0263-8223(95)00080-l

Composite Structuw~ 32 (1995) 621433 0 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0263-8223/95/$9.50

Advanced calculation of the room-temperature shapes of thin unsymmetric composite

laminates

M. Schlecht, K. Schulte Technical University of Hamburg-Harburg, Polymer/Composite Section. 21071 Hamburg, Germany

&

M. W. Hyer Virginia Polytechnic Institute and State Universi& Department of Engineering Science and Mechanics, Blacksburg VA

24061-0219, USA

It is well known that the room-temperature shapes of unsymmetric laminates do not always conform to the predictions of classical lamination theory. Instead of being saddle shaped, as classical lamination theory predicts, the room-temperature shapes of unsymmetrically laminated composites are often cylindrical in nature. In addition, a second cylindrical shape can sometimes be obtained from the first by a simple snap-through action. Hyer developed for the class of all square unsymmetric cross-ply laminates which can be fabricated from four layers, i.e., [OJ90], [OJ90/0], [O/90/0/90], [OJ90,], an extended classical lamination theory to predict whether these laminates have a saddle shape or one or two cylindrical shapes. The Finite Element Analysis (FEA) has just recently been used for the calculation of the room-temperature shapes of unsymmetric laminates, because more sophisticated finite element codes are now available and the calculations can be made in an acceptable time. The hope is to get more accurate results for the shape and the stresses and forces that occur during the snap through action. These results are needed for the development of active deformable composite structures based on unsynnnetric laminates and incorporated shape memory alloy wires [Schlecht, M. & Schulte, K., Development of active deformable structures due to thermal residual stresses and incorporating shape memory alloys. In Proc. ECCA4 Smart Composites Workshop, ECCM6, Bordeaux, 1993, pp. 11520.1 Results for different lay-ups are presented and compared.

1 INTRODUCTION

The room-temperature shapes of unsymmetric laminates are a result of residual thermal stres- ses. Residual thermal stresses (RTS) are induced in composite laminates due to the anisotropic expansion properties of the constitu- ents. During manufacturing, laminates are subjected to curing stresses. Because of the thermal expansion mismatch between plies with different fibre orientation, the RTS manifest themselves in warping unsymmetric laminates. The cured shapes of thin unsymmetric lami- nates do not conform to the predictions of classical lamination theory (CLT), as had been

reported by many investigators. Rather than being saddle shaped, as predicted by classical lamination theory, thin unsymmetric laminates are cylindrically-shaped or even exhibit a snap-through phenomenon with two room- temperature shapes (Fig. 1). Hyer conducted a series of experiments with several laminate lay- ups and developed a theory for the calculation of [Od90,] and [04/904 laminates.2-4

With the extended CLT is was possible to predict the shapes of the class of all square unsymmetric cross-ply laminates which can be fabricated from four layers, i.e., [O/90/0/90], [0,/90], [Od90/0], [02/902] laminates. The algebra associated with this family of laminates is sim-

627

628 M. Schlecht, K. Schulte, M. ?K Hyer

a) at curing tampsratura b) 1 = RT, saddle drape

c) T I RT, a cyllndriwl shape d)T=RT,theolh8fpouibh cylindrical shape

Fig. 1. Laminate shapes: (a) at the elevated curing tem- perature. At room-temperature, (b) saddle shape, (c)

cylindrical shape, (d) the other cylindrical shape.

pler than the algebra associated with other families, because some of the A,, B, and D, terms are zero in this family. One disadvantage is that the simplifying assumptions have to be made for each class of laminates. To generalize the calculations, the Finite Element Analysis was chosen. Well chosen boundary conditions should lead to good results independent of the laminate lay-up. The finite element code has to be able to deal with temperature dependent material properties, with the nonlinearities occurring, and the problem of multiple solu- tions. With the finite element code Marc, good results could be achieved in a very short time. The first calculations showed that the whole simulation is very sensitive to the material data used, the temperature steps for cooling down and the number of layers in the model. The final results are used to compare the calculated geometries and to prove that the FEA model is suitable to calculate the room-temperature shapes of different classes of laminates.

2 MATERIAL PROPERTIES

For the investigations BASF Rigidite @ 5250-2/M40B/6k-50B prepregs were used. The NARMCO resin system 5250-2 is a 177”-204” curing bismaleimid resin with a service tempera- ture range of - 59”-204”. The TORAY M40B-6k-50B carbon fibres are high modulus fibres (HM, E=392 GPa). The laminates were cured in a laboratory hot press. All laminates were cooled down below 50°C while holding pressure. The recommended standard post-cure has not been carried out.

20-

lo_

O_

-103’

-4- O”-laminate

- 9004aminate

150 190

Temperature (“Cl

Fig. 2. Temperature dependent thermal expansion coef- ficients in fibre direction and transverse to it.

2.1 Thermal properties

The temperature dependent thermal expansion coefficients were determined by the use of Digi- tal Speckle Pattern Interferometrie (DSPI). The specimens (20 x 50 mm2) were cut from plates with [O], respectively [9018 lay-ups. The speci- men surface was coated with white paint to obtain better images for the digital image pro- cessing. The results of the measurements are shown in Fig. 2.

The values for the thermal expansion coeffi- cients in fibre direction (ao) were calculated from the values measured for the 25” direction, because they are too small to be determined directly. There is no linear dependence between temperature and thermal expansion coefficients. In the case of the o”-lay-up the thermal expan- sion coefficient slightly increases in the temperature range from 30°C to 120°C and then decreases and becomes negative. In the case of the 90”~lay-up the thermal expansion coefficient increases relatively fast. These results show that the temperature-dependence of the thermal expansion coefficients has to be considered. For the calculations with the extended CLT con- stant values are used (a,=2*27 10m6/K, a2=2*60 10-5/K).

2.2 Mechanical properties

The mechanical properties were also measured temperature dependent in the range from 20°C to 170°C (Fig. 3). The mechanical properties at room-temperature are summarized in Table 1.

For the calculations with the extended CLT the material properties at room-temperature are used. The use of temperature dependent material data is possible but would not change the basic results.

Room-temperature shapes of composite laminates 629

E2 Wal

WOE+03

7,3OE+O3

7,3OE+O3

7,70E+O3

7,6OE+O3 \ -a- msssumd

- 0ItsdE”rvs

7soE+03 i Temperature [“Cl

Fig. 3. Transverse modulus (E,) vs. temperature.

Table 1. Mechanical properties at room-temperature

252 7.81 6-19 1629 22.97 51.0 O-32

3 CALCULATION OF ROOM- 3.2 Simulation with extended classical TEMPERATURE SHAPES lamination theory

The calculations of the room-temperature shapes were performed with the extended CLT and the FEA for following laminates: [02/902], [O/90,], [O/90/0/90], [90/O/90,], [OJ90,]. The cal- culated shapes will be compared. Some results of measured shapes are available and will also be presented. With the FEA other laminates were considered, namely, [O/ + 45/-45/O], [O,!-45,], [O/-45/90/0] and [O/+45/90/-45].

3.1 Simulation with finite element methods

For the simulation of the warping of the com- posite laminates the FEA-programme MARC (K5.2) was used. This programme is able to calculate the behaviour of laminated structures with anisotropic and temperature dependent properties.

The model has the same size as the manu- factured laminates (140 x 140 mm”). The whole mesh exists of 196 &node bilinear shell-ele- ments. Each element has a size of 10 x 10 mm’. For each layer of the real laminate two layers are used in the model. Different laminate lay- ups require different boundary conditions, because the expected deformations have to be allowed. The laminate is cooled down from 177°C (curing temperature) to 157°C in steps of 2°C followed by steps of 5°C down to 20°C (room temperature).

The convergence testing is based on strain energy, because this is an energetic problem. The use of residuals or displacements for con- vergence testing usually only leads to the saddle shape and not to cylindrical shapes. Large dis- placements are specified, so the geometric stiffness matrix and the initial stress matrix are calculated. The residual loads are corrected, such that the solution is always in equilibrium. For this correction the stresses at all integration points have to be stored. The solution of non- positive definite systems is forced.

For the simulation of the snap-through effect first one room-temperature shape is calculated. Then the four corner nodes are moved in the direction of the supposed second shape. After the snap-through the boundary conditions were changed to the old configuration. This is the proof as to whether the second shape is stable or not.

Hyer developed his theory for the calculation of the room-temperature shapes of unsymmetric laminates in two steps. In step one he only used [Od90,] and [OJ90,] laminates and a simple approach for the in-plane displacement fields. In step two he used all four-layer cross-ply lam- inates and a more sophisticated approach for the in-plane displacement fields. But the effect of the more sophisticated displacement fields on the predicted out-of-plane deformations and the in-plane strains is negligible, so the first model was used for this investigation.

Because Hyer’s theory is well documented in literature,2-4 only his basic ideas will be pre- sented in this paper. The classical lamination theory predicts the room-temperature shapes of all unsymmetric laminates to be a saddle with unique curvature characteristics. So this theory is unable to predict the shapes of thin unsym- metric laminates. To explain the behaviour of unsymmetric laminates, it was assumed that it would be necessary to incorporate a non-linear effect into classical lamination theory. The existence of two room-temperature shapes (i.e., the two cylinders) essentially ruled out a linear extension to the theory, since a linear extension would lead to the prediction of a unique shape, although perhaps not a saddle. Furthermore, since the out-of-plane deflections of the unsym- metric laminates were in the order of many

630 M. Schlecht, K. Schulte, M. W Hyer

laminate thicknesses, geometric nonlinearities were felt to be an important effect. So the clas- sical lamination theory was extended to include geometric nonlinearities through the strain-dis- placement relationship. Possible shapes are those with a minimum of the total potential energy, which leads to multiple solutions of this non-linear problem. Thus, for stable equilib- rium the second variation of the total potential energy, S’W, must be positive definite. The equation for the total potential energy, includ- ing the effects of thermal expansion, is given by:5

W= s

od Vol Vi31

(1)

where w equals the strain energy-density. The Cijkl are the elastic constants of the material and the flij are coefficients related to the elastic constants and the coefficients of thermal expan- sion of the material. The eti are the strains in the material and AT is the temperature change in the material due to cooling from curing. For a laminate the C,‘s can be related to the Qg’s, the reduced stiffnesses, and the pij’s can be related to the e,‘s and the a,, aY, aV, the lam- inate thermal expansion coefficients. Expanding eqn (2) yields (a,=0 for this family of lam- inates)

++ e22et2-(~211a,+el2~)ellAT

- (&cc, + &ay)edT. (3)

Hyer approximated the displacements in the main directions with the following equations:

w&y)=+ (a.x2+bY2) (4)

a2x3 abxy2 u”(x,y)=cx-----

6 4

b2y3 abx? v+,y)=dy-T--

4

(5)

At this point a, b, c and d are considered as generalized coordinates and are to be deter- mined. The problem of finding a minimum for the total potential energy, W, becomes a prob- lem of finding solutions to the values of a, b, c and d so that the first variation of W is zero.

3.3 Results

The aim of this investigation was to determine whether the FEA is able to predict the room- temperature shapes of thin unsymmetric laminates and what the differences are com- pared to the results of the extended CLT.

First of all, there are some obvious differ- ences. The calculations with the extended CLT can be performed with ‘Mathematics’ on an Apple Macintosh Quadra in a few minutes. As a result you get all possible shapes, and the information whether the calculated shapes are stable or not. A variation of the material prop- erties in a certain range only leads to a variation of the calculated curvatures but usually not to a change of the predicted stable shapes. To perform the simulation with the FEA you need several hours, even on a power- ful workstation. As a result you only get one possible stable shape. For the calculation of a second stable shape the snap-through has to be simulated. Slight changes of the material prop- erties can lead to the calculation of a saddle-shape instead of a cylindrical shape, especially if the material data are not com- pletely consistent. On the other hand, all stresses, strains and reaction forces are available without any additional effort. One gets more information about the laminate. Up to now it seems that the extended CLT is easier to use and causes less problems. The FEA takes much more time but one gets more information about the laminate. Therefore the results are com- pared for some laminates, to show the basic differences.

Figures 4 and 5 show the results for a [O.#O,] laminate calculated with the extended CLT and FEA. There is only a slight difference in the maximum displacements in z-direction. A change in the material properties used for the extended CLT could easily lead to the same displacements. The shape calculated with the extended CLT shows no edge effects. Only a unique curvatureyin one direction can be seen. The curvature in the second direction can be neglected. The edge with no major curvature

Room-temperature shapes of composite laminates 631

Displacements z [mm] 140 xA

P Y Z

Fig. 4. Room-temperature shape of a [0~90&aminate calculated with extended CLT.

$1 12-14

r3 l&l2

n 8-10

&a

n 4-6

w2-4

U&2

Displacements z [mm] 140 xA

c' Y Z

Fig. 5. Room-temperature shape of a [OJ90,]-laminate calculated with FEA.

seems to be straight. The shape calculated with FEA shows a clear edge effect. The edge with the smaller curvature is not straight. It has a slight saddle shape. The displacement in the z- direction in the corners is not as high as in the middle of the laminate.

The edge effect can clearly be seen in Fig. 6, where the difference between the two shapes was calculated. The greatest differences are at the corners. The calculated temperature at which the saddle shape converts to the cyhndri- cal shape is 169°C. The shape of the real laminate is very close to the shape calculated with FEA. First results prove that this laminate has a clear edge effect which can be clearly seen in the corners of the laminate.

The same comparative calculations were also performed for the other laminates mentioned in this chapter. The basic results are always the same. The shapes calculated with FEA show a

clear edge effect, those calculated with exten- ded CLT not. Both methods give the same results, whether or not there are one or more stable shapes.

The [O/90/90/90]-laminate has one stable cylindrical shape as predicted from the FEA and the extended CLT. The calculated maxi- mum displacements is in both cases 33.6 mm. The shape calculated with FEA shows a clear edge effect, but in this case there is a greater difference in the calculated curvature itself (Fig. 7). This seems to be a higher order effect, because Hyer only uses a second order equation for the out-of-plane displacements.

For the [Od90,]-laminate the FEA and the extended CLT both predict two stable shapes. The maximum calculated displacements in the z-direction for one cylindrical shape are 15.0 mm (extended CLT) respectively 15.9 mm (FEA), whereby the measured displacement is

632 M. Schlecht, K. Schulte, M. K Hyer

q 0,25-o&50

n o,oo-o,25

w5-wJ

n 450475

n 0,754 ,oo

0 -1,~1,25

Dlspiawments z [mm]

Fig. 6. Difference between the shapes calculated with extended CLT and FEA for a [0J902]-laminate.

- ext. CLT b FEA v diff.

35

31

27

23

displacements z ” [mm1 ,s

11

7

3

-1 0 140

xtoordlnate

Fig. 7. Difference between the shapes calculated with extended CLT and FEA for a [0/903]-laminate (y=70 mm).

16.0 mm. The manufactured laminate only has one stable cylindrical shape. Before one gets a second shape the laminate would crack. The second shape predicted from extended CLT is cylindrical with a maximum displacement in z- direction of 4 mm. But this second shape is only predicted below a temperature of 112°C. At higher temperatures only one stable shape is predicted. The second shape predicted with the FEA is a saddle shape with different curvatures in the two main directions. The forces needed to snap the laminate into this second shape are nearly three times as high as the forces needed to snap the [02/902]-laminate. Although both calculation methods predict two stable shapes there is a hint, that the second predicted shape is not as stable as the first one or that the second might be impossible.

The [90/O/90/90]-laminate has one stable cylindrical shape as predicted with both meth- ods. The calculated maximum displacements correspond with the measured ones. For the [O/90/0/90]-laminate two stable shapes are pre- dicted with both methods. The FEA results show a clear edge effect. For this laminate no experimental results are available.

The results for laminates with 45”-layers have only been calculated with the FEA, because the necessary algebra to calculate this class of lam- inates with the extended CLT has not yet been developed. All laminates calculated with FEA were manufactured subsequently to verify the results. Only the maximum displacements in the z-direction were measured. The [O/ - 45/ + 451 O]-laminate has a saddle shape. The [O/ -45/ 90/O]-laminate has one stable cylindrical shape

Room-temperature shapes of composite laminates 633

iI2632

124-20

n 20-24

n 16-20

n 12-16

n 3-12

w4-3

n o-4

Displacements 2 [mm]

‘2

Fig. 8. Room-temperature shape of a [O/ + 45/90/ - 451 laminate calculated with FEA.

and the [02/--45,]-laminate and the [O/+ 45/90/ -45]-laminate (Fig. 8) both have two stable cylindrical shapes. The experimental results and the measured maximum displacements in the z- direction correspond to the results of the FEA. This indicates that the used FEA model is able to deal with different classes of unsymmetric laminates.

4 CONCLUSIONS

The FEA as well as the extended CLT are able to predict the room-temperature shapes of thin unsymmetric laminates. The results have been proved experimentally. Hyer extensively calcu- lated the size influence on the reported effect, but there still is a lack of experimental data. The calculations with the FEA were performed for only one laminate size, but for laminate lay- ups which have not been studied before. The results for the laminates with 45”-layers demon- strate that the FEA is a well suited tool to calculate the room-temperature shapes of dif- ferent classes of laminates. The major difference between the results from both meth- ods are the displacements in the z-direction calculated for the edges. First experiments show that the measured shapes are closer to those calculated with FEA than to those calculated with extended CLT. Further, the FEA is able to handle more complex geometries without any assumptions regarding the room-temperature shape. In addition more information about the

internal stresses and forces needed to initiate a snap through action are raised, with the only drawback that a powerful workstation is needed and more time to perform the calculations.

ACKNOWLEDGEMENTS

The authors very much acknowledge the assis- tance of Frau Dipl.-Ing P. Aswendt (‘Fraunhofer Einrichtung fur Umformtechnik und Werkzeugmaschinen’, Chemnitz, Germany) for determining the thermal expansion coeffi- cients and the Deutscher Akademischer Austauschdienst for the funding of a part of this work.

REFERENCES

1.

2.

3.

4.

5.

Schlecht, M. & Schulte, K., Development of active deformable structures due to thermal residual stresses and incorporating shape memory alloys. In Proc. ECCM Smart Composites Workshop, ECCM 6, Bor- deaux, 1993, pp. 115-20. Hyer, M. W., Some observations on the cured shape of thin unsymmetric laminates. J. Comp. Mat., 15 (1981) 175-94. Hyer, M. W., Calculation of the room-temperature shapes of unsymmetric laminates. J. Comp. Mat., 15 (1981) 296-310. Hyer, M. W., The room-temperature shapes of four- layer unsymmetric cross-ply laminates. J. Comp. Mat,, 16 (1982) 318-40. Fung, Y. C., Foundations of Solid Mechanics. Prentice- Hall Inc., Englewood Cliffs, NJ, 1965.