f!) Pergamon 0045-7949(94)00355-6

Computers & Structures Vol. 54, No.4. pp. 689-703. 1995Copyright © 1995 Elsevier Science Ltd

Printed in Great Britain. All rights reserved0045-7949/95 $9.50 + 0.00

T~REE-DIMENSIONAL PLATE AND CONTACT/FRICTIONELEMENTS FOR LAMINATED COMPOSITE JOINTS

E. J. Barbero,t R. Luciano~ and E. Sacco~

tDepartment of Mechanical and Aerospace Engineering, West Virginia University Morgantown,WV 26506, U.S.A.

tDepartment of Industrial Engineering, University of Cassino, Italy

(Received 7 August 1993)

Abstract-A contact/friction isoparametric finite element for the analysis of connections betweenlaminated composite plates is presented. The element is compatible with a three-dimensional plate elementbased on layer-wise constant shear theory. The contact/friction element is based on a simple regularizationof the unilateral contact with an orthotropic Coulomb friction problem. Two different friction coefficientsin orthogonal directions and a constitutive law to model slip in any direction are used to account for theorthotropic surface texture of fiber-reinforced materials. The contact and friction constitutive law isdistributed over the entire surface in contact/friction. The integration scheme is chosen to accuratelyaccount for the partial contact and slip inside the element, thus avoiding unnecessary mesh refinement.Unlike nodal gap elements, the proposed element applies the contact and friction constitutive lawsuniformly over the contact/friction surface regardless of mesh refinement or element distortion. Theirreversibility of the process leads to unsymmetric secant stiffness matrix. However, an iterative procedureworking on symmetric matrices is proposed. Finally, numerical results are presented.

1. INTRODUCTION

Composite materials are very good candidates toreplace and complement conventional materials likesteel, aluminum or concrete, because of light weight,corrosion resistance, electromagnetic transparency,etc. Previous studies have demonstrated that connec­tions performance is one of the more importantlimiting factors for the application of composites.Bolted connections are preferred because they facili­tate construction, they can be disassembled, facilitaterehabilitation of damaged structures, and they areconsidered safer than bonded connections for longlife cycles. Experimental evidence [1] suggests thepossibility that friction plays an important role in thebolted and the bonded-bolted connections strength.The amount of the torque is found [1] to influence theultimate strength of both bolted and bonded-boltedconnections which may indicate that slip occurs atdifferent levels of load. Furthermore, experimentalevidence indicates that the rupture in compositematerial structures occurs suddenly with brittle be­havior, hence in a structure subjected to impact orearthquake loads, a large amount of energy can beabsorbed by the friction developed in the connec­tions. Furthermore, stresses are not transferred frombolt to hole until slip occurs. Hence, the frictionoccurring between jointed plates improves theefficiency of the connections. However, friction effectsare often neglected in the analysis of fastened connec­tions [2], because of the difficulty involved in itstreatment. On the contrary, it is our belief that it is

necessary to overcome these difficulties and to prop­erly account for the friction in bolted connections.Unilateral contact must be considered to properlytake into account the partial contact between jointedplates, because some separation in the region betweenthe joined parts always occurs due to the deformationunder load and bolt torque. In order to develop arealistic analysis of bolted connections it is thennecessary to consider appropriate plate and con­tact/friction models. Both a three-dimensional. (3D)plate element and a compatible interface element areintroduced next.

Many of the analysis tools for laminated compositeplates are based on the kinematic assumptions intro­duced by Reissner and Mindlin, which constitute thebasis for the first-order shear deformations theory(FSDT) [3]. FSDT produces excellent results forglobal response (e.g., deflections) but the accuracy ofthe stress distribution is not adequate for the analysisof bolted connections and other unilateral contactand friction problems of thick laminated compositesstructures. Higher order theories have been proposedin an attempt to improve the prediction of stresses,but only limited success has been possible with equiv­alent single-layer theories. All of the equivalentsingle layer theories have a common characteristic:the assumed distribution of the displacementsthrough the thickness has continuous derivatives withrespect to the thickness coordinate. This implies thatthe out-of-plane shear strains are continuousacross the material interfaces. As a result of thedifferent material properties, the out-of-plane stress

CAS 54/4-1689

690 E. J. Barbero el al.

components are discontinuous at the interfacesbetween layers, thus violating the equilibriumconditions.

To overcome the limitations of the. single-layertheories and to obtain· a good evaluation of thestresses, a new class of theories has emerged which isbased on a distribution of the displacement throughthe thickness that is continuous but with discontinu­ous derivatives at the interfaces between layers. If thedistribution of displacement is linear in the thicknesscoordinate for each layer, the· kinematics ·leads tolayer-wise constant shear (LCS) theories, which weredeveloped by several authors [4-12]. For the model­ing of joints, a disadvantage of 20 plate theories isthe need· to use multi-point constraints to joint themiddle surface of the two components of the joint.This problem could be avoided by using 3D contin­uum elements, but plate elements are universally usedto analyze laminated composite plates because of thecost of 3D analysis. The main disadvantage of con­ventional continuum 3D elements, when used toanalyze each layer of a composite plate individually,is the large number of degrees of freedom (OOF)involved, since the aspect ratio of the element limitsthe surface dimensions of the elements to no moretbatone order of magnitude larger than the thicknessofa layer.

A new element was developed by Barbero [12, 13]using the idea of the 3D shell element of Ahmadet al. [14] and using the kinematic constraints ofLCS theories. With the use of this special 3D plateelement, it is possible to analyze laminated compositeplates overcoming the difficulties of both the LCS 2Delements and the continuum 3D elements but retain-

ing the precise stress calculation. It is important tonote that, because in this element the position of themiddle surface is irrelevant, it is possible to modeljoint problems (e.g., lap joints) without using multi­point constraints, which are needed in conjunctionwith regular plate elements to link the middle surfacesof the connected plates.

The unilateral contact with Coulomb friction con­stitutive relationships between the contact stress andthe relative displacement vectors are governed bynondifferentiable, nonconvex and noncoercive func­tionals. In particular the nondifferentiability of thesegoverning functionals is due, for the unilateral prob­lem, to the requirement of absolute rigid contact ofthe two surfaces, and, for the friction problem, to thesliding or full adhesion of the contact surface [15, 16].Therefore, the contact/friction problem is complexboth from a theoretical and computational point ofview. On the other hand, experiments show that thisdescription of the contact is unrealistic. In fact, it hasbeen found that when surfaces are in contact theymay deform somewhat under normal pressure be­cause of the irregularities and asperities of the sur­faces. Moreover, tests on the friction phenomenaemphasize that no such line of separation of slidingand full adhesion exists. Researchers working ontribology proposed [17] an exponential equation. todescribe the normal behavior of the interface and apolynomial equation for the friction.

These facts suggest using approximate regularizedfunctionals for the unilateral and friction contactconstitutive relationships. Regularized constitutiveequations have been proposed [18, 19] for the staticand dynamic 2D analysis of monumental structures

<pi 12 15 11

4 8 1

<pi

D 2 dof

Ii 3 dof

Fig. 1. Three-dimensional plate element.

Contact/friction elements for laminated composite joints 691

2. PLATE ELEMENT

where U is the vector of the nodal displacementsdi = {u i

, Vi, wilT with i = 1, ... ,m, m is the number ofthe nodes in the element, and N is the matrix of the

Consider a laminated plate composed of n or­thotropic laminae, each arbitrarily oriented with re­spect to the global coordinate system (x and y in themiddle plane of the plate and z in the thicknessdirection). Each layer of the plate is discretized by 3Dplate elements. The displacement u = {u, v, w }T insidean element is given by

w

Fig. 2. Schematic representation of the degrees of freedomused in a three-layer model using 31? plate elements.

interpolation functions cp i. The order of interpolationcpi along the two directions on the surface of theelement (Fig. 1) can be chosen independently of theorder through the thickness. Linear or quadraticinterpolation of the displacements (u, v) and thegeometry (x, y) are commonly used. The order ofapproximation in the thickness direction correspondsto different kinematical assumptions in LCS theories.In this paper, and consistently with the findings ofBarbero [11], a linear variation is used. The quadraticelement has 18 nodes. Nodes 1-9 have three OOF (u,v and w), nodes 10-18 have two OOF (u and v) asshown in Fig. 1. The variables u and v correspond tothe in-plane displacements at the interfaces betweenelements (layers). The quadratic element has a totalof 45 DOF.

The transverse deflection is constant through thethickness by the incompressibility assumption. There­fore, the OOF of two nodes aligned through thethickness of an element can be reduced to a singleOOF (for example, w in the bottom filled squares inFig. 1). Then, at the element level the stiffness matrixis rearranged so that the OOF corresponding to thedisplacements u and v on the surface of the plate forall nodes (e.g., 18 nodes) are grouped first. Theremaining OOF corresponding to w-dis_placementsare assigned to a new set of nodes (e.g. nine nodes)called w-master nodes. The location of the w-masternodes through the thickness of the laminate does notaffect the results. The resulting element has 18 nodeswith two OOF (u and v) and nine nodes with oneOOF (w) per node.

If more than one layer is used to model thelaminate, another condensation must be done foreach vertical because the w-displacements are identi­cal. A three-layer laminate modeled with three el­ements is shown in Fig. 2. Since the transversedeflection w is constant through the thickness, asingle global node connects all the local w-nodes thatlie on a line perpendicular to the middle surface.

(1)u=NU,

made by superimposed blocks. An extension of theregularized constitutive equations is made herein for30 problem with orthotropic friction.

Once the constitutive equations for the contact/friction problem are introduced, the isoparametric30 .finite interface element can be derived. Severalinterface elements can be found in the literature.Most of the joint elements implement connectionsdirectly between the nodes defined by the finiteelement mesh, by means of ID springs. A 20 dis­placement based interface element was presented byGoodman et ale [20] to analyze the behavior ofjointed rocks. This element, to the best of the authors'knowledge, was the first full interface element, in thesense that it accounts for springs uniformly dis­tributed along one line which connects two surfacesin contact. The analysis was limited to 20 planestrain or plane stress problems and the interfacebehavior was essentially linear. Then, displacement­based interface elements for 3D nonlinear analysishas been proposed, for instance in [21,22].

The interface element proposed herein is based onan isoparametric formulation, is consistent with thestandard 3D continuum elements and with the new30 plate element, it can account for a trivial or nott~i-v-ialinitialgap distance between the two surfaces inconnection, and it can consider two different frictioncoefficients in orthogonal directions. Both the 3Dplate and the 3D interface finite elements are basedon the· displacement formulation. This is convenientfor its simplicity and ease of implementation incommercial finite element packages, which generallyuse spring connections between nodes to simulate thefriction/contact behavior of the interface. Nodalsprings induce a non-uniform regularization in thespace of the unilateral Coulomb law when the meshis not uniform. Therefore, both mesh refinement anddistorted elements create a model with non-uniformconstitutive interface law. It will be shown that theproposed plate and interface elements provide amore rational formulation of the problem at hand,using uniformly distributed regularization of the con­tact and friction applied directly on the contextdisplacement-based 3D plate elements.

692 E. J. Barbero et al.

The derivatives of the displacements with respect tothe global axes are obtained in the standard wayusing the isoparametric formulation. The numericalintegration is performed by Gaussian quadra­ture [25]. A two-point rule suffices in the z direction,while a minimum of three or four points are neededin the x and y directions for quadratic or cubicelement, respectively. Full integration is used in the xand y directions for all the terms, including the shearrelated terms. Otherwise, the penalty formulation ofthe unilateral contact may excite spurious modes (asshown in the numerical results). The integrationof the element stiffness matrix is performed as astandard I8-node element with three DOF per node(u, v, w) but with the appropriate shape functions cp i

described previously.The constitutive equations (4) are used to obtain

the components of stress at the Gauss points. Thedistribution of in-plane stresses tlx , tIx~" and til' islinear through the thickness, while the di'stribution ofinterlaminar stresses tIxz and tIyz obtained from consti­tutive equations is layer-wise constant. The new 3Dplate element (3DLCS) reduces to FSDT when onlyone element is used through the thickness of thelaminate. The 3DLCS element gives a very goodrepresentation of all the stress components except (1=

without the aspect ratio limitations of conventional3D continuum elements.

Then, only one w-master node (e.g., the one at thebottom of the laminate, solid squares in Fig. 2) mustbe retained. This condensation has been achieved byassigning the same global node number to all thew-master nodes through the thickness of the laminateat a given surface location (x, y). This techniqueeliminates the need for complex book-keeping toidentify individual sets of elements stacked to form alaminate. Furthermore, the present formulation elim­inates the need for elements with large number ofDOF per node, typical of 2D finite element im­plementations of LeS theories, which would result ifthe assembly through the thickness were performed apriori. This feature is particularly useful for im­plementation in commercial finite element programs.The strain vector is:

lx u,x

ly V,y

l= Yyz W,y + v,z (2)

Yxz w,x+ u,z

Yxy U,y + v,x

with the strain in the direction z neglected to beconsistent with the incompressibility of the normalsimplicit in the plate LCS theory. The layer-wiseconstant shear constraint is satisfied by virtue of thelinearity through the thickness of the displacements uand v in eqn (1) resulting from the choice of interp­olation functions cp i that are linear through thethickness (Fig. 1).

The constitutive equations for the kth layer (orelement) of an orthotropic material, oriented arbi­trarily with respect to the elemental coordinate sys­tem, are obtained in a similar way to that of amonoclinic material [23]. To overcome the lockingeffect that a vanishing transverse strain would have,the element constitutive matrix is a combination ofthe 2D rotated reduced stiffness terms Qij [23] and 3Dterms Cij [24]. In this way we obtain

Q11 Q12 0 0 0

Q12 Q~2 0 0 0j)k= 0 0 KC~ 0 0 (3)

0 0 0 KC~5 0

0 0 0 0 Q~6

where:

and Tk is the rotation matrix [23].The stiffness matrix is written as

K(e) = tBmB dx dy dz,

where the strain-displacement matrix is

lfJ:x 0 0

0 CP:y 0

B= 0 lfJ:z CP;y

CP:z 0 lfJ:x

CP~ lfJ:x 0

(5)

(6)

(7)

(8)

or in compact form

(4)

where k is the shear correction factor, (I' and l' arethe stress and strain vectors in the material coordi­nate system aligned with the fiber direction. Therelationship between stress and strain written withreference to the global coordinate system (x, y, z) is

3. CONTACT-FRICTION CONSTITUTIVE LAW

Let us consider two limited plane surfaces at aconstant distance g from each other. At each point P(see Fig. 3) of the surface (a) is associated a uniquepoint Q of the surface (b), obtained as orthogonalprojection of P on (b). The kinematics of the interfaceis defined by the relative displacement of the twosurfaces

Contact/friction elements for laminated composite joints 693

u (P)

surface (b)

surface (a)

Fig. 3. Kinematics of the interface.

Denoting by n the normal to the surface (a), therelative displacement is decomposed in its normalcomponent AUn = Au' n and tangential componentAv = Au - AUn n. Let p be the contact stress vector atthe interface, with normal and tangential componentsPn and t, respectively. Introducing the normal pen­etration b = AUn +g, the normal unilateral contactconstitutive equations are

Au(PQ) = u(Q) - u(P). (9) stiffness constant, k4 and k3 , according to the possi­bility or not of sliding, is proposed. Furthermore, theset of tangential stresses for which no slip occurs issupposed to be an ellipse with principal direction Xl

and X2' When the limit tangential stress is reached,sliding occurs in the same direction of the tangentialstress.

The limit tangential stress i, for which there is nosliding according to the Coulomb friction model, it isgiven by:

Pn = klAun, if b > 0Pn = (k2 - kl)g + k2Aun, if b ~ 0,

(10)(12)

where k I and k2 are constants representing the stiff­nesses in the normal direction at the iilterface (Fig. 4).

Orthotropic friction is governed by two frictioncoefficients JlI and· Jl2 relative to two orthogonaldirection x I and X2' respectively. Obviously, whenthere is no contact of the surfaces, no tangentialforces can be developed, then the limit tangentialstress is

with Jl friction coefficient in the direction of i. Takinginto account the proposed normal constitutiveequation (10) and the definition of b, eqn (12) reducesto:

(13)

The limit relative tangential displacement corre­sponding to the value of Ilill is then

i = 0, if b > O. (11)(14)

When the surfaces. are in contact, a multi-linearconstitutive relation, with low or high value of the with w = (k2b - k l g)/k3 •

.I.

I.I

k 2

--.--.-­.--_.---..-- -

perfect unilateral contact modelproposed modelpossible realistic behavior

Fig. 4. Constitutive law for unilateral contact.

694 E. J. Barbero et af.

\s \ 4 X 2

\

\_----.--~

\

\ I\ I\

\\

\

Fig. 5. Admissible set of no-slip displacement region.

The limit principal tangential relative displace­ments along the principal direction XI and X2 forwhich there is no sliding are given by

ordinate system r - s, with r orthogonal to s, areassumed to be

Let us denote by Av and AiT the total and the limittangential relative displacement in direction r (Fig. 5),respectively. Let us introduce the multiplier p as

if b ~ 0 and P< 1. (20)

Then, by taking into account eqns (15-17) we obtain

and let us write the equation of the ellipse that definesthe limit displacement before sliding as

PAv = At (16)Figure 6 depicts in schematic way this constitutiverelation. It can be pointed out that in the firstequation of (20) w ~ O. The term Avr/[(Avl /

IlI)2 + (AV2/1l2)2](1j2) ~ 0 represents the friction co­efficient in r direction. The second of equations (20)constrains the slippage in the s-direction.

The constitutive equations (20) can be rewritten inthe X 1-X2 coordinate system by using the rotationmatrix R, defined as

(18) (21)

If P~ 1 the slip does not occur, if p < 1 the slipdoes occur. It is worth noting' that p and, moreover,At depend upon Av, Ill' 112 and upon the normalpenetration b.

~quation (16) shows that when there is sliding,it occurs along the direction of At and t. This canbe seen as a nonstandard flow rule, in the planeof contact, while a standard flow rule wouldindicate slip along the normal to the limit surface(Fig. 5).

When the slip is not possible, the constitutiveequation is:

t = k 3 Av, if b ~ 0 and p~ 1. (19)

On the contrary, when the slip occurs, theconstitutive equations in the local Cartesian co-

with CI = AVI II Av II and C2 = AV2/IIAv II. The trans­formation performed by R is:

{ AVI} = R{AVr

}, {tl} = R{tr}. (22)

A~ A~ ~ ~

In the principal coordinate system, the tangentialconstitutive equation (20) is

if b ~ 0 and f3 > 1.

Contact/friction elements for laminated composite joints 695

Finally, the complete constitutive relations can bewritten in matrix form as

p = K(Au )Au + h(Au)w + i(Au)g (24)

(27)

(26)

where

p = K(Au)Au + m(Au)g,

or, in compact notation:

Recalling that w = (k2b - k)g)/k3 = [k2 Au" + (k2 ­

k)g ]/k3 , the constitutive equation (24) is equivalentto

(25)

with

If b > 0 (no contact) then

if b ~ 0 (contact) then if p ~ 1 (no slip) then

if P< 1 (slip) then

It is worth noting that when slip occurs (I12' ci4 and cisare not zeros and the new constitutive matrix givenin (26~)and (27) is not symmetric. The unsymmetry ofK precludes the existence of an energy functionalgoverning the interface constitutive relationship. Theinterface material can be seen as a Cauchy elasticmaterial, that is for a given Au a unique p can befound, but not as a Green elastic one. This means thatfor a closed 'deformation path, energy can be lost.

In Qrder to better understand the proposed model,iet us perform a simple experiment. Let us take rigidbrick resting on a rigid surface. The interface betweenthe block and the plane is modeled by the aboveproposed constitutive law. Let us then apply a normalforce -F on the block. Vertical displacementAUn = -F/k2 occurs. Then a tangential force T isapplied. Since F is fixed, the tangential limit stress isgiven. If T is greater than this limit tangential stress,

fS.vr

Coulomb model

proposed model

possible realisticbehavior

Fig. 6. Constitutive law for slip problem.

696

T

•o

E. J. Barbero et al.

energy lost

energy recoveredremoving T

T/ k 4 + I-lF(k 3- k 4) /k3k 4

Fig. 7. Model behavior for a closed displacement path.

~v

the horizontal displacement of the block is.1v = T/k4 + p.F(k3 - k4 )/(k3 k4 ). Once this conditionis reached, let us remove the force F. The verticaldisplacement goes to zero, while the horizontal oneincreases to the value T/k4 in order to satisfy theequilibrium equation. The energy related to the verti­cal effects is recovered completelY, while the oneasso~ia~~d to the horizontal displacement increases.Removing, now, the tangential force T only somepart of the spent energy is recovered as it is illustratedin Fig. 7..lt seems necessary to point out that the lostenergy is not transformed in heat or noise or mi­croplasticizations of the interfaces during the slip, asoccurs in the real phenomena. The model, in fact,does not consider inelastic strain. Thus it can be usedefficiently only for quasi static analysis of structuressubjected to a monotone increasing stress path.

The finite element formulation is carried out bystarting from the variational formulation of the equi­librium equation. Since no strain energy functionalcan be postu~ated, no potential energy approach ispossible, in contrast to the proposition of Beer [22].The weak form of the governing equation can beobtained by using the virtual displacement theoremas pointed out by Schafer[21]. The external virtualwork is performed by the forces q(b) and q(t} acting onthe bottom and top surfaces Q, times the virtualdisplacement ~U(b) and ~u(t)

The equilibrium condition for the external forcesasserts that the external work must be zero for anyvirtual rigid displace"ment (~U(b) = ~u(t) = ~u)

that is q(b) = _q(t) = q. Thus, the external virtual workcan be written correctly as

4. THE INTERFACE FINITE ELEMENT

A general isoparametric 3D I8-node interface el­ement is presented in the following. This element isdeveloped to be used with the new 3D plate elementdescribed previously. The thickness of the element isthe gap distance g, and it can be taken to be zerowhen surfaces are initially in contact. A schematicgeometry of the element is given in Fig. 8.

(30)

Fig. 8. Interface element.

Contact/friction elements for laminated composite joints 697

The internal virtual work is given, using the formula(24), by

Wj = In {(K(Au)Au + h(Au)w

+ i(Au)g)· ~Au} dX1 dX2' (31)

Finally, the virtual displacement theorem asserts

In q-t5AudxJ dx2= In {(K(Au)Au+h(Au)w

+ i(Au)g)'~Au} dx} dX2' (32)

According the finite element method, the relativedisplacements are assumed to be

L\v2 = (U~+ 9 - U~)cpi

(33)

where u~ and U~+9, with h = 1, 2, 3 andi = 1,2, ... ,9, are the nodal displacement of thebottom and the top of the element, respectively, in thethree directions. The Lagrangian interpolation func­tions used are cpi = cpi+9 with i'= 1, 2, ... ,9.

Substituting eqns (33) into the equilibriumequation (32) and taking into acco\lnt the definitionof W, we get

IFVIV V

2

g~II I~2

1 I2h

jFvj(a)

y

11

12

z x

13 (b)

11

12

(c)

Fig. 9. Geometry of a simple model for a single lap joint and discretization of one of the two componentsof the joint.

698 E. J. Barbero et al.

0/90 lambda= .99

90/0 lambda= .99

Fig. 10. Deformed shape of the joint for (0/90) and (90/0) stacking sequence for A. = 0.99.

(34)

with ~3k the Kronecher symbol. It is implicit that allthe quantities in the eqn (34) depend of L\u.

Then, setting K = (j - 1) x 3 + k and H = (i - 1)x 3 + h, and reordering matrices and vectors wehave

15.00 11.50 80.00 82.50 as.OO 81.50 90.00 92.50 15.00 87.50 100.0020.00 20.00

A=O.9

no-slip18.00

1&.00

14.00

12.00

10.00

8.00

8.00

4.00

2.00

no-contact

,,,,,,,,,.. _----, .. -"-"

18.00

1&.00

14.00

12.00

10.00

8.00

8.00

4.00

2.00

0.00 0.007S.00 71.50 80.00 82.50 85.00 87.50 90.00 92.50 ge.00 97.50 100.00

Fig. 11. Solid lines delimit the contact area with the value of the load inside the corresponding area. Brokenlines delimit the no-slip area with the value of the load inside the corresponding area.

Contact/friction elements for laminated composite joints 699

S(e) - r K i j d dxHK - J0 hk qJ qJ Xl 2

where U~) is the nodal displacement vector, Q~) is thenodal force vector, S<;)K is the secant symmetric partof the secant stiffness matrix, HWK is the secantunsymmetric part of the secant stiffness matrix, 1<;) issecant gap vector and the superscript (e) indicates anelemental quantity.

The numerical integration scheme used is veryimportant for the cortectdetertnination of thematrices and vectors involved in the interface finiteelement. In fact, Gauss quadrature can provide exactintegration as long as the element is fully in contact

Cb~

l ...~ -...

:J Q)V) '()

V) ~ --Cb l()l If)

Q (\)

.....,)

CJ«J

't..J

c-o\j

of fully separated and simultaneously is in completeno-slipping or complete sliping. Obviously, unless avery fine mesh is adopted, these cases are very rare.For an element in partial contact and slip, theconstitutive equations are very different for the con­tact and separated portions of the element. Therefore,the Gauss integration rule does not give a satisfactoryapproximation of the quantities defining the interfaceelement. In order to overcome this difficulty, thesimple and effective Simpson numerical integration isused, with a large number of Simpson integrationpoints. This integration technique requires some CPUtime for the evaluation of the interface elementmatrices and vectors. However, a large number ofintegration points produces a good evaluation of thecontribution to the nodes of the stiffness of theunilateral and friction element by representing cor­rectly the contact, no-contact, slip, and no-slip areas.The refined integration scheme proposed is an inex­pensive substitute for mesh refinement, which wouldbe necessary if nodal contact/friction springs wereused, or if the proposed element were to be integratedwith a three-point Gauss quadrature.

Finally, the equilibrium equation can be written inmatrix form as:

(35)

Fig. 12. Contact pressure between (0/0) plates for A= 0.0.

700

Q)~

l.. ~

:J '{)

V1 '{)

V1 ~Q)

~lQ ~

"'-J ~\}

~q)"'-J

~~

\3 ~

~~

~

~.........

~e~·ee

E. J. Barbero et al.

~ / ..... Q)CDee ",,<0.

Fig. 13. Contact pressure between (90/90) plates for A = 0.0.

s. NUMERICAL PROCEDURE

Once the stiffness matrix of the 3D plate elementand the discrete governing equation for the 3Dinterface element are defined, by assembling all theinterface and plate elements of the discretization, thefollowing nonlinear algebraic system of equations isobtained:

S(U)U = Q - H(U)U - I(U), (36)

where S(U) is the symmetric matrix of the stiffnessof the whole structure obtained by assembling thesymmetric stiffness matrices of the plate elements andthe symmetric part of the secant stiffness interfaceelement, Q is the global nodal force vector, H(U) isthe unsymmetric part of the secant stiffness matrix,I(U) is gap vector.

The nonlinear problem (36) is approached by usingan iterative procedure based on the direct iterationmethod. The solution at ith step is given in terms ofthat obtained at previous one by the formula

Vi = [S(Ui - 1)]-I{Q _ H(Ui - 1)Ui - 1_ I(Ui - 1)}.

(37)

The force vector {Q - H(Ui- 1)Ui - 1 - I(Ui

- 1)} isconstructed at element level. The matrix H(U i

-l ) and

the vector I(Vi- 1) are not assembled at the global

level.The convergence is reached when the error

Err = II Vi - Vi - 1 11/ II Vi II ~ '}', with'}' assigned admis­sible error. This kind of test may induce error in theevaluation of the approximate solution. In fact,during the iterations it may occur that although thevalue of Err is very small we are far from the solution.This happens because the displacements of the nodesduring two successive iterations are very close almosteverywhere except in only few nodes in the contactregion. Then, we propose a convergence test based ona new definition of Err = II U~ - U~-l II / II U~ II , wherep is a set of points at which the response is verysensitive to the contact and slip problem. The de­scribed iterative procedure is not based, like manyprocedures proposed in specialized literature, on thenumerical decoupling of two problems: the contactand the friction [25]. On the contrary, the numericalprocedure proposed here treats the coupled problem.

The contact/friction element has three DOF pernode on all the nodes. The 3D plate element has twoDOF per node on all the nodes except in the masternodes of the laminate, shown as solid squares inFig. 2. Each of the plates in contact has a set ofmaster nodes, but they are not necessarily located ina position adjacent to the contact/friction elements.Since the position of the master nodes through thethickness of the plate is irrelevant, the case of two

Contact/friction elements for laminated composite joints 701

plates in contact could be modeled by setting themaster nodes at the surfaces in contact. To model amore general case (e.g., double lap connection) it isrequired that contact/friction elements be connectedto nodes on both surfaces of the plate. This has beenaccomplished by implementing the capability toidentify nodes associated to interface elements, thenassigning the w-contribution of the interface elementto the stiffness of the master nodes.

6. NUMERICAL RESULTS

A simple problem is studied in the following toshow the capability of the 3D plate element, the 3Dinterface element and the numerical procedure devel­oped.Let us consider a bolted single lap joint withtwo identical components as shown in Fig. 9. Eachplate is composed by two layers with a thickness hand each layer is characterized by an orientation ofthe fibers B; (angle between the direction of the fibersand the x axis). The dimensions of the plates areI = 20 h, /1 = h, /2 = 3h, /3 = 5h and g = 0.0. Plate 1 isconstrained along the side x = 0 and the contactsurface between the two plates is represented by theshaded area in. Fig. 9(b).

'Twefr-iGtiollGGeflicie-ntsJll = /l2 = 0.5 characterizethe contact surface, and the elastic moduli of eachplate are: £1/£2 = 40, V12 = V23 = 0.25, GI2 /£2 = 0.5,where subscripts 1, 2 and 3 indicate the direction of

())

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the fiber, perpendicular to them, and perpendicular tothe plate, respectively. A vertical load Ft, distributedalong the perimeter of the rectangle a is appliedand an horizontal distributed force AFo is applied onthe plate 2 as shown in Fig. 9(a). The following valuesof the forces are used: Fv /E2h2 = 7 X 10-5 andFo= /lFt,·

Because of the symmetry of the structure, only onehalf of the structure is modeled in the finite elementanalysis with 24 elastic elements for each plate and sixinterface elements for the contact surface. The inter­face elements are integrated using 100 Simpsonequally spaced points in each direction. The meshused for of each layer in the plane x-y is shown inFig. 9(c) where /4 = 2h and /5 = h.

The following set of regularization parameters (seeFigs 4 and 6) is adopted in the computations:k I 2h/E2 =10-6

, k22h/E2 =1, k 3 =k2 and k4 =k1•

There are several constraints for the selection of thevalue of the contact stiffness k 2 • The value of k 2 islimited by the ability of the mesh size to capture thewavelength of the unilateral contact solution.Another limit is set by the difficulty in achievingconvergence of the numerical solution as the value ofk2 increases, because of the ill-conditioning of thesecant stiffness matrix at each iteration.. Finally, thecontact stiffness should allow for a certain degree oftransverse deformability of the two plates, which isabsent in the incompressible plate elements.

Fig. 14. Contact pressure distribution for joint with (0/0) stacking sequence, for A = 0.9.

702 E. J. Barbero et al.

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\.) ~

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Fig. 15. Contact pressure distribution for joint with (0/0) stacking sequence, for ). = 0.9, using reducedintegration for shear terms in the plate elements.

Four cases are considered, each characterized bythe orientation of the fibers in the two layers of eachplate. The first case has 9} = 0° and 92 = 0°, thesecond case 8} = 0° and 92 = 90°, the third case9} = 90° and 92 = 0° and the fourth case 9} = 90° and92 = 90°. For the (0/90) and (90/0) case, the defor­mation of the structure for a load parameter 2 = 0.99is presented in Fig. 10 with the displacements mag­nified 3000 times. In the second (0/90) and third(90/0) case, for the same value of the eccentrichorizontal force 2Fo, the curvature and the displace­ments of the structure are different, because of thedifferent bending-extension coupling stiffness of theplates.

For the case (0/0) the contact zone for 2 = 0.0, 0.5and 0.9 and the slip zone for 2 = 0.5 and 0.9 areshown in Fig. 11. Solid lines delimit the contact areawith the value of the load multiplier 2 inside thecorresponding area. Broken lines delimit the no-sliparea with the value of the multiplier load 2 inside thecorresponding area. The distribution of the contactzone changes by increasing the parameter A. Thecontact zone moves because of the inflection of theplates caused by the eccentricity of the horizontalforce 2Fo. The contact pressure distribution for thecase 2 = 0.0 is shown in Fig. 12. Next, in Fig. 13 thecontact pressure is plotted for the lamination (90/90)with A. = 0.0. It is worth noting the different shape of

the contact zone for the two cases % and 90/90 whenit = 0.0. In both cases the contact zone is oriented inthe direction of the fibers. For all cases studiedcomplete slip occurs exactly when 2Fo= JlF,., that isfor 2 = 1.

The contact pressure distribution for the case (0/0)is shown in Fig. 14 for 2 = 0.9. If reduced integrationof the shear terms is used, spureous modes may beexited by the penalization of the unilateral contactproblem. This may lead to lack of convergence, or ifconvergence can be achieved, to erroneous results.One such case is presented in Fig. 15 where thecontact pressure distribution for the same case andsame load condition used in Fig. 14 is shown. Theerratic behavior is caused by the use of the reducedintegration in combination with the unilateral penaltyapproach.

Since it is required to use full integration, a plateelement with minimum shear locking is required. Inthis work, this requirement is satisfied by the 3D plateelement presented by virtue of the following features.First, a nine-node Lagrangian interpolation is used.Second, the shear strain, thus the energy, is wellapproximated by a layer-wise constant distributionthrough the thickness. The accuracy of the approxi­mation to the shear energy increases as the numberof layers used to model the laminate increases. In facteven the need to use a shear correction factor k in the

Contact/friction elements for laminated composite joints 703

plate elements reduces as the number of elementsthrough the thickness of the laminate increases. Thiscan be explained for the case of an homogeneousplate as follows. An assemblage of 3DLCS elementsthrough the thickness represents the parabolic distri­bution of shear stress by a layer-wise constant ap­proximation. As the number of layers increases theerror reduces and no shear correction factor isneeded.

7. CONCLUSIONS

A combination of the 3D plate element and 3Dinterface element is presented for the analysis ofjointsof laminated composite plates. A simple and effectivenumerical procedure is presented for the solution ofthe nonlinear problem. The applicability of the modelis demonstrated by simple numerical examples. Thenumerical model presented is able to capture thesalient features of the problem even using a coarsediscretization. For the case of normal pressure, thecontact zone is oriented along the fiber direction asa result of the higher stiffness of the plates in thisdirection. When a horizontal load is applied, thecontact zone and the overall deflection of the platesiswe-llre-prese-nted including the e-ffeets of bending­extension coupling. Complete slip is predicted at thecorrect value of load according to Coulomb law. Theneed for using a fully integrated plate element isdocumented. Implementation of both the elementspresented. into a commercial finite element packageis underway, to be able to analyze more complexgeometries.

Acknowledgements-This work has been completed thanksto the financial support from Consiglio Nazionale delleRicerche in Italy, the Constructed Facilities Center at WestVirginia University, and NATO Grant CRG 910943.

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