Bifurcations in the regularized Ericksen bar model

0.34 MB·English

by  M. Grinfeld

Bifurcations in the regularized Ericksen bar model
M. Grinfeld G. J. Lord
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Department of Mathematics, Department of Mathematics and Maxwell Institute,
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The University of Strathclyde, Heriot-Watt University,
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J Glasgow, G1 1XH, UK. Edinburgh, EH14 4AS, UK.

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1 [email protected] [email protected]

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February 2, 2008
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Abstract
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WeconsidertheregularizedEricksenmodelofanelasticbaronanelasticfoundationon

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v anintervalwithDirichletboundaryconditionsasatwo-parameter bifurcationproblem.
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We explore, usinglocal bifurcation analysis andcontinuation methods, thestructureof
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2 bifurcations from double zero eigenvalues. Our results provide evidence in support of
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. Mu¨ller’s conjecture [18] concerning the symmetry of local minimizers of the associated
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1 energyfunctionalanddescribeindetailthestructureoftheprimarybranchconnections
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0 that occur in this problem. We give a reformulation of Mu¨ller’s conjecture and suggest
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v two further conjectures based on the local analysis and numerical observations. We
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conclude by analysing a “loop” structure that characterizes (k,3k) bifurcations.
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Keywords : microstructure, Lyapunov–Schmidt analysis, Ericksen bar model
AMS subject classification: 34C14, 74N15,37M20
1 Introduction
In the late eighties J. M. Ball suggested that an interesting and important question in
material science would be to understand the dynamical creation of microstructure [3]. A
1
model for creating microstructure would be a dynamical system with a Lyapunov functional
that does not reach its infimum value on, say, the set of W1,2(0,1) functions, the infimum
0
value being achieved instead by a gradient Young measure. Thus one might expect to
obtain microstructure dynamically, hoping that as the Lyapunov functional decreases along
trajectories, translates in time will form a minimizing sequence.
One of the candidates for such a process proposed by Ball et al. in [4] is Ericksen’s model
of an elastic bar on an elastic foundation [6]. This is given by
′
u = (W (u )+βu ) αu (1.1)
tt x tx x −
on the interval [0,1] with the Dirichlet boundary conditions
u(0,t) = u(1,t) = 0. (1.2)
Hereu := u(x,t)isthelateraldisplacement ofthebar,β measures thestrengthofviscoelastic
effects (the term βu provides the dissipation of energy mechanism) and α measures the
xxt
′
strength of bonding of the bar to the substrate. W (u ) is the (non-monotone) stress/strain
x
relationship; in what follows we specifically take the double well potential
1
2 2
W(z) = (z 1) . (1.3)
4 −
It is easily checked that
1
1 α
2 2
E = u +2W(u )+ u dx (1.4)
1 2 t x 2
0
Z
h i
is a Lyapunov function for (1.1).
Friesecke and McLeod [8] proved that (1.1) admits an uncountable family of steady states
that are energetically unstable but locally asymptotically stable. They also showed that
initial data evolves, roughly, to a saw-tooth pattern with the same lap number, (i.e minimum
number of non-overlapping intervals where the pattern is monotone) as the initial data. In
other words, as Friesecke and McLeod put it in the title of their paper [7], dynamics is a
mechanism preventing the formation of finer and finer microstructure. These results go some
way to explain the earlier numerical results of Swart and Holmes [19].
Mu¨ller [18] considered the regularized version of the Ericksen model,
′
u = (W (u )+βu γu ) αu (1.5)
tt x tx − xxx x −
2
on the interval [0,1] with the double Dirichlet boundary conditions
u(0,t) = u(1,t) = 0; u (0,t) = u (1,t) = 0. (1.6)
xx xx
The main thrust of Mu¨ller’s sophisticated analysis was to describe the global minimizer of
the associated energy functional,
1
1 α
2 2 2
E = u +γu +2W(u )+ u dx. (1.7)
2 2 t xx x 2
0
Z
h i
Before we continue, we need to define periodicity more precisely. Consider a stationary
solution u of (1.5). Take its odd extension to [1,2] and identify the points x = 0 and
0
x = 2. If the resulting function is D -periodic on the circle for some k Z, we say that
2k
∈
u is periodic. Then Mu¨ller’s result is that the global minimizer is a periodic function with
0
a precisely defined dependence of the period on α and γ. He also suggested the following
conjecture:
Mu¨ller’s Conjecture [18]: Local minimizers of E are periodic.
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Recently, Yip [26] has proved this conjecture for solutions of small energy where W has the
form
2
W(p) = ( p 1) .
| |−
In this case many calculations of energy of equilibria can be done explicitly. This case, with
more general boundary conditions, was also considered in [20, 24]. Nucleation and ripening
in the Ericksen problem with the above form of free energy density is considered from a more
thermodynamical point of view by Huo and I. Mu¨ller [15].
Finally, in a related paper [23], an extension of Ericksen’s model to system of two elastic bars
coupled bysprings asa modelformartensitic phasetransitionsis mainlystudied numerically.
The dynamics of the regularized Ericksen bar (1.5) was investigated in [16], where global
existence of solutions, existence of a compact attractor and convergence to equilibria was
proved. Furthermore, the case of α = 0 was investigated in detail and an almost complete
characterization of the structure of the attractor was given in that case. A. Novick–Cohen
has observed that if α = 0, the set of stationary solutions of (1.5) is precisely the same as
for the Cahn–Hilliard equation, which was thoroughly investigated in [10, 11]; more work
exploitingthisconnectionbetween(1.5)andtheCahn-Hilliardequationisinpreparation[12].
In particular, the stationary solutions of (1.5) for the double Dirichlet boundary conditions
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correspond to the Cahn-Hilliard equation with mass zero. As a consequence the bifurcation
diagram of the stationary solutions of (1.5) with α = 0 contains only supercritical pitchfork
bifurcationsfromthetrivialsolutions; only thebranches without internal zeroscanbestable.
Other studies of the dynamics of (1.5) include the work of Vainchtein and co-workers
[21, 22, 25], who considered, in particular, time-dependent Dirichlet boundary conditions
(loading/unloading cycles) in order to study hysteresis effects.
In this paper we would like to
present some evidence towards verifying Mu¨ller’s conjecture and reformulate it;
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explain how the situation for α = 0 for (1.5) can be reconciled with the result of
•
Friesecke and McLeod for (1.1) alluded to above.
In very recent related work, Healey and Miller [13], have considered the two-dimensional
version of the problem with hard loading on the boundary, using methods of global bifur-
cation theory and numerical continuation techniques, concentrating on primary bifurcating
branches and characterizing their symmetry.
Weusemethodsoflocalbifurcationtheory. Westartbyobtainingtheprimaryandsecondary
bifurcation points and presenting the results of numerical continuation using AUTO [5]. In
section 3 we apply directly the Lyapunov-Schmidt theory as detailed in [9]; this suggests
a mechanism for the restabilization of unstable solutions. The analysis has uncovered an
interesting patternof primarybranch connections which we analyse insection 4. Toconclude
we present two further conjectures, these are based on the local analysis backed up with the
numerical observations.
2 Preliminaries
We start by reviewing the bifurcation structure of the problem. As shown in [16], the
eigenvalues ν of the linearization of (1.5) around the trivial solution u = u = 0 satisfy
k t
1
ν = βπ2k2 β2π4k4 4(γπ4k4 π2k2 +α) . (2.8)
k
2 ± − −
(cid:16) p (cid:17)
Hence we have the following lemma.
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Lemma 2.1 The eigenvalues of the linearization of (1.5) around the trivial solution u =
u = 0 are generically simple, and pass through zero at points
t
π2k2 α
γ = i − , (2.9)
ki π4k4
i
where the integers k are ordered by their distance from the number
i
√2α
∗
k = .
π
For example, if α = 25, k = 2, k = 3, k = 1, k = n for n > 3. In other words at α = 25,
1 2 3 n
based on the eigenfunctions sin(kπx), the first bifurcating solution branch has one internal
zero, the second has two internal zeros, the third has no internal zeros and the nth has n 1
−
internal zeros.
As we will be working in the (α, 1/γ) plane, it is convenient to use (2.9) to define
2 2 4 4 2 2
Γ = (α, 1/γ) α [0, π k ], 1/γ = π k /(π k α) . (2.10)
k
{ | ∈ − }
ThusthecurvesΓ arethecurvesonwhichthelinearizationhasinitskerneltheeigenfunction
k
sin(kπx). Note that Γ has the vertical line α = k2π2 as an asymptote.
k
We can also determine the double zero eigenvalue points. The curves Γ and Γ will intersect
k l
at a point where
π2k2l2
α α = .
≡ k,l k2 +l2
The corresponding values γ can be found from
k,l
π2k2 α
k,l
γ = − .
k,l π4k4
We call these bifurcation points (k,l) bifurcations. Note that one can concoct any (k,l)
bifurcation point, but never a bifurcation point of multiplicity higher than two. From these
double zero eigenvalue points curves of secondary bifurcation points emanate and we denote
these by Γ .
kℓ
2.1 Numerical Evidence
We use AUTO [5] to investigate numerically the bifurcation diagram of equilibria of (1.5),
that is we look at Φ(u,α,γ) = 0 where
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Φ(u,α,γ) = γu +αu (u u ) , u(0) = u(1) = 0, u (0) = u (1) = 0. (2.11)
xxxx − x − x x xx xx
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For fixed α we can compute the bifurcation diagram in 1/γ and two examples are shown
in Fig. 1 for α = 7.5 and α = 33. The figure shows bifurcations from the trivial solutions
occurringatγ andthesecondarybifurcations(k,ℓ),plottingtheℓ2 normof(u,u ,u ,u )
ki x xx xxx
(an approximation of the H3 norm) of the solution as γ varies. For α = 7.5, (a)–(d) are
the branches of solutions with 0–3 internal zeros and sample solutions on these branches
are shown in Fig. 2. Sample solutions from the branches (e)–(h) that bifurcate from these
solution branches are also shown in Fig. 2. For α = 33 we have labeled the number of
internal zeros for the branches that bifurcate from the trivial solution.
α = 7.5 α = 33
L2 Norm L2 Norm
80. 125.
(d)
70.
100.
60. (c) (f) (g)
50. 75. 7
40. (b)
30. (h) 50. 3 6
(e) 5
20.
25. 2 4
10. 1
0. 0.
0. 50. 100. 150. 200. 250. 300. 0. 200. 400. 600. 800.
(a) 1/γ 100. 300. 500. 700. 1/γ
Figure 1: Two bifurcation diagrams for fixed values of α = 7.5 and α = 33. Figures show
the ℓ2 norm of (u,u ,u ,u ) as γ varies. In Fig. 2 sample solutions are shown from each
x xx xxx
of the branches for α = 7.5; the arrows give an indication where each the solution is taken
from. (a)–(d) show solution branches with 0,1,2 and 3 internal zeros. For α = 33 labels 1–7
show the number of internal zeros. The “loop” for α = 7.5 connects solutions with 0 and 2
internal zeros whereas for α = 33 it is between solutions with 1 and 5 internal zeros.
Wecanexploit thefactthat thebifurcations(k,ℓ)canbeidentified aslimit pointstoperform
two parameter continuation. The result of these computations is shown in the (α,1/γ) plane
in figure 3. We clearly see the curve Γ tending to the correct theoretical value of the
1
asymptote at α = π2. From the bifurcation (1,2) the Γ curve tends to infinite as α
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approaches zero as does Γ ; whereas the curves Γ , Γ , Γ appear to tend to infinite for
31 12 13 23
some α > 0.
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(a) (b) (c) (d)
0.000 0.15 0.075 0.050
---000...000752505 00..0150 00..002550 0000....000012340000
-0.100 0.00 0.000 0.000
---000...111752505 --00..1005 --00..005205 ----0000....000043210000
-0.200 -0.15 -0.075 -0.050
0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00
0.10 0.30 0.50 0.70 0.90 0.10 0.30 0.50 0.70 0.90 0.10 0.30 0.50 0.70 0.90 0.10 0.30 0.50 0.70 0.90
(e) (f) (g) (h)
0.090 0.025 0.15 0.125
00..007800 0.000 0.10 0.100
0.060 -0.025 0.05
00..004500 --00..007550 0.00 00..005705
0.030 -0.100 -0.05
00..001200 -0.125 -0.10 0.025
0.000 -0.150 -0.15 0.000
0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00 0.00 0.20 0.40 0.60 0.80 1.00
0.10 0.30 0.50 0.70 0.90 0.10 0.30 0.50 0.70 0.90 0.10 0.30 0.50 0.70 0.90 0.10 0.30 0.50 0.70 0.90
Figure 2: Sample solutions from the bifurcation diagram with α = 7.5. (a)–(d)
shows solutions on the branches bifurcating from the trivial solution with 1/γ =
123.87,111.34,123.78,218.2. (e)–(h) shows solutions on the secondary bifurcation branches
with 1/γ = 54.35,121.40,278.43,226.78.
3 (k,k + 1)-bifurcations and a mechanism for restabi-
lization
To present a plausible scenario for restabilization of unstable equilibria, we are interested in
the structure of stationary solutions of (2.11) in a neighbourhood of a (k,k+1) bifurcation
point, when dΦ(0,α,γ) has a double zero eigenvalue with eigenfunctions v = sin(kπx) and
k
v = sin((k+1)πx). To examine this we apply the Lyapunov-Schmidt theory as described
k+1
in [9].
Set
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X = u C ((0,1)), u(0) = u(1) = 0, u (0) = u (1) = 0. ,
xx xx
{ ∈ | }
and let Y = C0((0,1)). We let L denote the linearization :
L dΦ(0,α ,γ ).
k,k+1 k,k+1
≡
Letusexaminethesymmetries of(2.11). DefineonX twooperators,R andR byR u = u
1 2 1
−
and R u(x) = u(1 x). It is easily seen that the group I,R ,R ,R R is isomorphic to
2 1 2 1 2
− { }
7
300.
Γ
23
250. Γ
13
200.
Γ
1/γ 150. 32
Γ
31
100.
Γ
3
Γ
Γ 12
21 Γ
50. 2
Γ
1
0.
0. 5. 10. 15. 20. 25. 30. 35.
α
Figure 3: Two parameter continuation of the bifurcation points. For clarity only the curves
Γ k = 1,2,3 and the secondary curves Γ , k = 1,2,3, ℓ = 1,2,3 are plotted.
k k,ℓ
Z Z
, that Φ commutes with this group, i.e.
2 2
⊕
R Φ(u,α,γ) = Φ(R u,α,γ),
i i
and that if k is even,
R v = v while R v = v .
2 k k 2 k+1 k+1
−
Hence the theory of [9, Chapter X] is applicable.
In the Lyapunov-Schmidt framework (see [9, Chapter VII]),
X = sp v , v M,
k k+1
{ }⊕
where M is the orthogonal complement to kerL. Similarly,
Y = N rangeL,
⊕
where N is the orthogonal complement of the range of L. Since L is self-adjoint, N = kerL,
so we take (v , v ) to be a basis of both the kernel of L and of N.
k k+1
8
Z Z
By the theory of bifurcations with symmetry, the bifurcation equation will be of the
2 2
⊕
form
g(x,y,µ) = 0,
where
g (x,y,µ) Ax3 +Bxy2 +aµx
1
g(x,y,µ) = = .
" g2(x,y,µ) # " Cx2y +Dy3 +bµy #
From now on we fix α at a point α , and do not any longer indicate the dependence on
k,k+1
α. We take our distinguished parameter to be λ = 1/γ and let it vary through the critical
value 1/γ . Clearly,
k,k+1
∂2g dγ ∂2g 1 ∂2g
i i i
= = ,
∂λ∂x dλ∂γ∂x −γ2∂γ∂x
which means that the case of positive g , say, corresponds to the case of negative g , so
iγx iλx
we are in case (A) of [9, p. 430]. Then varying α will unfold the degenerate bifurcation.
By [9, Appendix 3] (see also (1.14) in [9, Chapter VII]), and since d2Φ 0 by oddness, we
≡
get, for example, for g ,
1
∂3g
1 3
= v , d Φ(0,λ)(v , v , v )
∂x3 h k k k k i
∂3g
1 3
= v , d Φ(0,λ)(v , v , v )
∂x∂y2 h k k k+1 k+1 i
∂2g
1
= v , d Φ(0,λ)(v ) ,
k γ k
∂x∂γ h i
with similar expressions holding for the partial derivatives of g .
2
Note that, for example,
3
∂ ∂ ∂
3
d Φ(v ,v ,v ) = Φ(v t )
k k k ∂t ∂t ∂t |t1=t2=t3=0 k i
1 2 3
i=1
X
3
∂ ∂ ∂
3 2
= ( (v ) t ) = 18[(v ) ] (v ) .
∂t ∂t ∂t |t1=t2=t3=0 − k x i x − k x k xx
1 2 3
i=1
X
Hence
1∂3g 1 3
1 4 4 4 2 4 4
A = = 3k π sin (kπx)cos (kπx) = k π .
6 ∂x3 8
0
Z
Similarly,
3
4 4
D = (k +1) π .
8
9
For a and b we have
1 1
4 4 4 4
a = v , (v ) = k π and b = v , (v ) = (k +1) π .
k k xxxx k+1 k+1 xxxx
h i 2 h i 2
Now to compute B and C:
∂ ∂ ∂
3
d Φ(v ,v ,v ) = Φ(t v +(t +t )v )
k k+1 k+1 ∂t ∂t ∂t |t1=t2=t3=0 1 k 2 3 k+1
1 2 3
∂ ∂ ∂
3
= [(v ) t +(t +t )(v ) ]
−∂t ∂t ∂t |t1=t2=t3=0 k x 1 2 3 k+1 x x
1 2 3
2
= 6 (v ) [(v ) ] +2(v ) (v ) (v ) .
k xx k+1 x k x k+1 xx k+1 x
−
Hence (cid:2) (cid:3)
1 ∂3g 1
1 2 2 4 2 2
B = = 3 k (k +1) π sin (kπx)cos ((k +1)πx)dx
2∂x∂y2
0
Z
1
1
3 4
+ k(k +1) π sin(2kπx)sin(2(k +1)πx)dx.
2
0
Z
Since the second integral is zero, we have that
1
3
2 2 2 2 4 2 2
B = C = 3k (k +1) sin (kπx)cos ((k +1)πx)dx = π (k +1) k .
4
0
Z
Now we can reduce the bifurcation equation to normal form. First note that
2
ǫ = sgn(A) = 1, ǫ = sgn(a)sgn( 1/γ ) = 1,
1 2
− −
2
ǫ = sgn(D) = 1, ǫ = sgn(b)sgn( 1/γ ) = 1,
3 4
− −
so that indeed we are in case (A) of [9, p. 430].
By Proposition 2.3 of [9, p. 424] the bifurcation diagram is determined by the modal pa-
rameters
b a
m = B and n = C.
Da Ab
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Now, (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(k +1)48(cid:12) (cid:12)1 3 (k +1)2
2 2
m = (k +1) k = 2
k4 3(k +1)44 k2
and
k4 8 1 3 k2
2 2
n = (k +1) k = 2 .
(k +1)43k44 (k +1)2
Hence for all k, m 2, but n can be either smaller or larger than one. If m > 1, n > 1, we
≥
are in region (1) of [9, p. 433]; if m > 1, n < 1, we are in region (2).
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