Nonlinear Analysis of Thermoviscoelasticity of Laminated Composites

The nonl inear thermovisc oelastic beha vior of composite thin plates is investigated. An experimental program covers an achievement of creep tests under different temperatures, dimensions of specimens and distribution loads to de scribe the eq uation of creep compliance. The stress relaxation is also determined from the experimental creep compliance. A new e quation of creep c ompliance fu nction D(t, s ,T) and relaxation modulus E (t, e ,T) w ere predicted from the e xperimental results to describe t he nonlinear thermoviscoelastic behavior of com posite thin plates. A good agreement has been observed between the proposed models of nonlinear behavior at different temperatures and experi mental resul ts and between both theoretical and FE M results. It was found that thu s the deflection is increasing at the beginning and the rate of increase is nearly constant and increase with increasing temperature from 30 C o to 60 C o with approximate rate (34.6%) for simply support plate at distributed lo ad (q=1.934E-3 N/mm 2 ), relative dimensions (a/b)=1.0 and time=15 min. The results indicate that the shear stress increases with rate (50.7%), so that strain in y-axis increases with rate (19%) as a result of increase the temperature from (30 C o to 60 C o ).

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Introduction
The tress-strain relations for creep are primarily empirical.Most of the equation were developed to fit the experimental creep curves obtained under constant stress levels and constant temperature.The actual behavior of the viscoelastic materials has shown that the strain at a given time depends on all of the values of the stress in the past.Thus, the creep phenomenon is affected by the magnitude and sequence of the stress or strain in a history of the material.Based on this fact, various methods have been suggested to present the time dependence or the viscoelastic behavior of this material [1].Viscoelastic material as other metallic and non-metallic materials can be affected by various factors such as temperature, humidity, history variables, etc.In this respect, literatures related to the viscoelastic behavior of composite materials under thermal effects are presented.Wenbo Luo et.al. ,2007 [2] study the creep behavior of commercial grade polycarbonate was investigated in this study.ten different constant stress ranges from 8 MPa to 50 MPa were applied to the specimen and the resultant creep strains were measured at room temperature.It was found that the creep could be modeled linearly below 15 MPa , and nonlinearly above 15 MPa.Different nonlinear viscoelastic models have been briefly reviewed and used to fit the test data.Marco Ferrari et. al. 2008 [3] the study aimed to estimate the effect of insertion length of posts with composite restoration on stress and strain distributions in central incisors and surrounding bone.The typical , average geometries were generated in FEA environment.Dentin was considered as an elastic orthotropic material and periodontal ligament was coupled with nonlinear viscoelastic mechanical properties.B. A. Sami and H. Naima ,2009 [4] is predicted the mechanical behavior of yarns under various levels of strain , by using only their technical parameters.The study of the yarn response to tensile test and relaxation test at different strain level has permitted us to propose an analytical model predicting the entire stressstrain response of yarn.Joannie chin et. al. , 2009 [5] The shape functions matrix [Κ] The stiffness matrix [Κ(ε)] Stiffness matrix as a function of strain.
PDF created with pdfFactory Pro trial version www.pdffactory.com The problems of bending of rectangular plates that have two opposite edges simply supported take the solution in the form of series solution as follows[6]: ...( 2 (3) ... The expression W 2 evidently has to satisfy the equation: ( And it must be chosen in such a manner as to make the sum Eq.( 3) satisfy all the boundary conditions of the plate .Taking w2 in the form of series (2) in which, from symmetry (m=1, 3, 4…) and substituting into Eq.( 5) we obtain : This equation can be satisfied for all values of (x) only if the function Ym satisfies the equation: The general integral of this equation can take the form: ...( 8 On the sides y = + b/2, we begin by developing expression of Eq.( 4) in a trigonometric series which gives : where: m=1, 3, 5 … The deflection series Eq.( 9) will be represented in the form : Substituting these values of constants in Eq.( 11) we obtain the equation of the plate surface deflection ,satisfying Eq.( 1) and the boundary conditions are given in the following form :

Numerical analysis
Consider a plate subjected to a distributed load (q) normal to its mid surface Fig.( 2).The stresses and strain produce work that is stored in the system as strain energy ( W ) [7] such that:  16) and Eq.( 17) in Eq.( 15) the strain energy ( W ) may be expressed as : The applied load (q) acting normal to the plate surface area produces the load potential energy (Ω ), that is: Substituting the strains and stresses from Eq. ( 19) and Eq. ( 20) into Eq.( 21) and performing the integration over plate thickness (t) gives: Where for the isotropic case, the elasticity matrix [D] can be written as in [8]: ...( 24) so that , the stress relaxation expressed as following : The curvature displacement Eqs.( 17) can be expressed in a matrix form as follows : Substituting Eq.( 29) and ( 30) into the strain nodal displacement Eq.( 28) , the matrix [B],Eq.(28), can be written in the expanded form : ...( 31 29) in Eq.( 28) and the result in Eq.( 23) the strain energy equation can be written as : 29) in ( 20), the load potential equation can be written as[8]: Since the total potential energy is given by[8]: ...( 34) Ω + = W U Substituting of Eqs.(32,33) into the total potential energy expression Eq.( 34) gives[8: According to the condition for equilibrium for the minimization of the total potential energy U., differentiation of Eq.( 35 where the element stiffness [ K ] e is given by : ...( 37 (40) .
The element stiffness matrix can be evaluated by substituting Eq.( 40) in Eq. ( 37), that is: Composite polyester was manufactured using one layer as, fiber in the polyester resin which was made by mixing the polyester resin with solidification (B).The percentage of type (B) was (0.8%), the final solidified material is rigid.Therefore flexible die is used to prevent the specimen cracking or fracture which may occur after removing the specimen from the die.The time required for solidification of this type of material depended on the room temperature , hence increasing this temperature reduces the required time for solidification.

The results and discussion
The central deflection of plate has been analyzed for composite polyester thin plates.The results are as shown in Figs.(12,13).The figures show the effects of three values of distributed load under thermal environment, and different relative dimension.The value of central point deflection of thin plate deflection has been tested for time range (0.05 < time <120 minute).In addition, those tests are performed at constant thickness (thick.=4 mm).
The main material properties used for the analysis of thin plate viscoelastic material and its composite are used in the experimental creep test data in the region of nonlinear behavior.Hence, the same value fraction as that for creep the test specimen is used to construct the plate material to ensure that the output data is correct for both the experimental and theoretical analysis.Furthermore, the material properties concluded from the creep test for different temperature are used to examine the plate behavior for that different temperature.
PDF created with pdfFactory Pro trial version www.pdffactory.comFig( 14) show the effect of relative dimension on the central deflection value represented by increasing the relative dimension (a/b) which will increase the deflection.The increasing in the temperature increases the central deflection, in addition the effect of temperature is greater than the relative dimension (a/b) if we compare between Fig( 16).c-Three Dimensional Plots Rotation About the x and y-axis: The rotation of plate surface represents the slope at certain Cartesian w.r.t the x-axis or y-axis.In Figs.(15,16) a symmetrical state is observed for both rotations ( θ x , θ y ) about the centerline which has zero slope.This specimen represents a simply supported plate at all four edges with a maximum value of slope (θ x ,θ y ) at the nodes of simply supported plate.So that,Figs.(15,16)show the effect of temperature that increases rotation of plate surface ( θ x , θ y ) and that increase appears clear where all four edges are fixed.The comparison among theoretical, experimental and FEM gives a good agreement with limited disparity percentage.The disparity range(3-6 %) is calculated between the theoretical and FEM

Conclusions:
The following conclusions can be drawn from the results of experimental, theoretical and FEM work for viscoelastic composite thin plate in nonlinear behavior: assumed that the sides (x=0 and x=a) as shown in Fig.(1) are simply supported.Hence each term of Eq.(2) satisfies the boundary conditions : It remains to determine the function Y m in such form as to satisfy the boundary conditions on the sides of y =+ b/2 and the equation of the deflection surface (1).The solution of Eq.(1) for uniform load is assumed to be in the form : ., W 1 represents the deflection of uniformly loaded strip parallel to the x-axis.It satisfies Eq.(1) and also the boundary conditions at the edges : x=0 and x=a.
Observe that the deflection surface of the plate is symmetrical with respect to the x-axis Fig.(1) .In expression of (8) only even functions of (y) are kept and the integration constants (Cm=Dm=0.0)are let.The deflection surface (3) is then represented by the following expressionsatisfies Eq.(1) and also the boundary conditions at the sides PDF created with pdfFactory Pro trial version www.pdffactory.com(x=0.0 and x=a) .It remains now to determine the coefficients of Eq.(9) (Am and Bm) in such a manner as to satisfy the boundary conditions: Substituting the boundary conditions from Eq.(10) in the expression of Eq.(11) and using the notation gaves:We obtain the following equation for determining the constant ( Am,Bm) : Using the assumption of the classical theory for plate bending, the strain displacement can be written as follows:PDF created with pdfFactory Pro trial version www.pdffactory.com The nodal displacements of the plate element with (n=4) (four node) is shown in Fig.(3).Each node gives the following three degrees of freedom The transverse deflection w , PDF created with pdfFactory Pro trial version www.pdffactory.comrotations about x-axis θx and rotations about y-axis θy.The element deformed shape can be approximated with a suitable set of shape functions Ni (x,y) : strain matrix [B] can be written as follows: Fig.(6,7) show a schematic and photograph of this specimen.The volume fraction is (0.26).The experimental results predict the creep compliance D(t, σ,T) as shows in fig.(8,9) and relaxation modulus E(t,ε,T) shows in fig.(10,11) to describe the nonlinear thermoviscoelastic behavior of rectangular thin plate.

Preparation of Creep Campsite Polyester Specimens :
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