VERIFICATION OF LAMINATE COMPOSITE PLATE SIMULATION UNDER COMBINED LOADINGS THERMAL STRESSES

This study deals with thermal cyclic loading phenomena of plates which were fabricated from composite materials (woven roving fiber glass + polyester) were exposed to (75 C°) temperature gradient thermal shock for ten times in different stage of conditioning times due to the effect of thermal fatigue using the method of Levy solution and compared these results with both results from experimental published work and (ANSYS Ver. 9) program. A composite laminate plate with fiber volume fraction (υ f =25.076%) is selected in this study and applying the combined loadings like bending moment (Mo), and in-plane force (Nxx) beside the effect of thermal fatigue. It involves multi theoretical and finite element fields; but the theoretical one contains the derived equation of stresses distribution and evaluating the normal deflection of a middle point for dynamic analysis applying different boundary conditions for heating and cooling. The main present numerical results for a composite plate with (80%) fiber volume fraction claim that the relative reduction in normal deflection and dynamic load factor are (78.593%) and (9.421%) during cooling to (-15 respectively


INTRODUCTION
Many researches before studied the effect of thermal fatigue in woven roving glass fibers with unsaturated polyester composite plate and its application on airspace experimentally.The effect of thermal cycles on tensile properties and coefficient of thermal expansion has been studied on graphiteepoxy 12-ply laminate configuration subjected to 5000 thermal cycles, [Fabmy A.A. and Cunningbam T.G., 1976]; but the method prediction applied on T300 graphite/934 epoxy under hygro-thermomechanical fatigue with two step procedure.The initial step consists of determining the composite ply strength associated with each type of cyclic load: mechanical, thermal and hygral.The next step is to determine the effect of combined cyclic loading [Ginty C.A. and Chamis C.C., 1988].It can be estimate the inter-laminar shear strength (ILSS) of glass/epoxy and glass/polyester composites of woven fabrics by changing the holding time at the holding temperature during the thermal fatigue and hydrothermal shock cycles [Ray B.C., 2005].In the other hand the inter-laminar shear strength (ILSS) of the thermally conditioned glass fibers of random orientation and epoxy resin laminates followed by ice-cold water quenching from the laminated composite was exposed to (50˚C) temperature and hydrothermal fatigue cycles varied weight fraction constituents of glass fiber reinforced (55,60, 65%), [Ray B.C., 2005], but in the field of cryogenic [Ray B.C., 2005] studied the effect for 55,60, and 65 weight percentages of E-glass fibers reinforced epoxy composites on inter-laminar shear strength during (sub-ambient) at -80˚C temperature in ultra low freezing chamber for 2 hours and subjected to an ambient at 30˚C temperature for 1 hour and investigate the effect of thermal shock on flexural modulus of Kevlar 49/epoxy laminates by thermally conditioned at a (80˚C) for (5, 10, and 20 min) to (-80˚C) for (5 min) or cryogenically conditioned at (-80˚C) for the same time periods to (80˚C) for (5 min).In the present work it can study the effect of combined loadings on the deflection combining with thermal fatigue effect of composite laminate plate.
So that the objectives of this research are:  In dynamic analysis derive the analytical solution to evaluate the stress-strain field and inter-laminar shear stress (ILSS) distributions for middle point using Levy solution for thermal cyclic under combined loadings. It will be compared the results of deflection in z-direction for middle point were obtained from thermal cyclic loading only with that obtained from (cyclic thermal loading + ), (cyclic thermal loading + ) and (cyclic thermal loading + ) as mentioned in

THE CLASSICAL LAMINATED PLATE THEORY (CLPT)
The classical laminate plate theory gives flexibility than the other methods of solution for composite plate depending upon the cyclic thermal loading, in-plane force , and the bending moment The stress and strain variation through the laminate thickness based on the classical lamination theory (CLPT) includes the following assumptions, [Reddy J.N. 2004]:  Transverse normal stresses remain straight before and after deformation (strain in z-direction equal to zero).
 Transverse normal stresses rotate and perpendicular to the mid-surface after deformation (strain in xz and yz plane equal to zero).

DISPLACEMENTS KINETIC RELATIONS
If a plate was established of a total thickness (h) composed of (N) orthotropic layers with the principal material coordinates ( of the kth lamina oriented at an angle to the laminate coordinate, x.The z-axis is taken positive downward from the mid plane.The kth layer is located between the points and in the thickness direction.The Kirchhoff hypothesis theory used instead of Mendlin theory by applying the principles of supper position displacements ( to be such that, [Reddy J.N. 2004] : (1) Where: : A displacements along the coordinate lines of a materials point on the x,y-plane.The strain-displacement matrix relations take the form, [Reddy J.N. 2004]: where: (2) For orthotropic material and layers axes oriented arbitrarily with respect to the laminate coordinates.The Hook's Law relations are [Reddy J.N. 2004]: And for especially orthotropic (that the material axes coincide with respect to laminates coordinates).
(4) Anyhow in Levy solution (dynamic analysis) the classical laminate plate theory (CLPT) is more suitable than other methods because it is linking the cyclic thermal loading with the combined loadings.

Methodology Discussions of Theoretical and Numerical Experiences
For thermal cyclic analysis the inputs in classical laminate plate theory program can be shown in Table (1):   Table (1 It can be taken the algebraic sum of stress field in (x, y, z) directions and deflection in zdirection for heating and cooling.Table (2) shows the verification test for dynamic analysis under different thermal loading of fiber volume fraction ( ) using (Fortran 90 and ANSYS Ver. 9) programs for the plate having aspect ratio (1.8).The four types of combined loading can be used in this section: (Thermal Fatigue), (Thermal Fatigue +Mo), (Thermal Fatigue +Nxx) and (Thermal Fatigue +Mo+Nxx) as mentioned in A triangular-shaped element may be formed by defining the same node number.After solving the four types of combined loadings it can be compared the last three type results with first type results to obtain the perfect design of this paper.It can be used both program of finite element analysis to find the deflection and dynamic load factor (D.L.F) which cannot applied this on different fiber volume fractions because absence the knowledge of the number of layers that gives the mass of fiber experimentally.The dynamic load factor in is higher than the dynamic load factor in because the central static deflection in is smaller than and the percentage error is acceptable on different thermal loadings.Tables (3,4,5) show the effect of different fiber volume fractions and different loading percent of combined loading on deflection.In Tables (3, 5) the deflection can be decreased with the increasing of fiber volume fractions for all the increasing of loading percent; but the deflection can be increased with the increasing of loading percent for the same fiber volume fraction.In Table (3) the deflection still constant with the increasing of loading percent of combined loading for the same fiber volume fraction; but the deflection can be decreased with the increasing of fiber volume fractions.4, 5) show the convergence of dynamic central deflection with total degree of freedom.There is a small sudden change increasing in dynamic deflection occurring between numbers of degree-of-freedom (DOF) 81 and 2187 which then reached the steady-state case between numbers of degree-of-freedom (DOF) 2187 and 27783 for different fiber volume fractions for ( .Figs. (6, 7) give the convergence of dynamic load factor (D.L.F) with total degree of freedom.There is a large sudden change increasing in dynamic load factor occurring between numbers of degree-of-freedom (DOF) 81 and 14739 which then reached the steady-state case between numbers of degree-of-freedom (DOF) 14739 and 27783 for different fiber volume fractions for ( .It can be noticed that the dynamic load factor decreased with the increasing of fiber volume fractions, while the fiber volume fractions increased with the increasing of fundamental natural frequency, then the dynamic load factor will be decreased.10) shows convergence of dynamic deflection with total degree of freedom under different combined loadings of for simply supported plate with four edges.For the curve concern (thermal cycling + here is a large sudden change increasing in dynamic deflection occurring between numbers of degree-of-freedom (DOF) 390 and 4998 which then reached the steadystate case between numbers of degree-of-freedom (DOF) 4998 and 7686; but the curve concern (thermal cyclic + there is a slow increasing of dynamic deflection occurring between numbers of degree-of-freedom (DOF) 390 and 1350 which then reached the steady-state case between numbers of degree-of-freedom (DOF)1350 and 7686.The curve concern (thermal cyclic + there is no clear change in dynamic deflection with numbers of degree-of-freedom (DOF).Figs.(11,12,13) illustrate the effect of different fiber volume fractions on dynamic deflection with total degree of freedom for (thermal cyclic + , (thermal cyclic + , and (thermal cyclic + with a plate with aspect ratio (1.8) of simply supported plate with four edges; therefore the best curve is the curve of for (thermal cyclic + because this curve gives a small value of dynamic deflection equal to (0.22426 e-6).15) gives the effect of different boundary conditions like (SSSS, CSSS, and CSCS) on inter-laminar shear stress varying with time of using classical laminate plate theory (CLPT) for the plate having aspect ratio (1.8).The inter-laminar shear stress can be decreased with the increasing of time for all three curves; but the best curve that curve concern (CSCS) boundary condition this curve gives small value of inter-laminar shear stress because to reduce the mismatch between the thermal expansions of the resin and the fiber can be caused the thermal cyclic loading.Figs.(17,18) illustrate the effect of different boundary conditions like (SSSS, CSSS, CSCS) on normal stresses in the (x, y) directions varies with time of under thermal cyclic loading using classical laminate plate theory (CLPT) for the plate having aspect ratio (1.8).In the two figures, the curves have (SSSS, CSSS) boundary conditions that the normal stresses decreasing with the increasing of time because the most value of coefficient of thermal expansion decreased with time and that effect on the decreasing of normal stresses; but the positive sign of the value of normal stresses because of the case of the particle of composite laminate plate is tension due to these boundary conditions.The curve (CSCS) boundary condition casing decrease in the value of normal stress with time increasing; but the negative sign because of the particle of composite plate suffer from compression.The conclusion of these curves the increasing in the clamped or fixed edges causing increasing in particle's compression.Figs.(19,20) show the effect of different boundary conditions on normal strain in the (x, y) directions varies with time of under thermal cyclic loading using classical laminate plate theory (CLPT) for the plate having aspect ratio (1.8).The comment is the same as in Figs.(17,18) because the stress is equal the multiplication of strain on material properties matrices and the positive, negative sign because the expansion and contraction on the particles of composite laminate plate.

CONCLUSIONS
This research conclude from its obtaining results under dynamic load (free vibration and dynamic response) of fiber-reinforced laminated plates comparing with analytical work, FEM and other investigations using different numerical and theoretical methods as follow :  the maximum dynamic central deflection is decreased with the increasing of different fiber volume fractions of heating and cooling and that due to due to the increasing of the value of natural frequency (free vibration analysis) for both thermal loadings ( . the increasing of fiber volume fractions causing decreasing in overshoot percent, rise time, and settling time; but all the plate with different fiber volume fractions have the same settling time and reached to the steady-state at the same time. The test of inter-laminar shear stress that the inter-laminar shear stress varies sinusoidal for both experimental, [3] and numerical solutions; but for analytical solution the inter-laminar shear stress reduced with time gradually. The effect of different percent of combined loadings and different fiber volume fractions on deflection.In the case of (thermal cyclic + and (thermal cyclic + the deflection can be decreased with the increasing of fiber volume fractions for all the increasing of combined loadings percent; but the deflection can be increased with the increasing of loading percent for the same fiber volume fraction; but the deflection still constant with the increasing of loading percent of combined loading for the same fiber volume fraction for (thermal cyclic + Fig. (1) to obtain perfect design of deflection. Applying different boundary conditions like (SSSS, CSSS, and CSCS) on composite plate under cyclic thermal loading.
Figure (1), used (SHELL 132) as given in Fig. (2).It has a threedimensional layered shell element having in-plane and thru-thickness thermal conduction capability.The element has eight nodes with up to 32 temperature degrees of freedom at each node.The conducting shell element is applicable to a three-dimensional, steady-state or transient thermal analysis."SHELL132" was switched element and to be analyzed structurally, the element should replaced by an equivalent structural element such as " SHELL 91".It is used for layered application of a structural shell model up to 100 different layers.The element is defined by eight nodes having six degrees of freedom at each node: translation in the nodal x, y, and z directions and rotation in the nodal x, y, and z directions to evaluate the stress field and deflection in z-direction.

Fig
Fig. (4) Convergence of dynamic central Fig. (5) Convergence of dynamic deflection with total degree of freedom deflection with total degree of freedom under .under .

Fig
Fig. (6) Convergence of dynamic load Fig. (7) Convergence of dynamic load factor with total degree of freedom under factor with total degree of freedom under . .

Fig
Fig. (8) Dynamic deflection variation Fig. (9) Dynamic deflection variation with time under different fiber volume with time under different fiber volume fractions of .fractions .

Fig. ( 10 )
Fig. (10) Convergence of dynamic deflection with total degree of freedom under combined loadings of .

Fig
Fig. (11) Convergence of dynamic deflection with total degree of freedom under thermal cyclic + bending moment for different fiber volume fractions.

Fig. ( 14
Fig. (14) shows comparisons between analytical solution (CLPT) and numerical solution (FEM) with the experimental published work [RayB.C.,2005] of for the plate having aspect ratio (1.8) that concern inter-laminar shear stress varies with time.It can be noticed that the inter-laminar shear stress varies sinusoidal for both experimental and numerical solutions; but for analytical solution the inter-laminar shear stress reduced with time gradually because it applied via Levy solution on analytical part.

Fig. ( 14 )
Fig. (14) Comparisons of inter-laminar shear stress varies with time under the effect of thermal cyclic of .

Fig. (
Fig. (15) gives the effect of different boundary conditions like (SSSS, CSSS, and CSCS) on inter-laminar shear stress varying with time of using classical laminate plate theory (CLPT) for the plate having aspect ratio (1.8).The inter-laminar shear stress can be decreased with the increasing of time for all three curves; but the best curve that curve concern (CSCS) boundary condition this curve gives small value of inter-laminar shear stress because to reduce the mismatch between the thermal expansions of the resin and the fiber can be caused the thermal cyclic loading.

Fig. ( 17 )
Fig. (17) Effect of different boundary conditions on normal stress varies

Fig. ( 18 )
Fig. (18) Effect of different boundary conditions on normal stress varies

Fig
Fig. (19) Effect of different boundary conditions on normal strain varies

Fig. ( 20 )
Fig. (20) Effect of different boundary conditions on normal strain varies