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____________________* K. Rah

Dept. of Materials Science and Engineering, Ghent University,Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium.

Phone: +32-09-264.95.17Fax: +32-09-264.35.87

Email address: [email protected]

EVALUATION OF DIFFERENT ADVANCED FINITE ELEMENT

CONCEPTS FOR DETAILED STRESS ANALYSIS OF LAMINATED

COMPOSITE STRUCTURES

K. Rah1*

, W. Van Paepegem1, A.M. Habraken

2, R. Alves de Sousa

3and R.A.F. Valente

3

1 Dept. of Materials Science and Engineering, Ghent University,

Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium2 ArGEnCo department Architecture, Geology, Environment & Constructions, University of Lige,

Chemin des Chevreuils 1 bt B52/3, B 4000 Lige, Belgium3 Department of Mechanical Engineering, University of Aveiro,

Campus de Santiago, 3810-193 Aveiro, Portugal

ABSTRACT: Despite their high specific stiffness and strength, laminated composite materials, e.g. fibre-reinforcedplastic plies stacked at different fibre orientations, are susceptible to damage. Damage can be divided into interalaminardamage and interlaminar damage. Delamination is a typical kind of interlaminar damage which occurs in laminatedcomposite materials, often accompanied with intralaminar damage, and may lead to a catastrophic structural collapse.

The first and most crucial step in the prediction of failure of Laminated Composite Structures (LCS) is to accuratelydetermine the stresses, particularly the three transverse stress components, also called the interlaminar stresses. It is

proposed in the present paper that the integration of a displacement based solid-shell formulation and partial-hybridstress formulation will lead to an accurate and robust solid-shell element, suitable for the efficient and detailedinterlaminar stress calculation.

KEYWORDS: specific stiffness and strength, fibre-reinforced plastic plies, interalaminar damage, interlaminardamage, delamination, laminated composite structures, solid-shell element , interlaminar stresses

1 INTRODUCTIONThe incompatibility between the overall dimensions ofthe LCS (up to 25-30 meters) and the thickness ofindividual composite plies (typically 150-200micrometers) makes solid-shell elements very attractivefor the stress analysis of LCS. All displacement basedsolid-shell formulations available in literature have some

issues that need to be addressed before their successfulemployment for the stress analysis of LCS. These issues,e.g. the transverse stress continuity through the laminate

thickness and the traction-free condition on the upperand/or lower surfaces of the laminate, can be addressedusing the hybrid/partial-hybrid stress formulation. The

main objective of this paper is to investigate differentFinite Element (FE) formulations available in theliterature for the stress analysis of LCS and to propose anew approach that can lead to efficient finite elementformulations suitable for the detailed stress analysis ofLCS.

2 DISPLACEMENT BASED SOLID-SHELL ELEMENT

Solid-shell elements form a class of finite element

models that is intermediate between thin shell andconventional solid elements. They can be referred to as

finite element models that have shell kinematicsassumption and possess no rotational dofs. They havethe same node and degrees-of-freedom configurations ofsolid elements but account for shell-like behaviour in the

thickness direction. They are useful for modelling shell-

like portions of a 3D structure without the need toconnect solid elements nodes to shell nodes. Solid-shellelement properties make them appropriate for themodelling of individual laminate plies whose surface andthickness dimensions are dimensionally incompatible,

and finally they can be stacked to model a laminate.

DOI 10.1007/s12289-009-0629-z

Springer/ESAFORM 2009

Int J Mater Form (2009) Vol. 2 Suppl 1:943947


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Figure 1: Solid-shell element [1]

2.1 ADVANTAGES OF SOLID-SHELLELEMENTS

Some advantages of solid-shell elements are as follows:a) they are simpler in their geometric and kinetic

description,

b) they perform better under high element aspect ratiosas compared to conventional 3D solid elements,

c) combined use of the solid and solid-shell elementsdo not need special solid-to-shell transitionelements, as both elements have only displacementsDegrees Of Freedom (DOF),

d) complications in handling the finite rotations, as inconventional shell elements, can be avoided,

e) full 3D constitutive law can be directly employed insolid-shell elements, hence there is no need for thedimensional reduction of constitutive law, as it isnecessary in conventional plane stress based shell

elements,f) they provide the direct calculation of thickness

variations and transverse shear and normal stresses,g) due to the presence of top and bottom surface nodes

in solid-shell elements, it is possible to considerautomatically the double-sided contact in sheet

forming simulations.

2.2 DISADVANTAGES OF SOLID-SHELLELEMENT

Apart the above mentioned benefits of solid-shell

elements they have the following disadvantages:a) they are prone to be plagued by different types of

locking, e.g. transverse shear locking, membranelocking, thickness locking, volumetric locking andtrapezoidal locking,

b) elements with high element aspect ratio can sufferfrom the ill-conditioning of the respective elementstiffness matrix,

c) they are computationally more costly than theconventional shell elements, especially when used tomodel layered structures, such as laminatedcomposites.

2.3 COMPARISON OF DIFFERENT SOLID-SHELL ELEMENTS

Many solid-shell element formulations have beenproposed by different researchers over the last few

decades. A comparison of the most establisheddisplacement based solid-shell formulations in literature

is shown in Table 1.Solid-shell formulations from Vu-Quoc and Alves de

Sousa are found to be more suitable for the stressanalysis of LCS. The main differences between these

formulations are as follows:a) Alves de Sousas element uses only one Enhanced

Assumed Strain (EAS) parameter while Vu-Quocselement uses seven EAS parameters,

b) reduced numerical integration scheme has beenemployed in Alves de Sousas element, with the

provision of user defined integration points throughthe element thickness, while full integration schemeis used in Vu-Quocs element,

c) a stabilization technique is introduced in Alves deSousas element to prevent the zero energy modes ofthe element caused by the reduced integration

scheme. On the other hand, no such stabilization isnecessary in Vu-Quocs element due to the use offull integration scheme,

d) due to less EAS parameters and reduced integrationscheme, Alves de Sousas element iscomputationally less costly than Vu-Quocs

element,e) Vu-Quocs element passes the bending patch test

exactly while Alves de Sousas element shows slighterror in this test,

f) Vu-Quocs element can be employed in higherelement aspect ratios (6667) than Alves de Sousaselement (1000).

2.4 DRAWBACKS OF DISPLACMENT BASEDFORMULATIONS FOR THE STRESS

ANALYSIS OF LCS

In developing finite elements for stress analysis of LCS,

the main requirement is to satisfy the continuityconditions on displacements and transverse stresses atinterlaminar surfaces, and traction-free condition on theupper and/or lower surfaces. It can be seen from Table 1

that no displacement based formulation is able to providethe transverse stress continuity. This is so since in

displacement based formulations stresses (secondaryvariables) are calculated from approximatedisplacements (primary variable) by using numericaldifferentiation at Gauss points. Stresses can vary from

element to element and from Gauss point to Gauss point.Owing to the discontinuity in material properties in

laminated composites, interlaminar surfaces are usuallylocations of large gradients of stresses. This leads to thedrastic dissimilarities in the stresses at interlaminarsurfaces, which are calculated by extrapolating the Gausspoint values, hence violating the transverse stress

continuity through the laminate thickness.Furthermore, the use of displacement based elementsrequires fine element meshes and extensive amount of

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____________________* K. Rah

Dept. of Materials Science and Engineering, Ghent University,Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium.

Phone: +32-09-264.95.17Fax: +32-09-264.35.87

Email address: [email protected]

Table 1: Comparison of various displacement based solid-shell formulations in literature

Bendi

ngpatch

testLockingfree

Absen

ceofrot.dofs/

disp.dofsonly

Highe

r-ordertermsin

thickn

esscoordinate

Absen

ceof

pre-in

tegration

Model

parameter-

space

dimension

Optim

alEAS

Geom

etrical

non-li

nearity

Mater

ial

non-li

nearity

Highe

staspectratio

Interlaminar

stress

contin

uityinthickness

direct

ion

Vu-Quoc [1, 3] Yes Yes Yes Yes Yes 3-D Yes Yes No 6667 No

Bischoff [4] Yes Yes Yes No No 2-D No Yes Yes - No

Hauptmann [5] Yes No Yes Yes Yes 3-D No Yes No - No

Betsch [6] Yes Yes No Yes Yes 2-D No Yes Yes - No

Miehe [7] No Yes Yes Yes No 3-D No Yes Yes 200 No

Reese [8] No Yes Yes Yes Yes 3-D Yes Yes Yes - No

De Sousa [9-11] slighterror

Yes Yes Yes Yes 3-D Yes Yes Yes 1000 No

computer space and time to be able to determine stressesand strains with a reasonable degree of accuracy. This isbecause of the fact that the convergence of displacementfinite element model for problems with large gradients of

stresses is slow [2].

3 HYBRID STRESS FINITE ELEMENTSThe development of hybrid stress finite elements ismotivated by attempts to overcome the above mentioned

disadvantages of displacement elements. The hybridstress formulations assume the stresses as independentvariables (primary variables) in the variational setup atthe beginning. Therefore, the degree of accuracy of the

stresses is the same as that of the displacements. This isdue to the fact that the stresses are obtained directly fromthe process of minimization and without having to gothrough the numerical differentiation of thedisplacements. Another advantage is that the continuity

condition of the primary variables can be imposed easily.A hybrid element can be formulated by many differenttechniques. Although most of the successful finiteelements were initially based on intuitive insight rather

than rigorous variational principles, researchers arealways keen on devising variational bases for the new

elements. Variational bases are considered to beimportant not only for legitimacy but also for theconfidence of the element users [2].

3.1 COMPARISON OF DIFFERENT HYBRIDSTRESS ELEMENTS

The first assumed stress element was suggested by Pian

[12], which was based on the complementary energyprinciple. Since then many hybrid stress elements wereproposed in literature. Tong and Pian [13] proposed amultilayer plate element that provided the desiredthrough-the-thickness transverse shear stress continuityat interlaminar surfaces and zero traction condition at the

top and bottom surfaces of a laminate. A 3D partial-

hybrid multilayer solid element was introduced by Hoaand Feng [2] that also provides the accurate interlaminarstress field through-the-thickness of a laminate, which isbased on the composite energy principle. Maindifferences between Pians element and Hoas elementare as follows:

a) Pians element is based on complementary energyprinciple while Hoas element on composite energyprinciple,

b) Pians element is a multilayer plate element butHoas element is a multilayer 3D solid element,

c) number of assumed stress parameters for Pians andHoas element are 20 and 16 respectively,

d) only Hoas element provides the transverse normalstress.

3.2 DISPLACEMENT BASED ELEMENTS VS.HYBRID STRESS ELEMENTS

In order to give the reader an overview of the transverseshear stress calculation using displacement basedelements and hybrid stress elements, for the stress

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analysis of LCS, the thick laminated beam example isdiscussed in the following. Figure 2 shows a simplysupported three-ply-laminate [0o/90o/0o] thick beam withsinusoidal loading. This problem is chosen as theanalytical solution of this problem is given by Pagano

[14]. A comparison of transverse shear stress distribution

in the thickness direction of the laminate at one end ofthe beam, employing different element formulations, isdepicted in Figure 3. It can be seen from Figure 3 thatthe transverse shear stress distribution given by thedisplacement based solid elements is inaccurate and

discontinuous through the laminate thickness and it doesnot fulfil the zero traction condition at the top and

bottom surfaces of the laminate. On the other hand, bothPians and Hoas elements (hybrid stress elements) notonly provide the stress continuity through the laminatethickness but also fulfil the zero traction condition at the

top and bottom surfaces of the laminate. This exampledemonstrates the suitability of hybrid stress formulation

for the transverse stress calculation in LCS.

Figure 2Simply supported three-ply-laminate [0o/90

o/0

o]

beam with sinusoidal loading. The fiber direction is alongx-axis for [0

o] layer and along y-axis for [90

o] layer [15]

Figure 3Transverse shear stress distribution in thicknessdirection of a [0

o/90

o/0

o] laminated simply supported

beam

4 SOLID-SHELL ELEMENT BASED ONTHE PARTIAL-HYBRID STRESS

FORMUALTION

The problem of delamination for LCS has been a greatchallenge for designers and researchers from thebeginning, being investigated during the past decades.

Many numerical techniques for the prediction ofdelamination in LCS account for the use of FE method.However, the problem has not yet been satisfactorilysolved. The main difficulty that arises is the calculationof transverse stresses efficiently and accurately. Without

efficient means to obtain accurate transverse stresses, it

is difficult to predict interlaminar failure.From the previous two sections it is evident that a solid-shell element can be a good choice for the modelling ofLCS. However, there are still many issues in solid-shellformulations presented in literature that need to be

addressed before they can be efficiently employed forthe transverse stress calculation in LCS. These issues

stem from the geometry, heterogeneity and layered setupof LCS, and can be addressed using a hybrid/partial-hybrid stress formulation. In literature no solid-shellelement based on hybrid/partial-hybrid stress

formulation can be found. This raises the need ofdevelopment of such an element that can significantly

contribute in FE technology in the context ofdelamination modelling in LCS.

5 CONCLUSIONAuthors believe that the integration of a displacement

based solid-shell formulation and partial-hybrid stressformulation can lead to an accurate and robust solid-shellelement, which could be efficiently employed for thedetailed stress analysis of LCS, which is the first crucialstep in the prediction of delamination. This integratedelement might have the following properties:

a) simpler geometric and kinetic description like 3Dsolid elements, making it possible to stack them tomodel a laminate,

b) accurate results in high element aspect ratios,making it suitable for the modelling of individualthin composite plies in a laminate,

c) accurate and efficient interlaminar stress calculationthrough the laminate thickness, thus fulfillinginterlaminar stress continuity through the laminatethickness and at the same time granting zero tractioncondition at the top and bottom surfaces of thelaminate,

d) use of the full 3D constitutive laws,e) automatic consideration of double-sided contact in

sheet forming simulations,f) direct and accurate calculation of thickness

variations.

REFERENCES

[1] Vu-Quoc, L. and X.G. Tan, Optimal solid shellsfor non-linear analyses of multilayercomposites. I. Statics. Computer Methods in

Applied Mechanics and Engineering, 192(9-10): p. 975-1016, 2003.

[2] Hoa, S.V. and W. Feng, Hybrid finite elementmethod for stress analysis of laminated

composites. 1998: Kluwer AcademicPublishers.

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[3] Vu-Quoc, L. and X.G. Tan, Optimal solid shellsfor non-linear analyses of multilayercomposites. II. Dynamics. Computer Methodsin Applied Mechanics and Engineering, 192(9-10): p. 1017-1059, 2003.

[4] Bischoff, M. and E. Ramm, Shear deformable

shell elements for large strains and rotations.International Journal for Numerical Methods inEngineering, 40(23): p. 4427-4449, 1997.

[5] Hauptmann, R. and K. Schweizerhof, Asystematic development of 'solid-shell' element

formulations for linear and non-linear analysesemploying only displacement degrees of

freedom. International Journal for NumericalMethods in Engineering, 42(1): p. 49-69, 1998.

[6] Betsch, P., F. Gruttmann, and E. Stein, A 4-node finite shell element for the implementation

of general hyperelastic 3D-elasticity at finitestrains. Computer Methods in Applied

Mechanics and Engineering, 130(1-2): p. 57-79,1996.

[7] Miehe, C., A theoretical and computationalmodel for isotropic elastoplastic stress analysis

in shells at large strains. Computer Methods inApplied Mechanics and Engineering, 155(3-4):

p. 193-233, 1998.[8] Reese, S., A large deformation solid-shell

concept based on reduced integration withhourglass stabilization. International Journal for

Numerical Methods in Engineering, 69(8): p.1671-1716, 2006.

[9] Cardodo, R.P.R., et al., Enhanced assumed

strain (EAS) and assumed natural strain (ANS)methods for one-point quadrature solid-shellelements. International Journal for NumericalMethods in Engineering, 75: p. 156-187, 2007.

[10] de Sousa, R.J.A., et al., A new one-point

quadrature enhanced assumed strain (EAS)solid-shell element with multiple integrationpoints along thickness: Part I - geometricallylinear applications. International Journal forNumerical Methods in Engineering, 62(7): p.952-977, 2005.

[11] de Sousa, R.J.A., et al., A new one-pointquadrature enhanced assumed strain (EAS)

solid-shell element with multiple integrationpoints along thickness - Part II: Nonlinearapplications. International Journal forNumerical Methods in Engineering, 67(2): p.

160-188, 2006.[12] Pian, T.H.H., Derivation of element stiffness

matrices by assumed stress distributions. AIAAJournal, 2: p. 1333-1336, 1964.

[13] Tong, S.T.M.P. and T.H.H. Pian, Finite elment

solutions for laminated thick plates. Journal ofComposite Materials, 6: p. 304-310, 1972.

[14] Pagano, N.J., Exact solutions for compositelaminates in cylindrical bending. Journal of

Composite Materials, 3: p. 398, 1969.

[15] Vu-Quoc, L., Solid-shell element for compositematerials . Internal communication, 2002.

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