Elasto-plastic stress analysis of aluminum metal-matrix composite laminated plates under in-plane loading

Elasto-plastic stress analysis of aluminum metal-matrixcomposite laminated plates under in-plane loading

O. Saymana,*, H. Akbuluta, C. Meric° b

aDepartment of Mechanical Engineering, Dokuz EyluÈl University, Bornova, Izmir, TurkeybDepartment of Mechanical Engineering, Celal Bayar University, Manisa, Turkey

Received 2 February 1998; accepted 3 March 1999

Abstract

The study presents an elasto-plastic stress analysis of symmetric and antisymmetric cross-ply, angle-ply laminatedmetal-matrix composite plates. Long stainless steel ®ber reinforced aluminum metal-matrix composite layer ismanufactured by using moulds under the action of 30 MPa pressure and heating up to 6008C. A laminated plate

consists of four metal-matrix layers bonded symmetrically or antisymmetrically. The ®rst-order shear deformationtheory and nine-node Lagrangian ®nite element is used. The in-plane load is increased gradually. # 2000 ElsevierScience Ltd. All rights reserved.

Keywords: Metal matrix composite; Elasto-plastic analysis; Finite element method; Laminated plate

1. Introduction

Metal-matrix composites consist of a ductile, usually

low strength matrix reinforced with elastic, brittle or

ductile and strong ®bers. The strength of the ®ber andthe ductility of the matrix provide a new material with

superior properties.

Plastic deformations and residual stresses are im-

portant in composite laminated plates. Residual stres-

ses are used to raise the yield point of the plate.Bahaei-El-Din and Dvorak [1] have investigated the

elastic±plastic behavior of symmetric metal-matrix

composite laminates for the case of in-plane mechan-ical loading. In this study, aluminum matrix is re-

inforced by boron ®bers. Metal-matrix composites

consist of a ductile, usually low strength matrix re-inforced with elastic brittle, or ductile strong ®bers

which provide a new material with superior properties

such as high strength and sti�ness, low density and re-sistance to corrosion, high creep and fatigue properties[2±5]. Karakuzu and OÈ zcan [6] have given an exact sol-

ution to the elasto-plastic stress analysis of an alumi-num metal-matrix composite beam reinforced by steel®bers.

Linear or nonlinear ®nite element method can beused to analyze the laminated composites [7±10]. Inthis study, aluminum metal-matrix composite lami-nated plates reinforced by steel ®bers are manufactured

and analyzed by using the ®nite element technique.

2. Mathematical formulation

The laminated plate of constant thickness is com-posed of orthotropic layers bonded symmetrically or

antisymmetrically about the middle surface of theplate. In the solution of this problem, the Cartesian

Computers and Structures 75 (2000) 55±63

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PII: S0045-7949(99 )00086-3

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* Corresponding author.

coordinates are used where the middle surface of the

plate coincides with the x±y plane, as shown in Fig. 1.Here we use the theory of plates with transverse

shear deformations theory which uses the assumptionthat particles of the plate originally on a line that is

normal to the undeformed middle surface remain on astraight line during deformations, but this line is notnecessarily normal to the deformed middle surface. By

using this assumption the displacement components ofa point with coordinates x, y, z for small deformationsare:

u�x,y,z� � u0�x,y� � zcx�x,y�

v�x,y,z� � v0�x,y� ÿ zcy�x,y�

w�x,y,z� � w�x,y� �1�

where u0, v0 and w are the displacements at any pointof the middle surface, and cx,cy are the rotations of

normals to midplane about the y and x axes, respect-

ively.The bending strains vary linearly through the plate

thickness, whereas shear strains are assumed to be con-stant through the thickness as

8<:exeygxy

9=; ��

@u0@x

@v0@y

@u0@y� @v0@x

�� z

��

@cx

@x

@cy

@y

@cx

@yÿ @cy

@x

��or

��exeygxy

�� e0xe0y

g0xy

�� z

��Kx

Ky

Kxy

��

�� gyzgxz

�� @w

@yÿ cy

@w

@x� cx

�� 2�

The total potential energy of a laminated plate under

static loadings is given as

P � Ub �Us � V �3�

where Ub is the strain energy of bending, Us is thestrain energy of shear and V represents potentialenergy of external forces. They are as

Fig. 1. Loading of laminated plate.

Fig. 2. The production of composite layer.

O. Sayman et al. / Computers and Structures 75 (2000) 55±6356

Ub � 1

2

�h=2ÿh=2

� �A

ÿsxex � syey � txygxy

�dA

�dz

Us � 1

2

�h=2ÿh=2

� �A

ÿtxzgxz � tyzgyz

�dA

�dz

V � ÿ�A

wp dAÿ�@R

��N

b

nu0n � �N

b

s u0s

�ds �4�

where dA � dx dy, p is the transverse loading per unitarea and �N

b

n and �Nb

s are the in-plane loads applied on

the boundary @R:

3. Production of laminated plates

The composite layer consists of stainless steel ®berand aluminum matrix. The production has been rea-

lized by using moulds which consist of upper andlower parts. Electrical resistance has been used to heatthe moulds and material which are insulated, as illus-trated in Fig. 2.

The hydraulic press has been used to obtain a press-ure of 30 MPa to the upper mould. Manufacturing sethas been heated to 6008C. In these conditions, the

yield strength of aluminum is exceeded and goodbonding between matrix and ®ber has been realized.The mechanical properties, yield points and plastic

parameters are given in Table 1. It is assumed that theyield point Z (in the z direction) is equal to the yieldpoint Y (in the y direction), the yield points of txz, tyzare equal to S. The von Mises and Tresca criteria are

used generally for isotropic materials. Huber±Misesyield criterion has been generalized by Hill for aniso-tropic metals [14]. It is more appropriate to use

Huber±Mises or Tsai±Hill failure criteria for anisotro-pic metals, since their yield points are di�erent inlongitudinal and lateral directions. The di�erence

between the numerical results for the residual stressesin the symmetric cross-ply ([08/908]2) laminated plate

obtained for Huber±Mises and Tsai±Hill criteria isfound as 0.8% as shown in Table 2. The Tsai±Hill cri-terion is used as a yield criterion [13]. Four layers have

been bonded to form a laminated plate symmetricallyor antisymmetrically by using a pressure of 30 MPaand heating up to 6008C. The stress±strain relation in

plastic region is given as

s � s0 � kenp

4. Finite element analysis

The symmetric or antisymmetric laminated plate iscomposed of four layers. A typical ®nite element for a

symmetric and an antisymmetric lamination is dividedinto eight imaginary layers for obtaining the ®nite el-ement results more accurately, as shown in Fig. 3.

The nine-node ®nite element is used in this study.The displacement ®eld can be expressed in the follow-ing matrix form:

�d� �

266666664

u0

v0

w

cx

cy

377777775 �Xni�1

26666664Ni 0 0 0 0

0 Ni 0 0 0

0 0 Ni 0 0

0 0 0 Ni 0

0 0 0 0 Ni

37777775266666664

u0

v0

w

cx

cy

377777775i

�5�

in which n is the total number of nodes and Ni is theshape function at node i.The relationship between strains and displacements

can be written in the matrix form:2666666664

e0xe0y

g0xyKx

Ky

Kxy

3777777775i

�

26666664Ni,x 0 0 0 00 Ni,y 0 0 0Ni,y Ni,x 0 0 00 0 0 Ni,x 00 0 0 0 ÿNi,y

0 0 0 Ni,y ÿNi,x

37777775i

2666664u0v0wcx

cy

3777775i

�6�

or symbolically as

febi g � �Bbi � �ui � �7�

in which i is the node number, and commas representpartial derivatives.The relationship between the transverse shear strains

Table 1

The measured mechanical properties and yield points of a

layer

Mechanical properties

E1 86 GPa

E2 74 GPa

G12 32 GPa

n12 0.30

Yield strengths and parameters

Axial strength, X 228.3 MPa

Transverse strength, Y 24.2 MPa

Shear strength, S 47.6 MPa

Hardening parameter, k 1254 MPa

Strain hardening parameter, n 0.7

O. Sayman et al. / Computers and Structures 75 (2000) 55±63 57

and displacement components can be written as

�gyzgxz

�i

��0 0 Ni,y 0 ÿNi

0 0 Ni,x Ni 0

�i

2666664u0v0wcx

cy

3777775i

�8�

or symbolically as

fesi g � �Bsi �fui g �9�

The sti�ness matrix of the plate element is obtained byusing the minimum potential energy method or theprinciple of virtual displacements [12]. Bending and

shear sti�ness matrices are

�Kb � ��A

�Bb �T �Db � �Bb � dA

�Ks � ��A

�Bs �T �Ds � �Bs � dA �10�

where

�Db � ��Aij Bij

Bij Dij

��Ds � �

�k21 A44 00 k22 A55

�

ÿAij,Bij,Dij

� � �h=2ÿh=2

Qij

ÿ1,z,z2

�dz �i,j � 1,2,6�

�A44,A55 � ��h=2ÿh=2�Q44,Q55 � dz �11�

jBbj6�45, jBsj2�45 and Db and Ds are the bending andshear parts of the material matrix, respectively. A45 is

negligible in comparison with A44 and A55 [11] andshear correction factors for rectangular cross sectionsare given as [11], k21 � k22 � 5=6:Once the nodal displacements are calculated, the

strain components of each layer can be found by usingEqs. (7) and (9); and the stress components can be cal-culated and used to check the yield state of the ma-

terial.Since the calculated stresses do not generally co-

incide with the true stresses in a nonlinear problem,

the unbalanced nodal forces and the equivalent nodalforces must be calculated. The equivalent nodal pointforces corresponding to the element stresses at each

iteration can be calculated as

fRgequivalent��

vol

�B�T�s� dA

��

vol

�Bb �T�sb � dA��

vol

�Bs �T�ss � dA �12�

When the equivalent nodal forces are known, theunbalanced nodal forces can be found by

fRgunbalanced� fRgappliedÿfRgequivalent �13�

These unbalanced nodal forces are applied for obtain-

Table 2

Residual stresses in the symmetric cross-ply, ([08/908]2), laminated square plate for 200 iterations

Orientation angle sx (MPa) sy (MPa) txy (MPa) txz (MPa) tyz (MPa)

The result for Tsai±Hill criterion

08 1.844 0.501 ÿ0.001 0.000 0.000

908 ÿ1.844 ÿ0.500 0.001 0.000 0.000

The result for Huber±Mises criterion

08 1.859 ÿ0.501 0.001 0.000 0.000

908 ÿ1.859 0.501 ÿ0.001 0.000 0.000

Fig. 3. A layered section for (a) symmetric and (b) antisymmetric lamination.


ing increments in the solution and must satisfy theconvergence tolerance in a nonlinear analysis. Thedi�erence between the plastic and elastic solution gives

the residual stresses. The residual stresses may increasethe possibility of failure of the laminated plates. In thissolution 216 nodes and 48 elements are used.

5. Numerical results and discussions

The laminated plate is assumed to be under uniformaxial in-plane loads along the rectangular edges and

the circular hole is unloaded. The laminated plates arecomposed of four orthotropic and generally orthotro-pic layers bonded in symmetric or antisymmetric form.

Loading is gradually increased up to plastic zone

which is not allowed to be very large. In the iterative

solution, the overall sti�ness matrix of the laminated

plate is the same at each loading step. The in-plane

load �Nx� is increased by 0.08 N/mm per step.

One quarter of the plate is enough to ®nd the expan-

sion of the plastic zone and the residual stresses in the

cross-ply symmetric laminated plate ([08/908]2) withouta hole. In this solution the plastic zone expands along

the layers of orientation angle which is 908 but the

other one is elastic. The in-plane load, Nx is increased

from 210.72 to 226.72 N/mm for 200 iterations. Re-

sidual stress components are given in Table 2.

The expansion of the plastic zone in the symmetric

and antisymmetric cross-ply, ([08/908]2), laminated

square plate with a hole under in-plane loading is illus-

Fig. 4. Expansion of plastic zone in cross-ply, ([08/908]2), laminated plates: (a) symmetric and (b) antisymmetric; and its distribution

across the cross-section for 800 iterations.

Fig. 5. Residual stress components sx (MPa) along AB for cross-ply ([08/908]2): (a) symmetric and (b) antisymmetric laminated

plates.


trated in Fig. 4, for simply supported condition. In the

layer of 908 orientation, for 400 iteration the residual

stress component sx along AB is given in Fig. 5.

The e�ect of orientation angle on the expansion

of plastic zone is presented in Fig. 6, for ([308/ÿ308]2) symmetric and antisymmetric angle-ply lami-

nated plates with simple edges. When the external

force Nx reaches 140.96 and 136.79 N/mm, yielding

occurs in the symmetric and antisymmetric layers, re-

spectively, and when it is further increased incremen-

tally the plastic zone expands around the hole. The

expansion of plastic zone in the layer of 308 orien-

tation angle is slightly di�erent from that of the layer

of ÿ308 orientation angle in each case.

The e�ect of orientation angles on the expansion

of plastic zone is shown in Fig. 7, for ([458/ÿ458]2)symmetric and antisymmetric laminated plates with

simple edges. When we increase the external force

gradually, the plastic zone expands around the hole.

The expansion of plastic zone in ([608/ÿ608]2)symmetric and antisymmetric laminated plates with

simple edges is shown in Fig. 8. When the external

force is increased gradually, the plastic zone spreads.

Plastic zones for the layers of orientation angles 608and ÿ608 are slightly di�erent in each case.

The yield points for symmetric and antisymmetric

laminates are given in Table 3. It is seen that the

in-plane load at the yield points for the symmetric

laminates is higher than that for the antisymmetric

laminates; because the antisymmetric laminates may

produce bending moments, and these moments may

cause yielding of the plates at lower external forces.

Elasto-plastic, elastic and residual stress com-

ponents for the symmetric cross-ply laminated plate

([08/908]2), at node A are given in Table 4. As seen

from this table the residual stress components are com-

pressive and tensile in the layers of 908 and 08 orien-

tation angles, respectively. When we increase the

iteration numbers the residual stress components

become greater.

The residual stress components for 800 iterations in

simply supported symmetric angle-ply, ([308/ÿ308]2),

Fig. 6. Expansion of plastic zone in (a) symmetric and (b) antisymmetric ([308/ÿ308]2) laminated plates; and its distribution across

the cross-section for 800 iterations.






Table 3

Yield points in symmetric and antisymmetric laminated plates

([08/908]2) ([308/308]2) ([458/458]2) ([608/608]2)

Symmetric Nx (N/mm) 80.56 140.96 109.76 88.16

Antisymmetric Nx (N/mm) 72.68 136.89 107.12 87.36

Table 4

Elasto-plastic, elastic and residual stress components for symmetric cross-ply laminated plate ([08/908]2), at node A for 200, 400

and 800 iterations

Iteration numbers Orientation angle sx (MPa) sy (MPa) txy (MPa) tyz (MPa) txz (MPa)

Elasto-plastic solution

200 908 24.213 2.793 ÿ0.675 0.000 0.000

08 37.223 5.729 ÿ0.6981 0.000 0.000

400 908 24.211 2.326 ÿ0.822 0.000 0.000

08 44.793 7.255 ÿ0.832 0.000 0.000

800 908 24.208 1.377 ÿ1.083 0.000 0.000

08 61.115 10.591 ÿ1.10 0.000 0.000

Elastic solution

200 908 29.248 3.901 ÿ0.684 0.000 0.000

08 34.212 4.638 ÿ0.684 0.000 0.000

400 908 34.260 4.570 ÿ0.801 0.000 0.000

08 40.075 5.433 ÿ0.801 0.000 0.000

800 908 44.285 5.907 ÿ1.036 0.000 0.000

08 51.801 7.023 ÿ1.036 0.000 0.000

Residual stresses

200 908 ÿ5.035 ÿ1.108 0.009 0.000 0.000

08 3.011 1.091 0.003 0.000 0.000

400 908 ÿ10.049 ÿ2.244 ÿ0.021 0.000 0.000

08 4.718 1.822 ÿ0.030 0.000 0.000

800 908 ÿ20.077 ÿ4.530 ÿ0.048 0.000 0.000

08 9.314 3.568 ÿ0.066 0.000 0.000


([458/ÿ458]2) and ([608/ÿ608]2) laminated plates are

given in Table 5. It is seen from this table that as weincrease the orientation angles in the symmetric angle-ply laminated plates the sx residual stress component

becomes greater. The maximum absolute value of thestress components in the plate is denoted in the table.The residual stress components for 800 iterations in

antisymmetric angle-ply, ([308/ÿ308]2), ([458/ÿ458]2)and ([608/ÿ608]2) and cross-ply ([08/908]2) laminatedplates are given in Table 6. It is seen that as we

increase the orientation angles in the antisymmetricangle-ply laminated plates the sx residual stress com-ponent becomes greater. The maximum intensity of thestress components in the plate is given in the table.

6. Conclusions

Elasto-plastic stress analysis has been carried out byusing the ®rst-order shear deformation theory in alu-

minium±steel ®ber laminated plates. The expansion ofthe plastic zone and residual stresses are obtained insymmetric and antisymmetric cross-ply and angle-ply

composite laminated plates.

1. In each case yielding occurs around the hole.2. For symmetric and antisymmetric cross-ply lami-

nates ([08/908]2), yielding occurs in the layers of 908orientation angle and the residual stress �Nx� iscompressive and tensile in layers of 908 and 08orientation angles, respectively, without a hole.

3. The orientation angle y a�ects the yield points oflaminated plates.

4. The yield point of symmetric laminated plates is

higher than that of the yield point of antisymmetriclaminated plates.

5. Load carrying capacity of the laminated plate can

be increased by means of residual stresses.

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Table 5

Residual stress components in the symmetric angle-ply laminated plates for 800 iterations

Laminated plate sx (MPa) sy (MPa) txy (MPa) tyz (MPa) txz (MPa)

([308/ÿ308]2) 308 ÿ4.622 ÿ5.330 2.960 0.000 0.000

ÿ308 ÿ4.012 ÿ4.580 ÿ2.551 0.000 0.000

([458/ÿ458]2) 458 ÿ8.684 ÿ6.436 4.247 0.000 0.000

ÿ458 ÿ7.924 ÿ5.827 ÿ3.818 0.000 0.000

([608/ÿ608]2) 608 ÿ12.292 ÿ5.628 4.488 0.000 0.000

ÿ608 ÿ11.630 ÿ4.900 ÿ3.967 0.000 0.000

Table 6

Residual stress components in the antisymmetric angle-ply and cross-ply laminated plates for 800 iterations

Laminated plate sx (MPa) sy (MPa) txy (MPa) tyz (MPa) txz (MPa)

([08/908]2) 908 ÿ18.806 ÿ3.954 ÿ0.111 ÿ0.128 ÿ0.12708 9.175 3.604 ÿ0.017 ÿ0.202 ÿ0.251

([308/ÿ308]2) 308 ÿ4.710 ÿ5.212 3.054 ÿ0.100 0.058

ÿ308 ÿ3.977 ÿ4.561 ÿ2.487 0.085 0.046

([458/ÿ458]2) 458 ÿ8.868 ÿ6.031 4.386 ÿ0.162 0.076

ÿ458 ÿ7.808 ÿ5.965 ÿ3.727 0.143 0.068

([608/ÿ608]2) 608 ÿ12.580 ÿ4.478 4.524 ÿ0.183 0.086

ÿ608 ÿ11.580 ÿ5.203 ÿ3.899 0.144 0.0622


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Elasto-plastic stress analysis of aluminum metal-matrix composite laminated plates under in-plane loading