Цилиндрический изгиб перекрестно армированных композитных пластин Текст научной статьи по специальности «Физика»

Математика. Физика

УДК 539.3

CYLINDRICAL BENDING OF ANGLE-PLY COMPOSITE PLATES G.M. Kulikov, S.V. Plotnikova

Department “Applied Mathematics and Mechanics”, TSTU; kulikov@apmath. tstu.ru

Key words and phrases: laminated composite plate; plane strain elasticity; sampling surfaces method.

Abstract: This paper presents an efficient method of solving the plane strain problem of elasticity for laminated composite plates. The method is based on the new concept of sampling surfaces developed recently by the authors. According to this concept, we introduce inside the nth layer In not equally spaced SaS parallel to the middle surface of the plate and choose displacements of these surfaces as plate unknowns. This fact gives an opportunity to derive the solutions of plane strain elasticity for thick and thin laminated composite plates with a prescribed accuracy by using a sufficiently large number of SaS, which are located at layer interfaces and Chebyshev polynomial nodes.

1. Introduction

A conventional way for developing the higher order discrete-layer plate theory accounting for thickness stretching is to utilize either quadratic or cubic series expansions in the transverse coordinate for each layer and to choose as unknowns the generalized displacements of layers [1-3]. Herein, we consider a new method of sampling surfaces (SaS) inside the plate body proposed recently by the authors [4, 5].

As SaS denoted by Q(n)1, Q(n)2,..., Q(n)/n , we choose outer surfaces and any inner surfaces inside the nth layer and introduce displacement vectors u(n)1, u(n)2,..., u(n)In of these surfaces as fundamental plate unknowns, where In is the total number of SaS chosen for each layer (In > 3). The index n identifies the belonging of any quantity to the nth layer and runs from 1 to N, where N is the number of layers. Such choice of displacements with the consequent use of Lagrange polynomials of degree (ln -1) in the thickness direction for each layer permits the representation of governing equations of the laminated plate theory developed in a very compact form.

However, the above polynomial interpolation implemented for equally located SaS [4, 5] does not work properly with Lagrange polynomials of high degree because Runge's phenomenon [6] can occur, which yields the wild oscillation at the edges of the

interval when the user deals with any specific functions. If the number of equally spaced nodes is increased then the oscillations become even larger. Fortunately, the use of Chebyshev polynomial nodes [7] can help to improve significantly the behaviour of Lagrange polynomials of high degree for which the error will go to zero as In ^ m. This fact gives an opportunity to derive the exact solutions of plane strain elasticity for thick laminated composite plates with a prescribed accuracy utilizing a sufficiently large number of not equally spaced SaS.

2. Three-dimensional description of laminated plate

Consider a thick laminated plate of the thickness h. Let the midsurface Q be referred to Cartesian coordinates x1 and x2, whereas the transverse coordinate x3 is oriented along the normal direction to the midsurface. The transverse coordinates of SaS inside the nth layer are defined as:

.(n)1 _ x[n-1]

”2 **"3 ^

x(n)In _ x[n].

x(n)mn _ _L(x[n-1]

AO -- I A’l

+ x

[n])

1

(

3 ) — hn cos 2

2mn - 3

2(In - 2)

(1)

where x3n-1] and x3n] are the transverse coordinates of the bottom and top surfaces

Q[n-1] and Q[n] of the nth layer (Fig. 1) such that x3°] _-h/2 and x3^] _ h/2;

hn _ x3n] - x3n-1] is the thickness of the nth layer; the index mn identifies the belonging of any quantity to inner SaS of the nth layer and runs from 2 to In -1 , whereas the indices in, jn , kn to be introduced in the next section for describing all SaS of the nth layer run from 1 to In.

It is important to note that transverse coordinates of inner SaS (1) coincide with the nodes of Chebyshev polynomials [7]. This fact has a great meaning for a convergence of the SaS method.

Fig. 1. Geometry of the laminated plate

71

2

2sj _ ui, j + uj,i, (2)

where u, are the displacements of the plate. Here and in the following developments, Latin tensorial indices i, j, k, l range from 1 to 3, whereas Greek tensorial indices a, p range from 1 to 2.

The strain components of SaS can be written as:

9P(n)in _ 2p (x(n)in ) _ u(n)in + u(n)in .

2Sap _ ap(x3 ) _ Ua,p + Up,a ;

2p(n)in _ 2p (x(n)in ) _ p(n)in + u(n)in . a3 _ 2Sa3(x3 ) _pa + u3,a ;

2s<fn _ 2e33(x3n)jn) _p3n)jn , (3)

where u(n)in (x1, x2) are the displacements of SaS of the nth layer; p(n)in (x1, x2) are the derivatives of displacements with respect to coordinate x3 at SaS, that is,

u(n)jn _ U, (x3n)jn); p(n)jn _ u,,3 (x3n)jn). (4)

It is convenient to introduce the displacements of bottom and top surfaces of the plate and layer interfaces as follows:

u,,(D1 _ u№; u(N)jn _u,[N];

U,(m)1m _ u,(m+1)1 _ uf[m], (5)

where u[m](x1,x2) are the displacements of layer interfaces Q[m] (m _ 1,2,...,N -1).

3. Displacement and strain distributions in thickness direction

Up to this moment, no assumptions concerning displacement and strain fields have been made. We start now with the first fundamental assumption of the proposed higher order layer-wise plate theory. Let us assume that the displacements are distributed through the thickness of the nth layer as follows:

u(n) _ V L(n)inu,(n)in ; xv *J < x3 < xx

U(n) _ V L(n)ln u(n),n; x3n-1] < x3 < x3n], (6)

in

where L(n)in (x3) are the Lagrange polynomials of degree (ln -1) expressed as

x - x(n) jn

L(n),n _ ^ _^L_x3--------------------------------------------------------. (7)

i x(n)in - x(n) jn Jn^ln **3 **-3

The use of relations (4) and (6) yields

p(n),n _ V M(n) jn (x3n),n )u(n)]n , (8)

n

where M(n)Jn = Jn are the derivatives of Lagrange polynomials. The values of these derivatives at SaS of the nth layer are calculated as:

( , \ 1 x(n)in _ x(n)kn

M(n)jn (X(n)in ) =__________________________________1__ n —3____________________________________3_ for J Ф i •

' 3 ' (n) jn _(n)i„ 11 (n) jn _(n)k„ J n ’

V' ' J tl ____ -y v 'ft y' ' J ri _ у

Л3 л3 kn *in, Jn 3 л3

M(n)n (x3n)in )=- £M(n)Jn (x3n)in). (9)

Jn

The latter formula is valid because a useful identity for derivatives of the Lagrange polynomials

£ M(n) ]n = 0 (10)

holds. Thus, the key functions p(n)in of the proposed higher order layer-wise plate theory are represented according to (8) as a linear combination of displacements of SaS of the nth layer u(n)Jn .

The following step consists in a choice of the correct approximation of strains through the thickness of the nth layer. It is apparent that the optimal solution of the problem is to choose the strain distribution, which is similar to the displacement distribution (6), that is,

8(n) = £ L(n)‘n e(p'n . x[n-1] < x3 < (11)

It is necessary to note that strain-displacement relationships (3) and (11) exactly represent all rigid-body motions of the laminated plate. A proof of this statement can be done following a technique developed in [8-12].

4. Total potential energy of laminated plate

Substituting strains (11) in the total potential energy of a laminated plate and introducing stress resultants

n

n

H (n)in =

J

J4n) L(n)indx3,

(12)

.[n _1]

n\

x

one obtains

П=Л

Q

1 ZZZ4n)in eF”

,2 n in i, J

■zX’[xx x]

-p!°40])

dxidx2 _ Ws,

(13)

where p[0] and p[N] are the loads acting on the bottom and top surfaces and

Q[N]; W2 is the work done by external loads applied to the boundary surface 2 .

For simplicity, we restrict ourselves to the case of linear elastic materials. The natural choice for constitutive equations is the generalized Hook’s law:

(n) _ vс(n)p(n)- x[n-1] < x < x[n]

ij ~ Z-,Cijk&kl ’ x3 -x3b x3 ■

(14)

k

Inserting stresses (14) in (12) and taking into account the strain distribution (11), we have

jn k

(n)injn c(n) jn ijkl bkl

(15)

where

DW,n]n _ j J L(n)ln L(n) jn dx3 .

x[n-1]

(16)

5. Exact solution for laminated composite plate

Consider a simply supported laminated plate in cylindrical bending subjected to the sinusoidally distributed transverse load

[ N ] • '

p3 J _ Po sin-

(17)

where a is the width of the plate.

To satisfy the boundary conditions, we search the analytical solution of the problem as follows:

Mi _ M10 cos

nxj

a

«2 _ «20 cos

nxj

a

"n sinnX1 ■ (18)

Substituting (17) and (18) into the total potential energy (13) with W2 = 0 and allowing for relations (3), (8) and (15), one finds

n_n(u(on)in )■

(19)

Invoking the principle of the minimum total potential energy, we arrive at the system of linear algebraic equations

дП

du

(n)in

i0

_ 0

(20)

(

of order 3

. The linear system (20) can be easily solved by using a

V n у

method of Gaussian elimination.

a

n]

X

a

a

The described algorithm was performed with the Symbolic Math Toolbox, which incorporates symbolic computations into the numeric environment of MatLab. The latter gave the possibility to derive the exact solutions of the plane strain problem of elasticity for angle-ply composite plates with a prescribed accuracy.

o o o

Consider a symmetric three-ply plate with a stacking sequence [30 / - 30 /30 ]. The mechanical and geometrical parameters of the simply supported plate are taken to be h = h3 = 0,25; h2 = 0,5; EL = 25ET; GLT = 0,5ET; GTT = 0,2ET; ET = 106;

v LT = v TT = 0,25, where subscripts L and T refer to the fiber and transverse directions of the ply. To compare the results derived with Pagano’s exact solution [13], the following dimensionless variables are introduced:

U3 = 100ETh3u3(a/2, z)/p0a4;

Sn = 10h2an(a/2, z)/p^^a2; S^ = 10h2CTj2(0, z)/P0a2;

Sa3 = 10h^a3(0, z)/P0a; S33 =^33(a/2, z)/P0, z = 03 /h (21)

Results for a three-ply plate

In U3(-0,5) U3(0) U(0,5) Sn(0,5) £12(0,5) £13(0) £23(0,3) £33(0,5)

a/h = 2

3 7,9961 8,6849 10,855 14,499 -6,6104 4,0347 -1,0567 1,0289

5 8,2414 8,9154 11,104 15,180 -6,9178 4,6995 -1,0003 1,0011

7 8,2432 8,9178 11,106 15,183 -6,9189 4,6296 -1,0106 1,0000

9 8,2432 8,9178 11,106 15,183 -6,9189 4,6330 -1,0106 1,0000

11 8,2432 8,9178 11,106 15,183 -6,9189 4,6329 -1,0106 1,0000

a/h = 4

3 3,1923 3,2531 3,2531 8,3971 -4,0226 4,3664 -1,2610 1,0327

5 3,2305 3,2900 3,4138 8,5205 -4,0811 5,0215 -1,2566 1,0003

7 3,2306 3,2901 3,4138 8,5205 -4,0811 5,0019 -1,2578 1,0000

9 3,2306 3,2901 3,4138 8,5205 -4,0811 5,0022 -1,2578 1,0000

11 3,2306 3,2901 3,4138 8,5205 -4,0811 5,0022 -1,2578 1,0000

a/h = 100

3 0,84662 0,84665 0,84662 6,0831 -3,2257 4,3804 -1,6241 1,0393

5 0,84663 0,84666 0,84663 6,0832 -3,2258 4,7758 -1,6199 1,0000

7 0,84663 0,84666 0,84663 6,0832 -3,2258 4,7758 -1,6199 1,0000

16 -8 0 8 Sn 16 -8 -4 0 4 S\2 8

а)

0 2 4 5,з 6 -2 -1 0 523 1

б)

Fig. 2. Distribution of tangential stresses Sn and S12 (a) and transverse stresses S13 and S23 (S) through the thickness of the three-ply plate for II = I2 = I3 = 7:

------- present analysis; O - Pagano

The data listed in Table show that the SaS technique permits the derivation of exact solutions of plane strain elasticity even for very thick plates with a prescribed accuracy using a large number of SaS. Fig. 2 present the distribution of stresses in the thickness direction for different values of the slenderness ratio a/h by choosing seven SaS for each layer. These results demonstrate convincingly the high potential of the proposed layer-wise plate formulation. This is due to the fact that boundary conditions on the bottom and top surfaces and continuity conditions at layer interfaces for transverse shear stresses are satisfied correctly in spite of applying constitutive equations (14). It is also necessary to mention that the proposed SaS method provides the uniform convergence that is impossible with equally spaced SaS.

This work was partially supported by Russian Ministry of Education and Science under Grant No. 1.472.2011.

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Цилиндрический изгиб перекрестно армированных композитных пластин

Г.М. Куликов, С.В. Плотникова

Кафедра «Прикладная математика и механика», ФГБОУ ВПО «ТГТУ»;

[email protected]

Ключевые слова и фразы: метод выборочных поверхностей; слоистая композитная пластина; плоская деформация.

Аннотация: Статья представляет эффективный метод решения плоской задачи теории упругости для слоистых композитных пластин. Метод основан на новой концепции выборочных поверхностей (8а8), разработанной недавно авторами. Согласно этой концепции внутри п-го слоя вводятся 1п произвольным

образом расположенных 8а8 параллельных срединной поверхности пластины и в качестве искомых функций выбираются перемещения этих поверхностей. Это дает возможность получать решения плоской задачи теории упругости для толстых и тонких слоистых композитных пластин с заданной точностью, используя достаточно большое число 8а8, которые размещаются на поверхностях раздела слоев и в узловых точках полинома Чебышева.

Zylindrische Biegung der kreuzbewehrten Kompositplatten

Zusammenfassung: Der Artikel prasentiert die effective Methode der Losung der Flachaufgabe der Elastizitatstheorie fur die geschichteten Kompositplatten. Die Methode basiert auf der von den Autoren vor kurzem auserarbeiteten neuen Konzeption der stichprobenartigen Oberflachen (SaS). Laut dieser Konzeption werden innen der n-Schicht von freiweise angeordneten SaS eingefuhrt und als Suchfunktionen werden die Umstellungen dieser Oberflachen gewahlt. Das gibt die Moglichkeit, die Losungen der Flachaufgabe der Elastizitatstheorie fur die dicken und dunnen geschichteten Kompositplatten mit der angegebenen Genauigkeit zu erhalten, dabei wird es die genug grope Zahl von SaS, die auf den Oberflachen des Schichtteils und in den Knotenpunkten des Polynomes von Chebyshev, angewandt.

Courbure cylindrique des plattes composites armees de la maniere croisee

Resume: L’article presente une methode efficace de la solution du probleme plat de la theorie de l’elasticite pour les plaques composites fueilletees. La methode est fondee sur une nouvelle conception des surfaces selectionnees (SaS), elaboree il n’y a pas longtemps par les auteurs. Suivant cette conception a l’interieur de la n-eme couche sont induits In des SaS situees de la maniere facultative paralleles a la surface moyenne de la plaque, et en qualite des fonctions recherchees sont choisis les deplacements de ces surfaces. Cela donne la possibilite d’obtenir les solutions du probleme plat de la theorie de l’elasticite pour les plaques composites fueilletees fines et grosses avec une precision donnee en utilisant une assez grande quantite de SaS qui sont situees sur les surfaces du domaine des couches et dans les points noueux du polynome de Tchebichev.

Авторы: Куликов Геннадий Михайлович - доктор физико-математических наук, профессор, заведующий кафедрой «Прикладная математика и механика»; Плотникова Светлана Валерьевна - кандидат технических наук, докторант кафедры «Прикладная математика и механика», ФГБОУ ВПО «ТГТУ».

Рецензент: Ярцев Виктор Петрович - доктор технических наук, профессор, заведующий кафедрой «Конструкции зданий и сооружений», ФГБОУ ВПО «ТГТУ».