Una proprieta di minimo nella cinematica delle deformazioni finite.
Boll. Un. Math. Ital., 2:252–255, 1940.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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 .
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The euclidean distance on GL+(n): only an extrinsic distance
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!
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The euclidean distance on GL+(n): only an extrinsic distance
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!
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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)
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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)
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Left GL(n)-invariant, right O(n)-invariant Riemannian metrics
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)
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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)))
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Left GL(n)-invariant, right O(n)-invariant Riemannian metrics
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
A new perspective: GL+(n) as a Riemannian manifold: intrinsic distance
SO(n)
11
R = polar(F )
GL+(n)
F
dist2euclid(F , SO(n))
= ||U − 11||2 = ||√
F TF − 11||2
dist2geod(F , SO(n)) = ?
Figure : The geodesic distance on GL+(n)
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
A new perspective: GL+(n) as a Riemannian manifold: intrinsic distance
SO(n)
11
R = polar(F )
GL+(n)
F
dist2euclid(F , SO(n))
= ||U − 11||2 = ||√
F TF − 11||2
dist2geod(F , SO(n)) = ?
Figure : The geodesic distance on GL+(n)
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
dist2euclid(F , SO(n))
= ||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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
dist2euclid(F , SO(n))
= ||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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
dist2euclid(F , SO(n))
= ||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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
dist2euclid(F , SO(n))
= ||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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
dist2euclid(F , SO(n))
= ||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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
dist2euclid(F , SO(n))
= ||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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Materially nonlinear extension of the isotropic Hencky energy
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Materially nonlinear extension of the isotropic Hencky energy
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
σH ?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The Hencky energy for large deformations
1 2 3 4 5 6 7λ
TBiot
Treloar 1944
Neo Hooke
Mooney-Rivlin
Ogden
Hencky
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Materially nonlinear extension of the isotropic Hencky energy
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Materially nonlinear extension of the isotropic Hencky energy
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 .
0.8 1 1.2 1.4
k = 2
WeH(x) = 1k e
k ln(x)2
WH(x) = ln(x)2
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Materially nonlinear extension of the isotropic Hencky energy
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 .
1 2 3 4 5
k = 1
k = 2
WeH(x) = 1k e
k ln(x)2
WH(x) = ln(x)2
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Materially nonlinear extension of the isotropic Hencky energy
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 .
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 <∞.
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
Materially nonlinear extension of the isotropic Hencky energy
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 .
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
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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) ?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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) ?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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) ?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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).
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
GL+ (n)
P
Q
SO(n)
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
GL+ (n)
P
Q
SO(n)
distSO(n)(P,Q) = µc ‖logQT1 Q2‖2
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
GL+ (n)
P
Q
SO(n)
distSO(n)(P,Q) = µc ‖logQT1 Q2‖2
distGL+(n)(P,Q) ?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
GL+ (n)
P
Q
SO(n)
distGL+(n)(P,Q) = µc ‖logQT1 Q2‖2 ?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
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 .
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
The geodesic distance on SO(n)
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?
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen
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!
Robert Martin Geodesics on GL(3) and nonlinear elasticity Faculty of Mathematics, Universitat Duisburg-Essen