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Comptes Rendus
Mécanique
Milan Batista
On stability of non-inflectional elastica
Volume 348, issue 2 (2020), p. 137-148.
<https://doi.org/10.5802/crmeca.2>
© Académie des sciences, Paris and the authors, 2020.
Some rights reserved.
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CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
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Comptes Rendus
Mécanique
2020, 348, n 2, p. 137-148
https://doi.org/10.5802/crmeca.2
On stability of non-inflectional elastica
Milan Batista a
a University of Ljubljana, Faculty of Maritime Studies and Transport,
Pot pomorščakov 4, 6320 Portorož, Slovenia.
E-mail: milan.batista@fpp.uni-lj.si.
Abstract. This study considers the stability of a non-inflectional elastica under a conservative end force sub-
ject to the Dirichlet, mixed, and Neumann boundary conditions. It is demonstrated that the non-inflectional
elastica subject to the Dirichlet boundary conditions is unconditionally stable, while for the other two bound-
ary conditions, sufficient criteria for stability depend on the signs of the second derivatives of the tangent an-
gle at the endpoints.
Keywords. Elasticity, Non-inflectional elastica, Stability.
Manuscript received 17th December 2018, revised 5th April 2019, accepted 2nd December 2019.
1. Introduction
In recent years, Lessinnes and Goriely [1], among others, have considered the stability of equi-
libria for initially uniformly curved weightless inextensible and unshearable rods (hereafter re-
ferred to as elastica), subject to Dirichlet and Neumann boundary conditions. For each bound-
ary condition, they defined an index, the value of which determines the stability or instability of
equilibria. This value can be determined by a geometric analysis of the phase-plane trajectories
of stationary solutions.
This study demonstrates that, for non-inflectional elastica [2]; that is, elastica bent in one
direction, the stability of equilibria may in certain cases be determined only from the signs of the
second derivative of the equilibria at endpoints. Using results from [1] we can also extend sign
criteria to prove the instability of a non-inflectional elastica subject to the Neumann boundary
condition.
2. Equilibria
A planar elastica with one end spatially fixed and another spatially free will be considered,
subject to a conservative force F (Figure 1). The potential energy Π of such elastica is commonly
expressed in the following dimensionless form:
Π[θ] =
∫ 1
0
[
1
2
(θ′−a)2 +ω2 cos(θ+α)
]
d s, (1)
where s ∈ [0,1] is the normalized arc length parameter, ( )′ = d( )/d s, θ = θ(s) is the angle between
the x-axis and the tangent to the curve,α is the angle between the force and the negative direction
ISSN (electronic) : 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/
138 Milan Batista
of the x-axis, a > 0 is the reference dimensionless curvature, and ω ≥ 0 is the load parameter
which is defined by
ω2 ≡ F`
2
E I
(2)
` is the elastica length and E I is its flexural rigidity. We seek an extremal θ = θ(s) of the functional
(1) subject to the condition
θ′(s) 6= 0, s ∈ [0,1] (3)
This condition means that the rod curvature does not change sign and is thus curved only in one
direction. This is a case of non-inflectional elastica; because we assume that initial curvature is
positive we can replace condition (3) by the condition
θ′(s) > 0, s ∈ [0,1]. (4)
To determine the equilibrium shape (or configuration or state) of the elastica, we make
δΠ[θ] = 0. In this manner, we deduce the equilibrium equation and boundary conditions for θ:
θ′′+ω2 sin(θ+α) = 0 (5)
(θ′−a)δθ|s=0 = 0, (θ′−a)δθ|s=1 = 0. (6)
where δθ is variation of θ. The solution of (5); that is, the equilibria ofΠ, subject to (4) is expressed
as [2–4]
θ =−α+2 am (kωs +C ,k−1), (7)
where am is Jacobi’s amplitude function [5], k ≥ 1 is the elliptic modulus, and C is an integration
constant. Because am is periodic with period 2K , C can always be selected to lie within the
interval
−K ≤C < K , (8)
where K = K (k−1) is the elliptic integral of the first type. The first and second derivatives of (7) are
θ′ = 2kωdn(kωs +C ,k−1) (9)
θ′′ =−2ω2sn(kωs +C ,k−1)cn(kωs +C ,k−1), (10)
where sn, cn, and dn are Jacobi’s elliptic functions [5].
The boundary conditions (6) for θ are now considered. Each can be satisfied in two manners,
as follows.
• If δθ = 0, and at the endpoint θ is prescribed, the endpoint can be said to be clamped.
• If θ′ = a, the endpoint can be said to be pinned or hinged. In this case, δθ′ = 0.
From the above, the following types of boundary conditions can be established.
• Dirichlet:
θ(0) = θ0, θ(1) = θ1. (11)
• Mixed:
(a) θ(0) = θ0, θ′(1) = a (12)
(b) θ′(0) = a, θ(1) = θ1. (13)
• Neumann:
θ′(0) = a, θ′(1) = a. (14)
This section can be concluded with the following lemma.
C. R. Mécanique, 2020, 348, n 2, 137-148
Milan Batista 139
Lemma 1. If a non-inflectional elastica is subject to Neumann boundary conditions, the second
derivatives of θ at the endpoints are either equal or opposite.
Proof. From (9) and (14), the equation obtained is
dn(C ,k−1) = dn(kω+C ,k−1). (15)
Because the function dn is periodic with period 2K and symmetric, this equation is identically
satisfied in two cases:
(a) kω= 2nK (16)
(b) kω=−2C +2nK , (17)
where n ≥ 0 is an integer. By substituting these solutions into expression (10) for the second
derivative of θ, and taking into account the symmetry and periodicity of the functions sn and
cn, we obtain
θ′′(1) = θ′′(0) (solution (a)) (18)
θ′′(1) =−θ′′(0) (solution (b)). (19)
This lemma implies that the situation in which θ′′ vanishes only at one endpoint is impossible
for Neumann boundary conditions; it can be zero only at both ends simultaneously. It is noted
that this is the case when the non-inflectional elastica forms closed loops.
3. Stability
To study the stability of solution (7), the second variation δ2Π[θ] must be examined. It is well
known [6] that the equilibrium is stable if δ2Π[θ] > 0 for any admissible variation δθ; that is, a
nontrivial variation satisfying given boundary conditions (see below) [7].
From (1), it can be seen that
δ2Π[θ] =
∫ 1
0
(δθ′2 −ω2 cos(θ+α)δθ2)d s. (20)
By differentiating (5) with respect to s, we obtain θ′′′+ω2 cos(θ+α)θ′ = 0. Owing to (4), this can
always be solved for cos(θ+α) and then the resulting expression can be substituted into (20).
Upon integration by parts, we obtain
δ2Π[θ] = θ
′′
θ′
δθ2
∣∣∣∣1
0
+
∫ 1
0
(
δθ′− θ
′′
θ′
δθ
)2
d x. (21)
For future use, this expression is rewritten as
δ2Π[θ] = B + J , (22)
where
B ≡ θ
′′
θ′
δθ2
∣∣∣∣
1
− θ
′′
θ′
δθ2
∣∣∣∣
0
and
J ≡
∫ 1
0
(
δθ′− θ
′′
θ′
δθ
)2
d x.
(23)
It is clear that J ≥ 0 for any δθ. Therefore, δ2Π > 0 is obtained when either of the following
conditions are fulfilled:
(a) J = 0 and B > 0 or
(b) J > 0 and B >−J .
Hereafter, the conditions imposed on B must be fulfilled for any admissible δθ.
C. R. Mécanique, 2020, 348, n 2, 137-148
140 Milan Batista
Figure 1. Geometry and load of the rod. Bold line correspond to deformed shape, tiny line
correspond to initial shape.
Figure 2. Bifurcation diagram of Dirichlet problem.
The question of the stability of equilibria is thus reduced to an investigation of the signs of B
and J, subject to the following boundary conditions.
• Dirichlet boundary conditions:
δθ(0) = δθ(1) = 0. (24)
C. R. Mécanique, 2020, 348, n 2, 137-148
Milan Batista 141
Figure 3. Stable forms of non-inflectional elastica corresponding to points 1 in Figure 2.
Figure 4. Bifurcation diagram of Neumann problem. Stable regions are non-shaded; un-
stable regions are shaded.
• Mixed boundary conditions:
(a) δθ(0) = 0, δθ′(1) = 0 (25)
(b) δθ′(0) = 0, δθ(1) = 0 (26)
• Neumann boundary conditions:
δθ′(0) = δθ′(1) = 0. (27)
C. R. Mécanique, 2020, 348, n 2, 137-148
142 Milan Batista
Figure 5. Stable forms of non-inflectional elastica corresponding to points 2 and 4 in
Figure 4.
We firstly consider J. From (23), it can be seen that J = 0 only when δθ′− (θ′′/θ′)δθ = 0; that is,
when
δθ = cθ′, (28)
where c is a constant [8, 9]. The boundary conditions (24) to (27) for which the variation (28) is
admissible are now considered.
Lemma 2. If the non-inflectional elastica has at least one endpoint clamped, δθ = c θ′ is not an
admissible variation.
Proof. Suppose that δθ = cθ′ is an admissible variation. By assumption, θ′ 6= 0; therefore,
the boundary condition δθ = 0 can be satisfied only if c = 0. However, in this case, (28) be-
comes δθ ≡ 0 (no variation), which is not an admissible variation. This contradiction proves the
lemma.
Lemma 3. If the non-inflectional elastica has at least one endpoint pinned, and if at that point
θ′′ 6= 0, δθ = cθ′ is not an admissible variation.
C. R. Mécanique, 2020, 348, n 2, 137-148
Milan Batista 143
Figure 6. Unstable forms of non-inflectional elastica corresponding to points 1 and 3 in
Figure 4.
Proof. Let δθ = cθ′ be an admissible variation; then, δθ′ = cθ′′. At the pinned endpoint, it is
required that δθ′ = 0. But this can be satisfied only if c = 0, because at that point θ′′ 6= 0 by
assumption. Therefore, δθ ≡ 0, which is not an admissible variation.
Lemma 4. If the non-inflectional elastica is subject to Neumann boundary conditions and if
θ′′(0) = θ′′(1) = 0, there exists an admissible variation so that δ2Π[θ] = 0.
Proof. Let δθ = cθ′. Then, δθ′ = cθ′′ and this is zero at the endpoints for any c 6= 0 because of the
assumption θ′′(0) = θ′′(1) = 0. Therefore, δθ = cθ′ is an admissible variation for which B = 0 and
J = 0. Hence, δ2Π[θ] = 0.
Thus, it has been inferred that J > 0 for all of the boundary conditions, except for the Neumann
boundary conditions with θ′′ = 0 at both endpoints, in which case J = 0 and B = 0 (see [1] for more
details for this case). The latter means that condition (a), i.e., J = 0 and B > 0, cannot occur. Thus,
only case (b) remains in the investigation; that is, with the condition B >−J . Now, in expression
(23) for B, the only signed variable is θ′′, because, by assumption, θ′ > 0. However, the sign of B
also depends on the magnitudes of (θ′′/θ′)δθ2 at each end, particularly the magnitude of δθ2.
However, the condition B > −J will certainly be fulfilled if B ≥ 0. The requirements J > 0 and
C. R. Mécanique, 2020, 348, n 2, 137-148
144 Milan Batista
Figure 7. Bifurcation diagram of mixed problem. Stable regions are non-shaded.
B ≥ 0 are therefore sufficient for the stability of equilibria. The above discussion is summarized
with the following corollary.
Corollary 1. If B ≥ 0, the equilibrium is stable for all of the boundary conditions except for the
Neumann boundary conditions, with θ′′(0) = θ′′(1) = 0.
The sufficient conditions for the shape stability of the non-inflectional elastica are thus
governed only by the sign of the boundary term B . These must be established separately for each
type of boundary condition. Based on Corollary 1, we can state the following theorems, which
can be trivially proven by verification.
Theorem 1. The equilibrium shapes of a non-inflectional elastica subject to the Dirichlet bound-
ary conditions are stable.
In this theorem, it is noted that there is no stability condition, so this is the ultimate result
concerning the Dirichlet boundary condition. It was provided for the initially straight rod by
Born [10]. The same conclusion can also be inferred from the Lessinnes and Goriely geometric
criteria [1]. In fact Theorem 1 is direct application of Theorem 1 of [1].
Theorem 2a. The equilibrium shape of a non-inflectional elastica subject to mixed end conditions
(12) is stable if
θ′′(1) ≥ 0. (29a)
Theorem 2b. The equilibrium shape of the non-inflectional elastica subject to mixed end condi-
tions (13) is stable if
θ′′(0) ≤ 0. (29b)
C. R. Mécanique, 2020, 348, n 2, 137-148
Milan Batista 145
Figure 8. Stable forms of non-inflectional elastica corresponding to points in Figure 7.
Theorem 3. The equilibrium shape of a non-inflectional elastica subject to Neumann boundary
conditions is stable if
θ′′(0) < 0 and θ′′(1) > 0. (30)
We note that Theorem 3 covers a case that is fully addressed in [1].
Several examples of stable non-inflectional elastica shapes, together with phase portraits and
bifurcation diagrams, are illustrated in Figures 2 to 10. In these figures,
q ≡ a/2π and β≡ (ω2/π), (31)
where q is the loops number andβ is the multiplier of the Euler critical force that buckles a hinged
straight rod. These theorems allow for easy identification of stability regions in the bifurcation
diagrams (Figures 4, 7, and 9); stability regions are determine by sign of (10). Figures 3, 5, 6, 8, 10
shows elastica shape correspond to points on the bifurcation diagrams and also a phase plane
diagram where one can decide stability of the shape according to Lessinnes-Goriely criterion [1]
(see below).
C. R. Mécanique, 2020, 348, n 2, 137-148
146 Milan Batista
Figure 9. Bifurcation diagram of mixed problem. Stable regions are non-shaded.
4. Instability
This section proves a theorem concerning the instability of a non-inflectional elastica subject to
the Neumann boundary condition. For this purpose, we use the Lessinnes-Goriely criterion [1]
(Theorem 3 therein): if V =−ω2 cos(θ+α), the equilibrium is stable if it crosses more maximum
than minimum of V, and it is unstable if it crosses more minimum than maximum. The case of
an equal number of maximum and minimum is not covered. In the case of the non-inflectional
elastica θ′ = 2ω
√
k2 − sin2(θ+α), the maximum and minimum of V correspond to the minimum
of θ′, and respectively the maximum of θ′. This observation allows us to apply the criteria in the
following form:
If, along a non-inflectional elastica, the number of minima of θ′ is less than the number of its
maxima, the equilibrium is unstable.
Theorem 4. If θ′′(0) > 0 and θ′′(1) < 0, the equilibrium shapes of a non-inflectional elastica subject
to the Neumann boundary conditions are unstable.
Proof. This case is covered by solution (17). If the variable s is changed toσ≡ kωs, ϑ(σ) ≡ θ(s(σ))
is defined for C ≤ σ ≤ −C + 2nK . Thus, ϑ′(σ) and ϑ′′(σ) become periodic with the period 2K
(Figure 11). Next, it is observed that the assumption ϑ′′(C ) > 0 is fulfilled if −K < C < 0. In this
case, we also have ϑ′′(−C +2nK ) =ϑ′′(−C ) < 0, as required by the theorem. The extremes of θ′ are
now examined for three intervals. The first interval is −K < C ≤ σ < 0. Here, ϑ′ has no extrema.
The next interval 0 ≤ σ < 2nK consists of n periods. In each period, there is one maximum and
one minimum of ϑ′. The last interval 2nK ≤ σ ≤ −C +2nK < K +2nK has one maximum of ϑ′.
The number of maxima is thus one greater than the number of minima. Hence, the elastica is
unstable.
C. R. Mécanique, 2020, 348, n 2, 137-148
Milan Batista 147
Figure 10. Stable forms of non-inflectional elastica corresponding to points in Figure 9.
We stress that Theorem 4 is not new: it is restatement of the Theorem 4 in [1]. We also note
that, by similar reasoning, Theorem 3 can be proved without appealing to Corollary 1.
5. Conclusion
This study demonstrated that the shape of a non-inflectional elastica subject to Dirichlet bound-
ary conditions is stable. For a non-inflectional elastica subject to mixed conditions, two theorems
that provide sufficient conditions for its stability were established. For a non-inflectional elastica
subject to Neumann boundary conditions that does not form closed periods, a sufficient condi-
tion for its stability as well as its instability was provided. We emphasize that the present Theo-
rems 1, 3, 4 on the Dirichlet and Neumann boundary conditions are not new; the main results are
Theorems 2a and 2b for mixed boundary conditions.
When investigating the stability of a non-inflectional elastica, the method proposed by this
study is slightly simpler than that suggested in [1] because no trajectory has to be examined
in the phase space; only the second derivative values at the endpoints must be calculated. This
also allows for simple identification of stability regions in bifurcation diagrams associated with
particular boundary conditions. However, the proposed method exhibits limitations compared to
that provided in [1]: it allows for establishing only sufficient conditions for stability, and cannot
be used to establish conditions for instability.
C. R. Mécanique, 2020, 348, n 2, 137-148
148 Milan Batista
Figure 11. Graph of function dn (solid line) and its normalized derivative (dotted line).
References
[1] T. Lessinnes, A. Goriely, “Geometric conditions for the positive definiteness of the second variation in one-
dimensional problems”, Nonlinearity 30 (2017), p. 2023-2062.
[2] A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th ed., Dover Publications, New York, 1944.
[3] M. Batista, “Analytical treatment of equilibrium configurations of cantilever under terminal loads using Jacobi
elliptical functions”, Int. J. Solids Struct. 51 (2014), p. 2308-2326.
[4] V. G. A. Goss, Snap Buckling, Writhing and Loop Formation in Twisted Rods, Center for Nonlinear Dynamics,
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[5] W. P. Reinhardt, P. L. Walker, “Jacobian elliptic functions”, in NIST Handbook of Mathematical Functions (F. W. J. Olver,
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[7] H. H. E. Leipholz, Stability Theory an Introduction to the Stability of Dynamic Systems and Rigid Bodies, 2nd ed.,
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C. R. Mécanique, 2020, 348, n 2, 137-148


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