Analytical solution for impact response of CFRP laminated composite plates.

Dogaru, Florin

1. INTRODUCTION

The CFRP material presents an increased susceptibility to damage due to impact. The weak behaviour of the polymeric composite material reinforced with carbon fibres is due to the reduced capacity to absorb the 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 the 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 (2005) studied deeper the impact response taking into account different models. Swanson (2000) showed that when the mass of the plate is much smaller than the projectile's mass the response is quasi-static. In this paper the author gives a simple solution for solving non-linear integral equation that results in the impact response case of the laminated composite plate using complete model based on Mindlin-Reissner's plate theory. The non-linear equation was solved iteratively using trapezoid formula and the results are obtained using Matlab program.

2. DESCRIPTION OF THE ANALYTICAL MODEL

The governing equations for a symmetric orthotropic laminated composite plate, subjected to lateral load, including transversal shear deformation with [B.sub.ij]=0, [A.sub.16]=[A.sub.26]=[D.sub.16]=[D.sub.26]=0, are given as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [D.sub.ij] and [A.sub.ij] are the stiffness of the laminated plate, h is the thickness, w is the traverse deflection, [rho] is the material density, [[psi].sub.x], [[psi].sub.y] are the shear rotations, [kappa] is the Mindlin shear correction factor, q is the lateral load per unit area, [I.sub.1]=[rho]h and [I.sub.3]=[rho][h.sup.3]/12. The solution is based on expansions of the displacements and rotations in double Fourier series which satisfy the edges boundary conditions for simple supported case. Each expression is assumed to be separable into a time dependent function and a function of position. In the same way, the lateral load is expressed in double Fourier series. The contribution of the rotary inertia terms are small and can be neglected. We obtain 3 independent equations for each set of the modal parameters (m,n):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Cristoforou & Swanson (2000) reduced the Eq.(2) to a single differential equation:

[[??].sub.mn](t) + [[omega].sup.2.sub.mn] [W.sub.mn](t) = [Q.sub.mn](t)/[rho]h. (4)

For zero initial displacements and velocity the solution of Eq.(4) is obtain using the convolution integral:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

After substitution of the Eq.(5) in Eq.(1) we obtain the plate's displacement of the point (x,y) in case of concentrate loading in point ([x.sub.1], [y.sub.1]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Considering the central impact case, 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]. From the equilibrium projectile's equation with initial conditions:

[w.sub.1(t=0)] = 0 [[??].sub.1(t=0)] = V, [w.sub.1] = [delta] + [w.sub.2], (7)

and through integration, the projectile's displacement becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (8)

where V and M1 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], (9)

where [[kappa].sub.c] is contact stiffness' coefficient. In case of central plate's impact, taking out [delta] from Eq.(7.3) and considering Eq.(6,8.2) after substituting in Eq.(9.1), we obtain the nonlinear integral equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

3. NON-LINEAR EQUATION SOLUTION

Trapezoid formula is used to solve Eq.(10). For particular case of Eq. (10), there are two kinds of functions under integral:

[f.sub.1]([tau]) = F([tau]) x (t - [tau]), [f.sub.2]([tau]) = F([tau]) x sin[[omega](t - [tau])] (11)

Solving separately and dividing the interval (0, t) in i equidistant intervals, [t.sub.0], t = [t.sub.0] x i. The results are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (12)

where F(i) = F(i x [t.sub.0]), and for the moment t=0 the two elements get into contact, F(0) = 0. Using Eq.(12) the Eq.(10) was solved iteratively with MATLAB (Dogaru & Baba 2008):

F(i +1) = [[kappa].sub.c][(V x [t.sub.0] x i - 1/[M.sub.1] A(i) - B(i)).sup.3/2], F(0) = 0. (13)

The dimensions of the composite plate were 125x75[mm.sup.2], 2.5mm thickness, 8 unidirectional laminae with[[0/45/-45/90].sub.s] stacking sequence and [M.sub.1] = 1.9kg. The characteristics of the lamina were: [E.sub.1]=54GPa, [E.sub.2]=[E.sub.3]=4.5GPa, [G.sub.12]=[G.sub.23]=1.65GPa, [[upsilon].sub.12]=0.3 with fibers volumetric ratio about 35%. The analytical response of the composite plate was calculated taking in account 5x5 modal parameters, for different cases of impact velocity. Figures (1,2) illustrate the variation related to time of the contact force, projectile's displacement and central plate's displacement in the contact point due to impact. Notice that the analytical solution is accurate only for small displacements, generally for maximum transversal displacement smaller than the half thickness of the plate. It is to note the concordance between the results obtained theoretically and experimentally (Dogaru & Baba 2008). It is noticed the maximum contact force and maximum displacement are simultaneously reached that means the impact is quasi static. In the future the authors intend to simulate the effect of the damages on the impact response.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

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

4. REFERENCES

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

Dogaru F. & Baba M., (2008). The Response Analysis of the CFRP Laminated Plates Due to Low Velocity Impact--The 19th International DAAAM SYMPOSIUM, ISSN 17269679, 22-25th October 2008, Trnava, Slovakia, pp.401-402

Olsson, R. (1992). Impact response of orthotropic composite plates predicted from a one-parameter differential equation, AAA Journal, Vol.30, No.6, pp.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, pp. 693-698

Swanson, S. R. (2000). Elastic impact stress analysis of composite plates and cylinders, Impact behaviour of fibre-reinforced composite materials and structures, Edited by Reid, S., R., Zhou, G., Woodhead Publishing Limited, Cambridge England, ISBN 1 85573 423 0, pp. 186-211