a model of a general elastic curved rod

MATHEMATICAL METHODS IN THE APPLIED SCIENCESMath. Meth. Appl. Sci. 2002; 25:835–854 (DOI: 10.1002/mma.315)MOS subject classi�cation: 65N 30; 34B 60; 74B 99

A model of a general elastic curved rod

Anca Ignat1;†, J�urgen Sprekels2;∗;‡ and Dan Tiba2;3;§

1Faculty of Computer Science; University “Al. I. Cuza”, str. Berthelot 16; RO-6600 Iasi; Romania2Weierstrass Institute for Applied Analysis and Stochastics; Mohrenstrasse 39; D-10117 Berlin; Germany

3Institute of Mathematics; Romanian Academy; P.O. Box 1-764; RO-70700 Bucharest; Romania

Communicated by W. Spr�o�ig

SUMMARY

We indicate a new approach to the deformation of three-dimensional curved rods with variable cross-section. The model consists of a system of nine ordinary di�erential equations for which we proveexistence and uniqueness via the coercivity of the association bilinear form. From the geometrical pointof view, we are using the Darboux frame or a new local frame requiring just a C1-parameterizationof the curve. Our model also describes the deformation occurring in the cross-sections of the rod.Copyright ? 2002 John Wiley & Sons, Ltd.

KEY WORDS: deformation of elastic rods; low geometrical regularity; variable cross-sections

1. INTRODUCTION

This work is devoted to the study of three-dimensional curved rods with non-constant thick-ness and with multiply connected cross-section. Under the action of body forces and of (out-side) surface tractions the rod will su�er deformations, and our approach allows to investigatedeformations of variable cross-sections as well.The model we propose is expressed as a system of nine ODEs and is derived from the

linear elasticity system under the minimal mechanical assumption that the transverse sectionsremain plane after the deformation, although their shape may change. We allow for shear andtorsion, and even for (pointwise) degeneracy of the sections to dimension one. However, weunderline that this last remark just means that our model does not impose other constraintsthan those coming from the validity requirements of the linear elasticity system.The literature concerning curved rods is very rich, and the books of Trabucho and Viano

[1], Lagnese et al. [2], Antman [3] give a comprehensive overview of the existing results and

∗Correspondence to: J�urgen Sprekels; Weierstrass Institute for Applied Analysis and Stochastics; Mohrenstrasse 39;D-10117 Berlin; Germany.†E-mail: [email protected]‡E-mail: [email protected], [email protected]§E-mail: [email protected]

Copyright ? 2002 John Wiley & Sons, Ltd. Received 16 November 2000

836 A. IGNAT ET AL.

methods, both in the stationary and the time-dependent cases. In the setting of asymptoticmethods, we quote the recent paper by Jurak et al. [4] and the very recent work of Muratand Sili [5] announces the obtaining of an ODE model. The works of Reddy [6], Chenaisand Paumier [7], Chapelle [8] discuss the ‘locking problem’ for the numerical approximationof arches and curved rods. An explicit solution for the Kirchho�–Love arches was found bySprekels and Tiba [9], and a detailed examination of optimization questions is due to Ignatet al. [10].An important ingredient in the study of curved rods is the description of their geometry

and the choice of a local frame. While in the scienti�c literature the classical Frenet frame is,usually, taken into account, in this paper we consider other variants with the aim of relaxingthe regularity assumptions or in connection with certain applications. In the case of shells,such regularity questions were discussed by Blouza and Le Dret [11], Blouza [12].In Section 2, we view the three-dimensional curve giving the line of centroids of the rod as

lying on a �xed surface, and we use the Darboux frame, Cartan [13]. This approach requiresa curve parameterization in W 2;∞ and �ts to the analysis of associated optimization problemswhere such a situation describes an important class of constraints. Section 3 introduces adirect method, based on a new choice of a Lipschitzian local frame for W 2;∞ rods.In all cases, we prove the coercivity of the corresponding bilinear form, and we obtain the

existence and the uniqueness of the solution. In the last section, we present some numericalsimulations based on our model, and we make a brief comparison with the already existingnumerical results.Finally, we point out that our way of working, by imposing the mechanical assumptions

directly in the linear elasticity system, may be compared with the work of Delfour and Zhao[14] in the case of shells.

2. THE DARBOUX FRAME

Let L¿0 be given, and let !(x3)⊂R2 be bounded domains, not necessarily simply connected,for any x3 ∈ [0; L]. We assume that !(x3)⊃!, a given open set in R2, such that

0=∫!x1 dx1 dx2 =

∫!x2 dx1 dx2 =

∫!x1x2 dx1 dx2 (1)

Hypothesis (1) on !(x3); x3 ∈ [0; L], is a slight relaxation of a similar condition used by Muratand Sili [5] and by Alvarez-Dios and Viano [15].We introduce the open set:

�=⋃

x3∈]0; L[(!(x3)×{x3})⊂R3 (2)

The curved rod � is given as a transformation of �:

(x1; x2; x3) = �x∈� �→F �x= x=(x1; x2; x3)

= ��(x3) + x1 �n(x3) + x2 �b(x3)∈ �; ∀�x∈� (3)

�= {x=F �x; �x∈�} (4)

Copyright ? 2002 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2002; 25:835–854

A MODEL OF A GENERAL ELASTIC CURVED ROD 837

Here ��∈W 2;∞(0; L)3, with ��′′ piecewise continuous is a three-dimensional curve, called theline of centroids, parameterized with respect to its arc length, and �n(x3); �b(x3)∈R3 are somevectors for any x3 ∈ [0; L], which are di�erent (in general) from the normal and binormalvectors associated to the Frenet frame. In the sequel, for any vector �a∈R3, we shall denoteby (a1; a2; a3) its components.We assume now that the curve ��(·) lies on a given surface S, de�ned by the equation

�(x1; x2; x3)=0 (5)

with �∈C2(R3), that is,�(�1(x3); �2(x3); �3(x3))=0 ∀x3 ∈ [0; L] (6)

We denote by �n(x3)= (n1(x3); n2(x3); n3(x3)) a unit normal to S in the points ��(x3)∈ S, that is,�(x3) �n(x3)=∇�( ��(x3)) for some real mapping �. We also denote by �t(x3)= (�′1(x3); �′2(x3);�′3(x3)) the unit tangent to the curve ��(·).Finally, we choose the unit vector �b(x3)= �t(x3)∧ �n(x3), the vectorial product between �t

and �n. Then, the local frame (�t; �n; �b) is positively oriented, and it is called the Darbouxframe.As �t; �n; �b are di�erentiable and |�t |2R3 = | �n|2R3 = | �b|2R3 = 1 ∀x3 ∈ [0; L], it follows that

〈�t; �t ′〉= 〈 �n; �n′〉= 〈 �b; �b′〉=0 in [0; L] (7)

The orthogonality relations (7) give the ‘equations of motion’ of the Darboux frame,

�t ′(x3) = a(x3) �b(x3) + �(x3) �n(x3)

�b′(x3) =−a(x3)�t(x3) + c(x3) �n(x3)�n′(x3) =−�(x3)�t(x3)− c(x3) �b(x3)

(8)

where the functions a(·); �(·); c(·) are piecewise continuous and, in that order, are calledthe geodesic curvature, the normal curvature, and the geodesic torsion, respectively, of thecurve ��(·), Cartan [11, p. 162].The curved rod � is clamped at both ends, and it is subjected to body forces f in � (weight,

electromagnetic �eld, etc.) and to surface tractions g on the lateral surface denoted �. Noticethat, on the ‘inside’ lateral face of � (i.e. corresponding to possible holes in the cross-section),we have g≡ 0.We denote by �u : �→R3 the corresponding displacement of each point x∈ � under the

action of the given forces. Our ‘mechanical’ assumption is that �u has the form

�u(x)= ��(x3) + x1 �N (x3) + x2 �B(x3) ∀x∈ � (9)

with �x=(x1; x2; x3)=F−1(x) and where ��; �N; �B∈H 10 (0; L)

3 are unknown functions.Relation (9) is a special simple case of the so-called polynomial approximation of the dis-

placement, used by Trabucho and Viano [1], Delfour and Zhao [14]. In particular, it says thattransverse sections of � (i.e. perpendicular on �t(x3); x3 ∈ [0; L]) remain plane or degenerate todimension one after the deformation. The vector ��(x3) describes the translation of the pointson the centroid line ��(·), and the vectors �N (x3) + �n(x3); �B(x3) + �b(x3) show the deformation


838 A. IGNAT ET AL.

of the orthogonal frame in the cross-section (which does not necessarily remain orthogonal tothe tangent of the new centroid line, i.e. to ��′(x3) + ��′(x3)). This allows for shear and lengthor volume changes after the deformation.By an obvious computation, we get the Jacobian of F , denoted by J (�x)=DF(�x), its deter-

minant, and its inverse (using (8)):

J (�x)=

n1(x3) b1(x3) t1(x3) + x1n′1(x3) + x2b

′1(x3)

n2(x3) b2(x3) t2(x3) + x1n′2(x3) + x2b′2(x3)

n3(x3) b3(x3) t3(x3) + x1n′3(x3) + x2b′3(x3)

(10)

J (�x)−1 =

n1 − ct1x21− �x1 − ax2 n2 − ct2x2

1− �x1 − ax2 n3 − ct3x21− �x1 − ax2

b1 +ct1x1

1− �x1 − ax2 b2 +ct2x1

1− �x1 − ax2 b3 +ct3x1

1− �x1 − ax2t1

1− �x1 − ax2t2

1− �x1 − ax2t3

1− �x1 − ax2

(11)

det J (�x)=1− �(x3)x1 − a(x3)x2; ∀�x∈� (12)

If !(x3) is contained in a ball with a su�ciently small radius with respect to the curvatures,then we may assume that

det J (�x)¿c¿0; ∀�x∈�We introduce the vectorial mapping �w : �→R3,

�w(�x)= ��(x3) + x1 �N (x3) + x2 �B(x3); ∀�x∈� (13)

and we notice that

�u(x)= �w(F−1(x)) (14)

We recall that

D �u(x)=D �w(F−1(x))DF−1(x)=D �w(F−1(x))J (F−1(x))−1; ∀x∈ � (15)

We simply denote:

DF−1(x)= (dij(x))i; j= 1;3; J (�x)−1 = (hij(�x))i; j=1;3 (16)

By (10)–(16), we can compute the symmetrized gradients eij of the displacement �u(x). Wehave

9ui9xj(x) =Ni(x3(x))d1j(x) + Bi(x3(x))d2j(x)

+ [�′i(x3(x)) + x1(x)N ′i (x3(x)) + x2(x)B

′i(x3(x))]d3j(x); i; j=1; 3 (17)

eij(x) =12

(9ui9xj

+9uj9xi

)(x); i; j=1; 3 (18)



Consequently, we get (without summation convention)

eii(x)=9ui9xi(x); i=1; 3 (19)

eij(x)= eji(x) = 12{Ni(x3(x))d1j(x) + Bi(x3(x))d2j(x)

+ [�′i(x3(x)) + x1(x)N′i (x3(x)) + x2(x)B

′i(x3(x))]d3j(x)

+Nj(x3(x))d1i(x) + Bj(x3(x))d2i(x)

+ [�′j(x3(x)) + x1(x)N ′j (x3(x)) + x2(x)B

′j (x3(x))]d3i(x)}; i; j=1; 3 (20)

If �; �¿0 denote the Lame constants of the material, the bilinear form governing the equationsof linear elasticity is de�ned on the subspace Z × Z ⊂H 1(�)3×H 1(�)3 of functions havingzero traces on the ‘bases’ of the rod �, i.e. on F(!(0))∪F(!(L)). We have (with summationconvention)

B(�u; �v)=∫�[�epp(�u)eqq( �v) + 2�eij(�u)eij( �v)] dx (21)

Notice that �u given by (9) belongs to Z , and we shall take the test functions �v∈ Z ina similar form by replacing the unknown mappings Ni; Bi; �i, respectively, by some arbi-trary mappings Mi;Di; �i ∈H 1

0 (0; L); i=1; 3. In this way, we perform a projection of B

onto the (in�nite dimensional) subspaces Z ⊂ Z of all mappings �u given by (9) with ‘co-e�cients’ ��; �N; �B∈H 1

0 (0; L)3. We underline that this is essentially the same procedure as in

the �nite element method (where �nite dimensional subspaces appear), and some convergenceand approximation results are proved by Trabucho and Viano [1] in a di�erent setting. Bythe above choice, it is obvious that the space Z can be identi�ed with H 1

0 (0; L)9. A standard

change of variables, and (9), (16), give

B(�u; �v) =∫F(�)

[�epp(�u)(F �x) · eqq( �v)(F �x) + 2�eij(�u)(F �x) · eij(�u)(F �x)] dx

= �∫�

3∑i; j=1

[Ni(x3)h1i(�x) + Bi(x3)h2i(�x) + (�′i(x3) + x1N′i (x3) + x2B

′i(x3))h3i(�x)]

× [Mj(x3)h1j(�x) +Dj(x3)h2j(�x) + (�′j(x3) + x1M ′j (x3) + x2D

′j(x3))h3j(�x)]

×|det J (�x)| d�x+�∫�

∑i �=j[Ni(x3)h1j(�x) + Bi(x3)h2j(�x) + (�′i(x3) + x1N

′i (x3)

+ x2B′i(x3))h3j(�x) + Nj(x3)h1i(�x) + Bj(x3)h2i(�x) + (�

′j(x3) + x1N

′j (x3)

+ x2B′j (x3))h3i(�x)][Mi(x3)h1j(�x) +Di(x3)h2j(�x) + (�′i(x3) + x1M

′i (x3)

+ x2D′i(x3))h3j(�x) +Mj(x3)h1i(�x) +Dj(x3)h2i(�x) + (�

′j(x3) + x1M

′j (x3)


840 A. IGNAT ET AL.

+ x2D′j(x3))h3i(�x)]|det J (�x)| d�x + 2�

∫�

3∑i=1[Ni(x3)h1i(�x) + Bi(x3)h2i(�x)

+ (�′i(x3) + x1N′i (x3) + x2B

′i(x3))h3i(�x)][Mi(x3)h1i(�x) +Di(x3)h2i(�x)

+ (�′i(x3) + x1M′i (x3) + x2D

′i(x3))h3i(�x)]|det J (�x)| d�x (22)

We have the following result.

Theorem 2.1Assume that 0¡c61− �(x3)x1 − a(x3)x26m; ∀�x∈�. Then, the bilinear form B is coerciveand bounded on H 1

0 (0; L)9.

The argument follows the same steps as in the proof given by Ciarlet [16] for the case ofplane arches.We start with the following new inequality:

Lemma 2.2There are c1¿0; c2¿0 such that

B(�u; �u)¿c1|�u|2H 10 (0; L)9 − c2|�u|2L2(0; L)9 (23)

where �u, given by (9), is identi�ed with the vector (�1; �2; �3; N1; N2; N3; B1; B2; B3)∈H 10 (0; L)

9.

ProofBy virtue of (22), (12), and since !(x3)⊃!; ∀x3 ∈ [0; L], we have

B(�u; �u)¿ �c∫!×[0; L]

∑i �=j[Ni(x3)h1j(�x) + Bi(x3)h2j(�x) + (�′i(x3) + x1N

′i (x3)

+ x2B′i(x3))h3j(�x) + Nj(x3)h1i(�x) + Bj(x3)h2i(�x) + (�

′j(x3) + x1N

′j (x3)

+ x2B′j (x3))h3i(�x)]

2 d�x + 2�c∫!×[0; L]

3∑i=1[Ni(x3)h1i(�x) + Bi(x3)h2i(�x)

+ (�′i(x3) + x1N′i (x3) + x2B

′i(x3))h3i(�x)]

2 d�x

Consequently, usual binomial inequalities imply that

1�c

B(�u; �u)¿12

∫!×[0; L]

∑i �=j[(�′i(x3) + x1N

′i (x3) + x2B

′i(x3))h3j(�x)

+ (�′j(x3) + x1N′j (x3) + x2B

′j (x3))h3i(�x)]

2+∫!×[0; L]

3∑i=1[(�′i(x3) + x1N

′i (x3)

+ x2B′i(x3))h3i(�x)]

2 d�x − C|�u|2L2(0; L)9 (24)

where we have used that hij ∈L∞(�); i; j=1; 3, which follows from (10)–(12), (16), andfrom the regularity of the curve ��(·) and of the Darboux frame.



We denote, for simplicity, zi= �′i + x1N′i + x2B

′i ; i=1; 3.

We make an algebraic computation:

12 [(z1h32 + z2h31)

2 + (z2h33 + z3h32)2 + (z1h33 + z3h31)2] + 32(z

21h231 + z

22h232 + z

23h233)

= 12 z

21h232 +

12 z

22h231 +

12 z

22h233 +

12 z

23h232 +

12 z

21h233 +

12 z

23h231 + z1z2h31h32

+ z2z3h32h33 + z1z3h31h33 + 32(z

21h231 + z

22h232 + z

23h233)

= 12(z

21 + z

22 + z

23)(h

231 + h

232 + h

233) +

12(z1h31 + z2h32)

2

+ 12(z1h31 + z3h33)

2 + 12(z2h32 + z3h33)

2

Hence, we can infer from (24) that

1c�

B(�u; �u)¿14

∫!×[0; L]

3∑i=1(�′i(x3) + x1N

′i (x3)

+ x2B′i(x3))

23∑i=1h23i(�x) d�x − C|�u|2L2(0; L)9 (25)

Examining (11), we see that, under the assumption of the theorem,

3∑i=1h23i =

1(1− �(x3)x1 − a(x3)x2)2

3∑i=1ti(x3)2 =

1(1− �(x3)x1 − a(x3)x2)2

¿ �¿0 (26)

for some �¿0, since |�t |R3 = 1.Let us also note that, owing to (1), we have

∫!×[0; L]

(�′i(x3) + x1N′i (x3) + x2B

′i(x3))

2 d�x

=meas(!)|�i|2H 10 (0; L)+∫!x21 dx1 dx2|Ni|2H 10 (0; L)+

∫!x22 dx1 dx2|Bi|2H 10 (0; L); i=1; 3 (27)

with meas(!) denoting the Lebesgue measure in R2 of the domain !.By combining (25)–(27), we end the proof.

Lemma 2.3If B(�u; �u)=0, then �u=0.

ProofIf B(�u; �u)=0, we obtain for a.e. �x∈� that

Ni(x3)h1i(�x) + Bi(x3)h2i(�x) + zih3i(�x)=0; ∀i=1; 3 (28)


842 A. IGNAT ET AL.

Ni(x3)h1j(�x) + Bi(x3)h2j(�x) + Nj(x3)h1i(�x) + Bj(x3)h2i(�x)

+ zih3j(�x) + zjh3i(�x)=0; i = j; i; j=1; 3 (29)

with zi as in the previous proof.Multiply (29) by h3i(�x) and (28) by h3j(�x) ( j is �xed!) and subtract. Then multiply (28)

by h3i(�x) for i= j, and add to the previous results, obtained for i = j. We get

( �N; �B) + zj3∑i=1h23i(�x)=0; ∀j=1; 3 (30)

with ( �N; �B) some linear mapping of the vectors �N; �B. Taking into account (11) and |�t |R3 = 1,(30) gives

�′i(x3) + x1N′i (x3) + x2B

′i(x3)=( �N; �B); ∀i=1; 3 (31)

with a clear modi�cation of .For any i, we give to (x1; x2)∈! three di�erent pairs of values, and we obtain a linear dif-

ferential system is normalized form with zero initial conditions, since �i; Ni; Bi ∈H 10 (0; L); ∀i=

1; 3). Then, the unique solution is identically zero and this gives that �u=0 in � as claimed.Here, we have again used that the coe�cients hij are bounded.

Proof of Theorem 2.1Assume, by contradiction, that for any ¿0 there is u=(�1; �

2; �

3; N

1 ; N

2 ; N

3 ; B

1; B

2; B

3)∈

H 10 (0; L)

9 such that

06B(u; u)6|u|2H 10 (0; L)9Without loss of generality, we may take |u|H 10 (0; L)9 = 1, and we denote by u the weak limitin H 1

0 (0; L)9, on a subsequence of {u}.

The weak lower semicontinuity of the quadratic form gives

06B(u; u)60

i.e. B(u; u)=0 and u=0 by Lemma 2.3.We get that u→ 0 weakly in H 1

0 (0; L)9 and, by compact imbedding, strongly in L2(0; L)9.

We use Lemma 2.2 for u, and we pass to the limit in the inequality (23) to obtain that

¿B(u; u)¿c1|u|2H 10 (0; L)9 − c2|u|2L2(0; L)9 = c1 − c2|u|2L2(0; L)9

Consequently, we obtain the contradiction

0¿c1

which ends the proof of Theorem 2.1.

According to the linear elasticity system and to our assumptions, the displacement �u, de�nedby (9), is obtained as the solution of the variational equation:

B(�u; �v) =∫�f‘(�x)(�‘(x3) + x1M‘(x3) + x2D‘(x3))|det J (�x)|d�x

+∫9�g‘(�x)(�‘(x3) + x1M‘(x3) + x2D‘(x3))|det J (�x)|

√i(�x)gij(�x)j(�x) d� (32)



where (i) is the unit outside normal to 9� and the summation convention is used. We recallthat (fi)i=1;3 and (gi)i=1;3 are the body forces, respectively, the surface tractions, acting onthe curved rod �, and we have denoted by �f=(f‘)‘=1;3 and �g=(g‘)‘=1;3 the quantities:

�f(�x)= f(F �x); �g(�x)= g(F �x) (33)

The coe�cients gij(�x) are obtained as (Ciarlet [17])

(gij(�x))i; j=1;3 = J (�x)TJ (�x) (34)

(gij(�x))i; j=1;3 = (gij(�x))−1i; j=1;3

(35)

The right-hand side in (32), and the relations (33)–(35), are due to a standard change ofvariable � �→� in the integrals∫

�fivi d x +

∫�givi d�; (vi)i=1;3 ∈H 1(�)3; vi=0 on F(!(0)∪!(L))

which express the action on the rod �.It should be noted that at least Lipschitz regularity is necessary for the part of 9� where the

tractions are non-zero, which represents a hypothesis on the ‘variation’ of the cross-sections!(x3). If ! is constant and 9! is smooth, then (32) is fully justi�ed.

Corollary 2.4Under the above assumptions, Equation (32) has a unique solution in H 1

0 (0; L)9.

3. A DIRECT APPROACH

In this section, we study general rods associated with the three-dimensional curve �� : [0; L]→R3; ��∈W 2;∞(0; L)3, with ��′′ piecewise continuous, parameterized with respect to the arclength which are no more assumed to lie on some given surface. In this setting, it is possibleto de�ne the Frenet frame with normal and binormal vectors in L∞(0; L)3. However, thisproperty is not enough for the study of the corresponding geometric transformation de�ningthe rod and of the associated di�erential equations, and that is why the standard assumptionin the literature is ��∈C3[0; L]3.We de�ne a new Lipschitzian local frame, simple to use in numerical computations, and

requiring just ��∈W 2;∞(0; L)3. Our construction also applies to curves from C1[0; L]3, but inthis case the local basis vectors are in C[0; L]3. In particular, we notice that all the vectorsof this basis have the same regularity as the tangent vector.For any s∈ [0; L], we de�ne as in Section 2, �t(s)= ��′(s), the unit tangent vector to ��, and

�t ∈W 1;∞(0; L)3.Let {si}; i=0; N + 1, be a partition of the interval [0; L] with s0 = 0 and sN+1 =L, and

with a su�ciently small norm in the sense that

|�t(si)− �t(si+1)|R3¡�; ∀i=0; N (36)


844 A. IGNAT ET AL.

with �¿0 ‘small’. This can be obtained due to the assumed uniform continuity of �t(·)in [0; L].For i=0; N + 1, we de�ne the vectors

n(si)=

(0;−�

′3(si); �

′2(si))=

√�′3(si)2 + �

′2(si)2 if (�′2(si); �

′3(si)) =(0; 0);

(0; �′1(si); 0)=|�′1(si)| otherwise(37)

We have:

|n(si)|R3 = 1; n(si)⊥ �t(si); i=0; N + 1 (38)

We can also assume that the angle between n(si), n(si+1) is acute, i.e. that

〈n(si); n(si+1)〉¿0; i=0; N (39)

This can simply be achieved by changing n(si+1) into −n(si+1), if necessary, and iterating theprocess from s1 to sN+1. We denote by �n(si) the thus obtained vectors.On each interval [si; si+1], we build the function

�mi(s)= �n(si) +s− sisi+1 − si [ �n(si+1)− �n(si)]; i=0; N (40)

Clearly, �mi(si)= �ni(si), �mi(si+1)= �ni(si+1), and the function �m(s)= �mi(s), s∈ [si; si+1], isLipschitz in [0; L]. Moreover, we have

| �mi(s)|R3¿√22; ∀s∈ [0; L]; ∀i=0; N (41)

Inequality (41) is a consequence of elementary geometric arguments in the triangle with twounit edges �n(si); �n(si+1) and with an acute angle, using (39). We just notice that | �mi(si)|R3 isthe length of the line segment connecting arbitrary points on the ‘basis’ of this triangle withits ‘top’ point, whence (41) easily follows.Assume, for the moment, that �m(s) is collinear with �t(s), for some s∈ [0; L]. Since |�t(s)|R3

= 1, we get from (40) that

�n(si) +s− sisi+1 − si [ �n(si+1)− �n(si)]=±| �mi(s)|R3 �t(s) (42)

where i is �xed such that s∈ [si; si+1]. We multiply (42) by �t(s) and apply (41), the triangleinequality, and the Cauchy–Schwarz inequality, to obtain that

√226| �mi(s)|R36

(1− s− si

si+1 − si

)|�t(s)− �t(si)|R3 +

s− sisi+1 − si |

�t(s)− �t(si+1)|R36� (43)

Here, we have also used (36) and (38).As (43) is a contradiction for � ‘small’, we infer that �m(s) and �t(s) are linearly independent

vectors for any s∈ [0; L]. We thus can apply the Schmidt orthogonalization process to them



and construct:

n(s) = �m(s)− 〈 �m(s); �t(s)〉�t(s) (44)

�n(s) = n(s)=|n(s)|R3 ; s∈ [0; L] (45)

Notice that n(s) =(0; 0; 0) due to the linear independence and, moreover, |n(s)|R3¿constant¿0due to its continuity.This shows that �n(s) as de�ned by (44) and (45) has the same smoothness as �t(s)

since �m(s) is Lipschitzian. We have proved:

Theorem 3.1If ��∈C1[0; L]3(W 2;∞(0; L)3) then �n∈C[0; L]3(W 1;∞(0; L)3), and | �n(s)|R3 = 1, 〈 �n(s); �t(s)〉=0,s∈ (0; L). The vector function �b= �t ∧ �n has the same regularity properties and completes thelocal frame.

RemarkThere are alternative ideas for the construction of normal vectors �n. For instance, starting witha smooth regularization �� of ��, to use the Frenet frame for �� and the Schmidt orthogonal-ization for the pair �t; �n, with �n the normal corresponding to ��. However, the proceduregiven in Theorem 3.1 seems to be more appropriate for numerical implementation.

RemarkIn the case ��∈C1[0; L]3, Theorem 3.1 provides a non-di�erentiable local frame, which isnot su�cient for the geometrical description and the solution of the di�erential equationsassociated to curved rods. However, it is simple to see that, even for rods with ‘corners’ (forinstance a rectangular thin frame of metal), one can provide a description of � as in (2)–(4)with �� smooth, since we allow the cross-sections !(·) to be non-constant—see (1). Takinginto account the di�culty of the asymptotic theory for curved rods, Trabucho and Viano [5,Section 38], when the thickness of the rod tends to zero, the reparameterization of a rod withcorners via smooth �� may become very complicated, or hypothesis (1) may be violated, etc.We shall analyse this question in a subsequent work.

RemarkWith the local basis provided by Theorem 3.1, the same arguments as in Section 2 work forproving the coercivity, and the deformation of the curved rod is computed in a similar way.It is only here that the piecewise continuity of ��′′ is necessary.

4. NUMERICAL EXAMPLES

We have tested both modelling approaches indicated in this paper on a large class of examplesinvolving various geometrical shapes for the line of centroids, various cross-sections andseveral types of applied forces.As three-dimensional curves, we have considered the spiral

��(t)=(cos

t√2; sin

t√2;t√2

)(46)


846 A. IGNAT ET AL.

lying on the cylinder x21 + x22 = 1 and clamped at the endpoints,

T1 = − �√2; T2 =

�√2

(47)

T1 = 0; T2 = 8�√2 (48)

Example (46) and (47) was also discussed by Arunakirinathar and Reddy [17], and byChapelle [8], by using the Frenet frame, while our method is based on the Darboux frame,as explained in Section 2.Another form of the line of centroids (a ‘decreasing’ spiral) is given by

��(t)=(− (6�− t)

2

20cos t;− (6�− t)

2

20sin t; t

); t ∈ [0; 4�] (49)

which is inspired by the modern access to the new dome-shaped cupola of the Reichstag inBerlin. In this case, we have tested the direct geometrical approach from Section 3, althoughit is clearly possible to de�ne a surface (a paraboloid) including �� as given by (49). Notice,as well, that the parameterization (49) is not with respect to the arc length, and the standardreparameterization is given by s(0)=0, s′= | ��′|R3 , ��1(t)= ��(s−1(t)). While this relation isdi�cult to integrate, in general, we remark that in the direct approach of Section 3, just��′1 = ��

′=| ��′|R3 is needed, which is very easy to obtain.Our choice of cross-sections includes:

(a) disk of radius 0.3 centred in the points of ��(·):

x21 + x2260:3

2

(b) elliptical crown, centred in the points of ��(·)

0:326x211+x221660:52

(c) rectangle with dimension 0:6× 0:2 centred in the points of ��(·).The curved rods described above were ‘subjected’ to distributed forces of the following

types:

• torsional force:

f(x1; x2; x3)= (−x2; x1; 0); x3 ∈[− �√

2;�√2

](50)

• torsional force in two opposite directions:

f(x1; x2; x3)=

(−x2; x1; 0); x3 ∈

[− �√

2; 0]

(x2;−x1; 0); x3 ∈[0;�√2

] (51)



• pushing in one or two opposite given directions:

f(x1; x2; x3) = (0; 0; 1); x3 ∈[− �√

2;�√2

](52)

f(x1; x2; x3) =

(0; 0; 1) x3 ∈

[T1;T1 + T22

]

(0; 0;−1) x3 ∈[T1 + T22

; T2

] (53)

f(x1; x2; x3) =

{(0; 0;−1000); x3 ∈ [2�; 3�](0; 0; 0) otherwise

(54)

• tangential forces:

f(x1; x2; x3)=

��′(x3); x3 ∈

[T1;T1 + T22

]

− ��′(x3); x3 ∈[T1 + T22

; T2

] (55)

• normal forces:f(x1; x2; x3)= �n(x3); (56)

where �n(·) is computed from the Darboux frame, i.e. is given by the normal to the cylindercontaining ��(·), in the points of the curve ��(·). We also mention that, in all cases where theDarboux frame was applied, the computation of �t; �n; �b and of the quantities det J (�x), hij(�x),a(x3), �(x3), c(x3) was performed exactly, by using Mathematica. The same holds for all thegraphical representations. For the method developed in Section 3, direct numerical calculationshave been done to obtain the above parameters in the discretization points.In Table I, we have collected details about the geometrical and mechanical data assumed

in the examples (Figures 1–15).The Lame constants were in all the examples �xed as �=50, �=100. For Figure 6, we

have also represented (in the last two �gures entitled Figures 16 and 17) the deformationof the cross-section in two points, corresponding to the division points i=50 and 100. Inall the examples, the interval [T1; T2] on which ��(·) is de�ned was divided in 200 equalsubintervals, i.e. i=50 and 100 correspond to the �rst quarter and to the middle of thecurved rod. The scalings used for the various �gures are intended to obtain a clearer graphicalrepresentation, and the procedure is equivalent to the multiplication of the force by the givenfactor, due to the linearity of the equations. In Figures 16 and 17 for Section 1 and Section2, the scalings are 20, respectively 10. Notice that the shear and the torsion e�ects are notrepresented, since this would require 3D �gures. Our aim in the numerical simulation wasto test the modelling approach, and all experiments have produced meaningful results from amechanical point of view. We have avoided the well-known ‘locking problem’ appearing in the


848 A. IGNAT ET AL.

Table I.

Example Interval Force Section Curve Scaling

Figure 1 (47) (50) (a) (46) 50Figure 2 (47) (51) (a) (46) 500Figure 3 (48) (53) (a) (46) 0.2Figure 4 (47) (53) (a) (46) 50Figure 5 (47) (52) (b) (46) 50Figure 6 (47) (50) (b) (46) 50Figure 7 (48) (55) (a) (46) 0.05Figure 8 (47) (55) (a) (46) 50Figure 9 (48) −(55) (a) (46) 0.05Figure 10 (47) −(55) (a) (46) 50Figure 11 (47) (56) (a) (46) 10Figure 12 (47) −(56) (a) (46) 10Figure 13 (48) −(56) (a) (46) 30Figure 14 (48) (56) (a) (46) 30Figure 15 (49) (54) (c) (49) 300

Figure 1. Example 1. Figure 2. Example 2.

numerical approximation of arches, rods and shells, Chenais and Paumier [7], Arunakirinatharand Reddy [18], Chapelle [8], Pitk�aranta and Leino [19], by choosing a very �ne division of[T1; T2] in comparison with the dimension of the cross-sections, i.e. by taking ‘large’ cross-sections.


850 A. IGNAT ET AL.




852 A. IGNAT ET AL.

Figure 15. Example 15.

Figure 16. Section 1.



Figure 17. Section 2.

Consequently, we have used standard piecewise linear, continuous �nite elements on [T1; T2].If Vm, m=200, is the discrete subspace associated to H 1

0 (T1; T2), then V9m is associated to

H 10 (T1; T2)

9, and its �nite element basis is constructed in a canonical way. The rigidity matrixis sparse, and the number of unknowns in the obtained linear algebraic system (correspondingto (32) was 1791. The integrals over the cross-sections, appearing in the coe�cients, werecomputed by a change of variable to polar coordinates, which reduced all the cases to variousrectangles and, then, by classical interpolation methods. Since the dimension was not very big,the algebraic linear system was solved by the Gaussian elimination method, which is known tobe very stable. In all the computed examples, we have noticed that the vectors �Nm((T1+T2)=2)and �Bm((T1 + T2)=2) (the deformations of the ‘normal’ and ‘binormal’ vectors, in the middleof the rod) are orthogonal.


854 A. IGNAT ET AL.

Finally, we point out that, although the theoretical results allow variable cross-sections, non-zero surface tractions, and not centered line of centroids (see (1)), these elements were notimplemented in the numerical examples in order to keep the complexity of the computationsat a reasonable level.

REFERENCES

1. Trabucho L, Viano JM. Mathematical modelling of rods. In: Handbook of Numerical Analysis, Ciarlet PG,Lions JL (eds), vol. IV. Elsevier: Amsterdam, 1996; 487–974.

2. Lagnese JE, Leugering G, Schmidt EJPG. Modeling, Analysis and Control of Dynamic Elastic Multi-linkStructures. Birkh�auser: Boston, 1994.

3. Antman SS. Nonlinear Problems of Elasticity. Springer: Berlin, 1995.4. Jurak M, Tambaca J, Tutek Z. Derivation of a curved rod model by Kirchho� assumptions. ZAMM 1999;79(7):455–463.

5. Murat F, Sili A. Comportement asymptotique des solutions du syst�eme de l’elasticite linearisee anisotropeheterog�ene dans les cylindres minces. CRAS Paris 1999; 328(I):179–184.

6. Reddy BD. Mixed �nite element methods for one-dimensional problems: a survey. Questiones Mathematicae1992; 15(3):233–259.

7. Chenais D, Paumier JC. On the locking phenomenon for a class of elliptic problems. Numerische Mathematik1994; 67:427–440.

8. Chapelle D. A locking-free approximation of curved rods by straight beam elements. Numerische Mathematik1997; 77:299–322.

9. Sprekels J, Tiba D. Sur les arches lipschitziennes. CRAS Paris 2000; 331(I):179–184.10. Ignat A, Sprekels J, Tiba D. Analysis and optimization of nonsmooth arches. SIAM Journal on Control and

Optimization 2001; 40:1107–1135.11. Blouza A, Le Dret H. Existence et unicite pour le mod�ele de Koiter pour une coque peu reguli�ere. CRAS

Paris 1994; 319(I):1127–1132.12. Blouza A. Existence et unicite pour le mod�ele de Nagdhi pour une coque pou reguli�ere. CRAS Paris 1997;

324(I):839–844.13. Cartan H. Formes di��erentielles. Hermann: Paris, 1967.14. Delfour MC, Zhao J. Intrinsic nonlinear models of shells. CRM Proceedings, Lecture Notes 1999; 21:

109–123.15. Alvarez-Dios JA, Viano JM. An asymptotic bending model for a general planar curved rod. In: Shells:

Mathematical Modelling and Scienti�c Computing, Bernadou M, Ciarlet PG, Viano JM (eds), Univ. Santiagode Compostela Press: Spain, 1997; 11–17.

16. Ciarlet Ph. The Finite Element Method for Elliptic Problems. North-Holland: Amsterdam, 1978.17. Ciarlet Ph. Introduction to Linear Shell Theory. Gauthier-Villars: Paris, 1998.18. Arunakirinathar K, Reddy BD. Mixed �nite element methods for elastic rods of arbitrary geometry. Numerische

Mathematik 1993; 64:13–43.19. Pitk�aranta J, Leino Y. On the membrane locking of the h–p �nite elements in a cylindrical shell problem.

International Journal for Numerical Methods in Engineering 1994; 37(6):1053–1070.


a model of a general elastic curved rod

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