2012 - M Montemurro - Design of the Elastic Properties of Laminates With a Minimum NUnber of Plies (MCM)

7/31/2019 2012 - M Montemurro - Design of the Elastic Properties of Laminates With a Minimum NUnber of Plies (MCM)

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DESIGN OF THE ELASTIC PROPERTIES OF LAMINATES

WITH A MINIMUM NUMBER OF PLIES

M. Montemurro,1,2 A. Vincenti,1 and P. Vannucci3*

Keywords: anisotropy, mechanical properties, laminate mechanics, numerical analysis

The design of laminates with a minimum number of layers for obtaining the given elastic properties is addressed in

the paper. The problem is treated and solved in the general caseno simplifying hypotheses are made about the type of theirstacking sequence. The problem is stated as a nonlinear programming problem, where a unique objective function includes all

the requirements to be satised by the solutions. The optimum solutions are found within the framework of the polar-genetic

approach, since the objective function is written in terms of polar parameters of the laminates, while a nonclassical genetic

algorithm is used as the optimization scheme. The optimization variables include the number of layers, layer orientations and

layer thicknesses. In order to include the number of plies among the design variables, certain modications of the genetic

algorithm have been done, and new genetic operators have been developed. Some examples and numerical results concerning

the design of laminates with a minimum number of layers for obtaining some prescribed elastic symmetries, bending-extension

uncoupling, and quasi-homogeneity are shown in the paper.

1. Introduction

The design of elastic properties is very important in many applications, e.g., for aircraft and space structures. Unlike

classical materials, composite laminates can be designed to obtain certain properties, such as bending-extension uncoupling,

in-plane and/or bending orthotropy, isotropy, and so on. This can be done mainly by a correct design of the stacking sequence

of laminate plies. The problem of tailoring a composite plate to realize a given elastic or hygrothermoelastic behavior has at-

tracted the attention of several researchers. A rather complete review of the state of the art of the problem, at least what concerns

the design with respect to stiffness, can be found in [1, 2]. The design of laminates considered as an optimization problem

is rather cumbersome and presents difculties in its solution due to the high nonlinearity and nonconvexity of the objective

Mechanics of Composite Materials, Vol. 48, No. 4, September, 2012 (Russian Original Vol. 48, No. 4, July-August, 2012)

1Institut dAlembert UMR7190 CNRS Universit Pierre et Marie Curie Paris 6, Case 162, 4, Place Jussieu, 75252 Paris

Cedex 05, France2Centre de Recherche Public Henri Tudor, av. 29, J.F. Kennedy, L-1855 Luxembourg-Kirchberg, Luxembourg3Universit de Versailles et St Quentin, 45 Avenue des Etats-Unis, 78035 Versailles, France*Corresponding author; e-mail: [email protected]

0191-5665/12/4804-0369 2012 Springer Science+Business Media, Inc.

Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 48, No. 4, pp. 539-570 , July-August,

2012. Original article submitted July 8, 2011; revision submitted April 13, 2012.


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function, which is caused by the fact that laminate properties depend on a combination of powers of the circular function of

layer orientations, the latter being natural design variables. As a consequence, designers generally limit the search for solu-

tions to a restricted class of laminates, usually to symmetric stacking sequences to ensure the bending-extension uncoupling,

or balanced sequences to have an in-plane orthotropy and so on.

The problem of designing laminates as an optimization problem received a general formulation, especially concern-

ing the design of elastic symmetries, with the works of Vannucci, Vincenti, and Verchery [3-5]. They have shown that it is

possible to construct, by using the so-called polar method, a unique objective function which is able to take into accountseveral design criteria, e.g., elastic properties, such as uncoupling, orthotropy, and many others, and given thermal responses

in extension and/or in bending. The general problem is therefore reduced to the classical nonlinear programming problem,

and its solutions are the minima of a nonlinear, nonconvex function in the design space of layer orientations. In these studies,

many optimum solutions were found for several different problems. In all the investigations made by these authors, the number

of plies was always xed a priori, and the design process was focused only on the geometry of stacking sequence, i.e., theonly design variables were layer orientation angles.

As a natural continuation of [3-6], the topic of this paper is a new formulation for the laminate design with a minimum

number of layers which satises assigned elastic symmetries. For this purpose, the number and orientations of plies, as wellas the thickness of each layer, are taken into account as design variables. More precisely, this paper tries to give an answer to

a question which is usually left apart by designers, but which is classical and fundamental in any mathematical problem, i.e.,

the question about the existence of a solution. In the case of laminate design, this question should be: what is the minimum

number of layers that guarantees the existence of at least one solution to a given problem of tailoring the elastic properties of

a laminate?

To the knowledge of the authors, only in one case the minimum number of layers to obtain some prescribed properties

is known exactly thanks to a theoretical result. This is the case of in-plane isotropy, solved by Werren and Norris [7]: at least

three unidirectional plies are needed to obtain a laminate that will be isotropic in tension, although with be membrane-bending

coupling. But as soon as one adds another or a different requirement, for instance, uncoupling or bending isotropy, the result

is unknown. Finding the minimum number of layers for which a given optimum laminate design problem can be solved is

actually a very difcult task. In fact, the number varies with the type of elastic requirements to be satised: the results arestrictly problem-dependent, and unfortunately in all the cases, the optimum solutions are unknown and there is no analytical

model which could describe their evolution with the number of layers. Therefore, a numerical investigation seems to be an

appropriate approach to the investigation.

It is worth noting that the optimum design of a laminate in terms of the number and properties (orientation, materials,

and thickness) of its layers is a combinatorial optimization problem, which is harder to solve for small numbers of layers. In

fact, the smaller the number of plies, the smaller becomes the design space, and the number of available solutions decreases.

However, solutions with a minimum number of plies are important when the problem of minimum weight of laminates is

addressed.

The way proposed in this paper is to formulate the problem in the form of search for the minimum of a positive

semidenite form, including the number of layers n among the variables. The function takes into account the numbern in theform of a penalization term in order to strongly drive the search for an optimum solution towards laminates with the lowest

number of layers.

The numerical technique adopted here is a genetic algorithm (GA), BIANCA, which stands for Biologically InspiredAnalysis of Composite Assemblages, created by the authors and already used in the previous research [3, 8]. This code has

been specially conceived for the optimum design of laminates; here, a modied version of this code, able to include the numberof layers into design variables, is used. In this way, the code BIANCA becomes a GA able to cross not only individuals, but

also species, see Sect. 5. In order to obtain an effective formulation of the problem, the polar formalism was used. It is based

upon an algebraic formulation making use of tensor invariants for representing planar tensors, see [9, 10], which has proven

to be rather effective in solving several design problems concerning laminates.


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The paper is organized as follows. In the rst section, the general equations of the Classical Laminated Plate Theory(CLPT) are recalled, and then the design problem is stated within the framework of the polar method and formulated as an

optimization problem. Further, an account of the numerical procedure, in particular, a concise description of the new genetic

algorithm for deriving solutions, is presented. Finally, several numerical examples are given in order to show the effectivenessof the approach proposed, and then some general considerations ends the paper.

2. General Equations

The general equations of the Classical Laminated Plate Theory (CLPT) (see for instance [11]) describing the behavior

of a composite laminate, in the case of zero hygrothermal loads, are

N

M

A B

B D

=

, (1)

where

N =

N

N

N

x

y

s

, M =

M

M

M

x

y

s

, =

x

y

s

, =

x

y

s

.

In Eq. (1), the classical Voigt notation is assumed, and all the quantities included are expressed in the global reference

frame of the laminate, i.e.,R = {O; x, y, z} (see Fig.1). N and M are the tensors of in-plane forces and bending moments, is

the tensor of in-plane strains for the midplane, and is the curvature tensor. The tensors A and D describe the in- and out-of-

plane behavior of the plate, but B is the coupling tensor. For example, ifB is not equal to zero, the in-plane forces applied to

the laminate cause both in-plane strains and curvatures of the midplane and vice versa. The tensors A, B, and D depend on

the mechanical and geometrical properties of layers. For a multilayer laminate with n plies, the expression ofA, B, and D are

A Q= ( ) ( )=

k kk p

p

k kz z 1 ,

B Q= ( ) ( )=

1

2

21

2k k

k p

p

k kz z , (2)

D Q= ( ) ( )=

1

3

3

1

3

k kk p

p

k kz z .

x

1

z = x3k

yx2k

x1k

dk

Fig. 1. Laminate reference systems.


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Here, Qk(

k) is the reduced stiffness tensor of a kth ply oriented at an angle

kwith respect to the global reference

frame of the laminate, see Fig. 1; n is the number of plies and is dened as

np

p=

+

2

2 1

if even

if odd

,

;

zk

andzk 1 are zcoordinates of the top and bottom surfaces of the kth layer. Figure 2 illustrates the denition ofz

kused in

this work. The homogenized stiffness tensors A*, B*, and D* of the laminate and its homogeneity tensorC are dened as

A A* =1

htot, B B* =

2

2htot

, D D* =12

3htot

, C A D* *= ,

where htot

is the total thickness of the laminate. The inverse of Eq. (1) is

a b

b d

N

MT

=

,

where a A BD B1= ( )

1

, b aBD 1= , d D BA B1= ( )

1

. (3)

In Eqs. (3), a, b, and d are the in-plane, coupling, and bending compliance tensors.

3. Polar Representation Method

The polar representation method is a mathematical technique introduced in 1979 by Verchery [9]. By using this method,

it is possible to represent a plane tensor of any order by its invariants. This technique has already been used to treat many

optimum design problems for laminates. A complete overview of the polar method can be found in [12]. There are different

ways to represent a tensor, and the Cartesian representation is the most-used one. The main drawback of the Cartesian methodis the fact that the components are frame-dependent.

The idea of Verchery is to replace these components with other parameters that are frame-independent, i.e., tensor

invariants. There are several ways to choose these invariants. The quantities introduced by Verchery are directly linked to the

elastic symmetries and to the decomposition of strain energy [12].

In the case of a plane fourth-rank elasticity tensorL, such as the layer stiffness tensorQ or the layer compliance

tensorS = Q1, the polar method ensures that the Cartesian components can be expressed in terms of four moduli T0, T

1,R

0,

andR1

and two polar angles 0

and 1. The components of the tensorL in the material frame {O;x

1,x

2,x

3} are expressed as

x

-p

a

z

h/2

h/2

zk

zk-1

p

-k

k

-1

1

0

b

x

z

h/2

h/2

zkzk-1

-p

p

-k

k

-1

1

Fig. 2. Sketch of laminate layers and interfaces.


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L T T R R

L R R

L

1111 0 1 0 0 1 1

1112 0 0 1 1

2 4 4 2

4 2 2

= + + +

= +

cos cos ,

sin sin ,

11122 0 1 0 0

1212 0 0 0

1222 0 0

2 4

4

4 2

= +

=

= +

T T R

L T R

L R

cos ,

cos ,

sin

RR

L T T R R

1 1

2222 0 1 0 0 1 1

2

2 4 4 2

sin ,

cos cos .

= + +

(4)

The norm of the tensorL as a function of polar parameters is

L = + + +T T R R0

2

1

2

0

2

1

22 4 . (5)

The inverse relations of Eqs. (4) are

8 2 4

8 2

8

0 1111 1122 1212 2222

1 1111 1122 2222

04

T L L L L

T L L L

R

= + +

= + +

,

,

eii

i

L iL L L iL L

R L

0

1

1111 1112 1122 1212 1222 2222

14

1

4 2 4 4

8

= + +

=

,

e 1111 1112 1222 22222 2+ + iL iL L .

The polar moduli T0, T

1,R

0, andR

1and the angular difference

0

1are invariants of the tensorL. In axes rotated

through an angle , the components of the tensorL are expressed as

L T T R R

L R

xxxx

xxxy

= + + +

=

0 1 0 0 1 1

0 0

2 4 4 2

4

cos ( ) cos ( ),

sin ( )

++

= +

=

2 2

2 4

4

1 1

0 1 0 0

0 0

R

L T T R

L T R

xxyy

xyxy

sin ( ),

cos ( ),

cos

(( ),

sin ( ) sin ( ),

c

0

0 0 1 1

0 1 0

4 2 2

2

= +

= + +

L R R

L T T R

xyyy

yyyy oos ( ) cos ( ).4 4 20 1 1 R

(6)

These relations show that the rotation changes the polar angles 0

and 1

into 0

and 1

and that T0

and

T1

describe the isotropic part of the tensor, whileR0

andR1

the anisotropic one. These are two reasons that make the polar

method very effective when it is employed to represent the layer stiffness and compliance tensors within the framework of

the CLPT theory, especially in design problems. In particular, the conditions for elastic symmetries are expressed in a simple

way when using polar invariants. A summary of these conditions is given in Table 1. We recall here that, in the plane case, the

square symmetry corresponds to the cubic syngony, i.e., it is characterized by the /2-periodicity of the elastic moduli, whiletheR

0-orthotropy is a special case of plane orthotropy, see [13] for more details.

TABLE 1. Conditions for Elastic Symmetries in Terms of Polar Invariants

Elastic symmetry Polar condition

Orthotropy 0 14

=K

R0- orthotropy R0 0=

Square symmetry R1

0=

Isotropy R R0 1

0= =


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In this paper, T0, T

1,R

0,R

1,

0, and

1will denote the polar components of the reduced stiffness tensor of the basic

layer, Q. It is also possible to deduce the polar components of the laminate stiffness tensors A, B, and D as functions of polar

components of the lamina stiffness tensorQ according to Eq. (2) of CLPT:

T T Tm

T z z

T T Tm

T

A B D

kk p

p

km

km

A B D

kk p

p

0 0 0 0 1

1 1 1 1

1

1

, , ,

, ,

= ( )

=

=

=

zz z

R R Rm

R

km

km

A i B i D i

k

iA B D k k

( )

=

+

1

04

04

04

040 0 0 0

1

,

, ,(

e e e e ))

,

, ,

k p

p

km

km

A i B i D i

k

i

z z

R R Rm

RA B D

=

( )

=

1

14

14

14

121 1 1

1e e e e

(( ).

11

k k

k p

p

km

km

z z+

=

( )

(7)

where TA

0 , TA

1 , RA

0 , RA

1 , F0A

, and F1A

stand for the polar components of the tensorA; TB0 , TB

1 , RB

0 , RB

1 , F0B

, and F1B

stand for the polar components of the tensorB; TD0 , TD

1 , RD

0 , RD

1 , F0D

and F1D

stand for the polar components of the ten-

sorD. In Eq. (7), m = 1, 2, 3 for the extension, coupling, and bending stiffness tensors, respectively.

From Eq. (7), it can be noticed that the symmetries of the laminate in terms of extension, coupling, or stiffness behavior

depend on its stacking sequence, layer properties, layer orientation and thickness, and of course on the number of plies. When

concerned with laminate design, a designer has to satisfy several conditions at the same time, including not only the com-

mon objectives, such as the buckling load or strength, but also the general properties of the elastic response of the laminate,such as uncoupling, extension orthotropy, bending orthotropy, and so on. In fact, it is not easy to take into account all these

aspects, and normally designers use some shortcuts (rules of thumb) to automatically get some properties, e.g., uncoupling or

the extension orthotropy. Vannucci and Vincenti showed in the previous studies (see [4-6]) that, within the framework of the

polar method, it is possible to formulate all problems of the optimum design of laminates, including the requirements on the

elastic symmetries; therefore, a general approach to the design of laminates is possible. The reader is referred to the works

cited previously for a deeper insight into the matter. The next section presents an important modication to this approach that

also includes the number of layers among the design variables.

4. Formulation of the Optimization Problem

In formulating the design of the elastic properties of a laminate as an optimization problem, the key point is the con-

struction of the objective function. For a laminate with n plies, the design variables can be the number of layers n, the vectorof layer orientation , and the vector of layer thickness h. In order to formulate the laminate design problem in a general way,

the objective functionf=f(n, , h) should include all design requirements, in particular, the elastic symmetries of the laminate.Vannucci and Vincenti [3-6] showed that, within the framework of the polar method, the laminate design for given

elastic properties can be reduced to the search for the minima of a positive semidenite function of polar elastic parameters ofthe laminate. In those works, the number of layers, layer thickness, and material properties were xed, while layer orientationswere assumed as the only optimization variables. The optimization problem was dened as

min

f( ) .

Since the objective functionf() is positive semidenite, as mentioned above, its minima are also zeros of the func-tion. For more details on the denition of this objective function for different combinations of elastic symmetries, see [3-6].


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As specied previously, the objective of this paper is the design of a laminate with assigned symmetries at a minimumnumber of layers. In such a case, the number of plies, and eventually the thickness of each layer, must belong to the set of

design variables, i. e., a modication of the objective function is necessary. The new, unconstrained optimization problem is

min , , ,

, , , , .

, ,n

s

F n

F n f n n

hh

h h

( )

( ) = ( )(8)

It can be noticed that the new objective function F(n, , h) is dened in such a way that it is a positive semidenitefunction whose zeros are solutions of our problem. The inuence of the number of layers n is introduced as a penalizationterm, wheres is a power whose value can be chosen from a certain range. The large number of numerical tests that the authors

have been conducted show that the best results are obtained whens is in the interval [1, 4].

Equation (8) formalizes the classical nonlinear programming problem without constraints, for which several numeri-

cal solving techniques can be used. Since f (n, , h) is a nonconvex function with several nonglobal minima, a suitable and

robust solving algorithm must be chosen.

In conclusion of this section, it can be noticed that the approach proposed is general, i.e., no simplifying assumptions

are introduced, such as, for example, restrictions to symmetric, balanced, or angle-ply laminates.

5. Search for Solutions by the Genetic Algorithm BIANCA

Some general considerations have determined the choice of a GA as the numerical technique for solving the optimization

problem expressed by Eq. (8). The key point is the nature of the objective function, which is highly nonlinear and nonconvex.Previous works, see for instance [14,15], showed that, due to the nonlinearity of the problem, in many cases, there are several

sets containing an innite number of solutions that describe geometrical sets as continuous hypersurfaces in the design space.Another very important aspect is the nonconvexity of the problem. In fact, in problems concerning the optimum design of

laminates, as a rule, there are many local and global minima, and the GA is able to look for a global minimum exploring the

entire research space in an effective way.

Another important point is that, in the case of laminates, problems often depend on continuous, discrete, or abstract

variables. For example, layer orientations can be considered as continuous or discrete variables the angles can take all

values in the range [90, 90] or they can be restricted to a limited and discrete set of xed orientations, such as 0/45/90.The same considerations can also be applied to the layer thickness. Finally, an abstract variable can represent a group of dif-

ferent variables, such as in the case of the constitutive material of layers when the material is chosen from a database: once a

particular material is associated with a ply, the entire set of properties of the layer are determined. Therefore, a suitable numeri-

cal technique should offer the same effectiveness for different types of design variables; GAs can deal with all these aspects

because they are basically the zeroth-order numerical strategies, which require only computations of the objective function.The GA is a metaheuristic method for solving optimization problems, which is based on two biologic metaphors the

natural evolution, according to the Darwinian principle of survival of the ttest, and the genetic coding of the phenotype. Awide and detailed presentation of genetic algorithms can be found in the books written by Goldberg [16] and Michalewicz [17].

Vannucci and Vincenti have developed a GA, BIANCA [8, 18], specially intended for solving many optimizationproblems for laminates. BIANCA has several original features, mainly concerning the representation of information, i.e.,

coding of the genotype of an individual. This program is applicable to all classical genetic operators, such as adaptation and

selection, crossover, mutation and elitism operators. In addition, specic strategies were developed within BIANCA in orderto take into account constrained optimization problems. We give here only some elements of the code BIANCA, more details

can be found in [8, 18].

Unlike most of the GAs dedicated to laminate problems, where generally a unique string of real variables composes a

sort of surrogate of the genetic code of the laminate, in BIANCA, the information concerning the laminate has an articulated

structure: a laminate is considered as an individualphenotype whose genomegenotype is stocked in a binary coded matrix.


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Each row of the matrix is a chromosome coding all data concerning a layer. In BIANCA, some chromosome corresponds to

each layer. In this way, the number of layers determines, in the biological metaphor, a species (in the sense that laminates with

a different number of layers belong to different biological species). Each chromosome is divided into genes; a gene codes a

particular quantity concerning the chromosome, i.e. the layer, such as, e.g., the orientation angle, material, thickness, etc. The

genes corresponding to various chromosomes form columns of the genome matrix. It is possible to code real and discrete

variables in the same genome at the same time. As programmed in BIANCA, the classical genetic crossover and mutation

operators mimic Nature very closely they act on each gene separately and independently.

5.1. The modifed genetic algorithm BIANCA

In order to include the design with respect to the number of layers in the objective function, we introduced some

modications in the code. As said previously, in BIANCA, the number of layers n determines the number of chromosomes,i.e., the biological species of layers. So, ifn has to be included among the variables and has to evolve during generations, a

crossover of the species has to be performed: the new version of BIANCA is a GA based upon the evolution of species and

individuals at the same time. The modications of the code BIANCA are substantially new genetic operators. We were partiallyinspired by the previous work of Park et al. [19]. In our case, their strategy was modied in order to adapt it to a scheme ofgenetic coding and genetic operations much more complicate and rich than those proposed in [19].

In particular, the classical reproduction phase, see Fig. 3, was changed by introducing new genetic operators called

chromosome shift operator, chromosome reorder, chromosome number mutation, and chromosome addition-deletion.

A brief description of these new operators and their use in the reproduction phase is given in the next paragraphs.

5.1.1. Structure of the individual in BIANCA. The rst modication concerns the structure of the generic individual

genotype. As briey mentioned previously, in BIANCA, the genotype of each individual is represented by the binary arrayshown in Fig. 4. In this gure, the quantity gij

k represents ajth gene of an ith chromosome of a kth individual. The letter e

stands for an empty location, i.e., there is no gene in this location, while n is the number of chromosomes of the kth individual.

In this paper, the number of layers and their orientation angles and thickness are assumed as optimization variables.

The new structure of the individual-laminate is shown in Fig. 5. In this case, the number of layers in the kth laminate is n, but

the orientation and thickness of the ith ply are iand h

i, respectively.

No

Yes

INITIAL

POPULATION

Random extractionof individualsN

ADAPTATION

Fitness evaluationof the individuals

SELECTION

Making parentscouples

CROSSOVER

Mixing of parentsgenotype

REPRODUCTION

BEST

INDIVIDUAL NEW

GENERATION

STOP

CRITERION

MUTATION

Stochastic alterationof individuals

genotypeFinal populationsaverage adaptation

Fig. 3. Standard GA scheme.


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The classical reproduction operators, such as crossover and mutation, act on the genotype of parents couples. However,to take into account the variable number of layers, i.e., the number of chromosomes, a new approach is required in executing

the reproduction phase. This new procedure is detailed below.

As mentioned previously, this new version of BIANCA is able to evolve species and individuals at the same time:

the best value ofn is the outcome of biological selection, and the most adapted species issues automatically as a natural

result of the Darwinian selection.

5.1.2. The modied crossover phase and the role of chromosome shift and reorder operators . To explain the way

whereby the reproduction phase takes place one can consider the following case. There are two parents,P1 andP2, with three

and ve layers, respectively, see Fig. 6. In this example, for the sake of synthesis, the maximum number of plies is assumedequal to six. The minimum number can be chosen arbitrary between 1 and 6. Before realizing the crossover between these two

individuals, it can be noticed that there are different ways to pass the information restrained in parents genotype to the next

generation, i.e., to their children. At the successive generation, two new individuals will be produced from this couple, one

with three and the other with ve layers. To improve the efciency of the genetic algorithm in terms of exploration and exploi-tation of the information on the design space, the concept of shift factor is introduced. The shift factor is sorted randomly,

with a given probabilitypshift

, in the range [0, nP1 nP2], where nP1 nP2 is the absolute value of difference between thenumbers of parents layers. With the shift factor, various combination of crossover are possible. In the example mentioned

before, the minimum shift factor is 0 and the maximum is 2. For example, if the shift factor is 1, all the genes ofP1 with a

smaller number of chromosomes are shifted up to down by a quantity equal to 1, as shown in Fig. 7.

( )g k11 ( )gk

12 ( )gmk

1 n

( )g k21 ( )gk

22 ( )g mk

2

( )gnk

1 ( )gnk

2 ( )gnmk

e e e e

Fig. 4. Structure of an individual genotype in BIANCA.

( )1k ( )h k1 n

( )2k

( )hk

2

( )nk ( )hn

k

e e

Fig. 5. Individual-laminate in BIANCA.


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After the shift operation, the real crossover phase takes place. The crossover operator acts on every single gene. The

position of crossover is chosen randomly for each gene of both individuals. This operator naturally involves all the chromo-

somes of the parent with a smaller number of plies, i.e., in the case shown in Fig. 8, all genes ofP1, while only the homologous

genes ofP2 undergo the action of crossover operator. At this point, two new individuals can be created, C1 and C2, that have

three and ve chromosomes-layers respectively, see Fig. 9. It can be noticed that the 1st and 5th chromosome ofP2 have not

e e 3 ( )12P ( )h P1

2 5

( )11P ( )h P1

1 ( )22P ( )h P2

2

( )21P ( )h P2

1 ( )32P ( )h P3

2

( )31P

( )hP

31

( )42P

( )hP

42

e e ( )52P ( )h P5

2

e e e e

Fig. 6. Structure of parents couple.

e e 3 ( )12P ( )h P1

2 5

( )11P ( )h P1

1 ( )22P ( )h P2

2

( )21P

( )hP

21

( )32P

( )hP

32

( )31P ( )h P3

1 ( )42P ( )h P4

2

e e ( )52P ( )h P5

2

e e e e

Fig. 7. Effect of the shift operator on parents couple.

e e 3 ( )1 2P ( )h P1 2 5

( )11P ( )h P1

1 ( )22P ( )h P2

2

( )21P ( )h P2

1 ( )32P ( )h P3

2

( )31P ( )h P3

1 ( )42P ( )h P4

2

e e ( )52P ( )h P5

2

e e e e

Fig. 8. Position of crossover for every gene.


11/22

379

undergone the crossover phase, so, according to the notation of Figs. 8 and 9, it is possible to write the equalities 1

2

1

2

( ) = ( )P C

,

5

2

5

2

( ) = ( )P C

, h hP C

1

2

1

2

( ) = ( ) , and h hP C

5

2

5

2

( ) = ( ) . At this point, before the mutation phase, a readjustment of chromo-somes position is required. The chromosome reorder operator achieves this phase by a translation of all chromosome down

to up in the structure of the individual with the smaller number of layers, see Fig. 10.

5.1.3. The modied mutation phase and the role of chromosomes number mutation and addition-deletion operators.

Mutation is a random process that is applied to the genotype to better explore the feasibility domain. Mutation is articulated

in two phases: it acts rst on the number of chromosomes-layers and then on the genes value.In the rst phase, the number of layers is changed arbitrarily by one at time for each individual with a given probabil-

ity pmut chrom

( ) ; then the chromosome addition-deletion operator acts on the genotype of both individuals by adding or delet-ing a chromosome-layer. The location of layer addition-deletion is also selected randomly. Naturally, if the number of layers

is equal to the maximum number of chromosomes, only deletion occurs. Similarly, if the number of plies is equal to the

minimum number, only addition is applied. In the case shown in Fig. 11, the number of layers ofC1 is increased by one, and

a new chromosome a

C

a

Ch( ) ( ){ }1 1, is randomly added in position 3, while the number of layers ofC2 is decreased by one

and the chromosome deletion is randomly done in position 2.

e e 3 ( )12C ( )h C1

2 5

( )11C

( )hC

11

( )22C

( )hC

22

( )21C ( )h C2

1 ( )32C ( )h C3

2

( )31C ( )h C3

1 ( )42C ( )h C4

2

e e ( )52C ( )h C5

2

e e e e

Fig. 9. Structure of new individuals after a crossover.

( )1 1C ( )h C1 1 3 ( )1 2C ( )h C1 2 5

( )21C ( )h C2

1 ( )22C ( )h C2

2

( )31C ( )h C3

1 ( )32C ( )h C3

2

e e ( )42C ( )h C4

2

e e ( )52C ( )h

C5

2

e e e e

Fig. 10. Effect of the chromosome reorder operator on the new individuals.


12/22

380

In the second phase, the mutation of gene values, i.e., orientations and thickness, takes place with a probability pmut

after the rearrangement of chromosome positions. In the example of Fig. 12, the mutation is undergone by the genes 2

1

( )C

and ha

C

( )1

of the individual C1 and by the gene 1

2

( )C

of the individual C2.

6. Sample Problems and Results

In order to demonstrate the effectiveness of the polar formulation and of the new genetic algorithm BIANCA in obtain-

ing composite laminates with a variable number of plies and with certain elastic properties, several calculations were carried

out, and a number of solutions that satisfy different combinations of design objectives was found. Among all possible designcases, the following ones are discussed in this paper:

1) uncoupling, total orthotropy withK= 0 and coincidence of axes, i.e., the in-plane and bending orthotropy with the

same axes;

2) uncoupling, total orthotropy withK=1 and coincidence of axes, i.e., the in-plane and bending orthotropy with the

same axes;

3) uncoupling and total isotropy , i.e., the in-plane and bending isotropy;

4) uncoupling and homogeneity, i.e., an identical behavior of the homogenized in-plane and bending stiffness tensors.

The corresponding polar conditions for the elastic symmetries are summarized in Table 1. Here, uncoupling is intended

to be the bending-extension uncoupling determined by the fact that the tensorB is zero.

( )11C ( )h C1

1 4 ( )12C ( )h C1

2 4

( )21C ( )h

C2

1 e e

( )aC1 ( )ha

C1 ( )32C ( )h C3

2

( )31C ( )h C3

1 ( )42C ( )h C4

2

e e ( )52C ( )h C5

2

e e e e

Fig. 11. Effect of the chromosome mutation and addition-deletion operators on the new individuals.

( )11C ( )h C1

1 4 ( )12

mC ( )h C1

2 4

( )2m ( )hC

21

( )32C

( )hC

32

( )aC1 ( )ham

C1 ( )42C ( )h C4

2

( )31C ( )h C3

1 ( )52C ( )h C5

2

e e e e

e e e e

Fig. 12. Effect of mutation operators on the new individuals.


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381

6.1. Sample problems

Before formulating the objective function F(n, , h) for each case, it is worth specifying its general properties. In allthe problems considered in the present section (cases 1 to 4), this function is a dimensionless, homogenized, convex function

of polar parameters of the tensors A, B and D, Eq. (7), while it is a highly nonlinear, nonconvex function of design variables,

i.e., of the number of layers, orientations, and thickness. In all the cases, the powers in Eq. (8) is assumed equal to 2.

The objective function for each problem is dened in such a way that the solutions, i.e., the minima, are also zerosof the function. Since each case corresponds to a given combination of elastic symmetries, the global objective function isconstructed as the sum of partial objective functions, and the values of each partial objective function are normalized betweenzero and unity.

As a conclusive remark, before expressing the objective function for each particular case, we should note that, when-ever a requirement on polar angles occurs in the expression of the objective function, these quantities are expressed in radians.

Cases n. 1 and n. 2. In order to obtain the elastic uncoupling, i.e., B = 0, the norm of the coupling tensorB must be

zero. To obtain orthotropy, the difference between the polar angles 0

and 1

must be a multiple of /4 (see the correspondingpolar condition in Table 1), both for the membrane and bending stiffness tensors. The last required elastic property considered

here is the coincidence of orthotropy axes, i.e., the angle 1

must be the same forA and D. The expression of the global objec-tive function including all these conditions is

F nK

A A A D

, ,

** * *

hB

Q( ) =

+

+

2

0 1

2

04

4

** * ** *

+

1

2

1 1

2

4

4 4

D DA DK

n2 . (9)

In Eq. (9), B* is the homogenized coupling tensor, and Q is one of layer stiffness tensors. Their norms are calculated

according to Eq. (5). All other polar parameters are referred to the norms of their respective homogenized tensors, i.e., A* and

D*. The constants KA

*

and KD*

can assume the values 0 or 1, depending upon the different kinds of orthotropy (case 1:

K KA D

* *

= = 0 ; case 2: K KA D

* *

= = 1 ).

Case n. 3. In this case, along with the elastic uncoupling, the requirement of total isotropy was formalized. The partialobjective function for uncoupling is expressed as in cases 1 and 2, see Eq. (9). In order to ensure the isotropy of in-plane andbending stiffnesses, the anisotropic part of the tensors A* and D* must be zero. Therefore, the global objective function hasthe form

F nR R

R R

R RA A

Q Q

D D

, ,

* * * * *

hB

Q( ) =

++

++

+2

02

12

02

12

02

14

4

422

02

12

2

4R Rn

Q Q+

. (10)

In Eq. (10), RA

0

*

, RA

1

*

and RD0

*

, RD1

*

are referred to the homogenized laminate in-plane and bending stiffness ten-

sors, respectively; RQ0 and R

Q1 are referred to the layer stiffness tensor, and they are used for the sake of normalization.

Case n. 4. In this case, the requirements are uncoupling and homogeneity, i.e., the laminate has the same behavior in

extension and bending. To realize the objective of homogeneity, the polar parameters T0, T

1,R

0,R

1,

0, and

1must assume

the same values for both the tensors A* and D*, and therefore the homogeneity tensor is equal to zero, C = 0. The objectivefunction is

F n n, , .

*

hB

Q

C

Q( ) =

+

22

2


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382

6.2. Numerical results

Since the elastic behavior of a laminate depends on the elastic properties of its elementary plies, the results must

refer to a given material; in this paper, a highly anisotropic unidirectional carbon/epoxy ply (T300/5208) [20] was chosen. Its

properties were as follows:E1

= 181 GPa,E2

= 10.3 GPa, G12

= 7.17 GPa, n12

= 0.28, T0

= 26.8804311 GPa, T1

= 24.7448933

GPa,R0

= 19.71043 GPa,R1

= 21.433122 GPa, and F0

= F1

= 0 deg.

For each of the cases from 1 to 4 mentioned, two kinds of simulations were performed. In the rst one, the thicknessof the elementary ply was assumed equal to 0.125 mm, and thus the design variables were only the number and orientations

of layers. In the second case, the thickness also was a variable of optimization.

We also considered laminates with layers of variable thickness in order to assess the inuence of such a variable on theminimum number of layers needed to obtain some specied elastic properties. In practice, this corresponds to increasing thenumber of design variables for the same kind of problem and should result in a better quality of results relative to those in the

cases of a xed thickness and perhaps in a lower nal minimum number of layers. Actually, it was so, which can be observedin the results shown below. Of course, in doing this, we do not consider the practical (e.g., manufacturing) aspects of such a

choice, because we were concerned merely with the theoretical solution of the mathematical problem of nding a laminatehaving the minimum number of layers to satisfy some elastic requirements; the practical aspects are not discussed in this paper,

at least for the cases of laminates with layers of different thickness. We considered only layers with the same elastic proper-

ties, but of different thickness, which implies the assumption that the volume fraction and arrangement of bers are constant.For each simulation, the number of plies n varied in the range [4, 16], while the orientation of each layer

k(k= 1,..,n)

could assume any integer value in the interval [90, 90] discretized by a step of 1. In simulations where the layer thickness

was also a design variable, the thickness hk

(k= 1,..,n) varied in a continuous way in the interval [0.1, 0.2] mm. The genetic

parameters common for all the cases were as follows: Nind

= 500, Ngen

= 500, pcross

= 0.85, pmut

= 1/Nind

, pshift

= 0.5, and

( )( ) ( )

.max minpn n

Nmut chrom

chrom chrom

ind

=

A remark regarding the mutation probabilitypmut

and the mutation probability of the

TABLE 2. The Best Stacking Sequences for Design Problems 1 to 4

Objective Stacking sequence (angles in degrees)No. of

pliesResidual

Fixed layer thickness

Case n. 1 [9/0/18/4/14/5] 6 3 5873 106

.

Case n. 2 [16/65/67/8/10/59] 6 1.7547 102

Case n. 3 [0/50/61/42/87/90/49/10/12/36/26/47/83] 13 1 6117 102

.

Case n. 4 [64/71/74/65/66/63/70] 7 1.7547 10 2

Variable layer thickness

Case n. 1[9/6/4/11/9/6]

[0.118/0.126/0.140/0.126/0.103/0.152 mm]6 3 2976 10

7.

Case n. 2[24/73/18/67]

[0.100/0.200/0.200/0.100 mm]4 7 3315 10 3.

Case n. 3[19/74/45/20/75/17/53/83/2/51]

[0.113/0.168/0.200/0.137/0.149/0.190/0.199/0.113/0.106/0.116 mm]10 2 0476 10

3.

Case n. 4[15/2/12/10/21/-6]

[0.156/0.188/0.158/0.151/0.111/0.182 mm]6 2 9945 10

5.


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383

number of chromosomes, (pmut

)chrom

: their values were chosen according to De Jongs study [21]. In this study, it was sug-

gested that the mutation probability be inversely proportional to the size of population. Hence, for all simulations, the mutation

probability was chosen according to this rule.

Table 2 shows the examples of stacking sequences for laminates satisfying the design criteria for cases n. 1 to n. 4 at

xed and variable layer thicknesses. The residual in the last column is the value of the global objective functionF(n, , h) for

the solution indicated aside in the adjacent column (recall that exact solutions correspond to zeros of the objective function). Asusual in a numerical technique, the real solution is found to within a small numerical tolerance; this tolerance is the residual.

A discussion on the importance of the numerical residual in this type of problems can be found in [3, 18].

Tables 3-6 show the values of polar parameters for all the stacking sequences found in both the cases of constant and

variable thicknesses of plies. As seen, all the laminates are extension-bending uncoupled, although the stacking sequences are

not symmetric. Actually, some of these sequences are antisymmetric (the condition that guarantees the bending orthotropy, but

not always the bending-extension uncoupling, see [15]). For instance, the sequence of case n. 1 with plies of constant thick-

ness can be reduced to the sequence [2/7/11/11/7/2], which is antisymmetric, simply by a rotation through 7. Actually,

such an angle (see Table 3) corresponds to the polar angle 1, and in this case, sinceK= 0, also to

0. In fact, for a given

elasticity tensorL, in the case of orthotropy withK= 0, the direction determined by the angle 1

corresponds to the strong

axis of orthotropy, i.e., to the direction of the highest value of the componentLxxxx

, as easily seen from Eq. (6) (see also [10]).

A similar result is valid also for case n. 2 if the plies have a constant thickness, as well as for cases n.1 and n.4 if the plies have

a variable thickness. Nevertheless, the condition of antisymmetrical stacks is a sufcient condition for obtaining the bendingorthotropy, which is valid only for laminates with identical layers; normally, this condition cannot be applied to laminates hav-

ing plies of variable thickness. In our calculations, we preferred not to x the orthotropy direction, because the properties thatwe were looking for were intrinsic, i.e., frame-independent. The use of the polar formalism allows one not only to x a frame,for instance, by imposing a given value of the polar angle

1, but also to do completely abstract from the frame whenever

intrinsic properties are sought for independently of any frame. Of course, a postprocessing operation of frame rotation, like

that described above, can always been done if one wishes to have the nal result in a particular frame.

TABLE 3. Polar Parameters for the Laminate in Case n.1

Elastic properties Tensor A* TensorB* TensorD*

At a constant ply thickness

T0, MPa 26.8804311 0 26.8804311

T1, MPa 24.7438933 0 24.7438933

R0, MPa 17.0334619 0.0195909 18.7724089R

1, MPa 20.6832749 0.00283 21.1735205

0, deg 7.00 - 7.00

1, deg 7.00 - 7.00

0

1, deg 0.00 - 0.00

At a variable ply thickness

T0, MPa 26.8804311 0 26.8804311

T1, MPa 24.7438933 0 24.7438933

R0, MPa 19.4379543 0.0025019 19.5838459

R1, MPa 21.3588283 0.0009081 21.3986655

0, deg 7.32 - 7.32

1, deg 7.32 - 7.32

0

1, deg 0.00 - 0.00


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384

Figures 13-20 show the directional plots of some elastic properties of laminates for cases n. 1-n. 4; for the sake of clarity

and brevity, not all elastic properties have been illustrated, but those presented here are sufcient to show that the propertiesprescribed have really been obtained.

TABLE 4.Polar Parameters for the Laminate in Case n.2



T0, MPa 26.804311 0 26.8804311

T1, MPa 24.7438933 0 24.7438933

R0, MPa 4.8733071 1.4079761 1.1229394R

1, MPa 13.0025339 0.1135579 14.6267514

0, deg 7.50 - 7.50

1, deg 37.50 - 37.50

0

1, deg 45.00 - 45.00


T0, MPa 26.8804311 0 26.8804311

T1, MPa 24.7438933 0 24.7438933

R0, MPa 4.0359329 1.3458389 1.0298979

R1, MPa 13.4207539 0.1588238 14.6732203

0, deg 0.50 - 0.50

1, deg 45.50 - 45.50

0

1, deg 45.00 - 45.00




T0, MPa 26.8804311 0 26.8804311

T1, MPa 24.7438933 0 24.7438933

R0

, MPa 0.1010911 0.5360837 0.1209914

R1, MPa 0.0093146 0.0907468 0.0636677

0, deg - - -

1, deg - - -

0

1, deg - - -


T0, MPa 26.8804311 0 26.8804311

T1, MPa 24.7438933 0 24.7438933

R0, MPa 0.0454948 0.2537272 0.0304769

R1, MPa 0.0293042 0.0465314 0.0068212

0

, deg - - -

1, deg - - -

0

1, deg - - -


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385

A detailed discussion of the results is presented below only for the laminates in cases n. 2 and n. 3, but similar con-

siderations can also be repeated verbatim in the other cases.

One can consider rst the requirement expressed by case n.2. From Table 4, it is seen that the laminate solution, bothat constant and variable thicknesses of plies, respects the design criteria (such verications are particularly simple in the polarformalism, which is frame-independent, see Table 1):

1. in-plane orthotropy with KA

*

= 1 :

a) plies of identical thickness, 0 1 7 50 37 50 45 00A A

* *

. . . = ( ) = ;

b) plies of nonidentical thickness, 0 1

0 50 45 50 45 00A A* *

. . . = ( ) = ;

2. bending orthotropy with KD*

= 1 :

a) plies of identical thickness, 0 1

7 50 37 50 45 00D D

* *

. . . = ( ) = ;

b) plies of nonidentical thickness, 0 1 0 50 45 50 45 00D D

* *

. . . = ( ) = ;

3. elastic uncoupling expressed by the polar condition B*

= 0 . The norm of tensor the B* is negligible compared

with that of the tensorA* orD*:

a) plies of identical thickness,B

A

*

*.= 0 0270 ;

b) plies of nonidentical thickness,B

A

*

*.= 0 0260 ;

4. coincidence of orthotropy axes, polar condition 1 1A D

* *

= :

a) plies of identical thickness, 1 1 37 50A D

* *

.= = ;

b) plies of nonidentical thickness, 1 1 45 50A D

* *

.= = .




T0, MPa 26.8804311 0 26.8804311

T1, MPa 24.7438933 0 24.7438933

R0, MPa 18.9297208 0.0064786 18.9216397R

1, MPa 21.2193737 0.0008899 21.2174539

0, deg 22.15 - 22.14

1, deg 67.85 - 67.85

0

1, deg 90.00 - -89.99


T0, MPa 26.8804311 0 26.8804311

T1

MPa 24.7438933 0 24.7438933

R0, MPa 18.0907546 0.0444380 18.0881731

R1, MPa 20.9823915 0.0192386 20.9836104

0, deg 10.05 - 10.05

1, deg 10.08 - 10.08

0

1, deg 0.03 - 0.03


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386

For case n. 2, it is important to note that, concerning laminates with identical layer thickness, the laminate which sat-

ises all requirements at a minimum number of plies involvs 6 layers, see Table 2. When the ply thickness becomes a designvariable, the solution found has many improvements, as shown in Table 3. First, the minimum number of layers decreases from

6 to 4, and this solution has a lower value of the residual, i.e., the laminate satises the requirements in a more satisfactoryway. Figures 15 and 16 show the polar diagrams of elastic properties of the laminate in this case.

From Table 5, it is seen that the laminate solution of problem n.3, both at constant and variable thicknesses of plies,

respects the design criteria:1. in-plane isotropy, expressed by the polar condition that the anisotropic part of the tensor A* must be zero, i.e.,

R RA A

0

2

1

24 0

* *

+ = . The ratio between the anisotropic and isotropic parts of the tensor is very close to zero, i.e., the anisotropic

components are negligible compared with the isotropic ones:

a) plies of identical thickness,R R

T T

A A

A A

0

2

1

2

0

2

1

2

4

2

0 0023

* *

* *.

+

+

= ;

A*11, MPa

D*11, MPa

B*11, MPa

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

180

150

210

120

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

ED11, MPa

GD12, MPaG

A12, MPa

EA11, MPa

Fig. 13. Polar variations for laminate n.1 at an identical ply thickness. (a) stiffness components, (b)

membrane Youngs modulusE11

and shear modulus G12

and (c) bending Youngs modulusE11

and

shear modulus G12

.

A*11, MPa

D*11, MPa

B*11, MPa

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

EA11, MPa

GA12, MPa

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

ED11, MPa

GD12, MPa

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

Fig. 14. The same for laminate n. 1 at a nonidentical ply thickness.


19/22

387

b) plies of nonidentical thicknessR R

T T

A A

A A

0

2

1

2

0

2

1

2

4

2

0 0017

* *

* *.

+

+

= ;

2. bending isotropy, expressed by the polar condition that the anisotropic part of the tensorD* must be zero, i.e.,

R RD D

0

2

1

24 0

* *

+ = . The ratio between the anisotropic and isotropic parts of the tensor is very close to zero, i.e., the anisotropic

components are negligible compared to the isotropic ones:

a) plies of identical thickness,R R

T T

D D

D D

0

2

1

2

0

2

1

2

4

2

0 0040

* *

* *.

+

+

= ;

b) plies of nonidentical thickness,R R

T T

D D

D D

0

2

1

2

0

2

1

2

4

2

0 0007

* *

* *.

+

+

= ;

150

210

120

180

240

9060

30

0

270300

330

1 0,0000

50,000

150,000

A*11, MPa

D*11, MPa

B*11, MPa

ED11, MPa

GD12, MPa

150

210

120

180

240

9060

30

0

270300

330

1 0,0000

50,000

150,000

EA11, MPa

GA12, MPa

150

210

120

180

240

9060

30

0

270300

330

80,000

60,000

100,000

40,000

20,000

Fig. 15. Polar diagrams for laminate n.2 at an identical ply thickness.

150

210

120

180

240

90

60

30

0

270300

330

150,000

100,000

50,000

A*11, MPa

D*11, MPa

B*11, MPa

EA11, MPa

GA12, MPa

150

210

120

180

240

90

60

30

0

270300

330

150,000

100,000

50,000

ED11, MPa

GD12, MPa

150

210

120

180

240

90

60

30

0

270300

330

150,000

100,000

50,000

Fig. 16. The same for laminate n. 2 at a nonidentical ply thickness.


20/22

388

3. the elastic uncoupling is expressed by the polar condition B* = 0 . The norm of the tensorB* is negligible com-

pared to that of the tensorA* orD*:

a) plies of identical thickness,B

A

*

*.= 0 0130 ;

b) plies of nonidentical thickness,

B

A

*

*

.= 0 0060.

In this case, concerning the laminate with identical layer thickness, the composite that satises all requirements at aminimum number of plies can be made from 13 layers, see Table 2. When the ply thickness becomes a design variable, the

best solution is really improved, as shown in Table 2: the minimum number of layers to obtain a solution decreases from 13

to 10, and the value of the residual is smaller, i.e., the laminate satises the requirements more accurately. Figures 17 and 18show the polar diagrams of elastic properties of the laminate in this case.

150

210

120

180

240

9060

30

0

270300

330

60,000

40,000

20,000

80,000

A*11, MPa

D*11, MPa

B*11, MPa

EA11, MPa

GA12, MPa

150

210

120

180

240

9060

30

0

270300

330

60,000

40,000

20,000

80,000

ED11, MPa

GD12, MPa

150

210

120

180

240

9060

30

0

270300

330

60,000

40,000

20,000

80,000


150

210

120

180

240

90

60

30

0

270300

330

60,000

40,000

20,000

80,000

A*11, MPa

D*11, MPa

B*11, MPa

EA11, MPa

GA12, MPa

150

210

120

180

240

90

60

30

0

270300

330

60,000

40,000

20,000

80,000

ED11, MPa

GD12, MPa

150

210

120

180

240

90

60

30

0

270300

330

60,000

40,000

20,000

80,000

Fig. 18. The same for laminate n.3 at a nonidentical ply thickness.


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389

7. Conclusions

The problem of determining the minimum number of layers ensuring some given elastic properties of a laminate has

been addressed. The approach proposed in the paper to solve this problem is to reduce it to the classical unconstrained optimiza-

tion problem of search for the minima of a semidenite positive function. This method is based on the so-called polar-geneticapproach and is totally general, i.e., no simplifying assumptions are required. The problem formulation is based on the polar

representation of plane tensors, while the numerical strategy for the search for solutions is a genetic algorithm, BIANCA,which has been modied to include the number of layers among the variables. The numerical results presented in this paper,which are completely new and nonclassical, show the effectiveness of the method proposed.

Aknowledgements. The rst author was supported by the National Research Fund (FNR) in Luxembourg throughAides la Formation Recherche Grant (PHD-09-139).

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

A*11, MPa

D*11, MPa

B*11, MPa

EA11, MPa

GA12, MPa

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000

ED11, MPa

GD12, MPa

150

210

120

180

240

9060

30

0

270300

330

150,000

100,000

50,000

200,000


150

210

120

180

240

90

60

30

0

270300

330

150,000

100,000

50,000

200,000

ED11, MPa

GD12, MPa

EA11, MPa

GA12, MPa

150

210

120

180

240

90

60

30

0

270300

330

150,000

100,000

50,000

200,000

A*11, MPa

D*11, MPa

B*11, MPa

150

210

120

180

240

90

60

30

0

270300

330

150,000

100,000

50,000

200,000

Fig. 20. The same for laminate n.4 at a nonidentical ply thickness.


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390

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2012 - M Montemurro - Design of the Elastic Properties of Laminates With a Minimum NUnber of Plies (MCM)