Efi Efrati In collaboration with Eran Sharon & Raz Kupferman September 2009 Frustrating geometry: Frustrating geometry: Non Euclidean plates Non Euclidean plates Racah Institute of Physics The Hebrew University of Jerusalem
Efi Efrati
In collaboration withEran Sharon & Raz Kupferman
September 2009
Frustrating geometry: Frustrating geometry: Non Euclidean platesNon Euclidean plates
Racah Institute of PhysicsThe Hebrew University of Jerusalem
Outline:
• Growth and geometric frustration.
• Incompatible 3D hyperelasticity.
• Non Euclidean plates: Examples.
• Reduced 2D elastic theory of non Euclidean plates.
• Non Euclidean plates: Analysis.
• An application: Almost minimal surfaces.
Outline:
• Growth and geometric frustration.
• Incompatible 3D hyperelasticity.
• Non Euclidean plates: Examples.
• Reduced 2D elastic theory of non Euclidean plates.
• Non Euclidean plates: Analysis.
• An application: Almost minimal surfaces.
Growth
Plant growth mechanics• Residual stress due to local growth.• Biological response to mechanical stress.• Manipulating growth regulation causes morphological changes.
Arabidopsis mutant Peach leaf curl CINCINNATA mutant, [Nath et al 2003]
Time scale separation
Plant growth (Arabidopsis):  Cell division hours 2 % area growth by expansion hours
 Acoustic time 1 μs Cantilever mode typical time 100 μs
ShapingElastic ττ <<
Spontaneous growth and geometric frustration
Why local growth is likely to result in residual stress ?
Given 8 points 28 rods of arbitrary lengths, you will (probably) not be able to connect every pair once.
“think in rod length rather than points”
Example: Isotropic nonhomogeneous growth.Every point initially at rest expands isotropically by a factor .
Spontaneous growth and geometric frustration
)(rrλ
Why local growth is likely to result in residual stress ?
Example: Isotropic nonhomogeneous growth.Every point initially at rest expands isotropically by a factor .
Spontaneous growth and geometric frustration
)(rrλ
Why local growth is likely to result in residual stress ?
The only expansion factor which does not result in residual stress : 2
0 )(
rr
Cr rrr
−=λ
Soft active deformations
Residual stress must be addressed when describing motion by autodeformation.
Making use of unused “residual work”.
Signature of geometric frustration
Tempered glass fragments cannot be “rejoined”When tempered glass shatters, each of its fragments deforms to relax internal stresses. The relaxed fragments do not fit one another.
Signature of geometric frustration
The helicoidal form of a Bauhinia pod is residually stressed. Slicing along its length generate helices of a higher pitch.
Outline:
• Growth and geometric frustration.
• Incompatible 3D hyperelasticity.
• Non Euclidean plates: Examples.
• Reduced 2D elastic theory of non Euclidean plates.
• Non Euclidean plates: Analysis.
• An application: Almost minimal surfaces.
Hyperelasticity
A measure of local deformation: Strain
Local elastic energy density as a function of the deformation
+
=Hyperelastic description
[Truesdell 1952]
∫∫∫= dVWE )(ε
ε∂∂
=W
SStress is obtained as a Frechet derivative of the elastic energy density.
A measure of deformation: The Strain
Stress free configuration
x
Deformed configuration
r)(xr
xru −=The displacement field
( ) ( )Irruuuu TTT −∇∇=∇∇+∇+∇=ε )()()( 21
21 Cauchy StVenant strain
A measure of deformation: The Strain
Stress free configuration
x
Deformed configuration
r)(xr
xru −=The displacement field
Rely on the existence of a stress free configuration
Differential geometry I : Three dimensions
)(xr• The metric tensor:
• Length element:
A 3 by 3 matrix is the metric of a body in Euclidean space.
jiij dxdxgds =2
jiijx
r
x
rg
∂∂
⋅∂∂
=rr
All components of the Riemann curvature tensor vanish.
Such metrics are called:
• Euclidean
• Compatible
• Embeddable
If two embeddable metrics are identical, the bodies they describe differ by a rigid motion.
The energy stored within a deformed elastic body is a volume integral of an elastic energy density which depends only on the local metric tensor and on tensors that characterize the body, but are independent of the configuration.
3D “incompatible” hyperelasticity
3D metric determine a configuration uniquely.
Our adaptation of Truesdell’s hyperelasticityprinciple is formulated in terms of metric.
• The energy density depends on the configuration through the metric.
• For every there exists a unique SPD tensor for which the energy density vanish, i.e.
• is called the reference metric.
• The CauchySt. Venant strain
current configuration metric reference metric
3D “incompatible” hyperelasticity
)( xgW ,
)(xgx
0))(( =xxgW ,
)(xg( )gg −= 2
1ε
0 0)( =⇔= εε xW ,
When a stress free configuration exists, we may set .
In such coordinates the CauchySt. Venant strain adopts the familiar form
3D “incompatible” hyperelasticity
Ig =
( ) ( ) ( )uuuuIrrIg TTT ∇∇+∇+∇=−∇∇=−=ε )()()( 21
21
21
When a stress free configuration exists, we may set .
In such coordinates the CauchySt. Venant strain adopts the familiar form
3D “incompatible” hyperelasticity
Lame coefficients
Ig =
( )3εεε OAW klijijkl += '
( ) ( ) ( )uuuuIrrIg TTT ∇∇+∇+∇=−∇∇=−=ε )()()( 21
21
21
For such a case the elastic energy density of an isotropic material is of the form
( )jkiljlikklijijklA δδδδμδλδ ++='
When a stress free configuration exists, we may set .
In such coordinates the CauchySt. Venant strain adopts the familiar form
3D “incompatible” hyperelasticity
Lame coefficients
Ig =
( )3εεε OAW klijijkl += '
( ) ( ) ( )uuuuIrrIg TTT ∇∇+∇+∇=−∇∇=−=ε )()()( 21
21
21
For such a case the elastic energy density of an isotropic material is of the form
( )jkiljlikklijijklA δδδδμδλδ ++='
A stress free configuration may not exist.
Locally (only at a point) we may set new coordinates in which
3D “incompatible” hyperelasticity
klijijklAW ''' εε2
1=
( )ijijij g δ−=ε '' 21
( )jkiljlikklijijklA δδδδμδλδ ++='
Ig ='
Locally (only at a point) we may set new coordinates in which
3D “incompatible” hyperelasticity
klijijklAW ''' εε2
1=
( )ijijij g δ−=ε '' 21
( )jkiljlikklijijklA δδδδμδλδ ++='
Ig ='
Tensor transformation rules (Covariance becomes valuable)
ikki xx ∂∂=Λ /'
ikki xx '/ ∂∂=)(Λ1
Locally (only at a point) we may set new coordinates in which
3D “incompatible” hyperelasticity
klijijklAW ''' εε2
1=
( )ijijij g δ−=ε '' 21
( )jkiljlikklijijklA δδδδμδλδ ++='
klijijklAW εε2
1=
( )ijijij gg −= 21ε
( )jkiljlikklijijkl ggggggA ++= μλ
Ig ='
Tensor transformation rules (Covariance becomes valuable)
ikki xx ∂∂=Λ /'
ikki xx '/ ∂∂=)(Λ1
Locally (only at a point) we may set new coordinates in which
3D “incompatible” hyperelasticity
klijijklAW ''' εε2
1=
( )ijijij g δ−=ε '' 21
( )jkiljlikklijijklA δδδδμδλδ ++='
klijijklAW εε2
1=
( )ijijij gg −= 21ε
( )jkiljlikklijijkl ggggggA ++= μλ
Ig ='
kllj
kiijg δΛΛ=
Tensor transformation rules (Covariance becomes valuable)
ikki xx ∂∂=Λ /'
ikki xx '/ ∂∂=)(Λ1
Recapitulation
3D “incompatible” hyperelasticity
ki
ik
kk
iiklij
ijklAW εμεεεεε λ +== 221( )ijijij gg −= 2
1ε
321 dxdxdxgWE ∫∫∫= 
Strain is measured with respect to a reference metric.
[Efrati Sharon & Kupferman, JMPS 2009]
Recapitulation
3D “incompatible” hyperelasticity
The reference metric need not be immersible.
ki
ik
kk
iiklij
ijklAW εμεεεεε λ +== 221( )ijijij gg −= 2
1ε
321 dxdxdxgWE ∫∫∫= 
Strain is measured with respect to a reference metric.
[Efrati Sharon & Kupferman, JMPS 2009]
Outline:
• Growth and geometric frustration.
• Incompatible 3D hyperelasticity.
• Non Euclidean plates: Examples.
• Reduced 2D elastic theory of non Euclidean plates.
• Non Euclidean plates: Analysis.
• An application: Almost minimal surfaces.
Environmentally responsive gels .
• NIsopropylacrylamide undergoes a large volume reduction
when heated above 33 co .
• The volume reduction ratio depends strongly on monomer
concentration.
• When injected cold with a catalyst, the gel polymerizes in
a few minutes, “freezing” the monomer concentration
gradient.
Experimental realization of Non Euclidean plates
[Klein Efrati & Sharon. Science 2007]
Experimental realization of Non Euclidean plates
[Klein Efrati & Sharon. Science 2007]
Does not shrink
Shrinkage homogeneous across the thickness
Shrinks considerably
Large thickness: Flat and strained.
Small thickness: Buckled.
Experimental realization of Non Euclidean plates
Does not shrink
Shrinkage homogeneous across the thickness
Shrinks considerably
Large thickness: Flat and strained.
Small thickness: Buckled.
Experimental realization of Non Euclidean plates
Experimental realization of Non Euclidean plates
Hyperbolic metric (saddle like everywhere)
Experimental realization of Non Euclidean plates
The only shaping mechanism is the Prescription of a NonEuclidean Metrics
Shaping by metric prescription
Crocheted segment of the hyperbolic plane [Taimina 1997].
Uncontrolled tearing
Controlled tearing
[Nath etal 2003]
[E. Coen etal 2003]
Leaf shaping Plastic deformation Crocheting & knitting
Computer simulations
[Seung & Nelson 1988]
[Koehl et al 2008]
Defects in flexible membranes with crystalline order
[Marder etal 2004]
Three layers of identical array of springs of spatially varying rest length.
Recent implementation to macroalgae blades
Outline:
• Growth and geometric frustration.
• Incompatible 3D hyperelasticity.
• Non Euclidean plates: Examples.
• Reduced 2D elastic theory of non Euclidean plates.
• Non Euclidean plates: Analysis.
• An application: Almost minimal surfaces.
+

Differential geometry II : Surfaces
The first quadratic form (metric)
The second quadratic form (curvatures)
rra βααβ ∂⋅∂=
Nrb ˆ⋅∂= αβαβ
( ) ( )βα=κ+κ= bTrH 21
2121 ( )βα=κκ= bK det21
),,( αβδγαβγαβ ∂∂∂= aaaFKGauss’ Theorema Egregium
The mean curvature The Gaussian curvature
+

Differential geometry II : Surfaces
The first quadratic form (metric)
The second quadratic form (curvatures)
rra βααβ ∂⋅∂=
Nrb ˆ⋅∂= αβαβ
( ) ( )βα=κ+κ= bTrH 21
2121 ( )βα=κκ= bK det21
),,( αβδγαβγαβ ∂∂∂= aaaFKGauss’ Theorema Egregium
The mean curvature The Gaussian curvature
),,,(0 2,1 αβγαβαβγαβ ∂∂= bbaaG
Peterson Mainardi Codazzi eq.
+

Differential geometry II : Surfaces
The first quadratic form (metric)
The second quadratic form (curvatures)
rra βααβ ∂⋅∂=
Nrb ˆ⋅∂= αβαβ
),,( αβδγαβγαβ ∂∂∂= aaaFKGauss’ Theorema Egregium
),,,(0 2,1 αβγαβαβγαβ ∂∂= bbaaG
Peterson Mainardi Codazzi eq.
Given two quadratic forms which satisfy all three GPMC equations, they define a surface uniquely.
A toy model
Too long
Similar lengths
Similar lengths
Small thickness: Buckled configuration. The trapezoid is bent to accommodate the excess length.
Large thickness: Flat configuration. The trapezoid is under compression.
Trapezoidal inclusion
• “Rest lengths” are continuous
• The body possess no internal structure across the thin dimension
No stress free configuration
A toy model
Neither plates nor shells
A plate may be considered as a stack of identical surfaces
Short
Long
A shell may be considered as a continuous collection of nonidentical surfaces
Similarly to plates: no structural variation across the thin dimension
Unlike plates: 2D metric of midsurface nonEuclidean
Neither plates nor shells
A plate may be considered as a stack of identical surfaces
Short
Long
A shell may be considered as a continuous collection of nonidentical surfaces
Similarly to plates: no structural variation across the thin dimension
Existing plate/shell theories do no apply
Unlike plates: 2D metric of midsurface nonEuclidean
Neither plates nor shells
denotes the coordinate along the thin dimension
NonEuclidean plate metric
23
2tt x ≤≤−
.0,10000
32221
1211
=∂⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛= αβaaa
aa
g
0)(0)( 32 ≠⇔≠ DijklD gRaK
Reduced 2D elastic energy
• Integrate out the thin dimension.
• Assumption: Faces of sheet are free of forces.
[ ]∫ ∫∫−
− ⎟⎟⎠
⎞⎜⎜⎝
⎛+
+=
2
2
32121)1(2
t
t
dxdxdxgY
E kk
ii
ik
ki εεεε
ν νν
Goal: Describe the elastic behavior, using only midsurface properties; First and second fundamental forms.
Reduced 2D elastic energy
Goal: Describe the elastic behavior, using only midsurface properties; First and second fundamental forms.
The first quadratic form (metric) of the midsurface
rra βααβ
The second quadratic form (curvatures) of the midsurface
∂⋅∂= Nrb ˆ⋅∂= αβαβ
( )γδαβννβδαγ
ναβγδ aaaaA Y
−+ += 11
tohdxdxabbAt
dxdxaaaaaAtE
..
))((21
2413
2181
+
+−−=
∫∫∫∫
γδαβαβγδ
γδγδαβαβαβγδ
Reduced 2D elastic energy
BS etetE 3+=
Stretching content Bending content
2181 ))(( dxdxaaaaaAeS ∫∫ γδγδαβαβ
αβγδ −−=
21241  dxdxabbAeB ∫∫ γδαβ
αβγδ=
Generalization of F.V.K and Koiter theories for large displacements and arbitrary intrinsic geometry.
Bending content measures the magnitude of curvatures.
Vanishes only for flat configuration .
Stretching content measures deviations from the given 2D metric.
Vanishes only if the midsurface is an isometric embedding of .a
Reduced 2D elastic energy
BS etetE 3+=
Stretching content Bending content
2181 ))(( dxdxaaaaaAeS ∫∫ γδγδαβαβ
αβγδ −−=
21241  dxdxabbAeB ∫∫ γδαβ
αβγδ=
Small thickness:
• Stretching term dominates
• 2D isometric solution
Large thickness:
• Bending term dominates
• Flat solutions (unbuckled)
Note: and are not independent variables.αβbαβa
[Efrati Shron & Kupferman JMPS 2009]
Reduced 2D elastic energy
[Cerda & Mahadevan 20034][Huang etal 2007]
Current literature: Elastic substrate, gravitational force, capillary forces, tension, confinement or some other mechanism necessitates buckling. Competition between the two energy terms yields the buckling length scales.
[Vella etal 2009]
Reduced 2D elastic energy
[Cerda & Mahadevan 20034][Huang etal 2007]
Current literature: Elastic substrate, gravitational force, capillary forces, tension, confinement or some other mechanism necessitates buckling. Competition between the two energy terms yields the buckling length scales.
No external force; Confinement is replaced by an embedding problem.
[Efrati Klein & Sharon. Physica D 2007]
[Vella etal 2009]
Outline:
• Growth and geometric frustration.
• Incompatible 3D hyperelasticity.
• Non Euclidean plates: Examples.
• Reduced 2D elastic theory of non Euclidean plates.
• Non Euclidean plates: Analysis.
 Buckling
 The vanishing thickness limit
 Equipartition
• An application: Almost minimal surfaces.
Hemispherical plate
We numerically minimize the elastic energyBS etteE 3+=
We use a reference metric of a spherical cap
The solution is assumed to be axially symmetric
⎟⎟⎠
⎞⎜⎜⎝
⎛=
)(001
12 xa
sin
1110 1 .. ≤≤ x
Hemispherical plate
BS etteE 3+=
Hemispherical plate
BS etteE 3+=
∞Ste
03Bet
Hemispherical plate
BS etteE 3+=
∞Ste
03Bet
Similar flat configurations
Almost isometric embedding
Hemispherical plate
BS etteE 3+=
∞Ste
03Bet
Similar flat configurations
Almost isometric embedding
Slightly buckled
Hemispherical plate
∞Ste
Similar flat configurations
0)( ≠aK
Analytic results:
• The unbuckled state must contain both compression and tension.
• If , there exists a finite buckling thickness.
• Classification in terms of the plane stress in the unbuckled configuration to supercritical and subcritical bifurcations.
[Efrati etal PRE 2009]
Hemispherical plate
We have proved that In the limit
i.e. an isometric embedding of midsurface.
0→t αβαβ aa =
Hemispherical plate
We have proved that In the limit
i.e. an isometric embedding of midsurface.
0→t αβαβ aa =
The isometry is (usually) not unique.
The limit configuration will minimize the bending content amongst all isometries.
Recent Gamma limit proof for non Euclidean plates directly from 3D [Lewicka & Pakzad 2009].
Hemispherical plate
BS etteE 3+=
27t∝
03Bet
00⎯⎯→⎯→tSe
00B
tB ee ⎯⎯→⎯→
No mutual zero to the two terms; No equipartition.
Hemispherical plate
( ) 2703 teet BB ∝−
The variation of the bending energy about its limit
BS ete && 2−=
BS etteE 3+=
00⎯⎯→⎯→tSe
00B
tB ee ⎯⎯→⎯→
Hemispherical plate
( ) 2703 teet BB ∝−
The variation of the bending energy about its limit
The power 7/2 comes from a boundary layer whose thickness scales as
.
k
tl LB ∝
[Efrati etal PRE 2009]
Outline:
• Growth and geometric frustration.
• Incompatible 3D hyperelasticity.
• Non Euclidean plates: Examples.
• Reduced 2D elastic theory of non Euclidean plates.
• Non Euclidean plates: Analysis.
• An application: Almost minimal surfaces.
Solving the vanishing thickness limit
Gaussian curvature Mean curvature
Provided there exists an isometric embedding with bounded energycontent, the vanishing limit configuration minimizes the Willmoreenergy among all isometric embeddings.
( ) ( )dAKHdAkkkk ∫∫∫∫ ν−−=ν++ 212
2122
214
1 2
21kkK = ( )2121 kkH +=
Solving the vanishing thickness limit
Gaussian curvature Mean curvature
[Willmore 1986]
Constrained minimization problem: The GMPC equations.
Provided there exists an isometric embedding with bounded energycontent, the vanishing limit configuration minimizes the Willmoreenergy among all isometric embeddings.
dAH∫∫ 2
21kkK = ( )2121 kkH +=
Differential geometry III: Isometric embedding of surfaces in 3D
The Gauss Peterson Mainardi Codazzi compatability conditions may be considered as evolution equations for the curvature form, .
• For elliptic set of P.D.Es .
• For hyperbolic set of P.D.Es .
K<0
0<K
αβb
2
91 −= cmK
2
91 −−= cmK
cmt 06.0=cmt 075.0= cmt 019.0=cmt 025.0=
Almost minimal surfaces
Minimal surfaces are surfaces of vanishing mean curvature
• Constitute trivial minima of the Willmore energy.
• Not every metric can be embedded as a minimal surface.
H = 0
dAH∫∫ 2
Almost minimal surfaces
Minimal surfaces are surfaces of vanishing mean curvature
• Constitute trivial minima of the Willmore energy.
• Not every metric can be embedded as a minimal surface.
H = 0
The mean curvature of any hyperbolic metric can vanish along a curve
dAH∫∫ 2
Almost minimal surfaces
We considering a thin narrow hyperbolic strip Lwt <<<<
dAHE
w
w
W ∫ ∫−
=2
2
2
w
L
Almost minimal surfaces
We considering a thin narrow hyperbolic strip Lwt <<<<
( )dxweweweeEW ......33
2210 ++++= ∫
w
L
Almost minimal surfaces
We considering a thin narrow hyperbolic strip
• On the midline of the strip the mean curvature vanishes.
• The resulting ribbons resemble minimal surfaces.
• Both chiralities possess the same energy, thus are equally probable.
Lwt <<<<
)( 5wOEW =
Almost minimal surfacesInspired by the result that every (narrow enough) hyperbolic metric can be embedded as an almost minimal surface, we found new family of pseudospherical embeddings (surfaces of constant Gaussian curvature: K=1).
Almost minimal surfacesInspired by the result that every (narrow enough) hyperbolic metric can be embedded as an almost minimal surface, we found new family of pseudospherical embeddings (surfaces of constant Gaussian curvature: K=1).
The new family of pseudospherical embeddings also admits embeddings of very large mean curvature.
Almost minimal surfaces
Shrinks
Do not Shrink
Prescribing a hyperbolic geometry on a strip using responsive gels
Almost minimal surfaces
Shrinks
Do not Shrink
Prescribing a hyperbolic geometry on a strip using responsive gels
• The mean curvature quadratic in the width coordinate.
Almost minimal surfaces
0.0 0.4 0.8 1.2 1.6 2.00.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
CenterEdge H2 /K
Cur
vatu
re S
quar
ed (m
m2)
Distance from the Edge (mm)
K H2
H2/K
Preliminary results
• The mean curvature quadratic in the width coordinate.
• Results become irrelevant for wide strips.
Almost minimal surfaces
0.0 0.4 0.8 1.2 1.6 2.00.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
CenterEdge H2 /K
Cur
vatu
re S
quar
ed (m
m2)
Distance from the Edge (mm)
K H2
H2/K
Preliminary results
Conclusion
• Residual stress inevitable in local shaping mechanisms.
• “Incompatible” elasticity is the theoretical tool to deal with residual stress.
• NonEuclidean plates:  Derivation from 3D elasticity.
 Buckling, Boundary layer.
 The isometric embedding problem.
• Application: Almost minimal surfaces.
Conclusion
• Residual stress inevitable in local shaping mechanisms.
• “Incompatible” elasticity is the theoretical tool to deal with residual stress.
• NonEuclidean plates:  Derivation from 3D elasticity.
 Buckling, Boundary layer.
 The isometric embedding problem.
• Application: Almost minimal surfaces.
• Natural extension: NonEuclidean shells.
Elastic theory of nonEuclidean plates and shells [submmited to Nonlinearity 2009]
• Geometric applications: Folding and controllability of non Euclidean plates.
The EndThe End