Residual Stresses in Composite Plates with Central Filleted Hole HAMIT AKBULUT Department of Mechanical Engineering Atatu¨rk University, Erzurum, Turkey MEHMET SENEL* Department of Mechanical Engineering Dumlup|nar University, Ku¨tahya, Turkey ABSTRACT: This article deals with the residual stresses in stainless steel fiber-reinforced aluminum composite plates under in-plane loading that have a central filleted hole. In many cases, the residual stresses in the plates increase ability to carry a larger load. The dimensions of the central hole vary from square to circle by increasing fillet radii at the corners. In the study, the effects of the hole dimension, the fillet radius, and the orientation angles on the residual stresses are investigated using the finite element method (FEM) software ANSYS. Results are presented in tabular and graphical forms. KEY WORDS: residual stresses, composite plate, filleted holes, finite element methods.
INTRODUCTION those containing more than one bonded material, each with different structural properties. Therefore they have been used widely in many structures such as aircraft, automobiles, sporting goods, and many consumer products. The important advantages of composite materials are the potential for a high ratio of stiffness to weight and a high ratio of strength to weight. The components such as glassepoxy, graphite-epoxy and boron-epoxy are used widely in the production of composites. However, aluminum matrices and stainless steel fibers have been used recently in some typical engineering applications because of their properties. There are many studies executed on composites in the literature. Chou et al. [1] examined the fiber-reinforced metal–matrix composites involving fabrication methods, mechanical properties, secondary working techniques, and interfaces. Yeh and Krempl [2] investigated the influence of cool-down temperature histories on the residual stresses in fibrous metal– matrix composites. Elastic-plastic finite element analysis of metal–matrix plates with edge notches was studied by Karakuzu et al. [3]. Elasto-plastic stress analysis of aluminum metal– matrix composite laminated plates under in-plane loading was studied by Sayman et al. [4]. This study presents an elastic-plastic stress analysis of symmetric and antisymmetric cross-ply,
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OMPOSITE MATERIALS ARE
*Author to whom correspondence should be addressed. E-mail:
[email protected] Figures 5 and 7 appear in color online: http://jrp.sagepub.com
Journal of REINFORCED PLASTICS
AND
COMPOSITES, Vol. 29, No. 3/2010
0731-6844/10/03 0409–13 $10.00/0 DOI: 10.1177/0731684408097781 SAGE Publications 2010 Los Angeles, London, New Delhi and Singapore
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angle-ply laminated metal–matrix composite plates. Karakuzu et al. [5] examined elastic-plastic behavior of woven-steel fiber-reinforced thermoplastic laminated plates under in-plane loading. Elastic-plastic stress analysis was carried out on simply supported and clamped aluminum metal–matrix laminated plates with a hole by Sayman and Aksoy [6]. Akbulut and Senel [7] studied residual stresses in stainless steel fiber-reinforced aluminum matrix composite plates with a central square hole. In the solution, FEM with twodimensional isoparametric rectangular nine node elements was employed. In the calculation of the residual stresses in the plate under uniform in-plane loads, the Newton–Raphson method (initial stress method) was exploited. Arslan et al. [8] investigated the prediction of the elastic-plastic behavior of thermoplastic composite laminated plates ([08/8]2) with a square hole. Akbulut [9] pointed out an optimization of a car rim using FEM. This study presents the optimization of an octopus-type car rim for which critical zones were found first and then optimum thickness was calculated using an elasto-plastic analysis. Elastic-plastic stress analysis and expansion of plastic zone in clamped and simply supported thermoplastic–matrix laminated plates with square hole was presented by Arslan et al. [10]. They used FEM and first-order shear deformation theory for small deformations. Bateman et al. [11] studied measurement of residual stress in thick section composite laminates using the deep-hole method, which is a method of measuring residual stress in large metallic components. Tang et al. [12] examined residual stresses and stress partitioning measurements by neutron diffraction in Al/Al–Cu–Fe composites. Correlation between matrix residual stress and composite yield strength in PM 6061Al–15 vol% SiC was examined by Pedro Ferna´ndez [13]. Residual stress and thermal expansion of graphite epoxy laminates subjected to cryogenic temperatures was indicated by Ifju et al. [14]. Residual stresses in composite cross-ply laminates depending on thermoelastic properties of the material and processing temperatures were conducted by Jeronimidis and Parkyn [15], in which their distribution in the various laminae is a function of stacking sequence and ply orientation. Ku¨c¸u¨k [16] investigated elasto-plastic stress analysis and residual stresses in metal–matrix-laminated plates under in-plane and transverse loading. In this study, steel fiber-reinforced aluminum metal–matrix composite-laminated plates were loaded under in-plane and transverse forces. FEM with Tsai–Hill yield criterion were used in the solution. The laminated plates were oriented symmetrically and antisymmetrically. In this study, the residual stresses in stainless steel fiber–reinforced aluminum metal– matrix composite plates with central filleted hole are investigated. The dimension of the hole varies from square to circle by increasing fillet radii at its corners. This study, based on Tsai–Hill yield criterion, follows that of Akbulut and Senel, in which the plates had a central square hole [7]. In the study, the effects of the hole dimension, the fillet radius, and the orientation angles on the residual stresses were explored using the commercial FEM software ANSYS. MATHEMATICAL FORMULATIONS Stress–Strain Relationship in an Orthotropic Plate The two-dimensional stress–strain relations in an orthotropic layer can be written as:
1
2
12
T
¼ ½Q "1
"2
12
T
ð1Þ
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where Qij are the reduced stiffness matrices for a plane stress state in the 1–2 plane in which 1–2 indicates the principal axis of the composite plates. For the orthotropic lamina, the Qij are: 2 3 Q11 Q21 0 6 7 ½Q ¼ 4 Q12 Q22 ð2Þ 0 5 0
0
Q66
where Q11, Q22, Q12, or Q21 and Q66 are elastic constants in terms of Young’s modules, Poisson’s ratio, and shear modulus, respectively. They are defined as: Q11 ¼ Q12
E1 1 12 21
Q22 ¼
9 > > =
E2 1 12 21
21 E1 12 E2 ¼ ¼ 1 12 21 1 12 21
> > ¼ G12 : ;
Q66
ð3Þ
The stresses in an x–y coordinate system can be transformed into the principal material direction (1–2 coordinate systems) as:
1
2
12
T
¼ ½T x
y
xy
T
ð4Þ
where [T ] is the transformation matrix and is written as: 2
c2 s 2 4 ½T ¼ s2 c2 sc sc
3 2sc 2sc 5 c2 s 2
ð5Þ
where c ¼ cos and s ¼ sin , and is the orientation angle between x and 1-axes (Figure 2(b)). Finally the stress–strain relations in the x–y coordinate system can be abbreviated as:
x
y
xy
T
¼ Q "x
"y
xy
T
in which Qij denotes the transformed reduced stiffness matrix as: 2 3 Q11 Q21 Q16 6 7 Q ¼ ½T 1 ½Q½T T ¼ 4 Q12 Q22 Q26 5 Q16 Q26 Q66
ð6Þ
ð7Þ
where the superscripts 1 and T denote the matrix inverse and the matrix transpose. Plastic Formulation The Tsai–Hill yield criterion used in the elasto-plastic solution is given as: 2 12 1 2 22 12 2 þ 2 þ 2 ¼ 1: 2 X X Y S
ð8Þ
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Upper mold pressure
Stainless steel fiber
Aluminum sheet
Lower pressure mold Figure 1. Production scheme for the composite plate.
Table 1. Mechanical properties of plate material. Elasticity modulus in 1-direction Elasticity modulus in 2-direction Shear modulus Poisson’s ratio Axial strength Transverse strength Shear strength
E1 E2 G12 12 X Y S
86 GPa 74 GPa 32 GPa 0.30 228.3 MPa 24.2 MPa 47.6
Equivalent stress in the first principal material direction is found by multiplying Equation (8) by X: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X2 X2 2 : ¼ 12 1 2 þ 2 22 þ 2 12 Y S
ð9Þ
The principal stresses are substituted into the Tsai–Hill equation to examine the yielding conditions of the composite plate. PRODUCTION OF COMPOSITE PLATE In this study, a production set-up was established with the intention of obtaining the mechanical properties of the composite plates consisting of stainless steel fiber and aluminum matrix (Figure 1). Composite plates were manufactured using a hydraulic press machine generating a pressure of 30 MPa at 6008C to the upper mold. Then the material properties of the composite plates (E1, E2 are Young’s modules; G12, 12 are the shear modules and Poisson’s ratio corresponding to the plane axis 1–2; X, Y, and S are axial, transverse, and shear yield strengths, respectively) were obtained using an Instron testing machine and are listed in Table 1. PROBLEM STATEMENT For the purpose of the present study, the square plates with a central filleted hole are considered. All the plates are assumed to be under in-plane loads along the edges and the central hole is free. The geometry and loading of the plate used in this analysis is illustrated in Figure 2. The plate dimensions are 200 200 mm and its thickness is 2 mm.
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(b)
y
y 1
2 q
C Nx
x L
d r
x (c)
d Fillets
L
Figure 2. (a) Geometry, coordinates and loading, (b) Axis and orientation angle, (c) Filleted hole.
Figure 3. Symbolic FEM model of whole plate.
While the hole dimensions are taken as 40 mm and 60 mm, the orientation angles are chosen as 0, 15, 30, 45, 60, 75, and 908. A symbolic finite element mesh generation of the plate with quadratic shell elements is shown in Figure 3. RESULTS AND DISCUSSION For the steel fiber-reinforced–aluminum matrix composite plates under in-plane load, the effects of the hole dimensions and the hole fillet radius on residual stresses have been investigated for varying hole dimensions and fillet radius. For this reason the residual stresses at point C, a critical point at the hole corner, are taken into consideration. In Figures 4–7, hole size effects on the residual stresses in the plate are presented in both
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txy
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(g)
r (mm) 20 10 0 0
5
10
15
20
25
−10 −20 −30
sx
sy
txy
−40 r (mm)
Figure 4. (a–g) Residual stresses at point C vs. fillet radius for d ¼ 40 mm and Nx ¼ 80 N/mm: (a) ¼ 08, (b) ¼ 158, (c) ¼ 308, (d) ¼ 458, (e) ¼ 608, (f) ¼ 758, (g) ¼ 908.
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(b) MN
Y
Y
MX X
Z
MX X
Z
.259789 15.623 30.986 46.348 61.711 7.941 23.304 38.667 54.03 69.393
(c)
.107779 12.722 25.336 37.95 50.565 6.415 19.029 31.643 44.257 56.872
(d) MN
Y
Y
MX
Z
X
Z
MX X
.226807 11.753 23.279 34.805 46.331 5.99 17.516 29.042 40.568 52.094
.16039 12.045 23.93 35.815 47.7 6.103 17.988 29.872 41.757 53.642
(e)
Y MX Z
X
.192181 12.839 25.487 38.134 50.781 6.516 19.163 31.81 44.457 57.105
Figure 5. Von Mises equivalent stress distribution vs. r in composite plate reinforced with 458 for d ¼ 40 mm: (a) r ¼ 0, (b) r ¼ 5 mm, (c) r ¼ 10 mm, (d) r ¼ 15 mm, (e) r ¼ 20 mm.
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M. SENEL
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sx
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Figure 6. (a–g) Residual stresses at point C vs. fillet radius for d ¼ 60 mm and Nx ¼ 80 N/mm: (a) ¼ 08, (b) ¼ 158, (c) ¼ 308, (d) ¼ 458, (e) ¼ 608, (f) ¼ 758.
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Residual Stresses in Composite Plates with Central Filleted Hole (a)
(b)
MX
Y Z
.379166 7.799
15.22
X
Z
X
30.06 22.64
Y
MX
44.901 37.481
.513763
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52.321
(c)
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9.261
67.162
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Y Z
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MN
MX Z
X
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45.44 52.902
60.364 67.826
.820828 7.608
14.396 21.183
X
27.971 34.758
41.546 48.333
55.121 61.908
Figure 7. Von Mises equivalent stress distribution vs. r in composite plate reinforced with 308 for d ¼ 60 mm: (a) r ¼ 0, (b) r ¼ 10 mm, (c) r ¼ 20 mm, (d) r ¼ 30 mm.
graphical and picture forms. For small plastic strain statement, the value of Nx does not exceed 80 N/mm (Nx ¼ 80 N/mm). In Figure 4, the hole dimension is taken as d ¼ 40 mm. In Figure 4(a), the residual stresses in the plate reinforced with ¼ 08 are presented. It is seen that x has 45 MPa, which is the highest negative value at r ¼ 0. Then, it becomes 0 at r ¼ 5 mm and later its curve varies around 0. However, the curves of y and xy do not fluctuate extremely up to r ¼ 20 mm. In Figure 4(b), when the plate is reinforced with ¼ 158 at r ¼ 0, x has 42 MPa, which is the highest negative value, while y has 21 MPa. Then, both stress components having 15 MPa coincide at r ¼ 5 mm. Afterwards, they nearly do not change. However, xy has positive values ranging from 13 to 10 MPa. Results show that the variation of the residual stresses mainly depends on the fillet radius and the orientation angle.
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In Figure 4(c), it is seen that when the plate is reinforced with ¼ 308 at r ¼ 0, x and y are 40 and 33 MPa, respectively. Then they gradually increase up to 15 MPa. However, while xy has 20 MPa at r ¼ 0, it reaches 21 MPa maximum value and decreases up to r ¼ 15 mm, then it becomes 10 MPa. Figure 4(d) indicates that at r ¼ 0 x ¼ 54 MPa and y ¼ 38 MPa, at r ¼ 5 mm x ¼ 30 MPa and y ¼ 22 MPa, then they vary slightly up to r ¼ 20 mm. After r ¼ 10 mm their values coincide. While xy has 30 MPa at r ¼ 0, it decreases 20 MPa at r ¼ 5 mm, then it does not fluctuate much up to r ¼ 20 mm. Figure 4(d) shows that when the plate is reinforced with ¼ 458, larger residual stresses occur. While the curves in Figure 4(e) are similar to those in Figure 4(d), the values of x and y do not coincide until r ¼ 20 mm. In Figure 4(f), in contrast to the former, the absolute value of y is larger than that of x at r ¼ 0. Then, the values of x are smaller than that of y up to r ¼ 20 mm. After r ¼ 3 mm, they do not coincide. While maximum value of xy is 15 MPa at r ¼ 5 mm, other values are about 12 MPa. It is deduced from Figure 4(f) that when the plate is reinforced with ¼ 758, y is more pronounced than x at r ¼ 0. In Figure 4(g), it is striking that all values of y are smaller than those of x. The curve of y is an exponentially decaying function of r. However, while x is 5 MPa at r ¼ 0, its minimum value occurs in between r ¼ 5 mm and r ¼ 10 mm and then it increases slightly. The xy curve takes its values at about 13 MPa. Residual stresses vs. orientation angles and hole fillet radius in composite plates at point C are presented in Table 2. It is understood from Table 2 that x, y, and xy are greater when ¼ 458. Figures 5(a–e) indicate the stress distributions in the composite plates (under in-plane load) with a hole whose geometry varies from square to circular form. It is clear from these figures that stresses become concentrated around the central hole, especially when the hole is of a square form where the residual stresses become more apparent at the hole corners. As the hole fillet radius gets larger, the residual stresses spread around the periphery of the hole first, and then to the other points of the plate. In Figure 6, the hole dimension in the center of the composite plate is taken as d ¼ 60 mm. In Figure 6(a), the residual stresses in the plate reinforced with ¼ 08 are given. x is 50 MPa, which is the highest negative value at r ¼ 0. Then, it becomes maximum Table 2. Residual stress components at point C for d ¼ 40 mm (stress unit is MPa).
r ¼ 0 mm
r ¼ 5 mm
r ¼ 10 mm
r ¼ 15 mm
r ¼ 20 mm
h
08
158
308
458
608
758
908
x y xy x y xy x y xy x y xy x y xy
44.7 5.75 8.248 0.248 5.96 2.414 1.435 0.348 1.079 2.121 1.709 2.032 2.429 2.105 2.346
42.086 21.596 14.323 14.6 14.172 13.186 11.557 11.515 11.235 10.482 10.504 10.385 10.073 10.137 9.993
40.236 33.355 19.863 26.934 22.433 20.09 16.478 15.246 15.392 12.799 12.502 12.549 12.342 12.265 12.222
53.993 37.918 28.923 27.623 22.377 20.53 21.443 21.122 21.685 19.451 19.052 19.138 17.491 16.929 17.075
52.081 32.354 23.826 27.503 16.898 15.345 19.619 15.997 15.759 16.766 14.192 15.215 14.955 14.332 14.434
21.976 33.056 12.311 22.689 13.874 16.669 17.366 13.575 13.176 16.226 14.167 13.775 15.457 13.301 13.754
5.715 31.71 14.159 10.252 18.092 15.43 8.877 14.185 13.34 7.756 13.647 12.327 6.991 13.864 11.756
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(15 MPa) at r ¼ 13 mm and later it approaches 0 at r ¼ 30 mm. However, all values of y are positive. xy varies from 15 MPa at r ¼ 0 to 14 MPa at r ¼ 13 mm, and then to 4 MPa at r ¼ 30. It is clear from Figure 6(b) that while x ¼ 52 MPa and y ¼ 18 MPa at r ¼ 0, their values coincide at r ¼ 10, 20 and 30 mm. However, xy decreases from 22 to 5 MPa at r ¼ 14 mm first, and then remains constant. In Figure 6(c), it is seen that when the plate is reinforced with ¼ 308 at r ¼ 0, x and y are 55 and 46 MPa, respectively. Then they increase exponentially up to 11 MPa and they remain coincident after r ¼ 20 mm. However, while xy is 24 MPa at r ¼ 0, it reaches its maximum value of 25 MPa at r ¼ 10 mm, and then it becomes 11 MPa at r ¼ 30 mm. Figure 6(d) indicates that while x ¼ 58 MPa and y ¼ 42 MPa at r ¼ 0, they remain coincident after r ¼ 10 mm at a value of 25 MPa. They later become 20 MPa at r ¼ 30 mm. However, while xy is 30 MPa at r ¼ 0, it decreases to 20 MPa at r ¼ 10 mm, then it does not change until r ¼ 30 mm. Figure 6(e) indicates that while x ¼ 54 MPa and y ¼ 45 MPa at r ¼ 0, and x ¼ 24 MPa and y ¼ 17 MPa at r ¼ 10 mm, x ¼ 17 MPa and y ¼ 15 MPa at r ¼ 20 mm, they coincide at a value of 15 MPa at r ¼ 30 mm. On the other hand, while xy is 28 MPa at r ¼ 0, it decreases 18 MPa at r ¼ 10 mm, then it does not change up to r ¼ 30 mm. In Figure 6(f), in contrast to the former cases, x and y are 34 and 42 MPa at r ¼ 0, respectively. After intercepting at r ¼ 5 mm, the values of x become greater than those of y. The curve of xy is similar to the former ones, xy ¼ 24 MPa at r ¼ 0. In Figure 6(g), it is also striking that all values of y are smaller than those of x. The curves of x and y possess an exponential variation with respect to radius r. Having 22 MPa at r ¼ 0, xy decreases to 15 MPa at r ¼ 10 mm. Then, its curve remains constant up to r ¼ 30 mm. Figures display that fillet radius is not of great influence on shearing stress after ¼ 458. At the point C of the composite plates subjected to in-plane load, residual stresses vs. orientation angles and hole fillet radius are presented in Table 3. It is understood from Table 2 that x, y, and xy are generally greater when ¼ 458. Obviously it is seen that the values of stress components are greatest at r ¼ 0 when the orientation angle is small, 758. Figures 7(a–d) display the stress distributions in the composite plates with a hole whose geometry varies from square to circle. The plates reinforced with 308 are subjected to an inplane load of 80 N/mm. It is clear from these figures that stresses are concentrated around Table 3. Residual stress components at point C for d ¼ 60 mm (stress unit MPa).
r ¼ 0 mm
r ¼ 10 mm
r ¼ 20 mm
r ¼ 30 mm
h
08
158
308
458
608
758
908
x y xy x y xy x y xy x y xy
50.265 14.292 14.449 12.182 15.486 13.149 6.14 6.475 6.124 3.072 3.246 3.039
52.803 17.304 22.815 6.346 5.575 5.949 4.499 4.514 4.549 4.892 4.951 4.938
55.096 46.089 25.101 27.91 25.716 26.224 16.625 16.274 16.539 12.848 12.765 12.834
57.224 42.544 29.208 24.471 23.574 21.362 18.648 19.855 19.792 18.228 17.921 18.096
53.68 45.253 27.667 23.237 17.416 16.344 17.303 15.762 15.349 16.059 15.338 15.533
34.94 42.261 24.321 21.671 15.864 15.948 16.722 14.544 14.356 14.447 13.76 13.718
21.154 30.53 22.174 13.826 15.519 15.505 11.086 14.723 14.388 11.113 14.264 13.811
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the central hole and around the diagonals of the plates, especially when the hole is of square form and the residual stresses are also pronounced at the hole corners. As the hole fillet radius increases, the residual stresses spread around the circle/hoop of the hole. It is seen in Figure 7 that the residual stresses spread in a wider zone in the plate than those of Figure 5. CONCLUSIONS In this study, elasto-plastic stress analysis has been performed in stainless steel fiber reinforced-aluminum composite plates with central filleted hole under in-plane loading. By means of residual stresses, load-carrying capacity of a plate can be increased. Cut-outs as a hole in a plate or anisotropy can yield residual stresses. Therefore, in this study, the effects of the hole dimension, the fillet radius, and the orientation angles on the residual stresses have been investigated using the software ANSYS based on FEM. The following results can be concluded: . In each case, larger residual stresses occur around the central hole, especially when the hole is of square form and the residual stresses are concentrated at the corners of the hole. When the plate is reinforced with 308, the residual stresses are concentrated around the diagonals of the plate. . At r ¼ 0, that is the central hole is of a square form, the largest residual stresses arise at the corners of the hole. . While x has larger absolute values than y for all cases, when the orientation angle is greater than 758, y has larger absolute values than x. . The enlargement of the hole fillet radius has a minor effect on the shear stress xy. . All stress components have their highest values when the orientation angle is 458.
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