Finite Elements in Analysis and Design 47 (2011) 373–377

Contents lists available at ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/finel

Use of unsymmetric finite element method in impact analysis ofcomposite laminates

Surendra Kumar

CSIR Centre for Mathematical Modelling and Computer Simulation, Council of Scientific and Industrial Research, NAL Belur Campus, Bangalore 560037, India

a r t i c l e i n f o

Article history:

Received 23 December 2009

Received in revised form

23 November 2010

Accepted 10 December 2010Available online 6 January 2011

Keywords:

Finite element analysis

Unsymmetric formulation

20-noded brick element

Polymer matrix composites

Laminates

Impact dynamics

4X/$ - see front matter & 2010 Elsevier B.V. A

016/j.finel.2010.12.016

ail addresses: surend_kr@yahoo.com, surendr

a b s t r a c t

The use of metric trial functions to represent the real stress field in what is called the unsymmetric finite

element formulation proposed recently is an effective way to improve predictions from distorted finite

elements. This approach works surprisingly well because the use of parametric functions for the test

functions satisfies the continuity conditions while the use of metric (Cartesian) shape functions for the

trial functions ensures that the stress representation during finite element computation can retrieve in a

best-fit manner, the actual variation in stress in the metric space. However the formulation so far has only

been applied to static problems in literature and a dynamic problem of composite structure has not been

addressed. In this paper, the impact response in composite laminate subjected to transverse impact by a

metallic impactor is investigated using three-dimensional layered twenty-noded hexahedral unsym-

metric finite element formulation. Some example problems of graphite/epoxy laminate (plate as well as

cylindrical shell) are considered and impact-induced response is analysed using the above formulation.

The performance comparison of this unsymmetric element is made vis-�a-vis conventional symmetric

8-noded and 20-noded hexahedral finite elements.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Conventional displacement type finite element formulationsuse identical trial and test functions (Galerkin elements) andperform well when used in regular meshes. When these meshesare distorted, their performance degrades rapidly and this has beenwell known in the literature [1–4]. Many efforts have been madeover several decades to improve mesh distortion sensitivity.

Recently, Rajendran and co-workers [5–9] introduced what theycalled the unsymmetric formulation. Here, two separate sets ofshape functions, viz. the so-called isoparametric shape functions(in natural space) enforcing compatibility (or continuity) require-ments and the so-called metric shape functions (in Cartesian space)enforcing completeness requirements are used. Numerical resultsfrom some test problems [5–9] have revealed that the unsym-metric elements can reproduce accurately displacement fieldsunder various types of admissible mesh distortions if the continuity

enforcing shape functions are based on isoparametric functions andare used as the test functions while the completeness enforcingshape functions are based on metric forms and are used as the trial

functions.Prathap et al. [10] showed recently that the reason why the

unsymmetric parametric-metric (PM) formulation has the greatestmesh distortion immunity is because the stress representation is

ll rights reserved.

a@cmmacs.ernet.in

managed in the metric (Cartesian) space. However, Prathap et al.[11] showed later that the unsymmetric PM formulation, eventhough it is practically very effective to meet the continuityrequirements and the best-fit stress recovery requirements simul-taneously in a distorted element, it is not strictly variationallycorrect. This is a very small price to pay for the great improvementin performance even under severe distortion. Later in a discussion,Rajendran [12] argued that the PM formulation is variationallycorrect although it does not give the best possible approximation ofdisplacement in the metric space. The idea mooted here was thatfor an unsymmetric formulation to be variationally correct; theformulation need not give the best-fit solution. Kumar and Prathap[13] also demonstrated that the use of a consistent definition of theconstrained strain field along with the use of the metric trialfunctions approach can ensure a lock-free solution in a 3-nodedTimoshenko beam element even when there is mesh distortion.

So far, the issue of the performance of unsymmetric elementshas been confined to the regime of static problems and a dynamicproblem of composite structure has not been specificallyaddressed. In this paper, therefore, a study is carried out on theimpact response in polymer matrix laminated composites sub-jected to transverse impact by a metallic impactor using three-dimensional layered 20-noded hexahedral unsymmetric finiteelement formulation. So far as the impact analysis of compositelaminate is concerned, a number of researchers have deployed a 3Dfinite element method for the solution of impact on laminatedcomposites, some of these works related to this paper can be cited

S. Kumar / Finite Elements in Analysis and Design 47 (2011) 373–377374

as Refs. [14–18]. However, the 3D finite element analysis so far isbased on the layered version of a conventional 8-noded brickelement [14–18] and a 20-noded layered hexahedral element usingunsymmetric formulation has not been investigated. In this study,some example problems of graphite/epoxy laminate (plate as wellas cylindrical shell) are considered and impact-induced response isanalysed and comparison of performance of this unsymmetricelement is made with respect to conventional symmetric 8- and20-noded hexahedral finite elements.

X3X2

X1

5

2

7

6

1

3

ξ

η ζ

9

10

11

12

14

15

16

13

17 18

19

8

4

20

Fig. 1. Layered version of 20-noded isoparametric brick element.

2. Finite element methodology

2.1. Governing equations

Finite element transient dynamic equilibrium equation forstructural analysis can be derived using Hamilton’s variationalprinciple. The equation for the case of no damping can be written as

½M�f €Ugþ½K�fUg ¼ fFAg ð1Þ

where [M] and [K] are structural mass and stiffness matrices, {U}and f €Ug are the nodal displacement and acceleration vectors, and{FA} is the applied load vector, which consists of contact forcebetween the impactor and the laminate.

The above finite element equation is integrated step-by-stepwith respect to time using the Newmark direct integration methodwith constant average acceleration (a¼0.5 and b¼0.25). Afterapplying this method, Eq. (1) can be evaluated at time tn + 1 to form

½K̂nþ1�fUnþ1g ¼ fF̂nþ1g ð2Þ

where

K̂nþ1

h i¼ Knþ1

� �þ

1

bðDtÞ2M½ � ð3aÞ

and

F̂nþ1

n o¼ FA

nþ1

n oþ FM

nþ1

� �¼ FA

nþ1

n o

þ M½ �1

bðDtÞ2Unf gþ

1

bðDtÞ_U n

n oþ

1�2b2b

€Un

n o !ð3bÞ

½K̂nþ1� and fF̂nþ1g can be referred to as effective stiffness matrix andeffective load vector at time tn +1. In Eq. (3b), fFA

nþ1g and fFMnþ1g are

referred to as external applied load vector and load vector causedby the inertia terms at time tn +1.

2.2. Unsymmetric formulation

Unsymmetric formulation employs two different sets of shapefunctions that seek to satisfy the minimum continuity require-ments and all completeness requirements in the test and trialfunctions, respectively. Rajendran and co-workers have given avery elaborate account of the formulation of the unsymmetricproblem [5–9]. For the sake of completeness, this is summarised asshown below.

In the unsymmetric formulation, we consider two different shapefunctions and their derivatives, viz. parametric and metric (inlinewith the phrase ‘metric interpolation’ by MacNeal [19]), to inter-polate the virtual displacement and strains, and the displacementand stress fields, respectively. The parametric shape functions [NP]are well described in standard FEM test books. The displacementfield interpolated with the metric shape functions is expressed as

fug ¼ ½NM�fUeg ð4Þ

where [NM] and {Ue} are the element metric shape function matrixand the nodal displacement vector for the element, respectively.

For a 20-node hexahedral element, the displacement field desired tobe reproduced is of the form

ui ¼ ðaiÞ1þðaiÞ2xþðaiÞ3yþðaiÞ4zþðaiÞ5x2þðaiÞ6xyþðaiÞ7y2þðaiÞ8yzþðaiÞ9z2

þðaiÞ10zxþðaiÞ11x2yþðaiÞ12xy2þðaiÞ13y2zþðaiÞ14yz2þðaiÞ15z2x

þðaiÞ16zx2þðaiÞ17xyzþðaiÞ18x2yzþðaiÞ19y2zxþðaiÞ20z2xy ð5Þ

where ui is the ith component of displacement vector {u}.The metric shape function matrix [NM] at any point (x,y,z) for

this element can be derived by solving a set of 20 linear algebraicequations written in a compact form as

X20

i ¼ 1

ðNMÞixpi yq

i zri ¼ xpyqzr ; p,q,r¼ 0,1,2 ð6Þ

where the terms xpi yq

i zri correspond to those present in the

displacement fields of Eq. (5) for the ith nodal point.The metric strain-displacement matrix [BM] involves the deri-

vatives of the metric shape function, which can be computed at thequadrature points directly by solving the derivative-version ofEq. (6). A typical derivative of Eq. (6) with respect to x can berepresented as

X20

i ¼ 1

ðNMÞi,xxpi yq

i zri ¼

@ðxpyqzrÞ

@x; p,q,r¼ 0,1,2 ð7Þ

Since the left hand side matrices of Eqs. (6) and (7) are the same,solutions of both metric shape functions and their derivativesrequire matrix inversion only once.

2.3. Element characteristics

In the present analysis, the layered version of the aboveunsymmetric 20-noded brick element (Fig. 1) is implemented tomodel the laminated structure. Layered version of element isdifferent from the isotropic brick element in the sense that itcan contain multiple plies inside it and accounts for changes inmaterial properties and orientation of the plies inside the element.Element stiffness and mass matrices for this element usingunsymmetric formulation are given as

½ke� ¼

Z þ1

�1

Z þ1

�1

Z þ1

�1½BP�

T ½C�½BM �det½J�dxdZdz

¼

Z þ1

�1

Z þ1

�1

Z þ1

�1½BP �

T ½T�T ½C�½T�½BM�det½J�dxdZdz ð8Þ

½me� ¼

Z þ1

�1

Z þ1

�1

Z þ1

�1½NP�

Tr½NM�det½J�dxdZdz ð9Þ

In the above equations, [NP] and [NM] are parametric and metricdisplacement interpolation matrices as discussed earlier, [BP] and [BM]are parametric and metric strain-displacement matrices and [J] is the

S. Kumar / Finite Elements in Analysis and Design 47 (2011) 373–377 375

Jacobian matrix. ½C� is the material property matrix relating the strainswithin the ply to the stresses in the ply coordinate system and [C] is thematerial property matrix transformed to the global directions. [T] is thetransformation matrix relating the strains in the ply principal directionsto those in the global reference axis.

Material density r, elasticity matrix ½C� and transformation matrix[T] in the above equations (8) and (9) depend on the material propertiesand the orientations of the plies through the thickness of the element.The element formulation was implemented in the computer pro-gramme in such a way that when the material properties and plyorientations are same through the thickness of the element, numericalintegrations of these equations using Gaussian quadrature are carriedout at the element level, otherwise they are accomplished from ply toply through the element thickness.

2.4. Calculation of impact force

When a composite laminate is impacted by a mass, local deforma-tion takes place in the contact region and in most cases must beaccounted for in the analysis if the contact force history is to bepredicted accurately. A practical approach consists of determining therelationship between contact force Fc with indentation depth a. In thisstudy, the modified version of Hertzian contact law proposed by Yangand Sun [20] is used. All the three phases of loading, unloading andreloading are considered. Since the contact area is generally small incomparison with the dimensions of the laminate, a point loadrepresenting the resultant contact force is assumed.

The contact force Fcnþ1 can be written in a general form of

contact law as

Fcnþ1 ¼fðanþ1Þ ¼fðdnþ1�wnþ1Þ

¼f dnþ_dnDt�

1

4

FcnþFc

nþ1

mðDtÞ2�wnþ1

� �ð10Þ

where dn + 1 is the displacement of the centre point of the impactorat the (n+1)th time-step and calculated by applying Newmark’smethod to the equation of motion of the impactor as shown in theabove equation. wn +1 is the displacement of the mid-surface of thelaminate at the impact point in the direction of impact and is notknown at the beginning of the (n+1)th time-step. Since the contactforce Fc

nþ1 is not known at the beginning of each time-step, thefollowing procedure is employed to implement the non-linearcontact law in the analysis.

Closely following an earlier paper by the author [18], the appliedload vector fFA

nþ1g in Eq. (3b) consists of time-varying contact forceFc

nþ1 at (n+1)th time-step and can be written as

fFAnþ1g ¼ Fc

nþ1fIg ð11Þ

where {I} is a column vector whose component corresponding tothe node at the contact point and in the direction of contact force isunity, all other components being zeroes.

The displacement vector {Un +1} in Eq. (2) can be decomposedinto two parts fUA

nþ1g and fUMnþ1g, so that

½K̂nþ1�fUAnþ1g ¼ fF

Anþ1g ¼ Fc

nþ1fIg ð12aÞ

and

½K̂nþ1�fUMnþ1g ¼ fF

Mnþ1g ð12bÞ

For a unit contact force, Eq. (12a) becomes

½K̂nþ1�fUinþ1g ¼ fIg ð13Þ

where fUinþ1g is the displacement caused by the unit contact

force at the (n+1)th time-step. From Eqs. (12a) and (13), it is seenthat

fUAnþ1g ¼ Fc

nþ1fUinþ1g ð14Þ

Thus wnþ1in Eq. (10) becomes

wnþ1 ¼wMnþ1þFc

nþ1winþ1 ð15Þ

In Eq. (15), wMnþ1 and wi

nþ1 are the components of the displace-ment vectors fUM

nþ1g and fUinþ1g corresponding to the node at the

centre of the mid-surface of the laminate and in the direction ofimpact. These displacement vectors are first calculated at eachtime-step using Eqs. (12b) and (13), since the terms on the righthand side of these equations are known at the beginning of the(n+1)th time-step. Having known all other terms in Eq. (10), thecontact force Fc

nþ1 can be evaluated by solving the equation usingthe Newton–Raphson root-finding algorithm. The displacementvector fUA

nþ1g is now evaluated from Eq. (14). Addition of fUAnþ1g

and fUMnþ1g gives the total displacement vector {Un + 1}.

2.5. Stress calculation

Although, the aim of the present paper is only to demonstratethe performance of unsymmetric formulation in impact responseanalysis, here we briefly mention for the sake of completeness, thestress calculation procedure in case of unsymmetric formulation.From the known displacements, the strains are calculated at theGauss points from metric strain-displacement matrix [BM]. Thesestrains are extrapolated at the vertex nodes of the element and aglobal smoothening of strain is performed. Since the element maycontain several plies of different orientations, the strains in eachindividual ply are calculated from element strains using theinterpolation procedure. The strains in each ply are then trans-formed in the material axes system using standard formula for thetransformation of strains. Finally the stresses in each ply areobtained using the stress–strain relation for the ply in the materialaxes system. This procedure is repeated for each time-step.

3. Numerical results and discussions

The above unsymmetric finite element formulation was imple-mented in an in-house computer code, which has been validatedsuccessfully for some benchmark problems in 2D and 3D domainusing both symmetric and unsymmetric formulations.

Having validated the formulation and the code, the impactresponse of graphite/epoxy laminates (plate as well as cylinder)subjected to transverse impact by a metallic impactor is investi-gated. The problem descriptions of impact on a general doublycurved laminate are depicted in Fig. 2. The material properties offiberite T300/976 graphite/epoxy composite are considered aslisted in [21]. First, a rectangular laminated plate of length (a)76.2 mm, width (b) 76.2 mm and thickness (c) 2.54 mm, and havinga ply orientation of [0/�45/45/90]2S, is investigated. The plateclamped on its edges is impacted by an aluminium sphere of12.7 mm diameter travelling at an initial velocity of 25.4 m/s. Thisimpact problem has earlier been studied by Wu and Chang [14]using a conventional (symmetric) 8-noded brick element withincompatible modes and the associated finite element formulationand analysis results have been reasonably validated by them withtheir experimental findings. In the present study, the problem issolved using a 20-noded unsymmetric layered brick element(referred to here as PM20) and the results are compared withthose obtained by 8-noded symmetric layered brick element withincompatible modes and 20-noded symmetric layered brick ele-ment (referred to here as PP8 and PP20, respectively). The resultsobtained by PP8 have been reported and validated in an earlierpaper by the author [18]. Taking advantage of symmetry, only onequarter of the plate is analysed. In the case of PP8, the finite elementmesh consists of 256 (8�8�4) elements while in the cases of PP20and PM20 the mesh consists of 32 (4�4�2) elements. The results

b

X3X2

X1

R1 R2

c

MASS, m NOSE RADIUS, rICONTACT VELOCITY, V0

a

Fig. 2. Problem description of impact on a general doubly curved shell.

0.0

0.5

1.0

1.5

2.0

2.5

Conta

ct F

orc

e (

FC),

kN

Time (t), µs

PP8 [18]

PP20

PM20

0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Dis

pla

ce

me

nt

(d),

mm

Time (t), µs

PP8 [18]

PP20

PM20

50 100 150 200 250 300

0 50 100 150 200 250 300

0.0

0.2

0.4

0.6

0.8

Dis

pla

ce

me

nt

(w),

mm

PP8 [18]

PP20

PM20

S. Kumar / Finite Elements in Analysis and Design 47 (2011) 373–377376

of contact force, impactor displacement and plate centre displace-ment are presented in Fig. 3. The results of PP20 and PM20 are ingood agreement with those of PP8. Since there is no distortion inelements, it is expected in Fig. 3 that the results of PP20 and PM20are in very close agreement to one another.

Next, T300/976 graphite/epoxy cylindrical shell with lay-up[904/08/904] and clamped on its edges is also considered. Withreference to Fig. 2, geometric properties of the cylindrical shell areas follows: a¼b¼100 mm; R1¼R¼a and R2¼N. The impactor is asteel mass having a half sphere head of 10 mm diameter having amass of 200 gm travelling at 5 ms�1. Here full shell is analysed andthe mesh size chosen is 16�16�4 elements in the PP8 case and8�8�2 elements in the PP20 and PM20 cases. Mesh density iskept higher at the centre than the sides in the curvilinear plane ofthe laminate. The results of contact force and shell centre dis-placement are presented in Fig. 4 using PP8, PP20 and PM20element formulations. It is seen that all the three results are in fairlygood agreement with each other. However unlike the plate, there isa mismatch between results obtained by PP20 and PM20 in the caseof the shell. In this particular case, both PP20 and PM20 haveprovided more stiff solution than PP8 and further PM20 gives lessstiff solution than PP20.

0

-0.4

-0.2

Time (t), µs

50 100 150 200 250 300

Fig. 3. Comparison of (a) contact force, (b) impactor displacement and (c) plate

centre displacement in a 76.2 by 76.2 mm T300/934 graphite/epoxy laminated plate

([0/�45/45/90]2S) with clamped edges impacted by 12.7 mm diameter aluminium

sphere at a velocity of 25.4 ms�1.

4. Conclusions

Several recent studies have demonstrated the robustness ofunsymmetric finite element formulation under mesh distortion.This element formulation is based on two different sets of shapefunctions, viz. parametric shape function for constructing the testfunctions and metric shape functions for constructing the trialfunctions. The former is required to satisfy the continuity require-ments and the latter is required to reproduce exactly the poly-nomial displacement fields desired inside the element.

In this paper, three-dimensional layered 20-noded hexahedralunsymmetric finite element formulation is implemented for ana-lysis of impact response in a composite laminate subjected to

transverse impact by a metallic impactor. A few example problemsof composite plate and cylindrical shell under impact are con-sidered and impact response is calculated by present formulation.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Conta

ct F

orc

e (

FC),

kN

PP8

PP20

PM20

0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Dis

pla

cem

ent (w

), m

m

Time (t), μs

PP8

PP20

PM20

200 400 600 800 1000 1200 1400 1600

0

Time (t), μs

200 400 600 800 1000 1200 1400 1600

Fig. 4. Comparison of (a) contact force and (b) shell centre displacement in graphite/

epoxy cylindrical shell ([904/08/904]; a¼b¼100 mm; R¼a), with clamped edges

and impacted by blunt-ended steel cylinder of nose radius 5 mm and mass 200 gm

having initial velocity of 5 ms�1.

S. Kumar / Finite Elements in Analysis and Design 47 (2011) 373–377 377

The performance of the present element was found to be similar tothat of classical 8-noded and 20-noded elements, when theelement geometry is regular. Since the unsymmetric 20-noded

element has been established in literature to have enhanced meshdistortion tolerance in comparison with its symmetric counterpart,it is expected that the same will hold true in the present transientdynamic problem of composite structures. A further study may betaken to investigate this.

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