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Elasto-plastic stress analysis of aluminum metal-matrix composite laminated plates under in-plane loading
Computers and Structures 75 (2000) 55±63
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Elasto-plastic stress analysis of aluminum metal-matrix composite laminated plates under in-plane loading O. Sayman a,*, H. Akbulut a, C. Meric° b a
Department of Mechanical Engineering, Dokuz EyluÈl University, Bornova, Izmir, Turkey b Department 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 laminated metal-matrix composite plates. Long stainless steel ®ber reinforced aluminum metal-matrix composite layer is manufactured 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 deformation theory and nine-node Lagrangian ®nite element is used. The in-plane load is increased gradually. # 2000 Elsevier Science 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 and the ductility of the matrix provide a new material with superior properties. Plastic deformations and residual stresses are important in composite laminated plates. Residual stresses 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 mechanical loading. In this study, aluminum matrix is reinforced by boron ®bers. Metal-matrix composites consist of a ductile, usually low strength matrix reinforced with elastic brittle, or ductile strong ®bers
* Corresponding author.
which provide a new material with superior properties such as high strength and stiness, low density and resistance to corrosion, high creep and fatigue properties [2±5]. Karakuzu and OÈzcan [6] have given an exact solution to the elasto-plastic stress analysis of an aluminum metal-matrix composite beam reinforced by steel ®bers. Linear or nonlinear ®nite element method can be used to analyze the laminated composites [7±10]. In this study, aluminum metal-matrix composite laminated 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 composed of orthotropic layers bonded symmetrically or antisymmetrically about the middle surface of the plate. In the solution of this problem, the Cartesian
0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 8 6 - 3
56
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Fig. 1. Loading of laminated plate.
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 assumption that particles of the plate originally on a line that is normal to the undeformed middle surface remain on a straight line during deformations, but this line is not necessarily normal to the deformed middle surface. By using this assumption the displacement components of a point with coordinates x, y, z for small deformations are: 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 point of the middle surface, and cx ,cy are the rotations of
normals to midplane about the y and x axes, respectively. The bending strains vary linearly through the plate thickness, whereas shear strains are assumed to be constant through the thickness as @ cx @ u0 @x 9 @x 8 < ex = @ v0 @ cy z ey @y : g ; @y xy @ cx @ cy @ u0 @ v0 @y ÿ @x @y @x e e0x Kx x 0 e ey z Ky y Kxy gxy g0 xy @w ÿ cy gyz @ y gxz @ w @ x cx
or
2
The total potential energy of a laminated plate under static loadings is given as P Ub Us V
Fig. 2. The production of composite layer.
3
where U b is the strain energy of bending, U s is the strain energy of shear and V represents potential energy of external forces. They are as
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Ub
1 2
Us
1 2
h=2 ÿh=2
A
h=2 ÿh=2
ÿ sx ex sy ey txy gxy dA dz
ÿ A
txz gxz tyz gyz dA dz
Vÿ
A
wp dA ÿ
@R
b b N n u0n N s u0s ds
4
where dA dx dy, p is the transverse loading per unit b b area and N n and N s are the in-plane loads applied on the boundary @ R:
3. Production of laminated plates The composite layer consists of stainless steel ®ber and aluminum matrix. The production has been realized by using moulds which consist of upper and lower parts. Electrical resistance has been used to heat the moulds and material which are insulated, as illustrated in Fig. 2. The hydraulic press has been used to obtain a pressure of 30 MPa to the upper mould. Manufacturing set has been heated to 6008C. In these conditions, the yield strength of aluminum is exceeded and good bonding between matrix and ®ber has been realized. The mechanical properties, yield points and plastic parameters are given in Table 1. It is assumed that the yield point Z (in the z direction) is equal to the yield point Y (in the y direction), the yield points of txz , tyz are equal to S. The von Mises and Tresca criteria are used generally for isotropic materials. Huber±Mises yield criterion has been generalized by Hill for anisotropic metals [14]. It is more appropriate to use Huber±Mises or Tsai±Hill failure criteria for anisotropic metals, since their yield points are dierent in longitudinal and lateral directions. The dierence Table 1 The measured mechanical properties and yield points of a layer Mechanical properties E1 E2 G12 n12 Yield strengths and parameters Axial strength, X Transverse strength, Y Shear strength, S Hardening parameter, k Strain hardening parameter, n
86 GPa 74 GPa 32 GPa 0.30 228.3 24.2 47.6 1254
MPa MPa MPa MPa 0.7
57
between the numerical results for the residual stresses in the symmetric cross-ply ([08/908]2) laminated plate obtained for Huber±Mises and Tsai±Hill criteria is found as 0.8% as shown in Table 2. The Tsai±Hill criterion is used as a yield criterion [13]. Four layers have been bonded to form a laminated plate symmetrically or antisymmetrically by using a pressure of 30 MPa and 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 is composed of four layers. A typical ®nite element for a symmetric and an antisymmetric lamination is divided into eight imaginary layers for obtaining the ®nite element 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 following matrix form: 2 3 2 3 u0 0 0 0 Ni 0 6 7 60 6 v0 7 Ni 0 0 0 7 7 n 6 6 7 X 6 7 7 60 d 6 0 Ni 0 0 7 6w 7 6 7 6 7 7 i1 6 6 cx 7 0 0 0 N 0 4 5 i 4 5 cy 0 0 0 0 Ni 2 3 u0 6 7 6 v0 7 6 7 6w 7 5 6 7 6 7 6 cx 7 4 5 cy i
in which n is the total number of nodes and Ni is the shape function at node i. The relationship between strains and displacements can be written in the matrix form: 2 3 32 3 2 e0 0 0 0 Ni,x 0 6 ex0 7 6 y 7 60 7 6 u0 7 Ni,y 0 0 0 6 7 6 7 6 v0 7 6 0 7 6 Ni,y Ni,x 0 0 76 7 0 6 gxy 7 6 7 6w 7 6 6 7 60 76 7 0 0 Ni,x 0 6 Kx 7 6 7 4 cx 5 6 7 40 0 0 0 ÿNi,y 5 4 Ky 5 cy 0 0 0 N ÿNi,x i i i,y Kxy i
or symbolically as febi g Bbi ui
7
in which i is the node number, and commas represent partial derivatives. The relationship between the transverse shear strains
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O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Table 2 Residual stresses in the symmetric cross-ply, ([08/908]2), laminated square plate for 200 iterations Orientation angle
sx (MPa)
The result for Tsai±Hill criterion 08 1.844 908 ÿ1.844 The result for Huber±Mises criterion 08 1.859 908 ÿ1.859
sy (MPa)
txy (MPa)
txz (MPa)
tyz (MPa)
0.501 ÿ0.500
ÿ0.001 0.001
0.000 0.000
0.000 0.000
ÿ0.501 0.501
0.001 ÿ0.001
0.000 0.000
0.000 0.000
and displacement components can be written as 2 3 u0 7 6 6 v0 7 ÿNi 6 7 0 0 Ni,y 0 gyz 6w 7 0 0 Ni,x Ni 0 gxz i 7 i6 4 cx 5 cy
A44 ,A55
8
i
or symbolically as fesi g Bsi fui g
9
The stiness matrix of the plate element is obtained by using the minimum potential energy method or the principle of virtual displacements [12]. Bending and shear stiness matrices are Kb Bb T Db Bb dA A
Ks
A
Bs T Ds Bs dA
10
Db
ÿ Aij ,Bij ,Dij
Bij Dij
h=2 ÿh=2
Ds
k21 A44 0
0 k22 A55
ÿ Qij 1,z,z2 dz i,j 1,2,6
Q44 ,Q55 dz
11
jBb j645 , jBs j245 and Db and Ds are the bending and shear parts of the material matrix, respectively. A45 is negligible in comparison with A44 and A55 [11] and shear correction factors for rectangular cross sections are given as [11], k21 k22 5=6: Once the nodal displacements are calculated, the strain components of each layer can be found by using Eqs. (7) and (9); and the stress components can be calculated and used to check the yield state of the material. Since the calculated stresses do not generally coincide with the true stresses in a nonlinear problem, the unbalanced nodal forces and the equivalent nodal forces must be calculated. The equivalent nodal point forces corresponding to the element stresses at each iteration can be calculated as fR gequivalent B T s dA
Aij Bij
ÿh=2
where
h=2
vol
vol
Bb T sb dA
vol
Bs T ss dA
12
When the equivalent nodal forces are known, the unbalanced nodal forces can be found by fR gunbalanced fR gapplied ÿfR gequivalent
13
These unbalanced nodal forces are applied for obtain-
Fig. 3. A layered section for (a) symmetric and (b) antisymmetric lamination.
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
59
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.
ing increments in the solution and must satisfy the convergence tolerance in a nonlinear analysis. The dierence between the plastic and elastic solution gives the residual stresses. The residual stresses may increase the possibility of failure of the laminated plates. In this solution 216 nodes and 48 elements are used.
5. Numerical results and discussions The laminated plate is assumed to be under uniform axial in-plane loads along the rectangular edges and the circular hole is unloaded. The laminated plates are composed of four orthotropic and generally orthotropic 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 stiness 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 expansion of the plastic zone and the residual stresses in the cross-ply symmetric laminated plate ([08/908]2) without a 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. Residual 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. 5. Residual stress components sx (MPa) along AB for cross-ply ([08/908]2): (a) symmetric and (b) antisymmetric laminated plates.
60
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
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.
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 eect of orientation angle on the expansion of plastic zone is presented in Fig. 6, for ([308/ ÿ308]2) symmetric and antisymmetric angle-ply laminated 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, respectively, and when it is further increased incrementally the plastic zone expands around the hole. The expansion of plastic zone in the layer of 308 orientation angle is slightly dierent from that of the layer of ÿ308 orientation angle in each case. The eect 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 608 and ÿ608 are slightly dierent 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 components 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 compressive and tensile in the layers of 908 and 08 orientation 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. 7. Expansion of plastic zone in (a) symmetric and (b) antisymmetric ([458/ÿ458]2) laminated plates; and its distribution across the cross-section for 800 iterations.
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
61
Fig. 8. Expansion of plastic zone in (a) symmetric and (b) antisymmetric ([608/ÿ608]2) laminated plates; and its distribution across the cross-section for 800 iterations.
Table 3 Yield points in symmetric and antisymmetric laminated plates
Symmetric Nx (N/mm) Antisymmetric Nx (N/mm)
([08/908]2)
([308/308]2)
([458/458]2)
([608/608]2)
80.56 72.68
140.96 136.89
109.76 107.12
88.16 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 Elasto-plastic solution 200 400 800 Elastic solution 200 400 800 Residual stresses 200 400 800
Orientation angle
sx (MPa)
sy (MPa)
txy (MPa)
tyz (MPa)
txz (MPa)
908 08 908 08 908 08
24.213 37.223 24.211 44.793 24.208 61.115
2.793 5.729 2.326 7.255 1.377 10.591
ÿ0.675 ÿ0.6981 ÿ0.822 ÿ0.832 ÿ1.083 ÿ1.10
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
908 08 908 08 908 08
29.248 34.212 34.260 40.075 44.285 51.801
3.901 4.638 4.570 5.433 5.907 7.023
ÿ0.684 ÿ0.684 ÿ0.801 ÿ0.801 ÿ1.036 ÿ1.036
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
908 08 908 08 908 08
ÿ5.035 3.011 ÿ10.049 4.718 ÿ20.077 9.314
ÿ1.108 1.091 ÿ2.244 1.822 ÿ4.530 3.568
0.009 0.003 ÿ0.021 ÿ0.030 ÿ0.048 ÿ0.066
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
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O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Table 5 Residual stress components in the symmetric angle-ply laminated plates for 800 iterations Laminated plate ([308/ÿ308]2) ([458/ÿ458]2) ([608/ÿ608]2)
308 ÿ308 458 ÿ458 608 ÿ608
sx (MPa)
sy (MPa)
txy (MPa)
tyz (MPa)
txz (MPa)
ÿ4.622 ÿ4.012 ÿ8.684 ÿ7.924 ÿ12.292 ÿ11.630
ÿ5.330 ÿ4.580 ÿ6.436 ÿ5.827 ÿ5.628 ÿ4.900
2.960 ÿ2.551 4.247 ÿ3.818 4.488 ÿ3.967
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
Table 6 Residual stress components in the antisymmetric angle-ply and cross-ply laminated plates for 800 iterations Laminated plate ([08/908]2) ([308/ÿ308]2) ([458/ÿ458]2) ([608/ÿ608]2)
908 08 308 ÿ308 458 ÿ458 608 ÿ608
sx (MPa)
sy (MPa)
txy (MPa)
tyz (MPa)
txz (MPa)
ÿ18.806 9.175 ÿ4.710 ÿ3.977 ÿ8.868 ÿ7.808 ÿ12.580 ÿ11.580
ÿ3.954 3.604 ÿ5.212 ÿ4.561 ÿ6.031 ÿ5.965 ÿ4.478 ÿ5.203
ÿ0.111 ÿ0.017 3.054 ÿ2.487 4.386 ÿ3.727 4.524 ÿ3.899
ÿ0.128 ÿ0.202 ÿ0.100 0.085 ÿ0.162 0.143 ÿ0.183 0.144
ÿ0.127 ÿ0.251 0.058 0.046 0.076 0.068 0.086 0.0622
([458/ÿ458]2) and ([608/ÿ608]2) laminated plates are given in Table 5. It is seen from this table that as we increase the orientation angles in the symmetric angleply laminated plates the sx residual stress component becomes greater. The maximum absolute value of the stress 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) laminated plates are given in Table 6. It is seen that as we increase the orientation angles in the antisymmetric angle-ply laminated plates the sx residual stress component becomes greater. The maximum intensity of the stress components in the plate is given in the table.
nates ([08/908]2), yielding occurs in the layers of 908 orientation angle and the residual stress Nx is compressive and tensile in layers of 908 and 08 orientation angles, respectively, without a hole. 3. The orientation angle y aects the yield points of laminated plates. 4. The yield point of symmetric laminated plates is higher than that of the yield point of antisymmetric laminated plates. 5. Load carrying capacity of the laminated plate can be increased by means of residual stresses.
References 6. Conclusions Elasto-plastic stress analysis has been carried out by using the ®rst-order shear deformation theory in aluminium±steel ®ber laminated plates. The expansion of the plastic zone and residual stresses are obtained in symmetric 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-
[1] Bahei-El-Din YA, Dvorak GJ. Plasticity analysis of laminated composite plates. Transactions of the ASME 1982;49:740±6. [2] Yeh NM, Krempl E. The in¯uence of cool-down temperature histories on the residual stresses in ®brous metal-matrix composites. Journal of Composite Materials 1993;27:973±95. [3] Doghri I, Jansson DS, Leckie FA, Lemaitre J. Optimization of coating layers in the design of ceramic ®ber reinforced metal-matrix composites. Journal of Composite Materials 1994;28:167±87.
O. Sayman et al. / Computers and Structures 75 (2000) 55±63 [4] Canumalla S, Dynan SA, Green DJ. Mechanical behavior of mullite ®ber reinforced aluminum alloy composites. Journal of Composite Materials 1995;29:653±69. [5] Nicolaou PD, Piehler HR, Saigal S. Experimental and ®nite element analytical guidelines for fabricating continuous ®ber (SCS-6) metal-matrix (Ti-6AI-4V) composites via the foil/®ber/foil tecnique. Journal of Composite Materials 1994;28:1694±722. [6] Karakuzu R, OÈzcan R. Exact solution of elasto-plastic stresses in a metal-matrix composite beam of arbitrary orientation subjected to transverse loads. Composite Science and Technology 1997;56:1383±9. [7] Yang HTY, He CC. Three dimensional ®nite element analysis of free edge stresses and delamination of composite laminates. Journal of Composite Materials 1994;28:1394±412. [8] Karakuzu R, Sayman O. Elasto-plastic ®nite element analysis of orthotropic rotating discs with holes. Computers and Structures 1994;51:695±703.
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[9] Hu HT, Schnobrich WC. Non-linear ®nite element analysis of reinforced concrete plates and shells under monotonic loading. Computers and Structures 1991;38:637±51. [10] Choi CK, Kwak HG. The eect of ®nite element mesh size in non-linear analysis of reinforced concrete structures. Computers and Structures 1990;36:807±15. [11] Lin CC, Kuo CS. Buckling of laminated plates with holes. Journal of Composite Materials 1989;23:536± 53. [12] Bathe KJ. Finite element procedures in engineering analysis. Englewood Ci, NJ: Prentice-Hall, 1982. [13] Jones RM. Mechanics of composite materials. Tokyo: McGraw-Hill Kogakusha, 1975. [14] Owen RJ, Figuerias JA. Anisotropic elasto-plastic ®nite element analysis of thick and thin plates and shells. International Journal for Numerical Methods in Engineering 1983;19:541±66.
www.elsevier.com/locate/compstruc
Elasto-plastic stress analysis of aluminum metal-matrix composite laminated plates under in-plane loading O. Sayman a,*, H. Akbulut a, C. Meric° b a
Department of Mechanical Engineering, Dokuz EyluÈl University, Bornova, Izmir, Turkey b Department 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 laminated metal-matrix composite plates. Long stainless steel ®ber reinforced aluminum metal-matrix composite layer is manufactured 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 deformation theory and nine-node Lagrangian ®nite element is used. The in-plane load is increased gradually. # 2000 Elsevier Science 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 and the ductility of the matrix provide a new material with superior properties. Plastic deformations and residual stresses are important in composite laminated plates. Residual stresses 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 mechanical loading. In this study, aluminum matrix is reinforced by boron ®bers. Metal-matrix composites consist of a ductile, usually low strength matrix reinforced with elastic brittle, or ductile strong ®bers
* Corresponding author.
which provide a new material with superior properties such as high strength and stiness, low density and resistance to corrosion, high creep and fatigue properties [2±5]. Karakuzu and OÈzcan [6] have given an exact solution to the elasto-plastic stress analysis of an aluminum metal-matrix composite beam reinforced by steel ®bers. Linear or nonlinear ®nite element method can be used to analyze the laminated composites [7±10]. In this study, aluminum metal-matrix composite laminated 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 composed of orthotropic layers bonded symmetrically or antisymmetrically about the middle surface of the plate. In the solution of this problem, the Cartesian
0045-7949/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 9 9 ) 0 0 0 8 6 - 3
56
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Fig. 1. Loading of laminated plate.
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 assumption that particles of the plate originally on a line that is normal to the undeformed middle surface remain on a straight line during deformations, but this line is not necessarily normal to the deformed middle surface. By using this assumption the displacement components of a point with coordinates x, y, z for small deformations are: 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 point of the middle surface, and cx ,cy are the rotations of
normals to midplane about the y and x axes, respectively. The bending strains vary linearly through the plate thickness, whereas shear strains are assumed to be constant through the thickness as @ cx @ u0 @x 9 @x 8 < ex = @ v0 @ cy z ey @y : g ; @y xy @ cx @ cy @ u0 @ v0 @y ÿ @x @y @x e e0x Kx x 0 e ey z Ky y Kxy gxy g0 xy @w ÿ cy gyz @ y gxz @ w @ x cx
or
2
The total potential energy of a laminated plate under static loadings is given as P Ub Us V
Fig. 2. The production of composite layer.
3
where U b is the strain energy of bending, U s is the strain energy of shear and V represents potential energy of external forces. They are as
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Ub
1 2
Us
1 2
h=2 ÿh=2
A
h=2 ÿh=2
ÿ sx ex sy ey txy gxy dA dz
ÿ A
txz gxz tyz gyz dA dz
Vÿ
A
wp dA ÿ
@R
b b N n u0n N s u0s ds
4
where dA dx dy, p is the transverse loading per unit b b area and N n and N s are the in-plane loads applied on the boundary @ R:
3. Production of laminated plates The composite layer consists of stainless steel ®ber and aluminum matrix. The production has been realized by using moulds which consist of upper and lower parts. Electrical resistance has been used to heat the moulds and material which are insulated, as illustrated in Fig. 2. The hydraulic press has been used to obtain a pressure of 30 MPa to the upper mould. Manufacturing set has been heated to 6008C. In these conditions, the yield strength of aluminum is exceeded and good bonding between matrix and ®ber has been realized. The mechanical properties, yield points and plastic parameters are given in Table 1. It is assumed that the yield point Z (in the z direction) is equal to the yield point Y (in the y direction), the yield points of txz , tyz are equal to S. The von Mises and Tresca criteria are used generally for isotropic materials. Huber±Mises yield criterion has been generalized by Hill for anisotropic metals [14]. It is more appropriate to use Huber±Mises or Tsai±Hill failure criteria for anisotropic metals, since their yield points are dierent in longitudinal and lateral directions. The dierence Table 1 The measured mechanical properties and yield points of a layer Mechanical properties E1 E2 G12 n12 Yield strengths and parameters Axial strength, X Transverse strength, Y Shear strength, S Hardening parameter, k Strain hardening parameter, n
86 GPa 74 GPa 32 GPa 0.30 228.3 24.2 47.6 1254
MPa MPa MPa MPa 0.7
57
between the numerical results for the residual stresses in the symmetric cross-ply ([08/908]2) laminated plate obtained for Huber±Mises and Tsai±Hill criteria is found as 0.8% as shown in Table 2. The Tsai±Hill criterion is used as a yield criterion [13]. Four layers have been bonded to form a laminated plate symmetrically or antisymmetrically by using a pressure of 30 MPa and 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 is composed of four layers. A typical ®nite element for a symmetric and an antisymmetric lamination is divided into eight imaginary layers for obtaining the ®nite element 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 following matrix form: 2 3 2 3 u0 0 0 0 Ni 0 6 7 60 6 v0 7 Ni 0 0 0 7 7 n 6 6 7 X 6 7 7 60 d 6 0 Ni 0 0 7 6w 7 6 7 6 7 7 i1 6 6 cx 7 0 0 0 N 0 4 5 i 4 5 cy 0 0 0 0 Ni 2 3 u0 6 7 6 v0 7 6 7 6w 7 5 6 7 6 7 6 cx 7 4 5 cy i
in which n is the total number of nodes and Ni is the shape function at node i. The relationship between strains and displacements can be written in the matrix form: 2 3 32 3 2 e0 0 0 0 Ni,x 0 6 ex0 7 6 y 7 60 7 6 u0 7 Ni,y 0 0 0 6 7 6 7 6 v0 7 6 0 7 6 Ni,y Ni,x 0 0 76 7 0 6 gxy 7 6 7 6w 7 6 6 7 60 76 7 0 0 Ni,x 0 6 Kx 7 6 7 4 cx 5 6 7 40 0 0 0 ÿNi,y 5 4 Ky 5 cy 0 0 0 N ÿNi,x i i i,y Kxy i
or symbolically as febi g Bbi ui
7
in which i is the node number, and commas represent partial derivatives. The relationship between the transverse shear strains
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O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Table 2 Residual stresses in the symmetric cross-ply, ([08/908]2), laminated square plate for 200 iterations Orientation angle
sx (MPa)
The result for Tsai±Hill criterion 08 1.844 908 ÿ1.844 The result for Huber±Mises criterion 08 1.859 908 ÿ1.859
sy (MPa)
txy (MPa)
txz (MPa)
tyz (MPa)
0.501 ÿ0.500
ÿ0.001 0.001
0.000 0.000
0.000 0.000
ÿ0.501 0.501
0.001 ÿ0.001
0.000 0.000
0.000 0.000
and displacement components can be written as 2 3 u0 7 6 6 v0 7 ÿNi 6 7 0 0 Ni,y 0 gyz 6w 7 0 0 Ni,x Ni 0 gxz i 7 i6 4 cx 5 cy
A44 ,A55
8
i
or symbolically as fesi g Bsi fui g
9
The stiness matrix of the plate element is obtained by using the minimum potential energy method or the principle of virtual displacements [12]. Bending and shear stiness matrices are Kb Bb T Db Bb dA A
Ks
A
Bs T Ds Bs dA
10
Db
ÿ Aij ,Bij ,Dij
Bij Dij
h=2 ÿh=2
Ds
k21 A44 0
0 k22 A55
ÿ Qij 1,z,z2 dz i,j 1,2,6
Q44 ,Q55 dz
11
jBb j645 , jBs j245 and Db and Ds are the bending and shear parts of the material matrix, respectively. A45 is negligible in comparison with A44 and A55 [11] and shear correction factors for rectangular cross sections are given as [11], k21 k22 5=6: Once the nodal displacements are calculated, the strain components of each layer can be found by using Eqs. (7) and (9); and the stress components can be calculated and used to check the yield state of the material. Since the calculated stresses do not generally coincide with the true stresses in a nonlinear problem, the unbalanced nodal forces and the equivalent nodal forces must be calculated. The equivalent nodal point forces corresponding to the element stresses at each iteration can be calculated as fR gequivalent B T s dA
Aij Bij
ÿh=2
where
h=2
vol
vol
Bb T sb dA
vol
Bs T ss dA
12
When the equivalent nodal forces are known, the unbalanced nodal forces can be found by fR gunbalanced fR gapplied ÿfR gequivalent
13
These unbalanced nodal forces are applied for obtain-
Fig. 3. A layered section for (a) symmetric and (b) antisymmetric lamination.
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
59
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.
ing increments in the solution and must satisfy the convergence tolerance in a nonlinear analysis. The dierence between the plastic and elastic solution gives the residual stresses. The residual stresses may increase the possibility of failure of the laminated plates. In this solution 216 nodes and 48 elements are used.
5. Numerical results and discussions The laminated plate is assumed to be under uniform axial in-plane loads along the rectangular edges and the circular hole is unloaded. The laminated plates are composed of four orthotropic and generally orthotropic 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 stiness 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 expansion of the plastic zone and the residual stresses in the cross-ply symmetric laminated plate ([08/908]2) without a 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. Residual 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. 5. Residual stress components sx (MPa) along AB for cross-ply ([08/908]2): (a) symmetric and (b) antisymmetric laminated plates.
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O. Sayman et al. / Computers and Structures 75 (2000) 55±63
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.
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 eect of orientation angle on the expansion of plastic zone is presented in Fig. 6, for ([308/ ÿ308]2) symmetric and antisymmetric angle-ply laminated 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, respectively, and when it is further increased incrementally the plastic zone expands around the hole. The expansion of plastic zone in the layer of 308 orientation angle is slightly dierent from that of the layer of ÿ308 orientation angle in each case. The eect 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 608 and ÿ608 are slightly dierent 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 components 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 compressive and tensile in the layers of 908 and 08 orientation 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. 7. Expansion of plastic zone in (a) symmetric and (b) antisymmetric ([458/ÿ458]2) laminated plates; and its distribution across the cross-section for 800 iterations.
O. Sayman et al. / Computers and Structures 75 (2000) 55±63
61
Fig. 8. Expansion of plastic zone in (a) symmetric and (b) antisymmetric ([608/ÿ608]2) laminated plates; and its distribution across the cross-section for 800 iterations.
Table 3 Yield points in symmetric and antisymmetric laminated plates
Symmetric Nx (N/mm) Antisymmetric Nx (N/mm)
([08/908]2)
([308/308]2)
([458/458]2)
([608/608]2)
80.56 72.68
140.96 136.89
109.76 107.12
88.16 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 Elasto-plastic solution 200 400 800 Elastic solution 200 400 800 Residual stresses 200 400 800
Orientation angle
sx (MPa)
sy (MPa)
txy (MPa)
tyz (MPa)
txz (MPa)
908 08 908 08 908 08
24.213 37.223 24.211 44.793 24.208 61.115
2.793 5.729 2.326 7.255 1.377 10.591
ÿ0.675 ÿ0.6981 ÿ0.822 ÿ0.832 ÿ1.083 ÿ1.10
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
908 08 908 08 908 08
29.248 34.212 34.260 40.075 44.285 51.801
3.901 4.638 4.570 5.433 5.907 7.023
ÿ0.684 ÿ0.684 ÿ0.801 ÿ0.801 ÿ1.036 ÿ1.036
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
908 08 908 08 908 08
ÿ5.035 3.011 ÿ10.049 4.718 ÿ20.077 9.314
ÿ1.108 1.091 ÿ2.244 1.822 ÿ4.530 3.568
0.009 0.003 ÿ0.021 ÿ0.030 ÿ0.048 ÿ0.066
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
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O. Sayman et al. / Computers and Structures 75 (2000) 55±63
Table 5 Residual stress components in the symmetric angle-ply laminated plates for 800 iterations Laminated plate ([308/ÿ308]2) ([458/ÿ458]2) ([608/ÿ608]2)
308 ÿ308 458 ÿ458 608 ÿ608
sx (MPa)
sy (MPa)
txy (MPa)
tyz (MPa)
txz (MPa)
ÿ4.622 ÿ4.012 ÿ8.684 ÿ7.924 ÿ12.292 ÿ11.630
ÿ5.330 ÿ4.580 ÿ6.436 ÿ5.827 ÿ5.628 ÿ4.900
2.960 ÿ2.551 4.247 ÿ3.818 4.488 ÿ3.967
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
Table 6 Residual stress components in the antisymmetric angle-ply and cross-ply laminated plates for 800 iterations Laminated plate ([08/908]2) ([308/ÿ308]2) ([458/ÿ458]2) ([608/ÿ608]2)
908 08 308 ÿ308 458 ÿ458 608 ÿ608
sx (MPa)
sy (MPa)
txy (MPa)
tyz (MPa)
txz (MPa)
ÿ18.806 9.175 ÿ4.710 ÿ3.977 ÿ8.868 ÿ7.808 ÿ12.580 ÿ11.580
ÿ3.954 3.604 ÿ5.212 ÿ4.561 ÿ6.031 ÿ5.965 ÿ4.478 ÿ5.203
ÿ0.111 ÿ0.017 3.054 ÿ2.487 4.386 ÿ3.727 4.524 ÿ3.899
ÿ0.128 ÿ0.202 ÿ0.100 0.085 ÿ0.162 0.143 ÿ0.183 0.144
ÿ0.127 ÿ0.251 0.058 0.046 0.076 0.068 0.086 0.0622
([458/ÿ458]2) and ([608/ÿ608]2) laminated plates are given in Table 5. It is seen from this table that as we increase the orientation angles in the symmetric angleply laminated plates the sx residual stress component becomes greater. The maximum absolute value of the stress 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) laminated plates are given in Table 6. It is seen that as we increase the orientation angles in the antisymmetric angle-ply laminated plates the sx residual stress component becomes greater. The maximum intensity of the stress components in the plate is given in the table.
nates ([08/908]2), yielding occurs in the layers of 908 orientation angle and the residual stress Nx is compressive and tensile in layers of 908 and 08 orientation angles, respectively, without a hole. 3. The orientation angle y aects the yield points of laminated plates. 4. The yield point of symmetric laminated plates is higher than that of the yield point of antisymmetric laminated plates. 5. Load carrying capacity of the laminated plate can be increased by means of residual stresses.
References 6. Conclusions Elasto-plastic stress analysis has been carried out by using the ®rst-order shear deformation theory in aluminium±steel ®ber laminated plates. The expansion of the plastic zone and residual stresses are obtained in symmetric 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-
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