transient analysis of laminated composite and sandwich plates with

ABSTRACT: Delamination is the most common damage type of laminar composites. It is of the great importance that perfectbonding between faces and the soft-core of sandwich plate remain intact under different types of dynamic loading. This paperexplains the transient response of laminated composite and sandwich plates with embedded delaminations. For this purpose,numerical model is derived using Reddy’s Generalized Laminated Plate Theory. This theory assumes layerwise linear variationof in-plane displacement components, while transverse displacement is constant through the thickness. Cross sectional warpingis accounted. Jump discontinuities in displacement field, which represent the delamination openings in three orthogonaldirections, are implemented using Heaviside step functions. Linear kinematic relations and Hooke’s constitutive law areconsidered. Equations of motion are derived using Hamilton’s principle. Numerical solution based on the proposed theory isobtained using the enriched finite elements with four or nine nodes. Governing partial differential equations are reduced to a setof ordinary differential equations in time using Newmark integration schemes. The equations of motion are solved usingconstant-average acceleration method using the originally coded MATLAB program. Effects of delamination size and positionthrough the plate thickness on transient response are commented. Also, illustrative comments are given about the influence ofshear deformation on transient response. Different forcing functions are investigated. After verification of the proposed modelfor the intact plates, the variety of new results for delaminated plates is presented as a benchmark for future investigations.

KEY WORDS: Composite plate; Sandwich plate; Delamination; Transient Analysis; GLPT.

1 INTRODUCTION

Laminar composites can be used as main load carryingmembers in the form of thick laminated and sandwich plates[1]. Their great stiffness-to-weight ratio makes them suitablefor different engineering purposes. Laminated plates whichare investigated in this paper are composed of severalorthotropic laminas. This orthotropic behavior comes from thehigh-strength fibers oriented in the arbitrary direction for eachlamina individually. It is of great importance to understand thefundamental dynamic characteristics of the laminate, such asnatural frequencies and mode shapes [2].

In the case of thick structural components, ESL theories arenot adequate for prediction of the plate response. The reasonis the neglection of transverse shear deformation. Laminarcomposites, exposed to different types of dynamic transientloading, are characterized by significant transverse sheardeformations. Also, ESL theories cannot account fordiscontinuities in transverse shear strains at layer interfaces[2]. Vuksanović investigated ESL plate theories of higherorder (HSDT) [3].

An extensive overview of theories of composite materials isgiven in detail in Refs. [4-7]. Soft-core sandwich panels, aswell as the honey-comb-core panels, are exploited by theaircraft industry. In the field of civil engineering, sandwichpanels are used as roof and wall panels to provide the thermalisolation of the building [18-19].

It is of the great importance that prefect bonding betweenthe layers in laminar composites remain intact during theservice life of the structure. Only if this is satisfied, the panelwill perform on the appropriate level. However, this is not

always satisfied, so delamination between the material layersoften occurs. This is the most common type of damage forlaminated composite plates, which usually occurs in theproduction process, or due to the impact forces.

In this paper, extended version of Generalized LayerwisePlate Theory (GLPT), proposed by Reddy, served as a basisfor the development of enriched finite elements [8-9]. Usingthe proposed model, transient response of the plate with thepresence of delaminations is calculated in this paper. In theprevious analyses, authors have derived both analytical andnumerical solutions for intact cross-ply laminated plates usingGLPT [10]. They have used the proposed solution to obtainthe transient response of cross-ply laminated composite platesunder different forcing functions.

GLPT allows independent interpolation of in-plane andtransverse displacement components, and includes possiblejump discontinuities at layer interfaces where delaminationexists. Piece-wise linear variation of in-plane displacementcomponents and constant transverse displacement (planestress state) are imposed. Using these assumptions, cross-sectional warping is taken into the calculation. Consistentmass matrix is imposed in a usual manner.

The goal of this paper is to present the comparison betweenthe transient responses of laminated composite and sandwichplates, with or without the presence of embeddeddelaminations. For all calculations, originally codedMATLAB program is used, and obtained results arecompared with existing data from the literature. The variety ofnew results for delaminated composite and sandwich plates ispresented.

Transient analysis of laminated composite and sandwich plateswith embedded delaminations using GLPT

Miroslav Marjanović1, Djordje Vuksanović1

1Faculty of Civil Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbiaemail: [email protected], [email protected]

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

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2 GENERALIZED LAYERWISE PLATE THEORY

We will analyze the laminated/sandwich plate composed of northotropic laminas. Global coordinate system is located in themid-plane of the laminate (see Figure 1), while localcoordinate system of each lamina coincides with the fiberdirection. N is the number of mathematical layers, while ND isthe number of delaminated interfaces. Lamina thickness isdenoted as hk, while h is overall thickness of the laminate.

Figure 1. 4-layer laminated plate in global coordinate system

GLPT is based on these assumptions: (1) all layers areperfectly bonded together, except in the delaminated area, (2)material is orthotropic, linearly elastic and follows Hooke’slaw, (3) strains are small (geometrically linear analysis isperformed) and (4) inextensibility of transverse normal isassumed (plane stress state).

2.1 Displacement field

Displacement field of the GLPT can be written as follows:

11 1

( , , ) ( , ) ( , ) ( ) ( , ) ( )N ND

I I I I

I I

u x y z u x y u x y z U x y H z

21 1

( , , ) ( , ) ( , ) ( ) ( , ) ( )N ND

I I I I

I I

u x y z v x y v x y z V x y H z

31

( , , ) ( , ) ( , ) ( )ND

I I

I

u x y z w x y W x y H z (1)

In Eq. (1), (u,v,w) are mid-plane displacement components,(uI,vI) are undetermined coefficients which describe thelayerwise expansion of the displacements, while (UI,VI,WI) aredisplacement jumps of the Ith delamination. The conditionWI≥0 should be adopted in all points to prevent layeroverlapping. The delamination front is defined by enforcingthe essential boundary condition UI=VI=WI=0 on the crackboundary. ΦI(z) are layerwise continuous functions of z-coordinate, while HI(z) are Heaviside step functions:

1 0 / 2( )

1 / 2 0

I

I

I

z hH z

h z(2)

In this paper, linear layerwise variation of in-planedisplacements is assumed, so in-plane displacements arepiece-wise continuous through the plate thickness in the intactregion. On the other hand, all displacement components arediscontinuous at delaminated interfaces in delaminated area.This leads to cross-sectional warping, shown in Figure 2. Notethat arbitrary number of delaminations can be incorporatedusing this numerical model (I = 1,2,…,ND).

Figure 2. Deformation of transverse normal, for u1

2.2 Strain field

After incorporation of assumed displacement field, we caneasily derive the strain field, using linear kinematic relations:

1 1

εI IN ND

I I

xI I

u u UH

x x x

1 1

εI IN ND

I I

yI I

v v VH

y y y

1 1

γI I I IN ND

I I

xyI I

u v u v U VH

y x y x y x

1 1

γI IN ND

I I

xzI I

w d Wu H

x dz x

1 1

γI IN ND

I I

yzI I

w d Wv H

y dz y(3)

As a result of the assumed inextensibility of transversenormal, transverse normal strain 0ε z in this case.

2.3 Constitutive relations of the single lamina

Constitutive equations for the kth orthotropic lamina, forlinearly elastic material that follows Hooke’s law, are firstlyderived in the local (material) coordinate system:

( ) ( ) ( )k k kσ εQ (4)

In Eq. (4), ( )kQ is the matrix of reduced stiffness components

for the plane stress case. Using the transformation matrices foreach layer independently, we derive the constitutive relationsfor kth lamina in the global coordinate system:

( )( ) ( )

11 12 16

12 22 26

16 26 66

55 45

45 44

0 0

0 0

0 0

0 0 0

0 0 0

σ εσ ετ γτ γτ γ

kk k

x x

y y

xy xy

xz xz

yz yz

Q Q Q

Q Q Q

Q Q Q

Q Q

Q Q

(5)

( ) ( ) ( )k k kσ ε= Q (6)

Reduced stiffness matrix in the global coordinate system isderived using the following relation:

( ) ( ) ( ) ( )-1k k k k= Τ ΤQ Q (7)

2.4 Equations of motion

When deriving the dynamic equilibrium of the virtual strainenergy (U), virtual work of external forces (V) and virtual

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

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kinetic energy (K), it is assumed that transverse loading q isacting in the middle plane of the plate [10]. Dynamic relationsfor U, V and K are given in Eqs. (8-10):

0

δ σ δε σ δε τ δγ τ δγ τ δγt

x x y y xy xy xz xz yz yz

V

U dVdt (8)

0

( , , )δ δt

V

V q x y t w dVdt (9)

1 1 2 2 3 3

0

δ ρ δ δ δt

V

K u u u u u u dVdt (10)

The dynamic version of virtual work statement is then derivedusing Hamilton’s principle, where δU + δV + δK = 0:

1

δ δ δ δ

δ δ

δ δ

δ δ

δδ δ

δ δ

δ δ

δ

x y xy

x y

I II I

x y

I INI

xyI

I I I I

x y

I II I

x y

I II

xy

II

x

u v u vN N N

x y y x

w wQ Q

x y

u vN N

x y

u vN

y xU

Q u Q v

U VN N

x y

U VN

y x

WQ

x

0

1

δ

t

ND

I

II

y

d dt

WQ

y

(11)

0

δ δt

V q w d dt (12)

0

1

, 1

1

1

δ δ δ

δ δ δ δ

δ δ

δ δ δδδ δ δ

δ δ

δ δ

NI I I I I

I

NIJ I I I I

I J

I I INDI

I I II

IJ I J I JND

JI I J I JJ

I u u v v w w

I u u v v u u v v

I u u v v

U u V v W wK Iu U v V w W

I u U v V

I U u V v

0

1

, 1

δ δ δ

t

N

I

NDIJ I J I J I J

I J

d dt

I U U V V W W

(13)

Using the integration of stresses through the thickness of theplate, we derive the stress resultants and inertia terms. Stressresultants are in detail given in [9]. Inertia terms are:

1( )

01

1( )

1

, , 1, ,

, ,

ρ

ρ

zknI IJ k I I J

k zk

zknIJ IJ k I J I J

k zk

I I I dz

I I H H H dz

(14)

3 NUMERICAL (FINITE ELEMENT) MODEL

Finite element model based on the previous considerationsconsists of the middle plane, I=1,2,…,N mathematical layersthrough the plate thickness (excepting the middle plane) andfinally I=1,2,…,ND numerical layers in which debonding mayoccur. In this work, isoparametric quadrilateral finite elementsare used. Two types of elements are used, with four or ninenodes in 2D plane. The model is in detail explained in [2].Here, only the main properties of the proposed model aregiven. Proposed finite elements require only C0 continuity ofgeneralized displacements on element boundaries. It is veryimportant to note that finite element mesh is generated in 2Dplane, and the adopted interpolation functions through theplate thickness are used for out-of plane interpolation ofunknown variables. This assumption allows independent in-plane and out-of-plane interpolation. Note that we are dealingwith rotational-free enriched layerwise finite element with full3D capacity. Using different combinations of Lagrangian in-plane interpolations ψi, and ФI(z) functions for through thethickness interpolation, allows the derivation of variety oflayerwise finite elements.

Interpolation of all generalized displacements is done usingthe same 2D Lagrangian interpolation functions, for the sakeof simplicity, in the following manner:

1

1

1

ψ

ψ

ψ

m

i ii

m

i ii

m

i ii

uu

v v

ww

, 1

1

ψ

ψ

mI

I i ii

I mI

i ii

uu

vv

(15)

1

1

1

ψ

ψ

ψ

mI

i iI i

mI I

i iiI

mI

i ii

UU

V V

WW

, 1

1

ψ

ψ

mI

I i ii

I mI

i ii

uu

vv

(16)

In Eqs. (15-16), m is the number of nodes per element. Byincorporation of assumed displacement field into virtual workprinciple, we obtain the mathematical model for the single,representative, layerwise finite element:

M d C d K d F (17)

In Eq. (17), [K], [C] and [M] are element stiffness, viscousdamping and consistent mass matrices, respectively; {F} isthe element force vector, while {d} is the displacement vector.Dots above the vector {d} denote the differentiation in time.Assembly procedure is done in a usual manner. Elementstiffness and consistent mass matrices are derived in thefollowing manner:


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1 1

1 , 1 1 1

1 1 1 , 1

e

N NDT T I T I

I I

N N N NDT I T IJ T IJ e

I I J I J

ND ND N NDT I T JI T IJ

I I J I J

B AB B B B B E B

B B B B D B B L B d

B E B B L B B F B

K (18)

01 1

1 , 1 1 1

1 1 1 , 1

e

N NDT I T I T

I I

N N N NDI T IJ T IJ T e

I I J I J

ND ND N NDT T JI T IJ T

I I J I J

I I I

I I I d

B I I

M

(19)

In Eqs. (18-19), [A], [BI], [DIJ], [EI], [LIJ] and [FIJ] areconstitutive matrices of the laminate (see Ref. [9]); [I0], [I

I],

[IIJ], IJI and IJI are inertia terms derived in Eq. (14),

while [B] and B are kinematic (strain) matrices. Matrices of

interpolation functions are given in Eq. (20). All terms in Eqs.(18-19) are derived using numerical integration over singlefinite element domain, denoted as Ωe. Selective integration isused for elimination of spurious shear stiffness fromcalculation (shear locking phenomenon).

1

1

1

1 2 21 3 3

1

1 2 3

0 00

0 0 , ,0

0 0

0 0

0 0

m

m

m

ψψ

ψψ

ψ

ψψ

Ψ Ψ

Ψ

(20)

4 TIME DISCRETIZATION

The governing partial differential equations of the problem,derived in the preceding sections, will be solved numericallyin this work, using Newmark’s integration schemes forsecond-order differential equations in time. These explicittime integration schemes are explained in Ref. [3] and work ofHinton and Vuksanović [11]. Interesting considerations oftransient response of intact sandwich plates using Newmark’sintegration are given in Refs [18-19]. In the Newmark’smethod [12], accelerations and velocities are approximatedusing truncated Taylor’s series [4]. The governing differentialequations are then satisfied only in discrete time points tn, sothe response of the structure is calculated exactly only in thesediscrete points in time. In this paper, constant averageacceleration method is chosen because of its numericalstability. Accelerations and velocities in tn+1 are:

2

1 1

1 1

4

2

n n n n n

n n n n

td d t d d d

td d d d

(21)

After the previous approximation of differential equationsin time, we obtain a system of algebraic equations at time

point tn+1, in terms of known values at tn. In this paper,homogenuous initial conditions are prescribed. If stiffness,damping and consistent mass matrices of the system areconstant in all time points, we obtain the solution by solvingthe following algebraic system (Δt is chosen time increment):

1ˆ ˆ

nK d F (22)

2

2 4ˆt t

K K C M (23)

1

2

2ˆ

4 4

n nn

nn n

t

t t

F F C d d

M d d d

(24)

5 TREATMENT OF THE DAMPING

The proportional damping model (Rayleigh damping) is oftenused in transient dynamic analysis of engineering structures[13]. In this model, damping matrix is proportional to massand stiffness matrices ([C] = α[M] + β[K]). However, if theexcitation is a single pulse, the effect of damping is usuallynot important, unless the system is highly damped [14].Damping has much more importance in controlling themaximum response of a structure to periodic or harmonicloads. Maximum response to an impulsive load will bereached in a very short time, before the damping forces canabsorb energy from the structure [15]. In further calculationswe will assume only undamped structural response, so theEqs. (22-24) will be used in reduced form.

6 NUMERICAL EXAMPLES AND DISCUSSION

In this chapter, some numerical examples based on previouslyexplained GLPT are presented. In all calculations, transientloading is divided into the loading phase and the subsequentfree vibration phase. Accuracy of the proposed model forintact plates is already verified in Ref. [10]. Free vibrationanalysis of delaminated composite and sandwich plates isperformed in [2]. In all calculations, it was assumed thatdynamic force lasts until t=T1 is reached (loading phase), andafter that it was observed how the structure behaves duringT1<t<T2 (free vibration phase). Different forcing functionswere used, as shown in Figure 3.

Figure 3. Forcing functions used in numerical examples

6.1 Transient analysis of delaminated composite plates

An 8-layer simply supported square composite plate withsymmetric (0/90/45/90)s stacking sequence is considered. All


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laminas are of equal thickness. Side length of the plate isa=b=250mm, while overall plate height is h=2.12mm. Eachlayer is made of orthotropic material with followingmechanical characteristics [16]:

E1=132GPa, E2=5.35GPa, G12=G13=2.79GPa,

υ12=υ13=0.291, υ23=0.30, ρ=1446.2kg/m3.

Figure 4. Finite element mesh with different sizes of centrallylocated square delamination

Figure 5. Different positions of delamination

Figure 6. Transient response of laminated plate under steppulse, for different sizes of delamination at Position 1

Transient response is obtained for different delamination sizesand positions (see Figures 4 and 5). Mesh of 20 20 4-nodelayerwise finite elements is used. Reduced integration isadopted to avoid spurious shear stiffness and rigid-body modeshapes. Forcing functions shown in Figure 3 are used.Uniformly distributed loading F0=1.0 over whole plate area isprescribed, and the motion of the plate center is plotted. Thefollowing time increments/durations are used in theNewmark’s integration: Δt=0.5ms, T1=25ms, T2=40ms. Thesevalues are obtained after calculation of the natural frequenciesof composite plate used in this example. Boundary conditionsused for nodes on simply supported (SS) edges are:

x = 0, a: v = w = vI = 0; y = 0, b: u = w = uI = 0.

From Figure 6 it is obvious that centrally located delaminationdoesn’t influence the transient response under step pulseseverely, until delamination size is small enough. The platestill oscillates globally, with a small increase of amplitude andperiod, both during the forced and free vibrations. Afterincrease of the delaminated area over 35%, (see Figure 4), theplate motion changes (Figure 6). It is obvious that during theforced motions, amplitudes and period are increased due-tothe reduced plate stiffness in presence of embeddeddelamination. In the free vibration phase, the delaminatedsegment of the plate continues to vibrate independently, withits own frequency and amplitude. This delaminated part isclamped in the intact rest of the plate, as can be seen from itsreduced amplitude and increased frequency.

Figure 7a. Transient response of intact rest of plate under steppulse, for different positions of Delamination 2

Figure 7b. Transient response of delaminated segment of plateunder step pulse, for different positions of Delamination 2


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Influence of the delamination position through thethickness of the plate is shown in Figures 7a and 7b. It isobvious that changing the position of the delaminationthrough the plate thickness, without the change ofdelamination size, doesn’t change severely the frequency ofoscillations of the intact rest of the plate. If the delamination iscloser to the top of the plate, the amplitude of the oscillationsis only slightly reduced, because the thickness of the intactrest of the plate increases. On the other hand, the response ofthe delaminated segment is more influenced by the position ofdelamination – if the delamination is closer to the top surface,the amplitude increases, as shown in Figure 7b.

Hypotheses stated above are valid for sine pulse, too, asshown in Figures 8 and 9.

Figure 8. Transient response of intact rest of plate under halfsine pulse, for different sizes of delamination at Position 1

Figure 9. Transient response of laminated plate under half sinepulse, for different positions of Delamination 2

6.2 Transient analysis of delaminated sandwich plates

A five layer simply supported square sandwich plate withanti-symmetric (0/90/core/0/90) stacking sequence isconsidered. The plate is composed from the rigid face sheetsand soft core, as shown in Figure 10, where tc/tf=10, whilea=250mm and h=2.5mm. Face sheets are made of Graphite-Epoxy T300/934 with following mechanical properties [17]:

E1=131GPa, E2= E3=10.34GPa, G12=G23=6.895GPa,

G13=6.205GPa, υ12=υ13=0.22, υ23=0.49, ρ=1627kg/m3.

Isotropic soft core has the following mechanical properties:

E=6.89MPa, G=3.45MPa, υ=0, ρ=97kg/m3.

Figure 10. Soft-core sandwich plate with different positions ofembedded delamination

Transient response is obtained for different delamination sizesand positions (see Figures 4 and 10). Mesh of 20 20 9-nodelayerwise finite elements is used. Reduced integration isadopted as in previous example. Forcing functions shown inFigure 3 are used. Uniformly distributed loading F0=1.0 overwhole plate area is prescribed, and the motion of the platecenter is plotted. The following time increments/durations areused in the Newmark’s integration: Δt=1.2ms, T1=25.2ms,T2=36.0ms.

Figure 11. Transient response of sandwich plate undertriangular pulse, for different sizes of delamination at Pos. 1

From Figure 11 it is obvious that sandwich plate is highlyvulnerable to embedded delamination between the soft-coreand rigid face sheets, when the plate is subjected to triangularstep pulse loading. After increase of the delaminated area over15% of plate area, (see Figure 4), the plate motionsignificantly changes, with increased amplitudes and reduced


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frequency due to the reduction of the plate stiffness (see Ref.[2]). We can see that small delaminated area (<5%) doesn’tinfluence the transient response severely.

Figure 12. Transient response of sandwich plate undertriangular pulse, for different positions of Delamination 2

From the Figure 12 it is obvious that in the case of sandwichplates subjected to triangular pulse loading, the position ofdelamination through the plate thickness is very important forthe transient response of the plate. It is shown that debondingbetween the soft-core and rigid face sheets (Position 1) ismore dangerous for the overall plate response, than thepresence of the embedded delamination in rigid face sheet(Position 2). In the first case, amplitudes are highly increased,as well as the period of oscillations, while debonding in theface sheet only slightly changes the overall plate response.

Figure 13. Transient response of sandwich plate under blastpulse, for different sizes of delamination at Position 1

Figure 14. Transient response of sandwich plate under blastpulse, for different positions of Delamination 2

Hypotheses stated above are valid for exponential blastpulse 0( ) tF t F e α , too, where α=200 is fictitious damping

factor chosen to simulate the blast loading that lasts during0<t<T1 for the chosen T1, as shown in Figure 3. Responses ofthe soft-core sandwich plate under blast loading are shown inFigures 13 and 14.

7 CONCLUSIONS

Finite element model based on the Generalized LaminatedPlate Theory (GLPT) is derived. For different types oftransient loading, the response of delaminated composite andsandwich plates is calculated using the layerwise finiteelements with four or nine nodes. The proposed model iscapable to accurately catch the motions of individual layers,as well as relative displacements of the laminas in thedelaminated area. In authors’ preliminary investigations, theproposed model is verified for intact plates using the availablenumerical and experimental data from the literature. In thispaper, the variety of new results for delaminated plates ispresented as a benchmark for future investigations in thisfield.

It is shown that transient response of soft-core sandwichplate is more influenced by the presence of relatively smallembedded delaminations, than in the case of typical laminatedcomposite plate. Debonding between the soft core and facesheets must be prevented, while the embedded delamination inthe face sheet only slightly changes the overall plate response.

Using the proposed model, local vibrations of thedelaminated segment of composite plate are calculated.Approximate critical areas of delamination which lead to theconsiderable change in plate response are given. It is shownthat position of delamination through the plate thickness in thecase of typical laminated composite plate is not of greatinterest for transient response.

In the further research, contact conditions for prevention oflayer overlapping, as well as geometrical nonlinearity, will be


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incorporated in the proposed finite element model, to calculatethe plate response with increased accuracy. Also,delamination propagation should be taken into account usingthe fracture mechanics criteria.

ACKNOWLEDGMENTS

This work was supported by Project TR 36048, funded byMinistry of Education and Science, Republic of Serbia.

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[3] Dj. Vuksanović, Linear analysis of laminated composite plates usingsingle layer higher-order discrete models, Composite Structures, 2000,Vol.48, 205-211.

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[14] A. K. Chopra, Dynamics of structures: theory and applications toearthquake engineering, Prentice Hall, 1995.

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transient analysis of laminated composite and sandwich plates with

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