On the Low Velocity Impact Response of Laminated Composite Plates Utilizing the p-Version Ritz Method

Y.P. Markopoulos and V. Kostopoulos

Applied Mechanics Laboratory, Department of Mechanical and Aeronautical Engineering,

University of Patras, GR-265 00 Patras, Greece e-mail: kostopoulos@mech.upatras.gr

ABSTRACT

Low energy impact in composite laminates is often a crucial and destructive loading condition since it leads to significant internal damage, undetectable by visual inspection. Low velocity impact upon a laminated plate imposes a complex stress state mainly due to the structural heterogeneity resulting by the ply orientation of the constituent laminas and the contact boundary conditions, which lead to a loading path that varies with the impact energy and the properties of the impactor-impacted plate system. The present work deals with the development of a numerical scheme for the calculation of the dynamic response of any type of laminated composite plates under low-velocity impact. The governing non-linear, second order differential equations are derived using p-Ritz admissible polynomial functions and the elastoplastic version of the Hertzian contact law. The dynamic response of fully clamped, cross ply and angle ply composite plates are investigated.

KEYWORDS: Low Velocity Impact, Normalised response, Clamped boundary conditions

1. INTRODUCTION

Low-velocity impact in composite laminates is often a destructive loading condition since it leads to significant internal damage, nondetectable by visual inspection. Numerous of researchers have made significant effort to model the impact response of composite plates and tried to facilitate the analysis and design for impact resistance, using simple linear spring-mass models or a combination of continuum mechanics models and contact laws. Many investigators have used simple engineering structures beams, plates and shells to demonstrate the impact response of composite structures.

The basic concept for modeling the impact of an object to a target was proposed by Timoshenko (1913). He introduced the procedure where the description of the interaction between the impactor and the structure (Euler beam) was implemented using the Hertzian contact law. This approach was further extended in isotropic plates and shells. Lee, 1940; Greszcuk, 1982; Lee et al., 1983 and Shivakumar et al, 1985, proposed simple models of the low-velocity impact problem.

Although many important contributions exist in the bibliography for the characterization of the impact response of laminated plates, analytical solutions are considered very few. Most of the models proposed are for analyzing specially orthotropic plates subjected to a local dominated impact. In addition, these models do not take into account the shear coupling terms of the bending stiffness matrix, e.g. D16 and D26.

Sun and Chattopadhyay (1975), Dobyns (1981), and Ramkumar and Chen (1983) employed the first order shear deformation theory developed by Whitney and Pagano (1970), and used it in conjunction with the Hertzian contact law or plasticity contact law for characterizing the impact of laminated composite plates. In their analysis they have studied the impact response of a simply supported orthotropic plate subjected to central impact using the lamination theory that includes transverse shear deformations. It is evident that the contact force history must be computed as part of the solution of the dynamic response problem solving the nonlinear integral equation. Christoforou and Swanson (1991) and Carvalho (1996), obtained an analytical solution of the impact problem using the Laplace transform technique. Qian and Swanson (1990) obtained analytical solutions by linearising the contact deformation law and compared this with a Rayleigh-Ritz approach with numerical integration in time.

The three dimensional finite element method with the explicit formulation where also used in the dynamic analysis of laminated plates subjected to low velocity impact. Moreover, many researchers have proposed simple models for characterizing the impact phenomenon. Shivakumar et.al. (1985) developed a simple model to predict the impact force and the duration of the impact phenomenon on composite plates. The composite plates were modeled as three springs while their stiffness was calculated by the plate properties and the contact parameters.

Other researchers have used instead of a classical Hertzian law the statical indentation law presented by

Yang and Sun (1982) and Tan and Sun (1985). The non Hertzian Contact impact was considered for combining the overall deformation of the structure to the local deformation in the contact area and was used to predict the impact response of transversely isotropic beams and plates. In general, it is impossible to obtain the exact solution for an impact problem except by ignoring the local contact deformation or pre-assuming the contact force.

The first step of understanding the problem is to predict the force applied by the projectile on the structure during impact. In order to characterize the contact force history, the model should account for the motion of the target, the effective stiffness of the projectile-target system and how this is affected by the interaction of the two materials. With the constitutive material equations used for the dynamic characterization of the plate, strain rate effects are negligible. Furthermore the low-velocity impact phenomenon frequency field, allows us to use the static contact laws for coupling the vibration of the plate problem and the external dynamic loading by an indenter-projectile.

The present work deals with the development of a numerical scheme for the calculation of the dynamic response of any type of laminated composite plates subjected to any type (and/or combination) of boundary conditions under low-velocity impact. Using the p-version Ritz polynomials any shape that can be represented in Cartesian coordinates can be formulated. The governing second order differential equations are derived and allow furnishing solutions with a variety of boundary conditions along the edges of the plate shape represented in Cartesian coordinates. Statically determined non-linear contact laws, for loading and unloading conditions, are coupled with the partial differential equations governing the dynamic response of the composite plate. These nonlinear governing equations of the contact-impact problem are decoupled according to the second order terms by the method of principal transformation and then solved numerically. The dynamic response of fully clamped cross-ply and a variation of simply support and fixed boundary conditions for angle ply and cross ply composite laminates was investigated. Rectangular plates where analyzed; with the dimensions used in the Airbus Industries Test Method (AITM 1.0010). Additionally the efficiency of the present method is investigated comparing the respective results with well-known benchmark problems and experimental results for the case of rectangular plates. The type of the response is intimately related to the ply thickness and orientation, geometrical configuration and boundary conditions, as well as the orthotropic material properties and the longitudinal, transverse and interlaminar strength both in compression and in tension.

These parameters are investigated within the frame of the present work and a normalization technique for the governing equations of the problem was proposed.

2. ANALYSIS

2.1. Governing Equations

As a first step the formulation of the equation of motion was conducted. A rectangular composite laminated plate consisted of k layers was considered and the origin of the Cartesian coordinate system assumed to be at the middle planeof the laminate having its x and y axis parallel to the plate edges of length a and b respectively as shown in Figure 1.

For the analysis of the plate, the Midlin approach was adopted, since it allows for accounting transverse shear deformations.

The displacement field is expressed by the relations:

(1)

where u0, v0, w0 are the displacement components of the midplane and x and y are the rotations of the normal to middle plane with respect to x-z and y-z planes respectively.

Introducing the cross sectional rotations

(2)

The strains throughout the laminated plate are given in rectorial form by the relation

0, ,

0, ,

0, 0, , ,

0, 0,

and0, ,

xx x x x

yy y y y

xy y x x y y x

zz xz x x yz y y

u

v z

u v

w w

= + + +

= = + = +

(3)

In equation (1) the in-plane displacements assumed to be negligible compared to the flexural part of the equations, thus pre-integrating the stresses for each lamina k along the thickness h of the plate we derive the bending moments and shear forces by the equations

(4)

It is well known that the lamination theory yields to a matrix constitutive relation providing the bending moments and in plane force resultants in the form

, i,j=1,2,6 (5)

where Aij, Bij and Dij are the extentional, conpling and bending stiffness matrices respectively.

For symmetric lay-up the coupling terms Bij of in plane and bending strain disappears. Additionally, we need to have input for the shear forces:

(6)

where:

(7)

Qij and k are the material constants for each lamina and the shear correction factor respectively.

2.1. Application of the Ritz Method

In the next we derive the equations of motion of an ab rectangular composite plate (Figure 1) subjected to the impact loading of a sphere of a radius R and mass mi.

Figure 1: Problem configuration

As a first step, the formulation of the homogeneous problem of the dynamic response of the composite plate is derived and then the implementation of the coupling terms originated by the impact of the composite plate by a given mass follows.

With no loss of generality and for convenience the following normalization of the coordinate system was introduced (Figure 2):

(8)

(a)

(b)

Figure 2: General plate and dimensional normalization, (a) original plate and (b) transformed system

For the derivation of the equations of motion of the plate impact problem, we introduce the admissible functions that will be used to derive the stiffness and mass matrices that govern the dynamic response of the composite plate. For the first shear deformation plate theory vibration problem, the p-version Ritz functions in rectangular coordinates for approximating the displacement field are defined as:

(9)

where ps, s=1, 2, 3 is a degree set of the complete polynomial space; m, Bm and Cm are the unknown coefficients while the subscript m is determined by the following relation:

(10)

whilst the total number of m, Bm and Cm is Nk, k=1,2,3 which depend on the degree set of the polynomial space, ps and given by:

(11)

In the present formulation we consider the same order of polynomials as the admissible functions for the evaluation of the displacement field of equation (9), e.g. p1=p2=p3 thus N1=N2=N3. The functions and that are used to interpolate the corresponding

displacement variables are formed as follow:

(12)

The functions and control the values of the displacement field along the boundary. These boundary equations can be furnished in such a way as to characterize the shape of the composite plate edges and the corresponding boundary conditions.

The boundary functions are formed as:

(13)

where ne is the number of the plate edges, is the boundary equation of the jth edge (Liew (1998)):

0 Free (F) or simply supported (S) in y-direction

1 Simply supported (S) in x-direction or clamped (C)

0 Free (F) or simply supported (S) in x-direction

1 Simply supported (S) in y-direction or clamped (C) 0 Free (F)

1 Simply supported (S) or clamped (C)

(14)

The sequence of the admissible orthogonal functions is depicted in Figure 3. It must be stressed that all the calculations will be performed using polynomials of rate 15 in order to achieve the required accuracy in our calculations.

Figure 3: Taylor triangle

2.2. Equations of motion

The strain energy due to bending of the plate is given by the surface integral:

(14)

The kinetic energy produced by the bending terms is expressed as:

(15)

At this point, we must refer to the Ritz approximation procedure for dynamic analysis of the composite plate. According to this theory, we approximate a displacement function by the finite linear combinations of the p-version polynomials:

(16)

i are obtained by determining the extremum of the energy functional :

(17)

and reduce our problem into the classical eigenvalue problem:

! "

(18)

Substituting equations (4), (5), (6) into (14) and (15), rewriting using the admissible p-Ritz functions of equations (9), we obtain the stiffness and mass matrices and then equation (18) can be written in the form

(19)

where m is the number of the polynomials used in the approximation in our case Nk=135. The stiffness and mass matrices are given explicitely in Appendix I.

2.3. The dynamic coupled problem

According to classical mechanics Meirovich (1967) the equation of motion of the plate under the effect of a dynamic load exerted to the plate may be written as:

! " ! " ! "

(20)

where ! " are the generalized non-conservative forces. According to Meirovich and classical mechanics theory these generalized forces can be expressed in terms of the actual forces and the admissible functions by means of the virtual work performed by these forces. Using the principle of virtual work, it is easily concluded that the virtual work of a concentrated force at a point of interest within the composite plate is given by

! " (21) The problem of a composite plate with the application of a dynamic load at the center concludes to the following differential equation:

"

#

(22)

where FC is the contact load applied at .

For the derivation of the coupled contact-impact problem we need to realize the mechanics of contact and the parameters involved (Figure 4).

Figure 4: Central transverse impact of a mass m, with the composite material and associated parameters.

The contact force FC is governed by the following differential equation:

" " " "

" "

$

%

(23)

where mi and wi are the mass and the displacement of the impactor respectively and FC is the contact force.

2.4. Contact Mechanics

Numerous investigators have proposed several linear and non-linear versions of the contact law for composite laminates and a deformable spherical-impactor. Among others, Yigit and Christoforou (1995) have proposed a linear version for the elastoplastic contact of the indenter and the composite plate. According to the authors opinion, this contact law facilitates the solution of the differential equation (22) and the normalization scheme they proposed appears to be extremely useful for the effective grouping of several impact cases of composite materials under low-velocity impact. According to this approach

" (24)

where is the local indentation (Figure 4) defined as the difference between the impactor displacement wi(t) and the plate deflection w(xC,yC,t). Thus:

(25) The contact stiffness Ky in equation (24) is given as:

(26) where Kh is the Hertzian stiffness and is the critical indentation for local yield to occur, Christoforou (1998).

#$

#

$

(27)

E* is the effective modulus of the contact system in the impact front expressed as:

%$ $$

(28)

here and $ are the Poisson ratio and Youngs modulus of the impactor, %$ is the transverse Youngs modulous which according to Christoforou (1998) is taken to be equal to that of the ply. Sy is the yield strength of the softer material and for the case of laminated composites is taken to be Sy=2Su where Su is the shear strength of the laminate.

2.4. Solution Method

For solving the governing equations of the problem of equation (22), the second derivative terms must be decoupled. Solving the eignenfrequency problem of equation (19) we obtain the frequency of the laminated plate with any boundary condition selected, and the mode shape matrix, & . The orhogonality condition implies the following:

& &

& &

(29)

Additionally we apply the transformation of the principal scheme, Meirovich (1967), using the following transformation:

&

&

&

(30)

where is the principal coordinate of the laminate dynamic system and is a function of time like the constants of the admissible functions.

The constants of the admissible p-Ritz functions are shown in the next related to the principal coordinates as

&

&

&

(31)

Substituting equations (30) into (22) and taking into account equation (29) we conclude to the following system of differential equations:

&

"

&

& '() ( ) () (()* +

(32)

where

and

.

Assuming that the laminated plate is initially at rest the initial conditions of the problem are:

(33)

The system of equations (32) can be solved numerically using the Runge-Kutta-Nystrom numerical integration scheme Bogacki (1989).

3. RESULTS

3.1. Numerical Tests

For the verification of the capability of solving the impact problem of composite plates we compare the solutions for simply supported composite plates as given by Pierson (1996). The lay-up and the two material configurations to be used for the numerical simulations are presented in Table 1.

For the implementation of the simply supported and the clamped boundary conditions case we present in Table 2 the boundary equations and applied.

Table 1: Properties of the laminate configurations used for the numerical tests

Configuration A B

Material T800H/39002 CFRP T300/934

CFRP Dimensions 127 x 76.2 mm 200 x 200 mm

Lay-up [45/90/-45/0]3S [0/90/0/90/0]S E11 (GPa) 129 120 E22(GPa) 7.5 7.9

0.33 0.33 G12(GPa) 3.5 5.5 G13(GPa) 3.5 5.5 G23(GPa) 2.6 5.5 (kg/m3) 1540 1580 h (mm) 4.65 2.69

Table 2: Boundary Equations for the two cases under investigation

An interpretation of the boundary equations presented at Table 2 fulfill the simply supported boundary conditions that correspond with the following:

(34)

It is evident that the above set of equations (34) is the hard type of simply supported boundary conditions. For the case of fixed boundary condition in the four edges we have:

(35)

In order to investigate the accuracy of the present method we compare analytical solutions and experimental data provided by Pierson and Vaziri (1996) to the simulations performed using the present analysis. As reported, it is evident from the study of the formulation performed by them; their model is capable of modeling only the special orthotropic laminations, only 0O and 90O lay-ups, and only simply supported boundary conditions. In our formulations apart from the capability of modeling different shapes and mixed boundary conditions, the above limitation of the lay-up angles does not exist.

Pierson and Vaziri (1996) have performed simulations for the configuration A of Table 1 in order to much some experimental results reported for this configuration. The inability of their formulation to handle the coupling terms D16 and D26 gave some significant differences on the response reported by the experiments and the simulations. Also it must be

stressed that the majority of the experiments are performed in fixed boundary conditions, for which an adequate formulation is not possible to implement using the trigonometric admissible functions, as presented by the aforementioned authors and others, (Dobyns (1981), Christoforou (1998)). For this reason we present two cases of impact in the center of the model plate of configuration A. In these cases, we provide the fixed and simply supported boundary conditions solution, according to the boundary admissible functions of Table 2 in order to present the significant difference of the results. It is surprising that the experimental results reported by Delfosse (1993) show that the authors of this work were able to apply the simply supported boundary conditions instead of the fixed ones in an gas-gun impact test experiment, and this is verified also by the results of the present work.

0.0 0.2 0.4 0.6 0.8 1.0 1.20

2

4

6

8

10

12 Experiment

No Target Damage Fixed Boundary

Conditions Simply Supported

Boundary Conditions

Conta

ct Fo

rce

(kN)

Time (msec)

Figure 5. Contact force history for 314g and 7.7 m/sec impact of the configuration A laminate; Experimental

results reported by Delfosse (1993)

0.0 0.2 0.4 0.6 0.8 1.0 1.20

2

4

6

8

10

12

14

16 Experiment

Target Damaged Fixed Boundary

Conditions Simply Supported

Boundary Conditions

Conta

ct Fo

rce

(kN)

Time (msec)

Figure 6. Contact force history for 314g and 11.85 m/sec impact of the configuration A laminate;

Experimental results reported by Delfoss (1993)

For the above experiments, a 25.4 mm diameter hemisphere was used. Also for the impactor a density of

7960 kg/m3 was used while the Youngs modulus and Poisson ratio were taken to be 200GPa and 0.3 respectively, Sy was chosen to have a value of 202MPa. Using equations (26)-(28) the contact stiffness was calculated to have a value of 1.33x109.

From the above presented results it is evident that the simulation of the configuration A agrees extremely well with the experimental results in all the stages of the impact phenomenon. Not only in terms of predicting the overall duration of the phenomenon but also in terms of predicting the maximum contact force exerted to the composite plate and the features of the response, e.g. the prediction accuracy of features involving the structural oscillation during impact and the interaction of the impactor with the plate. It is evident that the type of the response may be characterized as a transitional one, Christoforou (1998), where continuously we have an oscillatory interaction between the impactor and the composite target. Additionally we present the respective solution for clamped boundary in order to verify whether the plate was fixed according to this condition. The divergence of the clamped simulation from the experimental data clarifies the fact that the experiments in both cases (Figure 5 and Figure 6) were carried out in purely simple support conditions. Additionally we are quite confident that the results of the clamped case justify the quality of our method due to two reasons. First of all we see that up to 100 sec the two boundary condition cases present the same response, which is quite evident because at this time we have local response e.g. the boundaries up to this time do not affect the solution and the pressure wave hasnt reached the boundary. Secondly, the duration of the impact phenomenon for the clamped case is smaller while the maximum force is greater than the simply supported case.

The above investigation of the method justifies the accuracy of our present approach. Another way to verify the accuracy of our method is the solution accuracy of the natural eigenfrequencies. Pierson (1996) reports the estimated eigenfrequencies with their method. Keeping in mind that they account only for the uncoupled terms of the bending stiffness matrix and a close investigation of the differences of the eigenfrequencies could give estimation of how much stiffer is the plate with the incorporation of the coupling terms D16 and D26.

Table 3. Calculated natural frequencies for the laminate configurations defined in Table 1.

Mode no.

Present method p-Ritz

(rad/sec)

Pierson (1996)

(rad/sec)

CONFIGURATION A Simply Supported

1-1 18300 18122 5-1 620455 504501 10-1 1800126 1115277

CONFIGURATION B Simply Supported

1-1 1898 1902 5-1 132015 140042 10-1 467528 493687

From the results reported in Table 3 it is evident that the plate stiffness of Configuration A, neglecting the coupling terms, provides a solution corresponds to a more flexible plate and this justifies our claim that neglecting D16 and D26 , even they have small values, it gives rise to a dynamic response of a more flexible structure (as expected). In conjunction with this claim configuration B due to the fact that comprises only from 0O and 90O layers provides eigen frequency results quite close to the present model with the only difference that the natural frequencies calculated by the present method are slightly lower. This justifies the fact that Piersons solution used 160 modes for the calculation of the eigenvalue solution while our solution scheme used 408 (Figure 3).

Furthermore, we present two other cases characterized as local and global response as they have categorized by Christoforou (1998). In Figure 7 and Figure 8 a local impact response is presented using an impactor with a diameter of 12.7 mm, assuming the same parameters as used in the previous we obtain a contact stiffness to be 6.64x106. It is evident that up to 180 sec the two boundary condition cases present the same response, which is quite evident because at this time we have local response, the boundaries up to this time do not affect the solution, and the pressure wave hasnt reached the boundary. Then we observe that after the boundary plays a significant role in the response we have re-impact at different times. Due to the fact that the clamped case provides an overall stiffer system the impactor re-impacts faster to the target due to the fact that oscillates with a lower period, Figure 7. In addition, the difference of the transient response is quite evident if we observe Figure 8 where the central displacement history is presented and exhibits after 180- completely different response.

0 200 400 600 800 1000 12000

50

100

150

200

250

300

350

400

[0/90/0/90/0]S T800H/934 CFRP

Simply Supported Fixed

Con

tact

Fo

rce

(N)

Time (sec)

Figure 7. Contact force history for 8.5g and 3 m/sec impact of the configuration B laminate.

0 200 400 600 800 1000 1200

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5 [0/90/0/90/0]S T800H/934 CFRP

Simply Supported Fixed

Disp

lace

men

t (mm

)

Time (sec)

Figure 8. Central Displacement history for 8.5g and 3 m/sec impact of the configuration B laminate.

In Figure 9 and Figure 10 a global large-mass impact response is presented using an impactor with a diameter of 25.4 mm, assuming the same parameters as used in the previous we obtain the contact stiffness to be 1.33x109. Here the response is dominated by the impactor which due to its large mass the target follows its movement. This type of impact is characterized as a global response. Here the boundaries play a quite significant role due to the high energy transferred to the structure. The clamped case provides, as expected, lower duration and higher contact force while the displacement for the clamped case is significantly lower because the overall stiffness of the target system is higher than the simply supported case.

0 1 2 3 4 50.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

[45/90/-45/0]3ST800H/3900-2 CFRP

Simply Supported Fixed

Con

tact

Fo

rce

(kN)

Time (msec)

Figure 9. Contact force history for large mass impact 6.15g and 1.76 m/sec impact of the configuration A

laminate.

0 1 2 3 4 50.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

[45/90/-45/0]3ST800H/3900-2 CFRP

Simply Supported Fixed

Conta

ct Fo

rce

(kN)

Time (msec)

Figure 10. Central Displacement history for 6.15g and 1.76 m/sec impact of the configuration A laminate.

3.2. Normalized Response The impact response of composite plates have been analyzed in detail. The breakthrough on the analysis of the impact response was the normalization schemes presented by Olsson (1992) and Christoforou (1998). In the present section, we choose to apply the normalization scheme used by Christoforou (1998) to the mathematical system of equation (32). The same non-dimensional parameters were applied and presented in Appendix II. Applying the normalization scheme equation (32) becomes:

&

&

& '() ( ) () (()* +

(36)

The goal in the present section is the derivation of the functional relationship between the normalized maximum impact force and the loss factor , for different cases of where is the ratio of the static and the contact stiffness:

(37)

The benefit of the present p-Ritz method is that Kst may be calculated by equation (22) omitting the time dependent matrices and solving for Ci. Then using the Ci in equation (9) we may calculate the static deflection for an arbitrary force applied to the plate center, solving the following linear system:

"

#

(38)

and " .

Equation (38) is quite useful because since there are no closed form analytical solutions for clamped boundary conditions and/or non-special orthotropic plates. Thus we can calculate Kst for several configurations of shapes and boundary conditions.

According to Olsson (1992), the response of an infinite plate is governed by the parameter , which is called the loss factor. In Appendix II one may notice that this parameter depends upon the effective stiffness D* which can be calculated only for simply supported conditions and special orthotropic plates by a simple formula. To facilitate the derivation of the normalized curve we calculate effective stiffness as to be the stiffness of a special orthotropic plate with the same dimensions.

Using Whitneys (1987), closed form relation for the first eignenfrequency we estimate the effective stiffness of the plate by the following formula:

'

(39) The response of the infinite plate is calculated solving the following first-order differential equation, Olsson (1992):

(40) Equation (40) provides the normalised impact force for different values of . Note that is characterized as the loss factor due to the fact that the damping mechanism in the phenomenon studied is the energy lost or transferred by the impactor to the composite plate.

The normalized graph that depicts the response of composite rectangular plates with clamped boundary conditions is presented in Figure 11. The normalised curves were obtained by applying several cases of resulting and calculated from different configurations of composite plates and impactor characteristics. Figure 11 includes the solution of equation (40) for infinite plate response e.g. with no size effects. To verify the categorization we have applied for the cases solved (local, transition and global), the graph includes the normalized impact parameters as calculated in the several impact cases and presented in Figures 5-10.

0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 5, =.645 - Transition Figure 6, =.645 - Transition Figure 7, =.006 - Local Figure 9, =1.3 - Global

!"

!

!#

!$ !

$!%

&'(

Figure 11. The Normalized Impact Response curve as calculated from different composite material configurations

for clamped boundary conditions.

4. ACKNOWLEDGEMENTS The authors would like to thank Professor Andreas Christoforou for the fruitful cooperation and discussions they had on the Low Velocity Impact response of Composite laminates, during his stay in University of Patras.

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APPENDIX I

Each sub-matrix and

have a

dimension of mxm where m is the degree of the polynomial admissible function.

The stiffness and mass matrices were calculated as presented in section 2.2 and are given at the following:

Stiffness matrix

and

Mass matrix

and

,

Note that the following notation is used:

'

For generality and convenience, the coordinates are normalised with respect to the maximum length dimensions, equation (8), and the integration of the stiffness and mass equations are calculated at [-1,1]. Thus in case of geometrical normalisation we have:

' '

APPENDIX II

Non Dimensional Parameters

In this section the non-dimensional parameters used for the derivation of the functional relationship between the maximum impact force and the loss factor are presented according to the scheme followed by Christoforou (1998). Space and time variables are normalised as follows:

where:

, the linear contact frequency.

, the maximum indentation obtained from the half space solution.

The normalised impact force and the loss factor are given by:

""

where is the material density, h the plate thickness and D* is the effective laminate bending stiffness as discussed in section 3.2.