influence of invariant material parameters on the flexural optimal design of thin anisotropic...

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International Journal of Mechanical Sciences 51 (2009) 192–203

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International Journal of Mechanical Sciences

0020-74

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Influence of invariant material parameters on the flexuraloptimal design of thin anisotropic laminates

Paolo Vannucci �

Universite Versailles St. Quentin, Institut Jean Le Rond d’Alembert – UMR CNRS 7190, Tour 55-65, Case 161 – 4, Place Jussieu, 75252 Paris Cedex 05, France

a r t i c l e i n f o

Article history:

Received 19 September 2008

Received in revised form

7 January 2009

Accepted 16 January 2009Available online 25 January 2009

Keywords:

Flexural properties

Laminates

Composites

Polar method

Optimization

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a b s t r a c t

A general discussion of the optimal solutions to the problems of maximum stiffness, buckling load and

modal frequencies for rectangular laminates is proposed. The study is based upon some invariant

dimensionless material parameters. The conditions giving rise to the same objective function for the

three cases are discussed and a new physical meaning of the objective as a stiffness modulus is

proposed. The geometrical and material conditions giving rise to non-trivial optimal solutions are

also discussed.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The topic of this paper is the influence of some geometrical andinvariant material parameters on the solution of some typicalproblems concerning the optimal design of thin laminates. Thereare various material parameters able to describe the elasticbehaviour of anisotropic layers and it is not easy to determine,among them, those having a direct influence on the optimalsolution and what their influence is. This is essentially due to thefact that, usually, Cartesian components of the elasticity stiffnesstensor or the so-called technical moduli are used in this kind ofinvestigations. Unfortunately, all these quantities are not invar-iants and there are too many of these quantities, at least four,to obtain a simple assessment of their importance: their influenceon the existence and value of an optimal solution can reveal tobe not decisive and in any case their use can hide some otherphenomena.

In some previous researches, from the earliest works of Bertand Chen [1], for buckling and of Bert [2], for vibrations, to thelater papers of Pedersen [3], Muc [4], for biaxial buckling, Avalleand Belingardi [5], for bending stiffness, the influence of somematerial parameters on the optimal design of laminates has beenconsidered; however, these papers do not deal with all thepossible cases of flexural design, or make use of some non-invariant material parameters, such as the so-called Tsai and

ll rights reserved.

Pagano invariants [6], which are intended as material constantsfor the design of a laminate with identical plies, but not asmaterial invariants.

In this paper a new approach, common to all the classical casesof flexural design, is proposed, using invariant dimensionlessmaterial parameters, derived from the so-called polar constants

proposed as early as 1979 by Verchery [7]. The influence ofanisotropy is reduced in this way to only two parameters whosephysical meaning is precisely determined. In this paper, it isshown that the dependence of the final elastic properties of theoptimal laminate upon such parameters is somewhat unexpectedand that the use of these dimensionless parameters reveals someeffects (e.g. a bifurcation of the optimal solution as a functionof a material parameter) due to the material properties thatare unlikely to be evident using other tensor or technical materialparameters.

The reduction of the number of parameters of concern in thetheoretical evaluation of the optimal solution is completed using aunique dimensionless parameter accounting for both the geome-trical and mode aspects. A new interpretation of the objectivefunction, common to the above three cases, is also proposed,which shows its link with the material properties of the layerscomposing the laminate.

In order to have a qualitative assessment of the importance ofthe different geometrical or material design parameters, the basiccase of a simply supported rectangular laminate, uncoupled andorthotropic in bending, with the orthotropy axes parallel to theplate’s side, is addressed. Generalizing a well-known concept, andfor the sake of brevity, such a plate will be briefly indicated as

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b

ax

yh

z

n p

1

...

Fig. 1. General sketch of a rectangular simply supported laminate: in plane and side views.

P. Vannucci / International Journal of Mechanical Sciences 51 (2009) 192–203 193

flexural specially orthotropic laminate (FSOL). For such a plate,simple exact analytical Navier’s type solutions for a given modeare known. The way such a laminate can be obtained is beyondthe scope of this paper. Some special solutions can be found inValot and Vannucci [8].

The general sketch used in this paper for a FSOL of sides a and b

is shown in Fig. 1.

2. Recall of the basic equations

The basic equation of the mechanics of laminated plates is thecelebrated equation linking internal actions and kinematicalquantities:

N

M

� �¼

A B

B D

� � eo

j

� �, (1)

where N ¼ {Nx, Ny, Ns}T and M ¼ {Mx, My, Ms}

T are, respectively, thetensors of in-plane forces and bending moments (here and inthe following, we have used the Voigt’s notation, see for instanceLekhnitskii [9]; the subscript s stands for xy), while thekinematical quantities are represented by the two tensors eo,the strain tensor of the middle plane, and j, the tensor of thecurvatures of this plane.

The three tensors A, B and D describe, in the order, the in-plane, coupled and bending behaviour of the plate; in the case oflaminates composed of np identical plies (this assumption isessential if general solutions are sought for, and it will be made inthe rest of the paper), they are given by the following relations:

A ¼h

np

Xnp

j¼1

Q ðdjÞ; B ¼1

2

h2

n2p

Xnp

j¼1

bjQ ðdjÞ,

D ¼1

12

h3

n3p

Xnp

j¼1

djQ ðdjÞ (2)

with

bj ¼ 2j� np � 1; dj ¼ 12jðj� np � 1Þ þ 4þ 3npðnp þ 2Þ

andPnp

j¼1

dj ¼ n3p :

(3)

The tensor Q(dj) is the reduced stiffness tensor of the jth layer,oriented at the angle dj with respect to the laminate’s x-axis.

In the framework of the classical laminated plates theory(CLPT), the flexural equations for a FSOL can be written as (see forinstance, [10–13])

½L33 þ ðcl � 1ÞLl þ ð1� coÞLo�ðwÞ ¼ clcop, (4)

where the bending equation is obtained for cl ¼ 1, co ¼ 1, thebuckling equation for cl ¼ 0, co ¼ 1 and the natural frequencies

equation for cl ¼ 1, co ¼ 0. The differential operators appearing inEq. (4) are

L33 ¼ Dxx@4

@x4þ 2ðDxy þ 2DssÞ

@4

@x2@y2þ Dyy

@4

@y4,

Ll ¼ Nx@2

@x2þ 2Ns

@2

@x@yþ Ny

@2

@y2,

Lo ¼ m @2

@t2. (5)

In the above equations, w is the deflection of the plate’s middleplane, p the distributed load, m the mass per unit of area of theplate and t the time.

Decomposing w and p into double Fourier series,

pðx; yÞ ¼X1

m;n¼1

pmn sinmpx

asin

npy

b,

wðx; yÞ¼X1

m;n¼1

amn sinmpx

asin

npy

b½coð2cl�1Þ2þð1� coÞsinomnt�,

(6)

a Navier’s type solution can be found, which is exact for a givenmode (m, n). The amplitude pmn will be rewritten as

pmn ¼ cP

abp�mn (7)

with P the resultant of the load, c a numerical coefficientdepending upon the load type and pmn* a dimensionlessparameter depending upon the mode.

The polar method is an alternative method to represent polaranisotropy for plane tensors of any order and type. It has beenused successfully to handle different problems concerning theanalysis and design of laminates; for a deeper insight into themethod and a wider bibliography see [14]. Here, we will recallonly the features of the method of interest in the rest of this paper.

For an elastic tensor T, the Cartesian components can bewritten as

Txx ¼ T0 þ2T1 þR0 cos 4F0 þ4R1 cos 2F1;

Txs ¼ R0 sin 4F0 þ2R1 sin 2F1;

Txy ¼ �T0 þ2T1 �R0 cos 4F0;

Tss ¼ T0 �R0 cos 4F0;

Tys ¼ �R0 sin 4F0 þ2R1 sin 2F1;

Tyy ¼ T0 þ2T1 þR0 cos 4F0 �4R1 cos 2F1:

(8)

The four polar constants T0, T1, R0, R1, as well as the difference ofthe polar angles F0–F1, constitute a complete set of independentinvariants for T. The choice of one of the two polar angles fixes theframe; the usual choice is F1 ¼ 0. Computing the components of Tin a frame rotated of an angle y is easily done, it is sufficient tosubtract the rotation angle y from the two polar angles in Eq. (8).

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P. Vannucci / International Journal of Mechanical Sciences 51 (2009) 192–203194

The inverse of Eq. (8) are, in complex form,

8T0 ¼ Txx �2Txy þ4Tss þTyy;

8T1 ¼ Txx þ2Txy þTyy;

8R0e4iF0 ¼ Txx �2Txy �4Tss þTyy þ4iðTxs � TysÞ;

8R1e2iF1 ¼ Txx �Tyy þ2iðTxs þ TysÞ:

(9)

The previous equations can be used to find the polarcomponents of a given tensor known by its Cartesian components,namely of Q, the reduced stiffness tensor of the basic layercomposing a given laminate. The polar invariants are linked to theelastic symmetries of T:

�
T is orthotropic if and only if F0–F1 ¼ k p/4, kA{0, 1} [16]; � T is R0-orthotropic if and only if R0 ¼ 0 [8,17]; � T is R1-orthotropic if and only if R1 ¼ 0 [8]; � T is isotropic if and only if R0 ¼ R1 ¼ 0 [8].
It is apparent from Eqs. (8) that the parameters R0 and R1

account for the anisotropic part of T, while T0 and T1 account forits isotropic part. R1-orthotropy is a well-known type oforthotropy: it is, in the plane case, the corresponding of the cubicsyngony; sometimes it is called square symmetry, because in thistype of orthotropy there are two couples of orthotropy axes,turned of p/4, and the two axes of a couple are equivalent: eachelastic property is periodic of p/2, not of p, as it is apparent if inEq. (8) one puts R1 ¼ 0.

We introduce here three dimensionless parameters of interest.First of all, we set

r ¼ R0

R1; t ¼ T0 þ 2T1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R20 þ R2

1

q (10)

for reasons given above, we will call r the anisotropy ratio and tthe isotropy-to-anisotropy ratio. Due to the bounds on the polarparameters [14], rA[0,N) and t41. In particular, r ¼ 0 corre-sponds to R0-orthotropic materials, while r ¼N corresponds tosquare-symmetric layers.

The third non-dimensional parameter of importance in the restof the paper is the number k; actually, for the same set of polarinvariants, two kinds of orthotropic materials can exist: thosewith k ¼ 0 and those with k ¼ 1. Vannucci [14], has shown thatmaterials with k ¼ 1 are more rare than those with k ¼ 0 but theycan exist and that the case k ¼ 0 corresponds to what Pedersen[18], has called a low shear modulus orthotropy, while the casek ¼ 1 corresponds to the case of high shear modulus orthotropy.We will see in the following the importance of this parameter onthe optimal design of laminates.

The rules of composition of the tensors A, B, and D (Eq. (2)),apply to the polar components as well (Eq. (9)). In particular, for alaminate composed of identical plies, the polar components of Dare readily found:

TD0 ¼

h3

12T0,

TD1 ¼

h3

12T1,

RD0 e4iFD

0 ¼1

12

h3

n3p

R0e4iF0Xnp

j¼1

dje4idj ,

RD1 e2iFD

1 ¼1

12

h3

n3p

R1e2iF1Xnp

j¼1

dje2idj . (11)

In the above equations, T0, T1, R0, R1, F0, and F1 are the polarcomponents of the basic layer composing the laminate; as amatter of fact, all the layers are orthotropic and, choosing theangle F1 ¼ 0 (this is always possible and corresponds to choosing

the strongest orthotropy axis of the ply as the x1-axis of the layer’smaterial frame), through Eqs. (8) and (11) the Cartesian compo-nents of tensor D can be finally expressed as functions of the polarparameters of the basic layer and of the so-called lamination

parameters xi, introduced by Tsai and Hahn [19] (see also [20,21]):

Dxx

Dyy

Dxy

Dss

Dxs

Dys

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;¼

h3

12

1 2 x0 4x1

1 2 x0 �4x1

�1 2 �x0 0

1 0 �x0 0

0 0 x2 2x3

0 0 �x2 2x3

26666666664

37777777775

T0

T1

ð�1ÞkR0

R1

8>>><>>>:

9>>>=>>>;

. (12)

The four lamination parameters for D are given by

x0 ¼1

n3p

Xnp

j¼1

dj cos 4dj; x1 ¼1

n3p

Xnp

j¼1

dj cos 2dj;

x2 ¼1

n3p

Xnp

j¼1

dj sin 4dj; x3 ¼1

n3p

Xnp

j¼1

dj sin 2dj: (13)

For a FSOL, the only two lamination parameters entering in thecalculation of D are x0 and x1 (Dxs ¼ Dys ¼ 0); in addition (see[13,20,21]), it is

�1px1p1; 2x21 � 1px0p1. (14)

In Eq. (12), the matrix completely accounts for the geometry ofthe laminate, while the column vector for its mechanical proper-ties. Once the material chosen, the optimal design acts only uponthe lamination parameters. A classical technique in optimal designof laminates is to consider x0 and x1 as the design variables and todetermine after, by some supplementary considerations, the truedesign variables, which are the layer orientations dj [13].

If Eq. (12) is used in the expression of the differential operator,Eq. (51) and then in Eq. (4), this last can be rewritten as

h3

12ðT0 þ 2T1ÞDDw

þ ½L33 þ ðcl � 1ÞLl þ ð1� coÞLo�ðwÞ ¼ clcopz, (15)

where the first operator is the classical double Laplacian, affectingthe isotropic part of the laminate: it is an easy task to verify thatT0+2T1 ¼ E/(1�n2), E and n being, respectively, the mean values ofYoung’s modulus and of Poisson’s coefficient of the basic layer.The term

L33 ¼h3

12½ð�1ÞkR0x0 þ 4R1x1�

@4

@x4

(� 6ð�1ÞkR0x0

@4

@x2@y2

þ½ð�1ÞkR0x0 � 4R1x1�@4

@y4

)(16)

is the operator affecting the anisotropic part of the laminate; itscoefficients depend upon the anisotropic part of the stiffnesstensor of the basic layer, i.e. upon R0, R1 and k, and upon thegeometry of the sequence, through the lamination parameters x0

and x1.

3. A flexural optimization problem common to bending,buckling and vibrations

For a FSOL, the bending stiffness can be measured through thecompliance (see [22–24]), i.e. through the functional JD defined as

JD ¼

Z a

0

Z b

0pw dx dy. (17)

The design leading to the highest bending stiffness of thelaminate coincides with the minimization of JD, under the implicit

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condition of a fixed number of layers (otherwise the problemwould be meaningless, as the stiffest laminate would simply bethe one having an infinite number of layers). The equilibriumequation (4) with cl ¼ co ¼ 1, is satisfied if in the expression ofw(x,y) (Eq. (62)), it is

amn ¼1

p4

pmn

Dxxa2 þ 2ðDxy þ 2DssÞabþ Dyyb2

,

with a ¼ m2

a2; b ¼

n2

b2(18)

using this result in Eq. (17), along with Eqs. (7), (10) and (12), oneobtains

JD ¼3c2Zp4

P2a2

h3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

0 þ R21

q X1m;n¼1

p�2mn

m4ð1þ w2Þ2jðx0; x1Þ

, (19)

where

jðx0; x1Þ ¼ tþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ð�1Þkrx0w4 � 6w2 þ 1

ð1þ w2Þ2þ 4x1

1� w2

1þ w2

" #.

(20)

In the above equations, we have introduced some dimensionlessparameters: the mode ratio g, the aspect ratio Z and the wavelength

ratio w, respectively, defined as

g ¼ m

n; Z ¼ a

b; w ¼ Z

g¼

na

mb¼

ffiffiffiba

r. (21)

In this way, all the dimensional part of JD is condensed in thepositive scaling factor multiplying the dimensionless summationin Eq. (19), the true objective function. Whenever the materialcomposing the basic layer is chosen a priori, this is the case we arehere interested in, the design variables are the ply orientations,condensed in the two lamination parameters x0 and x1, while theelastic properties of the basic layer, represented by the two ratiosr and t and by the number k, are the state parameters, along withthe ratio w.

A step further in the simplification of the bending stiffnessoptimization problem can be done if a sinusoidal load is selected,i.e. if a precise couple (m, n) is given (not necessarily m ¼ n ¼ 1).In such a case, the objective function is simply reduced to j(x0,x1)(Eq. (20)), and the corresponding problem of maximizing thebending stiffness for given dimensions and material evidentlybecomes

maximise jðx0; x1Þ,

subjected to � 1px1p1; 2x21 � 1px0p1 (22)

the objective function is then linear, but the problem remains anon-linear one, because one of the constraints imposed on thetwo variables is quadratic.

Let us now consider the problem of maximizing the bucklingload of a given mode (m, n); the solution of the buckling equation,Eq. (4) with cl ¼ 0 and co ¼ 1, in the case of in-plane actions ofthe type N ¼ l (Nx, Ny, 0), l being a load multiplier, is non-trivialfor the mode (m, n) if the corresponding load multiplier lmn is(see for instance [12]),

lmn ¼ p2 Dxxa2 þ 2ðDxy þ 2DssÞabþ Dyyb2

Nxaþ Nyb(23)

we want to maximize lmn, the buckling load multiplier for themode (m, n). Introducing again the polar constants and thelamination parameters, along with the force ratio

n ¼Ny

Nx, (24)

one gets easily

lmn ¼p2m2h3

12a2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

0 þ R21

N2x þ N2

y

vuut ð1þ w2Þ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ n2p

1þ nw2jðx0; x1Þ. (25)

So, for given dimensions, material and applied forces, theproblem of maximizing the buckling load of a mode is reduced tothe same problem (22), as in the case of the bending stiffness.

Finally, let us consider the problem of maximizing the naturalfrequency of a given mode (m, n); the natural frequencies are stillobtained by Eq. (4) with cl ¼ 1 and co ¼ 0. Then, a non-trivialsolution corresponds to the following value of the naturalfrequency, omn:

o2mn ¼

p4

m ½Dxxa2 þ 2ðDxy þ 2DssÞabþ Dyyb2� (26)

through the same steps as the ones followed above, one gets

o2mn ¼

p4m4h3

12ma4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

0 þ R21

qð1þ w2Þ

2jðx0; x1Þ (27)

once more, the maximization of the natural frequency of a modefor given dimensions and material corresponds to problem (22);hence, the optimal flexural design of a FSOL has the commonobjective function j(x0,x1) to be maximized.

Some considerations on j(x0,x1) can be done; first of all, j canbe split into two parts: the first one, t, depends only upon theisotropic part of the basic material and of course, once this lastchosen, it cannot be modified by the optimization process. Thesecond part, depending upon the anisotropic part of the basicmaterial, i.e. upon r, k, and upon the wavelength ratio w, can beoptimized by a proper choice of the two lamination parameters, x0

and x1, which in turn affect the anisotropy of the final laminate.So, by the use of the polar parameters, the isotropic andanisotropic parts are separated, both for the contribution of thebasic layer and for the final laminate.

A question arises about the choice of the two ratios t and r; infact, several combinations of the polar parameters can be used toobtain dimensionless quantities. Actually, the choice of r isjustified by the fact that this parameter seems to have a certainimportance in the qualitative description of plane anisotropy. Inparticular Verchery [25], has shown that for rp1 the componentTxx of an orthotropic elasticity tensor, Eq. (81), has its absolutemaximum and minimum on the material axes and there are noother stationary points for Txx, while for r41 the absoluteminimum, if k ¼ 0, or maximum, if k ¼ 1, are along a skewdirection. In other words, a value of r greater than one determinesthe existence of an absolute off-axis minimum or maximum forTxx. This fact determines a change in different plane anisotropicproblems, which can be regarded as bifurcations of the solution(see [14]). We will see in the rest of the paper that this is the casealso for the three flexural problems studied in this paper.Concerning t, the physical meaning of T0+2T1 has already beenintroduced; being R0 and R1 representative of the anisotropic part,t represents the ratio between the modulus entering the classicalequations of isotropic plates and a mean anisotropic modulus. Inaddition, t is defined also for the two extreme cases of anisotropy:square symmetry, corresponding to R1 ¼ 0, and R0-orthotropy,corresponding to R0 ¼ 0.

The Cartesian expressions for r and t can be computedsubstituting Eq. (9) into Eq. (10); for an orthotropic layer, one gets

r ¼ Txx þ Tyy � 2ðTxy þ 2TssÞ

Txx � Tyy,

t ¼3ðTxx þ TyyÞ þ 2ðTxy þ 2TssÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4ðTxy þ 2TssÞðTxy þ 2Tss � Txx � TyyÞ þ 2ðT2xx þ T2

yyÞ

q . (28)

ARTICLE IN PRESS


The Cartesian expression of k can still be obtained throughEqs. (9) and (10) remembering (Section 2), that for orthotropiclayers F0–F1 ¼ k p/4:

k ¼4

p1

4arctan

4ðTxs � TysÞ

Txx þ Tyy � 2ðTxy þ 2TssÞ

� �

�1

2arctan

2ðTxs þ TysÞ

Txx � Tyy

�. (29)

So, if we put, as usual, F1 ¼ 0, it is

k ¼ 0 if Txx þ Tyy42ðTxy þ 2TssÞ,

k ¼ 1 if Txx þ Tyyo2ðTxy þ 2TssÞ. (30)

Another remark concerning the objective function j(x0,x1) isthat also the problem opposite to that stated in Eq. (22) can beconsidered:

minimise jðx0; x1Þ,

subjected to � 1px1p1; 2x21 � 1px0p1 (31)

of course, the solutions to this last problem give, in the commonsense, the worst laminate, i.e. the least stiff laminate for a givensinusoidal load, and the laminate having the lowest buckling loadand natural frequency corresponding to a given mode. Despite thefact that problem (31), which for the sake of brevity we will callanti-optimal, is likely to have very few applications, it cannonetheless be considered.

4

3.5

3

2.5

2

1.5

10 2 4 6 8 10 12 14

�

qmax (k = 1), - qmin (k = 0)

qmax (k = 0), - qmin (k = 1)

aa

a a

Fig. 2. Diagrams of the function qamaxðrÞ and �qa

minðrÞ.

4. Polar discussion of Qxx(h)

For physical reasons, compliance, buckling load multipliers andnatural frequencies are positive quantities and being positivefactors in front of j(x0,x1) in Eqs. (19), (25) and (27), also j(x0,x1)is positive definite. This can be easily shown by a comparison withthe value of Qxx(y), the normal stiffness component for the basicmaterial in a given direction y. Let us then discuss Qxx (y); this willgive a new physical meaning for the quantity j(x0,x1). The generalpolar expression of Qxx(y) can be obtained from the first of Eqs. (8),

QxxðyÞ ¼ T0 þ 2T1 þ ð�1ÞkR0x0 þ 4R1x1, (32)

where now

x0 ¼ cos 4y ¼ 2x21 � 1; x1 ¼ cos 2y. (33)

The same symbols of the lamination parameters have been usedin Eqs. (32) and (33); the reason for this choice is apparent,though in this case the above functions represent circularfunctions of the rotation angle y, not a combination of layerorientations.

We introduce the dimensionless quantity

qxxðyÞ ¼QxxðyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2

0 þ R21

q ¼ tþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ½ð�1Þkrx0 þ 4x1�, (34)

whose resemblance with the objective function j(x0,x1) (Eq. (20)),is apparent. In order to compare the objective function with qxx(y),let us discuss here the extreme values qmax and qmin of qxx(y);using Eqs. (33) and (34), it is easily recognized that these can be(recall that x1 ¼1 corresponds to 01 while x1 ¼ �1–901)

q11 ¼ qxxðx1 ¼ 1Þ ¼ tþ ð�1Þkrþ 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ,

q22 ¼ qxxðx1 ¼ �1Þ ¼ tþ ð�1Þkr� 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ,

qyy ¼ qxxðx1 ¼ �ð�1Þk=rÞ

¼ t� ð�1Þkr2 þ 2

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ; only for r41. (35)

A simple inspection of Eqs. (35) shows that the inequalityq11Xq22 is always satisfied and that qyy is a minimum if k ¼ 0,a maximum if k ¼ 1. So, comparing these three values, one cansee that

�
for k ¼ 0, qmax ¼ q11 8r and qmin ¼ q22 if rp1, while qmin ¼ qyyif r41; � for k ¼ 1, qmin ¼ q22 8r and qmax ¼ q11 if rp1 while qmax ¼ qyy
if r41.

The role played by k is even clearer if we introduce the quantity

qaxx ¼ qxx � t (36)

which represents the only anisotropic part of qxx; looking atEqs. (35) we see that

qaminðk ¼ 1Þ ¼ �qa

maxðk ¼ 0Þ,

qamaxðk ¼ 1Þ ¼ �qa

minðk ¼ 0Þ. (37)

Hence, k changes maxima to minima, and vice versa, changingthe sign of qa

xx. In addition, the values of the minimum andmaximum of qa

xx are the same for both k ¼ 0 and 1 for the lowestand highest values of r:

qamin ¼ �4; qa

max ¼ 4 for r ¼ 0,

qamin ¼ �1; qa

max ¼ 1 for r ¼ 1. (38)

The diagrams of qamaxðrÞ and �qa

minðrÞ are shown in Fig. 2.Let us now consider the sign of qxx(y); of course, it must be

positive, qxx(y) being a dimensionless normal stiffness. None-theless, we can prove this using the thermodynamic bounds onpolar constants, which are, for orthotropic materials [14]

T04R0,

T1½T0 þ ð�1ÞkR0�42R21 (39)

To prove that qxx(y)408y, it is sufficient to show that qmin40,which corresponds to prove that q22 and qyy are positive. To showthis, let us write Q22 ¼ Qxx(x0 ¼ 1,x1 ¼ �1)40 (Eq. (32)):

Q22 ¼ T0 þ 2T1 þ ð�1ÞkR0 � 4R140 (40)

from Eqs. (40) and (392), one obtains, thanks to Eq. (391), thefollowing sufficient condition for Q2240:

½T0 þ 2T1 þ ð�1ÞkR0�248T1½T0 þ ð�1ÞkR0�

3½T0 � 2T1 þ ð�1ÞkR0�240 (41)

ARTICLE IN PRESS


the last condition being identically satisfied, Q2240 and hence

q22 ¼Q22ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R20 þ R2

1

q ¼ tþ ð�1Þkr� 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p 40

8ðt;r; k ¼ 1Þ and 8ðt;rp1; k ¼ 0Þ. (42)

For k ¼ 0 and r41, we must show qyy40; this is easily proved ifwe consider that qyy(r ¼ 1) ¼ q22(r ¼ 1) and that @qyy=@r40, thismeaning that qyymin ¼ q22(r ¼ 1)40. So, qxx(y)40 8(y, t, r, k).Eq. (42) along with the corresponding one concerning qyy give thebounds on the value of t

tmin ¼4� ð�1Þkrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r2p 8ðr; k ¼ 1Þ and 8ðrp1; k ¼ 0Þ,

tmin ¼2þ r2


p 8ðr41; k ¼ 0Þ. (43)

In particular, for k ¼ 0 maxðtminÞ ¼ tminðr ¼ 1=4Þ ¼ffiffiffiffiffiffi17p

; whilefor k ¼ 1 maxðtminÞ ¼ tminðr ¼ 0Þ ¼ 4.

5. Analysis of the objective function u(n0,n1)

Let us discuss in this section the objective function j(x0,x1);our goal is to find the optimal solutions and their dependence,along with that of their value and uniqueness, upon thedimensionless polar parameters t, r and k of the basic layer, aswell as the influence of the wavelength ratio w.

First of all, let us rewrite j(x0,x1), as

jðx0; x1Þ ¼ tþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ½ð�1Þkrc0ðwÞx0 þ 4c1ðwÞx1�,

with c0ðwÞ ¼w4 � 6w2 þ 1

ð1þ w2Þ2

; c1ðwÞ ¼1� w2

1þ w2(44)

and ponder the existence of singular solutions. The two functionsc0(w) and c1(w) account for the influence of w, i.e. of the geometryand of the mode; they are defined 8wX0 and are bounded in theinterval [�1,1]; their diagrams are shown in Fig. 3; it is of someimportance to note that c0(w) ¼ 0 for w ¼

ffiffiffi2p� 1, while c1(w) ¼ 0

for w ¼ 1.As j(x0,x1) is linear with respect to x0 and x1, it cannot have

stationary points and its maxima and minima, when they exist,

1

0.5

0

-0.5

-1

0 1 2 3 4 5 6�

c0 (�)

c1 (�)

12-1 2+1

Fig. 3. Functions c0(w) and c1(w).

are located on the boundary of the feasible domain, defined by theinequalities in Eqs. (33) and (43). Nonetheless, before examiningthe value of j(x0,x1) on the boundary, it is useful to check itsgradient,

rjðx0;x1Þ ¼@j@x0

;@j@x1

� �¼

ð�1Þkrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p c0ðwÞ;4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r2p c1ðwÞ

!,

(45)

in order to find eventual singular situations. Actually, it is easy tosee that the gradient is null in two cases:

�

Figorth

r ¼ 0 and w ¼ 1: this is the case of laminates made ofR0-orthotropic materials and with equal wavelength of themode along x and y, which can be obtained, for instance, forsquare plates and modes with m ¼ n;ffiffiffip
� r ¼N and w ¼ 2� 1: this is the case of laminates made of
square-symmetric materials (R1 ¼ 0), i.e. reinforced by ba-lanced fabrics, and, if for instance m ¼ n, having an aspect ratioZ ¼

ffiffiffi2p� 1.

In these two circumstances, it is not possible to optimize thelaminate, because the objective function is constant and reducesto only its isotropic part, t. Actually, in such cases, thecontribution of the anisotropic part disappears, due to specialcombinations of geometry, mode and anisotropy properties of thelayer: the laminate behaves like it was made of isotropic layersand any possible stacking sequence give the same result.

We continue now to consider the points on the boundary ofthe feasible domain where the objective function, beinglinear, takes its maximum and minimum. Such a domain isdefined by the conditions (14) and is a sector of parabola, shownin Fig. 4.

It is well known (see for instance [13]), that the FSOLrepresented by a lamination point on the curved perimeter ofthe feasible domain are angle-ply laminates, i.e., laminatescomposed by an even number of plies and having for each plyat the orientation d a ply at the orientation �d. The segmentbetween the two points A and C in Fig. 4 represents cross-ply

laminates, that is laminates having layers at 01 or 901, in relativequantities depending upon the location of the lamination point onthe segment. So, optimal and anti-optimal, especially orthotropiclaminates can be only of two types: angle-ply or cross-ply.

Let us first consider cross-ply optimal solutions, determined byx0 ¼ 1, �1px1p1: considering the components of rj (Eq. (45)),they can exist if and only if

rjðx0;x1Þ ¼ ða2;0Þ; a 2 R� f0g (46)

1-1 0

1

-1

AC

B

Arc of the angle-plylaminates

Segment of the cross-ply laminates

Feasible domain

�0

�1

. 4. The feasible domain in the space of the lamination parameters for a special

otropic laminate.

ARTICLE IN PRESS

3

31

0

1

2

3

4

5

6

0 1 2 3 4 5 6

2 − 1

2 + 1

�

�

�4 (�)

�3 (�)

�2 (�)

�1 (�)

Fig. 5. Domain of the plane (r,w) where jdd exists (shaded).


these conditions are satisfied for

ð�1Þkc0ðwÞ403k ¼ 0 and w 2 ½0;

ffiffiffi2p� 1Þ or w4

ffiffiffi2pþ 1;

k ¼ 1 andffiffiffi2p� 1owo

ffiffiffi2pþ 1

8<:

c1ðwÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ¼ 03w ¼ 1 or r ¼ 1. (47)

For the anti-optimal problem, it is sufficient in the first ofEq. (47) to change k ¼ 0 into 1 and vice versa, i.e., as already for qxx,k changes maxima into minima and vice versa.

We can notice that cross-ply solutions exist only in thepresence of a generalized square symmetry: of the material,condition r ¼N, or of the geometry and mode, condition w ¼ 1(for instance, m ¼ n and a square plate). The values of thesolutions are

for w ¼ 1; j ¼ t� ð�1Þkrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ,

for r ¼ 1; j ¼ tþ ð�1Þkc0ðwÞ, (48)

so that in the first case r influences the extreme values, while w inthe second:

jmax12 ¼ max t� ð�1Þkrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r2p

" #k¼1;r¼1

¼ max½tþ ð�1Þkc0ðwÞ�ðk¼0;w¼f0;1gÞor ðk¼1;w¼1Þ

¼ tþ 1,

jmin12 ¼ min t� ð�1Þkrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r2p

" #k¼0;r¼1

¼ min½tþ ð�1Þkc0ðwÞ�ðk¼1;w¼f0;1gÞor ðk¼0;w¼1Þ

¼ t� 1. (49)

We note that, when m ¼ n, w ¼ 1 corresponds to a square plate,while w ¼ 0 or N to an infinite strip, respectively, along the axis y

or x.It is also interesting to notice that the maximum value of the

minimum and the minimum of the maximum are identical andequal to t and that they are obtained for the same conditions asthose discussed above: r ¼ 0 and w ¼ 1 or r ¼N and w ¼

ffiffiffi2p� 1,

i.e. when the laminate behaves as if it was composed of isotropiclayers.

A last remark concerns the uniqueness of the optimal and anti-optimal cross-ply solutions: as x1 disappears from the differentexpressions above, solutions are not unique and any laminatemade by a combination of layers at 01 and at 901 and respectingthe prescription on uncoupling (i.e., with B ¼ O, such assymmetric stacking sequences or quasi-trivial uncoupled lami-nates (see [15])) is an optimal, or anti-optimal, solution whenconditions (47) are satisfied.

Let us now consider the general case of angle-ply laminates; inthis case, the two lamination parameters are linked by the relation

x0 ¼ 2x21 � 1 (50)

and the objective function becomes

jðx1Þ ¼ tþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ½ð�1Þkrc0ðwÞð2x21 � 1Þ þ 4c1ðwÞx1�, (51)

whose maxima and minima can be only

j11 ¼ jðx1 ¼ 1Þ ¼ tþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ½ð�1Þkrc0ðwÞ þ 4c1ðwÞ�,

j22 ¼ jðx1 ¼ �1Þ ¼ tþ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ½ð�1Þkrc0ðwÞ � 4c1ðwÞ�,

jdd ¼ jðx1 ¼ xst1 Þ ¼ t� ð�1Þkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r2p r2c2

0ðwÞ þ 2c21ðwÞ

rc0ðwÞ(52)

with

@j@x1

xst

1

¼ 0) xst1 ¼ �

ð�1Þk

rc1ðwÞc0ðwÞ

(53)

which are results already found in [2–4]. The laminate being anangle-ply, the corresponding orientation angles d are easilycomputed by Eqs. (132) and (33):

x1 ¼ 1 correponds to d ¼ 0,

x1 ¼ � 1 correponds to d ¼p2

,

x1 ¼ xst1 correponds to d ¼ 1

2arccosxst1 . (54)

Here, only the third case corresponds to a true angle-plysolution, while the first two determine unidirectional laminates,the first with all the layers oriented at 01 and the second at 901.In addition, it is easy to verify (Eqs. (52)–(54)), that two plateswith reciprocal wavelength ratios have complementary solutionangles d.

The angle-ply solution cannot exist always, because of bounds�1pxst

1 p1, which give the following conditions, linking theinfluence of the material part to that of the mode on the existenceof such solutions:

�1pð�1Þk

rc1ðwÞc0ðwÞ

p1. (55)

It is not difficult to verify that conditions (55) are satisfied onlyfor

r41 and 0owpw1ðrÞ,rX0 and w2ðrÞowpw3ðrÞ,r41 and wXw4ðrÞ (56)

functions wi(r), i ¼ 1,y,4, are the solutions of the equationsxst

1 ¼ �1, with k ¼ 0 or 1:

w1ðrÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3r�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8r2 þ 1

prþ 1

s; w2ðrÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3r�


pr� 1

s,

w3ðrÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3rþ


prþ 1

s; w4ðrÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3rþ


pr� 1

s. (57)

ARTICLE IN PRESS

0

1

2

3

4

5

6

0 1 2 3 4 5 6 0

1

2

3

4

5

6

0 1 2 3 4 5 6

cross - ply

45°<δ≤90°

2+12+1

2−12−1

k = 0: optimal sol. k = 0: anti-optimal sol.

0

1

2

3

4

5

6

0 1 2 3 4 5 6 0

1

2

3

4

5

6

0 1 2 3 4 5 6

cross-ply

2+12+1

2−12−1

k = 1: anti-optimal sol. k = 1: optimal sol.

45°<�≤90°

0°<�<45°

ϕ 11

ϕ δδ

��

��

� �

� �

�11 � = 0°0°<�≤45°

45°<δ≤90°

� = 90°

�22

�22

� = 90°��<δ≤45°<δ≤90°

� = 0°

�11

��

0°<�<45°

45°<δ≤90°��

45°<�<90°

�22

� = 90°

� = 0°�� 0°<δ≤45° � = 90° �22

��

�11

� = 0°

Fig. 6. Map of the optimal and anti-optimal solutions.


Fig. 5 shows these functions and the regions detailed byconditions (56), shaded in the figure. In these regions, conditions(55) are satisfied and by consequence the orientation angle dcorresponding to xst

1 is between 0 and p/2; outside these regions,the layers must be placed at 0 or at p/2, i.e. the optimal solution isan unidirectional one and, in a sense, optimization (or anti-optimization) loses its meaning, as the search for a preciseoptimal orientation angle d is no more needed. We note thatcross-ply solutions lay on the horizontal line w ¼ 1 or at r ¼N.

Computing the second derivative of j(x1) we get

@2j@x2

1

¼ 4ð�1Þkrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ r2p c0ðwÞ (58)

Therefore, once more k changes a maximum into a minimum,i.e. an optimal into an anti-optimal solution. Namely, through theexpression of c0(w), we can easily see that

�
if ðr41; 0owpw1ðrÞÞ or ðr41; wXw4ðrÞÞ, thenjdd ¼ jmax for k ¼ 1, jdd ¼ jmin for k ¼ 0; � if ðrX0; w2ðrÞowpw3ðrÞÞ, then jdd ¼ jmax for k ¼ 0,jdd ¼ jmin for k ¼ 1.
Just as for qxx(y), we consider now the hierarchy of the valuesj11, j22 and jdd; using Eqs. (52), we can easily see that:

j114j223c1ðwÞ413w41,

jdd4j11

j22

3ð�1Þk�1

½ð�1Þkrc0ðwÞ � c1ðwÞ�240;

c0ðwÞ40;

8<:

3

ffiffiffi2p� 1owo

ffiffiffi2pþ 1 if k ¼ 0;

0owoffiffiffi2p� 1 or w4

ffiffiffi2pþ 1 if k ¼ 1:

8<: (59)

Comparing these last results with conditions (56), defining theregions where jdd exists, one can trace a map in the plane (r, w) ofpoints corresponding to different types of optimal or anti-optimalsolutions (Fig. 6), where the range of the value of the angle d arealso indicated for solutions of the type jdd, value that can becomputed by Eqs. (53) and (543); the layer orientations of thedifferent solutions are represented by simple schemes.

We note that for k ¼ 1 and r41, the optimal solutionscorresponding to w ¼ 0 or N, i.e. to an infinite strip, if m ¼ n,are not given by a unidirectional solution with all the layersoriented transversely to the infinite side, as one could expect, butby an angle-ply solution. For k ¼ 0, this happens for the anti-optimal solutions.

6. Effectiveness of the optimal and anti-optimal angle-plysolutions

We have seen above that angle-ply solutions are not theoptimal solution 8(r, w); nevertheless, in some sense, they are thetrue optimal solution, i.e., the solution to be found by anoptimization procedure; in all the other cases, the optimalsolution is a unidirectional one, with the direction of the fibresparallel to the shortest side, which in a sense is the intuitive

optimal solution, or in some very special cases, a cross-ply solution.Hence, it is interesting to evaluate the gain of the optimal solutionwith respect to the intuitive one, i.e. the ratio

zmaxðt;r;wÞ ¼jddjii

; i ¼ 1 if wo1; i ¼ 2 if w41 (60)

this ratio is shown in Fig. 7 for the case t ¼ 5.Using Eqs. (52) it can be easily seen that this ratio, as apparent in

Fig. 7, gets its maximum value for r-N and w ¼ 0, w ¼ 1 or w-N,

ARTICLE IN PRESS

1.5

1.4

1.3

1.21.1

10 1 2 3 4 5 6

6

543

21

0

�

�

0 1 2 3 4 5 66

54

321

0

�

�

� max

� min

0.9

0.8

0.7

Fig. 7. Ratios zmax and zmin for t ¼ 5.


i.e., if m ¼ n, for square plates or infinite strips composed by square-symmetric layers (R1 ¼ 0):

maxðzmaxÞ ¼tþ 1

t� 1. (61)

In the same way, we can consider the ratio between the anti-optimal solution and the intuitive anti-optimal solution, i.e. the oneobtained by aligning the fibres orthogonally to the shortest side:

zminðt;r;wÞ ¼jddjii

; i ¼ 2 if wo1; i ¼ 1 if w41 (62)

this ratio too is shown in Fig. 7 for t ¼ 5.Once again using Eqs. (52) it can be easily seen that zmin, as

apparent in Fig. 7, gets its minimum value for r-N and w ¼ 0,w ¼ 1 or w-N, i.e., for the same previous conditions determiningmax(zmax):

minðzminÞ ¼t� 1

tþ 1, (63)

so that

minðzminÞmaxðzmaxÞ ¼ 1. (64)

Actually the optimal and anti-optimal problems are reciprocal.Finally, it is the ratio t that determines how much effective the

optimal solution can be and how much worst the anti-optimalsolution can be; in particular, max(zmax) decreases with t and, asthe minimum theoretical value of t for r-N is 1,(Eqs. (43)),max(zmax) is not upper bounded, at least theoretically. Actually, tis always greater than 1, because T1 is strictly positive (see [14]).This renders also min(zmin) strictly greater than zero, as it must be.In addition,

maxðzminÞ ¼minðzmaxÞ ¼ 1, (65)

as it can be easily verified.A final remark about this point: the ratio q11/qyy, or q22/qyy

(Eqs. (35)), for the case r ¼N corresponds to max(zmax) if k ¼ 0and to min(zmin) if k ¼ 1; in other words, the limits max(zmax) andmin(zmin) are, respectively, equal to the ratio between the normalstiffness at 01 and 451 or at 451 and 01 of a square-symmetricmaterial having the same values of T0, T1 and R1 of the givenmaterial.

7. Some considerations on real materials

Apart from square-symmetric layers, which have r ¼N, themost part of composite layers have ro1 and k ¼ 0; the data ofsome materials are shown in Table 1. This means that, in practice,only the left part of the diagram in Fig. 5 is usually of concern. It isinteresting to note that for materials having ro1, the range of w

where optimization of the orientation angle is meaningful, i.e.where the solution is not 01 or 901, increases with r and, for r ¼ 1,it lies between w ¼ 1=

ffiffiffi3p

andffiffiffi3p

. As the most part of materialshave a value of r close to 1, optimization of the orientation anglefor m ¼ n is meaningful, roughly speaking, only for plates wherethe longest to shortest side ratio is less than

ffiffiffi3p

, whilst if it isgreater, the highest bending stiffness is obtained when the layerfibres are parallel to the shortest side. The above value of

ffiffiffi3p

confirms and improve the value already found in [3].For materials with r ¼ 1 and k ¼ 0, we get from Eq. (43)

tmin ¼ 3=ffiffiffi2p

, value that inserted in Eq. (60) gives zmax ¼ 2; inother words, for current materials (rE1, k ¼ 0), the value of theobjective function for the true optimal solution is at most twicethe one corresponding to the intuitive solution (this happens foran angle-ply laminate with d ¼ p/4, i.e. for a cross-ply rotated of451 with respect to the axes of the plate).

Materials with r41 are not so common, but they do exist;two examples are those of unidirectional fibre reinforced layersS-glass-epoxy S2-449/SP 381 (see [26]), and boron-epoxy B(4)-55054 (see [19]), reported in Table 1. A more exotic, but natural,example is that of a sheet of ice, of course not interesting toconstruct structural laminates, but showing that real and naturalmaterials having these characteristics do exist. The ice whoseproperties are shown in Table 1 is that of the Mendenhall glacierat a temperature of 270 K (see [27]). This kind of ice has ahexagonal syngony in three dimensions and the data reportedhere concern a sheet of ice having the x1-axis in the direction ofthe hexagonal symmetry axis.

Also materials with k ¼ 1 are rare; nonetheless, three examplesare shown in Table 1. The first one is a layer of titanium-boride,TiB2, a hard ceramic compound (see again [27]). Also in this case,the syngony of the material in three dimensions is hexagonal andthe same procedure used for the case of the ice has been applied.The two remaining examples, this time of composite materials,are those of carbon-epoxy braided layers, studied by Falzon andHerszberg [28], and respectively, named by the authors BR45a andBR60. The first one has the 26.1% of fibres at 01 and the rest inequal quantities at 7451, while the second one has the 20% offibres at 01 and the rest in equal quantities at 7601; in this lastcase, x2 is the strong axis, see the values of E1 and E2 in Table 1; forthis reason, F1 ¼ 901 and strictly speaking, k ¼ �1; nevertheless,the sign of k is inessential (see also Eqs. (8)).

8. Sensitivity of umax and umin to material and geometricparameters

We consider in this section the influence of material para-meters, r and t, and of the geometrical parameter w, on the value

ARTICLE IN PRESS

Table 1Technical constants, Cartesian components of the reduced stiffness tensor, polar parameters, anisotropy ratio, isotropy-to-anisotropy ratio and volume fraction, Vf, of

various materials (moduli in GPa).

Material Fir wood Ice Titanium-

boride, TiB2

Boron-epoxy

B(4)-55054

Carbon-epoxy

T300-5208

Kevlar-

epoxy 149

S-Glass-epoxy

S2-449/SP 381

Glass-epoxy

balanced fabric

Braided carbon-

epoxy BR45a

Braided carbon-

epoxy BR60

Reference Lekhnitskii

[9]

Cazzani and

Rovati [27]

Cazzani and

Rovati [27]

Tsai and Hahn

[19]

Tsai and Hahn

[19]

Daniel and

Ishai [29]

MIL-HDBK-17-2F

[26]

Daniel and Ishai

[29]

Falzon and

Herszberg [28]

Falzon and

Herszberg [28]

E1 10 11.75 387.60 205.00 181.00 86.90 47.66 29.70 40.40 30.90

E2 0.42 9.61 253.81 18.50 10.30 5.52 13.31 29.70 19.60 42.60

G12 0.75 3.00 250.00 5.59 7.17 2.14 4.75 5.30 25.00 14.00

n12 0.01 0.27 0.44 0.23 0.28 0.34 0.27 0.17 0.75 0.34

Q11 10 12.51 445.62 206.00 181.81 87.54 48.65 30.58 55.56 36.76

Q22 0.42 10.22 291.80 18.59 10.35 5.56 13.59 30.58 26.96 50.68

Q66 0.75 3.00 250.00 5.59 7.17 2.14 4.75 5.30 25 14.00

Q12 0.004 2.78 130.12 4.27 2.89 1.89 3.67 5.20 20.22 17.23

T0 1.68 3.65 184.65 29.80 26.88 12.23 92.38 8.99 17.76 13.62

T1 1.30 3.54 124.71 29.14 24.74 12.11 86.97 8.94 15.37 15.24

R0 0.93 0.65 65.35 24.21 19.71 10.09 44.86 3.70 7.24 0.38

R1 1.19 0.28 19.23 23.42 21.43 10.25 43.82 0 3.57 1.74

F0 0 0 p/4 0 0 0 0 0 p/4 p/4

F1 0 0 0 0 0 0 0 0 0 p/2

k 0 0 1 0 0 0 0 0 1 (�)1

r 0.78 2.32 3.40 1.03 0.92 0.98 1.02 N 2.03 0.22

t 2.83 15.16 6.37 2.61 2.62 2.53 4.25 7.26 6.01 24.76

Vf – – – 0.50 0.70 0.60 0.50 0.45 0.60 0.60

6

5

4

3

2

1

00 1 2 3 4 5 6

�

�

�4

�6

�2�5

�4 �1

�1

�2

�3

�3

2+1

2+ 3

2- 3

3

32

31/

2-1

S

st

st

st

st

st

st

Fig. 8. Curves w1–w4 and wst1 � wst

6 .


of jmax and jmin. In particular, we consider the circumstancesproducing the highest possible value of an optimal solution and,in the same way, when an anti-optimal solution is the lowest one.In other words, we look for the maximum possible value of a givenjmax and the possible minimum value of a given jmin; both jmax

and jmin can be only j11, j22 or jdd (Eqs. (52)), so we look for theconditions leading to jmax

11 , to jmax22 and to jmax

dd , as well as to jmin11 ,

jmin22 and to jmin

dd .In particular, we will consider only r and w, i.e. the anisotropy

ratio and the wavelength ratio; in fact, j11, j22 and jdd dependlinearly upon the isotropy-to-anisotropy ratio t, so we seeimmediately that their maximum value increases or decreaseswith t. As t is lower bounded by Eqs. (43), a given jmin is lowerbounded, but, unlike this case, jmax is not upper bounded,because of course t has not upper bounds: the value of theobjective function increases linearly, for given anisotropy moduliR0 and R1, with the isotropy moduli T0 and T1, i.e., bendingstiffness, buckling loads and natural frequencies increases linearlywith the isotropic part of the ply material.

For these reasons, we consider only the influence of r and wupon jmax and jmin, i.e., we look for the curves w ¼ w(r) in theplane (r, w) where the surfaces j11, j22 and jdd have a local orabsolute maximum (minimum) with respect to w; equations

@jii

@w ¼ 0; i ¼ 1;2; d (66)

give the curves where jmax or jmin are stationary with respect tow. Using Eqs. (52), it is not difficult to find that, remembering themap of Fig. 6, the only curves carrying stationary points of jii,i ¼ 1, 2, d, with respect to w are:

w ¼ 0; w ¼ 1; w!1; wst1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffir� 1

rþ 1

s; wst

2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffirþ 1

r� 1

s,

wst3 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3rþ 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8rðrþ 1Þ

pr� 1

s; wst

4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3r� 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8rðr� 1Þ

prþ 1

s,

wst5 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3r� 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8rðr� 1Þ

prþ 1

s; wst

6 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3rþ 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8rðrþ 1Þ

pr� 1

s. (67)

The curves in Eqs. (67) are shown in Fig. 8, along with curves w1

to w4, given by Eqs. (57). The value of the second derivatives alongeach one of these curves determines if it is a maximum orminimum curve; going, for the sake of brevity, directly to theresults, we have (to abridge, in the following formulae, jmax

ii andjmin

ii indicate, respectively, the optimal and anti-optimal value ofj, obtained in correspondence of a given point of the plane (r, w)by jii, i ¼ 1, 2, d:

for w ¼ 0 :

maxjmax11 ¼ q11 8ðr; k ¼ 0Þ and 8ðro1; k ¼ 1Þ;

minjmin22 ¼ q22 8ðr; k ¼ 1Þ and 8ðro1; k ¼ 0Þ;

maxjmaxdd ¼ qyy 8ðr41; k ¼ 1Þ;

minjmindd ¼ qyy 8ðr41; k ¼ 0Þ;

8>>>><>>>>:

ARTICLE IN PRESS

543210

12

34

56 6

54

32

10

�

�

�a

5

4

3

2

1

01 2 3 4 5 6 6

5432

10

�

�

�a

01

23

45

6 01

23

45

0-1-2-3-4�

� �a

012

3

456 0 1 2 3 4 5

0-1-2-3-4

�

�

�a

Fig. 9. Diagrams of the anisotropic part ja(r, t) of the objective function. (a) k ¼ 0, optimal solution, (b) k ¼ 0, anti-optimal solution, (c) k ¼ 1, optimal solution and (d)

k ¼ 1, anti-optimal solution.


for w ¼ 1 :

maxjmin11 ¼ maxjmin

22 ¼ qs1 8ðr; k ¼ 0Þ;

minjmax11 ¼ minjmax

22 ¼ qs2 8ðr; k ¼ 1Þ;

minjmaxdd ¼ qs2 8ðro1; k ¼ 0Þ;

maxjmaxdd ¼ qs2 8ðr41; k ¼ 0Þ;

maxjmindd ¼ qs1 8ðro1; k ¼ 1Þ;

minjmindd ¼ qs1 8ðr41; k ¼ 1Þ;

8>>>>>>>>>><>>>>>>>>>>:

for w!1 :

maxjmax22 ¼ q11 8ðr; k ¼ 0Þ and 8ðro1; k ¼ 1Þ;

minjmin11 ¼ q22 8ðr; k ¼ 1Þ and 8ðro1; k ¼ 0Þ;

maxjmaxdd ¼ qyy 8ðr41; k ¼ 1Þ;

minjmindd ¼ qyy 8ðr41; k ¼ 0Þ;

8>>>><>>>>:

for w ¼ wst1 :

maxjmax11 ¼ qyy8ðr41; k ¼ 1Þ;

minjmin22 ¼ qyy8ðr41; k ¼ 0Þ;

(

for w ¼ wst2 :

maxjmax22 ¼ qyy8ðr41; k ¼ 1Þ;

minjmin11 ¼ qyy8ðr41; k ¼ 0Þ;

(

for w ¼ wst3 and w ¼ wst

6 :maxjmin

dd ¼ qd18ðr41; k ¼ 0Þ;

minjmaxdd ¼ qd28ðr41; k ¼ 1Þ;

(

for w ¼ wst4 and w ¼ wst

5 :maxjmin

dd ¼ qd3 8ðr41; k ¼ 1Þ;

minjmaxdd ¼ qd4 8ðr41; k ¼ 0Þ:

(

Hereon, it is

qs1 ¼ t� rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p ; qs2 ¼ tþ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ r2

p (68)

and

qd1 ¼ t� 2rþ 1


p ; qd2 ¼ tþ 2rþ 1


p ,

qd3 ¼ tþ 2r� 1


p ; qd4 ¼ t� 2r� 1


p . (69)

A special point of the plane (r, w) is the point S ¼ (1, 1): here, theoptimal, for k ¼ 0, and anti-optimal, for k ¼ 1, values of j aregiven by jdd, and by Eq. (523) one gets

rjdd

S¼ ð�1Þk

ffiffiffi2p

4;0

!; rðrjddÞ

S¼�ð�1Þk 3

ffiffi2p

8 0

0 0

" #, (70)

where S is a saddle point, in correspondence of which there is abifurcation of minjmax

dd for k ¼ 0 and of maxjmindd for k ¼ 1, from

the curve w ¼ 1 if ro1 to the curves w ¼ wst4 and w ¼ wst

5 if r41.The diagrams in Fig. 9 show the surfaces corresponding to the

optimal and anti-optimal value of ja¼ j�t, i.e. of the only

anisotropic part of the objective function, for k ¼ 0 and 1. Theregions where the optimal or anti-optimal value is obtained byj11, j22 or jdd are shadowed in different ways; on the samesurfaces, we have also traced in white the curves w ¼ 1 and wst

1 towst

6 : the bifurcation in correspondence of the point S is apparent inboth the cases of optimal, k ¼ 0, and anti-optimal solution, k ¼ 1.

ARTICLE IN PRESS


9. Comparison of u(n0, n1) with qxx(h)

Let us now consider the optimal or anti-optimal solution j(x0,x1) (Eq. (52)), in comparison with the highest or lowest value ofqxx(y) (Eq. (35)): their similarity is apparent. First of all, in both thecases the isotropic part, t, is identical and does not affect theoptimization, it only affects the final value of j or qxx. The twoanisotropic parts, ja and qa

xx, differs only for the two functionsc0(w) and c1(w), that affect ja, representing the influence of thegeometry and mode on the value of ja. Actually, the combinedeffect of c0(w) and c1(w) does not allow the functions j11, j22 andjdd to exceed q11, q22 and qyy, both for the optimal and anti-optimal case. In other words, q11, q22 and qyy, bound to the optimaland anti-optimal values of j(x0, x1), j11, j22 and jdd; in particular(see also the previous section), j11 and j22 can take the extremevalues q11 and q22 only for w ¼ 0 or w-N, while j11, j22 and jdd

take the extreme value qyy for w ¼ 0, w-N, w ¼ wst1 and wst

2 if r41.The other values qs1, qs2 and qd1 to qd4, that can be taken only onthe curves w ¼ 1 and wst

3 –wst6 , are intermediate values. Therefore,

global maxima and minima can be taken only for w ¼ 0, w-N,w ¼ wst

1 and wst2 , while local maxima and minima only for w ¼ 1 and

on wst3 –wst

6 ; in addition, w ¼ 1 and wst3 –wst

6 can be absolute maximaof the anti-optimal solution and absolute minima of the optimalsolution (see also Fig. 9). The fact that q11, q22 and qyy, bound tothe optimal and anti-optimal values of j(x0, x1) implies also, forthe reasons given in Section 4, that the function j(x0, x1) is strictlypositive, as it must be in view of its physical meaning.

Finally, the optimal and anti-optimal values of the objectivefunction, can be interpreted as a sort of normal stiffness, thattakes its highest or lowest possible value in some special cases,namely when it coincides perfectly with the highest or lowestnormal stiffness qxx of the material composing the laminate.Nevertheless, the direction that maximizes or minimizes qxx doesnot necessarily coincide with the direction of the layers giving thehighest optimal or lowest anti-optimal solutions.

10. Final remarks

In this paper, we have seen how dimensionless invariantmaterial, t, r, k, and geometric, w, parameters influence the flexuraloptimal design of rectangular plates made of anisotropic layers.Namely, the value of r is of importance in the existence of somesolutions: only for r41, i.e. only when there are off-axisstationary values of the material normal stiffness, some possibleoptimal, or anti-optimal, solutions exist. The same parameter rdetermines also the value of the optimal orientation angle whenangle-ply solutions are optimal. The orthotropy parameter k

transforms systematically an optimal into an anti-optimal solu-tion, and vice versa, whilst the isotropy-to-anisotropy ratio tdetermines the maximum value of the effectiveness of angle-plyoptimal solutions.

The objective function j is also bounded by the dimensionlessnormal stiffness of the material, so it can be interpreted as a sortof normal stiffness, modified in its value by geometry and mode,

and taking its highest possible values only in some circumstances,when it is exactly equal to the dimensionless normal stiffness.

All the analysis being based upon dimensionless parameters,the same results apply to different laminates, regardless of thenumber of plies.

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