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Analysis and optimisation of multistable composites under residual stresses
Composite Structures 55 (2002) 319±327
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Analysis and optimisation of multistable composites under residual stresses W. Hufenbach, M. Gude
*
Institut fur Leichtbau und Kunststotechnik (ILK), Technische Universitat Dresden, 01062 Dresden, Germany
Abstract During the manufacturing process of multilayered ®bre-reinforced composites with variable ®bre orientations, residual stresses build up due to the directional expansion of the unidirectionally reinforced single layers. Dependent on the laminate lay-up, these inhomogeneous residual stresses, which are caused by thermal eects, moisture absorption and chemical shrinkage, can lead to large multistable out-of-plane deformations in case of unsymmetric laminates. Instead of avoiding these laminate's curvatures, they can be advantageously used for technical applications such as near-net-shape technologies or adaptive structures. In order to adjust the deformations to technical requirements, a dimensioning tool consisting of an advanced stability and failure analysis in combination with a novel optimisation procedure has been developed. This tool, based on a non-linear calculation method, enables the purposeful adaption of the laminate lay-up dependent on the loading and process parameters and is furthermore the basis for the design of novel adaptive structures. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Multistable composites; Residual stresses; Failure analysis; Optimisation; Adaptive structures
1. Introduction In multilayered ®bre-reinforced composites with variable ®bre orientations, the directional expansion of the unidirectional-reinforced (UD) single layers due to thermal eects, moisture absorption and chemical shrinkage leads to a discontinuous residual stress ®eld over the laminate thickness. In case of unsymmetric laminate plates, these residual stresses can cause dierent multistable out-of-plane deformations. This paper focuses on the purposeful adaption of the residual stresses dependent on the manufacturing process in order to realise either laminates with de®ned (multistable) deformation states or to design novel adaptive structures. For the adjustment of laminate curvatures to technical requirements, new optimisation methods using genetic algorithms are developed, that can eciently be applied to ®nd an optimal laminate lay-up dependent on material properties, loading conditions and process parameters. Furthermore, a realistic failure analysis must be carried out in order to assess the resulting residual stress states with respect to ®rst-ply failure.
*
Corresponding author. Tel.: +49-351-463-38153; fax: +49-351-46338143. E-mail address: [email protected] (M. Gude).
The new laminate design method has successfully been veri®ed by experiments and numerical calculations on unsymmetric glass- (GFRP) and carbon-®bre-reinforced plastics (CFRP) and was applied to the design of multilayered curved hybrid structures. Some earlier papers deal with the calculation of the purely thermal out-of-plane deformations with the help of extended non-linear displacement±strain relations (e.g. [1±4]). However, the important in¯uence of the chemical shrinkage and moisture absorption on the laminate deformation is not considered, although its contribution to residual stresses can be higher than the thermal contribution. Therefore the in¯uence of shrinkage and moisture cannot be neglected in advanced design processes [5±7]. Considering extended stress± displacement relations, the principle of minimising the elastic potential in combination with the Rayleigh±Ritz method results in a system of non-linear equations for the calculation of the multistable laminate deformations. Dependent on the size and lay-up of the unsymmetric laminate, the equation system leads to dierent solutions of stable and unstable deformation states like saddle shapes or cylindrical shapes [2±5]. Most of the papers are further restricted to describing the deformation states without considering the resulting stresses, even though the residual stresses can induce a premature failure. Only some work has been done to
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 5 3 - 2
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calculate the residual stress states and to assess the stresses with respect to fracture by conventional failure criteria [8,9]. A failure analysis of multistable composites using realistic, physically based failure criteria has not yet been carried out.
2. Non-linear deformation theory of unsymmetric composites For unsymmetric laminates, the hygro-thermally and chemically caused directional deformations of the UD single layers result in large out-of-plane deformations. To take into account the large multistable deformations, which are often many times over the laminate thickness, the linear strain±displacement relations must be extended by non-linear terms (see e.g. [10]): 2 o2 w ou0 1 ow o2 w 0 ex ex z 2 z 2; ox ox ox 2 ox 2 2 0 o w ov 1 ow o2 w ey e0y z 2 z 2; oy oy oy 2 oy 0 2 0 ow 1 ou ov ow ow o2 w ; exy e0xy z z oxoy 2 oy ox oy oxoy ox 1 where the index 0 refers to the laminate reference plane. The applied theory is based on the principle of minimising the total potential energy, which is given here by Z 1 P Q ei ej gTi ei DT gMi ei DM gSi ei dV 2 2 ij V ij are the reduced transwith (i; j 1; 2; 6), where Q formed stinesses, DT and DM are the dierences in temperature and relative media concentration with respect to the reference state and gTi , gMi , gSi are related to the elastic constants and to the ij aj , · thermal expansion coecients aj gTi Q · swelling coecients bj gMi Qij bj or ij sj , respectively [5]. · shrinkage sj gSi Q Based on the total potential energy, the Rayleigh± Ritz method is applied to obtain approximate solutions for the displacement ®elds. Therefore, general approaches in the form of polynomials are used dependent on the laminate lay-up.
Fig. 1. Basic shapes of square 0n =90n laminates: (a) reference state at elevated curing temperature; (b)±(d) saddle and cylinder shapes at room temperature.
cylindrical shapes (Figs. 1(c) and (d)). For 0n =90m laminates with n m or n m however only one stable cylindrical shape occurs. Geometrical assumptions for the out-of-plane displacements of cross-ply laminates lead to the general second-order approach 1 w x; y a0 x2 b0 y 2 ; 2
where the coecients a0 and b0 de®ne the laminate curvatures along the x and y axes [4]. For the description of the in-plane deformations, several approximations can be found in the literature [1,3,4,9]. The following geometrical assumptions: u0 x; y u0 x; y; u0 0; y 0; 0
v x; y
For a square cross-ply 0n =90n laminate the occurring basic shapes are illustrated in Fig. 1. Starting from the plane shape, which is considered as the reference state (Fig. 1(a)), the residual stresses lead ± dependent on the laminate dimensions and non-mechanical loads ± to a saddle shape (Fig. 1(b)) or to either of the two stable
u0 x; y
u0 x; y;
v0 x; y v0 x; y; 0
v x; y;
4
0
v x; 0 0
are ful®lled by the general approaches u0 x; y x a1 a2 x2 a3 y 2 a4 x2 y 2 ; v0 x; y y b1 b2 y 2 b3 x2 b4 x2 y 2 :
5
The assumptions that e0x is independent on x, e0y is independent on y and the consideration of especially the non-linear terms in (1) reduce the amount of coecients and ®nally lead to the approaches u0 x; y a1 x 0
2.1. Cross-ply laminates
3
v x; y b1 y
1 2 3 a x a3 xy 2 ; 6 0 1 2 3 b y b3 yx2 : 6 0
6
Using the displacement approximations (3) and (6) in the strain displacement relations (1) and substituting the resulting expressions into (2), the total potential energy of the laminate becomes a function dependent on the coecients ak , bk (k 0; 1; 3). The principle of the minimum total potential energy requires the ®rst variation to be zero, which means
W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327
Fig. 2. Basic shapes of angle-ply laminates: (a) reference state at elevated curing temperature; (b)±(d) twisted saddle and twisted cylindrical shapes at room temperature (x, y, z: global coordinate system; ~x, y~, ~z: principal curvature coordinate system).
dP
oP oP dak dbk 0: oak obk
7
To satisfy this condition, every summation in (7) must be zero, which results in a coupled non-linear algebraic equation system in ak , bk . Dependent on the laminate lay-up and the laminate size, more than one solution can be obtained. These solutions have to be checked for their stability by means of d2 P, which has to be positive for a stable deformation state. 2.2. Angle-ply laminates The achieved semi-analytical solutions for cross-ply laminates can be extended for general angle-ply /=/ laminates. In contrast to cross-ply laminates, the directions of the main curvatures of angle-ply composites generally do not coincide with the global axes of the laminate. Thus, also twisted shapes occur in addition to the above mentioned shapes (Fig. 2). The angle h between the direction of the principal curvatures and the global coordinate system is strongly dependent on the ®bre orientations of the single layers. Thus it is advantageous for the stability analysis to formulate the displacement approaches in the principal curvature system ~x; y~; ~z at ®rst and to carry out a transformation into the global coordinate system (x; y; z) according to the transformations 0 01 0 10 0 1 cos h sin h 0 u u~ @ v0 A @ sin h cos h 0 A@ v~0 A: 8 0 0 1 ~ w w
321
Fig. 3. Curvatures of a 04 =904 CFRP-laminate dependent on laminate edge length L.
04 =904 CFRP laminate are shown in Fig. 3 dependent on the laminate edge length. Branch AB shows the curvature of the stable saddle shape (a0 b0 ), which occurs in the case of small laminates. The critical edge length for the saddle shape is where the saddle shape becomes unstable, which de®nes the bifurcation point (B). After this bifurcation point, the saddle shape occurs only theoretically as an unstable equilibrium state but not in reality, which is indicated by the dashed line BC. Instead of the stable saddle shape, now two equivalent stable solutions are calculated (BD and BE). Branch BD represents the curvature a0 of the cylindrical shape in Fig. 1(d) and the curvature b0 of the cylindrical shape in Fig. 1(c). Branch BE represents the secondary curvature b0 of the cylindrical shape in Fig. 1(d) and the secondary curvature a0 of the cylindrical shape in Fig. 1(c), which asymptotically approach zero with an increasing edge length L. In Fig. 4 the curvatures of GFRP and CFRP 02 =902 laminates cured at 125 °C are compared dependent on the laminate's edge length. It can clearly be seen, that for this special stacking sequence the GFRP composite has a higher curvature than the CFRP laminate. However,
3. Analysis of unsymmetric composites Applying the above-mentioned theory, the solutions expressed in terms of the curvatures of an exemplary
Fig. 4. Curvatures of dierent cross-ply laminates dependent on the laminate edge length L.
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several theoretical and experimental investigations on dierent unsymmetric composites have shown, that a general statement about the comparison of curvature magnitudes of laminates made of dierent materials cannot be made because of the signi®cant in¯uence of many dierent material parameters like Ek , E? , mk? , Gk? , ak , a? and lay-up parameters like the number of layers, layer thickness and layer arrangement. Instead, it has to be distinguished in each single case dependent on the lay-up, whether for example the GFRP or the CFRP laminate shows a higher curvature. Moreover, the comparison of Figs. 2 and 3 illustrates that in general thinner laminates show a higher curvature than thicker laminates of the same material. Further investigations on angle-ply /=/ laminates have shown a deformation behaviour analogous to cross-ply laminates. The course of the curvature curves given in the principal curvature co-ordinate system dependent on the edge length L are similar to the cross-ply laminates (Fig. 5). The highest laminate principal curvatures result for orthotropic laminates (/ 45° and they reduce with decreasing ®bre orientations / 30° and / 15°. 3.1. Experimental and numerical veri®cation Within the current research, several experimental investigations have been carried out on laminates made of CFRP and GFRP. As an example, the results of CFRP laminates are discussed here. In Fig. 6, CFRP laminates with the stacking sequence 02 =902 (single layer thickness: 0.125 mm) and a laminate size of 500 mm 500 mm, which have been processed at a curing temperature CT 125 °C and slowly cooled down to room temperature RT 20 °C, are shown. After manufacture and during the experimental investigations, the laminates have been stored in dry conditions, so that moisture eects are not expected.
Fig. 5. Curvatures of dierent angle-ply laminates dependent on the laminate edge length L.
Fig. 6. Shapes of cross-ply 02 =902 laminates.
Besides the experimental veri®cation of the developed semi-analytical calculation method, numerical calculations of multistable composites with the help of a ®nite element analysis (FEA) have been carried out. Here it has been found that in many cases it is only possible to calculate the correct deformation state by applying imperfections such as small initial, temporary forces or deformations, which initiate the correct deformation during the non-linear computation [11]. The semi-analytically and numerically (FEA) calculated results and the measured curvatures for the 02 =902 laminates are shown in Fig. 7. It can be seen that the calculated curvatures of the laminates and the experimentally determined curvatures are in good agreement. 3.2. Mono- and multistable deformation behaviour It has been shown that the deformation behaviour of unsymmetric composites of large dimensions due to residual stresses is characterized by multistable states of equilibrium. These states need not be equivalent. In case of cross-ply laminates, the equivalence of bi-stable cylindrical shapes depends on the ratio of the 0° and 90° plies. For a highly uneven ratio, only one single (monostable) cylindrical shape is observed.
Fig. 7. Theoretical and experimental results of cross-ply CFRP laminates.
W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327
Fig. 8. Curvatures of a CFRP laminate with variable layer ratio.
The dierent states of stability due to residual stresses are exempli®ed by a quadratic cross-ply laminate [0/90] made of CFRP-T3,6 (300 mm 300 mm 1 mm). The thickness ratio of the single layers is varied, while the total laminate thickness is constant. Starting from a purely 0° UD laminate (ratio h1 =h 0), which is plane, ®rstly a monostable cylindrical shape occurs as soon as the thickness of the 90° layer increases (Fig. 8). Further increasing the 90° layer thickness leads to a higher curvature b0 until a maximum value is achieved for a ratio of about 0.2. For small ratios, also a second unstable solution (cylindrical shape 2) is calculated. The associated unstable equilibrium state turns into a stable equilibrium state for a thickness ratio h1 =h 0:3. For a ratio between 0.3 and 0.7, two stable cylindrical shapes with generally dierent curvatures (a0 and b0 ) exist. Only for a ratio of 0.5, these shapes are equivalent with a0 b0 otherwise either of the two cylindrical shapes preferably appears. For a ratio of 0.7, the ®rst cylindrical shape becomes unstable and the theoretical curvature tends to zero for higher ratios. The second cylindrical shape reaches its maximum value for a ratio of 0.8 and tends to zero for a purely 90° composite. The investigations of unsymmetric cross-ply laminates demonstrate that dierent stability states as well with equivalent as with non-equivalent shapes can be realized dependent on the laminate lay-up, which will purposefully be used in the following. A similar stability behaviour can be observed for /=/ laminates.
323
is therefore necessary to carry out a realistic failure analysis. Many failure criteria for the analysis of the failure behaviour of anisotropic composites exist. They are based on a micro-, meso- or macromechanical point of view. The realistic description of the failure of the UD-layer, which is the basic element of technical composites, is of special interest for the structural design of composite structures. For this element, it has to be distinguished between the basic failure modes ®bre failure (FF) and inter-®bre failure (IFF) with its submodes tensile, compressive and shear failure. Each of these modes is assigned one strength coecient (Fig. 9). For the failure analysis of unsymmetric laminates strained by residual stresses, the physically based failure mode concept (FMC) of Cuntze [12] has been applied. In contrast to the well-known failure criterion of Hashin/Puck, where an extreme value problem has to be solved, the FMC features an easier mathematical formulation using invariants. In contrast to the well-known general invariants, which are independent on the choice of the coordinate system for a given state of stress, special invariants Ii (i 1; . . . ; 5) are used for the formulation of the FMC. These are only invariant with respect to any rotation around one axis. The rotating axis is chosen with respect to a main material axis, which in this case is the ®bre orientation. By using these invariants, realistic failure conditions can be formulated for each fracture mode according to Fig. 9. In the case of a transversally isotropic continuum, e.g., for a UD basic element, the material structure is invariant to rotation around the ®bre (x1 ) axis, which leads to the following special invariants [12]: I1 r1 ; I2 r2 r3 ; I3 s231 s221 ; I4 r2
r3
I5 r2
r3
9 2
4s223 ; s231 s221
4s23 s31 s21 :
4. Failure analysis of multistable composites The described process-related residual stresses in unsymmetric laminates are sometimes critical and can induce a premature failure. Within the design process, it
Fig. 9. Failure modes and strength coecients.
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W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327
Using these invariants, Cuntze gives ®ve fracture conditions according to the dierent fracture modes. For FF there are the conditions I2 Fkr 1 2 1; Rzk
Fks
I1 1; Rdk
10
where the ®rst condition can be replaced by a numerically simpler linear term, which considers the ®bre tensile stress in formulating the failure condition and not r1 [12]. The failure conditions for IFF are: 3=2
I3 I2 I3 I5 b?k 1; 3 R?k R3?k p I2 I4 F?r 1; 2Rz? I2 bs? I4 bs?k I3 1; F?s bs? 1 d 2 R? Rd?
5. Adjustment of residual stresses using genetic algorithms
F?k
11
where the coecients b?k , bs? and bs?k are parameters, which have to be determined from multiaxial fracture tests [12]. kr From the failure conditions, the reserve factors fRes ks ?k ?r and fRes for the single fracture modes of FF and fRes , fRes ?s and fRes for the modes of IFF can be calculated. Based res on these reserve factors, the resultant reserve factor fRes can be determined m_ m_ m_ res kr ks ?r fRes fRes fRes fRes
?s fRes
m_
?k fRes
m_
1=m_
;
posites together with the fracture curve according to the FMC in the r2 ; s21 plane. It can be seen that in the case of the cross-ply laminate all stress combinations are within the fracture curve and a resultant reserve factor res fRes 5:28 can be calculated. For the angle-ply com res posite however a reserve factor fRes 0:9 is calculated due to high tensile stresses perpendicular to the ®bres in combination with shear stresses s21 , so that a ®rst ply failure is induced here.
12
where the term m_ describes the smoothing exponent [12]. As examples, assessment of critical states of stress due to residual stresses (curing at 125 °C) in unsymmetric square composites (CFRP-T3,6-02 =902 cross-ply and 302 =302 angle-ply) with an edge length L 300 mm has been carried out. Fig. 10 shows the maximum stress combinations at the layer interfaces of the cross-ply and angle-ply com-
Fig. 10. Stress combinations in cross-ply and angle-ply laminates and corresponding fracture curve.
Due to the high number of material-, geometry- and load-speci®c parameters, which in¯uence the residual stresses of unsymmetric laminates, it is necessary to develop a powerful optimisation procedure in order to adjust the curvature to the technical requirements. The calculation of the curvature requires an iterative procedure and the solution cannot be given in a closed analytical form due to the application of a numerical energy principle. Thus, conventional optimisation methods like the hill-climbing methods are not suitable to ®nd an optimum, because the function's derivatives are missing. Next, the investigations have shown that the relationship between the laminate's lay-up and the laminate's curvature is often described by a multimodal function with several local and global maximums, so that conventional optimisation procedures like hill-climbing methods or simulated annealing can fail in ®nding an optimal solution since they are likely to stack in a local optimum [13]. This can be depicted by a criterion surface of basic examples shown in Fig. 11. These criterion surfaces are built by the single curvature values of square CFRP laminates cured at 125 °C (edge length L 400 mm) with three single layers of varying thickness ratios (total height h h1 h2 h3 2:5 mm) and lay-up sequences. It can be clearly noticed that for this basic example ± considered as simple ± the optimisation procedure already has to search a multimodal, nondierentiable surface in order to ®nd the optimum laminate lay-up. For the adjustment of residual stresses to technical requirements, genetic algorithms (GA) in combination with the above mentioned non-linear calculation method oer special advantages and enable to ®nd the optimal combination of material, load and process parameters. These optimisation methods are robust and can be used to eciently scan the multidimensional search space and they do not need any derivatives of the function to be optimised [12]. The algorithms are applied to determine the optimum laminate lay-up either in order to maximise the laminate curvature or to adjust the curvature to de®ned limits by changing the number of layers, layer thickness, layer orientation, layer mate-
W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327
(a)
325
(b)
Fig. 11. Curvature dependence on single layer thickness ratio (h1 : ®rst layer thickness, h2 : second layer thickness) for dierent lay-up sequences: (a) lay-up 0h1 =90h2 =0h3 ; (b) lay-up 0h1 =90h2 =90h3 .
Fig. 12. Lay-up of CFRP and GFRP cross-ply laminates resulting in maximum curvatures.
rial, ®bre volume content, the laminates size and the process parameters like temperature or moisture. As examples, the developed genetic algorithm was used to optimise the laminate's lay-up in terms of layer arrangement, layer thickness and layer orientation, so that the main cylindrical curvature of dierent cross-ply laminates (400 mm 400 mm), consisting either of GFRP or CFRP with dierent ®bres, is maximised. It has been found that dependent on the laminate material the maximum curvature has to be realised by dierent lay-ups. The left bar graph in Fig. 12 shows the optimum single layer thickness and the layer arrangement of the investigated composites with a global height of 2.5 mm. While the curvature of a GFRP laminate can be achieved by manufacturing a simple 0h1 =90h2 lay-up with the ratio shown in Fig. 12, in case of the investigated CFRP laminates the maximum curvatures are realised by dierent 0h1 =90h2 =0h3 lay-ups. Furthermore it is evident from the right bar graph in Fig. 12, that in contrast to the curvatures of the 02 =902 laminates shown in Fig. 4 now the CFRP laminates have higher curvature values. It was furthermore observed, that independent of the total height of the laminates the ratios of the single layer thicknesses which result in the maximum curvature are
the same, so that once the optimum lay-up ratio is found for a special material, it can be transferred to any laminate of the same material with a varying total thickness. Furthermore, the developed genetic algorithm enables the ecient determination of stacking sequences to ful®l technical requirements. In order to obtain a de®ned target curvature a0target with respect to given material properties, laminate thickness and process parameters for instance, the GA calculates a choice of stacking sequences, which result in the de®ned curvature. In Fig. 13
Fig. 13. Stacking sequences resulting in a curvature a0target 0:002 mm 1 0:12%.
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W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327
for example, the calculated stacking sequences of CFRP-laminates with a total laminate thickness of 2.5 mm and a target curvature a0target 0:002 mm 1 0:12% are shown. This choice of lay-ups oers the design engineer the possibility to choose the lay-up, which best meets further requirements with regard to reserve factors or easy-to-build-strategies for instance.
NiSnap NiT NiM NiS ; MiSnap MiT MiM MiS :
13
The new adaptive structures can be adjusted to individual technical requirements by arranging a loadadapted composite structure and they oer a high potential for transfer of forces or for usage as components in further adjusting devices, valve ¯aps or active membranes.
6. Adaptive multistable structures The combination of the described multistable composites with actuators enables the design of novel adaptive structures. By advantageously using the existence of dierent stable deformation states, large adaptive deformations and forces can be realised by a short electrical impulse in contrast to conventional adaptive structures, which need a continuous adaptive support to realise only small de¯ections. Thus, the dierent stable deformation states of a cross-ply laminate for instance can be changed by snap-through from cylindrical shape 1 to cylindrical shape 2 (Fig. 14). The novel adaptive structures consist of passive composite layers which are arranged in that way that a bi-stable laminate results due to residual stresses. The second components are smart materials, which are embedded in the laminate as it is shown for cross-ply laminates in Fig. 15. The necessary mechanical and adaptive properties of the composite and the adaptive components can be calculated based on the developed semi-analytical method. To initiate the snap-through, an adaptive stress state has to be superposed to the residual stress state according to the relations:
7. Conclusions The residual stress ®eld, which builds up during the manufacturing process of multilayered ®bre-reinforced composites with variable ®bre orientations can be purposefully used to design unsymmetric composites with a de®ned deformation state and curvature. With the help of a non-linear stability analysis in combination with the failure mode concept, the resulting deformation states and resulting states of stresses can be predicted and assessed with respect to failure. Due to the high number of parameters which in¯uence the curvature like the number, thickness, orientation of layers, the layer material, ®bre volume content, the laminates size and the process parameters, genetic algorithms have been developed, which enable an ecient scanning of the resulting multidimensional search space of dierent laminate lay-ups to realise the desired curvature. It has been shown, that the new optimisation procedures enable the ecient design of multilayered composites and the adjustment of the curvature to technical demands. This does not only meet the single technical requirements within a near-netshape technology but also helps to reduce production costs by replacing tools of complex shape by plane tools and furthermore enables the design of novel adaptive structures. References
Fig. 14. Snap-through between stable deformation states.
Fig. 15. Integration of smart alloys in multistable composite.
[1] Hyer MW. Calculation of the room-temperature shapes of unsymmetric laminates. J Compos Mater 1981;15:296±310. [2] Hyer MW. The room-temperature shapes of four layer unsymmetric cross-ply laminates. J Compos Mater 1982;16:318±40. [3] Jun WJ, Hong CS. Eect of residual shear strain on the cured shape of unsymmetric cross-ply thin laminates. Compos Sci Technol 1990;38:55±67. [4] Peeters LJB, Powell PC, Warnet L. Thermally-induced shapes of unsymmetric laminates. J Compos Mater 1996;30:603±26. [5] Hufenbach W, Kroll L, Gude M. Deformation states of unsymmetric ®bre-reinforced composites dependent on residual stresses. In: Proceedings of SAMPE EUROPE/JEC Conference, 1999 April 13±15; Paris, France. p. 341±52. [6] Hufenbach W, Kroll L, Gude M. Temperatur- und schwindungsbedingte Deformationszustande bei unsymmetrischen Mehrschichtverbunden. In: Proceedings of 11th Workshop, Composite-Forschung in der Mechanik, 1998 December 8±9; Paderborn.
W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327 [7] Hufenbach W, Kroll L. Design and construction of hygrothermally stable shells. Mech Compos Mater 1995;31(5):677±83. [8] Norwood DS. An analysis of interlaminar stresses in unsymmetrically laminated plates. PhD Thesis, Virginia Polytechnic Institute and State University, 1992. [9] Kam TY, Sher HF. Nonlinear and ®rst ply failure analyses of laminated composite cross-ply plates. J Compos Mater 1995;29(4):463±82. [10] Lekhnitskii SG. Anisotropic plates. New York: Gordon and Breach; 1968.
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[11] Gude M. Zum nichtlinearen Deformationsverhalten multistabiler Mehrschichtverbunde mit unsymmetrischem Strukturaufbau. Dissertation, TU Dresden, 2000. [12] Cuntze R. Progressive failure of 3D-stressed laminates: multiple nonlinearity treated by the failure mode concept (FMC). In: Proceedings of the Duracosys Conference, 1999; Br ussel. [13] Hufenbach W, Gude M, Kroll L. Design of unsymmetrical composites using genetic algorithms. In: Seventh Annual International Conference on Composite Engineering (ICCE/7), July 2± 8. Colorado: Denver; 2000.
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Analysis and optimisation of multistable composites under residual stresses W. Hufenbach, M. Gude
*
Institut fur Leichtbau und Kunststotechnik (ILK), Technische Universitat Dresden, 01062 Dresden, Germany
Abstract During the manufacturing process of multilayered ®bre-reinforced composites with variable ®bre orientations, residual stresses build up due to the directional expansion of the unidirectionally reinforced single layers. Dependent on the laminate lay-up, these inhomogeneous residual stresses, which are caused by thermal eects, moisture absorption and chemical shrinkage, can lead to large multistable out-of-plane deformations in case of unsymmetric laminates. Instead of avoiding these laminate's curvatures, they can be advantageously used for technical applications such as near-net-shape technologies or adaptive structures. In order to adjust the deformations to technical requirements, a dimensioning tool consisting of an advanced stability and failure analysis in combination with a novel optimisation procedure has been developed. This tool, based on a non-linear calculation method, enables the purposeful adaption of the laminate lay-up dependent on the loading and process parameters and is furthermore the basis for the design of novel adaptive structures. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Multistable composites; Residual stresses; Failure analysis; Optimisation; Adaptive structures
1. Introduction In multilayered ®bre-reinforced composites with variable ®bre orientations, the directional expansion of the unidirectional-reinforced (UD) single layers due to thermal eects, moisture absorption and chemical shrinkage leads to a discontinuous residual stress ®eld over the laminate thickness. In case of unsymmetric laminate plates, these residual stresses can cause dierent multistable out-of-plane deformations. This paper focuses on the purposeful adaption of the residual stresses dependent on the manufacturing process in order to realise either laminates with de®ned (multistable) deformation states or to design novel adaptive structures. For the adjustment of laminate curvatures to technical requirements, new optimisation methods using genetic algorithms are developed, that can eciently be applied to ®nd an optimal laminate lay-up dependent on material properties, loading conditions and process parameters. Furthermore, a realistic failure analysis must be carried out in order to assess the resulting residual stress states with respect to ®rst-ply failure.
*
Corresponding author. Tel.: +49-351-463-38153; fax: +49-351-46338143. E-mail address: [email protected] (M. Gude).
The new laminate design method has successfully been veri®ed by experiments and numerical calculations on unsymmetric glass- (GFRP) and carbon-®bre-reinforced plastics (CFRP) and was applied to the design of multilayered curved hybrid structures. Some earlier papers deal with the calculation of the purely thermal out-of-plane deformations with the help of extended non-linear displacement±strain relations (e.g. [1±4]). However, the important in¯uence of the chemical shrinkage and moisture absorption on the laminate deformation is not considered, although its contribution to residual stresses can be higher than the thermal contribution. Therefore the in¯uence of shrinkage and moisture cannot be neglected in advanced design processes [5±7]. Considering extended stress± displacement relations, the principle of minimising the elastic potential in combination with the Rayleigh±Ritz method results in a system of non-linear equations for the calculation of the multistable laminate deformations. Dependent on the size and lay-up of the unsymmetric laminate, the equation system leads to dierent solutions of stable and unstable deformation states like saddle shapes or cylindrical shapes [2±5]. Most of the papers are further restricted to describing the deformation states without considering the resulting stresses, even though the residual stresses can induce a premature failure. Only some work has been done to
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calculate the residual stress states and to assess the stresses with respect to fracture by conventional failure criteria [8,9]. A failure analysis of multistable composites using realistic, physically based failure criteria has not yet been carried out.
2. Non-linear deformation theory of unsymmetric composites For unsymmetric laminates, the hygro-thermally and chemically caused directional deformations of the UD single layers result in large out-of-plane deformations. To take into account the large multistable deformations, which are often many times over the laminate thickness, the linear strain±displacement relations must be extended by non-linear terms (see e.g. [10]): 2 o2 w ou0 1 ow o2 w 0 ex ex z 2 z 2; ox ox ox 2 ox 2 2 0 o w ov 1 ow o2 w ey e0y z 2 z 2; oy oy oy 2 oy 0 2 0 ow 1 ou ov ow ow o2 w ; exy e0xy z z oxoy 2 oy ox oy oxoy ox 1 where the index 0 refers to the laminate reference plane. The applied theory is based on the principle of minimising the total potential energy, which is given here by Z 1 P Q ei ej gTi ei DT gMi ei DM gSi ei dV 2 2 ij V ij are the reduced transwith (i; j 1; 2; 6), where Q formed stinesses, DT and DM are the dierences in temperature and relative media concentration with respect to the reference state and gTi , gMi , gSi are related to the elastic constants and to the ij aj , · thermal expansion coecients aj gTi Q · swelling coecients bj gMi Qij bj or ij sj , respectively [5]. · shrinkage sj gSi Q Based on the total potential energy, the Rayleigh± Ritz method is applied to obtain approximate solutions for the displacement ®elds. Therefore, general approaches in the form of polynomials are used dependent on the laminate lay-up.
Fig. 1. Basic shapes of square 0n =90n laminates: (a) reference state at elevated curing temperature; (b)±(d) saddle and cylinder shapes at room temperature.
cylindrical shapes (Figs. 1(c) and (d)). For 0n =90m laminates with n m or n m however only one stable cylindrical shape occurs. Geometrical assumptions for the out-of-plane displacements of cross-ply laminates lead to the general second-order approach 1 w x; y a0 x2 b0 y 2 ; 2
where the coecients a0 and b0 de®ne the laminate curvatures along the x and y axes [4]. For the description of the in-plane deformations, several approximations can be found in the literature [1,3,4,9]. The following geometrical assumptions: u0 x; y u0 x; y; u0 0; y 0; 0
v x; y
For a square cross-ply 0n =90n laminate the occurring basic shapes are illustrated in Fig. 1. Starting from the plane shape, which is considered as the reference state (Fig. 1(a)), the residual stresses lead ± dependent on the laminate dimensions and non-mechanical loads ± to a saddle shape (Fig. 1(b)) or to either of the two stable
u0 x; y
u0 x; y;
v0 x; y v0 x; y; 0
v x; y;
4
0
v x; 0 0
are ful®lled by the general approaches u0 x; y x a1 a2 x2 a3 y 2 a4 x2 y 2 ; v0 x; y y b1 b2 y 2 b3 x2 b4 x2 y 2 :
5
The assumptions that e0x is independent on x, e0y is independent on y and the consideration of especially the non-linear terms in (1) reduce the amount of coecients and ®nally lead to the approaches u0 x; y a1 x 0
2.1. Cross-ply laminates
3
v x; y b1 y
1 2 3 a x a3 xy 2 ; 6 0 1 2 3 b y b3 yx2 : 6 0
6
Using the displacement approximations (3) and (6) in the strain displacement relations (1) and substituting the resulting expressions into (2), the total potential energy of the laminate becomes a function dependent on the coecients ak , bk (k 0; 1; 3). The principle of the minimum total potential energy requires the ®rst variation to be zero, which means
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Fig. 2. Basic shapes of angle-ply laminates: (a) reference state at elevated curing temperature; (b)±(d) twisted saddle and twisted cylindrical shapes at room temperature (x, y, z: global coordinate system; ~x, y~, ~z: principal curvature coordinate system).
dP
oP oP dak dbk 0: oak obk
7
To satisfy this condition, every summation in (7) must be zero, which results in a coupled non-linear algebraic equation system in ak , bk . Dependent on the laminate lay-up and the laminate size, more than one solution can be obtained. These solutions have to be checked for their stability by means of d2 P, which has to be positive for a stable deformation state. 2.2. Angle-ply laminates The achieved semi-analytical solutions for cross-ply laminates can be extended for general angle-ply /=/ laminates. In contrast to cross-ply laminates, the directions of the main curvatures of angle-ply composites generally do not coincide with the global axes of the laminate. Thus, also twisted shapes occur in addition to the above mentioned shapes (Fig. 2). The angle h between the direction of the principal curvatures and the global coordinate system is strongly dependent on the ®bre orientations of the single layers. Thus it is advantageous for the stability analysis to formulate the displacement approaches in the principal curvature system ~x; y~; ~z at ®rst and to carry out a transformation into the global coordinate system (x; y; z) according to the transformations 0 01 0 10 0 1 cos h sin h 0 u u~ @ v0 A @ sin h cos h 0 A@ v~0 A: 8 0 0 1 ~ w w
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Fig. 3. Curvatures of a 04 =904 CFRP-laminate dependent on laminate edge length L.
04 =904 CFRP laminate are shown in Fig. 3 dependent on the laminate edge length. Branch AB shows the curvature of the stable saddle shape (a0 b0 ), which occurs in the case of small laminates. The critical edge length for the saddle shape is where the saddle shape becomes unstable, which de®nes the bifurcation point (B). After this bifurcation point, the saddle shape occurs only theoretically as an unstable equilibrium state but not in reality, which is indicated by the dashed line BC. Instead of the stable saddle shape, now two equivalent stable solutions are calculated (BD and BE). Branch BD represents the curvature a0 of the cylindrical shape in Fig. 1(d) and the curvature b0 of the cylindrical shape in Fig. 1(c). Branch BE represents the secondary curvature b0 of the cylindrical shape in Fig. 1(d) and the secondary curvature a0 of the cylindrical shape in Fig. 1(c), which asymptotically approach zero with an increasing edge length L. In Fig. 4 the curvatures of GFRP and CFRP 02 =902 laminates cured at 125 °C are compared dependent on the laminate's edge length. It can clearly be seen, that for this special stacking sequence the GFRP composite has a higher curvature than the CFRP laminate. However,
3. Analysis of unsymmetric composites Applying the above-mentioned theory, the solutions expressed in terms of the curvatures of an exemplary
Fig. 4. Curvatures of dierent cross-ply laminates dependent on the laminate edge length L.
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several theoretical and experimental investigations on dierent unsymmetric composites have shown, that a general statement about the comparison of curvature magnitudes of laminates made of dierent materials cannot be made because of the signi®cant in¯uence of many dierent material parameters like Ek , E? , mk? , Gk? , ak , a? and lay-up parameters like the number of layers, layer thickness and layer arrangement. Instead, it has to be distinguished in each single case dependent on the lay-up, whether for example the GFRP or the CFRP laminate shows a higher curvature. Moreover, the comparison of Figs. 2 and 3 illustrates that in general thinner laminates show a higher curvature than thicker laminates of the same material. Further investigations on angle-ply /=/ laminates have shown a deformation behaviour analogous to cross-ply laminates. The course of the curvature curves given in the principal curvature co-ordinate system dependent on the edge length L are similar to the cross-ply laminates (Fig. 5). The highest laminate principal curvatures result for orthotropic laminates (/ 45° and they reduce with decreasing ®bre orientations / 30° and / 15°. 3.1. Experimental and numerical veri®cation Within the current research, several experimental investigations have been carried out on laminates made of CFRP and GFRP. As an example, the results of CFRP laminates are discussed here. In Fig. 6, CFRP laminates with the stacking sequence 02 =902 (single layer thickness: 0.125 mm) and a laminate size of 500 mm 500 mm, which have been processed at a curing temperature CT 125 °C and slowly cooled down to room temperature RT 20 °C, are shown. After manufacture and during the experimental investigations, the laminates have been stored in dry conditions, so that moisture eects are not expected.
Fig. 5. Curvatures of dierent angle-ply laminates dependent on the laminate edge length L.
Fig. 6. Shapes of cross-ply 02 =902 laminates.
Besides the experimental veri®cation of the developed semi-analytical calculation method, numerical calculations of multistable composites with the help of a ®nite element analysis (FEA) have been carried out. Here it has been found that in many cases it is only possible to calculate the correct deformation state by applying imperfections such as small initial, temporary forces or deformations, which initiate the correct deformation during the non-linear computation [11]. The semi-analytically and numerically (FEA) calculated results and the measured curvatures for the 02 =902 laminates are shown in Fig. 7. It can be seen that the calculated curvatures of the laminates and the experimentally determined curvatures are in good agreement. 3.2. Mono- and multistable deformation behaviour It has been shown that the deformation behaviour of unsymmetric composites of large dimensions due to residual stresses is characterized by multistable states of equilibrium. These states need not be equivalent. In case of cross-ply laminates, the equivalence of bi-stable cylindrical shapes depends on the ratio of the 0° and 90° plies. For a highly uneven ratio, only one single (monostable) cylindrical shape is observed.
Fig. 7. Theoretical and experimental results of cross-ply CFRP laminates.
W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327
Fig. 8. Curvatures of a CFRP laminate with variable layer ratio.
The dierent states of stability due to residual stresses are exempli®ed by a quadratic cross-ply laminate [0/90] made of CFRP-T3,6 (300 mm 300 mm 1 mm). The thickness ratio of the single layers is varied, while the total laminate thickness is constant. Starting from a purely 0° UD laminate (ratio h1 =h 0), which is plane, ®rstly a monostable cylindrical shape occurs as soon as the thickness of the 90° layer increases (Fig. 8). Further increasing the 90° layer thickness leads to a higher curvature b0 until a maximum value is achieved for a ratio of about 0.2. For small ratios, also a second unstable solution (cylindrical shape 2) is calculated. The associated unstable equilibrium state turns into a stable equilibrium state for a thickness ratio h1 =h 0:3. For a ratio between 0.3 and 0.7, two stable cylindrical shapes with generally dierent curvatures (a0 and b0 ) exist. Only for a ratio of 0.5, these shapes are equivalent with a0 b0 otherwise either of the two cylindrical shapes preferably appears. For a ratio of 0.7, the ®rst cylindrical shape becomes unstable and the theoretical curvature tends to zero for higher ratios. The second cylindrical shape reaches its maximum value for a ratio of 0.8 and tends to zero for a purely 90° composite. The investigations of unsymmetric cross-ply laminates demonstrate that dierent stability states as well with equivalent as with non-equivalent shapes can be realized dependent on the laminate lay-up, which will purposefully be used in the following. A similar stability behaviour can be observed for /=/ laminates.
323
is therefore necessary to carry out a realistic failure analysis. Many failure criteria for the analysis of the failure behaviour of anisotropic composites exist. They are based on a micro-, meso- or macromechanical point of view. The realistic description of the failure of the UD-layer, which is the basic element of technical composites, is of special interest for the structural design of composite structures. For this element, it has to be distinguished between the basic failure modes ®bre failure (FF) and inter-®bre failure (IFF) with its submodes tensile, compressive and shear failure. Each of these modes is assigned one strength coecient (Fig. 9). For the failure analysis of unsymmetric laminates strained by residual stresses, the physically based failure mode concept (FMC) of Cuntze [12] has been applied. In contrast to the well-known failure criterion of Hashin/Puck, where an extreme value problem has to be solved, the FMC features an easier mathematical formulation using invariants. In contrast to the well-known general invariants, which are independent on the choice of the coordinate system for a given state of stress, special invariants Ii (i 1; . . . ; 5) are used for the formulation of the FMC. These are only invariant with respect to any rotation around one axis. The rotating axis is chosen with respect to a main material axis, which in this case is the ®bre orientation. By using these invariants, realistic failure conditions can be formulated for each fracture mode according to Fig. 9. In the case of a transversally isotropic continuum, e.g., for a UD basic element, the material structure is invariant to rotation around the ®bre (x1 ) axis, which leads to the following special invariants [12]: I1 r1 ; I2 r2 r3 ; I3 s231 s221 ; I4 r2
r3
I5 r2
r3
9 2
4s223 ; s231 s221
4s23 s31 s21 :
4. Failure analysis of multistable composites The described process-related residual stresses in unsymmetric laminates are sometimes critical and can induce a premature failure. Within the design process, it
Fig. 9. Failure modes and strength coecients.
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Using these invariants, Cuntze gives ®ve fracture conditions according to the dierent fracture modes. For FF there are the conditions I2 Fkr 1 2 1; Rzk
Fks
I1 1; Rdk
10
where the ®rst condition can be replaced by a numerically simpler linear term, which considers the ®bre tensile stress in formulating the failure condition and not r1 [12]. The failure conditions for IFF are: 3=2
I3 I2 I3 I5 b?k 1; 3 R?k R3?k p I2 I4 F?r 1; 2Rz? I2 bs? I4 bs?k I3 1; F?s bs? 1 d 2 R? Rd?
5. Adjustment of residual stresses using genetic algorithms
F?k
11
where the coecients b?k , bs? and bs?k are parameters, which have to be determined from multiaxial fracture tests [12]. kr From the failure conditions, the reserve factors fRes ks ?k ?r and fRes for the single fracture modes of FF and fRes , fRes ?s and fRes for the modes of IFF can be calculated. Based res on these reserve factors, the resultant reserve factor fRes can be determined m_ m_ m_ res kr ks ?r fRes fRes fRes fRes
?s fRes
m_
?k fRes
m_
1=m_
;
posites together with the fracture curve according to the FMC in the r2 ; s21 plane. It can be seen that in the case of the cross-ply laminate all stress combinations are within the fracture curve and a resultant reserve factor res fRes 5:28 can be calculated. For the angle-ply com res posite however a reserve factor fRes 0:9 is calculated due to high tensile stresses perpendicular to the ®bres in combination with shear stresses s21 , so that a ®rst ply failure is induced here.
12
where the term m_ describes the smoothing exponent [12]. As examples, assessment of critical states of stress due to residual stresses (curing at 125 °C) in unsymmetric square composites (CFRP-T3,6-02 =902 cross-ply and 302 =302 angle-ply) with an edge length L 300 mm has been carried out. Fig. 10 shows the maximum stress combinations at the layer interfaces of the cross-ply and angle-ply com-
Fig. 10. Stress combinations in cross-ply and angle-ply laminates and corresponding fracture curve.
Due to the high number of material-, geometry- and load-speci®c parameters, which in¯uence the residual stresses of unsymmetric laminates, it is necessary to develop a powerful optimisation procedure in order to adjust the curvature to the technical requirements. The calculation of the curvature requires an iterative procedure and the solution cannot be given in a closed analytical form due to the application of a numerical energy principle. Thus, conventional optimisation methods like the hill-climbing methods are not suitable to ®nd an optimum, because the function's derivatives are missing. Next, the investigations have shown that the relationship between the laminate's lay-up and the laminate's curvature is often described by a multimodal function with several local and global maximums, so that conventional optimisation procedures like hill-climbing methods or simulated annealing can fail in ®nding an optimal solution since they are likely to stack in a local optimum [13]. This can be depicted by a criterion surface of basic examples shown in Fig. 11. These criterion surfaces are built by the single curvature values of square CFRP laminates cured at 125 °C (edge length L 400 mm) with three single layers of varying thickness ratios (total height h h1 h2 h3 2:5 mm) and lay-up sequences. It can be clearly noticed that for this basic example ± considered as simple ± the optimisation procedure already has to search a multimodal, nondierentiable surface in order to ®nd the optimum laminate lay-up. For the adjustment of residual stresses to technical requirements, genetic algorithms (GA) in combination with the above mentioned non-linear calculation method oer special advantages and enable to ®nd the optimal combination of material, load and process parameters. These optimisation methods are robust and can be used to eciently scan the multidimensional search space and they do not need any derivatives of the function to be optimised [12]. The algorithms are applied to determine the optimum laminate lay-up either in order to maximise the laminate curvature or to adjust the curvature to de®ned limits by changing the number of layers, layer thickness, layer orientation, layer mate-
W. Hufenbach, M. Gude / Composite Structures 55 (2002) 319±327
(a)
325
(b)
Fig. 11. Curvature dependence on single layer thickness ratio (h1 : ®rst layer thickness, h2 : second layer thickness) for dierent lay-up sequences: (a) lay-up 0h1 =90h2 =0h3 ; (b) lay-up 0h1 =90h2 =90h3 .
Fig. 12. Lay-up of CFRP and GFRP cross-ply laminates resulting in maximum curvatures.
rial, ®bre volume content, the laminates size and the process parameters like temperature or moisture. As examples, the developed genetic algorithm was used to optimise the laminate's lay-up in terms of layer arrangement, layer thickness and layer orientation, so that the main cylindrical curvature of dierent cross-ply laminates (400 mm 400 mm), consisting either of GFRP or CFRP with dierent ®bres, is maximised. It has been found that dependent on the laminate material the maximum curvature has to be realised by dierent lay-ups. The left bar graph in Fig. 12 shows the optimum single layer thickness and the layer arrangement of the investigated composites with a global height of 2.5 mm. While the curvature of a GFRP laminate can be achieved by manufacturing a simple 0h1 =90h2 lay-up with the ratio shown in Fig. 12, in case of the investigated CFRP laminates the maximum curvatures are realised by dierent 0h1 =90h2 =0h3 lay-ups. Furthermore it is evident from the right bar graph in Fig. 12, that in contrast to the curvatures of the 02 =902 laminates shown in Fig. 4 now the CFRP laminates have higher curvature values. It was furthermore observed, that independent of the total height of the laminates the ratios of the single layer thicknesses which result in the maximum curvature are
the same, so that once the optimum lay-up ratio is found for a special material, it can be transferred to any laminate of the same material with a varying total thickness. Furthermore, the developed genetic algorithm enables the ecient determination of stacking sequences to ful®l technical requirements. In order to obtain a de®ned target curvature a0target with respect to given material properties, laminate thickness and process parameters for instance, the GA calculates a choice of stacking sequences, which result in the de®ned curvature. In Fig. 13
Fig. 13. Stacking sequences resulting in a curvature a0target 0:002 mm 1 0:12%.
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for example, the calculated stacking sequences of CFRP-laminates with a total laminate thickness of 2.5 mm and a target curvature a0target 0:002 mm 1 0:12% are shown. This choice of lay-ups oers the design engineer the possibility to choose the lay-up, which best meets further requirements with regard to reserve factors or easy-to-build-strategies for instance.
NiSnap NiT NiM NiS ; MiSnap MiT MiM MiS :
13
The new adaptive structures can be adjusted to individual technical requirements by arranging a loadadapted composite structure and they oer a high potential for transfer of forces or for usage as components in further adjusting devices, valve ¯aps or active membranes.
6. Adaptive multistable structures The combination of the described multistable composites with actuators enables the design of novel adaptive structures. By advantageously using the existence of dierent stable deformation states, large adaptive deformations and forces can be realised by a short electrical impulse in contrast to conventional adaptive structures, which need a continuous adaptive support to realise only small de¯ections. Thus, the dierent stable deformation states of a cross-ply laminate for instance can be changed by snap-through from cylindrical shape 1 to cylindrical shape 2 (Fig. 14). The novel adaptive structures consist of passive composite layers which are arranged in that way that a bi-stable laminate results due to residual stresses. The second components are smart materials, which are embedded in the laminate as it is shown for cross-ply laminates in Fig. 15. The necessary mechanical and adaptive properties of the composite and the adaptive components can be calculated based on the developed semi-analytical method. To initiate the snap-through, an adaptive stress state has to be superposed to the residual stress state according to the relations:
7. Conclusions The residual stress ®eld, which builds up during the manufacturing process of multilayered ®bre-reinforced composites with variable ®bre orientations can be purposefully used to design unsymmetric composites with a de®ned deformation state and curvature. With the help of a non-linear stability analysis in combination with the failure mode concept, the resulting deformation states and resulting states of stresses can be predicted and assessed with respect to failure. Due to the high number of parameters which in¯uence the curvature like the number, thickness, orientation of layers, the layer material, ®bre volume content, the laminates size and the process parameters, genetic algorithms have been developed, which enable an ecient scanning of the resulting multidimensional search space of dierent laminate lay-ups to realise the desired curvature. It has been shown, that the new optimisation procedures enable the ecient design of multilayered composites and the adjustment of the curvature to technical demands. This does not only meet the single technical requirements within a near-netshape technology but also helps to reduce production costs by replacing tools of complex shape by plane tools and furthermore enables the design of novel adaptive structures. References
Fig. 14. Snap-through between stable deformation states.
Fig. 15. Integration of smart alloys in multistable composite.
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[11] Gude M. Zum nichtlinearen Deformationsverhalten multistabiler Mehrschichtverbunde mit unsymmetrischem Strukturaufbau. Dissertation, TU Dresden, 2000. [12] Cuntze R. Progressive failure of 3D-stressed laminates: multiple nonlinearity treated by the failure mode concept (FMC). In: Proceedings of the Duracosys Conference, 1999; Br ussel. [13] Hufenbach W, Gude M, Kroll L. Design of unsymmetrical composites using genetic algorithms. In: Seventh Annual International Conference on Composite Engineering (ICCE/7), July 2± 8. Colorado: Denver; 2000.