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Elastic–plastic stress analysis and expansion of plastic zone in clamped and simply supported aluminum metal–matrix laminated plates
Composite Structures 49 (2000) 9±19
www.elsevier.com/locate/compstruc
Elastic±plastic stress analysis and expansion of plastic zone in clamped and simply supported aluminum metal±matrix laminated plates Cesim Atas *, Onur Sayman Department of Mechanical Engineering, Dokuz Eyl ul University, Bornova, Izmir, Turkey
Abstract An elastic±plastic stress analysis and the expansion of plastic zone in layers of stainless steel ®ber-reinforced aluminum metal± matrix laminated plates are studied by using Finite Element Method and First-order shear deformation theory for small deformations. The plate is meshed into 64 elements and 289 nodes with simply supported or clamped boundary conditions. Laminated plates of constant thickness are formed by stacking four layers bonded symmetrically or antisymmetrically. It is assumed that the laminated plates are subjected to transverse uniform loads. Loading is gradually increased from yield point of the plate as 0.0001 MPa at each load step. Load steps are chosen as 100, 150 and 200. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Elastic±plastic stress analysis; Aluminium metal±matrix composite; Laminated plate; Residual stress; Plastic zone; Finite element method
1. Introduction Composite materials are those formed by combining more than one bonded material, each with dierent structural properties. The main advantages of composite materials are the potential for a high ratio of stiness to weight, strength, corrosion resistance, thermal properties, wear resistance, and fatigue life. Metal±matrix composites have been used in many structures and commercial applications for a long time, because of their high speci®c stiness and high temperature performance. In recent times, many manufacturing techniques have been developed. The objective is usually to make a material which is strong and sti, often with a low density. Elastic±plastic and residual stresses are very important in failure analysis of metal±matrix laminated plates. Permanent deformations and residual stresses occur in laminated plates when the yield point of the laminate is exceeded. These residual stresses can be used to raise the yield points of the plates. Chou et al. [1] studied the ®ber-reinforced metal± matrix composites involving fabrication methods, me*
Corresponding author.
chanical properties, secondary working techniques and interfaces. Kang and Kang [2] reviewed the work on short ®ber-reinforced aluminum composite material, fabricated by the stirring method and extruded at high temperatures at various extrusion ratios. In that study, the tensile strength of the hot extrusion billet was experimentally determined for dierent extrusion ratios, and compared with the theoretically calculated strength. Metal±matrix composites provide a relatively new way of strengthening metals [3]. Kwon and Yoon [4] examined the eect of SiCp and Al2 O3 on the high temperature ¯ow stress of particulate-reinforced aluminum composites. Nan and Clarke [5] investigated the in¯uence of particle size and particle fracture on the elastic / plastic deformation of metal±matrix composites. In that paper, a methodology was introduced for calculating the deformation response of particulate-reinforced metal± matrix composites in terms of an eective medium approach combined with the essential features of dislocation plasticity. A feature of that methodology is that the eect of particle cracking during deformation can be incorporated quantitatively. The in¯uence of particle volume fraction, shape and aspect ratio on the behavior of particle-reinforced metal±matrix composites was investigated through numerical modeling, which was performed using axisymmetric unit cell models, at high rates of strain [6]. Dutta et al. [7] worked on the particle
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C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
redistribution and matrix micro-structure development during hot extrusion of cast SiCp -reinforced aluminum alloy matrix composites and the in¯uence of process parameters on these two key micro-structural features. The thermal performance of six commercially available particulate-reinforced aluminum-based metal±matrix composites [8] and superplasticity in aluminum-based metal±matrix composites [9] were investigated. Yeh and Krempl [10] examined the in¯uence of cool-down temperature histories on the residual stresses in ®brous metal±matrix composites. Lorentzen and Clarke [11] measured thermomechanically induced residual strains in Al/SiCp metal±matrix composites by using neutron diraction at room and elevated temperatures to monitor the eects of in situ uniaxial plastic deformations. Lee and Sun [12] reviewed the work on the ¯exural behavior of clamped elliptic composite laminates for both symmetric and antisymmetric laminates. In that work, closed form solutions were discussed for laminates under uniform transverse loading. The apparent bending isotropy was identi®ed for clamped elliptic laminates. Vlot and Ingen [13] studied delamination resistance of post-stretched ®ber metal laminates consisting of bonded alternating thin aluminum and ®ber / epoxy layers. Ananth et al. [14] investigated the application of push out test to characterize the mechanical behavior of interfaces in metallic and intermetallic matrix composites. In that study, ®nite element technique was used. In addition, the eect of dierent material and testing variables on the experimental data was de®ned. Pomies and Carlsson [15] carried out micro-mechanical ®nite element analysis of rigidity and strength of transverse tension loaded continuous ®ber composites and compared their results with experimental data for dry and wet glass±epoxy, carbon±epoxy composites. In that paper, transverse tensile strength predictions employed the combined mechanical and residual stress states at the ®ber±matrix interface and in the matrix, in conjunction with debonding and matrix failure criteria. Bahei-El-Din and Dvorak [16] investigated the elastic±plastic behavior of symmetric metal±matrix composite laminates for the case of in-plane mechanical loading. Boron ®ber-reinforced aluminum metal±matrix laminates were used in their work. Linear or non-linear ®nite element method can be used to analyze the laminated composites [17± 19]. Owen [20] studied elasto-plastic ®nite element analysis of anisotropic plates and shells. Theo [21] investigated the stresses around rectangular holes in orthotropic plates. It was shown by Chen and Beraun [22] that a non-quadratic plasticity model is suitable for metal±matrix composites. Their work based on the assumption that uniform dilatation has negligible eect on the yield point. A high-order yield function of non-equal stress exponents in conjunction with a non-associated ¯ow rule was proposed to formulate 3-D plastic constitutive equations for ®brous metal±matrix composites.
It was also shown that the uniform dilatation assumption is adequate for other ®ber composites. Karakuzu and Ozcan [23] gave an exact solution to the elastoplastic stress analysis of an aluminum metal±matrix composite beam reinforced by steel ®bers. In order to manufacture continuous stainless steel reinforced aluminum matrix composites, the die casting process was used. Elasto-plastic ®nite element analysis of orthotropic rotating discs with holes in steel-aluminum composite was carried out by Karakuzu and Sayman [24]. Karakuzu et al. [25] de®ned the elastic±plastic ®nite element analysis of metal±matrix plates with edge notches. Sayman [26] investigated elasto-plastic stress analysis in stainless-steel ®ber-reinforced aluminum metal±matrix laminated simply supported plate which is subjected to transverse loading by using ®nite element technique. Also, the expansion of the plastic zone and residual stresses for dierent oriented angle-ply and cross-ply laminated plates for small deformations were determined. In this paper, stainless steel ®ber-reinforced aluminum metal±matrix laminated simply supported or clamped plates are loaded transversely. Elastic±plastic, elastic and residual stresses and the expansion of the plastic zone are carried out by using ®nite element method for small deformations. In the solution, the Tsai-Hill theory is used as a yield criterion and a ninenode isoparametric quadrilateral element is used for the elastic±plastic analysis of the laminated plate. The load steps are chosen as 100, 150 and 200.
2. Mathematical formulation The stainless steel ®ber-reinforced aluminum metal± matrix laminated plates of constant thickness are formed by stacking four layers bonded symmetrically or antisymmetrically about the middle surface. Geometry of the plate in Cartesian coordinates and boundary conditions are illustrated in Figs. 1 and 2, respectively. Considering transverse shear deformations in the solution of laminated plate elements, the constitutive equations for an orthotropic layer are given as
Fig. 1. Geometry of the plate in Cartesian coordinates.
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
1 Ub 2
Z
ÿh2
Z
1 Us 2
h 2
h 2
ÿh2
2 3 Z ÿ 4 rx ex ry ey sxy cxy dA5dz; A
2 3 Z ÿ 4 sxz cxz syz cyz dA5dz;
Fig. 2. Boundary conditions for simply supported (a) and clamped (b) laminated plates.
8 9 2 38 Q11 Q12 Q16 > > = < rx > < eX 6 7 ry 4 Q21 Q22 Q26 5 eY > > ; : > : sxy Q16 Q26 Q66 cXY sYZ Q44 Q45 cYZ ; sXZ Q45 Q55 cXZ
9 > = > ;
; 2:1
u x; y; z u0 x; y zwx x; y; v x; y; z v0 x; y ÿ zwy x; y;
2:2
w x; y; z w0 x; y; where u0 ; v0 and w0 denote the displacements of a point on the middle surface, and wx ; wy denote rotations about the normal to the y- and x-axes, respectively. The bending strains vary linearly through the plate thickness and are given by the curvatures of the plate, whereas the transverse shear strains are assumed to be constant through the thickness. To obtain the equilibrium equations of the element, the total potential energy of a laminated plate under static loading is given as P Ub Us V ;
Mx
Nx
6 4 Ny (
Nxy Qx
3
7 My 5 )
Qy
Z b b N n u0n N s u0s ds; oR
Mxy Z
h 2
ÿh2
"
Z
2 h 2
ÿh2
sxz syz
rx
3
6 7 4 ry 5 1; z dz;
#
sxy
2:5
dz:
Equilibrium requires that P is stationary, i.e., dP 0, which may be regarded as the principle of virtual displacement for the plate element [28].
3. Finite element model In order to obtain the yield points of laminates, the spread of plastic zone and the residual stresses in the laminated plates, a nine-node Lagrangian ®nite element was used. The plate is meshed into 64 elements and 289 nodes with both simply supported or clamped boundary conditions. Laminated plates of constant thickness are formed by stacking four layers bonded symmetrically or antisymmetrically. Bending and shear stiness matrices of the plate are obtained by using the minimum potential energy principle as Z Kb
2:3
where Ub is the strain energy of bending, Us the strain energy of shear and V represents the potential energy of external forces. They are as follows:
wp dA ÿ
where dA dx dy; p is the transverse load per unit area, and Nnb and Nsb are the in-plane loads applied to the boundary oR. The resultant forces, moments and shear forces per unit length of the cross section of the laminated plate are obtained by integration of the stresses through the thickness as [27] 2
where Qij are the transformed reduced stinesses given in terms of the orientation angle and the engineering constants of the material. The ®rst-order shear deformation theory (FSDT) was used in this work. It is assumed that the particles of the plate, originally on a line that is normal to the undeformed mid-plane, remain on a straight line during deformation; whereas this line is not necessarily normal to the deformed middle surface in the theory. This theory is based on the displacement ®eld, for small deformations,
A
2:4
A
Z V ÿ
11
Z Ks where
A
A
Bb T Db Bb dA; 3:1 T
Bs Ds Bs dA;
12
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Aij Bij Db ; Bij Dij " # k12 A44 0 ; Ds 0 k22 A55 Z h2 Qij 1; z; z2 dz Aij ; Bij ; Dij Z A44 ; A55
ÿh2
h 2
ÿh2
3:2 i; j 1; 2; 6;
Q44 ; Q55 dz:
in the plastic region, therefore, PoissonÕs ratio in the plastic region is taken as 0.5. Fiber volume fraction in the plastic region for small deformations can be taken as 0.15 the same as in the elastic region. The calculated stresses do not generally coincide with the true stresses in a non-linear solution; because of this reason external forces are applied incrementally. Therefore, the equivalent nodal forces and the unbalanced nodal forces must be found; for each load step, Z Z T T fRgequivalent Bb rb dA Bs rs dA; 3:7 vol
vol
Db and Ds are the bending and shear parts of the material matrix, respectively. k12 and k22 denote the shear correction factors. For rectangular cross sections they are given as k12 k22 5=6 [29]. In this solution, Tsai-Hill theory is used as a yield criterion due to the same yield values in tension and compression in this aluminum metal±matrix composite. In the elastic±plastic solution, the tangential modular matrix is used instead of the elasticity matrix. The yield function f is
These unbalanced nodal forces represent the increments in the solution and must satisfy the convergence tolerance in a non-linear analysis. A widely used iteration procedure is the modi®ed Newton iteration method [28]. This iterative solution can be derived from the Newton± Raphson method. The equations used in the modi®ed Newton iteration are, for i 1; 2; 3; 4; . . .
f re ÿ r0 0;
t
3:3
fRgunbalanced fRgapplied ÿ fRgequivalent :
iÿ1
KDU i f Rgunbalanced tDt fRgapplied ÿ tDt fRgequivalent ;
where re and r0 are the eective stress and yield stress, respectively. The tangential modular matrix is obtained as [30]
tDt
Dt DI ÿ aaT D= H aT Da;
with the initial condition,
3:4
where H is the local slope of the uniaxial stress / plastic strain curve and can be determined experimentally and D, I and a are the elastic matrices which are used instead of Db and Ds , unit matrix and ¯ow vector, respectively. The ¯ow vector a is found by using the Prandtl±Reuss equations as a of =or:
3:5
The stress and strain curve of the composite layer is obtained in the principal material, ®ber, direction x. It is given by the Ludwik equation as r r0 kenp ;
U i tDt U iÿ1 DU i 3:9
tDt
U 0 t U ;
tDt
0 Req t Req ;
3:10
where t, Dt, U and K are the time, time interval, overall displacement vector and the stiness matrix, respectively. These equations were obtained by linearizing the response of the ®nite element system about the conditions at time t. In each load step, the unbalanced nodal forces are calculated, and the iteration is continued until the unbalanced nodal forces (load vector) or the displacement increments, DU i are suciently small.
3:6
where k and n are the plasticity constant and strain hardening exponent, respectively, as given in Table 1. It is assumed that the composite material is incompressible Table 1 Mechanical properties and yield points of a layer E1 (GPa) E2 (GPa) G12 (GPa) Axial yield value, X (MPa) Transverse yield value, Y (MPa) Shear yield value, S (MPa) Plasticity constant, k (MPa) Strain hardening exponent, n PoissonÕs ratio, m12 Fiber volume fraction, Vf
3:8
85 74 30 230.0 24.0 48.9 1254 0.7 0.30 0.15
4. Fabrication of laminated plates A layer of stainless steel ®ber-reinforced aluminum metal±matrix laminate was manufactured by setting under 30 MPa and at 600°C temperature [24]. In order to fabricate laminates, moulds, which were heated by electrical resistance and insulated by glass-®bers, were used. Under these conditions the steel ®ber and the aluminum matrix were bonded, provided aluminum exceeds the yield strength. Tensile test specimens were obtained from this layer and loaded in principal material directions by Instron tensile machine to get mechanical properties and yield strengths of a layer. Strain-gauges were used during these tests. Laboratory test results are given in Table 1.
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
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Table 2 The uniform transverse force at the yield points of the simply supported and clamped laminated platesa
a
0°=90°2
30°= ÿ 30°2
45°= ÿ 45°2
60°= ÿ 60°2
Simply supported S (MPa) AS (MPa)
0.0407 0.0401
0.0408 0.0406
0.0407 0.0405
0.0408 0.0406
Clamped S (MPa) AS (MPa)
0.0581 0.0572
0.0690 0.0685
0.0857 0.0847
0.0690 0.0685
S: Symmetric laminate, AS: Antisymmetric laminate.
5. Results and discussion
5.1. Simply supported laminates
Symmetrically or antisymmetrically oriented laminated plates of four layers, subjected to uniform transverse loads, were used to get an elastic±plastic solution and the spread of plastic region with boundary conditions of simply supported or clamped edges. The transverse load was increased, 0.0001 MPa at each step, from the yield point for 100, 150 and 200 load steps, incrementally. The uniform transverse force at the yield points of both simply supported and clamped laminated plates is given in Table 2. It is clearly seen that the transverse uniform loads which cause the beginning of the yield for the symmetrically laminated plates are higher than those for the antisymmetrically oriented laminates for both simply supported and clamped plates. For both boundary conditions given above, the expansion of the plastic zone and intensity of stress components at the upper and lower surfaces of symmetric oriented laminates are the same, so they are given in the tables and ®gures only for upper surfaces. However, they are generally dierent from each other at the upper and lower surfaces in antisymmetric stacking sequences. Symmetric or antisymmetric laminated plates of four layers are oriented as cross-ply laminated plate, 0°=90°2 , and angle-ply laminated plates, 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 .
The expansion of the plastic zone for symmetric cross-ply, 0°=90°2 , laminated plate is shown in Fig. 3. Elastic±plastic, elastic and residual stresses at the midpoint (at the center of the plate, denoted by A) of upper surface of the symmetric cross-ply laminated plate for 200 load steps are given in Table 3. They are given in Table 4 for antisymmetric orientation of 0°=90° laminated plate at the upper and lower surfaces. In Table 5, the residual stress components for symmetric angle-ply, laminated 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 plates are given. The distribution of the plastic regions
Table 3 Elastic±plastic, elastic and residual stress components for symmetric cross-ply 0°=90°2 laminated plate at the center of the laminate (node A) for 200 load steps in simply supported plates rx (MPa) Elastic± plastic stresses Elastic stresses Residual stresses
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
39.587
23.858
)0.049
)0.035
)0.079
40.025
35.776
)0.039
)0.061
)0.072
)0.438
)11.917
)0.010
0.026
)0.007
Table 4 Elastic±plastic, elastic and residual stress components for antisymmetric cross-ply, 0°=90°2 , laminated plate at the center of the laminate (node A) for 200 load steps in simply supported platesa sxy (MPa)
syz (MPa)
sxz (MPa)
Elastic±plastic stresses 0° 37.140 23.888 90° )23.889 )37.050
)0.034 0.054
)0.065 )0.065
)0.065 )0.065
Elastic stresses 0° 39.003 90° )35.990
36.003 )38.987
)0.035 0.043
)0.066 )0.066
)0.066 )0.066
Residual stresses 0° )1.863 90° 12.101
)12.114 1.938
0.000 0.011
0.001 0.001
0.001 0.001
rx (MPa)
Fig. 3. The expansion of the plastic zone for symmetric cross-ply 0°=90°2 simply supported laminated plate at the upper surface.
a
ry (MPa)
0°: For upper surface, 90°: For lower surface.
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C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Table 5 Residual stress components for symmetric angle-ply laminated plates at the center of the laminate (node A) for 200 load steps in simply supported plates 30°= ÿ 30°2 45°= ÿ 45°2 60°= ÿ 60°2
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
)2.226 )6.344 )10.226
)10.234 )6.359 )2.236
4.262 5.554 4.268
)0.019 )0.034 )0.004
)0.004 )0.034 )0.019
Fig. 4. The expansion of the plastic zone for symmetric angle-ply 30°= ÿ 30°2 simply supported laminated plate at the upper surface.
Fig. 6. The expansion of the plastic zone for symmetric angle-ply 60°= ÿ 60°2 simply supported laminated plate at the upper surface.
laminates for the same orientation angle at the upper surfaces. However, the plastic region expands along diagonals of the square plates in dierent directions at the upper and lower surfaces for antisymmetric laminated plates such as 30°= ÿ 30°2 angle-ply laminated plate, as shown in Fig. 7. 5.2. Clamped laminates
Fig. 5. The expansion of the plastic zone for symmetric angle-ply 45°= ÿ 45°2 simply supported laminated plate at the upper surface.
for symmetric angle-ply, 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 , laminated plates at the upper surface are given in Figs. 4, 5 and 6, respectively. For antisymmetric angle-ply stacking sequences, the residual stress components are given in Table 6 for upper and lower surfaces. There is a little dierence between the expansion of the plastic region in symmetric and antisymmetric
Residual stress components for symmetric cross-ply and angle-ply laminated plates at nodes A, B and C for 200 load steps are given in Table 7. Here, nodes of A, B and C denote the center of the plate and mid-points of the edges in axial and transverse directions, respectively, as shown in Fig. 2. The spread of the plastic zone for symmetric cross-ply 0°=90°2 and angle-ply 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 laminated plates is given in Figs. 8, 9, 10 and 11, respectively. The residual stress components for antisymmetric cross-ply laminated plate at nodes A, B and C of upper and lower surfaces for 200 load steps are given in Table 8 and for antisymmetric angle-ply laminated plates, they are given in Tables 9 and 10. The expansion of the plastic zone for antisymmetric cross-ply 0°=90°2 and angle-ply 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 laminated plates at upper and lower surfaces are given in Figs. 12, 13, 14 and 15, respectively. It is clearly seen from these
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
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Table 6 Residual stress components for antisymmetric angle-ply laminated plates at the center of the laminate (node A) for 200 load steps in simply supported plates Laminated Plates
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
At the upper surface 30°= ÿ 30°2 45°= ÿ 45°2 60°= ÿ 60°2
)2.493 )6.366 )10.089
)10.079 )6.344 )2.514
4.351 5.628 4.340
)0.006 )0.027 )0.010
)0.010 )0.048 )0.005
At the lower surface 30°= ÿ 30°2 45°= ÿ 45°2 60°= ÿ 60°2
2.435 6.239 10.037
10.031 6.457 2.455
4.369 5.574 4.360
0.005 )0.030 )0.051
)0.050 )0.051 0.006
Fig. 7. The expansion of the plastic zone for antisymmetric angle-ply 30°= ÿ 30°2 simply supported laminated plate at the upper (a) and lower (b) surfaces.
Table 7 Residual stress components for symmetric cross-ply and angle-ply laminated plates at nodes A, B and C for 200 load steps in clamped plates Stacking sequence
Nodes
rx (MPa)
ry (MPa)
sxy (MPa)
0°=90°2
A B C
0.285 )0.363 2.469
0.066 )0.095 8.269
0.000 0.000 )0.011
0.003 0.000 0.890
)0.004 )0.021 0.000
30°= ÿ 30°2
A B C
)0.177 )0.215 3.231
)0.663 )0.060 5.299
0.254 )0.007 )2.199
)0.015 0.000 0.449
)0.007 )0.012 )0.020
45°= ÿ 45°2
A B C
)3.340 2.945 3.408
)3.345 3.408 2.953
2.217 )1.993 )1.995
)0.010 )0.019 0.343
)0.009 0.343 )0.019
60°= ÿ 60°2
A B C
)0.663 5.292 )0.060
)0.178 3.232 )0.215
0.254 )2.200 )0.007
)0.007 )0.020 )0.012
)0.015 0.449 0.000
®gures that the expansions of the plastic zone at the upper and lower surfaces are dierent from each other for antisymmetric laminates, although they are the same for symmetrically oriented laminates.
syz (MPa)
sxz (MPa)
6. Conclusions In order to carry out the elastic±plastic stress analysis in clamped or simply supported steel ®ber-reinforced
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C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Fig. 8. The expansion of the plastic zone for symmetric cross-ply 0°=90°2 clamped laminated plate.
Fig. 11. The expansion of the plastic zone for symmetric angle-ply 60°= ÿ 60°2 clamped laminated plate.
Table 8 Residual stress components for antisymmetric cross-ply, (0°/90°)2 , laminated plate at nodes A, B and C for 200 load steps in clamped plates Nodes
Fig. 9. The expansion of the plastic zone for symmetric angle-ply 30°= ÿ 30°2 clamped laminated plate.
Fig. 10. The expansion of the plastic zone for symmetric angle-ply 45°= ÿ 45°2 clamped laminated plate.
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
At the upper surface A 0.105 B 0.134 C 2.509
0.274 0.045 8.439
0.000 0.001 )0.014
)0.001 )0.001 0.907
)0.001 )0.266 )0.001
At the lower surface A )0.274 B )8.439 C )0.045
)0.105 )2.509 )0.134
0.000 0.020 )0.001
)0.001 )0.001 )0.266
)0.001 0.907 )0.010
aluminum metal±matrix composite laminated plates under transverse loading, the ®rst order shear deformation theory was used. The expansion of the plastic zone and elastic±plastic, elastic, and residual stress components are investigated in symmetric and antisymmetric cross-ply and angle-ply laminated plates. Conclusions obtained from this investigation can be summarized as follows: For both clamped and simply supported plates: · The yield points in symmetric cross-ply and angle-ply laminated plates are higher than those in antisymmetric cross-ply and angle-ply laminates. · The yield point for cross-ply laminated plates is always smaller than that for angle-ply laminated plates. · The yield loads in clamped laminated plates are higher than those of the simply supported plates (almost between the range of 42% and 110% , depending on the orientation angle). · The expansion of the plastic zone and the intensity of the stress components are the same at the upper and
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
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Table 9 Residual stress components for antisymmetric angle-ply laminated plates at nodes A, B and C of upper surface for 200 load steps in clamped plates Stacking sequence
Nodes
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
30°= ÿ 30°2
A B C
)0.149 )0.221 3.409
)0.590 )0.063 5.534
0.224 )0.021 )1.980
)0.011 0.000 0.451
)0.006 )0.013 )0.001
45°= ÿ 45°2
A B C
)3.146 3.145 3.628
)3.151 3.628 3.153
2.199 )1.823 )1.825
)0.012 )0.002 0.341
)0.012 0.341 )0.002
60°= ÿ 60°2
A B C
)0.590 5.527 )0.062
)0.150 3.410 )0.221
0.224 )1.981 )0.021
)0.006 )0.001 )0.013
)0.010 0.451 0.000
Table 10 Residual stress components for antisymmetric angle-ply laminated plates at nodes A, B and C of lower surface for 200 load steps in clamped plates Stacking sequence
Nodes
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
30°= ÿ 30°2
A B C
0.149 0.225 )3.419
0.583 0.064 )5.589
0.227 )0.020 )1.985
)0.002 0.000 0.449
)0.018 )0.013 )0.001
45°= ÿ 45°2
A B C
3.127 )3.174 )3.649
3.130 )3.649 )3.182
2.202 )1.842 )1.843
)0.013 )0.002 0.342
)0.013 0.342 )0.002
60°= ÿ 60°2
A B C
0.583 )5.581 0.063
0.148 )3.420 0.225
0.227 )1.986 )0.020
)0.018 )0.001 )0.013
)0.002 0.449 0.000
Fig. 12. The expansion of the plastic zone for antisymmetric cross-ply 0°=90°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
lower surfaces in symmetric laminated plates. However, they are dierent from each other at upper and lower surfaces in antisymmetrically oriented plates. · The intensity of the residual stress components, because of the major dierences in material properties in the principal material directions, is maximum for symmetric and antisymmetric cross-ply 0°=90°2
laminated plate in comparison with the other angleply laminated plates. For simply supported plates: · In simply supported laminates, plastic zone expands along the ®ber directions. · By increasing the load steps, yielding begins also at the corners which are in the transverse direction to the ®bers because of great shear stresses.
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C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Fig. 13. The expansion of the plastic zone for antisymmetric angle-ply 30°= ÿ 30°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
Fig. 14. The expansion of the plastic zone for antisymmetric angle-ply 45°= ÿ 45°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
Fig. 15. The expansion of the plastic zone for antisymmetric angle-ply 60°= ÿ 60°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
For clamped plates: · Yielding begins at the center and / or at the mid-point of the edges of the laminated plates related to the orientation angle. At the corner of the plate yielding does not occur. · The intensity of the residual stress components at the edges, at which yielding occurs, reaches the greatest values and are too large in comparison with other edges and nodes. · In all the stacking sequences of 45° orientation angle, yielding starts both at all edges and the center of the laminated plate. References [1] Chou TW, Kelly A, Okura A. Fibre-reinforced metal±matrix composites. J Composite Mater 1985;16:187±206. [2] Kang CG, Kang SS. Eect of extrusion on ®bre orientation and breakage of aluminar short ®bre composites. J Composite Mater 1994;28:155±65. Onel [3] C ocen U, K, Ozdemir I. Microstructures and age hardenability of Al-5% Si-0.2% Mg based composites reinforced with particulate SiC. Composite Sci Technol 1997;57:801±8. [4] Kwon H-C, Yoon E-P. Eect of SiCp and Al2 O3 on the high temperature ¯ow stress of particulate reinforced aluminium composites. J Mater Sci Lett 1996;15:1205±1211. [5] Nan CW, Clarke DR. The in¯uence of particle size and particle fracture on the elastic / plastic deformation of metal matrix composites. Acta Mater 1996;44(9):3801±11. [6] Li Y, Ramesh KT. The in¯uence of particle volume fraction, shape and aspect ratio on the behavior of particle-reinforced metal±matrix composites at high rates of strain. Acta Mater 1998;46(16):5633±46. [7] Dutta B, Samajdar I, Surappa MK. Particle redistribution and matrix microstructure evolution during hot extrusion of cast SiCp reinforced aluminum alloy matrix composites. Mater Sci Technol 1998;14:36±46. [8] Pitcher PD, Shakeshe AJ, Lord JD. Aluminium based metal matrix composites for improved elevated temperature performance. Mater Sci Technol 1998;14:1015±23. [9] Pilling J. Superplasticity in aluminium base metal matrix composites. Scripta Metallurgica 1989;23:1375±80. [10] Yeh NM, Krempl E. The in¯uence of cool-down temperature histories on the residual stresses in ®brous metal±matrix composites. J Composite Mater 1993;27:973±95. [11] Lorentzen T, Clarke AP. Thermomechanically induced residual strains in Al/SiCp metal matrix composites. Composites Sci Technol 1998;58:345±53.
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[12] Lee S-WR, Sun CT. On the apparent bending isotropy in clamped elliptic composite laminates. J Composite Mater 1995; 29(12):1601±21. [13] Vlot A, Van Ingen JW. Delamination resistance of poststretched ®bre metal laminates. J Composite Mater 1998; 32(19):1784±805. [14] Ananth CR, Chandra N. Numerical modeling of ®bre push-out test in metalic and intermetalic matrix composites-mechanics of the failure process. J Composite Mater 1995;29:1488±514. [15] Pomies F, Carlsson LA. Analysis of modulus and strength of dry and wet thermoset and thermoplastic composites loaded in transverse tension. J Composite Mater 1994;28:22±35. [16] Bahei-El-Din YA, Dvorak GJ. Plasticity analysis of laminated composite plates. Trans ASME 1982;49:740±6. [17] Yang HTY, He CC. Three dimensional ®nite element analysis of free edge stresses and delamination of composite laminates. J Composite Mater 1994;28:1394±412. [18] Hu HT, Schnobrich WC. Nonlinear ®nite element analysis of reinforced concrete plates and shells under monotonic loading. Comput Struct 1991;38:637±51. [19] Choi CK, Kwak HG. The eect of ®nite element mesh size in nonlinear analysis of reinforced concrete structures. Comput Struct 1990;36:807±15. [20] Owen DRJ. Anisotropic elasto-plastic ®nite element analysis of thick and thin plates and shells. Int J Numer Methods Eng 1983;19:541±66. [21] Theo J. Stress around rectangular holes in orthotropic plates. J Composite Mater 1981;15:311±28. [22] Chen JK, Beraun JE. A non-quadratic plasticity model suitable for metal matrix composites. J Composite Mater 1998; 32(23):2156±76. [23] Karakuzu R, Ozcan R. Exact solution of elasto-plastic stresses in a metal±matrix composite beam of arbitrary orientation subjected to transverse loads. Composite Sci Technol 1997;56:1383±9. [24] Karakuzu R, Sayman O. Elasto-plastic ®nite element analysis of orthotropic rotating discs with holes. Comput Struct 1993;51:695± 703. [25] Karakuzu R, Ozel A, Sayman O. Elasto-plastic ®nite element analysis of metal±matrix plates with edge notches. Comput Struct 1997;63:551±8. [26] Sayman O. Elasto-plastic stress analysis in stainless steel ®ber reinforced aluminum metal matrix laminated plates loaded transversely. Composite Struct 1998;43:147±54. [27] Jones RM. Mechanics of Composite Materials. Tokyo: McGrawHill Kogakusha, 1975. [28] Bathe KJ. Finite Element Procedures in Engineering Analysis. Englewood Cli, NJ: Prentice-Hall, 1982. [29] Lin CC, Kuo CS. Buckling of laminated plates with holes. J Composite Mater 1989;23:536±53. [30] Owen DRJ, Hinton E. Finite Elements in Plasticity: Theory and Practice. Swansea, UK: Pineridge Press, 1980.
www.elsevier.com/locate/compstruc
Elastic±plastic stress analysis and expansion of plastic zone in clamped and simply supported aluminum metal±matrix laminated plates Cesim Atas *, Onur Sayman Department of Mechanical Engineering, Dokuz Eyl ul University, Bornova, Izmir, Turkey
Abstract An elastic±plastic stress analysis and the expansion of plastic zone in layers of stainless steel ®ber-reinforced aluminum metal± matrix laminated plates are studied by using Finite Element Method and First-order shear deformation theory for small deformations. The plate is meshed into 64 elements and 289 nodes with simply supported or clamped boundary conditions. Laminated plates of constant thickness are formed by stacking four layers bonded symmetrically or antisymmetrically. It is assumed that the laminated plates are subjected to transverse uniform loads. Loading is gradually increased from yield point of the plate as 0.0001 MPa at each load step. Load steps are chosen as 100, 150 and 200. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Elastic±plastic stress analysis; Aluminium metal±matrix composite; Laminated plate; Residual stress; Plastic zone; Finite element method
1. Introduction Composite materials are those formed by combining more than one bonded material, each with dierent structural properties. The main advantages of composite materials are the potential for a high ratio of stiness to weight, strength, corrosion resistance, thermal properties, wear resistance, and fatigue life. Metal±matrix composites have been used in many structures and commercial applications for a long time, because of their high speci®c stiness and high temperature performance. In recent times, many manufacturing techniques have been developed. The objective is usually to make a material which is strong and sti, often with a low density. Elastic±plastic and residual stresses are very important in failure analysis of metal±matrix laminated plates. Permanent deformations and residual stresses occur in laminated plates when the yield point of the laminate is exceeded. These residual stresses can be used to raise the yield points of the plates. Chou et al. [1] studied the ®ber-reinforced metal± matrix composites involving fabrication methods, me*
Corresponding author.
chanical properties, secondary working techniques and interfaces. Kang and Kang [2] reviewed the work on short ®ber-reinforced aluminum composite material, fabricated by the stirring method and extruded at high temperatures at various extrusion ratios. In that study, the tensile strength of the hot extrusion billet was experimentally determined for dierent extrusion ratios, and compared with the theoretically calculated strength. Metal±matrix composites provide a relatively new way of strengthening metals [3]. Kwon and Yoon [4] examined the eect of SiCp and Al2 O3 on the high temperature ¯ow stress of particulate-reinforced aluminum composites. Nan and Clarke [5] investigated the in¯uence of particle size and particle fracture on the elastic / plastic deformation of metal±matrix composites. In that paper, a methodology was introduced for calculating the deformation response of particulate-reinforced metal± matrix composites in terms of an eective medium approach combined with the essential features of dislocation plasticity. A feature of that methodology is that the eect of particle cracking during deformation can be incorporated quantitatively. The in¯uence of particle volume fraction, shape and aspect ratio on the behavior of particle-reinforced metal±matrix composites was investigated through numerical modeling, which was performed using axisymmetric unit cell models, at high rates of strain [6]. Dutta et al. [7] worked on the particle
0263-8223/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 9 9 ) 0 0 1 1 9 - 1
10
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
redistribution and matrix micro-structure development during hot extrusion of cast SiCp -reinforced aluminum alloy matrix composites and the in¯uence of process parameters on these two key micro-structural features. The thermal performance of six commercially available particulate-reinforced aluminum-based metal±matrix composites [8] and superplasticity in aluminum-based metal±matrix composites [9] were investigated. Yeh and Krempl [10] examined the in¯uence of cool-down temperature histories on the residual stresses in ®brous metal±matrix composites. Lorentzen and Clarke [11] measured thermomechanically induced residual strains in Al/SiCp metal±matrix composites by using neutron diraction at room and elevated temperatures to monitor the eects of in situ uniaxial plastic deformations. Lee and Sun [12] reviewed the work on the ¯exural behavior of clamped elliptic composite laminates for both symmetric and antisymmetric laminates. In that work, closed form solutions were discussed for laminates under uniform transverse loading. The apparent bending isotropy was identi®ed for clamped elliptic laminates. Vlot and Ingen [13] studied delamination resistance of post-stretched ®ber metal laminates consisting of bonded alternating thin aluminum and ®ber / epoxy layers. Ananth et al. [14] investigated the application of push out test to characterize the mechanical behavior of interfaces in metallic and intermetallic matrix composites. In that study, ®nite element technique was used. In addition, the eect of dierent material and testing variables on the experimental data was de®ned. Pomies and Carlsson [15] carried out micro-mechanical ®nite element analysis of rigidity and strength of transverse tension loaded continuous ®ber composites and compared their results with experimental data for dry and wet glass±epoxy, carbon±epoxy composites. In that paper, transverse tensile strength predictions employed the combined mechanical and residual stress states at the ®ber±matrix interface and in the matrix, in conjunction with debonding and matrix failure criteria. Bahei-El-Din and Dvorak [16] investigated the elastic±plastic behavior of symmetric metal±matrix composite laminates for the case of in-plane mechanical loading. Boron ®ber-reinforced aluminum metal±matrix laminates were used in their work. Linear or non-linear ®nite element method can be used to analyze the laminated composites [17± 19]. Owen [20] studied elasto-plastic ®nite element analysis of anisotropic plates and shells. Theo [21] investigated the stresses around rectangular holes in orthotropic plates. It was shown by Chen and Beraun [22] that a non-quadratic plasticity model is suitable for metal±matrix composites. Their work based on the assumption that uniform dilatation has negligible eect on the yield point. A high-order yield function of non-equal stress exponents in conjunction with a non-associated ¯ow rule was proposed to formulate 3-D plastic constitutive equations for ®brous metal±matrix composites.
It was also shown that the uniform dilatation assumption is adequate for other ®ber composites. Karakuzu and Ozcan [23] gave an exact solution to the elastoplastic stress analysis of an aluminum metal±matrix composite beam reinforced by steel ®bers. In order to manufacture continuous stainless steel reinforced aluminum matrix composites, the die casting process was used. Elasto-plastic ®nite element analysis of orthotropic rotating discs with holes in steel-aluminum composite was carried out by Karakuzu and Sayman [24]. Karakuzu et al. [25] de®ned the elastic±plastic ®nite element analysis of metal±matrix plates with edge notches. Sayman [26] investigated elasto-plastic stress analysis in stainless-steel ®ber-reinforced aluminum metal±matrix laminated simply supported plate which is subjected to transverse loading by using ®nite element technique. Also, the expansion of the plastic zone and residual stresses for dierent oriented angle-ply and cross-ply laminated plates for small deformations were determined. In this paper, stainless steel ®ber-reinforced aluminum metal±matrix laminated simply supported or clamped plates are loaded transversely. Elastic±plastic, elastic and residual stresses and the expansion of the plastic zone are carried out by using ®nite element method for small deformations. In the solution, the Tsai-Hill theory is used as a yield criterion and a ninenode isoparametric quadrilateral element is used for the elastic±plastic analysis of the laminated plate. The load steps are chosen as 100, 150 and 200.
2. Mathematical formulation The stainless steel ®ber-reinforced aluminum metal± matrix laminated plates of constant thickness are formed by stacking four layers bonded symmetrically or antisymmetrically about the middle surface. Geometry of the plate in Cartesian coordinates and boundary conditions are illustrated in Figs. 1 and 2, respectively. Considering transverse shear deformations in the solution of laminated plate elements, the constitutive equations for an orthotropic layer are given as
Fig. 1. Geometry of the plate in Cartesian coordinates.
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
1 Ub 2
Z
ÿh2
Z
1 Us 2
h 2
h 2
ÿh2
2 3 Z ÿ 4 rx ex ry ey sxy cxy dA5dz; A
2 3 Z ÿ 4 sxz cxz syz cyz dA5dz;
Fig. 2. Boundary conditions for simply supported (a) and clamped (b) laminated plates.
8 9 2 38 Q11 Q12 Q16 > > = < rx > < eX 6 7 ry 4 Q21 Q22 Q26 5 eY > > ; : > : sxy Q16 Q26 Q66 cXY sYZ Q44 Q45 cYZ ; sXZ Q45 Q55 cXZ
9 > = > ;
; 2:1
u x; y; z u0 x; y zwx x; y; v x; y; z v0 x; y ÿ zwy x; y;
2:2
w x; y; z w0 x; y; where u0 ; v0 and w0 denote the displacements of a point on the middle surface, and wx ; wy denote rotations about the normal to the y- and x-axes, respectively. The bending strains vary linearly through the plate thickness and are given by the curvatures of the plate, whereas the transverse shear strains are assumed to be constant through the thickness. To obtain the equilibrium equations of the element, the total potential energy of a laminated plate under static loading is given as P Ub Us V ;
Mx
Nx
6 4 Ny (
Nxy Qx
3
7 My 5 )
Qy
Z b b N n u0n N s u0s ds; oR
Mxy Z
h 2
ÿh2
"
Z
2 h 2
ÿh2
sxz syz
rx
3
6 7 4 ry 5 1; z dz;
#
sxy
2:5
dz:
Equilibrium requires that P is stationary, i.e., dP 0, which may be regarded as the principle of virtual displacement for the plate element [28].
3. Finite element model In order to obtain the yield points of laminates, the spread of plastic zone and the residual stresses in the laminated plates, a nine-node Lagrangian ®nite element was used. The plate is meshed into 64 elements and 289 nodes with both simply supported or clamped boundary conditions. Laminated plates of constant thickness are formed by stacking four layers bonded symmetrically or antisymmetrically. Bending and shear stiness matrices of the plate are obtained by using the minimum potential energy principle as Z Kb
2:3
where Ub is the strain energy of bending, Us the strain energy of shear and V represents the potential energy of external forces. They are as follows:
wp dA ÿ
where dA dx dy; p is the transverse load per unit area, and Nnb and Nsb are the in-plane loads applied to the boundary oR. The resultant forces, moments and shear forces per unit length of the cross section of the laminated plate are obtained by integration of the stresses through the thickness as [27] 2
where Qij are the transformed reduced stinesses given in terms of the orientation angle and the engineering constants of the material. The ®rst-order shear deformation theory (FSDT) was used in this work. It is assumed that the particles of the plate, originally on a line that is normal to the undeformed mid-plane, remain on a straight line during deformation; whereas this line is not necessarily normal to the deformed middle surface in the theory. This theory is based on the displacement ®eld, for small deformations,
A
2:4
A
Z V ÿ
11
Z Ks where
A
A
Bb T Db Bb dA; 3:1 T
Bs Ds Bs dA;
12
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Aij Bij Db ; Bij Dij " # k12 A44 0 ; Ds 0 k22 A55 Z h2 Qij 1; z; z2 dz Aij ; Bij ; Dij Z A44 ; A55
ÿh2
h 2
ÿh2
3:2 i; j 1; 2; 6;
Q44 ; Q55 dz:
in the plastic region, therefore, PoissonÕs ratio in the plastic region is taken as 0.5. Fiber volume fraction in the plastic region for small deformations can be taken as 0.15 the same as in the elastic region. The calculated stresses do not generally coincide with the true stresses in a non-linear solution; because of this reason external forces are applied incrementally. Therefore, the equivalent nodal forces and the unbalanced nodal forces must be found; for each load step, Z Z T T fRgequivalent Bb rb dA Bs rs dA; 3:7 vol
vol
Db and Ds are the bending and shear parts of the material matrix, respectively. k12 and k22 denote the shear correction factors. For rectangular cross sections they are given as k12 k22 5=6 [29]. In this solution, Tsai-Hill theory is used as a yield criterion due to the same yield values in tension and compression in this aluminum metal±matrix composite. In the elastic±plastic solution, the tangential modular matrix is used instead of the elasticity matrix. The yield function f is
These unbalanced nodal forces represent the increments in the solution and must satisfy the convergence tolerance in a non-linear analysis. A widely used iteration procedure is the modi®ed Newton iteration method [28]. This iterative solution can be derived from the Newton± Raphson method. The equations used in the modi®ed Newton iteration are, for i 1; 2; 3; 4; . . .
f re ÿ r0 0;
t
3:3
fRgunbalanced fRgapplied ÿ fRgequivalent :
iÿ1
KDU i f Rgunbalanced tDt fRgapplied ÿ tDt fRgequivalent ;
where re and r0 are the eective stress and yield stress, respectively. The tangential modular matrix is obtained as [30]
tDt
Dt DI ÿ aaT D= H aT Da;
with the initial condition,
3:4
where H is the local slope of the uniaxial stress / plastic strain curve and can be determined experimentally and D, I and a are the elastic matrices which are used instead of Db and Ds , unit matrix and ¯ow vector, respectively. The ¯ow vector a is found by using the Prandtl±Reuss equations as a of =or:
3:5
The stress and strain curve of the composite layer is obtained in the principal material, ®ber, direction x. It is given by the Ludwik equation as r r0 kenp ;
U i tDt U iÿ1 DU i 3:9
tDt
U 0 t U ;
tDt
0 Req t Req ;
3:10
where t, Dt, U and K are the time, time interval, overall displacement vector and the stiness matrix, respectively. These equations were obtained by linearizing the response of the ®nite element system about the conditions at time t. In each load step, the unbalanced nodal forces are calculated, and the iteration is continued until the unbalanced nodal forces (load vector) or the displacement increments, DU i are suciently small.
3:6
where k and n are the plasticity constant and strain hardening exponent, respectively, as given in Table 1. It is assumed that the composite material is incompressible Table 1 Mechanical properties and yield points of a layer E1 (GPa) E2 (GPa) G12 (GPa) Axial yield value, X (MPa) Transverse yield value, Y (MPa) Shear yield value, S (MPa) Plasticity constant, k (MPa) Strain hardening exponent, n PoissonÕs ratio, m12 Fiber volume fraction, Vf
3:8
85 74 30 230.0 24.0 48.9 1254 0.7 0.30 0.15
4. Fabrication of laminated plates A layer of stainless steel ®ber-reinforced aluminum metal±matrix laminate was manufactured by setting under 30 MPa and at 600°C temperature [24]. In order to fabricate laminates, moulds, which were heated by electrical resistance and insulated by glass-®bers, were used. Under these conditions the steel ®ber and the aluminum matrix were bonded, provided aluminum exceeds the yield strength. Tensile test specimens were obtained from this layer and loaded in principal material directions by Instron tensile machine to get mechanical properties and yield strengths of a layer. Strain-gauges were used during these tests. Laboratory test results are given in Table 1.
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
13
Table 2 The uniform transverse force at the yield points of the simply supported and clamped laminated platesa
a
0°=90°2
30°= ÿ 30°2
45°= ÿ 45°2
60°= ÿ 60°2
Simply supported S (MPa) AS (MPa)
0.0407 0.0401
0.0408 0.0406
0.0407 0.0405
0.0408 0.0406
Clamped S (MPa) AS (MPa)
0.0581 0.0572
0.0690 0.0685
0.0857 0.0847
0.0690 0.0685
S: Symmetric laminate, AS: Antisymmetric laminate.
5. Results and discussion
5.1. Simply supported laminates
Symmetrically or antisymmetrically oriented laminated plates of four layers, subjected to uniform transverse loads, were used to get an elastic±plastic solution and the spread of plastic region with boundary conditions of simply supported or clamped edges. The transverse load was increased, 0.0001 MPa at each step, from the yield point for 100, 150 and 200 load steps, incrementally. The uniform transverse force at the yield points of both simply supported and clamped laminated plates is given in Table 2. It is clearly seen that the transverse uniform loads which cause the beginning of the yield for the symmetrically laminated plates are higher than those for the antisymmetrically oriented laminates for both simply supported and clamped plates. For both boundary conditions given above, the expansion of the plastic zone and intensity of stress components at the upper and lower surfaces of symmetric oriented laminates are the same, so they are given in the tables and ®gures only for upper surfaces. However, they are generally dierent from each other at the upper and lower surfaces in antisymmetric stacking sequences. Symmetric or antisymmetric laminated plates of four layers are oriented as cross-ply laminated plate, 0°=90°2 , and angle-ply laminated plates, 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 .
The expansion of the plastic zone for symmetric cross-ply, 0°=90°2 , laminated plate is shown in Fig. 3. Elastic±plastic, elastic and residual stresses at the midpoint (at the center of the plate, denoted by A) of upper surface of the symmetric cross-ply laminated plate for 200 load steps are given in Table 3. They are given in Table 4 for antisymmetric orientation of 0°=90° laminated plate at the upper and lower surfaces. In Table 5, the residual stress components for symmetric angle-ply, laminated 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 plates are given. The distribution of the plastic regions
Table 3 Elastic±plastic, elastic and residual stress components for symmetric cross-ply 0°=90°2 laminated plate at the center of the laminate (node A) for 200 load steps in simply supported plates rx (MPa) Elastic± plastic stresses Elastic stresses Residual stresses
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
39.587
23.858
)0.049
)0.035
)0.079
40.025
35.776
)0.039
)0.061
)0.072
)0.438
)11.917
)0.010
0.026
)0.007
Table 4 Elastic±plastic, elastic and residual stress components for antisymmetric cross-ply, 0°=90°2 , laminated plate at the center of the laminate (node A) for 200 load steps in simply supported platesa sxy (MPa)
syz (MPa)
sxz (MPa)
Elastic±plastic stresses 0° 37.140 23.888 90° )23.889 )37.050
)0.034 0.054
)0.065 )0.065
)0.065 )0.065
Elastic stresses 0° 39.003 90° )35.990
36.003 )38.987
)0.035 0.043
)0.066 )0.066
)0.066 )0.066
Residual stresses 0° )1.863 90° 12.101
)12.114 1.938
0.000 0.011
0.001 0.001
0.001 0.001
rx (MPa)
Fig. 3. The expansion of the plastic zone for symmetric cross-ply 0°=90°2 simply supported laminated plate at the upper surface.
a
ry (MPa)
0°: For upper surface, 90°: For lower surface.
14
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Table 5 Residual stress components for symmetric angle-ply laminated plates at the center of the laminate (node A) for 200 load steps in simply supported plates 30°= ÿ 30°2 45°= ÿ 45°2 60°= ÿ 60°2
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
)2.226 )6.344 )10.226
)10.234 )6.359 )2.236
4.262 5.554 4.268
)0.019 )0.034 )0.004
)0.004 )0.034 )0.019
Fig. 4. The expansion of the plastic zone for symmetric angle-ply 30°= ÿ 30°2 simply supported laminated plate at the upper surface.
Fig. 6. The expansion of the plastic zone for symmetric angle-ply 60°= ÿ 60°2 simply supported laminated plate at the upper surface.
laminates for the same orientation angle at the upper surfaces. However, the plastic region expands along diagonals of the square plates in dierent directions at the upper and lower surfaces for antisymmetric laminated plates such as 30°= ÿ 30°2 angle-ply laminated plate, as shown in Fig. 7. 5.2. Clamped laminates
Fig. 5. The expansion of the plastic zone for symmetric angle-ply 45°= ÿ 45°2 simply supported laminated plate at the upper surface.
for symmetric angle-ply, 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 , laminated plates at the upper surface are given in Figs. 4, 5 and 6, respectively. For antisymmetric angle-ply stacking sequences, the residual stress components are given in Table 6 for upper and lower surfaces. There is a little dierence between the expansion of the plastic region in symmetric and antisymmetric
Residual stress components for symmetric cross-ply and angle-ply laminated plates at nodes A, B and C for 200 load steps are given in Table 7. Here, nodes of A, B and C denote the center of the plate and mid-points of the edges in axial and transverse directions, respectively, as shown in Fig. 2. The spread of the plastic zone for symmetric cross-ply 0°=90°2 and angle-ply 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 laminated plates is given in Figs. 8, 9, 10 and 11, respectively. The residual stress components for antisymmetric cross-ply laminated plate at nodes A, B and C of upper and lower surfaces for 200 load steps are given in Table 8 and for antisymmetric angle-ply laminated plates, they are given in Tables 9 and 10. The expansion of the plastic zone for antisymmetric cross-ply 0°=90°2 and angle-ply 30°= ÿ 30°2 ; 45°= ÿ 45°2 ; 60°= ÿ 60°2 laminated plates at upper and lower surfaces are given in Figs. 12, 13, 14 and 15, respectively. It is clearly seen from these
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
15
Table 6 Residual stress components for antisymmetric angle-ply laminated plates at the center of the laminate (node A) for 200 load steps in simply supported plates Laminated Plates
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
At the upper surface 30°= ÿ 30°2 45°= ÿ 45°2 60°= ÿ 60°2
)2.493 )6.366 )10.089
)10.079 )6.344 )2.514
4.351 5.628 4.340
)0.006 )0.027 )0.010
)0.010 )0.048 )0.005
At the lower surface 30°= ÿ 30°2 45°= ÿ 45°2 60°= ÿ 60°2
2.435 6.239 10.037
10.031 6.457 2.455
4.369 5.574 4.360
0.005 )0.030 )0.051
)0.050 )0.051 0.006
Fig. 7. The expansion of the plastic zone for antisymmetric angle-ply 30°= ÿ 30°2 simply supported laminated plate at the upper (a) and lower (b) surfaces.
Table 7 Residual stress components for symmetric cross-ply and angle-ply laminated plates at nodes A, B and C for 200 load steps in clamped plates Stacking sequence
Nodes
rx (MPa)
ry (MPa)
sxy (MPa)
0°=90°2
A B C
0.285 )0.363 2.469
0.066 )0.095 8.269
0.000 0.000 )0.011
0.003 0.000 0.890
)0.004 )0.021 0.000
30°= ÿ 30°2
A B C
)0.177 )0.215 3.231
)0.663 )0.060 5.299
0.254 )0.007 )2.199
)0.015 0.000 0.449
)0.007 )0.012 )0.020
45°= ÿ 45°2
A B C
)3.340 2.945 3.408
)3.345 3.408 2.953
2.217 )1.993 )1.995
)0.010 )0.019 0.343
)0.009 0.343 )0.019
60°= ÿ 60°2
A B C
)0.663 5.292 )0.060
)0.178 3.232 )0.215
0.254 )2.200 )0.007
)0.007 )0.020 )0.012
)0.015 0.449 0.000
®gures that the expansions of the plastic zone at the upper and lower surfaces are dierent from each other for antisymmetric laminates, although they are the same for symmetrically oriented laminates.
syz (MPa)
sxz (MPa)
6. Conclusions In order to carry out the elastic±plastic stress analysis in clamped or simply supported steel ®ber-reinforced
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C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Fig. 8. The expansion of the plastic zone for symmetric cross-ply 0°=90°2 clamped laminated plate.
Fig. 11. The expansion of the plastic zone for symmetric angle-ply 60°= ÿ 60°2 clamped laminated plate.
Table 8 Residual stress components for antisymmetric cross-ply, (0°/90°)2 , laminated plate at nodes A, B and C for 200 load steps in clamped plates Nodes
Fig. 9. The expansion of the plastic zone for symmetric angle-ply 30°= ÿ 30°2 clamped laminated plate.
Fig. 10. The expansion of the plastic zone for symmetric angle-ply 45°= ÿ 45°2 clamped laminated plate.
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
At the upper surface A 0.105 B 0.134 C 2.509
0.274 0.045 8.439
0.000 0.001 )0.014
)0.001 )0.001 0.907
)0.001 )0.266 )0.001
At the lower surface A )0.274 B )8.439 C )0.045
)0.105 )2.509 )0.134
0.000 0.020 )0.001
)0.001 )0.001 )0.266
)0.001 0.907 )0.010
aluminum metal±matrix composite laminated plates under transverse loading, the ®rst order shear deformation theory was used. The expansion of the plastic zone and elastic±plastic, elastic, and residual stress components are investigated in symmetric and antisymmetric cross-ply and angle-ply laminated plates. Conclusions obtained from this investigation can be summarized as follows: For both clamped and simply supported plates: · The yield points in symmetric cross-ply and angle-ply laminated plates are higher than those in antisymmetric cross-ply and angle-ply laminates. · The yield point for cross-ply laminated plates is always smaller than that for angle-ply laminated plates. · The yield loads in clamped laminated plates are higher than those of the simply supported plates (almost between the range of 42% and 110% , depending on the orientation angle). · The expansion of the plastic zone and the intensity of the stress components are the same at the upper and
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
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Table 9 Residual stress components for antisymmetric angle-ply laminated plates at nodes A, B and C of upper surface for 200 load steps in clamped plates Stacking sequence
Nodes
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
30°= ÿ 30°2
A B C
)0.149 )0.221 3.409
)0.590 )0.063 5.534
0.224 )0.021 )1.980
)0.011 0.000 0.451
)0.006 )0.013 )0.001
45°= ÿ 45°2
A B C
)3.146 3.145 3.628
)3.151 3.628 3.153
2.199 )1.823 )1.825
)0.012 )0.002 0.341
)0.012 0.341 )0.002
60°= ÿ 60°2
A B C
)0.590 5.527 )0.062
)0.150 3.410 )0.221
0.224 )1.981 )0.021
)0.006 )0.001 )0.013
)0.010 0.451 0.000
Table 10 Residual stress components for antisymmetric angle-ply laminated plates at nodes A, B and C of lower surface for 200 load steps in clamped plates Stacking sequence
Nodes
rx (MPa)
ry (MPa)
sxy (MPa)
syz (MPa)
sxz (MPa)
30°= ÿ 30°2
A B C
0.149 0.225 )3.419
0.583 0.064 )5.589
0.227 )0.020 )1.985
)0.002 0.000 0.449
)0.018 )0.013 )0.001
45°= ÿ 45°2
A B C
3.127 )3.174 )3.649
3.130 )3.649 )3.182
2.202 )1.842 )1.843
)0.013 )0.002 0.342
)0.013 0.342 )0.002
60°= ÿ 60°2
A B C
0.583 )5.581 0.063
0.148 )3.420 0.225
0.227 )1.986 )0.020
)0.018 )0.001 )0.013
)0.002 0.449 0.000
Fig. 12. The expansion of the plastic zone for antisymmetric cross-ply 0°=90°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
lower surfaces in symmetric laminated plates. However, they are dierent from each other at upper and lower surfaces in antisymmetrically oriented plates. · The intensity of the residual stress components, because of the major dierences in material properties in the principal material directions, is maximum for symmetric and antisymmetric cross-ply 0°=90°2
laminated plate in comparison with the other angleply laminated plates. For simply supported plates: · In simply supported laminates, plastic zone expands along the ®ber directions. · By increasing the load steps, yielding begins also at the corners which are in the transverse direction to the ®bers because of great shear stresses.
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C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
Fig. 13. The expansion of the plastic zone for antisymmetric angle-ply 30°= ÿ 30°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
Fig. 14. The expansion of the plastic zone for antisymmetric angle-ply 45°= ÿ 45°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
Fig. 15. The expansion of the plastic zone for antisymmetric angle-ply 60°= ÿ 60°2 clamped laminated plate at the upper (a) and lower (b) surfaces.
C. Atas, O. Sayman / Composite Structures 49 (2000) 9±19
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