Elasto-plastic Stress Analysis in Laminated Thermoplastic Composite Plates with an Elliptic Hole

In this paper, elasto-plastic stress analysis in laminated thermoplastic composite plates having an elliptic hole in the middle is examined by using finite element method. Composite plates consist of four orthotropic laminations and bonded symmetrically [ 0 /- 0 ]. Uniform loadings in vertical direction are applied to the selected composite plates. The loading and reinforcement angle are gradually increased from the yield point of the plate. The load steps increased as 0.0001 MPa at each iteration. Iteration numbers are chosen 25, 50, 75 and 100. A quarter of the plate is taken into consideration due to symmetry. Elasto-plastic stresses are obtained according to load steps and orientation angles.


INTRODUCTION
Thermoplastic composites have many advantages, such as a high specific stiffness and specific strength, increased impact resistance and improved fracture toughness began to increase their importance.In recent years, applications of reinforced thermoplastic have increased in different fields such as marine equipments, construction, automobile, etc. Composite materials are those formed by combining more than one bonded material, each with different structural properties.An productive way is to increase the load capacity of the plastic material by using reinforcement with steel or glass fibers.Elasto-Plastic stresses are important in failure analysis of thermoplastic matrix laminated plates.When the yield point of the plate, is exceeded the residual stresses can be used to raise the yield points of the plates.
Yapıcı et al. [1], searched elasto-plastic stress created in hole-side spacing in fiber reinforced laminated thermoplastic composite plates.Bahei-El-Din et al. [2], have developed a three dimensional finite element code for the elasto-plastic analysis of fiber-reinforced composite materials and structures.Arslan et al., Arslan and Çelik, [3,4] studied an elasto-plastic stress analysis in symmetric and antisymmetric wovenreinforced polyethylene thermoplastic matrix composite that using finite element method and first-order shear deformation theory for small deformations.Gür et al. [5], analyzed elasto-plastic stress analysis in planar plate produced by a composite material and with a rectangle hole in the middle and being exposed to uniform spring strain forces.Karakuzu et al. [6], examined elasto-plastic stress analysis by finite element method in metal matrix plates having spiral dents on the sides.Karakuzu et al. [7], researched of the elasto-plastic stress analysis and residual stresses in woven steel fiber reinforced thermoplatic laminated composite plates for transverse uniform loads.Sayman [8], researched elasto-plastic stress analysis under vertical forces in stainless steel woven fibers aluminum metal matrix laminated plates.Sayman et al. [9], studied an elasto-plastic stress analysis is carried out on simply supported and clamped crossply and angle-ply aluminum metal-matrix composite laminated plates with a circular hole.Sayman et al. [10], studied an elasto-plastic stress analysis of metal-matrix composite plates.Atas et al. [11], studied an elasto-plastic stress analysis and the expansion of plastic zone in layers of stainless steel fiber-reinforced aluminum metalmatrix laminated plates using F.E.M..
In this study, stainless steel fiber-reinforced thermoplastic matrix laminated, simply supported plates with elliptic hole are selected (Fig. 3a).Elastic, elasto-plastic, residual stresses and expansion of the plastic zone are obtained bu using finite element method for small deformations.The Tsai-Hill theory is used as a yield criterion.The loading is increased by 0.0001 Mpa increments at each loading step and iteration numbers are chosen as 25, 50, 75, 100.

MATHEMATICAL FORMULATION
The woven steel fiber-reinforced a low density homogeneous polyethylene thermoplastic laminated plates of constant thickness are formed by stacking four layers bonded symmetrically about the middle surface.The plate in cartesian coordinates (x,y,z) is illustrated in Fig. 1 and Fig. 2a.The stress-strain relations for an orthotropic layer can be written as [12],  Where Q ij are the transformed reduced stiffnesses given in terms of the orientation angle and the engineering constants of the material.Theory of plates is used with transverse shear deformations included.The transverse shear deformation theory 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.Therefore, the displacement components of a point of coordinates x,y,z for deformations are [8,12].
    0 , , , w x y z w x y  where u 0 , v 0 and w 0 are the displacements at any point of the middle surface and , xy  are the rotations of normals to midplane about the y and x axis, as shown Fig. 2a.The bending strains vary linearly through the plate thickness and are given by the curvatures of the plate, whereas the transverse shear strains are assumed to be constant through the thicknes as, The total potential energy of a laminated plate under static loadings is given as [13].).The resultant forces, moments and shear forces perunit length of the cross section of the laminated plate are obtained by integration of the stresses through the thickness as [11,12].
Equilibrium requires that  is stationary, i.e., 0  , which may be regarded as the principle of virtual displacement for the plate element [11,13].

FINITE ELEMENT MODEL
Finite elements method is used in the analysis.In finite elements solution, nine node isoparametric finite element type is used.Due to symmetry, the quarter of the plate is taken and shown Fig. 3b.The quarter of plate, meshed into 13 elements and 69 nodes.This is illustrated in  b)Dimensions of ¼ plate an elliptic hole and border conditions [17].
Bending and shear stiffness matrices of the plate are obtained by using the minimum potential energy principle as; where  

, ( , )
D b and D s are the bending and shear parts of the material matrix, respectively. 2  1 k and 2 2 k denote the shear correction factors.For rectangular cross sections they are given as [16].In this solution, Tsai-Hill Theory is used as a yield criterion due to the same yield values in tension and compression in this thermoplastic composite.In the elasto-plastic solution, the tangential modular matrix is used instead of the elasticity matrix [11].
where e  and 0  are the effective stress and yield stress, respectively.The tangential modular matrix is obtained as [12].

T t T aa D D d I H a Da
     (14) where H is the local slope of the uniaxial stress/plastic strain curve and can be determined experimentally and D, I and a are the elastic matrices which are used instead of D b and D s , unit matrix and flow vector, respectively.The flow vector is found by using the Prandtl-Reuss equations as [11].
The stress and strain curve of the composite layer is obtained in the principal material, fiber, direction x.It is given by the Ludwik equation [15] as, where k and n are the plasticity constant and strain hardening exponent, respectively, as given in Table 1.It is assumed that the composite material is incompressible in the plastic region, therefore, Poissons ratio in the plastic region is taken as 0.5 [11].The calculated stresses do not generally coincide with the true stresses in a non-linear solution; because of this reason external forces are applied incrementally [13].Therefore, the equivalent nodal forces and the unbalanced nodal forces must be found; for each load step, These unbalanced nodal forces represent the increments in the solution and must satisfy the convergence tolerance in a non-linear analysis.A widely used iteration procedure is the Modified Newton Iteration Method [13].This iterative solution can be derived from the Newton-Raphson Method.The equations used in the modified Newton Iteration are, for i = 1, 2, 3, 4, . . .
where t, t, U and K are the time, time interval, overall displacement vector and the stiffness matrix, respectively.These equations were obtained by linearizing the response of the finite element system about the conditions at time t.In each load step, the unbalanced nodal forces are calculated, and the iteration is continued until the unbalanced nodal forces (load vector) or the displacement increments, U (i) are sufficiently small, 0.0001 [4,11].
Table 1.The measured mechanical properties and yield points of a woven-reinforced thermoplastic composite layer [4].

DESCRIPTION OF THE PROBLEM
As matrix material thermoplastic (PVC (poliyvinilyl cloride)) is selected.Fiber is selected woven, according to different angles.Each layers of the laminated plate as a constant thickness, 2.1 mm.By combining 4 of them, laminated composite plates are made, illustrated Fig. 1.In this study, Fig. 3a-b are illustrated, dimensions of square shaped whole plate, 1/4 of plate and boundary conditions, stainless steel fiber-reinforced thermoplastic matrix laminated simply supported plates with elliptic hole.For ¼ of plate; the mesh generation of the laminated plate is performed by using the prepared computer program that shown is Fig. 4.c.[17].As shown in Fig. 4a., the plates have symmetrical permutation in this study.They are connected symmetrically in the middle

ELASTO-PLASTIC STRESS COMPONENTS OBTAINED ACCORDING AS ITERATION NUMBER
Obtained stresses in the solution are elastic, plastic and residual stresses.Residual stresses can be found by subtracting elastic stresses from elasto-plastic stresses.The changes of elasto-plastic stress components obtained according to orientation angles for 25, 50, 75 and 100 iterations are shown in the diagrams.σ x , σ y , τ xy values of elastoplastic stress components are shown as Pxx, Pyy, Pxy in the diagram.
As seen in Fig. 6, the sequence of layer [45 0 /-45 0 ], when at 25 iteration, value of σ x was occured 14.325 MPa.When at 100 iteration, this value increased to16.664MPa.However, when at 25 iteration, value of σ y was obtained 17.840 Mpa.When at 100 iteration, this value increased to 20.376 MPa.As seen in Fig. 7, the sequence of layer [30 0 /-30 0 ], when at 25 iteration, the value of σ x occured 13.219 Mpa.When at 100 iteration, this value reached to 15.230 MPa.Similar trend was observed for the value of σ y .The value of σ y, at 25 and 100 iterations were occured 18.733 Mpa and 22.218 Mpa, respectively.However, the value of τ xy , when at 25 iteration were occured -7.076 Mpa.When at 100 iteration was determined -8.740 Mpa.
As seen in Fig. 8, the sequence of layer [60 0 /-60 0 ], value of σ x, when at 25 iteration was found 17.545 Mpa.When at 100 iteration, this value was observed 20.534 MPa.However, when at 25 iteration, value of σ y was obtained 18.555 Mpa.When at 100 iteration, this value increased to 20.780 MPa.However, the value of τ xy , at 25 and 100 iterations were occured -8.565 Mpa, -9.562 Mpa, respectively.
In this study, the highest yield value have [60 0 /-60 0 ] laminated plate occurred elasto-plastic, elastic and residual stress values are given below as an example in Table 3.

DISCUSSION OF RESULTS
The results of the analysis can be summarized as follows: In plates loaded by vertical loading [0 0 ] 4 , [60 0 /-60 0 ] 2 , [45 0 /-45 0 ] 2 and [30 0 /-30 0 ] 2 symmetrical sequence laminated plates, different yield values are obtained.It is found that in [45 0 /-45 0 ] 2 laminated plates, load value starting yielding is lower compared to the other selected angles.The highest yield points are obtained in [60 0 /-60 0 ] 2 , [30 0 /-30 0 ] 2 , [0 0 ] 4 , [45 0 /-45 0 ] 2 laminated plates, respectively.Elasto-plastic stress values emerging in lower and upper laminations are identical in terms of absolute value.The maximum elasto-plastic stress components occurred in upper and lower surfaces.It is found that stress components in inter laminations have lower values than the stresses in lower and upper surfaces.As it is obvious from the diagrams, elasto-plastic stress values increase depending iteration numbers.In symmetrically sequence laminations, it is observed that in four different orientation angle, stress values are identical in terms of absolute value in the 1 st -8 th , 2 nd -7 th , 3 rd -6 th , 4 th -5 th laminations.In 100 iteration σ x value which is one of the components of elasto-plastic stress is maximum in [60 0 /-60 0 ] laminated plate and minimum in [45 0 /-45 0 ] laminated plate.In 100 iteration, σ y value is maximum in [0 0 /0 0 ] laminate plate and minimum in [45 0 /-45 0 ] laminated plate.Again in 100 iteration, τ xy value which is one of the components of shear stress is maximum in [45 0 /-45 0 ] laminated plate and minimum in [0 0 /0 0 ] laminated plate.At the same time, all sequence layers were found to be identical value for τ yz ve τ xz values.
However, it is found that results obtained in this study have good agreement with available literatures.

Figure 2 .
Figure 2. a) Displacements of the middle surface b) Loading of laminated plate.

5 )
Where U b is the strain energy of bending, U s the strain energy of shear and V represents the potential energy of external forces.They are as follows; where dA = dxdy, p is the transverse load per unit area and b n N and b s N are the in-plane loads applied on the boundary ( R  ) (Fig.2.a,b Fig 4.c.For ¼ of plate, with simply supported boundary conditions shown in Fig. 3b.The symmetric laminated plate is composed of four layers.

Figure 3 .
Figure 3. a) Dimensions of square shaped whole plate and border conditions.b)Dimensions of ¼ plate an elliptic hole and border conditions [17].
observed for the value of τ xy .The value of τ xy , at 25 and 100 iterations were occured -9.616 Mpa and -11.720Mpa, respectively.

Figure 6 .
Figure 6.The variation of elasto-plastic stress components with iteration numbers (upper surface).

Figure 7 .
Figure 7.The variation of elasto-plastic stress components with iteration numbers (upper surface).

Figure 8 .
Figure 8.The variation of elasto-plastic stress components with iteration numbers (upper surface).

Figure 9 .
Figure 9.The variation of elasto-plastic stress components with iteration numbers (upper surface).
x  y  xy  yz  xz 