# The polar method in optimal design of laminates

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The polar method in optimal design of laminates
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).

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.

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

Recall of the Classical Lamination Theory
The Classical Lamination Theory provides the constitutive law for a thin laminate under extension and bending actions: z p h/2 zk-1 zk 1 k -1 -k -p n=2p n=2p

Recall of the Classical Lamination Theory
The normalized tensors are also useful: 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,

Recall of the Classical Lamination Theory

Recall of the Classical Lamination Theory

Recall of the Classical Lamination Theory
The coefficients bk, ck and dk are It is of some importance to remark that the bk's are odd, while the ck's and dk's are even.

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 The uncoupling problem is ruled by only the two equations at left.

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):

Recall of the Classical Lamination Theory
with the normalized thermo-elastic tensors given by The thermo-elastic homogeneity tensor can be also introduced:

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 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.

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 R1-orthotropic layers (also different) it will be automatically R0- or R1-orthotropic, 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).

Some exact solutions to simple design problems
6-layers designed to be quasi-homogeneous, composed by R1-orthotropic layers (complete solution found numerically).

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 quasi-homogeneous 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 quasi-homogeneity (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

Some exact solutions to simple design problems
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.

Some exact solutions to simple design problems
An example: an 18-layers laminate: q-h, q-t solution (unsymmetric!) 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.

Some exact solutions to simple design problems
For the second question, some applications of q-t solutions, among the possible ones, are shown here.

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!

Some exact solutions to simple design problems
If layers form an anti-symmetric sequence, i.e. if 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 anti-symmetric laminates, reduce to 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.

Some exact solutions to simple design problems
The figure shows the geometrical locus of the anti-symmetric solutions in the space (d2, d3, d4) for 9-ply laminates; in this case the previous equation becomes Some plane sections of the surface in the figure aside: a) Planes d 4 = 0° and 90° ; b) Plane d 4 = 30° ; c) Plane d 4 = 45°

Some exact solutions to simple design problems
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)

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 with Nx Ny y x b a

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: d and –d). So, we have a problem with only one unknown, d. In such a case the previous equation simplifies to lmin(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 d; this is determined as the solution of a 2nd degree equation, hence in closed form.

Some exact solutions to simple design problems
The figure shows the case of a plate with a/b= 2, Nx=Ny in carbon-epoxy (T0= GPa, T1= GPa, R0= GPa, R1= GPa, K=0 ). d dopt =70.65°

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; quasi-homogeneity 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.

Some exact solutions to simple design problems
Number of plies Orientations Stacking sequence 18 0= –60° 1= 0° 2= 60° 24 0= –45° 2= 45° 3= 90° 27 30 0= 0° 1= 72° 2= 144° 3= 216° 4= 288°

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.

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: 2 L = 120 mm h a b 16 plies: [-a/a2/-a/a/-a2/a]s 26 plies: [0/a/-a/02/-a/0/a/02/a/-a/0]s

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 a and –a  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 quasi-homogeneous 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.

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:

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) So, the problem is reduced to posing ax(q)=0, and this leads to if t1= r1, ax=0 for if t1<r1, ax=0 for and ax<0 between these two directions; if t1= 0, ax=0 for the two orthogonal directions isochoric piezo-electric in-plane response; if r1= 0, ax= ay and as=0 : isotropic piezo-electric in-plane response.

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: An example: T300/5208 carbon-epoxy layers and PZT-4 actuators. q1 q2 qmax p/4 h q Solutions are in q1 ( 14,14°)≤q≤ q2 ( 39.05°). For q=qmax ( 28.54°), h= hmax= So, if q=qmax the highest quantity of piezoelectric layers can be used (the 14.6% in thickness) to maximize the piezoelectric action.

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.

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 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

A general statement for the optimal design of laminates
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.

A general statement for the optimal design of laminates
C= O B= O 2

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.

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: Here, x is the vector of design variables (orientations, thicknesses etc.). find x such that f(x)= min with I[Pk(x)]=0 and H corresponding to the desired elastic symmetries

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: 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 multi-purpose optimization problems find x such that f(x)+ l I [Pk(x)]= min with H corresponding to the desired elastic symmetries

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 non-convex, highly multimodal and with non-isolated 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.

Numerical strategy for the search of solutions
Input : Fitness operator Selection operator Cross-over population of n individuals no Output: New generation best individual, mean fitness of the population yes Mutation Stop condition Elitism 111001 100011 2 parents 2 children point of cross-over Original gene Mutated gene position of mutation 1 Cross-over Mutation

Numerical strategy for the search of solutions
We have made a genetic algorithm specially conceived for laminate problems, BIANCA (BIo ANalyse de Composites Assemblés). Characteristics haploid structure multi-chromosome elitism gene-based cross-over Boolean operators

Numerical strategy for the search of solutions
Some examples. A 12-plies designed to be quasi-homogeneous and square symmetric. Global objective Square symmetry B= O C= O 10-8 10-6 10-4 10-2 1 2000 4000 f p Solution Orientations (°) I (Pk) residual BIANCA [0/62.46/ - 53.44/81.56/ 15.80/ 75.75/66.59/0/ 0.54/46.07/ 28.12/ 88.94] 2.27 x 10 5 approximated [0/62/ 53/82/ 16/ 76/67/0/ 1/46/ 28/ 89] 7.84 [0/ / / / / / /1.0953/ 2.5155/ / / ] Gradient 52/83/ 18/ 78/65/1/ 2 /45/ 30/90] 8.56 1.09 13

Numerical strategy for the search of solutions
In the table, some results obtained by the code BIANCA.

Numerical strategy for the search of solutions
An example of practical design (constrained optimization): a 12-plies laminate made of carbone-epoxy T , designed to have B=O, A orthotropic and such that: Emmax≥100 GPa (0.55 E1); Emmin≥40 GPa (3.88 E2); orientations d{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°]. 10 20 30 40 50 generation E max min A 100 150 GPa Am Emmax Emmin Ef(q) Em(q)

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] Elastic properties In-plane, tensor A Bending, tensor D Coupling, tensor B (MPa) 27218 27805 25240 26100 3550 6781 224 122 138 564 (°) 16 -1 = Piezoelastic criteria In-plane, tensor a Bending, tensor d (V-1) 1.88·10-7 1.25·10-8 1.97·10-9 1.47·10-10

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 24743 97 5370 427 243 11227 119 (°) = -18.27 -18.19 Thermal properties In-plane, tensor u Bending, tensor w (°C-1) 1.56·10-6 2.74·10-6 3.19·10-8 2.58·10-6 71.5

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] E Average in-plane Young's modulus Gxy generation

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

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.

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…!).

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

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.

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.

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