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The effect of ply number, orientation angle and bonding type on residual stresses of woven steel fiber reinforced thermoplastic laminated composite plates subjected to transverse uniform load
Composites Science and Technology 64 (2004) 1049–1056 www.elsevier.com/locate/compscitech
The effect of ply number, orientation angle and bonding type on residual stresses of woven steel fiber reinforced thermoplastic laminated composite plates subjected to transverse uniform load Ramazan Karakuzua,*, Zu¨leyha Aslanb, Buket Okutanb . Department of Mechanical Engineering, Dokuz Eylu¨l University, Bornova-Izmir, Turkey b Department of Mechanical Engineering, Cumhuriyet University, Sivas, Turkey
a
Received 23 August 2001; received in revised form 26 August 2003; accepted 3 September 2003
Abstract This paper is concerned with the numerical results of the elasto-plastic stress analysis and residual stresses in woven steel fiber reinforced thermoplastic laminated composite plates for transverse uniform loads. The effects of ply number, orientation angle and bonding type on the residual stresses of laminated composite plates are investigated. Elasto-plastic stress analysis is carried out in the laminated plate by using the finite element technique. The finite element code ANSYS is used to perform the numerical analyses using an eight-node layered shell element. Yielding loads and residual stresses are obtained for symmetric and anti-symmetric laminated plates with simply supported boundary conditions. Different stacking sequences of laminated composites are used in analysis and the results are compared with each other. Three load steps are carried out for each analysis consecutively. In the first load step, the yielding transverse load is applied. Secondly, a series of load increments is added until the load reaches ‘‘Yielding Load+0.005 MPa’’. In the last step the external load is released to obtain the residual stress components. # 2003 Elsevier Ltd. All rights reserved. Keywords: C. Residual stress; A. Layered structures; B. Mechanical properties; B. Plastic deformation; Thermoplastic composite plate
1. Introduction The increasing interest in thermoplastic matrix composites operating in extremely severe mechanical and thermal environments has brought more attention to the design of structures made of these materials. They offer high specific stiffness and specific strength, improved fracture toughness, and increased impact resistance. Moreover thermoplastic composites possess the unique characteristic that they may be remelted, reprocessed and reformed. They are easily repaired and they can be remelted to repair the local cracks and delaminations. Therefore, the characterization of the elasto-plastic response is indispensable in the limit analysis stage, which must be performed along the reliable design processes for composite structures involving plasticity effects in the nonlinear behavior.
* Corresponding author. E-mail address: [email protected] (R. Karakuzu). 0266-3538/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2003.09.014
The study of the optimal design of thermoplastic composite structures is an emerging research field and few authors have presented results in this area. BaheiEl-Din et al. [1] have developed a three-dimensional finite element code for the elasto-plastic analysis of fiber-reinforced composite materials and structures. They have obtained the solution of the nonlinear equilibrium equations with a Newton–Raphson type iteration technique. Owen and Figueiras [2] have presented the anisotropic elasto-plastic finite element analysis of thick and thin plates and shells. Jeronimidis and Parkyn [3] have investigated the residual stresses in APC-2 cross-ply laminates. Predictions based on classical laminate theory were compared to measure levels of residual stress obtained from a number of experimental techniques. The analysis of the results showed that accurate predictions can be made provided that the changes in thermoelastic properties of the materials with temperature are taken into account. Vaziri et al. [4] presented a new finite element program for in plane loading of FRC laminates. The
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R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
program is based on a new comprehensive constitutive model incorporating yielding and failure in a unified manner. Sayman and Zor [5] studied on a low-density polyethylene (LDFE, F.2.12) thermoplastic composite beam reinforced by woven steel fibers loaded by a uniformly distributed force on the upper surface. A closed form solution is performed by satisfying both equations of equilibrum and boundary conditions. Karakuzu and Ozcan [6,7] presented an analytical elasto-plastic stress analysis for a metal matrix composite beam of arbitrary orientation subjected to a single transverse force applied to the free end of the beam or uniform distributed load applied to upper surface. As the value of the orientation angle decreases the length of the beam increases. Akay and O¨zden [8] investigated the influence of residual stresses in carbon fiber-thermoplastic matrix laminates. Bahei-El-Din and Dvorak [9] studied the elastic plastic behavior of symmetric metal-matrix composite laminates for the case of in-plane mechanical loading. Local stresses, hardening parameters, and strains are found in each lamina and in the fiber and matrix phases within each lamina. Comparisons of analytical results with experimental measurements were made for certain laminated plates. Karakuzu and Sayman [10] and Karakuzu [11] worked on the elasto-plastic solutions for a fiber reinforced metal matrix composite cantilever beam. In the elasto-plastic solution, the material is assumed perfectly plastic. At distant sections from the free end, the elastic and the residual stresses are found to be high. Residual stresses are highest at the fixed end. By producing residual stresses, the beam can be strengthened. If it is loaded again by the external forces in the same direction, the cantilever can carry greater external forces than in the elastic case. On the other hand, a lot of works have been done using numerical and analytical methods to predict elastic–plastic behavior of composite structures [12–19]. Cervenka [20] has given advantages and disadvantages of thermoset and thermoplastic matrices for continues fibre composites. The present study is about the determination of elasto-plastic and residual stresses of woven steel fiber reinforced thermoplastic laminated composite plates with different lamination schemes under transverse uniform loads. A three-load step is carried out for each analysis consecutively. The effects of ply number, orientation angle and bonding type on the residual stress components of laminated composite plates are investigated numerically.
sequence and each layer of the composite. Though the finite element method is a well established tool of computer aided engineering, the quality of the simulated data is fully dependent on the models used for approximating the real behavior of the investigated component. Modeling for conventional materials is well studied, but the building of a proper model for composite structures requires more effort. In order to predict the properties of a laminated composite, the properties of the single woven lamina are of great importance. The lamina is manufactured from low-density polyethylene (LDPE) and plane woven Cr–Ni steel fibers. For matrix materials, the thermoplastic granule is melted by electrical resistance up to 160 C without pressure in 5 min. Then, the material is held for 5 min under a pressure of 2.5 MPa by moulds and the temperature is dicreased to 30 C under a pressure of 15 MPa in 3 min and polyethylene layer is manufactured. Subsequently the plane woven Cr–Ni steel fibers are put into two polyethylene layers and processed in the same way described, as shown in Fig. 1. The mechanical properties of the composite lamina given in Table 1 are measured by Instron-1114 Tensile Testing Machine of 50 kN capacity at a ratio of 0.5 mm/ min. To obtain the modulus of elasticity in the weft direction (E1), Poisson’s ratio (12) and the tensile strength in the weft direction (F1t) a specimen whose weft direction coincides with the loading direction is taken and two strain gauges perpendicular to each other are stuck on. One of them is in the weft direction, the other is in the warp direction. A uniaxial tension test is used in the weft direction and the specimen is loaded step by step up to rupture. Because of the woven structure E1 and F1t are equal to E2 and F2t respectively. For all steps, strains in the weft directions ("1) and strains in the warp directions ("2) are measured by indicator. To define G12, a specimen whose principal axis is on 45 is
2. Properties of the lamina material Static mechanical characterization has been carried out before numerical study, in order to evaluate the mechanical constants for both the homogenized
Fig. 1. Producton of composite lamina.
R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
taken and a straingauge is stuck on loading direction of the lamina. The specimen is loaded by the Instron Tensile Machine and Ex is obtained by using one straingauge in the tensile direction. Where subscript x shows the loading axis. G12 is calculated by using the values of Ex, E1, E2, 12 [21]. To find the shearing strength S, a shear test set up is used. A flat lamina, which has a T-shaped is taken as shown in Fig. 2 and loaded up to rupture. Minimum clearance is left to prevent bending and to provide contact between specimen and metal support. The load at rupture (Fmax) is taken and shear strength is calculated by the following equation. S¼
Fmax 2:h:tT
ð1Þ
The dimensions of the specimen are chosen as h=5 mm, LT=80 mm, a=10 mm and b=40 mm. The thickness of the plate tT is 2.5 mm.
3. Simulation with finite element methods The ANSYS finite element analysis program [22] is used to simulate elastic, elasto-plastic and residual stress components of woven steel fiber reinforced thermoTable 1 The measured mechanical properties and yield strengths of a lamina Mechanical properties
Magnitudes
Longitudinal modulus (E1) Transverse modulus (E2) In-plane shear modulus (G12) Major Poisson’s ratio (n12) Longitudinal strength (F1) Transverse strength (F2) In-plane shear strength (S) Plastic constant (K) Strain hardening parameter (n)
12 GPa 12 GPa 0.67 GPa 0.35 23 MPa 23 MPa 8.5 MPa 100 MPa 0.675
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plastic laminated composite plates. However some software packages, i.e. ANSYS, additionally provide a full three-dimensional approach. This program is able to calculate the behavior of laminated structures with anisotropic nonlinearities. Eight-node shell element-shell91 is used for layered applications and modeling structures with nonlinear materials. Anisotropic failure criterion at ANSYS is chosen for woven steel fiber reinforced thermoplastic laminated composite plate. The anisotropic option uses Hill’s yield criterion, which accounts for differences in yield strengths in orthogonal directions and allows for different stress–strain behavior in the x,y and z material directions. 3.1. Empirical equation for stress–strain curve It is sometimes useful to represent the stress–strain curve of a given material by an equation obtained empirically by fitting the experimental data. Eq. (2) is such an equation which will frequently fit most of a given stress–strain curve. The Ludwick stress–strain relation in plastic region is given as, ¼ 0 þ K"np
ð2Þ
where 0 , "p, n and K are the yield stress, the true strain, the strain hardening parameter and the plastic constant, respectively. For finite element analysis the material behaviour is described by the uniaxial stress–strain curves in three orthogonal directions and the engineering shear stress– shear strain curves in the corresponding directions. A bilinear response in each direction is assumed. It is assumed that stress–strain diagrams in tension and compression are the same. So mechanical properties of the composite are obtained in the tension. The initial slope of the curve is taken as the elastic modulus of the material. At the specified yield stress, the curve continues along the second slope defined by the tangent modulus (having the same units as the elastic modulus) as seen in Fig. 3. To find the tangent modulus from the empirical Eq. (2), the following operations are done: "T ¼
0 0 þ E ET
ð3Þ
"e ¼
E
ð4Þ
"p ¼ "T "e ¼
0 0 þ E E ET
ð5Þ
or "p ¼ ð 0 Þ
Fig. 2. Shear test set up.
E ET EET
and stress–strain equation is obtained as
ð6Þ
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R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056 Table 2 Real constants that are used in the analysis
Fig. 3. The uniaxial stress–strain curve.
¼ 0 þ
EET "P E ET
ð7Þ
if n has been taken as n=1 in Eq. (2) and Eq. (2) is equal to Eq. (7) K¼
EET E ET
Total number Orientation angle of layers of weft direction
Layer thickness Element total (mm) thickness (mm)
4 8 12 16 8 8 8
2.5 1.25 0.83 0.625 1.25 1.25 1.25
[0/45]2 [0/0/45/45]2 [0/0/0/45/45/45]2 [0/0/0/0/45/45/45/45]2 [0/0/15/15]2 [0/0/30/30]2 [0/0/45/45]2
10 10 10 10 10 10 10
residual stresses. In the first analysis, transverse load has been increased as a series of little load increments from zero to the point at which the first plastic strain occurs. In the second analysis, three load steps were applied transversely and consecutively as shown in Fig. 5. This figure illustrates a typical load history for a nonlinear analysis. In the load step 2, the Newton– Rhapson Iteration Method is used in the each load increment.
ð8Þ 4. Numerical results and discussion
and ET is obtained as ET ¼
KE KþE
ð9Þ
where, "T, "e and ET are the total strain, the elastic strain and the tangent modulus respectively. 3.2. Finite element model and boundary conditions The model size is 400400 mm and the whole mesh exists of 331 node and 100 element. The type of element chosen allows the calculation of the three normal stresses and the three shear stresses. The laminated plate is assumed simply supported and bonded symmetrically or anti-symmetrically about the middle surface of the plate. Cartesian coordinates (x,y,z) are used for the plate coordinates where x–y plane coincides with the middle surface of the plate and z axes denotes the transverse axis. Material directions (weft and warp) and plate directions (x, y) are shown in Fig. 4. In this figure, is the orientation angle of weft direction relative to the x-direction. Total number of layers, the orientation angle of weft direction and layer thickness are shown in Table 2. The laminate is loaded transversely. It is assumed that the uniform pressure acts on the upper surface of the laminated composite. In this paper two separate analyses were performed. The first one is to find the transverse yielding load and the second one is to obtain the
The aim of this investigation is to determine numerically the residual stress of woven steel fiber reinforced thermoplastic laminated for both symmetrically and anti-symmetrically bonded composite plates. To be sure from the results, an analysis, compare with reference [18] has been carried out in ANSYS. The results are compared to see the differences as seen in Table 3. There are acceptable differences between the present study and the reference [18]. The residual stress components of simply supported, symmetrically and anti-symmetrically bonded laminated composite plates are given in Table 4. The values of the residual stresses x and y are the same and the maximum at the mid-point of the composite plate. The other residual stress components are zero. 4.1. The effect of ply number on the yielding load of laminated composite plates The laminated composite plates of 4, 8, 12 and 16 layers that are oriented as angle ply plates [0/45], are used to see the effect of ply number on their yielding load values for two different situations in the same graph as seen in Fig. 6. The number of the layer does not have significant effect on the values of the yielding load of laminated composite plates except for the 16 layer anti-symmetric laminate as seen from Fig. 6. Yielding loads in anti-symmetric cases are greater than in the symmetric cases for all configurations.
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R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
Fig. 4. Material directions and the orientaion angle of weft direction.
Fig. 5. Load history for nonlinear analysis. Table 3 The results of simply supported symmetrically bonded [0/90]2 laminated composite plates from both the reference [18] and present study Results
Stresses
x
y
z
xy
yz
xz
Reference [18]
Elastic–plastic stresses Residual stresses Elastic–plastic stresses Residual Stresses
39.587 0.438 40.811 1.971
23.858 11.917 24.146 14.221
0.000 0.000 0.000 0.000
0.049 0.010 0.000 0.000
0.035 0.026 0.000 0.000
0.079 0.007 0.000 0.000
Present study
4.2. The effect of orientation angle and bonding type on the yielding load of laminated composite plates Each bar in the graph as shown in Fig. 7 has represents the different configuration of laminates and the graph represents the different cases of laminates according to their bonding type. The change in the orientation angle also changes the yielding load of laminated plates. The yielding load increases as the orientation angle increases in symmetric cases. The
yielding load is the maximum for the configuration [ 0/ 0/30/30 ]2 in the anti-symmetric case. The yielding loads in anti-symmetric cases are greater than in the symmetric ones for all configurations. 4.3. The effect of ply number on residual stress components The laminated composite plates of 4, 8, 12 and 16 layers that are oriented as angle ply plates [0/45], are
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R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
Table 4 The residual stress components of simply supported, symmetrically and anti-symmetrically bonded laminated composite plate (in the upper surface) Orientation angle Bonded type
x
y
z
xy
yz
xz
[0/0/15/15 ]2
3.343 3.200 3.214 1.985 3.181 0.039
3.343 3.200 3.214 1.985 3.181 0.039
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
[0/0/30/30 ]2 [ 0/0/45/45 ]2
Symmetric Anti-symmetric Symmetric Anti-symmetric Symmetric Anti-symmetric
Fig. 8. Residual stress comparison graph for simply supported symmetrically bonded [0/45] laminated plates.
Fig. 6. Comparisons of yielding loads of laminated plates having different ply number and cases.
Fig. 9. Residual stress comparison graph for simply supported anti-symmetrically bonded [0/45] laminated plates.
model’s x axis in Cartesian coordinates are represented in these figures. As seen from Fig. 8, the residual stresses had their maximum values in the midpoints of the plates and their minimum values near the supporting edges. As seen from Fig. 9 the number of plies are more effectual on anti-symmetric plates against the symmetrical ones and increasing ply number has decreased the residual stresses for anti-symmetrically bonded simply supported laminated composite plates. 4.4. The effect of orientation angle and bonding type on residual stress components Fig. 7. Comparisons of yielding loads of simply supported laminated plates.
used to see the effect of ply number on the residual stress ( x) for four different situation in the same graphic. The residual stress comparison graphs for 4, 8, 12 and 16 layer symmetrically and anti-symmetrically laminated composite plates are shown in Figs. 8 and 9. The residual stresses versus the distance that passes through the mid-point and parallel to the finite element
The residual stress–distance graphs are sketched for eight layer angle ply laminated plates to see the effect of bonding type on the residual stress changing according to distance as shown in Figs. 10–12. The orientation angle has not affected the residual stress components significantly for the plates, which are bonded symmetrically. In the anti-symmetric plates, residual stresses decrease as the orientation angle increases and residual stress reaches 0.039 MPa in configuration [0/0/45/45]2. Residual stresses are greater in symmetric plates than in anti-symmetric ones.
R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
3.
4.
5. Fig. 10. The residual stress–distance graph for angle ply laminated plate [0/0/15/15]2.
6.
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plates does not affect the yielding load significantly. For symmetrically bonded plates, ply number does not affect the residual stress components. There are not so much differences between the laminates, which have different ply number. The numbers of plies are more effectual on antisymmetric plates against the symmetric ones. Increasing ply number has decreased the residual stress components for simply supported plates. In anti-symmetrically bonded, oriented as [0/0/ 15/15]2, [0/0/30/30]2 and [0/0/45/45]2 laminated plates, residual stresses decrease as the orientation angle increases. Yielding loads in anti-symmetric cases are greater than in symmetric cases for all configuration. But residual stresses are reverse.
References
Fig. 11. The residual stress–distance graph for angle ply laminated plate [0/0/30/30]2.
Fig. 12. The residual stress–distance graph for angle ply laminated plate [0/0/45/45]2.
5. Conclusion The following conclusions from elasto-plastic analysis have been obtained: 1. The orientation angle in plies affects the yielding load and the formation of plastic region. 2. The number of plies in laminated composite
[1] Bahei-El-Din YA, Dvorak GJ, Utku S. Finite element analysis of elastic–plastic fibrous composite structures. Comput Struct 1981; 13:321–30. [2] Owen DRJ, Figueiras JA. Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells. Int J Numer Meth Eng 1983;19:541–66. [3] Jeronimidis G, Parkyn AT. Residual stresses in carbon fiber– thermoplastic matrix laminates. J Comp Mater 1988;22:401–15. [4] Vaziri R, Olson MD, Anderson DL. Finite element analysis of fibrous composite structures: a plasticity approach. Comput Struct 1992;44:103–16. [5] Sayman O, Zor M. Elastic-plastic stress analysis and residual stresses in a woven steel fiber reinforced thermoplastic composite cantilever beam loaded uniformly. J Reinf Plast Comp 2000;19: 1078–92. [6] Karakuzu R, Ozcan R. Exact solution of elasto-plastic stresses in a metal–matrix composite beam of arbitrary orientation subjected to transverse loads. Comp Sci Technol 1996;56:1383–9. [7] Karakuzu R, Ozcan R. Elasto plastic stress analysis in metal matrix composite beam of arbitrary orientation subjected to transverse single force. Modelling, Measurement and Control, B 2000;69(1):33–44. [8] Akay M, O¨zden S. Influence of residual stresses on mechanical and thermal properties of injenction moulded polycarbonate. Plastics, Rubber and Composites Processing and App 1996;25(3): 138–44. [9] Bahei-El-Din YA, Dvorak GJ. Plastic analysis of laminated composite plates. Journal of Applied Mechanics 1982;49:740–6. [10] Karakuzu R, Sayman O. Elasto-plastic analysis in metal matrix composite beam. Modelling, Measurement and Control, A 1999; 72(1):57–68. [11] Karakuzu R. Analytical solution of residual stresses in aluminum matrix–steel fiber composite beam subjected to transverse uniform distributed load. Modelling, Measurement and Control, B 2000;69(1):45–58. [12] Ifju PG, Kilday BC, Niu X, Liu SC. A novel method to measure residual stresses in laminated composites. J Comp Mater 1999;33: 1511–24. [13] Sayman O, Kayry´cy´ M. An elastic-plastic stress analysis in a thermoplastic composite cantilever beam. Comp Sci Technol 2000;60:623–31. [14] Karakuzu R, O¨zel A, Sayman O. Elastic–plastic finite element
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R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056 analysis of metal matrix plates with edge notches. Comput Struct 1995;63(3):551–8. Ersoy N, Vardar O¨. Measurement of residual stresses in layered composites by compliance method. J Comp Mater 2000;34:575– 98. Karakuzu R, Sayman O. Elasto-plastic finite element analysis of orthotropic rotating discs with holes. Comput Struct 1993;51(6): 695–703. Sayman O, C¸ally´odlu H. An elastic–plastic stress analysis of thermoplastic composite beams loaded by bending moments. Comput Struct 2000;50:199–205. Atas C, Sayman O. Elastic-plastic stress analysis and expansion
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[20]
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of plastic zone in clamped and simply supported aluminum metal–matrix laminated plates. Comput Struct 2000;49:9–19. Karakuzu R, Atas C, Akbulut H. Elastic-plastic behavior of woven-steel fibre reinforced thermoplastic laminated plates under in-plane loading. Comp Sci Technol 2001;61(10):1475–83. Cervenka A. Advantages and disadvantages of thermoset and thermoplastic matrices for continues fibre composites. Mechanics of Composite Materials and Structures NATO Science Series, Series E: Applied Science 1998;361:289–97. Jones RM. Mechanics of composite materials. Philadelphia: McGraw-Hill; 1999. ANSYS user’s manuel, 2000.
The effect of ply number, orientation angle and bonding type on residual stresses of woven steel fiber reinforced thermoplastic laminated composite plates subjected to transverse uniform load Ramazan Karakuzua,*, Zu¨leyha Aslanb, Buket Okutanb . Department of Mechanical Engineering, Dokuz Eylu¨l University, Bornova-Izmir, Turkey b Department of Mechanical Engineering, Cumhuriyet University, Sivas, Turkey
a
Received 23 August 2001; received in revised form 26 August 2003; accepted 3 September 2003
Abstract This paper is concerned with the numerical results of the elasto-plastic stress analysis and residual stresses in woven steel fiber reinforced thermoplastic laminated composite plates for transverse uniform loads. The effects of ply number, orientation angle and bonding type on the residual stresses of laminated composite plates are investigated. Elasto-plastic stress analysis is carried out in the laminated plate by using the finite element technique. The finite element code ANSYS is used to perform the numerical analyses using an eight-node layered shell element. Yielding loads and residual stresses are obtained for symmetric and anti-symmetric laminated plates with simply supported boundary conditions. Different stacking sequences of laminated composites are used in analysis and the results are compared with each other. Three load steps are carried out for each analysis consecutively. In the first load step, the yielding transverse load is applied. Secondly, a series of load increments is added until the load reaches ‘‘Yielding Load+0.005 MPa’’. In the last step the external load is released to obtain the residual stress components. # 2003 Elsevier Ltd. All rights reserved. Keywords: C. Residual stress; A. Layered structures; B. Mechanical properties; B. Plastic deformation; Thermoplastic composite plate
1. Introduction The increasing interest in thermoplastic matrix composites operating in extremely severe mechanical and thermal environments has brought more attention to the design of structures made of these materials. They offer high specific stiffness and specific strength, improved fracture toughness, and increased impact resistance. Moreover thermoplastic composites possess the unique characteristic that they may be remelted, reprocessed and reformed. They are easily repaired and they can be remelted to repair the local cracks and delaminations. Therefore, the characterization of the elasto-plastic response is indispensable in the limit analysis stage, which must be performed along the reliable design processes for composite structures involving plasticity effects in the nonlinear behavior.
* Corresponding author. E-mail address: [email protected] (R. Karakuzu). 0266-3538/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2003.09.014
The study of the optimal design of thermoplastic composite structures is an emerging research field and few authors have presented results in this area. BaheiEl-Din et al. [1] have developed a three-dimensional finite element code for the elasto-plastic analysis of fiber-reinforced composite materials and structures. They have obtained the solution of the nonlinear equilibrium equations with a Newton–Raphson type iteration technique. Owen and Figueiras [2] have presented the anisotropic elasto-plastic finite element analysis of thick and thin plates and shells. Jeronimidis and Parkyn [3] have investigated the residual stresses in APC-2 cross-ply laminates. Predictions based on classical laminate theory were compared to measure levels of residual stress obtained from a number of experimental techniques. The analysis of the results showed that accurate predictions can be made provided that the changes in thermoelastic properties of the materials with temperature are taken into account. Vaziri et al. [4] presented a new finite element program for in plane loading of FRC laminates. The
1050
R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
program is based on a new comprehensive constitutive model incorporating yielding and failure in a unified manner. Sayman and Zor [5] studied on a low-density polyethylene (LDFE, F.2.12) thermoplastic composite beam reinforced by woven steel fibers loaded by a uniformly distributed force on the upper surface. A closed form solution is performed by satisfying both equations of equilibrum and boundary conditions. Karakuzu and Ozcan [6,7] presented an analytical elasto-plastic stress analysis for a metal matrix composite beam of arbitrary orientation subjected to a single transverse force applied to the free end of the beam or uniform distributed load applied to upper surface. As the value of the orientation angle decreases the length of the beam increases. Akay and O¨zden [8] investigated the influence of residual stresses in carbon fiber-thermoplastic matrix laminates. Bahei-El-Din and Dvorak [9] studied the elastic plastic behavior of symmetric metal-matrix composite laminates for the case of in-plane mechanical loading. Local stresses, hardening parameters, and strains are found in each lamina and in the fiber and matrix phases within each lamina. Comparisons of analytical results with experimental measurements were made for certain laminated plates. Karakuzu and Sayman [10] and Karakuzu [11] worked on the elasto-plastic solutions for a fiber reinforced metal matrix composite cantilever beam. In the elasto-plastic solution, the material is assumed perfectly plastic. At distant sections from the free end, the elastic and the residual stresses are found to be high. Residual stresses are highest at the fixed end. By producing residual stresses, the beam can be strengthened. If it is loaded again by the external forces in the same direction, the cantilever can carry greater external forces than in the elastic case. On the other hand, a lot of works have been done using numerical and analytical methods to predict elastic–plastic behavior of composite structures [12–19]. Cervenka [20] has given advantages and disadvantages of thermoset and thermoplastic matrices for continues fibre composites. The present study is about the determination of elasto-plastic and residual stresses of woven steel fiber reinforced thermoplastic laminated composite plates with different lamination schemes under transverse uniform loads. A three-load step is carried out for each analysis consecutively. The effects of ply number, orientation angle and bonding type on the residual stress components of laminated composite plates are investigated numerically.
sequence and each layer of the composite. Though the finite element method is a well established tool of computer aided engineering, the quality of the simulated data is fully dependent on the models used for approximating the real behavior of the investigated component. Modeling for conventional materials is well studied, but the building of a proper model for composite structures requires more effort. In order to predict the properties of a laminated composite, the properties of the single woven lamina are of great importance. The lamina is manufactured from low-density polyethylene (LDPE) and plane woven Cr–Ni steel fibers. For matrix materials, the thermoplastic granule is melted by electrical resistance up to 160 C without pressure in 5 min. Then, the material is held for 5 min under a pressure of 2.5 MPa by moulds and the temperature is dicreased to 30 C under a pressure of 15 MPa in 3 min and polyethylene layer is manufactured. Subsequently the plane woven Cr–Ni steel fibers are put into two polyethylene layers and processed in the same way described, as shown in Fig. 1. The mechanical properties of the composite lamina given in Table 1 are measured by Instron-1114 Tensile Testing Machine of 50 kN capacity at a ratio of 0.5 mm/ min. To obtain the modulus of elasticity in the weft direction (E1), Poisson’s ratio (12) and the tensile strength in the weft direction (F1t) a specimen whose weft direction coincides with the loading direction is taken and two strain gauges perpendicular to each other are stuck on. One of them is in the weft direction, the other is in the warp direction. A uniaxial tension test is used in the weft direction and the specimen is loaded step by step up to rupture. Because of the woven structure E1 and F1t are equal to E2 and F2t respectively. For all steps, strains in the weft directions ("1) and strains in the warp directions ("2) are measured by indicator. To define G12, a specimen whose principal axis is on 45 is
2. Properties of the lamina material Static mechanical characterization has been carried out before numerical study, in order to evaluate the mechanical constants for both the homogenized
Fig. 1. Producton of composite lamina.
R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
taken and a straingauge is stuck on loading direction of the lamina. The specimen is loaded by the Instron Tensile Machine and Ex is obtained by using one straingauge in the tensile direction. Where subscript x shows the loading axis. G12 is calculated by using the values of Ex, E1, E2, 12 [21]. To find the shearing strength S, a shear test set up is used. A flat lamina, which has a T-shaped is taken as shown in Fig. 2 and loaded up to rupture. Minimum clearance is left to prevent bending and to provide contact between specimen and metal support. The load at rupture (Fmax) is taken and shear strength is calculated by the following equation. S¼
Fmax 2:h:tT
ð1Þ
The dimensions of the specimen are chosen as h=5 mm, LT=80 mm, a=10 mm and b=40 mm. The thickness of the plate tT is 2.5 mm.
3. Simulation with finite element methods The ANSYS finite element analysis program [22] is used to simulate elastic, elasto-plastic and residual stress components of woven steel fiber reinforced thermoTable 1 The measured mechanical properties and yield strengths of a lamina Mechanical properties
Magnitudes
Longitudinal modulus (E1) Transverse modulus (E2) In-plane shear modulus (G12) Major Poisson’s ratio (n12) Longitudinal strength (F1) Transverse strength (F2) In-plane shear strength (S) Plastic constant (K) Strain hardening parameter (n)
12 GPa 12 GPa 0.67 GPa 0.35 23 MPa 23 MPa 8.5 MPa 100 MPa 0.675
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plastic laminated composite plates. However some software packages, i.e. ANSYS, additionally provide a full three-dimensional approach. This program is able to calculate the behavior of laminated structures with anisotropic nonlinearities. Eight-node shell element-shell91 is used for layered applications and modeling structures with nonlinear materials. Anisotropic failure criterion at ANSYS is chosen for woven steel fiber reinforced thermoplastic laminated composite plate. The anisotropic option uses Hill’s yield criterion, which accounts for differences in yield strengths in orthogonal directions and allows for different stress–strain behavior in the x,y and z material directions. 3.1. Empirical equation for stress–strain curve It is sometimes useful to represent the stress–strain curve of a given material by an equation obtained empirically by fitting the experimental data. Eq. (2) is such an equation which will frequently fit most of a given stress–strain curve. The Ludwick stress–strain relation in plastic region is given as, ¼ 0 þ K"np
ð2Þ
where 0 , "p, n and K are the yield stress, the true strain, the strain hardening parameter and the plastic constant, respectively. For finite element analysis the material behaviour is described by the uniaxial stress–strain curves in three orthogonal directions and the engineering shear stress– shear strain curves in the corresponding directions. A bilinear response in each direction is assumed. It is assumed that stress–strain diagrams in tension and compression are the same. So mechanical properties of the composite are obtained in the tension. The initial slope of the curve is taken as the elastic modulus of the material. At the specified yield stress, the curve continues along the second slope defined by the tangent modulus (having the same units as the elastic modulus) as seen in Fig. 3. To find the tangent modulus from the empirical Eq. (2), the following operations are done: "T ¼
0 0 þ E ET
ð3Þ
"e ¼
E
ð4Þ
"p ¼ "T "e ¼
0 0 þ E E ET
ð5Þ
or "p ¼ ð 0 Þ
Fig. 2. Shear test set up.
E ET EET
and stress–strain equation is obtained as
ð6Þ
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R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056 Table 2 Real constants that are used in the analysis
Fig. 3. The uniaxial stress–strain curve.
¼ 0 þ
EET "P E ET
ð7Þ
if n has been taken as n=1 in Eq. (2) and Eq. (2) is equal to Eq. (7) K¼
EET E ET
Total number Orientation angle of layers of weft direction
Layer thickness Element total (mm) thickness (mm)
4 8 12 16 8 8 8
2.5 1.25 0.83 0.625 1.25 1.25 1.25
[0/45]2 [0/0/45/45]2 [0/0/0/45/45/45]2 [0/0/0/0/45/45/45/45]2 [0/0/15/15]2 [0/0/30/30]2 [0/0/45/45]2
10 10 10 10 10 10 10
residual stresses. In the first analysis, transverse load has been increased as a series of little load increments from zero to the point at which the first plastic strain occurs. In the second analysis, three load steps were applied transversely and consecutively as shown in Fig. 5. This figure illustrates a typical load history for a nonlinear analysis. In the load step 2, the Newton– Rhapson Iteration Method is used in the each load increment.
ð8Þ 4. Numerical results and discussion
and ET is obtained as ET ¼
KE KþE
ð9Þ
where, "T, "e and ET are the total strain, the elastic strain and the tangent modulus respectively. 3.2. Finite element model and boundary conditions The model size is 400400 mm and the whole mesh exists of 331 node and 100 element. The type of element chosen allows the calculation of the three normal stresses and the three shear stresses. The laminated plate is assumed simply supported and bonded symmetrically or anti-symmetrically about the middle surface of the plate. Cartesian coordinates (x,y,z) are used for the plate coordinates where x–y plane coincides with the middle surface of the plate and z axes denotes the transverse axis. Material directions (weft and warp) and plate directions (x, y) are shown in Fig. 4. In this figure, is the orientation angle of weft direction relative to the x-direction. Total number of layers, the orientation angle of weft direction and layer thickness are shown in Table 2. The laminate is loaded transversely. It is assumed that the uniform pressure acts on the upper surface of the laminated composite. In this paper two separate analyses were performed. The first one is to find the transverse yielding load and the second one is to obtain the
The aim of this investigation is to determine numerically the residual stress of woven steel fiber reinforced thermoplastic laminated for both symmetrically and anti-symmetrically bonded composite plates. To be sure from the results, an analysis, compare with reference [18] has been carried out in ANSYS. The results are compared to see the differences as seen in Table 3. There are acceptable differences between the present study and the reference [18]. The residual stress components of simply supported, symmetrically and anti-symmetrically bonded laminated composite plates are given in Table 4. The values of the residual stresses x and y are the same and the maximum at the mid-point of the composite plate. The other residual stress components are zero. 4.1. The effect of ply number on the yielding load of laminated composite plates The laminated composite plates of 4, 8, 12 and 16 layers that are oriented as angle ply plates [0/45], are used to see the effect of ply number on their yielding load values for two different situations in the same graph as seen in Fig. 6. The number of the layer does not have significant effect on the values of the yielding load of laminated composite plates except for the 16 layer anti-symmetric laminate as seen from Fig. 6. Yielding loads in anti-symmetric cases are greater than in the symmetric cases for all configurations.
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R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
Fig. 4. Material directions and the orientaion angle of weft direction.
Fig. 5. Load history for nonlinear analysis. Table 3 The results of simply supported symmetrically bonded [0/90]2 laminated composite plates from both the reference [18] and present study Results
Stresses
x
y
z
xy
yz
xz
Reference [18]
Elastic–plastic stresses Residual stresses Elastic–plastic stresses Residual Stresses
39.587 0.438 40.811 1.971
23.858 11.917 24.146 14.221
0.000 0.000 0.000 0.000
0.049 0.010 0.000 0.000
0.035 0.026 0.000 0.000
0.079 0.007 0.000 0.000
Present study
4.2. The effect of orientation angle and bonding type on the yielding load of laminated composite plates Each bar in the graph as shown in Fig. 7 has represents the different configuration of laminates and the graph represents the different cases of laminates according to their bonding type. The change in the orientation angle also changes the yielding load of laminated plates. The yielding load increases as the orientation angle increases in symmetric cases. The
yielding load is the maximum for the configuration [ 0/ 0/30/30 ]2 in the anti-symmetric case. The yielding loads in anti-symmetric cases are greater than in the symmetric ones for all configurations. 4.3. The effect of ply number on residual stress components The laminated composite plates of 4, 8, 12 and 16 layers that are oriented as angle ply plates [0/45], are
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Table 4 The residual stress components of simply supported, symmetrically and anti-symmetrically bonded laminated composite plate (in the upper surface) Orientation angle Bonded type
x
y
z
xy
yz
xz
[0/0/15/15 ]2
3.343 3.200 3.214 1.985 3.181 0.039
3.343 3.200 3.214 1.985 3.181 0.039
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
[0/0/30/30 ]2 [ 0/0/45/45 ]2
Symmetric Anti-symmetric Symmetric Anti-symmetric Symmetric Anti-symmetric
Fig. 8. Residual stress comparison graph for simply supported symmetrically bonded [0/45] laminated plates.
Fig. 6. Comparisons of yielding loads of laminated plates having different ply number and cases.
Fig. 9. Residual stress comparison graph for simply supported anti-symmetrically bonded [0/45] laminated plates.
model’s x axis in Cartesian coordinates are represented in these figures. As seen from Fig. 8, the residual stresses had their maximum values in the midpoints of the plates and their minimum values near the supporting edges. As seen from Fig. 9 the number of plies are more effectual on anti-symmetric plates against the symmetrical ones and increasing ply number has decreased the residual stresses for anti-symmetrically bonded simply supported laminated composite plates. 4.4. The effect of orientation angle and bonding type on residual stress components Fig. 7. Comparisons of yielding loads of simply supported laminated plates.
used to see the effect of ply number on the residual stress ( x) for four different situation in the same graphic. The residual stress comparison graphs for 4, 8, 12 and 16 layer symmetrically and anti-symmetrically laminated composite plates are shown in Figs. 8 and 9. The residual stresses versus the distance that passes through the mid-point and parallel to the finite element
The residual stress–distance graphs are sketched for eight layer angle ply laminated plates to see the effect of bonding type on the residual stress changing according to distance as shown in Figs. 10–12. The orientation angle has not affected the residual stress components significantly for the plates, which are bonded symmetrically. In the anti-symmetric plates, residual stresses decrease as the orientation angle increases and residual stress reaches 0.039 MPa in configuration [0/0/45/45]2. Residual stresses are greater in symmetric plates than in anti-symmetric ones.
R. Karakuzu et al. / Composites Science and Technology 64 (2004) 1049–1056
3.
4.
5. Fig. 10. The residual stress–distance graph for angle ply laminated plate [0/0/15/15]2.
6.
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plates does not affect the yielding load significantly. For symmetrically bonded plates, ply number does not affect the residual stress components. There are not so much differences between the laminates, which have different ply number. The numbers of plies are more effectual on antisymmetric plates against the symmetric ones. Increasing ply number has decreased the residual stress components for simply supported plates. In anti-symmetrically bonded, oriented as [0/0/ 15/15]2, [0/0/30/30]2 and [0/0/45/45]2 laminated plates, residual stresses decrease as the orientation angle increases. Yielding loads in anti-symmetric cases are greater than in symmetric cases for all configuration. But residual stresses are reverse.
References
Fig. 11. The residual stress–distance graph for angle ply laminated plate [0/0/30/30]2.
Fig. 12. The residual stress–distance graph for angle ply laminated plate [0/0/45/45]2.
5. Conclusion The following conclusions from elasto-plastic analysis have been obtained: 1. The orientation angle in plies affects the yielding load and the formation of plastic region. 2. The number of plies in laminated composite
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