Estimation of contact force on composite plates using impact-induced strain and neural networks

PII: S 1359-8368(98)00003-1

ELSEVIER

Composites Part B 29B (1998) 363-370 © 1998 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/98/$19.00

Estimation of contact force on composite plates using impact-induced strain and neural networks

K. Chandrashekhara, A. Chukwujekwu Okafor and Y. P. Jiang Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, Rolla, MO 65409, USA (Received 15 January 1997; accepted 21 December 7997) A method of determining the contact force on laminated composite plates subjected to low velocity impact is developed using the finite element method and a neural network. The backpropagation neural network is used to estimate the contact force on the composite plates using the strain signals. The neural network is trained using the contact force and strain histories obtained from finite element simulation results. The finite element model is based on a higher order shear deformation theory and accounts for von-Karman non-linear strain-displacement relations. The non-linear time dependent equations are solved using a direct iteration scheme in conjunction with the Newmark time integration scheme. The training process consists of training the network with strain signals at three different locations. The effectiveness of different neural network configurations for estimating contact force is investigated. The neural network approach to the estimation of contact force proved to be a promising alternative to more traditional techniques, particularly for on-line health-monitoring system. © 1998 Elsevier Science Limited. All rights reserved (Keywords: contact force; A. plates; neural networks)

INTRODUCTION Advanced fiber-reinforced composite materials have been used extensively in aerospace and other applications. The effect of low-velocity impact on laminated composite structures has been the subject of much study over the last two decades 1'2. Impact-induced damage may arise during manufacture, maintenance and service operation. Impact analysis includes the study of contact laws, damage from impact and estimating contact force. The determination of the contact force on laminated composite plates subjected to low velocity impact is the focus of the present study. Doyle 3 and Prasad et al. 4 have determined the contact force history from strain gauge measurements. Chang and Sun 5 determined the dynamic impact forces on a composite laminate by using experimentally generated Green's functions and signal deconvolutions. Wu et al. 6 presented an analytical method for identification of impact force histories exerted at multiple locations on a laminated composite plate. However, due to ill-posed characteristics of Green's function, this inverse problem has difficulties in achieving accurate results. In recent years, applications of neural networks have attracted increasing attention owing to their capabilities for pattern recognition, classification and function approximation. Some of the key features of neural networks are their processing speeds which are due to massive parallelism, their proven ability to be trained, to produce instantaneous

and correct responses from noisy or partially incomplete data, and their ability to generalize information over a wide range. The applications of neural networks for damage detection have been studied by Kudva et al. 7 Szewczyk and Hajela 8 and Okafor et al. 9 With the availability of neural networks, new powerful tools for contact force estimation can be developed. In the present study, a methodology is developed to determine the contact force on laminated composite plates using finite element analysis and neural networks. The finite element model (see Schroeder and Chandrashekharal°), which is based on a higher-order shear deformation theory, is used to calculate the strain pattern and contact force for low-velocity impact of the composite plate. The neural network is trained using the results from the finite element analysis. Once trained, the network can be used for on-line estimation of contact force if the impact-induced strain pattern can be obtained experimentally or numerically. The study of this type of problem represents one technique that offers promise for incorporation into smart structures as part of subsystems for on-line structural health monitoring.

FINITE ELEMENT MODEL FOR IMPACT ANALYSIS In impact analysis, where the load is essentially a point load, the errors due to the neglect of shear deformation effects

363

Contact force on composite plates: K. Chandrashekhara et al. o 3Wo "~yz = ~/y -]- ~-y ' ~yz ~---3dpy • °z = Cx + OWo

o ~ ' ~xz = 34,x

o

>Y

+_/ >/

OUo

OVo . OWo OWo

7x

xy=

,,Txy=-ffy The laminate constitutive equation can be written as

{M} {P}

Figure 1 Laminated plate geometry and coordinates systems

=[/)1

{Q} will be severe. Another complication of laminated composite is the influence of bending-twisting coupling in the constitutive behavior. The complications introduced by the foregoing factors, as well as other issues such as non-linear behavior, prevent the use of closed form analyses of composite laminates subject to impact. A non-linear finite element model based on a modified Hertzian contact law is used for impact analysis. A brief description of the finite element model is as follows (for details, see Schroeder and Chandrashekharal°). Consider a laminated plate as shown in Figure 1. Any point within the plate can be described with the following higher-order displacement field:

u(x,y,z,t)=Uo(x,y,t)+ZCx(X,y,t)+z3rkx(X,y,t,)

(1)

{r} {~/}

(4)

{7}

where {N} = {NxNyNxy}r; {M} = {MzMyMxy} r

(5)

{P} = {PxPyPxy}r; {Q} = {QyzQxzRyzRxz} r f oo o~T. {e} : 16xey'J'xyl , {K} : {KxKyKxy} r {n} ~---{nxnynxy}T;

{~} = {'~yz'~°z~yz~xz} T

and [/3] is the laminate stiffness coefficient matrix. A nine-noded isoparametric quadrilateral element with 63 degrees of freedom is implemented. The generalized displacements are interpolated over the element and are given by 9

{fie(x'Y 't)} = Z

v(x, y, Z, t) = Vo(X,y, t) + Z~y(X, y, t) + Z3~y(X, y, t) W(X, y, Z, t) = Wo(X, y, t)

(2)

[M]{A} + [K{A}I{A} = {F}

"~xy : "~°xy~- ZKxy -~ Z3~xy where

miwi = - Fc

OCx

:x = 7 x * 2 \ ox ] ' Kx = o x ' ~ = ox o 0Y0 1 : aW0~ 2 OCy O~y Ey : ~y "q- 2 ~k Oy ] ' Ky = Oy , 7qy = -~y

(8)

(9)

where Fc is the applied contact force at the center of the plate. For a rigid impactor, the equation of motion can be expressed as

o _~_Z2~ 7xz = ~xz ~xz

Ouo 1: Owo 2

(7)

and N~/ are the Lagrange interpolation functions, [/] is a 7 x 7 identity matrix, and Uoi, voi, etc., are the nodal values. The assembled equations can be written as

{F} = {0,0 ..... O,F~,O ..... 0} r

2

7yz = "~yz "~-Z ~yz

384

{ ~le } ~- { Uoil)oiWoi ~xil~yi(~xi~y i } T

where [/14] is the mass matrix, [K{A}] is the non-linear stiffness matrix. The force vector {F} is defined as

ey = eyo + ZKy + Z3"qy o

(6)

where

where u, v and w are displacements along the x-, y- and z-axes; Uo, v0 and w0 are the midplane displacements; and Cx, Cy, q~x and thy are the rotations. The non-linear strain-displacement relations, based on the von Karman assumptions, can be written as

ex = e° + ZKx + Z3nx

[Ne(x'Y)]{tte(t)}

i=1

(3)

(10)

In order to solve eqn (9) and eqn (10), the contact force between the laminated plate and impactor must be known. It is assumed that during loading and unloading, the contact force distribution follows the Hertzian contact law for loading and an expression proposed by Yang and Sun 11 for

Contact force on composite plates: K. Chandrashekhara e t al. unloading. These relations are given by

NEURAL NETWORK

Fc = KO~1'5 loading

(ll)

] 2.5 unloading

f c : F m I 0o /~- o

/ C~r,-- C¢o/ where Ot : Wi(t ) -- Wp(t)

(12)

and K is the modified Hertzian contact stiffness given by K=

4

(13)

3 [(1 - .~)/EI + 1/E2]

where E2 is the modulus of elasticity transverse to the fiber direction, ri, ul and E1 are the radius, Poisson's ratio and modulus of elasticity of the impactor, respectively. In the unloading equation, F m is the maximum contact force reached during loading, o~m is the maximum indentation which corresponds to Fm and ao is the permanent indentation from the loading/unloading cycle. Expressions for the permanent indentation are oto = 0 for Otm < ffcr

Oto = Otm 1 - k ~mm!

(14)

J for Otm ~ Otcr

where acr is the critical indentation. To solve eqn (8), an iterative procedure in conjunction with the Newmark constant acceleration scheme is used. Numerical results for the contact force and strain histories can be obtained for various impactor conditions, plate geometry and boundary conditions. For training the network, the strain signals are determined at three different locations of the plate (see Figure 2).

e s~ -- e - si

•

Impact location • Strain location All dimensions are in mm

I

f ( s i ) = tanh(si) =

25.4

127

-,._1 I-I

63.5

127 Figure 2

Strain and impact locations

e si + e - s~

(15)

where si is the summed weighted signal for the ith processor in the current layer andf(si) ranges from - 1 to 1. Okafor and Adetona ]2 have shown that the hyperbolic tangent function gives a better performance than the sigmoid and sine functions. Training consists of providing a set of known inputoutput pairs to the network. The network iteratively adjusts the weights of each of the nodes so as to obtain the desired outputs. The normalized cumulative delta rule is used to train the network. During the backward propagation, weights in each node are adjusted based on the errors using the following gradient descent equation:

I]1 25.4 ~]'<-"'!

63.5

Artificial neural networks (ANN) are systems with inputs and outputs composed of a large number of interconnected parallel processing units. Although various network architectures exist, a backpropagation network is used in the present study, based on its applicability to many different types of prediction and pattern recognition tasks. A backpropagation network is a multilayered architecture and is made up of one or more hidden layers placed between the input and output layers. Each layer consists of a number of processing units known as neurons. Inputs are passed through weighted connections to the first hidden layer, then from the hidden layer to the output layer. The hidden layers permit non-linear modeling of the input-output relationships. The performance of the network, which depends on the hidden layers, is problem-dependent. There are no established procedures for choosing the optimal number of hidden layers and nodes per layer. The current choices are made after trying out several alternatives. In the present work, a backpropagation neural network with one input layer, three hidden layers and one output layer is used to estimate the contact force. A schematic of the neural network is shown in Figure 3. The input layer has six processing units. The inputs to the input layers are inplane normal strains at three different locations of the plate (see Figure 2). The first hidden layer has 20 processing units, the second and third layers have 10 processing units each. The output layer has one processing unit which predicts the contact force. The network is designed and trained using the Neural Works Professional II package. The operation of this network consists of two phases: forward activation and backward error propagation. For a given set of input/output pairs, called a training set, forward activation starts in the neuron from the input layer. In the present study, the hyperbolic tangent function is used as the activation function and is given by

>1

w i j ( n --[- 1) = w i j ( n ) q- A w i j ( n -J¢- l)

(16)

X >

where Awij(n + 1) is the current change in weight between node i and j at the (n + 1)th cycle, and can be

365

Contact force on composite plates: K. Chandrashekhara et al. Hidden layer 1

Hidden layer 2

Hidden layer 3

Output layer

Contact force

Figure 3 Neural network for contact force prediction

3500

3000

Linear Nonlinear

2500 v

2000

1500

1000

500

100

200

300

400

500

Time (psec) Figure 4 Contact force histories using linear and non-linear plate theories

expressed as

Awij(n -I- 1) = rlSiYj -F otAwij(n )

(17)

where ~? is the learning rate, c¢ is the momentum coefficient, 5i is the error to be propagated back for the ith output layer processor, and Yj is the output from the jth processor in the lower layer. The training and learning of a neural network is an iterative process. At the beginning of the training, the network will produce outputs which most likely differ significantly from the desired

366

values. The procedure is repeated until convergence is achieved.

RESULTS AND DISCUSSION A computer code is developed based on the finite element model presented. The reduced integration is used to evaluate the coefficients associated with the shear energy terms. For all the examples considered, a full plate is discretized with a

Contact force on composite plates: K. Chandrashekhara et al. O.OEO

-5.0E-4

g ._o -1.0E-3

o~ E3 "~ Q} -1.5E-3 C.)

-

-

.....

,'

Linear

',

Nonlinear /

~x.

-2.0E-3

-2.5E-3

1O0

200

300

400

500

Time (psec) Figure 5

Center deflection histories using linear and non-linear plate theories

~2500

Filv

1500

~ /

~ ~J~

,ooo1

Ifl /

A

,

.

0

1O0

"

^AA

0

Figure 6

~

200

I

Time (ttsec)

,300

t

400

,

500

Comparison of non-linear contact force histories

4 × 4 uniform mesh of nine noded isoparametric elements. The impactor considered is a steel sphere having a mass density of 7870 kg m -3. Table 1 Composite material properties

Linear and non-linear analysis Figures 4 and 5 show the contact force and center deflection histories of a four-layer symmetric cross-ply [0°/900/90°/0 °] Gr/Ep plate with simply supported boundary conditions. The material properties of Gr/Ep composite are shown in Table 1. The plate dimensions used are a = b = 254 mm, and h ---- 2.54 mm. The impactor diameter is

Properties

Graphite/epoxy (Gr/Ep)

E-Glass/epoxy (GVEp)

E~ E2 G12 G13 G23 ~'tz

144.8 GPa 9.65 GPa 7.10 GPa 7.10 GPa 5.92 GPa 0.30 1389.2 kg m -3 8.03 X 10 -z mm

42.75 GPa 11.72 GPa 4.14 GPa 4.14 GPa 3.45 GPa 0.27 1901.5 kg m 3 10.16 × 10 2 mm

p otc,

367

Contact force on composite plates: K. Chandrashekhara

et

al.

O.OEO.

- - •

-.

-5.0E-4

DYNA3D Present

E

," -1.0E-3 0 0

~e ' - -1.5E-3 0

-2.0E-3

100

0

200

300

400

500

Time (p.sec)

Figure 7 Comparison of non-linear center deflection histories 1500

•

[

[

i,00[

^

/ \

/\ I \/

/~

I

~

,,

~ A

T~t,~,,i~

. . . .

/ v ,, v /

F'miteelemtntsimulation

.o

O

0

10

20

30

~

40

50

60

70

80

90

100

Time (ITS)

Figure 8 Contact force prediction by the trained ANNI

12.7mm and is traveling at 3 0 m s -~. The difference between linear and non-linear theories is shown in Figures 4 and 5. During the initial impact, it appears that no appreciable difference in contact force prediction exists between the two theories. The second contact occurs at an earlier time for the non-linear theory than the linear case. Linear theory predicts a larger center deflection after approximately 100/zs compared with that for the non-linear case. A non-linear finite element simulation of the impact analysis for the above composite plate is also performed using the explicit LS-DYNA3D code. This code was

368

adopted owing to its strong contact algorithms. The composite plate is modeled using Belytschko-Tsay fournoded quadrilateral shell elements with single-point quadrature per lamina. Four hundred shell elements were used to model the composite plate. In the contact area, very fine mesh was used because the contact pressure changed dramatically in the contact area. Figure 6 shows the contact force histories using LS-DYNA3D simulation. Figure 7 shows center deflection histories using LS-DYNA3D simulation. The results from LS-DYNA3D simulations are in good agreement with the present non-linear theory prediction.

Contact force on composite plates: K. Chandrashekhara et al. 1500

/\ / \/

10001

XA

---

Finiteelementsimulation

O500

0

10

20

30

40

50

60

70

80

1 O0

90

Time (p.s)

Figure 9 Contact force predictionby the trained ANN2

. . . .

i

. . . .

I

'

'

/~

t,

/

'

I

. . . .

I

. . . .

~'

/ / I \1

i

. . . .

i

. . . .

i

. . . .

I

. . . .

I

. . . .

O,=0.6,11=0.2,trainingcycles=350699

l

•

'~ A

....

Finim element ,imulation

1000

g

,/

5OO

4

0

,.

0

1

10

. . . .

I

. . . .

20

I

30

. . . .

I

,

40

.

50

60

70

80

90

100

Time (p-s)

Figure 10 Contactforce predictionby the trainedANN3 Training and testing of the neural network For training the network, a set of data which contains the strains as the inputs and the corresponding contact force as the desired output is used. In the present study, a 12-layer [0°/90°/0°/90°/0°/90°] s laminated GI/Ep plate with clamped edges and central impact loading is analyzed using the finite element code. The impactor diameter is 7.9 m m and is traveling at 23.4 m s -1. The plate dimensions are a ----b = 127 m m and h = 2.3 mm. The material properties of G1/Ep considered are shown in Table 1. Training data sets of non-linear strain and contact force histories are generated from the finite element simulation results with a time step of 0.5/~s for contact duration of

100/zs. The data sets at 10/xs interval are kept for testing and are not used for training. The inplane normal strains are computed at three locations (see Figure 2) on the bottom surface of the plate. There are 180 i n p u t - o u t p u t data sets for training the neural network. To investigate the effect of training parameters, four neural networks are considered: A N N I : ~/= 0.2, c~ = 0.2; ANN2: ~/= 0.2, c~ = 0.6; ANN3; ~/ = 0.6, c~ = 0.2; and ANN4: ~ / = 0.6, a = 0.6. The performance of the trained network is investigated by inputting the strain signals obtained at a time step of 10/~s. Thus there are 10 data sets for testing the trained network. These strain signals are generated from the finite element simulation results and are not included in the original training sets. The threshold for convergence check is taken

369

Contact force on composite plates: K. Chandrashekhara et al. 1500

.

.

.

.

.

'

'

'

I

'

'

'

1

I

'

'

~

1

'

'

l

.

.

.

.

I

.

.

.

.

I

.

.

.

.

l

.

.

.

.

I

.

.

.

.

qx---0.6,rl=0.6, training cycles=571725

~ ~

• A

. . . .

10o0

Test results Finite element simulation rk prediction

A

z

~3 500 I

10

20

30

40

50 Time (ITS)

60

70

80

90

100

Figure 11 Contact force prediction by the trained ANN4

as 0.05 for normalized contact force output. Figures 8-11 show the predicted and actual contact force histories. The networks ANN1, ANN3 and ANN4 converged after 763010, 350699 and 571725 training cycles, respectively. The network ANN2 failed to converge to the threshold. It can be seen that the estimation of contact forces obtained by the trained networks ANN1, ANN3 and ANN4 are in good agreement with finite element simulation results. Among the networks considered, ANN3 needs least training cycles for convergence.

Missouri Department of gratefully acknowledged.

1. 2. 3.

CONCLUSIONS 5. 6. 7. 8. 9.

10.

ACKNOWLEDGEMENTS This work is supported by the Office of Naval Research under grant #ONR N00014-94-1200 with Dr Thomas M. McKenna as the technical monitor. Partial support from the

370

Development

is

REFERENCES

4.

Application of a backpropagation neural network for contact force estimation has been described. The training data sets, namely strains and contact force histories, for the composite plate are obtained using non-linear finite element analysis. For the case illustrated, three hidden layers are needed for the accurate prediction of the contact force. The learning rate and momentum coefficient have significant effects on the training process. The present study demonstrates that the trained network can be used for on-line prediction of contact force if impact induced strain patterns can be obtained experimentally or numerically.

Economic

11. 12.

Cantwell, W. J. and Morton, J., The impact resistance of composite materials--a review. Composites, 1991, 22, 347-362. Abrate, S., Impact on laminated composites: recent advances. Applied Mechanics Review, 1994, 47, 517-544. Doyle, J. F., Determining the contact force during the transverse impact of plates. Experimental Mechanics, 1987, 27, 68-72. Prasad, C. B., Ambur, D. R. and Starnes, J. H. Jr., Response of laminated composite plates to low-speed impact by different impacturs. AIAA Journal, 1994, 32(6), 1270-1277. Chang, C. and Sun, C. T., Determining transverse impact on a composite laminate by signal deconvolution. Experimental Mechanics, 1989, 29, 414-419. Wu, E., Yeh, J. C. and Yen, C. S., Identification of impact forces at multiple locations on laminated plates. AIAA Journal, 1994, 32, 2433-2439. Kudva, J. N., Munir, N. and Tan, P. W., Damage detection in smart structures using neural networks and finite element analyses. Smart Materials and Structures, 1992, 1, 108-112. Szewczyk, Z. P. and Hajela, P., Damage detection in structures based on feature-sensitive neural networks. ASCE Journal of Computing in Civil Engineering, 1994, 8, 163-179. Okafor, A. C., Chandrashekhara, K. and Jiang, Y. P., Delamination prediction in composite beams with built-in piezoelectric devices using modal analysis and neural network. Smart Materials and Structures, 1996, 5, 338-347. Schroder, T. and Chandrashekhara, K., Nonlinear impact response of laminated plates using a higher order theory. European Journal of Mechanics: A/Solids, 1994, 13, 833-855. Yang, S. H. and Sun, C. T., Indentation law for composite laminates. NASA CR-165460, 1981. Okafor, A. C. and Adetona, O., Predicting quality characteristics of end-milled parts based on multi-sensor integration using neural networks: Individual effects of learning parameters and rules. Journal oflntelligent Manufacturing, 1995, 6, 389-400.