Prediction of thermal expansion coefficients of sandwiches using finite elements methods validated by experimental test results

Acla Astronautica VoL 10, No. 5-6, pp. 409-427. 1983 Printed in Great Britain.

0094-5765183 $3.00+.00 Pergamon Press Ltd.

PREDICTION OF THERMAL EXPANSION COEFFICIENTS OF SANDWICHES USING FINITE ELEMENTS METHODS VALIDATED BY EXPERIMENTAL TEST RESULTS M. MARCHET17 Istituto di Tecnologia Aerospaziale, University of Rome, via Eudossiana 16, 00184 Rome and F. MORGANTI Space Division, Soc. Selenia, Rome, Italy

(Received 6 January 1983) Abstract--The new generation of telecommunicationantennas for space applications working at frequencies above 20 GHz (which will be generally manufactured by sandwich structures) needs an accurate thermostructural design due to the envisaged increase of the dimensions of the reflectors and the related decrease of the required RMS thermal distortion. In an integrated design which takes into account the reciprocal influence of thermal and structural aspects, the thermal expansion coefficient(C.T.E.) of the structures which are part of the reflector assembly, is a very important parameter. A good prediction of this value is a contribution to an accurate estimate of the electrical performance of the antenna system in the space environment. This work describes a prediction method to evaluate the C.T.E. of a sandwich validated by some experimental results and checked by an analytical study. Numerical results illustrate the procedures; graphs and appendices complete the work.

1. INTRODUCTION The new generations of transmitter and receiver antennas utilized for telecommunication satellites require a large employment of working frequencies in the bandwidth up to 20-30 GHz, to avoid bandwidth saturation and interference problems. Italy has pioneered the use of high frequencies in this field [1], starting from the SIRIO project, with experiments in the 12 and 18 GHz bands. The success of this program has encouraged the promotion of the development of a satellite for national purposes, ITALSAT, with communication systems operating in the 20-30 GHz band and the possibility to explore the 40--50 GHz band [2]. The National Space Program is obviously complementary to the international ones, mainly to the ESA (European Space Agency) programs, to which Italy contributes. Actually, under ESA commitment, the telecommtinication satellite LSAT is under development, with an experimental payload at 20-30 GHz. ESA have in parallel finalized some advanced activities concerning the main critical areas regarding the use of such frequencies. Selenia have been and are actually involved, in the frame of the ASTP programs, in several study activities concerning the electrical [3] and the structural aspects. These activities, that are still under development, will consist in a finalized electrical and mechanical design of a 20-30 GHz multibeam antenna system with maximum size 4m (note that LSAT will have two 20-30GHz reflectors with maximum size 1.2 m and ITALSAT two reflectors with maximum size 2 m).

The increase of the operating frequencies has in general a large impact on the reflector design, either from manufacturing or from thermostructural behaviour [4] point of view, and these aspects become more important for the future applications, for which is foreseen an increase of the reflector dimensions, necessary to obtain high gains to overcome the atmospheric attenuation. The problem is to estimate if the current technology is still valid to ensure, also in this case, the correct performance of the antennas. The study performed for the ASTP program has shown that the traditional technology employing sandwich structures made with C.F.R.P. [5] skins and aluminium honeycomb core is yet capable, even if marginally, to reach the necessary requirements with a good degree of reliability. The critical areas which have been found are essentially two: --manufacturing process in order to have the maximum degree of accuracy in the reflecting surface minimizing the related residual RMS, ---optimization of the reflector design taking into account both thermal and structural aspects and in particular minimizing the expected distortion RMS due to the space environment. The present work will emphasize some aspects regarding the second point, which is actually retained as the most important. In effect, the expected in orbit distortion RMS of such antennas is larger than the manufacturing one; this latter can be in fact improved by ground procedures, while the former can be only in part 409

410

M. MARCHETT[and F. MORGANT!

improved utilizing antenna pointing mechanisms (APM). In particular it has been found that the distortion RMS value is very sensitive to the ratio between the Thermal Expansion Coefficients (C.T.E.) of the reflector dish and its backing structure. So that, in an optimised thermostructural design, it is fundamental to find the best

compromise between the worst temperature distribution and the C.T.E. of the coupled structures. A good prediction of this value is therefore very important to define the worst design case and the related RMS, The C.T.E. values which are requested are generally very low (in general very close to 10-6°C-1)

Prediction,of C.T.E. of sandwich panelsI

Numerical approach (Finite ElementMethod)

"1

ra oson

Various Model

I

Analytical approach

II

Definition of the Model

Manufacturingof Sandwichesspecimens

Choiceof the model

Analysison two referencecases

I

Analysison two referencecases

Testing activities

t l Checkingof the prediction i methods

i

l

ii

Analysison sandwichesspecimens

-{

Fitting betweenprediction methods and test results

i

1

V l

Setof TradeOtison Various Sandwiches Fig. 1. Flowdiagram.

]

Prediction of thermal expansion coefficients and that is the reason why it is necessary to have available precise analytical or numerical methods of prediction, eventually validated by experimental results to increase their reliability. The fitting between numerical and test results depends on the analytical approach, the input data to it, the test procedure and the choice of the specimens. This work has been conceived to cope these thematics, mainly concerning the C.T.E. of sandwich structures which are foreseen for large antennas applications [6]. The inputs of the prediction methods are the experimental thermomechanical properties of the skins and of the adhesive. The output will be validated by the results of suitable tests on sandwich specimens. The numerical prediction method concerns sandwiches having 6.35 and 12.7 mm core thickness, utilizing some reference values for the skins properties, but it is easily generalizable for other cases. A NASTRAN model of the sandwich is utilized with boundary conditions schematizing the continuity. In parallel to this numerical approach an analytical one is developed mainly to identify some physical aspects of the problem, which cannot be easily interpretated by the finite element method. The validation is made by testing sandwich specimens from -120 to + 100°C by means of an high precision dilatometer of the "Istituto di Tecnologia Aerospaziale" of Rome University. The specimens have been provided by Society Selenia of Rome having 6.35mm aluminium core thickness with C.F.R.P. skins 0/90190/0 plied up. In Fig. 1 the Flow Diagram briefly describes the connection among all the aforesaid activities.

411

J Fig. 2. Sandwichportion schematizedby finite elements. Suitable boundary conditions and internal constraints have been considered to impose the symmetries (i.e. the continuity of the material) and the congruence at the interface between honeycomb and skins. The membrane elements have been chosen because the structure, in its nominal configuration, when undergoing an homogeneous variation of temperature, is interested in plane stress-strain fields without bending effects. The constant thermal load has been given to all grid points by means of the TEMPD card. The model is detailed enough to put in evidence the concentration of stresses in the structure. 2.2 Choice of materialst The materials chosen for the trade-offs, considering the constraints given by the concept of a larger reflector design, are the following:

2. PREDICTION OF C.T.E. IN SANDWICH PANELS: NUMERICALAPPROACH

The basic concept that has been developed is to build up a detailed finite element model to be mathematically validated by an analytical method. By the finite element model it is possible to put in evidence the main physical aspects which characterize the behaviour of sandwich structures working under plane stress and strain fields [7]. A suitable set of trade-offs has been carried out to this purpose. 2.1 Finite element model The NASTRAN computer program has been utilized as finite element method [8]. Considering the two symmetry planes of the honeycomb cell, only the portion of sandwich having these planes at the edge has been considered (Fig. 2). The structure has been schematized with membrane elements for C.F.R.P. skins and aluminium honeycomb and with rod elements for the adhesive. The overall mathematical model consists of 161 grid points, ll2 elements CQDMEM1, 15 elements CTRMEM, and 15 elements CROD. The mesh of the model is shown in Figs. 3(a, b). tNomenclature is given in Appendix 1 at end of paper

Fig. 3(a) Finite element mesh of sandwich honeycombcore.

412

M. MARCHETTIand F. MORGANTI

----

].....

(c) Adhesive. The current adhesive for this application is CIBA Redux 312L or UL, with the following average thermostructural characteristics: Young modulus EAx = EAy = EA = 2940 MN/m 2 C.T.E. aAx = SAy = aA = 50-- 60 X 10-6°C t Active section after polimerization (average value) AA = 0.50 mm 2.

_ ~:

j ! _

_

-j,"

Fig. 3(b). Finite element mesh of sandwich skins.

(a) Reflector skins. The kind of skin which reaches a good compromise among weight, stiffness and C.T.E. requirements is the 019019010laminate made by ultra high modulus fiber layers. The current thermostructural parameters which can be considered for this laminate are in the following ranges: Young Modulus Poisson ratio Shear modulus C.T.E. Thickness (one skin)

Esx = Es~ = Es = 112,700 - 156,800 MNlm 2 Us = 0.02 - 0.04 Gsxy = Gs = 4704 - 7840 MN/m 2 t~sx = asy = t~s = 0.2 - 0.5 x 10-6~C s = 0.25 - 0.3 mm.

The Young Modulus, Poisson ratio and C.T.E. values result from analytical results validated by test performed in Selenia and in "Instituto di Tecnologia Aerospaziale". The shear modulus are referred only to analytical results. (b) Honeycomb. Two kinds of aluminium honeycomb can be considered reaching a good compromise between the shear strength and C.T.E. of the sandwich, with the following characteristics: a=3/16in, a = 3/8 in. Young modulus Poisson C.T.E.

t=0.01778mm t = 0.0254 mm

c=6.35mm c = 12.70 mm

EHx = Eny = En = 71540 MN/m 2 un = 0.34 aHx = any = an = 23× 10 6~C-I.

2.3 Trade-offs philosphy The trade-offs have been conceived with the purpose of putting in evidence the influence of some parameters on the sandwich behaviour and to check the effect of some physical assumptions on which the schematization of the structure is based. Regarding this second aspect, it is well known that the bonding between honeycomb and skins impacts very much on the behaviour of the structure, so this is a very important area to investigate. Two main aspects have been considered to this intent: (a) to find the best modelization which schematizes the bonding between honeycomb and skins, taking into account the implications on the congruence inside the structure, (b) to evaluate the main parameters which characterize and determine the bonding effects. The aim of this work is not a detailed study of the mechanics of the bonding, so that only the implications on the overall effects at structural level will be taken into account. It is clear that there are several second order aspects, mainly resulting from the technological process, such as the not homogeneous adhesive distribution on the bonding areas, the effect of internal stresses in the structure due to the bonding itself, microcreeps in the adhesive at the interface between skins and honeycomb, which actually are not considered. Another physical aspect which is important to investigate concerns the effects of the various deformation shapes of the honeycomb, respecting the internal congruence, on the sandwich behaviour. This aspect is connected in part with the previous one, because the distribution of the adhesive has a large influence in constraining the deformation of the honeycomb. Having verified the validity of the physical assumption which are on the basis of the structural schematization, it is possible to devise a suitable model which will be used to run a set of trade-offs in which the thermostructural characteristics of the honeycomb and the-skins will be varied. The outputs of this activity are the previsional C.T.E. values of some sandwiches of more interest. 2.4 Bonding schematization What is required for the schematization is to provide the congruence between the skins and the honeycomb and exact stresses in the structure, in particular inside the adhesive. These latter are given by two main contributions: --the forces which arise at the interfaces with the skin and the honeycomb due to the congruence, that are practically shear forces on the adhesive (Fig. 4),

Prediction of thermal expansioncoefficients

413

With the purpose of evaluating the impact of another schematization on the final results, another modelization has been considered in which the congruence has been obtained by means of very stiff rod elements, in parallel to those utilized to take into account the properties of the adhesive inside the structure. The trade-off has shown results in accordance with the previous approach, either in terms of C.T.E. of the sandwich, or regarding the stresses in the adhesive. The structural elements utilized to schematize the adhesive are the NASTRAN CROD, which are assumed the best interpretation of the shape of the adhesive film after the polymerization.

Fig. 4. Shear forces on the adhesive.

--the forces which arise inside the adhesive due to its own coefficient of thermal expansion (Fig. 5). Note that the former ones are independent from the characteristics of the adhesive, and can be calculated by only imposing the congruence inside the structure. This fact points out that the best way to schematize the bonding is to impose the congruence without utilizing structural elements (employing the NASTRAN program, this can be obtained by MPC cards) putting them in the model only to take into account the contribution of the adhesive in the C.T.E. of the sandwich. In effect this approach has been preferred, even if it does not directly provide the shear stress on the adhesive, which can be derived from the forces acting on the honeycomb and on the skins.

2.5 Constraints on the honeycomb deformation A dedicated set of trade-offs has been carried out in order to evaluate the impact of the deformation of the honeycomb on the sandwich behaviour. Some limit cases have been modelized and analyzed, as described below: (a) Perfect congruence of the displacements of all interfacing grid points of the honeycomb and of the skins. Furtherly, the constance of the displacements of all grids lying on each honeycomb cross section has been considered, constraining the deformed shape to be parallel with respect to the indeformed one. The displacements are sketched in Fig. 6. (b) Perfect congruence of the displacements at the honeycomb/skins interface, as in (a), but without imposing the constance of the displacements in the honeycomb cross sections. The displacements are sketched in Fig. 7. In this case the honeycomb deformation leads to a distortion of the elements with a consequent bending of the cross sections, symmetrically with respect to the symmetry plane of the sandwich. This fact does not occur at the edge of the model, due to the symmetry conditions. Note that the oblique side of the honeycomb has an antisymmetric deformation with respect to the

i

J

Fig. 5. Thermal forces inside the adhesive.

Fig. 6. Constant displacementin each honeycomb section.

414

M. MARCHETrland F. MORGANTI (d) As (c), but imposing the constance of the displacements to the grids of the honeycomb corner sides. (e) Congruence only at the edge of the model. In this case only the symmetry congruence conditions are respected and the honeycomb can have, inside the sandwich element which has been considered, a deformed shape completely independent from the skins one Fig. 9). This is not a realistic case but it is useful to interpret some results carried out from the analytical approach (See Section 3.3). (f) As point (a), but considering the honeycomb with all cell sides with the same thickness. Also this trade-off is useful to interpret some physical implication of the problem (see Section 3.3). In Table 1 a matrix which describes the degree of freedom (D.O.F.) constrained to be constant in each honeycomb section is provided, referring to the examined cases. The symmetry conditions are also included between brackets. The condition of congruence which have been imposed are indicated by the letter identifying the section.

Fig. 7. Not constant displacement in each honeycomb section.

2.6 Choice of the model reference sandwich configura-

antisymmetry plane of the sandwich which is defined in Fig. 8. (c) Perfect congruence of the displacements at the honeycomb/skins only in correspondance to the corner side of the honeycomb and without imposing any constance of displacements in it, as in (b). The congruence at the edge of the model is also included because it results from the symmetry conditions, which must be in any case respected.

tions

The choice of the model to be run for the trade-offs has been made on the basis of the analysis results concerning the limit cases described in Section 2.5. To F

" E I

--r

--[--

i

i

÷ !

,

__r

i

Fig. 9. Deformed shape.

Fig. 8. Identificationof the antisymmetry plane. Table 1. Model matrix

F G a Congruence

b

ABCD

ABe)

EFC

EFG

c ACEG

d ACEG

e AG

f ABCD EFG

D.O.F. 1

2

(ABCEFG) IABCEFG) IABCEFG) [ABCEFG) IABCEFG) ABCEFG) D D D (AG) BCDEF

(AG)

(AG)

(AG)

(AG)

(AG)

CE

BCDEF

BCDEF

415

Prediction of thermal expansion coefficients

symmetry plane in addition to the other two. Under these conditions only a deformation without distortion of the hexagon shape is possible. The most representative model of the physical problem under concern lies certainly between the cases (a) and (b). On the other hand, considering that the deformation of the honeycomb is not so free as it has been schematized in (b), due to the presence of the bonding, which has not zero thickness and undergoes a greater concentration just in the corner areas, the model (a) is preferred, which provides also more conservative C.T.E. values. It must be noted that cases (b) and (e) are rather similar, i.e. making free in some way the deformations of the corner sides of the honeycomb, a quasi-isotropic behaviour of the sandwich results. The tendency to a thermal isotropy of the sandwich is coupled with a quick decrease of the shear stresses in the skins. This effect will be examined also in Section 3.4.

this purpose, two suitable types of sandwiches with some skins and different honeycombs have been considered, which shall constitute the so called "reference configurations" (Table 2). The bonding has been schematized as described in point 2.4, i.e. using multi-point constraints to achieve the congruence between skins-honeycomb and element rods to simulate the presence of the adhesive. The honeycomb deformation has been constrained considering the limit cases of Section 2.5. The C.T.E. results are shown in Table 3 referring to the two configurations. They show that the most influent physical effects on the thermal expansion behaviour of the sandwich are fundamentally two: --presence of the congruence conditions at the skins/honeycomb interface in correspondence to the corner side of the latter, ---constraints on the deformation of the honeycomb in the corner sections. Moreover, observing the equivalence of the results of the cases (a), (d) and (b), (c), it is evident that the presence of additional congruence conditions at the skins/honeycomb interface (except obviously the extreme sections of the element) has no influence on the C.T.E. values. The case (e) provides an isotropic behaviour of the sandwich (Section 3.4) averaging the extreme C.T.E. values previously obtained, but it is not of interest because it is too little accurate. It is only useful to check the good working of the model. Also the case (f) provides an isotropic behaviour of the sandwich and it is interesting, even if not applicable, to evaluate the effect of the double thickness ribbon of the honeycomb, which is practically the only cause of the difference between ax and ay, providing that quasi-isotropic skins are considered. In this case in fact, having the ribbon the same thickness as the other sides, the antisymmetry plane of the structure (Fig. 8) turns into a

2.7 Trade offs results The trade-offs have been run utilizing the model 2.5(a) considering the variation of some parameters as already anticipated in Section 2.2. The trade-offs matrix is shows in Table 4, in which are identified three different cases A, B,C. Table 4. Trade-offs matrix Honeycomb type Trade-otis

Case A Case B

c = 12.7 mm a = 3/8 in. t = 0.0254 mm

c = 6.35 mm a = 3/16 in. t = 0.01778 mm

E~ = 156,800MN/m2 G~ = 7840 MN/m2 s = 0.25 mm E~ = 112,700 MN/m2 Gs = 4704 MN/m2 s = 0.30 mm

Es = 156,800MN/m2 Gs = 7840 MN/m2 s = 0.25 mm E~ = 112,700MN/m2 Gs = 4704 MN/m2 s = 0.30 mm Es = 112,700MN/m2 G~ = 4704 MN/m2 s = 0.25 mm

G~

Case C

Table 2. Reference configurations Honeycomb

C

E

(ram)

(MN/mz)

vn

(1) ~"/505210.0007 in/2

6.35

71,540

(2) ~-~/5052/0.001 in 1.6

12.7

71540

t

a

(mm)

(10 6*C-I)

s

0.34

0.01778

23

0.34

0.0254

23

( r a m ) (MN/m2)

AA

(mm2)

Skins 019019010 laminate

156,800

0.5

(GY70/Code 96 Resin) Adhesive CIBA Redux 312

2940

50

0.25

7840 0.5

Table 3. Numerical results Conf.

(a) ~x

1

2

~y

1.14 1.37 1.28 1.58

(b) ~tx

~ty

1.21 1.25 1.36 1.42

(c) Otx

~ty

1.21 1.25 1.36 1.42

(d) £1~x

ay

1.14 1.37 1.28 1.58

(e) Otx

Oty

1.21 1.23 1.34 1.35

(f) ~x

O~y

1.13 1.13 1.31 1.31

416

M. MARCHETrland F. MORGANTI C.T.E., x and y components, versus the skins Young modulus (Es) and C.T.E. (a~) (case A, C). Figure 11 shows 6.35 mm and 12.7 mm core thickness sandwich C.T.E., x component, vs the skins C.T.E. (as) in the case A for both values of aA. Figure 12 shows sandwich C.T.E., x and y components, vs the core thickness (c) and the adhesive

The C.T.E. results are synthetized in Table 5 for each type of sandwich. Note that the results concerning the reference configurations are marked by *. In order to better understand which parameters have the stronger impact on the C.T.E. values, the following graphs would be helpful. Figure 10 shows 6.35mm core thickness sandwich

(7. • I O - 6 " C - ' %.

~

\A.

0

.

.....

\

\

\

1.6 -

\

s = O.2.10-6oc

-'

(::ts = O . 5 . 1 0 " 6 ° C

~

MOD \

6.35

\ \ %`

\

%` \

%` \ %` %`%

1.4"~.

\%`

1,2-

1.OaA_-50,

I O ' 6 ' C -~

AA:O.50

mm

2

I

1

112700

156800

E I N/r"9

Fig. 10. C.T.E. of sandwich vs Es: numerical results. Table 5. Numerical results of trade-offs Trade-offs

Numerical results of sandwich C.T.E, (10-6~C i) Model 12.7

Adhesive

1 II [II IV V VI VII VIII

Skins

AA

etA

O~S

0.2 0.2 0.2 0.2 0.5 0.5 0.5 0.5

50 50 60 60 50 50 60 60

0.2 0.5 0.2 0.5 0.2 0.5 0.2 0.5

Case A ~

0.93 1.22 0,94 1,23 0.99 1.28' 1.01 1.30

Olv

1.24 1.53 1.25 1.54 1.29 1.58" 1.31 1.60

Model 6.35

Case B Ot~

fly

1.04 1.32 i.04 1.33 1.10 1.39 1.12 1.41

1.39 1.67 1.39 1.68 1.45 1.73 1,47 1,75

Case A 0~

0.74 1.03 0.75 1.05 0.85 1.14" 0.88 1.18

O~v

0.97 1.26 0.99 1.28 1.08 1.37' 1.12 1.41

Case B ~

0.82 1.11 0.83 1.12 0.94 1.23 0.99 1.28

Case C

Oly

Otx

Otv

1.08 1.37 1.10 1.39 1.21 1.49 1.25 1.54

0.94 1.23 0.96 1.25 1.09 1.38 1.14 1.43

1.24 1.53 1.26 1.55 1.39 1.67 1.43 1.72

417

Prediction of thermal expansion coefficients

a.

1Ge1~-1 a A : 5 0 . l d ~ ' O "~

/ /

~"

aA= 6 0 - 10-e "O"1

1.3

{12.7

//{6,35

// /

1,1

:/

-

, J/

0.9,

0:2

o's

a s 10"6"C "1

Fig. 11. C.T.E. of sandwich vs as: numerical results.

a - 10"6"C-1

-

-

.....

MOD.

12.7/A

MOD. 635/A _._4F__.~I

Y

1.2

~Y J

/ 1

/ /

1,0

x

..A I

/ x

J J J

0.8

1

/

O2 2

/

J

aA= 5 0 . 1 0 6 C "1

t

as= O. 2. I06 C -~ 0.'5

Fig. 12. C.T.E. of sandwich v s AA: numerical results.

A, (m r~ ~)

M. MARCHETTIand F. MORGANTI

418

active section (AA) for given skins C.T.E. (as) and adhesive C.T.E. (aA) in the case A. Figure 13 shows sandwich C.T.E., x and y components, (for 6.35 mm core thickness) vs the skin thickness (ts) and C.T.E., (as) (case B, C). 2.8 Evaluation of the results The trade-offs have shown a great sensitivity of the sandwich C.T.E. vs the Young modulus and the C.T.E. of the skins, taking into account the possible range of variation of these parameters. The sensitivity effects are generally more important in the 6.35 mm core thickness sandwich rather than in the 12.7 mm one, particularly regarding the influence of the adhesive C.T.E. and its active section. Note that the difference between the values of the two sandwich C.T.E. components ax and ay decreases vs the increase of the skins Young modulus Es. This fact is given, moreover, by the simultaneous increase of the skins shear modulus Gs (see Section 3.4).

date the mathematical model and to have more sensitivity on the impact that each parameter will have on the sandwich behaviour.

3.1 Analytical schematizations The analytical approach to determine the C.T.E. of sandwich composites will essentially consist in the solution of the classical equation of equilibrium of a nothomogeneous body which undergoes an homogeneous variation of temperature. This equation, taking into account the simplification that is given by the presence of a symmetry plane in the sandwich that eliminates bending effects, can be written as follows: {N} = [K] {a},

(1)

with {N} = .F.,,{N,}, [K] = E,[k,] and

3. PREDICTION OF C.T.E. IN SANDWICH PANELS: ANALYTICAL APPROACH

An analytical study of the sandwich under constant variation of temperature has been developed in order to put in evidence some effects which do not immediately result from the numerical data. It is also useful to vali-

{N,} = [kd{a,}, where i identifies the ith layer of the composite. Relationships such as (I) shall be written for each

O. " I ()6 =C -1

O,S= O. 2 - 1 0 6 oC-1 1.70

.....

(~S = O . 5 . 1 0 6 ° O -1

M O D 6.35

1.50-

¥

1.3G

Y 1.10-

aA-- 5 0 " 10 ~ s~ ~ AA-- 0 . 5 0 m r ra2 x I

0.2 5

0:30 Fig. 13. C.T.E. of sandwich vs ts: numericalresults.

t s (m m l

Prediction of thermal expansion coefficients composite portion in which the congruence of the strains of each layer must be respected. Various methods can be developed to determine {N} and [K] differing from each other in the way which is used to impose this internal congruence in the structure. The element considered in the schematization is corresponding to one honeycomb cell as shown in Fig. 14, considering that it is the smallest one that does not undergo any distortion, as supposed also in the numerical approach. Three kinds of schematization can be in principle considered, providing different accuracy in te results: (a) Impose the congruence between honeycomb and sandwich only at the edge of the cell element. In this case the effect of the bonding inside the element is lost because the honeycomb can have a deformation shape independent from the skins behaviour. This model reflects the numerical case 2.5(d) analyzed before. (b) Consider the element divided in three sub-elements as shown in Fig. 15 and impose the congruence at the edge of the single sub-element. This schematizaton better reflects the actual bonding condition, (c) Consider the element divided as before and in

419

addition take into account the Poisson's effects in the honeycomb. This schematization brings one to solve a congruence problem also in the direction orthogonal to the plane of the structure and provides the best accuracy. Note that the first two schematizations are referring to a bi-dimensional problem, while the third is a tridimensional one, which is very close to the numerical modelization performed by NASTRAN package. The adhesive will be taken into account considering it as rods, conforming the NASTRAN model. 3.2 Determinationof the stiffness matrices 3.2.1. Referring to the approach described in point 3. l(a) the following stiffness matrices can be written for each layer: skins:

honeycomb:

kH=EH06[I I] a

adhesive:

2t

3.2.2 The schematization concerning the point 3.1(b) will be solved by assembling the stiffness matrices of the three sub-elements I, II, I (Fig. 15). The assembling procedure depends on the way used to impose the congruence between the sub-elements themselves and it is described, with the calculation method, in Appendix 2. The following stiffness matrices have been found for the sub-element I: skins:

t

Fig. 14. Element considered for mathemeatical schematization.

(5)

1 - v s Ix/ --PS

t

honeycomb:

Km=2ctEr~[~

I

00]

(6)

adhesive:

¢__

For the sub-element H: skins: I (8) Fig. 15. Identificationof sub-elements.

M. MARCHETI'Iand F. MORGANTI

420

adhesive:

honeycomb:

kuA = EAA ku. = E,cI

(9)

2____

~ 0 3 3X/3

(16)

8

0 adhesive:

3.2.3 The schematization concerning the point 3.1(c) will be solved with the same procedrue as before. The following matrices have been found for the subelement I:

EsaS

3

X/3 0

(11)

honeycomb:

10 0 : l kls=i-S~-~]

x/3a

0

adhesive:

krA = EAA

0 0

.

(13)

For the sub-element H skins:

k.s .B-~ I~3

o lpS

(14)

- -

= l - vs L~O 0

3.3 Applications o[ the analytical method The analytical method, as described before, has been utilized to evaluate the C.T.E. of the 6.35 mm and 12.7mm core thickness sandwiches in the reference configurations (see Section 2.6) and some other parameters of interest (such as the internal stresses). These values will be put in comparison with the results of the numerical analysis performed by NASTRAN. 3.4 C.T.E. Results The C.T.E. values related to the schematization 3.1(a) can be calculated directly by means of (1), utilizing the stiffness matrices at layer level (2), (3), (4) which are referred to the cell element. The C.T.E. values related to the schematizations 3.1(b) and 3.1(c) can be evaluated by the method described in Appendix 2 utilizing the stiffness matrices (5)-(16) for each layer and for each sub-element. The results are synthetized in Table 6 for 6.35 and 12.7mm honeycomb thickness; the corresponding NASTRAN values are also indicated. Note that the x and y directions of the analytical approach are inverted with respect to the NASTRAN model. In showing the results, this latter convention is used for homogeneity. The results which are carried out utilizing the method 3.1(a) show an isotropic behaviour of the sandwich, as the numerical schematization 2.5(e). As can be observed, the methods (b) and (c) provide the same results. The honeycomb Poisson's ratio has in fact, in the examined cases, very negligible effects and does not produce any variation on the C.T.E. values in the plane of the sandwich. It has otherwise a strong effect on the C.T.E. value perpendicularly to the plane of the sandwich (az), as shown in Table 7 and consequently produces crz stresses in the honeycomb.

honeycomb:

Table 7. Numericalresults

I k"" = 1---=~./_~3- 3V3 V32~H Eu ct

4 !

1

4 3

l

T vl-i ~ 1

(15)

31

L~v- c V-~c

Honeycomb thickness (mm)

(a) (b)

(c)

NASTRAN

az

a=

a.,

6.35 12.7

23 23

30.28 30.16

30.24 30.12

Table 6. Numericalresults Honeycomb thickness (mm)

(a)

(b)

(c)

NASTRAN

ex

ay

~x

ay

ax

~v

ex

ay

6.35 12.7

1.22 1.34

1.22 1.34

1.12 1.21

1.42 1.62

1.12 1.21

1.42 1.62

1.14 1.28

1.37 1.58

421

Prediction of thermal expansioncoefficients The fitting between the NASTRAN calculation and the analytical results is very good, but, unless a specific calculation of the az is necessary, it is preferable to apply the method (b) which is simpler to use than the (c) one. The little differences between the NASTRAN and analytical prediction on ax and ay are given essentially by the shear effects inside the element under distortion. In effect the analytical schematization does not produce any shear effect inside the element, because the skins of the sub-elements I and II are constrained to maintain their rectangular shape. On the contrary, by NASTRAN model it is possible to find out that the sub-element II is subjcted to a shear effect that leads the skins to an antisymmetric deformation shape with respect to the anti-symmetry plane of the sandwich. This deformation shape is compatible with the congruence conditions which link the displacements of the skins to those of the honeycomb, in the bonding areas. The shear stresses are negligible with respect to the tensile ones (Table 9), but are sufficient to modify the thermal distortion behaviour of the element. To verify tis assumption, some check runs with NASTRAN model have been made either constraining all the sub-elements to maintain their rectangular shape, or increasing the shear modulus of the skins. The results are synthetised in Table 8 (respectively columns 2, 3). There is an exact correspondence between the analytical calculations (method (b)) and the results obtained via NASTRAN model in column 2 and 3. This confirms that the increase of the shear stiffness of the skins leads to distortion behaviour similar to that predicted by the analytical model. The shear deformability, on the other hand, makes the C.T.E. values closer in x and y directions. Another correspondence which is important to check concerns

the half thickness ribbon case (see 2.5(t3 and 2.6) which provides an isotropic behaviour of the sandwich. The analytical results confirm exactly the NASTRAN ones, showing also that their values are completely independent from the internal congruence conditions. This is given by the present symmetries in the structure which, constraining the distortions to maintain parallel the honeycomb hexagon sides, provide new conditions which override the congruence ones. 3.5 Internal stresses results Some values of the internal stresses have been calculated, (only for sandwich with 6.35 mm honeycomb core thickness) which shall be put in comparison with those computed by NASTRAN. The Table 9 synthetizes the results (the calculations have been performed with the analytical approach b) by means of the method shown in Appendix 3 [9, 10]). The previous values are referred to a temperature increase of 10°C. The fitting between the anaytical and the average NASTRAN stresses is very satisfactory, showing that the Poisson's ratio of the honeycomb has not any impact on the results. The most significant differences are in the skins of the sub-element II, but it must be remembered that in this zone the maximum shear stresses have been found by NASTRAN. Constraining the sub-element H to maintain its rectangular shape, the numerical results are practically coincident with the analytical ones. It must be noted the concentration of the stresses in the skins, near the bonding area with the honeycomb (NASTRAN maximum values), which cannot be evaluated by this analytical procedure. 3.6 Bonding stress A fundamental parameter to be evaluated is the bonding shear stress on the adhesive.

Table 8. Numerical resuts Honeycomb i (b)2 NASTRAN 3 NASTRAN thickness (mm) a~ ay ax ar 6.35 12.7

1.12 1.21

1.42 1.62

1.12 1.21

1.42 1.62

(Gsxy = 68600MN/m2) ax ay

1.13 1.21

1.39 1.6

Table 9. Internal stress Stress (kg/mm2)

Sub element

Analytical value

asx

I

asy an aA

I I I

Tsxy

II

0

asx asy an O'A

H // H /it

0.104 0.104 1.53 0.147

0.105 0.17 1.5 0.145

NASTRAN value (average) 0.108 0.16 1.4 0.14 0.012 0.109 0.12 1.6 0.15

NASTRAN max value 0.12 0.29 1.4 0.14 0.032 0.12 0.18 1.6 0.15

422

M. MARCHETrland F. MORGANT!

The ~rA value previously shown is concerning only the effect given by the thermal expansion coefficient which characterizes the adhesive itself. It is also stressed by th skins and the honeycomb, being the interface between them and the path through which the equilibrium forces are exchanged. Considering, e.g. a bonding area for the sub-element I by about 3.0mm 2, a shear stress in the adhesive of 0.14kg/mm 2 is found, due to the bonding effect, for a variaton of temperature of 10°C. 3.7 Evaluation o[ the results The analytical results have confirmed that the presence of shear stresses and strain in the skins generally corresonds to a not isotropic behaviour of this kind of sandwich structures. The more the shear deformation is constrained, the greater difference between ax and ay. The analytical model provides the maximum difference, showing that ay is the more sensitive to these effects. 4. EXPERIMENTAL TESTS

For the experimental test an electronic dilatometer Adamel Lhomargy DL 10.2 has been used. Figure 16 shows the test chamber hole and the silica support head with the specimen. The test speciens (Fig. 17) measured 50 mm in length 10ram in width and have been cut from two types of samples (Fig. 18), manufactured using the following technologies. (A) C.F.R.P. GY70 skins--CODE69 Resin, autoclave cured, 0/90/90/0 plied up facing Hexcell honeycomb 3/16 in/5052/0.0007 in/2; each skin 0.2 mm thick.

W

380

:

~ t6.3..=,.n6

Fig. 18. C.T.E. sandwich sample. (B) C.F.R.P. GY70 skins--CODE69 Resin, oven cured, 0/90/90/0 plied up facing Hexcell honeycomb 3/16 in/505210.0007 in/2; each skin 0.30 mm thick. Given the test chamber dimension, only the 3/16in cell size honeycomb sandwich could have been tested. A particular care in cutting the specimens has been used in order to reduce causes of results dispersion: (a) the specimen surfaces used as reference planes for the dilatometer have been cut very carefully, parallel to each other and perpendicular to the specimen axis, (b) the honeycomb has been carefully centered with respect to the specimens axis in order to have the cells symmetric with respect to it. In Table 10 are reported the physical characteristics of the matrix and he fibers (at room temperature) of the laminate. The two different technologies provide a skin C.T.E. by 0.2 10-roC-t for the sample A and by 0.5 10-6°C 1 for the sample B. Both results were obtained by I.T.A. dilatometric tests. The C.T.E. tests have been performed on 20 specimens for each sample specified below: --n. 10 specimens cut along 0° direction of fibers, i.e. along the ribbon direction (L direction) of the honeycomb, --n. 10 specimens cut along 90° directon of fibers, i.e. along W direction of the honeycomb.

4.1 Description o[ tests and results

Fig. 16. Dilatometerhead.

Each specimen has been heated to 90°C at the approximate rate of 1.3°C/min; after that it has been cooled at - 120°C at the same rate and heated again to 100°C. The value of the C.T.E. has been obtained in the experimental curve between - 120 and + 100°C. The thermal cycle is plotted in Fig. 19. Table 10. Physical characteristic of fibers and matrix Property

Code 69

GY70

Young modulus

3,37 GPa

500 GPa

C.T.E.

65 10-~C-I

Poisson ratio

0.38

(L) - 0.6 10 6~C I

Fig. 17. Test specimens.

(T)+I710 6~C t 0.35

Prediction of thermal expansion coefficients

423

T °C

1OO

A 50.

\

,I/ \\

\

\

\

,/

1'o

t/h)

\

-50

\\\

-100.

Fig. 19. Thermal cycle. The experimental results are reported in Figs. 20 and 21 in which the C.T.E. of the samples manufactured with the different technologies are put in comparison versus the temperature. Each C.T.E. is concerning the mean value on 10 specimens for a given temperature. An experimental difference between aL and aw can be observed, as predicted by the analyses, being aL > aw. The number of the tests is sufficient to make the results statistically representative.

cases are shown in Table 11 concerning the only 6.35 mm core thickness sandwich. In order to have significant test C.T.E. to be put in comparison with the numerical ones, the average values, between - 50 and 50°C, have been calculated considering that (see Figs. 20 and 21) in this temperature range the specimens show a rather constant behaviour. Table 11. Trade-off cases Adhesive AA aA

4.2 Fitting with the prediction methods The C.T.E. values obtained from the tests can be fitted with the analysis results on sandwiches having the same characteristics as the specimens. The applicable trade-off

0.5 0.5

50 50

Skins as

Trade-off case

Type of sample

A B

A B

0.2 0.5

a ~- 10-6 °C-~

A

m

~

0 ~

sample A sample

B

1.5.

~.........~

Z~.___..~ ..-- - ~ ' f

~ _ . . . ~

.5-

0 -100

-go

-2b

6

~o

go

Fig. 20. Experimental results (*0 direction (L)).

T°C

~60

424

M. MARCHETTIand F. MOR6ANTI -6

ct: 10 °C

"

|

-1

Li-l-l-i_l--L-Li-~-~J

'

~

'

~

"

~

c~

.--

zx ~ _ _ _ _

sample

A

o

sample

B

O

1.,

.5.

-6

1oo

~-~6-

6

2b

5b

T°C

160 ~--

Fig. 21. Experimental results (900 direction (W)). Table 12. Comparison between analytical and experimental results C.T.E.(10 6°C-~) ax ay

Case A Analytical Experimental 0.85 1.08

In Table 12 the C.T.E. values to be compared for the fitting are represented, related to A and B cases (or samples): The correspondence between the analytical and test results can be considered satisfactory. The experimental values show a smaller difference between ax and ay than the numerical ones. This fact is ~due in some way to the dispersion of the results and be the physical consideration of Section 2.6 concerning the deformation of the honeycomb inside the structure. In any case it is evident that the fitting is also influenced by the envisaged C.T.E. values at skin level, which are input to the previsional model. Since they are obtained by tests, they are affected by their own dispersion. 5. CONCLUSIONS

The various activities performed while developing this work have shown that it is possible to have reliable methods to foresee the C.T.E. methods to have reliable methods to foresee the C.T.E. of sandwich panels. These methods are very helpful to have an immediate feeling on the distortion behaviour of the structures, considering that the obtained values can be directly utilized as design input. The fitting with the test results allows a better understanding of some physical aspects of the problem and to check the assumptions considered in the analytical modelization. Selenia and ITA are actually engaged in the prosecu-

0.88 1.01

Case B Analytical Experimental 1.23 1.49

1.19 1.35

tion of the activities concerning these thematics, mainly in the testing area. The manufacturing procedures will be carefully investigated in order to quantify the impact of some of their aspects on the thermomechanical characteristics of the composite materials. In this frame, dedicated tests to evaluate the influence of the moisture desorbtion on the sandwich C.T.E. during ageing will be performed. In parallel other tests are foreseen in order to have a very large spectrum of data of C.T.E. and Young modulus values of several kinds of laminates, which are the most important input parameter to the previsoinal methods of the sandwich characteristics. Acknowledgements--The authors are grateful to European Space Agency (E.S.A.) and Soc. Selenia S.p.A. who have given the opportunity to perform this work and to Prof. P. Santini for having encouraged them in the development of these activities.

REFERENCES 1. G. Manoni and S. Ferri Italy in millimeter waves activities for space communication. XXXIInd Cong. of the Int. Atrronautical Federation, Paper 81-59, Rome, Italy (1981). 2. G. Manoni and G. Perrotta, Requirements for a domestic communication satellite. 8TH AIAA Cong. Orlando, Florida (1980). 3. M. Lopriore and G. Manoni, The design of a 30/20GHz regenerative payload for satellite application. 5th Int. Con[. on Digital Satellite Communication, Genoa, Italy (1981).

425

Prediction of thermal expansion coefficients 4. P. Santini, Thermostructural problems of high accuracy antennas mounted on telecommunication satellites. XXVIII th Cong. of the Int. Astronautical FederatMn, Paper 77-237, Prague, Czechoslovakia (1977). 5. K. Keen, P. Molette, B. Pieper, C. M. Herkert and W. Schaefer Development and testing of a new C.F.R.P. antenna reflector for communication satellites. Raumfahrtforschung Heft 4, 173-181 (1976). 6. S. Ahmed, R. A. Russel, T. G. Campbell, R. E. Freeland and J. F. Clemmet, Requirement, design and development of large space antenna structures. AGARD Rep. No 676 (1979). 7. D. Engrand and J. Bordas, Calcul des coques en materiaux multicouches et sandwiches par le m6thode des 616ments finis. La Recherche A~rospatiale 2, 109-118 (1973). 8. R. H. Mac Neal, The NASTRAN theoretical manual. N A S A SP 221, (1972). 9. L. Broglio and P. Santini, High temperature effects in aircraft structures-thermal stresses. AGARDograph No. 28 (1958). 10. F. J. Plantema, Sandwich Construction Wiley, New York (1966).

l

Y

a

Fig. A1. Identification of the sub-elements.

APPENDIX 1

Nomenclature

AA active section of the adhesive EA young modulus of the adhesive En young modulus of the honeycomb Es young modulus of the skins Gs shear modulus of the skins /, H sub-elements K total stiffness matrix k stiffness matrices of each layer N totalthermal force N thermal force of each layer T temperature a cell size c core thickness I length of the honeycomb cell side s thickness of skins t thickness of honeycomb a thermal expansion coefficient of the composite (sandwich) ax x component of a ay y component of a az z compocent of a a thermal expansion coefficient of the layer aA thermal expansion coefficient of the adhesive an thermal expansion coefficient of the honeycomb as thermal expansion coefficient of the skins e thermal expansion coefficient of sub-elements vn Poisson ratio of the honeycomb vs Poisson ratio of the skins L longitudinal T transversal C.T.E. thermal expansion coefficient D.O.F. degree of freedom I.T.A. Istituto di Technologia Aerospaziale APPENDIX 2

For the sub-elements I and II (Fig. AI), the congruence relationships (related to one degree of temperature) will be given by: kls(as + ets) = kiss1,

(AI)

kin(an + era) = kmel,

(A2)

kaA(aA + etA) = kiA~t,

(A3)

kll.(Otl-I + Eli.) = k111t~2,

(A5)

knA(aA + euA) = koAe2.

(A6)

Taking into account the equilibrium of the forces in each subelement: kls~is + kmEm + kIAelA = O, kns~us + knn~.n + knA~nA = O,

the previous relationships will be written as follows: klsas + kman + kv~aA = (kls + k m + klA)O, knsas + knnan + knAaA = (kns + knn + klla)e2.

Ni = k1~1,

(A9)

N2 = k2~2,

(AI0)

where ~ and ~2 are the thermal expansion coefficients of the sub-elements I and II. Let now define the relationships of equilibrium and congreuence between the sub-elements I and II. The congruence can be expressed in general by: klal.

(All)

k2(~2+ Bi2) = k2a2;

(AI2)

k l ( o + A~t) =

taking into account that the component of a~ and a2 are connected to the components of i~ and i2, in a bi-dimensional theory, as follows: I~lx

-- /Jlxy~ly

{~:;} = lt.,2r ' 2 x J/ + { - ' 2 ,

The matrices A and B are expressed by: Kns(as + Ens) = knsE2,

(A4)

(A8)

Defining kl and k2 as the stiffness matrices, while Nt and N2 as total thermal forces, in free expansion condition, of each subelement, (A7) and (A8) can be synthetized:

V2xy ~:2y

and:

(AT)

7 ] ,=[X

}'

(AI4)

426

M. MARCHETTIand F. MORGANTI

The congruence along the y direction of the interface of the sub-elements will be respected if

a~ =

2elx + e2x 2 3 ] pAey,

Gtly = ~2y,

ay

i.e. if:

~2y + 2qely 2q+ 1 '

where: ely + ~ly = E2y + ~2y,

A2klab -AIk2ab P = 2&kzoa + A2klaa"

or:

(A15)

~ly -- ~2y = e2y -- ely = Aey.

The previous relationships can be written in the assumption that the coupling between the sub-elements I a n d / / d o e s not imply any force exchange in the x direction, along which they are free to move. For this reason the contribution in this direction to the congruence is the Poisson's effect given by the displacements in the y direction. The equilibrium of the forces of the overall element I + H + 1 will be given by: 2klA~l + k2B~.2 = 0,

(A16)

So that, the C.T.E. vector of the sandwich can be expressed: {a} : [M,]{~,}+ [Mzl{e2},

(A21)

with [Md

=[2/3

2/3p]

[0 and

vfl.

i.e. developing the terms of the stiffness matrices:

2(-klaaPIxy + klab)gly + ( - k2aaV2xy + k2ab)~2y = O, 2 ( - klb.vl~y + klbb)~ly + ( - k2baV2xy+ k2bb)~2y

=

0.

From the first relationship, both terms are equal to zero become no force in the x direction is present at the interface between the sub-elements, so that:

klab

k2ab

vl~y = k-~-~; v2xy= k-~-~"

ax ar

\ (a18)

This ratio defines the degree of orthotropy of the sandwich. Let now introduce the three-dimensional approach to the problem. In this case the Poisson's effect in the honeycomb core will be considered. The solution of the problem is conceptually the same as before, only changing the dimensions of the stiffness matrices of the sub-elements and of the strain vectors. The congruence will be now expressed as follows: ~l,q

Calling At and A2 the determinants of the matrices kl and k2, the unknown strain g~y and i2y will be determined solving the system given by (A15) and (Al8):

EIx~

LauJ

LOd

~:,/=1,2d+ // / | 1

~2,

~2zJ

~2z

A~y

=

- AI k2aa

AE~ - 2q + 1'

,., AI k2aa _ = - 2q~ly.

L

,

(m3)

so that the (AI 1) (A12) expressions are still valid, considering the matrices A and B expressed as follows: (Al9)

zGk-/S+l 2 laa --/" A'-~kla---~El y

L~2:J

z]

~lz

L

2 kl~a ' l y q- k2~a '2y = 0,

~ly

A= (A20)

The components of the C.T.E. of the sandwich are given by: 2~lx + ~2x 2Vlxvely+ P2xy~2y 3 3 '

Along y, in fact, due to the congruence, ay = aly = ff2y, while in the x direction the C.T.E. is expressed by the mean value related to the three sub-elements. Taking into account the (AI7), (A19) and (A20) these relationships, can be written as follows:

1 0

and B =

ax =

-- Plxyl~ly --

t:ly - ~2y = AG, from which:

~2y

261x+ ~2x 2e~r+E2y

(AI7)

The independent equilibrium relationship is therefore the second one'.

2(-klba klab q- klbb) ~-lyac(-k2ba k2ab " klaa

Since the value of q is very close to 1, and the value of p to 0, the ratio between the two C.T.E. components a~ and ay mainly depends on the ratio of the mean C.T,E. value of the subelements:

I!

01

-- F2x~ 1

.

0

The equilibrium expression (A16) now provides (with evident symbology):

klab vl,y

=

k,,~ kl.c

Flxz ~" klaa

Prediction of thermal expansion coefficients

427

vector of the sandwich can be evaluated, considering that, conforming to the bi-dimensional approach, the y-z components will result:

k2ab

v2xy = k2~--~ k2ac

P2xz = ~2aa

2 ~ala ~ly + ~ala ~2y' ~ A I 2 - '

Oty = ~tly = O~2y,

A22--0

(A24)

"r L k,a-'~aElZ * k2a-'-~aE2z 2~1y+~.~23-

AI4_

A24- =0

(A25)

~tz = Otlz = Ot2z,

while the x component is given by the mean value related the three subelements: 2alx ~x =

+ t3t2x 3

with: klaa

klab

kiaa

A . = d e t kb~ kibb 'Ai2=det k ~ A,3=det kk::: kk:::l,A,4=detlkk:,a:

APPENDIX 3

kiac .

kbc'

The evaluation of the thermal stesses inside the subelements I, II is performed utilizing the general approach which is described in Appendix 2. Once the thermal expansion coefficients of the sub-elements have been found, having considered the congruence at their interface, the internal thermal stresses can be calculated as follows (referring to a temperature variation of I°C):

kk::: ;

i = 1.2. The congruence expressions are: t:ty - ~2~ = ~2y- ~ty = A~y,

(A26)

Kij(al - aj) = Nij,

t:t: - ~2z = ~2z - ~lz = A~:.

(A27)

where K~j is the stiffness matrix of the layer j in the sub-element i, a~ is the thermal expansion coefficient of the sub-element i (Fig. AI) and a~ is the thermal expansion coefficient of the layer j.

The system formed by (A24), (A25), (A26) and (A27) can be solved in ~y, *2r, ~ , ,2~. From these values, the global C.T.E.

(A28)