Transcript Diapositive 1

Università di Pisa
Facoltà di Ingegneria
13 luglio 2007
The polar method in
optimal design of laminates
P. Vannucci
UVSQ - Université de Versailles et Saint-Quentin-en-Yvelines
Foreword

This seminar deals with some results obtained in optimal design of
laminates by the use of the polar method.

The advantages given by the polar method in this field are
essentially the fact that the rotation formulae are expressed in a
simple way and that the material characteristics appear through
invariants expressing the elastic symmetries.

For these reasons, the polar method has proven to be rather
effective in all those problems concerning the elastic design of a
laminate.

The originality of these researches consists in having considered the
design of the elastic symmetries as a part of the design phase.

This is usually discarded by other authors, who search the optimal
solution in a class of laminates automatically giving some desired
elastic symmetries (for instance balanced and symmetric
sequences).
2
Foreword

Unfortunately, this classical approach tightens so much the design
space that almost every time the solutions so found are not true
optimal solutions.

Our approach can be distinguished into three phases:



research of as much as possible exact solutions;
research of a general formulation for the optimal design of laminates;
research of a numerical strategy for the search of the solutions.

This presentation will briefly show these phases in the order.

All what will be said concerns laminate made of identical plies; this
is a necessary assumption to have general solutions.
3
Content

Recall of the Classical Lamination Theory

Some exact solutions to simple design problems

A general statement for the optimal design of laminates

Numerical strategy for the search of solutions

Conclusions and perspectives

An unconventional historical note
4
Recall of the Classical Lamination Theory

The Classical Lamination Theory provides the constitutive law for a
thin laminate under extension and bending actions:
z
z
p
h/2
p
k
1
zk
h/2
k
zk-1
1
0
h/2
-1
zk
-1
h/2
-k
-p
-k
-p
n=2p+1
zk-1
n=2p
 N   A B  ε 
 
 ,

M B D  χ 
5
Recall of the Classical Lamination Theory
1 p
A,B,D  k  - p Q k ( k ) ( zkm - zkm-1 ),
m
m  1 for A, 2 for B, 3 for D.

The normalized tensors are also useful:
A*  A / h, B*  2B / h 2 , D*  12D / h 3 .

A laminate is said uncoupled if B=O and quasi-homogeneous if, in
addition, also the homogeneity tensor
C=A*-D*=O.

When translated in polar form, the previous formulae give, in the
case of n identical plies,
6
Recall of the Classical Lamination Theory
tensor A* : T0  T0 ,
T1  T1,
R0 4i 0 p
4 i k
e
e
,

k

p
n
R
p
 1 e 2i1 k  - p e 2i k ;
n
R0 e 4i 0 
R1 e 2i1
tensor B* : Tˆ0  0,
Tˆ1  0,
R0 4i 0 p
4 iˆ 0
4 i k
ˆ
R0 e
 2 e
b
e
,

k - p k
n
R1 2i1 p
2iˆ1
2i k
ˆ
R1 e
 2 e
b
e
;

k - p k
n
7
Recall of the Classical Lamination Theory

tensor C : T0  0,

T1  0,
 4i0
1
p
R0 e
 3 R0 e 4i 0 k  - p c k e 4i k ,
n
 2i1
1
p
R1 e
 3 R1 e 2i1 k  - p c k e 2i k ;
n
~
tensor D* : T0  T0 ,
~
T1  T1,
~ 4i~0 R0 4i 0 p
4 i k
R0 e
 3 e
d
e
,

k - p k
n
~ 2i~1 R1 2i1 p
2i k
R1 e
 3 e
d
e
.

k - p k
n
8
Recall of the Classical Lamination Theory

The coefficients bk, ck and dk are
2k
if n  2 p  1,

bk  
k
2k - , b0  0 if n  2 p;

k
4( p 2  p - 3k 2 )
if n  2 p  1,
ck   2
2
if n  2 p;
4 p - 3k  3 k - 1 , c 0  0


 12k 2  1
if n  2 p  1,
dk  
2
12k - 12 k  4 , d 0  0 if n  2 p.

It is of some importance to remark that the bk's are odd, while the
ck's and dk's are even.
9
Recall of the Classical Lamination Theory

It is important to notice that for laminates with identical plies, only
the anisotropic behavior can be designed: so, you have only two
polar equations for each tensor.

Quasi-homogeneous laminates are not only uncoupled, but they
show the same elastic behavior in extension and in bending in each
direction.

So, the polar equations of quasi-homogeneity are

p
2i k
b
e
k  - p k
p
b
k - p k

e 4i k  0,
 0,

p
2i k
c
e
k  - p k
p
c
k - p k
e 4i k  0,
 0.
The uncoupling problem is ruled by only the two equations at left.
10
Recall of the Classical Lamination Theory

These are 8 real equations; in fact, 4 equations, concerning the
isotropic part of B and C, are identically satisfied.

This is immediately recognized in polar, but not with Cartesian
coordinates, when 12 equations, with only 8 independent, are to be
solved.

The above equations have not, in the general case, a complete
analytical solution.

Another point deserves attention: elastic quasi-homogeneous
solutions are also thermo- and hygro-elastic quasi-homogeneous.

This is easily recognized if one considers the laminates constitutive
law considering also the thermal effects (for the moisture absorption
results are similar):
11
Recall of the Classical Lamination Theory
N   A B ε o 
U t  V 

t
  
  o   -  ,

M B D  κ 
V  h W 
with the normalized thermo-elastic tensors given by
U
R
p
: T *  T , R * e 2i  e 2i k  - p e 2i k ;
h
n
V
1
p
ˆ
V *  2 2 : Tˆ *  0, Rˆ * e 2i  2 R e 2i k  - p bk e 2iδk ;
h
n
W
1
p
~
~ 2i~
W *  12 3 : T *  T , R * e
 3 R e 2i k  - p d k e 2iδk .
h
n
U* 

The thermo-elastic homogeneity tensor can be also introduced:

 2i
1
p
Z  U* - W* : T  0, R e
 3 R e 2i k  - p c k e 2i k .
n
12
Recall of the Classical Lamination Theory

It is suddenly recognized that the conditions for thermo-elastic
quasi-homogeneity, i.e. to have a laminate that has the thermal
expansion coefficients identical in extension and bending in each
direction, is to have
VO

ZO

e 2iδk  0,

p
2i k
c
e
 0.
k  - p k
p
b
k - p k

These two complex equations are just a part of those giving elastic
quasi-homogeneity.

This means that a quasi-homogeneous laminate for the elastic
properties is also quasi-homogeneous for the thermo-elastic
behavior, but the converse is not, generally speaking, true.
13
Some exact solutions to simple design problems

Some simple problems concerning the design of laminates for
different purposes can be solved analytically. Some of them are
shown here.

Laminates composed by R0 or R1- orthotropic materials

In this case the problem is simpler, car one polar equation is
identically satisfied.

It is worth noticing also that if a laminate is composed by R0- or R1orthotropic layers (also different) it will be automatically R0- or R1orthotropic, for all the tensors.

Some complete solutions, analytical or numerical, concern
laminates designed to be uncoupled or quasi-homogeneous, with a
small number of layers (4, 5 or 6).
14
Some exact solutions to simple design problems

6-layers designed to be quasi-homogeneous, composed by R1orthotropic layers (complete solution found numerically).
e 4iδ1 - e 4iδ -1  3e 4iδ2 - 3e 4iδ - 2  5e 4iδ3 - 5e 4iδ -3  0,
 4iδ1
 4e 4iδ -1  e 4iδ2  e 4iδ - 2 - 5e 4iδ3 - 5e 4iδ -3  0;
4e
30
 -2
20
 -1
10
( 3- -3)/2
0
1
-10
-20
 3+ -3=0
-30
-15
-10
-5
0
5
2
10
15
15
Some exact solutions to simple design problems

A special class of laminates: the quasi-trivial solutions

Quasi-trivial solutions are a particular class of uncoupled or quasihomogeneous laminates.

A quasi-trivial solution has the particularity that the solution is exact
and depends only on the stacking sequence but not on the
orientations: this is rather useful when other properties (stiffness,
strength and so on) must be optimized.

Actually, though the general problem of solving the quasihomogeneity (or simply the uncoupling) equations has not a unique
analytical solution, a particular class of laminates satisfying these
equations can be found exploiting a fundamental property of the
coefficients bk and ck: their sum is null.

So, a quick glance at the quasi-homogeneity equations
16
Some exact solutions to simple design problems

p
k - p bk e 2i k
p
b
k - p k
e 4i k  0,
 0,
e 4i k

p
k - p ck e 2i k
p
c
k - p k
 0,
 0.
show that a sufficient condition to have a solution is to dispose
groups of layers with the same orientation, no matter of its value, in
such a way that the sum of the coefficients for each group is zero.

Such groups are called saturated and the solutions quasi-trivial,
because they are obtained without solving explicitly the previous
equations.

As coefficients bk and ck are integer, the solutions so found are
exact.

It is worth noting that the very well known symmetric solutions for
uncoupling are just a subset of the quasi-trivial solution to the
problem B=O.
17
Some exact solutions to simple design problems

An example: an 18-layers laminate: q-h, q-t solution (unsymmetric!)
k -9 -8
group 0 1
b k -17 -15
c k -136 -88

-7
2
-13
-46
-6
0
-11
-10
-5
1
-9
20
-4
2
-7
44
-3
2
-5
62
-2
2
-3
74
-1
1
-1
80
1
1
1
80
2
0
3
74
3
0
5
62
4
1
7
44
5
0
9
20
6
0
11
-10
7
2
13
-46
8 9
2 1
15 17
-88 -136
Two questions arise:


how much q-t solutions do exist?
when they can be useful?

The first question: just a look at the diagram below, showing the
number of q-h q-t solutions as a function of the ply number (in
brackets: the symmetric solutions).

The number of q-t is rapidly increasing with the ply number and
gives a practically unlimited quantity of different possibilities for
applications.
18
10000
5902 (3)
8000
7000
6000
5000
4000
3000
2000
1000
0
1 (1)
1
1
1
3 (2)
1
3
2 (1)
4
8 (1)
23
5
52
40
44 (2)
130 (3)
594 (1)
167
2352
1495 (7)
1282 (1)
Nombre de solutions
9000
6146 (3)
45441
Some exact solutions to simple design problems
7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

couches
For the second question,Nombre
some de
applications
of q-t solutions, among
the possible ones, are shown here.
19
Some exact solutions to simple design problems

Generally speaking, working on the set of q-t solutions of q-h type
allows the designer to look for optimal solutions of bending
properties working on the extension properties, which is much
simpler, and with a lower number of unknowns, the orientation of the
saturated groups in place of those of the layers.

This strategy can be applied to a number of different problems;
some examples are shown here.

Fully orthotropic laminates

A fully orthotropic laminate is orthotropic in extension and in bending
and is uncoupled.

Unlike extension orthotropy, rather easy to be obtained, bending
orthotropy is very difficult to be obtained, so that in most researches
a laminate is considered orthotropic in bending also if it is not!
20
Some exact solutions to simple design problems

If layers form an anti-symmetric sequence, i.e. if
 n -k 1  - k
then extension and bending orthotropy are assured, but not B=O, in
general.

A strategy consists in looking for anti-symmetric sequences that are
also uncoupled.

It is not difficult to show that uncoupling polar equations, for antisymmetric laminates, reduce to
4
 p

  bk sin 2 k  - b12
 k 2


2
 p

  bk sin 2 k   b12
 k 2

 p

  bk sin 2 k cos 2 k 
 k 2

2
 0.
This equation holds the problem of finding uncoupled anti-symmetric
orthotropic laminates. For a small number of layers, solutions can be
found analytically or completely described numerically and traced on
a graph.
21
Some exact solutions to simple design problems

The figure shows the geometrical locus of the anti-symmetric
solutions in the space (2, 3, 4) for 9-ply laminates; in this case the
previous equation becomes
(3 sin 2 2  2 sin 2 3  sin 2 4 ) 4 - 16 (3 sin 2 2  2 sin 2 3  sin 2 4 )2 
 16 (3 sin 2 2 cos 2 2  2 sin 2 3 cos 2 3  sin 2 4 cos 2 4 )2  0.
Some plane sections of the surface in
the figure aside:
a) Planes  4 = 0° and 90° ; b) Plane
 4 = 30° ; c) Plane  4 = 45°
22
Some exact solutions to simple design problems
Ply number

2
3
4


7
8
9
10
11
12
7 plies
1

-
-
0


-
/
/
/
/
/
8 plies
2

-
-

-


-
/
/
/
/
3

-
-

0
-


-
/
/
/
4

-
0
-
0

0

-
/
/
/
5
0

-
-
0


-
0
/
/
/
6

-
-

0
0
-


-
/
/
7

-
0
-

-

0

-
/
/
8

0
-
-
-



0
-
/
/
9
0

-
-

-


-
0
/
/
10

-
-

0
0
0
-


-
/
11

-
0
-

0
-

0

-
/
12

-
0
0
-
0

0
0

-
/
13

0
-
-
0
0
0


0
-
/
14
0

-
-

0
-


-
0
/
15
0

-
0
-
0

0

-
0
/
16
0
0

-
-
0


-
0
0
/
17

-

-
-
-



-

-
18

-
-

0
0
0
0
-


-
19

-
0
-

0
0
-

0

-
20

-
0
0
-

-

0
0

-
21

0
-
-
0

-
0


0
-
22

0
-
0
-
-


0

0
-
23
0

-
-

0
0
-


-
0
24
0

-
0
-

-

0

-
0
25
0

0
-
-
-



0
-
0
26
0
0

-
-

-


-
0
0
9 plies
10 plies
11 plies
12 plies

Another possibility is to
look
for
in-plane
orthotropic solutions in
the set of q-h, q-t
laminates.

The results from 7 to 12
layers are presented in
the table on the left.

These solutions can be
used, for instance, in
exact optimization of the
buckling
load
of
rectangular plates (all the
solutions in the literature
are approximated)
23
Some exact solutions to simple design problems

Let us consider the case of a rectangular simply supported plate,
orthotropic in bending, with the axes of orthotropy parallel to the
sides of the plate. If the plate is not subjected on its boundary to
tangential loads, the expression of the buckling load multiplier, in
polar form, is

 h
2 3

  cos 4 j  4R1
nT0  2T1      - 1  - 6   R0
2
K
2
2
p
2
j - p
-
2
  cos 2 j
p
j - p
,
N x   N y
12n
N
Nx y
with
2
p
    ,
a
2
q
    ,
b
p x
q y
w  c pq sin
sin
.
a
b
y
b
x
a
N
Ny x
24
Some exact solutions to simple design problems

It is well known that the optimal solution to the above problem must
be looked for in the class of the angle-ply laminates (having only two
possible orientations:  and –). So, we have a problem with only
one unknown, .

In such a case the previous equation simplifies to
2
K
 2 h 3 nT0  2T1      - 1  2 - 6   2 R0 cos 4  4R1  2 -  2 cos 2

.
12n
N x   N y

min(p,q) must be maximized. This can be done easily and exactly if
a q-h - q-t orthotropic solution is used:



an in-plane orthotropic q-h - q-t solution can be easily selected;
the laminate will be also orthotropic in bending and uncoupled, so the
only parameter to be chosen is the lamination angle ;
this is determined as the solution of a 2nd degree equation, hence in
closed form.
25
Some exact solutions to simple design problems
  2h3 

 
 12n 

The figure shows the
case of a plate with a/b=
2, Nx=Ny in carbon-epoxy
(T0= 26.88 GPa, T1=
24.74 GPa, R0= 19.71
GPa, R1= 21.43 GPa,
K=0 ).

opt
=70.65°
26
Some exact solutions to simple design problems

Fully isotropic laminates

The hard problem of finding fully isotropic laminates has been
addressed by several authors.

For what concerns exact fully isotropic solutions, a strategy is to
apply the Werren and Norris rule to q-h q-t solutions.

So, we have looked for q-h q-t laminates having at least 3 saturated
groups with an equal number of layers in each group; if the groups
are equally spaced, the solution is in-plane isotropic; quasihomogeneity ensures also fully isotropy.

In this way we have found the exact fully isotropic solutions with the
least number of layers: 5 unsymmetrical laminates with 18 layers
(before, the number of layers was 36!).

In the next table, some solutions of exact fully isotropic laminates.
27
Some exact solutions to simple design problems
Number
of plies
Orientations
Stacking sequence
18
0= –60°
1= 0°
2= 60°
0
0
0
0
0
24
0= –45°
1= 0°
2= 45°
3= 90°
0 1 2 3 2 3 1 3 0 2 0 1 0 1 3 1 2 0 2 3 2 3 0 1
0= –60°
1= 0°
2= 60°
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
2
2
1
1
2
1
2
2
2
2
2
1
2
2
2
1
2
0
2
1
2
1
2
0
0
2
1
1
1
1
2
2
2
2
2
1
2
2
2
1
1
1
1
1
1
0
0
2
0
0
0
0
1
1
1
0
0
0
2
2
0
0
1
1
2
0
0
0
0
0
1
1
0
0
0
2
2
1
1
1
2
1
0
2
2
0
0
2
0
0
1
2
0
0
0
1
2
0
0
0
2
1
1
1
2
1
2
2
2
1
2
1
1
1
2
0
0
2
2
1
0= 0°
1= 72°
2= 144°
3= 216°
4= 288°
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
3
0
3
4
3
4
3
4
4
4
4
4
0
0
0
1
4
3
2
2
4
4
2
2
2
0
0
0
0
0
0
1
1
1
1
3
1
1
3
3
1
2
4
3
4
2
2
2
1
3
2
4
4
3
4
2
2
2
3
1
2
4
2
4
2
4
4
4
1
4
4
2
4
2
4
3
1
1
4
1
1
2
1
1
3
3
3
3
2
2
4
3
3
3
1
0
0
0
0
0
0
1
1
1
1
2
2
2
4
1
3
1
3
2
3
1
4
3
3
3
3
3
0
0
0
1
1
1
2
2
1
1
2
4
2
4
3
4
1
4
2
0
0
0
0
0
0
0
0
0
2
0
0
0
0
1
3
3
1
1
0
0
3
4
1
4
4
4
3
4
0
0
2
3
2
3
1
1
2
2
1
2
4
2
4
0
0
0
0
0
3
3
1
1
3
0
0
0
0
0
4
3
0
0
0
2
2
2
4
3
0
4
1
1
3
2
2
2
4
3
3
27
30
1
1
1
1
1
2
2
2
2
1
0
2
2
2
2
1
0
0
1
2
2
1
1
0
2
2
1
1
2
0
2
2
2
0
0
1
0
0
1
2
1
1
2
1
1
0
2
1
0
0
0
0
0
2
0
1
2
1
1
1
0
0
0
2
1
0
1
2
0
1
2
0
0
0
2
2
1
2
2
2
1
2
1
1
0
4
2
2
2
1
1
1
2
2
1
2
3
4
1
4
3
4
1
4
4
1
4
3
4
3
4
3
3
1
2
28
Some exact solutions to simple design problems

Optimal design of test specimens.

Experimental testing of composite materials and laminates is not as
easy as for classical materials, because more mechanical properties
are concerned and more mechanical effects must be accounted for.

So, the optimal design of test specimens has a great importance for
obtaining good results from testing.

We have worked on two problems:



the optimal design of a unique laminate specimen for the elastic testing
of the basic layer elastic properties, by tension, bending and anticlastic
bending tests;
the optimal design of a specimen for the fracture propagation tests.
Let us briefly consider this last case: the goal of the experimental
research was to measure some delamination parameters, as a
function of the lamination angle, in the absence of parasite effects,
such as twisting-bending coupling and change of the elastic
properties in the separated sub-laminates.
29
Some exact solutions to simple design problems

Properties imposed to the specimen (of angle-ply type):






uncoupling of the entire specimen and of the sub-laminates;
same elastic properties for the entire laminate and the two sub
laminates;
same behavior in extension and in bending;
no coupling terms of the type D16 and D26;
possibility of varying the lamination angle without altering the above
properties.
This problem has been solved
using q-h q-t sequences:
b
16 plies:
[-/2/-//-2/]s
26 plies:
[0//-/02/-/0//02//-/0]s
h
h
a
30
Some exact solutions to simple design problems

These laminates have the following properties:






they are obtained as the symmetric superposition of two anti-symmetric
identical q-h q-t sequences;
they have the same number of plies at  and –  A orthotropic 
A16=A26=0
the laminate and the sub-laminates are quasi-homogeneous  B=O
and C=O  D orthotropic  D16=D26=0;
A*=D* for the laminate and the two sub-laminates, as they are quasihomogeneous and the layers volume fraction is the same;
quasi-trivial solutions  the lamination angle can be varied without
altering the above properties.
In this way the delamination parameters, such as the fracture
toughness, can be measured without parasite effects.
31
Some exact solutions to simple design problems

Laminates with null piezo-electric deformations in some
directions.

Patches of piezo-electric actuators acts in the same way in each
direction. In some cases it can be interesting to have a laminate
which, under the action of a piezo-electric actuator, has in-plane and
bending deformations null in one direction.

This problem can be solved in closed form for the in-plane strains,
and for a standard laminate:
32
Some exact solutions to simple design problems

It can be shown that the components of the in-plane piezo-electric
strains are (t1, r1 an j1 are some of the polar components of A-1)
 x (q )  4 eˆ3 [t1  r1 cos 2(j1 - q )],
 y (q )  4 eˆ3 [t1 - r1 cos 2(j1 - q )],
 s (q )  8 eˆ3 r1 sin 2(j1 - q ).

So, the problem is reduced to posing x(q)=0, and this leads to




if t1= r1, x=0 for q  j1 -

;
2
t 
 1
if t1<r1, x=0 for q  j1   - arccos 1  and x<0 between these
r1 
2 2
two directions;

if t1= 0, x=0 for the two orthogonal directions q  j1  ;
4
piezo-electric in-plane response;
isochoric
if r1= 0, x= y and s=0 q: isotropic piezo-electric in-plane response.
33
Some exact solutions to simple design problems

Let us concentrate on the second case, the most interesting.

The corresponding condition can be stated using the stiffness polar
components, and a closed-form solution can be found for the case
of an angle-ply laminate:
2hA 2R1M cos 2q - T0M - ( -1)k R0M cos 4q
0  

.
A
hM
T0


An example: T300/5208 carbon-epoxy layers and PZT-4 actuators.
Solutions are in q1 ( 14,14°)≤q≤
q2 ( 39.05°). For q=qmax (
28.54°), = max= 0.146. So, if
q=qmax the highest quantity of
piezoelectric layers can be used
(the 14.6% in thickness) to
maximize
the
piezoelectric
action.

q
0
q1
qmax
q2
/4
34
A general statement for the optimal design of laminates

Further, we have looked for a general approach to the optimal
design of laminates, where the basic elastic properties, such as
uncoupling, orthotropy and so on, are a part of the design process.

This means also that the search for a basic elastic property is seen
itself as an optimization problem.

We have worked in two steps:



in the first step, we have given a general formulation of all the design
problems of basic elastic properties and we have used this formulation
to solve some design problems of laminates;
in the second step, we have given a completely and classical unified
formulation of the optimal design of laminates with respect to a given
objective function including basic elastic properties.
In all the cases, a suitable numerical approach is needed; this will
be discussed in the next section.
35
A general statement for the optimal design of laminates

In the framework of the polar method, it is possible to give a unified
formulation of all the problems concerning the design of laminates
with respect to their basic elastic properties.

To this purpose, let us introduce the quadratic form of R18
I(Pk )  P  H P  Hij Pi Pj ,

i , j  1,...,18, H  HT ,
P is the vector of all the polar parameters of the laminate (for A*, B*
and D*), divided by a given factor M to work with non-dimensional
quantities, for instance
M
1 n
i 1 T02 i  2T12 i  R02 i  4R12 i .
n
36
A general statement for the optimal design of laminates
T0
R
T
R
, P2  1 , P3  0 , P4  1 , P5   0 , P6   1,
hM
hM
hM
hM
2 Tˆ0
2 Rˆ 0
2 Tˆ1
2 Rˆ 1
P7  2 , P8  2 , P9  2 , P10  2 , P11  ˆ 0 , P12  ˆ1,
h M
h M
h M
h M
~
~
~
~
12 T0
12 R0
12 T1
12 R1
P13  3 , P14  3 , P15  3
, P16  3 , P17  ~0 , P18  ~1,
h M
h M
h M
h M
P1 

The solutions are the minima of the quadratic form I(Pk).

The advantage is that the value of these minima is known a priori
(usually it is zero)

The choice of the matrix H determines the problem to be solved.
37
A general statement for the optimal design of laminates
B= O
i 7 Pi2  0
10
C= O

6
i 1
Pi - Pi 12 2  0
R1  0
P42  0
1

1

1


1

2
1

A-A
A-B

1


1

1


1

1



B-B
B-A

-1

 -1

-1

-1


-1

D-B
D-A
-1

-1


-1

-1


-1

-1 
A-D
- 1









B-D


1

1


1

1


1

D-D
1 
38
A general statement for the optimal design of laminates

A general, unified and totally free from simplifying assumptions
formulation for the optimal design of laminates with respect to
current and important objectives (minimum weight, highest stiffness
and/or strength, highest buckling load and so on) can also be
obtained in the framework of the polar method.

In this case, the previous general approach to the basic elastic
properties must be integrated into a more general formulation of an
optimum problem, becoming in this framework a constraint
condition.

What is new, is the fact that in this way a completely general
approach to the optimum design of laminates can be obtained.
39
A general statement for the optimal design of laminates

In fact, let us suppose that a laminate must be designed to minimise
a certain objective function f, but with some basic elastic properties
to be respected, e.g. uncoupling and membrane orthotropy.

Then, the problem can be stated as follows:
find x such that f(x)= min
with
I[Pk(x)]=0
and H corresponding to the
desired elastic symmetries

Here, x is the vector of design variables (orientations, thicknesses
etc.).
40
A general statement for the optimal design of laminates

By the technique of Lagrange multipliers, this problem can be put in
the form of an unconstrained optimization problem:
find x such that f(x)+  I [Pk(x)]= min
with H corresponding to the desired elastic symmetries

This is the most general form of the mono-objective design problem
of a laminate.

The challenge, in this case, concerns much more the construction of
an effective algorithm for the solution of hard constrained and multipurpose optimization problems
41
Numerical strategy for the search of solutions

The search of the solutions is a delicate
point: generally speaking the problems
formulated in the previous section are nonconvex, highly multimodal and with nonisolated minima.

In addition, the number of the design
variables is often rather great (at least n-1,
n being the number of layers).

Also, the design variables can be of different type: continuous,
discrete, grouped (i.e., representing more than one quantity).

For these reasons, we decided to use a genetic algorithm (Holland,
1965).

The general structure of a classical genetic algorithm is sketched in
the following figure.
42
Numerical strategy for the search of solutions
Input :
population of n
individuals
Fitness
operator
Selection
operator
Crossover
New
generation
Mutation
no
Output:
best individual,
mean fitness of
the population
yes
Stop condition
Elitism
Cross-over
1000110111001001
111001 0111001001
1110010101100010
100011 0101100010
point of cross-over
2 parents
Mutation
1110010101100010
0
position of mutation
Original gene
2 children
1110110101100010
1
Mutated gene
43
Numerical strategy for the search of solutions
We have made a genetic algorithm specially conceived for laminate
problems, BIANCA (BIo ANalyse de Composites Assemblés).
k
4
3
2
1

Characteristics





haploid structure
multi-chromosome
elitism
gene-based cross-over
Boolean operators
chr. n
chr. k
chr. 4
chr. 3
chr. 2
chr. 1
chromosome k
n layers
1
0
0
1
0
n
genome with n chromosomes

gene of
the
material
1
0
1
1
1
gene of the
orientation
1
0
0
1
0
1
0
0
6 genes of
components
44
Numerical strategy for the search of solutions

Some examples.

A 12-plies designed to be quasihomogeneous
and
square
symmetric.
1
f
Global objective
Square symmetry
B= O
C= O
10-2
EA and ED
10-4
10-6
10-8
0
Solution
BIANCA
BIANCA
approximated
2000
p
4000
Orientations
(°)
[0/62.46/- 53.44/81.56/-15.80/- 75.75/66.59/0/- 0.54/46.07/-28.12/-88.94]
I (Pk) residual
-5
2.27 x 10
-5
[0/62/-53/82/-16/-76/67/0/-1/46/-28/-89]
7.84 x 10
Gradient
[0/61.7640/- 52.1221/82.6706/- 18.2096/-78.3146/
64.6143/1.0953/- 2.5155/44.6293/- 29.8974/-89.6532]
1.09 x 10
Gradient
approximated
[0/62/-52/83/-18/-78/65/1/-2 /45/-30/90]
8.56 x 10
-13
-5
45
Numerical strategy for the search of solutions

In the table, some results obtained by the code BIANCA.
Numéro du
problème
Type de problème
Type de
couche
Ensemble de
définition des
orientations
n
N
p
t
f
Solution (°)
1
23
Découplage
R1=0
]-45°, 45°]
5
200
200
7
0
[4.5735/0/0/1.5364/6.1092]
2
28
]-90°, 90°]
10 200
500
5
2.53 x 10-5
[0/-17.16/-2.69/5.05/-14.60/-5.97/
-14.45/5.34/1.65/-12.95]
3
28
Orthotropie K=1 pour A et
K=0 pour D, avec axes
coïncidents et découplage
quelconque
]-90°, 90°]
12 200
500
5
4.74 x 10-5
[0/44.67/15.70/-39.98/-25.46/
-37.21/59.04/54.28/36.92/-38.16/18.58/-5.23]
4
28
Orthotropie K=1 pour A et
K=0 pour D, avec axes
coïncidents et découplage
quelconque
]-90°, 90°],
pangle=10°
12 200
300
3
5.08 x 10-4
[0/10/40/-40/-20/50/-20/30/-40/30/10/-10]
5 13+14+27
Isotropie de A, orthotropie
K=0 pour D, découplage
quelconque
]-90°, 90°]
12 200
500
7
9.40 x 10-6
[0/75.99/-31.45/-67.48/37.97/31.97/-38.49/
-76.87/57.69/89.31/14.15/-23.88]
6 13+14+27
Isotropie de A, orthotropie
K=0 pour D, découplage
quelconque
]-90°, 90°],
pangle=10°
12 200
500
7
1.13 x 10-4
[0/60/70/10/-60/-50/-60/-50/60/0/10/70]
2.27 x 10-5
[0/62.46/-53.44/81.56/-15.80/-75.75/66.59/0/
-0.54/46.07/-28.12/-88.94]
Orthotropie K=0 pour A et D,
avec axes coïncidents et
quelconque
découplage
7
24
quasi-homogénéité avec
symétrie carrée
quelconque
]-90°, 90°]
12 1000 4000 70
8
18
isotropie totale
quelconque
]-90°, 90°]
12 2000 3000 100 3.46 x 10-4
[0/51.58/-51.49/85.83/-51.34/85.04/24.09/
-19.08/30.94/-11.16/63.28/-65.21]
46
Numerical strategy for the search of solutions

An example of practical design (constrained optimization): a 12plies laminate made of carbone-epoxy T300-5208, designed to have
B=O, A orthotropic and such that:
Emmax≥100 GPa (0.55 E1);
Emmin≥40 GPa (3.88 E2);
orientations {0°, 15°, 30°, 45°, 60°, 75°, 90° etc.}.

A solution found by BIANCA
[0°/30°/–15°/15°/90°/–75°/0°/45°/–75°/0°/–15°/15°].
80
150
GPa
Ef(q)
100
0
Am
A
EEmmax
max
50
Em(q)
A
m
Emin
E
min
0
0
10
20
generation
30
40
50
-80
-160
-80
0
80
160
47
Numerical strategy for the search of solutions

A 12-plies T300/5208 carbone-epoxy laminate designed to have
R1=0 in extension and bending, B=O and isotropic piezoelectric
response (PZT4 actuators).

Solution found by BIANCA:
[0/90/44.98/-41.80/-74.53/40.47/0/-71.92/34.36/-45/-1.86/85.08]
Variation des rigidités pour la membrane, la flexion et le couplage
100000
D11
A11
In-plane, tensor
A
Bending, tensor
D
Coupling, tensor B
(MPa)
27218
27805
0
(MPa)
25240
26100
0
(MPa)
3550
6781
224
(MPa)
122
138
564
Elastic properties
(°)
16
-1
=
(°)
=
0
=
Piezoelastic criteria
In-plane, tensor a
Bending, tensor d
(V-1)
1.88·10-7
1.25·10-8
(V-1)
1.97·10-9
1.47·10-10
Direction 90° (GPa)
50000
B11
0
-50000
-100000
-100000
-50000 des rigidités
0 pour la membrane,
50000 la flexion
100000
Variation
et le couplage
Direction 0° (GPa)
0.0000002
1
Direction 90° (GPa)
0.0000001
1
0
-0.0000001
-0.0000002
-0.0000002
-0.0000001
6 = 0 et 6 = 0
0
Direction 0° (GPa)
0.0000001
0.0000002
48
Numerical strategy for the search of solutions

A 12-plies T300/5208 carbone-epoxy laminate designed to be
isotropic in extension, K=0 orthotropic in bending, B=O, with
isotropic in-plane thermo-elastic response and one direction of zero
bending thermal coefficient due to a temperature gradient through
the thickness.

Solution found by BIANCA:
[0/-29.97/44.3/-61.88/89.3/61.83/
31.56/-89.12/33.4/-71.72/-11.6/-28.13]
Elastic properties
In-plane, tensor A
Bending, tensor D
Coupling, tensor B
(MPa)
26880
26880
0
(MPa)
24743
24743
0
(MPa)
97
5370
427
(MPa)
243
11227
119
(°)
=
-18.27
=
(°)
=
-18.19
=
Thermal properties
In-plane, tensor u
Thermal
properties
Bending, tensor w
(°C-1)
1.56·10-6
(°C-1)
2.74·10-6
(°C-1)
3.19·10-8
(°C-1)
2.58·10-6
(°)
=
(°)
71.5
49
Numerical strategy for the search of solutions

A 10-plies T300/5208 carbon-epoxy laminate designed to maximize
the Young's modulus along an orthotropy axis, K=1 in-plane
orthotropic and uncoupled.

Solution found by BIANCA (residual: 4.6210-4):
[0./54.50/-44.67/87.19/-33.75/87.19/26.90/16.60/78.75/-31.31]
Average In-plane Young modulus vs. Generations
Modules d'Young et de cisaillement pour le stratifié non end
membrane
70 000
80000
65 000
60000
E
60 000
Average in-plane
Young's modulus
50 000
45 000
20000
Direction 2
Average In-plane Young modulus
40000
55 000
40 000
0
Gxy
-20000
35 000
-40000
30 000
-60000
25 000
generation
20 000
1
100
199
298
397
496
595
694
793
892
991
-80000
-1E+05 -80000 -60000 -40000 -20000
0
20000 40000 60000 80000 100000
Generations
Direction 1
50
Numerical strategy for the search of solutions

Some final considerations about the use of genetic algorithms in the
mechanical context.

Bio-inspired metaheuristics mark the entrance of biological laws in
various sectors of knowledge, also in hard sciences like mechanics.

This, in a sense, is the recognition that laws of the living world have
a wider validity than that they have in their own biological context.
La loi de l’évolution est la plus importante
de toutes les loi du monde; elle a présidé à
notre naissance, a régi notre passé et, dans
une large mesure, contrôle notre avenir.
Y. Coppens
51
Numerical strategy for the search of solutions

A mathematical interpretation can be given to these laws (see The
simple genetic algorithm, by M. D. Vose) but, actually, the proof of
their effectiveness is a matter of fact.

Genetic algorithms are able to well manage complexity; in the
treatment of some inverse problems, the organization and
management of complexity are, sometimes, a way to success.

The basic question is: when to simplify is the good choice?

Actually, in the nature, it is complexity which dominates biological
systems (sexual reproduction, diploids and dominance, redundancy
in the stocking of genetic data etc.).

In a sense, it is just the way we have followed with BIANCA, which
is a genetic algorithm completely different from those used in
laminates optimization.
52
Conclusions and perspectives

The use of the polar method has proven to be rather effective in
some laminate design problems.

When coupled with a genetic algorithm, some hard problems can be
solved with a sufficient accuracy.

A further step will be the inclusion of the ply number among the
design variables. This will allow for weight optimization.

We believe that this can be done by some special genetic
operations, i.e., we think that there must be a genetic way for the
optimal design of the ply number (still to be verified: work in
progress…!).
53
Conclusions and perspectives

A promising way of action is the use of another metaheuristic, the
PSO (Particle Swarm Optimization, Eberhart & Kennedy, 1995).

This seems to be a very effective and robust numerical method for
the solution of non convex optimization problems in Rn.

An example: the search of an in-plane isotropic 4-ply laminate,
formulated with the unified polar approach.

The algorithm finds quickly one of the Werren and Norris solutions.
fmean
step
54
An unconventional historical note

The use of bio-inspired metaheuristics in the solution of hard
numerical problems in mechanics are only the last of a long sequel
of points of contact between these two sciences, and demonstrates
once more the usefulness of transversal knowledge.

Actually, it is a little bit curious to know that at the origin of modern
mechanics the contacts with biology have been in the mind and in
the work of three great scientists.

G. Galilei, in Discorsi e dimostrazioni matematiche intorno a due
nuove scienze… (Leiden, 1638) makes some speculations of
strength of materials concerning biological structures.
55
An unconventional historical note

R. Hooke, in Micrographia
(London, 1665) is the first to
publish systematic observations
of biological tissues made by
himself with a microscopy that
he fabricated. He is the father of
the word cell, that he proposed
after the observation of the
texture of the cork.

P. L. M. de Maupertuis, in Venus Physica (Paris, 1745) rigorously
demonstrates the genetic transmission of characters from the father
and the mother. In De universali naturae systematae (Erlangen,
1751) he is the first to make the hypothesis that mutation is a cause
of biodiversity. He published also some papers about his naturalistic
observations.
Thank you very much for your attention.
56