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Comportamento macroscopico e microscopico di un
sottile nastro elastico.
Giuseppe Tomassetti
Arezzo, 25 gennaio 2019
Collaborazione con Riccardo Barsotti e Roberto Paroni
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Controlled shape change in thin elastic strips
From: S. Armon et al., Science, 2011.
Thin elastic strips are interesting because
they exhibit radical shape changes upon
small changes of texture or environmental conditions.
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Elastic ribbons
A ribbon is body having the shape of a rectangular parallelepiped with
thickness ε << width h << length `
The body prefers to accommodate constraints, loads, or geometrical incompatibilitiesby undergoing bending rather than stretching.
For this reason, most effects can be captured by modeling elastic ribbons as unstretchableplates.
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Elastic ribbons
A ribbon is body having the shape of a rectangular parallelepiped with
thickness ε << width h << length `
The body prefers to accommodate constraints, loads, or geometrical incompatibilitiesby undergoing bending rather than stretching.
For this reason, most effects can be captured by modeling elastic ribbons as unstretchableplates.
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Elastic ribbons
A ribbon is body having the shape of a rectangular parallelepiped with
thickness ε << width h << length `
The body prefers to accommodate constraints, loads, or geometrical incompatibilitiesby undergoing bending rather than stretching.
For this reason, most effects can be captured by modeling elastic ribbons as unstretchableplates.
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Application example: a deployable structure
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Michael Sadowsky
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Sadowsky’s quest for the Mobius band
In a 1930 paper∗ Sadowsky addresses the question whether a Mobius band could be actuallyrealized from a sheet of paper.
• he models paper as an elastic plate which can bend but cannot stretch;
• by assembling pieces of flat and cylindrical surfaces, he constructs a constrained Mobiusband;
From: Schwarz, G. E. (Am. Math. Mon. 1990) The Dark Side of the Moebius Strip.
• Sadowsky poses the problem of what would be the shape of an unconstrained Mobiusband.
∗ See also: D.F. Hinz and E. Fried (2014) Translation of Michael Safowsky’s Paper “An Elementary Prooffor the Existence of a Developable Mobius Band and the Attribution of the Geometric Problem to a VariationalProblem.
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What is the shape of an unconstrained Mobius band?
Sadowsky assumes that in the absence of loads or constraints the shape of the band can befound by solving
min∫ `
0
[∫ +h/2
−h/2(tr K)2dx2
]dx1,
where ` is the length, h is the width, and K is the second fundamental form.
Considering this formulation too hard to attack, he turns his attention to an infinitesimally-narrow band, characterized by:
h/` << 1.
He argues that for such a band the equilibria can be found by solving:
min∫ `
0WS(τ, κ)dx1,
where τ is the torsion and κ is the principal curvature of the midline, and where
WS(τ, κ) =(τ2 + κ2)2
κ2 if κ 6= 0,
and WS(τ, κ) = 0 otherwise.
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What is the shape of an unconstrained Mobius band?
Sadowsky assumes that in the absence of loads or constraints the shape of the band can befound by solving
min∫ `
0
[∫ +h/2
−h/2(tr K)2dx2
]dx1,
where ` is the length, h is the width, and K is the second fundamental form.
Considering this formulation too hard to attack, he turns his attention to an infinitesimally-narrow band, characterized by:
h/` << 1.
He argues that for such a band the equilibria can be found by solving:
min∫ `
0WS(τ, κ)dx1,
where τ is the torsion and κ is the principal curvature of the midline, and where
WS(τ, κ) =(τ2 + κ2)2
κ2 if κ 6= 0,
and WS(τ, κ) = 0 otherwise.
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What is the shape of an unconstrained Mobius band?
Sadowsky assumes that in the absence of loads or constraints the shape of the band can befound by solving
min∫ `
0
[∫ +h/2
−h/2(tr K)2dx2
]dx1,
where ` is the length, h is the width, and K is the second fundamental form.
Considering this formulation too hard to attack, he turns his attention to an infinitesimally-narrow band, characterized by:
h/` << 1.
He argues that for such a band the equilibria can be found by solving:
min∫ `
0WS(τ, κ)dx1,
where τ is the torsion and κ is the principal curvature of the midline, and where
WS(τ, κ) =(τ2 + κ2)2
κ2 if κ 6= 0,
and WS(τ, κ) = 0 otherwise.
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What is the shape of an unconstrained Mobius band?
Sadowsky assumes that in the absence of loads or constraints the shape of the band can befound by solving
min∫ `
0
[∫ +h/2
−h/2(tr K)2dx2
]dx1,
where ` is the length, h is the width, and K is the second fundamental form.
Considering this formulation too hard to attack, he turns his attention to an infinitesimally-narrow band, characterized by:
h/` << 1.
He argues that for such a band the equilibria can be found by solving:
min∫ `
0WS(τ, κ)dx1,
where τ is the torsion and κ is the principal curvature of the midline, and where
WS(τ, κ) =(τ2 + κ2)2
κ2 if κ 6= 0,
and WS(τ, κ) = 0 otherwise.
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Isometric deformations of the band
The band initially occupies a rectangular domain of length ` and width h, in the (x1, x2)plane:
ω = (0, `)× (−h/2, h/2).
The typical configuration of the strip is specified by the deformation χ : ω → R3.
In-plane inextensibility yields∂αχ · ∂βχ = δαβ.
For n = ∂1χ ∧ ∂2χ and F = ∇χ, the second fundamental form is
K = −F>∇n.
Theorema Egregium implies thatdet K = 0.
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Isometric deformations of the band
The band initially occupies a rectangular domain of length ` and width h, in the (x1, x2)plane:
ω = (0, `)× (−h/2, h/2).
The typical configuration of the strip is specified by the deformation χ : ω → R3.
In-plane inextensibility yields∂αχ · ∂βχ = δαβ.
For n = ∂1χ ∧ ∂2χ and F = ∇χ, the second fundamental form is
K = −F>∇n.
Theorema Egregium implies thatdet K = 0.
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Isometric deformations of the band
The band initially occupies a rectangular domain of length ` and width h, in the (x1, x2)plane:
ω = (0, `)× (−h/2, h/2).
The typical configuration of the strip is specified by the deformation χ : ω → R3.
In-plane inextensibility yields∂αχ · ∂βχ = δαβ.
For n = ∂1χ ∧ ∂2χ and F = ∇χ, the second fundamental form is
K = −F>∇n.
Theorema Egregium implies thatdet K = 0.
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Isometric deformations of the band
The band initially occupies a rectangular domain of length ` and width h, in the (x1, x2)plane:
ω = (0, `)× (−h/2, h/2).
The typical configuration of the strip is specified by the deformation χ : ω → R3.
In-plane inextensibility yields∂αχ · ∂βχ = δαβ.
For n = ∂1χ ∧ ∂2χ and F = ∇χ, the second fundamental form is
K = −F>∇n.
Theorema Egregium implies thatdet K = 0.
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Isometric deformations of the band
The band initially occupies a rectangular domain of length ` and width h, in the (x1, x2)plane:
ω = (0, `)× (−h/2, h/2).
The typical configuration of the strip is specified by the deformation χ : ω → R3.
In-plane inextensibility yields∂αχ · ∂βχ = δαβ.
For n = ∂1χ ∧ ∂2χ and F = ∇χ, the second fundamental form is
K = −F>∇n.
Theorema Egregium implies thatdet K = 0.
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Isometric deformations of the band
The band initially occupies a rectangular domain of length ` and width h, in the (x1, x2)plane:
ω = (0, `)× (−h/2, h/2).
The typical configuration of the strip is specified by the deformation χ : ω → R3.
In-plane inextensibility yields∂αχ · ∂βχ = δαβ.
For n = ∂1χ ∧ ∂2χ and F = ∇χ, the second fundamental form is
K = −F>∇n.
Theorema Egregium implies thatdet K = 0.
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The building blocks of Sadowsky’s construction
The constraint det K = 0 implies that K is proportional to
K(θ) = I− b(θ)⊗ b(θ).
whereb(θ) = cos θe1 + sin θe2,
for some θ.
From: D. Hinz and E. Fried (J. Elasticity 2015).
In Sadowsky’s construction
θ = ±π
3.
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The material frame
By the inextensibility constraint, the spatial curve defined by
r(x1) := χ(x1, x2)
is the arc-length parametrization of the midline in the deformed configuration.
In-plane inextensibility also implies that the material frame
d1(x1) = r′(x1), d2(x) = ∂2χ(x1, 0), d3(x) = n(x1, 0)
constitute an orthonormal triad.
The material frame uniquely identifies the bending-twist vector k by the property
d′i = k ∧ di.
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The Frenet frame is material
The components of k are:
k1 = d′2 · d3 twist;
k2 = d′3 · d1 curvature around d2;
k3 = d′1 · d2 curvature around d3.
Unstretchability impliesk3 = 0,
Thus, if k2 6= 0, then d3 coincides with the Frenet normal and
τ2 = k21, κ2 = k2
2.
Sadowsky refers to this property by writing that the Frenet frame of a band is material.
Moreover, he uses this property as a kinematic characterization of a band.
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The Frenet frame is material
The components of k are:
k1 = d′2 · d3 twist;
k2 = d′3 · d1 curvature around d2;
k3 = d′1 · d2 curvature around d3.
Unstretchability impliesk3 = 0,
Thus, if k2 6= 0, then d3 coincides with the Frenet normal and
τ2 = k21, κ2 = k2
2.
Sadowsky refers to this property by writing that the Frenet frame of a band is material.
Moreover, he uses this property as a kinematic characterization of a band.
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The Frenet frame is material
The components of k are:
k1 = d′2 · d3 twist;
k2 = d′3 · d1 curvature around d2;
k3 = d′1 · d2 curvature around d3.
Unstretchability impliesk3 = 0,
Thus, if k2 6= 0, then d3 coincides with the Frenet normal and
τ2 = k21, κ2 = k2
2.
Sadowsky refers to this property by writing that the Frenet frame of a band is material.
Moreover, he uses this property as a kinematic characterization of a band.
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The Frenet frame is material
The components of k are:
k1 = d′2 · d3 twist;
k2 = d′3 · d1 curvature around d2;
k3 = d′1 · d2 curvature around d3.
Unstretchability impliesk3 = 0,
Thus, if k2 6= 0, then d3 coincides with the Frenet normal and
τ2 = k21, κ2 = k2
2.
Sadowsky refers to this property by writing that the Frenet frame of a band is material.
Moreover, he uses this property as a kinematic characterization of a band.
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Sadowsky’s energy
The restriction K◦ to the midline of second fundamental form K is determined by k1, k2,except for γ := K◦22:
K◦ =(−k2 k1k1 γ
).
In particular,(tr K◦)2 = (−k2 + γ)2.
If k2 6= 0 then
det K◦ = 0 ⇔ −γ =k2
1k2
,
and
(tr K◦)2 = k22 +
k41
k22+ 2k2
1 =(k2
1 + k22)
2
k22
=(τ2 + κ2)2
κ2 .
If k2 vanishes also k1 does.
Thus, Sadowsky strips cannot twist.
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Sadowsky’s energy
The restriction K◦ to the midline of second fundamental form K is determined by k1, k2,except for γ := K◦22:
K◦ =(−k2 k1k1 γ
).
In particular,(tr K◦)2 = (−k2 + γ)2.
If k2 6= 0 then
det K◦ = 0 ⇔ −γ =k2
1k2
,
and
(tr K◦)2 = k22 +
k41
k22+ 2k2
1 =(k2
1 + k22)
2
k22
=(τ2 + κ2)2
κ2 .
If k2 vanishes also k1 does.
Thus, Sadowsky strips cannot twist.
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Sadowsky’s energy
The restriction K◦ to the midline of second fundamental form K is determined by k1, k2,except for γ := K◦22:
K◦ =(−k2 k1k1 γ
).
In particular,(tr K◦)2 = (−k2 + γ)2.
If k2 6= 0 then
det K◦ = 0 ⇔ −γ =k2
1k2
,
and
(tr K◦)2 = k22 +
k41
k22+ 2k2
1 =(k2
1 + k22)
2
k22
=(τ2 + κ2)2
κ2 .
If k2 vanishes also k1 does.
Thus, Sadowsky strips cannot twist.
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Sadowsky’s energy
The restriction K◦ to the midline of second fundamental form K is determined by k1, k2,except for γ := K◦22:
K◦ =(−k2 k1k1 γ
).
In particular,(tr K◦)2 = (−k2 + γ)2.
If k2 6= 0 then
det K◦ = 0 ⇔ −γ =k2
1k2
,
and
(tr K◦)2 = k22 +
k41
k22+ 2k2
1 =(k2
1 + k22)
2
k22
=(τ2 + κ2)2
κ2 .
If k2 vanishes also k1 does.
Thus, Sadowsky strips cannot twist.
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Sadowsky’s energy
The restriction K◦ to the midline of second fundamental form K is determined by k1, k2,except for γ := K◦22:
K◦ =(−k2 k1k1 γ
).
In particular,(tr K◦)2 = (−k2 + γ)2.
If k2 6= 0 then
det K◦ = 0 ⇔ −γ =k2
1k2
,
and
(tr K◦)2 = k22 +
k41
k22+ 2k2
1 =(k2
1 + k22)
2
k22
=(τ2 + κ2)2
κ2 .
If k2 vanishes also k1 does.
Thus, Sadowsky strips cannot twist.
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Sadowsky’s energy
The restriction K◦ to the midline of second fundamental form K is determined by k1, k2,except for γ := K◦22:
K◦ =(−k2 k1k1 γ
).
In particular,(tr K◦)2 = (−k2 + γ)2.
If k2 6= 0 then
det K◦ = 0 ⇔ −γ =k2
1k2
,
and
(tr K◦)2 = k22 +
k41
k22+ 2k2
1 =(k2
1 + k22)
2
k22
=(τ2 + κ2)2
κ2 .
If k2 vanishes also k1 does.
Thus, Sadowsky strips cannot twist.
12
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Equilibria
We incorporate the constraint det K◦ = 0 in the (renormalized) strain energy by letting
W(K◦) :={
(tr K◦)2 if det K◦ = 0,+∞ otherwise.
Neglecting self-contact, we formulate an equilibrium problem as follows:
• find a curve r(x1), an orthonormal frame di(x), and an internal parameter γ(x1) whichsolve
min∫ `
0W([−k2 k1k1 γ
])where k1 = d′2 · d3 and k2 = d′3 · d1,
under the constraints d1 = r′ and d′1 · d2 = 0,
and
r(0) = r(`), d1(0) = d1(`),∫ `
0k1dx1 = π.
Observe that W is nonconvex.
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Equilibria
We incorporate the constraint det K◦ = 0 in the (renormalized) strain energy by letting
W(K◦) :={
(tr K◦)2 if det K◦ = 0,+∞ otherwise.
Neglecting self-contact, we formulate an equilibrium problem as follows:
• find a curve r(x1), an orthonormal frame di(x), and an internal parameter γ(x1) whichsolve
min∫ `
0W([−k2 k1k1 γ
])where k1 = d′2 · d3 and k2 = d′3 · d1,
under the constraints d1 = r′ and d′1 · d2 = 0,
and
r(0) = r(`), d1(0) = d1(`),∫ `
0k1dx1 = π.
Observe that W is nonconvex.
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Equilibria
We incorporate the constraint det K◦ = 0 in the (renormalized) strain energy by letting
W(K◦) :={
(tr K◦)2 if det K◦ = 0,+∞ otherwise.
Neglecting self-contact, we formulate an equilibrium problem as follows:
• find a curve r(x1), an orthonormal frame di(x), and an internal parameter γ(x1) whichsolve
min∫ `
0W([−k2 k1k1 γ
])where k1 = d′2 · d3 and k2 = d′3 · d1,
under the constraints d1 = r′ and d′1 · d2 = 0,
and
r(0) = r(`), d1(0) = d1(`),∫ `
0k1dx1 = π.
Observe that W is nonconvex.
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Equilibria
We incorporate the constraint det K◦ = 0 in the (renormalized) strain energy by letting
W(K◦) :={
(tr K◦)2 if det K◦ = 0,+∞ otherwise.
Neglecting self-contact, we formulate an equilibrium problem as follows:
• find a curve r(x1), an orthonormal frame di(x), and an internal parameter γ(x1) whichsolve
min∫ `
0W([−k2 k1k1 γ
])where k1 = d′2 · d3 and k2 = d′3 · d1,
under the constraints d1 = r′ and d′1 · d2 = 0,
and
r(0) = r(`), d1(0) = d1(`),∫ `
0k1dx1 = π.
Observe that W is nonconvex.
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Equilibria
We incorporate the constraint det K◦ = 0 in the (renormalized) strain energy by letting
W(K◦) :={
(tr K◦)2 if det K◦ = 0,+∞ otherwise.
Neglecting self-contact, we formulate an equilibrium problem as follows:
• find a curve r(x1), an orthonormal frame di(x), and an internal parameter γ(x1) whichsolve
min∫ `
0W([−k2 k1k1 γ
])where k1 = d′2 · d3 and k2 = d′3 · d1,
under the constraints d1 = r′ and d′1 · d2 = 0,
and
r(0) = r(`), d1(0) = d1(`),∫ `
0k1dx1 = π.
Observe that W is nonconvex.
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Equilibria
We incorporate the constraint det K◦ = 0 in the (renormalized) strain energy by letting
W(K◦) :={
(tr K◦)2 if det K◦ = 0,+∞ otherwise.
Neglecting self-contact, we formulate an equilibrium problem as follows:
• find a curve r(x1), an orthonormal frame di(x), and an internal parameter γ(x1) whichsolve
min∫ `
0W([−k2 k1k1 γ
])where k1 = d′2 · d3 and k2 = d′3 · d1,
under the constraints d1 = r′ and d′1 · d2 = 0,
and
r(0) = r(`), d1(0) = d1(`),∫ `
0k1dx1 = π.
Observe that W is nonconvex.
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An example of microstructure
Nonconvexity of the integrand can imply non-existence of a minimizer.
As an example, consider the problem
min|y′ |=1
∫ 1
0y2(x)dx. (1)
We incorporate the constraint |y′| = 1 in the integrand by defining the non-convex function
w(y′) =
{0 if |y′| = 1;+∞ otherwise.
We can then replace (1) with
min∫ 1
0[y2 + w(y′)]dx
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There exists a
minimizing sequence (yn)
such that |y′n| = 1 andyn → y = 0.
We construct yn by alternating on a scale 1/n the slopes +1 and −1 in such a waythat yn is as close as possible to 0 on average.
the limit y does not comply with the constraint |y′| = 1.
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There exists a
minimizing sequence (yn)
such that |y′n| = 1 andyn → y = 0.
We construct yn by alternating on a scale 1/n the slopes +1 and −1 in such a waythat yn is as close as possible to 0 on average.
the limit y does not comply with the constraint |y′| = 1.
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Convexification
We replace w by its the largest convex minorant:
w∗∗(y′) =
{0 if |y′| ≤ 1+∞ otherwise
The limit y = 0 solves
min∫ 1
0[y2 + w∗∗(y′)]dx
We interpret
y2 + w∗∗(y′) as the macroscopic energy density
because it captures the macroscopic behavior of minimizing sequences.
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The macroscopic energy of a ribbon
Instead of solving
min∫ `
0W([−k2 k1k1 γ
]),
we solve
min∫ `
0W∗∗
([−k2 k1k1 γ
]).
After the relaxation the kinematical constraint between γ, k1, and k2 is lost.
Thus we can minimize first with respect to γ to obtain the effective macroscopic energy
min∫ `
0W(k1, k2) where W(k1, k2) = min
γW∗∗
([−k2 k1k1 γ
]).
This procedure is supported by rigorous results obtained by Γ-convergence.∗
∗L. Freddi, P. Hornung, M.G. Mora, R. Paroni (J. Elasticity, 2016) A corrected Sadowsky functional forinextensible elastic ribbons.
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The relaxed version of Sadowsky’s energy
The microscopic bending energy is
(tr K)2 = |K|2 + 2 det K.
The macroscopic bending energy turns out to be:
W∗∗(K) = (tr K)2 + 4(det K)− = |K|2 + 2|det K|.
In particular,
W(k1, k2) = minγ
W∗∗([−k2 k1k1 γ
])=
(k2
2 + k21)
2
k22
= WS(k1, k) if |k2| > |k1|,
4k21 otherwise.
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Identikit of the relaxed Sadowsky energy
Since W∗∗(K) is isotropic, it depends only on the eigenvalues of K, thus there exists w∗∗ :R2 → R such that:
w∗∗(κ1, κ2) = W∗∗(K) whenever K = κ1b1 ⊗ b1 + κ2b2 ⊗ b2,
with bα · bβ = δαβ.
Graph of w∗∗.
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Identikit of the relaxed Sadowsky energy
Since W∗∗(K) is isotropic, it depends only on the eigenvalues of K, thus there exists w∗∗ :R2 → R such that:
w∗∗(κ1, κ2) = W∗∗(K) whenever K = κ1b1 ⊗ b1 + κ2b2 ⊗ b2,
with bα · bβ = δαβ.
Graph of w∗∗.
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Explicit form of the convex envelope: laminations
Theorem 1. ∗ For Q : R2×2sym → R a quadratic form, let
W(K) =
{Q(K) if det K = 0,+∞ otherwise.
Then for each K there exist A, B satisfying det A = det B = 0 and t ∈ [0, 1] such that
W∗∗(K) = (1− t)Q(A) + tQ(B), with K = (1− t)A + tB.
Explicit formulas for A, B and t are available.
∗R. Paroni and G. Tomassetti (J. Elasticity, 2018) Macroscopic and microscopic behavior of narrow elasticribbons.
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Twisted states: macroscopic solution
We conside the following twisted state:
r(x1) = x1e1,d2(x1) = + cos(Θx1/`)e2 + sin(Θx1/`)e3,d3(x1) = − sin(Θx1/`)e2 + cos(Θx1/`)e3.
In this state the ribbon is uniformly twisted under a controlled terminal rotation Θ.
The twist isk1 = Θ/`,
while k2 = 0 and k3 = 0.
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Twisted states: microscopic solution
On the midline, the second fundamental form is
K◦(γ) =(
0 k1k1 γ
).
SinceW∗∗(K) = |K|2 + 2|det K|,
the macroscopic energy density is
minγ
W∗∗ (K◦(γ)) = W∗∗([
0 k1k1 0
]).
It turns out that the lamination is(0 k1k1 0
)=
12(A + B) with A = k1
(−1 11 −1
), B = k1
(1 11 1
).
and that
W(k1, 0) =12(Q(A) + Q(B)).
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Interpretation
For a twisted state the second fundamental form associated to the macroscopic defor-mation has non-vanishing determinant, hence is microscopically inaccessible.
At the microscopic level, this deformation can be generated by alternating, on a finescale, two deformations with null Gaussian curvature.
The developability constraint survives only as a constraint that the ribbon cannot bendabout the normal d3
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Interpretation
For a twisted state the second fundamental form associated to the macroscopic defor-mation has non-vanishing determinant, hence is microscopically inaccessible.
At the microscopic level, this deformation can be generated by alternating, on a finescale, two deformations with null Gaussian curvature.
The developability constraint survives only as a constraint that the ribbon cannot bendabout the normal d3
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Interpretation
For a twisted state the second fundamental form associated to the macroscopic defor-mation has non-vanishing determinant, hence is microscopically inaccessible.
At the microscopic level, this deformation can be generated by alternating, on a finescale, two deformations with null Gaussian curvature.
The developability constraint survives only as a constraint that the ribbon cannot bendabout the normal d3
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A closer look at the microscopic states
A microscopic state is must be a multiple of
K(θ) = I− b(θ)⊗ b(θ), with b(θ) = cos θe1 + sin θe2.
Twisted states are a fine mixture of
A = k1K(π/2), B = k1K(−π/2).
flat state
→ state A
state B
The actual construction requires some work because A and B are kinematically incompatible.
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A closer look at the microscopic states
A microscopic state is must be a multiple of
K(θ) = I− b(θ)⊗ b(θ), with b(θ) = cos θe1 + sin θe2.
Twisted states are a fine mixture of
A = k1K(π/2), B = k1K(−π/2).
flat state
→ state A
state B
The actual construction requires some work because A and B are kinematically incompatible.
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Assembling the microscopic states
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General case
One period of the function fn.
Let θA and θB such thatA = K(θA), B = K(θB).
We letθ(n)(x1) = fn(nx1)θA + (1− fn(nx2))θB,
andK(n)(x1) = K(θ(n)(x1)).
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Thanks for your attention
References
G.T., V. Varano (Meccanica, 2017) Capturing the helical to spiral transitions in thin ribbonsof nematic elastomers.
R. Paroni, G.T. (J. Elasticity, 2018) Macroscopic and Microscopic Behavior of Narrow ElasticRibbons.
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