CHAPTER3

NEURAL NETWORK- GENETIC ALGORITHM

APPROACH FOR THE DETERMINATION OF

FRACTURE TOUGHNESS OF COMPOSITE

LAMINATES

3.1 INTRODUCTION

This chapter deals with how the vast test data on notched strength I intralaminar

fracture toughness of multilayered laminates available in literature can be utilized to

predict the toughness value of other type of laminate made of different material and

lay-ups. The intralaminar fracture toughness of fibre reinforced composite laminate

is experimentally evaluated using standard test specimens and analytically predicted

using the well-known MCCI approach corresponding to the failure strength of the

laminate (Parhizgar, et al., 1982 and Ramesh Kumar et al., 2002). Analytical approach

has certain limitations in the evaluation of fracture toughness of multi layered laminates

due to the intricacies involved in the determination of failure modes (Poe, 1984 ). In

other words, in the absence of analytical method to determine the crack initiation

direction in multilayered laminates, neural network is followed in the prediction of

fracture toughness of laminates for other material system.

The analytical prediction of failure strength of composite laminates with flaws

was very well established based on the Whitney-Nuismer (W-N) fracture criteria

m1hitney and Nuismer, 197 4). However, the characteristic distances ( d and a ) which \ \'Y O o

are apriori for the analytical prediction are to be experimentally determined for each

39

type of laminates which involves effort and time. It is interesting to note that the

notched strength depends on the lay-up sequence oflaminates.

If it is possible to predict the fracture toughness/notched strength of different

laminate configurations and material systems based on the available test data, then

such an effort can be used for the quick assessment of residual strength of the

laminates.

In the present study in view oflarge amount of test data on fracture toughness/

notched strength of multilayered composite reported in literature (Poe, 1984;

Aronsson, 1986; Tan, 1988; Khatibi et al.; Awerbuch and Madhukar, 1985), artificial

neural network (ANN)-genetic algorithm (GA) approach is used for predicting the

toughness value oflaminates with different material systems and lay up sequences.

3.2 NOTCHED STRENGTH PREDICTION BASED ON NEURAL NETWORK

GENETICALGORITHMAP PROACH

The notched strength prediction of composite laminates is highly significant in

the design of structures and the experimental data available on composite fracture arc

not in a suitable form for a design engineer. The present study aims at predicting

notched strength of [0/±a]5

and [±a/0]5 class oflaminates with hole or crack using

neural network-genetic algorithm approach following WN failure criteria based on

the test data available in literature.

Large amount test data on notched strength of ASl/3501-6 graphite/epoxy

laminates with different lamination sequences available in literature (Awerbuch and

Madhukar, 1985) is given in Table 3 .1. It may be noted that for the three classes of

laminates, namely [0/± aL, [±a/Ot and [ +a/0/-aL , both un-notched and notched

strengths are different for the same a and the deviation is more as a increases. The

40

deviation on the un-notched strength is around 30% and in the presence of a hole the

variation is.found to be even up to 45%.

3.2.1 Neural Network Model

In this study, notched strength of [0/±a]5 and [±a/0]

5 class of composite laminate

is predicted using a neural network-genetic algorithm approach. The symmetric

laminates considered in this study are having six layers with fibre orientations varying

from -90° to 90°. The proposed neural network model predicts the characteristic

dimension of various classes of laminates which can be used for the notched strength

computations.

Table 3.1 Test data on the failure strength of ASl/3501-6 laminates (Awerbuch and

Madhukar, 1985)

Lay-up do ao CTo KT- CTN- (PSC)

mm mm MP a MPa.Jmm MP a

[±5/0]s 1.92 9.82 1134.0 6.08 824.71

[0/±5Js 1.28 5.46 1288.3 6.08 758.29

[0/±lO]s 1.06 3.80 1140.1 5.58 587.67

[±10/0] s1.24 5.56 1016.0 5.58 575.07

[±15/0]5 1.42 4.74 998.0 5.00 591.63

[0/±15] 50.9 3.64 1003.0 5.00 468.06

[±20/0]5 1.83 5.99 1063.5 4.46 682.39

[±25/0]51.36 3.78 1061.9 3.99 588.87

[±30/0Js1.04 2.98 855.0 3.60 421.71

[0/±30] 50.57 1.45 945.0 3.60 381.44

[±45/0]51.36 3.76 668.0 3.00 354.94

[0/±45]50.58 1.40 787.0 3.00 332.28

[±60/0]s 1.86 5.72 698.0 3.00 413.28

[0/±60] 50.86 2.57 814.0 3.00 377.02

[±75/0]5 2.34 7.16 672.0 3.49 444.41

[0/±75]51.48 6.00 733.0 3.49 410.33

[0/902]5 3.31 13.92 732.0 4.14 558.39

41

The symmetric laminate having six layers are modeled using a neural network

with three input neurons and one output neuron (Figure 3 .1 ). The inputs to the network

are the fibre orientations in the three layers of symmetric laminates which varies

. from -90° to 90°. Typically, for (0/±45)5 laminate, the inputs to the network are 0°,

+45° and -45°. These input values are normalized to a value between zero and one

using the mapping function;

I Input

nput = -.========== nonnalizod /r2 I2 12 '\J I+ 2 + .... + n

. where 11, 1

2 etc. are the values of input data.

Figure 3.1 Neural Network with Nine Hidden Layer

(3.1)

42

The network output is the characteristic dimension (a0 or d) for each class of

laminates. The target outputs are also normalized using Eq.3.1. The connection weight

between input and hidden layer is W 1 .. (i= 1,3 and j= 1,9) and the weight between hidden lj

and output layer is W2 po (p= 1, 9). The bias values of the hidden neurons are B/ (j= 1, 9)

and that for the output neuron is B 0

• The slashing function used in the network is

tanh(x) function. The flow diagram for the notched strength prediction of composite

lami�ate using ANN is given in Figure 3.2. The network training is achieved by

· following either back-propagation algorithm or genetic algorithm using input data.

The two methods are compared based on the network error. Once the network is

trained, it represents a neural network model for the prediction of notched strength

· •. of a composite laminate.

Input Data 0.1 ,a.2 & 0.3 for Testing

Input Data

Neural Network

do* Based on

Testing data

BP or GA

do* Based on

Training data

-·

CTN

Based on W-N criterion

Figure 3.2 Flow diagram for the prediction of crN -* with Neural Network 43

3.2.2 Back Propagation Parameters

The feed forward back-propagation algorithm is a well known supervised training

model in neural networks. It does not have feed back connections, but the errors are

back-propagated during training. Errors in the output layer determine hidden layer

errors, which are used as the basis for adjustment of connection weights between the

input and hidden layers. Adjusting the two sets of weights between the pairs of layers

and recalculating the outputs is an iterative process that is carried on until the errors

fall below a tolerance level. Learning rate and momentum parameters scale the

adjustments to weights.

The available data on characteristic distances and laminate strength given in

Table 3.1 has fibre orientations of three layers varying from -90° to 90° at intervals of

5°. These input parameters are scaled to a value between zero and one, using the

mapping function {Beale and Jackson, 1990) given in Eq. 3.1. The slashing function

used in the present study is tanh( x). A learning parameter of value 0.01 and momentum

parameter0.001 are used for the back-propagation algorithm in the present study.

3.2.3 Genetic Algorithm Parameters

In the network training using GA, the connection weights and node bias values

are coded into chromosomes. The present neural network modelis having three input

neurons, one hidden layer with nine neurons and one output neuron. Each neuron in

the hidden layer has a bias value and three connections weights from the input neurons.

The neuron in the output layer is having a bias value and nine connection weights from

the hidden neurons. Real number coded chromosomes are employed to train the

. network with a chromosome length of 46 (i.e. (3+1)*9 + (9+1)*1, Figure 3.1). A

population of hundred chromosomes is considered in this study. The neural network

specific cross over is adopted with a low cross over probability of 0.01 and real

number mutation with a high mutation probability of 0.85 is used.

44

3.3 FRACTURE TOUGHNESS PREDICTION BASED ON

NEURAL NETWORK GENETIC ALGORITHMAPPROACH

Intralaminar fracture toughness of multilayered laminate with defect in the form

of crack is predicted using a neural network with five input neurons, one hidden layer

with twelve neurons and one output neuron, as shown in Figure3.3. The test data on

fracture toughness of carbon epoxy material of grades T300/5208, T300/914C,

T300/1034, AS4/3502 and glass epoxy Scotchply 1002 laminates with different lay

up sequences available in literature are used for training and testing the neural network

developed. The material properties are given in Appendix A. The number of neurons

in the hidden layer is obtained by a convergence study based on the network error. The

inputs to the network are chosen such that they are all represented with values less

than unity so as to avoid further mapping.

Figure 3.3 Neural network with twelve hidden neurons for the fracture toughness

45

i.'

The five inputs to the neural network are the aspect ratio (a/w), the Poisson's

ratio (n), and the ratios of elastic constants v IE, n IE and n IG . The neural network xy x xy y -�\' ,\'.\'

output is the fracture toughness (K) of the laminate in MPa�. The aspect ratio

(a/w) is taken as a network input because fracture toughness of a laminate, though

considered as a material property, is essentially dependent on the a/w ratio ( crack

length to plate width) to the extent of about 5 to 33 % (Figure 3.4).

3.3.1 Neural network and Genetic algorithm parameters

A real number coded chromosomes having a length of 85 ( = ( 5 input+ 1 bias)*

12 neuron+ (12 neuron+ 1 bias), Figure 3.3) are used to represent the network with

a population size of 100. A neural network specific cross over with a high crossover

probability of 0.9 and real number mutation with a low mutation probability of 0.01 is

used as the genetic operators in the present study.

4000

,n ,n

3000

2000

r==:

[ 45/90/-45/90]

. L0/90/0/90/45/90/-45/90]

----::::=::=:: z/ : 1000 I . T

1 2 3 4

a/w

Figure 3.4 Variation of fracture toughness with a/w for T300/5208 laminaes [Poe, 1984]

46

3.4RESULTS AND DISCUSSION

A neural network model with three inputs neurons, nine hidden neurons and one

output neuron is used to predict the notched strength of laminates. Both back

propagation and genetic algorithm has been used to train the network and established

that training using GA gives better result in the prediction of characteristic distance

d for (0/±a) and (±a/0) class of laminates considered. The characteristic distance 0 S S

a for (0/±a) and (±a/0) laminates is also predicted for tbe class of laminates 0 S S

considered.

3.4.1 Notched Strength Prediction

3.4.1.1 Neural network training

Initially, to establish convergence on number of neurons in the hidden layer,

the network is trained using the training set of experimental data on d for (0/±a) and I) s

(±a/0) class of graphite/epoxy laminates. For this purpose, one hidden layer with s

three, six, nine, twelve and fifteen neurons is tested. A variation of network error with

number of hidden neurons is shown for ANN with BP and GA in Figure 3.5. It may be

noted that with nine neurons in the hidden layer both algorithms converge with minimum

network error.

0.13

0.12

0.11

W 0.10

l 0.00

Z 0.(8

Genetic Algorithm 0.07 -·

o.re

2 4 6 8 10 12

Number of Hidden Neurons

14 16

Figure 3.5V ariation of network error with number of hidden neurons for the neural network training using GA and BP. 47

The error is least with genetic algorithm than back propagation algorithm. Typically

for [±10/0]s laminate, the error in the predicted failure strength (crooN*) is 16.3% for

GA and 18.3% for BP (Table-3.2). For all other (±a /0) laminates are 3.7% and 6.5% s

for GA as well as 6.5% and 9% for BP (Table-3.2). It is obvious that for the (±10/0) s

laminate there may be some experimental error in determining the d0

• Similar values

for (0/±a) laminates, percentage error varies from minimum value of 0.5 to a s

maximum value of 3.6 while using GA and it is 1.7% and 6.8% with BP respectively.

Table-3.2 Neural network training data for the notched strength prediction _of AS 1

3501-6 Laminate based on Point stress criterion.

Lay-up

[O I ±5]5

[0 I ±lO]s

[O I ±15]s

[0 I ±45]s

[O I ±60]s

[0 I ±75]s

[O I 902]s

[± 5 I O]s

[± 10 I O]s

[± 20 I O]s

[± 25 I O]s

Back-propagation

do" � *

O' N

1.34 779.3

1.03 576.2

0.79 436.1

0.63 338.l

0.96 389.2

1.37 399.3

2.89 537.4

1.73 787.4

1.71 680.5

1.55 637.9

1.40 596.7

Genetic Algorithm

% Error on do� - *

% Error onO' N

strength strength-2.8 1.32 772.7 -1.9

1.9 1.07 590.5 -0.5

6.8 0.87 460.7 1.6

-1.7 0.56 329.5 0.8

-3.2 0.90 380.8 -1.0

2.7 1.48 409.8 0.1

3.8 2.90 538.0 3.6

4.5 1.65 771.0 6.5

-18.3 1.65 668.8 -16.3

6.5 1.57 641.1 6.1

-1.3 1.49 610.6 - 3.7

For the prediction of notched strength of (0/±a) and (±a /0) class of laminates s s

based on average stress criterion, a network having the same structure with lay-up

sequence as the inputs and characteristic distance (a0) as the output is implemented.

Genetic algorithm is used for training the network with (0/±30)5

, (0/±45\, (0/± 75)5

,

(±30/0) , (±45/0) and (± 7 5/0) class of laminates as the training data. The trained s s s

network shows a variation of percentage error on predicted notched strength from a

minimum value of 1.1 to a maximum value of 9.9 for the (0/± a) laminates. In the s

48

case of (±a /0) laminates, the percentage error varies from a minimum value of 0.01 s

to a maximum value of 6.9 (Table- 3.3).

Table-3.3 Neural network training data for the notched strength prediction of

ASl/3501-6 Laminate based on Average stress criterion.

Lay-up �

= *% Error onao O' N

strength

[0/±30]s1.34 382.0 -2.00

[0/±45]51.48 342.3 1.10

[0/±75]58.57 501.3 9.90

[±30/0]s3.11 438.4 1.30

[±45/0]s3.66 363.0 0.01

[±75/0]s11.06 469.2 6.90

3.4.1.2 Testing of Neural Network

A class of (0/±a) and (±a /0) laminates namely, (0/±30), (±15/0), (±30/0), s s s s s

(± 75/0) and (902/0) laminates whose data on d and cr00

N available in literature

S S 0

(Table 3.1) are used for testing the accuracy of the trained network. Table 3.4 gives a

comparison of predicted strength based on Genetic algorithm and Back propagation

approaches with test data. It can be observed that the percentage error in predicted

strength using GA with test data varies from 0.7% to 8.5%. Similar comparison shows

variation of error from 2.1 % to 11.3% for BP. Based on the available test data on

strength of the laminate, for (0/±30\, the % error on strength is obtained as 0.74

(Table 3.4) and for (±a /0)5 lamiriates with a= 15°, 30°, 75° and 90°, the % errors are

-6.4, -7.5, -8.5 and 6.27 respectively. One of the reasons for the large error observed

. for certain: class of laminates is the limited number of test data available for the

network training.

The trained neural network model for the notched strength prediction of

(0/±a) and (±a/0) composite laminates based on average stress criterion is tested s s

49

using (0/±5)5, (0/±15)5, (0/±60\, (±15/0\ and (±60/0)s laminates whose data on a0 is

available in literature (Table 3.5). The% error on the predicted notched strength shows

aminimum ofO.Olfor [±15/0]5 laminate and a maximum of 11.6 for [0/±15t laminate.

Table 3.4 Neural network testing data on failure strength of AS 1/3501-6 laminate

based on point stress criteriol

Lay-up Back-propagation Genetic Algorithm

do •00 • % Error on d/ ·X , * % Error on O' N O' N

strength strength [0/±30]s

0.5 367.4 3.7 0.56 378.62 0.7

[±15/0]s1.7 635.2 -7.4 1.62 629.51 -6.4

[±30/0]s1.2 446.8 -4.5 2.90 477.74 -7.5

[±75/0]s2.5 453.6 -2.1 1.37 464.18 -8.5

[902/0]s 2.4 469.9 11.3 2.85 496.27 6.3

Table-3.5 Testing data on failure strength ofASl/3501-6 laminate based on average stress criterion.

Lay-up .

00 • % Error on ao O' N

strength [0/±5] s

4.90 742.3 -3.5

[0/±15]s5.07 588.8 11.6

[0/±60]53.89 449.8 10.8

[±5/0] s8.21 764.4 -4.5

[±15/0]s4.84 577.3 0.01

[±60/0]s7.54 462.3 7.6

3.4.1.3 Empirical Relationship for Characteristic Distance

The IJ.etwork used in the present study is having three input neurons, nine neurons

in the hidden layer and one output neuron. The values of connection weights and bias

values of the trained neural network are given in Table 3.6.

50

From the values of connection weights and bias given in Table 3 .3, an empirical

relationship (Eq. 3.2) for predicting the characteristic distance (d) is developed and

1sg1venas;

y -y

Characteristic distance, ( = e - e e +e-y

e';-e-x; (3 ) 0. = and x. = ""'Wl.. * a. + B.

I Xj -Xj J � IJ I J e +e ;�1

(3.2)

oci are the ply angles, Wl ii, W2 pa ,Bi and B0 are the connection weights and node bias

values. Similiar relations may be developed for ao also .

Table 3.6 Connection weights and bias of the neural network for the notched strength

prediction.

Connection weights Bias Bi Wlii Wh Wh W210 -0.135 -0.269 -0.085 -0.279 -0.1022.618 1.459 -3.408 0.0643 -2.638-0.182 -0.405 0.0602 0.173 -0.391-0.063 -0.007 1.600 0.408 -0.7900.861 0.146 0.139 -0.478 0.977 -0.724 -0.205 -0.047 0.434 -0.891-0.502 -1.455 1.1463 0.588 0.864 -0.474 2.5545 -0.737 1.542 2.450 -0.036 1.944 1.074 -1.436 -1.194

Bias of output Neuron B0= 1.411

3.4.2 Fracture Toughness Prediction

The test d ata on fracture toughness o f carbon-epoxy material of grades

T300/�208, T300/914C, T 300/1034, AS4/3502 and glass-epoxy Scotchply 1002

laminates with different lay up sequences available in literature are used for training

and testing the neural network developed (Table 3. 7). It may be noted that the mode of

51

failure associated with these laminates are all through fibre breakage. This is because

both o0 and 90° fibre orientations are present in all the laminates considered except

for one laminate out of AS4/3502.

Table 3.7 Neural network input data for fracture toughness prediction

Laminate Material Lamination Ex Ey Gxy 'Uxy Sequence (0Pa) (0Pa) (0Pa)

A [90z/0/90]s 40.6 100 5.7 0.03 B [902/45/90z/- 17.9 102.5 12.6 0.096

45/902Js

C [0/90/0/90]5 70.4 70.4 5.7 0.05 D T300/5208 [0/90/0/90/45/90/- 47.3 76.2 12.6 0.133

[Poe, 1984] 45/90]5

E [ 45/90/-45/90]5 23.3 75.1 19.6 0.201 F [±45/90/±45/90]5 25.3 56.6 24.3 0.30 G [ 45/90/-45/0]s 51.2 51.2 19.6 0.31 H [0/45/0/-45]s 75.1 23.3 19.6 0.648 I [±45b 19.6 19.6 33.6 0.73 J T300/914C [0/90/0/90]5 72.0 72.0 6.54 0.039 K [Peters, 1983] [0/±45/0]5 78.8 23.5 20.6 0.632 L Scotchply 1002 [0/±45/90Jzs 18.9 18.9 7.44 0.27 M [Khatibi, 1996] [0/90]4s 23.5 23.5 4.1 0.09 N T300/1034 [0/90/±45]5

[Aronsson, 1986] 52.8 52.8 19.8 0.33 0 AS4/3502 [O]s 11.86 143.92 6.69 0.33 p [Tan, 1989] [±30]s 14.15 56.46 29.7 0.30

3.4.2.1 Network training for fracture toughness prediction

Initially, to establish convergence on the number of neurons in the hidden

layer, the network is trained using GA for a set of training data given in Table 3.8. A

variation of network r.m.s error with number of hidden neurons is shown in Figure3 .6.

It may be noted that a network with twelve hidden neurons converges with minimum

error.

The network error during training on fracture toughness for different laminates

is given in Table 3.8. A comparison of toughness obtained with test data shows that

the maximum error on the prediction is 19.8% and the minimum 0.4%.

52

Table 3.8 Neural network training for the fracture toughness prediction

Laminate a Fracture toughness MP&

w Test Prediction %

error 0.102 4860 4921 1.3

0.203 5710 5686 0.4

A 0.305 6290 7193 14.4

0.508 5210 6061 16.2

B 0.102 2710 2753 1.6

0.203 2910 3126 7.4

0.305 3260 3862 18.4

0.508 3270 3811 16.5

C 0.102 1310 1223 6.6

0.203 1350 1373 1.7

0.305 1510 1819 19.8

0.508 1410 1474 4.6

0.102 1310 1230 6.1

D 0.203 1390 1189 14.5

0.305 1390 1487 7.0

0.508 1380 1453 5.3

E 0.102 1400 1411 0.7

0.203 1550 1498 3.3

0.305 1710 1699 0.6

0.508 1580 1527 3.3

0.102 1330 1428 7.4

F 0.203 1490 1444 3.1

0.305 1470 1230 16.4

0.508 1530 1570 2.6

0.102 1260 1385 9.9

G 0.203 1320 1347 2.1

0.305 1390 1350 2.8

0.508 1320 1277 3.2

0.102 1380 1361 1.4

H 0.203 1110 1065 4.1

0.305 1020 985 3.5

0.508 920 860 6.6

0.102 1110 1014 8.6

I 0.203 1150 1126 2.1

0.305 1120 1107 1.1

0.508 960 933 2.8

53

. . I ..

0.7

0.6

e o.s

� 0.4

� 0.3

i 0.2

0.1

0 8 10 12 14 16

Number of hidden neurons

Figure 3.6 Variation of network rm error with number of neurons in the hidden layer.

· . 3.4.2.2 Network Testing for fracture toughness prediction

Accuracy of the trained neural network is tested using the test data available in

literature. Table 3.9 gives a comparison of predicted fracture toughness with test

data. It can be observed that the percentage of error in predicted toughness for most

of the laminate varies from 0.2 % to 14.9%. However, for [0/90]4s laminate (laminate

M) with aspect ratio 0.339, the percentage error on predicted value is 15.2%. It may

be noted that the toughness value of T300/5208 with lay up sequence of [0/90/0/90] s

used for the training is 1410 MPa.Jmm while the prediction for T300/914C with a

lay up sequence of [0/90/0/90]5

is 3642 MPa.Jmm as against the average test value

of3520 MPa.Jmm shows a good agreement within 3.5 %. It may be concluded that

prediction for different material system is possible as toughness is related to. the

stiffness properties of material for energy required to produce unit crack extension;

in other words related to the strain energy release rate.

54

Table 3.9 Neural network testing for the fracture toughness prediction

Laminate a/w Fracture toughness

Neural Test network %

prediction Dev. J 0.40 3520 3642 3.5

K 0.40 3720 3692 0.7

0.306 604 646 7.1

L 0.291 677 666 1.4

0.326 674 672 0.2

0.103 557 639 14.9

M 0 .. 288 686 765 11.6

0.294 822 782 4.8

0.339 743 856 15.2

0.350 3643 3688 1.2

N 0.450 2884 2943 2.1

0.550 2384 2393 0.4

0.650 1972 1964 0.4

0.099 2084 1916 8.1

0 0.229 2880 3065 6.5

0.300 4660 4358 6.5

0.312 4488 4896 9.1

0.336 6482 6306 2.7

0.210 673 542 19.5

p 0.305 1198 1242 3.8

0.327 1834 2080 13.4

· · · 3.4.2.3 Empirical reiationship for fracture toughness prediction

Using the neural network connection weights and bias values given in Table 3.10,

. an empirical relationship is formulated for the prediction of fracture toughness of I.·' multilayered composite laminates as: f:: .. i '

ey -e-y i Fracture toughness , K = --­ey +e-y

I.·

·. ( 20

J e'P -e-·"

where y = �p=I o p * W2rx, + Bo Op = L.J e'''+e-•p

(3.3)

55

a V V V

I. are the input data, w ' v, E 'E' GJ X y xy

WI.. ,W2 , B. and B0are the connection weights and node bias values.

IJ po I

Table 3. IO Connection weights and bias values of the neural network used for the

fracture toughness prediction

Bias Connection Weight Bli Wl1(il Wl2(i) Wh(il Wl4(il Wls(il W21(0) 6.102 -17.391 3.991 -0.807 3.372 0.952 2.644 -0.400 1.100 -1.693 3.279 -4.912 -15.70 4.697 -5.368 8.582 5.213 -3.967 2.014 2.266 0.317 -0.556 3.954 -0.034 -7.193 5.977 -3.881 -0.835-0.301 -0.518 2.977 3.347 5.105 -0.935 -0.344-0.342 -5.828 -0.998 -1.933 -0.564 0.811 -2.6054.828 -13.957 3.526 -2.224 6.172 0.352 -2.4960.132 1.732 3.210 0.351 0.000 0.121 -1.511-3.674 9.949 2.813 -6.837 0.951 -4.015 0.627 -0.423 0.495 -4.081 3.697 1.927 -0.870 -2.129-5.556 6.020 5.023 0.106 0.821 3.146 -0.1961.081 0.390 -0.449 2.011 1.853 -1.389 -1.047Bias of output Neuron B0

= 3.086

3.5 CONCLUSION

The analytical prediction of failure strength I fracture toughness of composite

laminates with defect in the form of hole or crack is very well established using the

limited number of training data. In the analytical prediction of notched strength of

laminates using W-N failure criterion, the characteristic dimensions d0 or a

0 must be

apriori known, which is obtained from experimental results. In the analytical

prediction of fracture toughness, the mode of failure and the failure stress of the

laminate should be known in advance. In the present work notched strength I fracture

toughness of a laminate is predicted from the large amount oftest data available in

literature using artificial neural networks.

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Forthe notched strength prediction, an optimum network architecture with nine

neurons in the hidden layer is established using the training set ofexperimental data

on do for (O/±a)s and (±a /O)s class of graphite-epoxy laminates. Genetic algorithm

used to train the network in the prediction ofnotched strength of(O/± a)s and (±a/O)s

ASI/3501-6 graphite-epoxy laminates with circular holes gives minimum error

compared to back propagation algorithm.

For most of the laminates of class (±a/O) , the percentage network trainings

error in the predicted failure strength based on point stress criterion, varies from a

minimum value of3.7 to a maximum value of6.5 except for [±10/0]s laminate for

which theremay be some experimental error in determining the do. Similar values for

(O/± a)slaminates are 0.1% and 3.6% respectively. However, in the case of average

stress criterion the variations are 1.1 to 9.9 for the (O/±a)s laminates and 0.01 to 6.9

for (±alO)s laminates. The trained network predicts the characteristic distances do or

aofordifferent lay-ups which can be used for the notched strength prediction. The

maximum error on notched strength prediction based on point stress criterion is 8.5%

and that for average stress criterion is 11.6% respectively.

The intra-laminar fracture toughness of multilayered laminates of different

composite material systems ofT3001914C, T300/l034, AS4/3502 and glass-epoxy

Scotchply 1002 based on the toughness values ofT300/5208 carbon-epoxy for a/w

ratios varying from 0.1 to 0.65 are predicted using neural-GA approach. A reasonably

good agreement is observed between the prediction and the test data for different

material system. It is interesting to note that even for glass-epoxy laminate that has

about one third of the modulii ratio of carbon-epoxy laminate, could give a good

prediction on toughness value within 0.2 to 15%. It may be concluded that such a

prediction for different material system is possible as toughness is related to the

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stiffness properties of material for energy required to produce unit crack extension;

in other words related to the strain energy release rate.

Empirical relationships useful for the prediction of characteristic distance

· and fracture toughness of multilayered laminates is also developed using the

� connection weights and bias values of the trained network. � � 'ft

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