The response analysis of the CFRP laminated plates due to low velocity impact.

Dogaru, Florin ; Baba, Marius Nicolae

1. INTRODUCTION

The CFRP material presents an increased susceptibility to damage due to impact. The weak behavior of the polymeric composite material reinforced with carbon fibers is due to the reduced absorption capacity of kinetic energy of the projectile.

There are many solutions in literature that were presented for different impact cases. Sun & Chattopadhyay (1975) used a complete model for studying a rectangular plate in the case of impact influenced by the wave propagation when the maximum displacement of the plate is out of phase with the projectile's displacement. Olsson (1992) gave an approximate solution for case of impact influenced by the wave propagation. Abrate (1998, 2005) studied deeper the impact response taking into account different models proposed by researchers in past years. Swanson (1997) showed that when the mass of the plate is much smaller than the projectile's mass the response is quasistatic.

In this paper the authors give a simple solution using complete model on the basis of Kirchhoff's plate theory concerning the impact response of CFRP laminated plate.

2. DESCRIPTION OF THE ANALYTICAL MODEL

A complete model takes into consideration the structure's dynamic, the projectile's dynamic and the contact force, (Sun 1975, Abrate 2005). In case of simply supported plate at the edges, Navier resolution can be used to obtain a closed-form solution to the transitory response. In accordance with the classic plate theory, for a symmetrically laminated plate, without bending-twisting coupling ([D.sub.16]=[D.sub.26]=0), the displacement equation of the plate loaded by a concentrated force may be written in the form:

[D.sub.11] [[partial derivative].sup.4]w/[partial derivative][x.sup.4] + 2([D.sub.12] + 2[D.sub.66]) [[partial derivative].sup.4]w/[partial derivative][x.sup.2] [partial derivative][y.sup.2] + [D.sub.22] [[partial derivative].sup.4]w/[partial derivative][y.sup.4] + + [I.sub.1] [??] = F(t), (1)

where [D.sub.ij] stands for bending and torsion stiffness, [I.sub.1]=ph, [rho] is density of the plate, h thickness of the plate, F(t) contact force and w stands for the transversal displacement of the plate.

Considering a plate with axb dimensions we obtain the transversal plate's displacement of point (x,y), for the impact case in point ([x.sub.1], [y.sub.1]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The projectile's displacement, [w.sub.1], is the sum of the local displacement due to contact, [delta] and plate's displacement, [w.sub.2]. Considering the equilibrium projectile's equation with initial conditions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and so, through integration, the projectile's displacement becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where V and [M.sub.1] are the velocity and the mass of the projectile. In order to model the contact, Hertz's contact law was used. It was supposed that the projectile has spherical shape and the Hertz's contact law is valid for the dynamic case as well:

F(t) = [[kappa].sub.c][[delta].sup.3/2], [[kappa].sub.c] = 4/3[E.sub.3][R.sup.0.5], [E.sub.3] = [E.sub.2], (4)

where [[kappa].sub.c] is contact stiffness' coefficient. Taking out [delta] from Eq.(3.1) and considering Eq.(3.3, 2) after substituting in Eq.(4.1), the non-linear integral equation, for central plate's impact becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

3. NON-LINEAR EQUATION RESOLUTION

Trapezoid formula is used to solve Eq.(5). For particular case of equation (5), two kinds of functions under integral exist:

[f.sub.1]([tau]) = F([tau]) x (t - [tau]), [f.sub.2]([tau]) = F([tau]) x sin[[omega](t - [tau])] (6)

Solving separately dividing (0, t) in i equidistant intervals, [t.sub.0], t = [t.sub.0] x i. The result is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where F(i) = F(i x [t.sub.0]), and for the moment t=0 the two elements get into contact, F(0) = 0 . Quite in the same way, for the second function, taking in account m x n participating mode of the plate, the result is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Using Eq.(7, 8) the Eq.(5) was solved iteratively with MATLAB program (Dogaru et al., 2005).

4. RESULTS AND CONCLUSIONS

The analytical and experimental analyses were conducted on a composite plate made of epoxy vinyl ester matrix (Derakane 470-30-S) reinforced with carbon fibers with dimensions 150x100[mm.sup.2] and 2.5mm thickness, 8 unidirectional laminae and symmetric orientation [[0/-45/+ 45/90].sub.s]. The characteristics of lamina's plate were: (E.sub.1)=54GPa, [[upsilon].sub.12]=0.3, [E.sub.2]=[E.sub.3]=4.5GPa, [G.sub.12]=[G.sub.23]=1.65GPa. The plates used to cut off the specimens were manually manufactured and the resin was impregnated by brushing-on action and the fibers volumetric ratio was about 35%.

The impact test was done by the use of a device designed for this particular study, having the energy capacity of 1-50J obtained by adjusting the height and/or the weight. The specimen was simply supported at the edges against a metal plate (30mm thickness) with interior cutting-out of 125x75[mm.sup.2] by the intermediary of a wooden plate (6mm thickness) in order to avoid the specimen crushing at ends. The specimen was fixed at the edges during the impact at four points with screws having rubber disposed on the tip and manually screwed. The projectile had a 16-mm diameter semispherical head made of alloyed steel with increased hardness, 1.9kg weight and the impact was targeted at the plate's center. Behind the projectile, an accelerometer was attached (screwed), in order to measure the projectile's acceleration and then to calculate the contact force during the impact. The results were recorded by using an acquisition plate NI USB 6251 BNC. The velocity and the displacement of the projectile during the impact were calculated through integration of the measured acceleration curve using LabVIEW program (Dogaru et al., 2005).

A new static study was done because the impact is quasi static, it meant that the maximum force and the maximum displacement are reached simultaneously (Swanson 1997), using FEM (Ansys), in which the force was applied statically on the projectile which is considered in contact with the laminated plate. A 1000 elements were used, SHELL181 elements for plate's simulation, SOLID187 for projectile's simulation and CONTA174, TARGE170 for contact's simulation. The force applied on projectile was the maximum value recorded in dynamic experimental investigation, F=3750N. The analysis is done on quarter of the model due to the symmetry considering the large displacements and contact effects.

The analytical response of the composite plate was calculated using analytical solution, taking in account 5x5 modal parameters, for different cases of impact velocity.

Fig.(1) illustrates the variation of the maximum contact force due to impact related to the central maximum transversal displacement of the plate obtained analytically and experimentally. For comparison, the static solution obtained with FEM is presented, too. The solution due to Eq.(5) is accurate only for small displacements, generally for maximum transversal displacement smaller than the thickness of the plate fg.(1).

[FIGURE 1 OMITTED]

Notice that for a projectile initial velocity higher than 1.5m/s, the results obtained using dynamical experiment are quite scattered and the mean value of the forces recorded, is between linear analytical solution and nonlinear numerical solution obtained with FEM. This occurred because at this loading level, the damages are introduced in the plate and they reduce the stiffness and also the contact force value. Also notice the concordance, for small displacement case, between the results obtained analytically, numerically using FEM static solution and experimentally. In the future the authors intend to investigate the damage introduced by impact, the level of the contact force that causes the damage and its effect on the residual properties. On the basis of these observations, the concept of a new equivalent model shall be also considered in the future in order to simulate the damage effects occurred in the composite plate, on the residual properties and impact response.

Acknowledgements. This research was done with financial support of MECT and ANCS, contract PN II--IDEI, ID_187, 110 / 1.10.2007.

REFERENCES

Abrate, S. (1998). The Dynamics of Impact on Composite Structure, In: Impact Response and Dynamic Failure of Composites and Laminate Materials, Part 2, editors J.K.Kim and T.,X., Yu, Trans Tech Publications, ISBN-13: 978-0878497690, Switzerland.

Abrate, S. (2005). Impact on Composite Structures, Cambridge University Press, ISBN-13: 978-0521018326, United Kingdon.

Dogaru F. & Curtu I., et al., (2005). Analytical And Experimental Investigation Concerning Response Of The CFRP Laminated Plates Due To Low Velocity Impact--9th International Research/Expert Conference "Trends in Development of Machinery and Associated Technology", ISBN 9958-617-28-5, TMT 2005, 26-30.09, Antalya, Turkey.

Olsson, R. (1992). Impact response of orthotropic composite plates predicted from a one-parameter differential equation, AIAA Journal, Vol.30, No.6, p.1587-1596.

Sun, C., T., Chattopadhyay, S. (1975). Dynamic Response of Anisotropic Laminated Plates under Initial Stress to Impact of a Mass, Journal of Applied Mechanics, p. 693-698.

Swanson, S. R. (1997). Introduction to Design and Analysis with Advanced Composite Materials, Pretice Hall, Upper Saddle River, ISBN-13: 978-0024185549, New Jersey.