Non-polynomial Zig-Zag and ESL shear deformation theory to study advanced composites

J.L. MANTARI , I.A. RAMOS, J.C. MONGE

a Faculty of Mechanical Engineering, Instituto de investigacio′n en ingenier?′a naval (IDIIN), National University of Engineering (UNI), Lima 15333, Peru

b Department of Mechanical Engineering, University of New Mexico, Albuquerque, NM 87131, USA

c Faculty of Mechanical Engineering, Universidad de Ingenier?′a y Tecnolog?′a (UTEC), Lima 15063, Peru

KEYWORDS Analytical solution;Composite materials;CUF;ESL;Plate;Trigonometric functions;Zig-Zag effects

Abstract The mechanical behavior of advanced composites can be modeled mathematically through unknown variables and Shear Strain Thickness Functions (SSTFs). Such SSTFs can be of polynomial or non-polynomial nature and some parameters of non-polynomial SSTFs can be optimized to get optimal results. In this paper, these parameters are called ‘‘r” and ‘‘s” and they are the argument of the trigonometric SSTFs introduced within the Carrera Unif ied Formulation(CUF). The Equivalent Single Layer (ESL) governing equations are obtained by employing the Principle of Virtual Displacement (PVD) and are solved using Navier method solution. Furthermore, trigonometric expansion with Murakami theory was implemented in order to reproduce the Zig-Zag effects which are important for multilayer structures. Several combinations of optimization parameters are evaluated and selected by different criteria of average error. Results of the present unif ied trigonometrical theory with CUF bases conf irm that it is possible to improve the stress and displacement results through the thickness distribution of models with reduced unknown variables. Since the idea is to f ind a theory with reduced numbers of unknowns, the present method appears to be an appropriate technique to select a simple model. However these optimization parameters depend on the plate geometry and the order of expansion or unknown variables. So, the topic deserves further research.

1. Introduction

Laminated composite materials are utilized in several areas,such as marine, aerospace, civil, biomedical and other industries.Their good mechanical properties (high specific stiffness,excellent fatigue strength, resistance to corrosion, etc.) have been the reason for an increase in demand and consequently the substitutes of traditional engineering materials such as

Nomenclature

CPT Classical Plate Theory

CUF Carrera Unif ied Formulation

ESL Equivalent Single Layer

FSDT First-order Shear Deformation Theory

GUF Generalized Unif ied Formulation

HSDTs Higher-order Shear Deformation Theories

LW Layerwise

PVD Principle of Virtual Displacement

SSDT Sinusoidal Shear Deformation Theory

SSTFs Shear Strain Thickness Functions steel, aluminum, concrete, etc. Therefore, in order to understand the behavior of composite structures under different kinds of loads and boundary conditions, several theoretical and experimental studies were performed.

Classical Plate Theory (CPT) for metallic structures based on the Kirchhoff’s assumptions, which neglects the shear deformation, was extended to laminated plates in order to study thin plates. First-order Shear Deformation Theory(FSDT)based on Reissner1and Mindlin2was introduced considering deformations in the section.A shear correction factor is used to correct the constant transverse shear strain component obtained with this theory.However,correction factor values depend on the material coefficients, geometry, boundary conditions and loading conditions,which are difficult to calculate. Therefore, Higher-order Shear Deformation Theories(HSDTs) were developed in order to overcome the limitation of FSDT.

The HSDTs can be developed using polynomial Shear Strain Thickness Functions (SSTFs), which can be seen from,for example, the articles by Reddy and Liu3, Reddy4, Levinson5and Librescu6. Further, non-polynomial shape functions were developed by Levy7, Stein8, Touratier9, Soldatos10,Karama11, Mantari and Guedes Soares12, Zenkour13-16and Mantari et al.17-20. An optimization of a HSDT based on trigonometric function for the bending analysis of functionally graded shells was proposed by Mantari and Guedes Soares21,where parameters‘‘r”and‘‘s”were introduced in the displacement field in order to produce results close to 3D elasticity solutions.

Several investigators developed HSDTs to study advanced composites. Houari et al.22presented a new simple higherorder theory with only three unknowns considering sinusoidal thickness function on the transverse shear stress for solving bending and free vibration problems.Belabed et al.23proposed a higher-order theory with 5 unknowns which considered the stretching effect based on hyperbolic shear strain shape functions.Belkorissat et al.24developed a new nonlocal hyperbolic model to study free vibration of functionally graded nano plates based on Mori-Tanaka scheme. Boukhari et al.25proposed a deformation theory for wave propagation of an infinite functionally graded plates in the presence of thermal environment. Bennoun et al.26presented five unknown variables based on a hyperbolic expansion to study functionally graded sandwich plates. Bourada et al.27studied a refined trigonometric higher-order beam theory with three unknown variables for the bending and free vibration of functionally graded beams. Mahi et al.28calculated the free vibration for functionally graded plates using orthogonal polynomials associated with Ritz method. Tounsi et al.29presented a new HSDT of three unknown variables for the study of buckling and free vibration of sandwich functionally graded plates.Ait Yahia et al.30presented an interesting approximate model for functionally graded plates with porosity phases to study the wave propagation of plates.In fact,this paper can be used as a reference for ultrasonic inspection techniques and structural health monitoring. Hebali et al.31improved the plate theory proposed by El Meiche et al.32by studying the effect of considering nonzero transverse normal strain.Draiche et al.33studied a sinusoidal higher-order deformation theory for studying the f lexure of angle and cross ply laminated composite plates.

Carrera Unif ied Formulation (CUF) was developed for composite laminated plates and shells34-36using originally Taylor’s expansions of N-order.A Sinusoidal Shear Deformation Theory (SSDT) within the CUF framework was developed by Ferreira et al.37for static and free vibration analysis of laminated shells.Additionally,Generalized Unif ied Formulation (GUF), presented as a more general approach of CUF,was introduced for the first time by Demasi38for a single isotropic plate and extended for multilayered composite plate in Refs.39-43.Several non-polynomial expansions were developed by Filippi et al.44and Mantari et al.45. Furthermore, trigonometric expansion for laminated plate subject to thermal load was developed by Ramos et al.46. In order to obtain HSDTs with low computational cost, asymptotic/axiomatic technique was developed by Miglioretti et al.47,48and Carrera and Petrolo49,50, where the effectiveness of each term of the displacement field is evaluated by means of elimination and comparison to a reference solution. The genetic algorithm to evaluate the importance of each displacement variable for finite element plate models was developed by Carrera and Miglioretti51.

Fig.1 Material coordinate(1,2,3)and plate coordinate(x,y,z).

Table 1 Comparison between exact, LD4 and ED4 solution.

The present work uses an optimized method52,53based on two different optimized parameters which are‘‘r”and‘‘s”that are related to the in-plane and transverse displacement,respectively.The described parameters are contained in the argument of trigonometric SSTFs in order to solve the analytical static problem of laminated plate by means of a compact formulation. Trigonometric expansion with Equivalent Single Layer(ESL) and Zig-Zag (Murakami) approach was introduced by using the Principle of Virtual Displacement (PVD) to obtainthe unif ied governing equations, which are solved by employing Navier method solution. Several combinations of optimization parameters are evaluated and selected by different criteria of average error. Improvement in the def lection and stresses results for low-order expansion are presented for especial values of the parameters(‘‘r”and‘‘s”)obtained by particular criteria of average error. However the optimization parameters depend on the plate geometry and the order of expansion of the utilized displacement field.

Table 2 Average errors with their respective weight.

2. Analytical modeling

2.1. Model kinematics

The through-the-plate-thickness unif ied displacement field developed by Carrera34for any order of expansion is expressed as follows:

Fig. 2 Variation of percent errors for sin displacement as a function of parameters r and s （N=2,a/h=4）.

Fig. 3 Variation of percent errors for sin displacement as a function of parameters r and s （N=5,a/h=4）.

where FN,GNand HNare functions through the thickness and N is the order of expansion.These equations can be grouped as follows:

and in compact form:

where the subscript indicates summation according Einstein notation. Fτand uτare given as

This paper studies an axiomatic compact theory with Zig-Zag effect by exploring trigonometrical shear strain shape function (see Refs.8-10for single and Ref.54for multilayered plates) in a systematic manner. The results are compared with polynomial axiomatic expansions.

Polynomial, trigonometric, hyperbolic, exponential functions and different combinations of them are studied for beam in Ref.55and for plates in Refs.44,45within CUF framework.In the present work,optimized trigonometric functions have been implemented and compared with the classical Taylor expansion.

Trigonometric expansion (sin) developed in Ref.46is modified as follows:

In order to reduce the number of optimized parameters,only 2 optimization parameters (‘‘r” and ‘‘s”) are included in the trigonometric SSTFs, where the parameter ‘‘r” is related to in-plane deformation(x and y directions)and the parameter‘‘s” to the transverse deformation, i.e. thickness direction (z direction). The selection of these parameters within the SSTF was carried out with the idea to obtain close to 3D solution.

Fig. 4 Variation of percent errors for sin Z displacement as a function of parameters r and s （N=2,a/h=4）.

Additionally, the Mukarami Zig-Zag function is implemented in the trigonometric displacement field (sin Z).

where -1（ ）kζkis the Mukarami Zig-Zag function.

2.2. Elastic stress-strain relations

The stress σkand εkvectors of the k th layer are grouped according to CUF:

where the subscript n is related to the in-plane components,while p to the out-of-plane components, and the straindisplacement relationships are given as

The stress-strain relationships in the material coordinates(see Fig. 1) of the k th layer can be expressed as

Fig. 5 Variation of percent errors for sin Z displacement as a function of parameters r and s （N=5,a/h=4）.

and

According to Eq.(7),the material matrices can be grouped as

2.3. Principle of virtual work

Considering the static version of the principle of virtual works,the following expression holds:

where εkand σkare the strain and the stress vectors of the k th layer and Nlstands for number of layers.

Using Eqs. (3) and (8), the expression becomes

The subscript z indicates partial derivation with respect to z. By applying integration by parts (see Eq. (17)) to Eq. (16)

Table 3 Non-dimensionless results for optimized trigonometric expansion (sin).

the plate boundary and internal governing equations for the case problem are obtained:

On the other hand, the external virtual workdue to a load qzapplied on the top face （z =h/2） can be expressed in the following manner:

where

Finally, the governing equations are

2.4. Analytical solution

Navier type closed form solution can be applied in simply supported plates.So,the displacement variable and the distributed load can be expressed in the following Fourier series:

Table 4 Non-dimensionless results for optimized trigonometric expansion (sin Z).

where

Therefore, the governing equation becomes

where

Fig.6 Through-the-thickness distribution of dimensionless stress for a(0°/90°)square laminated plate for sin displacement field and N=2, 3, 4, 5.

The explicit expressions of the fundamental stiffness nuclei,, are unique for any combination of optimized parameter.In contrast to classical polynomial expansions,the exact Gauss integration in the thickness direction of the trigonometric expansion is not possible and the accuracy of this must necessarily be controlled. The formulation of the present stiffness nuclei was written in MATLAB and the integrations were obtained by quadrature function incorporated in MATLAB with 10-6as tolerance.

3. Numerical results and discussion

Benchmark proposed by Demasi43,which consists of two-layer asymmetric cross-ply simply supported plate (0°/90°), is utilized.Although the material properties do not have a practical use, the difficulty of this benchmark allows comparing several theories and evaluating different combinations of optimization parameters. A sinusoidal static load qz=sin αx（ ）sin βy（ ）applied at the top surface （m =n=1） of thick square plate（a/h=4,a=b） is considered in this paper. The mechanical properties of each layer are

Layer 1 (Bottom):

The following non-dimensional quantities are used:

Table 1 presents ED4 results obtained by using polynomial ESL theory of order 4 and layerwise (LW) theory LD456, and these results are compared with exact 3D solution provided in Ref.43. LD4 values present very close results for def lection（z =h/4） and shear stress（z =h/4） with errors equal to 0.005% and 0.106%, respectively. Furthermore, for normal stressz=h/4（ ）, the obtained error is 0.914%. In order to facilitate the calculation, LD4 was employed as reference result.In Table 1,ED4 and LD4 problem results are calculated by using classical form of Hook’s law.

The percent error of def lectionz=h/4（ ）, normal stressz=0（ ）and shear stress calculated by means of equilibrium equationat z=-h/4 are obtained for several parameters r and s introduced in the trigonometric SSTFs(Eqs.(5)and(6)).The errors were computed using the following formula:

Fig.7 Through-the-thickness distribution of dimensionless stress σ-xz for a(0°/90°)square laminated plate for sin displacement field and N=2, 3, 4, 5.

where R refers to def lection or stress and RLD4is the reference value obtained by LD4 theory. Additionally, average errors are given as

where wiis arithmetic weights, i.e. def lection or stress values can be weighted (following a defined optimization criterion which needs to be further developed in future works)to impact or alleviate,depending on the case study,the average error and thus get the parameter of optimization r,s（ ）. Five assumed optimization criteria are presented in Table 2 with their respective weight to calculate the average error Av X（ ）, so X adopt values from 1 to 5.

Parameters r and s were selected to get close to 3D solution of displacements and stresses,and in this way less percent error was produced according to weighted arithmetic means presented in Table 2. The weights depend on the relevance of the results to be evaluated.For example,normally,Av4(equal weights) is chosen in the literature to calculate the average errors.Averages‘‘Av1-Av3”represent independently the error of def lection, normal stressand shear stress, respectively. Such criterion perhaps can be of interest in some applications where, for example, it is just relevant to knowing def lections. Additionally, an optimization criterion, with arbitrary weights to calculate the average error(Av5),is presented.

The variations of the average errors with the optimized parameters r and s considering an order of expansion of N=2 and N=5 for sin expansions are shown in Figs. 2 and 3, respectively. In the expansion N=2, a great influence of the parameter s on the def lectionerror(Av1)and average error (Av4) can be noticed. The low error values (blue zone)are for parameter s less than 5. The parameter r is important for the errors of the stressesand(Av2 and Av3). The areas with high errors(red zones)can be interpreted as incompatibility or instability of the SSTF. Fig. 3 shows low average error and a pronounced dependence on both optimized parameters. However, there are some instability areas with high percent error (see Av1 and Av2).

Figs.4 and 5 show the variation of the various kinds of average error with the parameters r and s for sin Z expansion when N=2 and N=5, respectively. In contrast to sin theory of order N=2 (see Fig. 2), sinZ with order N=2 presents pronounced dependence on both optimized parameters.Additionally, it shows several zones with low (blue) and high (red)average error. Although sinZ of order N=2 presents lower percentage of errors than sin theory, the sinZ theory for N=2 presents some instabilities produced perhaps for low performance of the SSTFs.On the other hand,the parameter s presents a great influence on the errors for sinZ expansion for the order N=5 (see errors Av1, Av2and Av4 （average error） in Fig. 5). Additionally, stability and low percent error are obtained for values of parameter s less than 6.

Fig. 8 Through-the-thickness distribution of dimensionless def lection for a (0°/90°) square laminated plate for sin displacement field and N=2, 3, 4, 5.

Optimized parameters are presented in Tables 3 and 4,where ‘‘sin-opt Y” (Y adopts values from 1 to 5) is the theory and the parameters are obtained by average error ‘‘Av X”. In these tables, results of dimensionless def lectionz=h/4（ ）,normal stress（z =0） and shear stress（z =-h/4） are compared with LW theory by means of CUF (LD4)34for the side-to-thickness ratio a/h=4. The errors in def lection(Av1) is lower for sin-opt1, but the average errors, such as Av4 or Av5, do not present signif icant improvement, as can be observed for N=2. On the other hand, sin-opt4 and sinopt5 present low percent of average error, but these reduce the precision in the def lection, as can be seen for N=3 and 4.In Fig. 6,close values between LD4 and optimized trigonometric theory(opt4 and opt5),which overcome the Taylor and classical trigonometric displacement fields, are observed for N=3 and 4 for normal stressevaluated in z=0 (bottom layer). The through-the-thickness distribution of shear stress,calculated by means of equilibrium equation,does not present better approximation than the classical trigonometrical expansion (without optimized parameter) (see Fig. 7). On the other hand, the through-the-thickness distribution of def lectionsfor opt4 and opt5, showed in Fig. 8, present a light improvement for N=2 and 5, but these lose precision for N=3 and 4. Fortunately, the idea is to select a model with reduced number of known variables.

Table 4 presents results for optimized trigonometric expansion with Zig-Zag model (sin Z-opt X) for different orders of expansion N compared with LD4 results. Moderate reduction of about 2% in the def lection error is obtained for sin Z-opt5 N=2（ ） which is even less than Taylor and classical trigonometric expansion (without optimization); the reduction in average error (Av4) from approximately 5% to 1.8% can be achieved.Moreover,a considerable reduction in the Av4 from approximately 3.82% to 0.08% can be observed for N=3.Nevertheless, for N=4 and 5, very low reductions are achieved(up to 0.15%).The high number of SSTFs allows better accuracy in results,but these reduce the dependence on the optimization parameters r and s. Figs. 9-11 present the through-the-thickness distribution of normal stress, shear stressand def lection. A good improvement with respect to non-optimization trigonometrical expansion(sin Z)is shown for the def lectionwith sin Z-opt5 and sin Z-opt4 for N=2 and 3. However, minimal improvement of the def lection is obtained for N=4 and 5.

The practical application of this work relies on the optimization strategy to select a theory that best suits what is desired to achieve in a case problem,i.e.displacements,stresses or both. The method consists of incorporating (according to what is desired to get in terms of accuracy) several arithmetic weights in transverse displacements, normal stress and transverse shear stress. Consequently, a shear deformation theory that best suits with a particular experimental case problem can be persuaded. For example, one could be just interested in obtaining vertical def lection or good accuracy in terms of shear stresses of advanced composite structures.This work can be a reference study to new numerical and experimental investigators by showing them that ‘‘case dependent problem” is a fact and so a shear deformation theory can be selected by weighting the desired output to best benef it the experiment validations through computational mechanics.

Fig. 9 Through-the-thickness distribution of dimensionless stress for a (0°/90°) square laminated plate for sin Z displacement field and N=2, 3, 4, 5.

Fig. 10 Through-the-thickness distribution of dimensionless stressfor a (0°/90°) square laminated plate for sin Z displacement field and N=2, 3, 4, 5.

Fig. 11 Through-the-thickness distribution of dimensionless def lection for a (0°/90°) square laminated plate for sin Z displacement field and N=2, 3, 4, 5.

4. Conclusions

The present work introduces an optimized method in order to f ind reduced theories with better results than displacement field without optimization. Trigonometrical SSTFs with optimized parameters r and s contained in the in-plane and transverse displacements field are respectively developed by means of the CUF for analytical static problem of asymmetric laminated plate (0°/90°).

Governing equations are obtained by employing the PVD and are solved using Navier method solution. Several values and combination of optimization parameters are evaluated and selected by different criteria of average error.

A general displacement field was presented in this paper to use different displacement expansions,where Fτ,Gτand Hτare the SSTFs for the displacements in all the directions. In order to reduce the number of optimized parameters, trigonometrical displacement field with parameter r in×and y directions and parameter s in z direction are taken into account. Future studies are necessary in order to evaluate the influence of different non-polynomial SSTFs for the displacements in all the directions.

Improvement in the def lection and stress results for low order of expansions N are obtained by using an optimization procedure. However the optimization parameters depend on the plate geometry and the utilized order of expansion.Analytical axiomatic/asymptotic evaluation of the non-polynomial SSTFs needs to be performed to obtain reduced theories and evaluate the sensibility of the results by the variation of the optimized parameters r and s.

Acknowledgement

This paper was written in the context of the project:‘‘Disen~o y optimizacio′n de dispositivos de drenaje para pacientes con glaucoma mediante el uso de modelos computacionales de ojos” founded by Cienciactiva, CONCYTEC, under the contract number N° 008-2016-FONDECYT. The authors of this manuscript appreciate the f inancial support from the Peruvian Government.