Form ing modeling and sensitivity analysis of sandwich composites

Xi-wang HE，Yu-bing BAO，Hui-ping WANG，Wei SUN，Xue-guan SONG*

（1 School of Mechanical Engineering，Dalian University of Technology，Dalian 116024，China）

（2 Beijing Institute of Aerospace Systems Engineering，Beijing 100076，China）

Abstract：In this paper，through analyzing the resin state，the numerical simulation is built for the Sandwich and the surrogatemodel is used to explore the influence of the thermal expansion coefficientof themold and panel，the elasticmodulus of the panel，the honeycomb size and other parameters on the shrinkage and roundness error during curing process.First of all，the finite elementmethod is used to simulate the stress and deformation of the sandwich during curing process.Among them，the honeycomb equivalentmodel is established and the life and death cell technology is applied to perform equivalentmodeling of the resin state change.The simulation results match the actual.The results show that the state change of the resin duringmolding is themain factor affecting the shrinkage of the sandwich structure.Secondly，the relationship between the shrinkage rate or roundness error and the influencing parameters is established by using the radial basis function（RBF）.Finally，the sensitivity of the influence parameters of the shrinkage and roundness error is discussed.The results show that the thermal expansion coefficient of themold and the thermal expansion coefficient of the panel are themost sensitive to the shrinkage of the sandwich structure and themost sensitive factor affecting the roundness error of the sandwich structure is the coefficient of thermal expansion of resin.

Key words：Sandwich composite，FEA，Resin equivalentmodel，Surrogatemodel，Global sensitivity analysis

1 Introduction

The honeycomb sandwich structure is a special structural composite material，which has the characteristics of lightweightand high axial rigidity.It is also one of the most popular structural innovation projects in the compositematerial industry and has been widely used in transportation，automobile，aerospace，rail and other industries.The sandwich consists of three parts，the honeycomb of the core layer and the upper and lower panels，which are glued by resin.The panel ismade of relatively strongmaterial，which gives high rigidity and high strength to bear the high load.Due to the light weight of the core material，it must have sufficient shear strength to withstand the transverse shear stress and sufficient thickness to provide high shear stiffness to resist the buckling of the panel［1-3］.In the aerospace industry，the honeycomb sandwich is a common structure of a rocket shell.The rebound of the sandwich caused by the thermal expansion of thematerial and the shrinkage of the resin during the curing process will affect the decline of the rocket shell assembly accuracy.Therefore，the research on the rebound of the honeycomb sandwich during the curing process have a great significance to the accuracy improvement in the assembly process of the aerospace structure.

Aerospace monolithic components are generally complex，and has thin and large size.When the autoclavemolding is formed，the partwill deform and rebound after forming due to the residual stress generated in the thin wall［4］.During the manufacturing process of the T-shaped composite material，there is curing deformation after demolding due to the mismatch of the thermal expansion coefficient between the layers and the chemical heat shrinkage effect of the resin［5-7］.Liand Yao［8］studied the curing process of the T-shaped monolithic component and found that the interaction caused by themismatch of the thermal expansion coefficient between the mold and the compositematerial is an important factor affecting the curing deformation of the structure.For the analysis of the residual stress and deformation of the resin matrix composite，many studies have been conducted on the thermal deformation in the cooling stage after the curing.It is believed that the residual stress of thematerial is very small before reaching the glass transition temperature.Among them，the foreign Lange experiment［9］analyzed the curing process and founded that there was neither thermal stress nor chemical shrinkage stress before reaching the glass transition temperature.It is considered that the generation of residual stress occurred during the process of falling from the glass transition temperature to room temperature.At present，there are twomain researches on composite.One is advanced resin-based composite material.It considers the composite effectof resin and carbon fiber layering and calculates the curing kinetic model of carbon fiber and resin structure［8，10-11］.However，the calculation cost in the analysis process is high due to the complexity of the resin constitutive model，whichmakes thismethod not suitable for structural optimization analysis.On the other hand，it is themodeling and simulation of composite honeycomb sandwich［12-13］.The type of composite does not take into account the effect of resin on the curing of the composite material in simulation.In the analysismethods of the above two composite，the large-scale problems of the structure are not considered.Numerical simulation and analytical derivation arewidely used to predict residual stress and curing deformation.As a representative numerical simulation technique，the robustness and accuracy of the finite element method（FEM）have been significantly improved in recent years due to consideration of factors such as constitutive relations，mold-part interaction and multi-field coupling analysis［14-15］.The life and death element technology is a technique commonly used in finite elements，which is realized by multiplying the stiffness matrix and massmatrix of themodel by a small factor［16］.Ma［17］combined with the method of life and death element to numerically simulate the fiber pull-out behavior of glass fiber/cement-based composite material grafted with epoxy resin.Huang［18］used the life and death element technology to simulate the process of laser irradiation of carbon fiber/epoxy laminates.Therefore，it is an accurate and efficientmethod to realize the equivalent modeling of the resin model by using the life and death element technology during the resin curing process.

In view of the above existing problems，this paper takes the semi-cylindrical sandwich composite as an example.Based on the curingmechanism of the resin，an equivalent resinmodelwas established by using the life and death element technology.The curing process of the sandwich composite was simulated by the finite elementmethod and the deformation and stress distribution of the sandwich structurewere analyzed.Based on the above，the radial basis function（RBF）was used to establish the relationship between the shrinkage of the sandwich structure and the roundness error of the sandwich with different parameters，and the sensitivity of the influence parameters was analyzed.

2 Analysis p rocess

The sensitivity analysis process of sandwich compositematerialmolding proposed in this paper is as follows：

（1）The geometricmodel of the sandwich structure and mold was established using CATIA，in which the core layer aluminum honeycomb of the sandwich structure is equivalent to an orthotropic entity.

（2）Import the geometry files of the sandwich structure and the mold into the ANSYS workbench software for thermal-mechanical coupling calculations.According to the temperature change，the resin molding process is equivalently processed by the life and death element technology.After the calculation，the deformation value and stress distribution of the sandwich structure are obtained，then calculate the roundness error of itsmold.

（3）Nine parameters that have a greater impact on the shrinkage and roundness errors of the sandwich structure are selected to construct a surrogate model，and the sensitivity of each parameter is analyzed and calculated.

Fig.1 shows the process of forming sensitivity analysis of sandwich composite structure.

Fig.1 Sensitivity ana Iysis p rocess of sandw ich composite

3 Geom etric model of sandw ich

Combined with the structure of a certain spacecraft，the shape and size of the model were determined.In this simulation process，the mold width isH=2 115 mm，themodel diameter isD=4 992 mm.According to the given conditions，the semi-cylindrical sandwich structure is formed by overlapping sixteen substrates.There are 8 substrates in the circumferential direction and two substrates in the axial direction.The overlap between the substrate and the substrate is connected by resin Fig.2（a）.In order to avoid too many grids and computational complexity due to excessive structural size during the simulation process，themodelwas simplified Fig.2（b）.

Fig.2 Schem atic d iag ram of sim u Iation struc tu re

The sandwich structure is a new type of structure made by filling the sandwich core between two layers of wall panels.It is formed by high temperature curing，and the two layers of aluminum alloy are formed by resin“gluing”and“hot pressing”curing［19］.It is composed of five parts，the upper panel，resin，honeycomb，resin，and lower panel，as shown in Fig.3.The resin thickness of the panel is 0.4 mm and 0.1 mm，the height of the honeycomb aluminum is 28 mm，and the side length and wall thickness are 5mm and 0.5 mm.Honeycomb structure is a regular hexagon.Thematerial properties of each part are shown in Table.1.

Fig.3 Schem atic diagram of sandw ich

Tab Ie 1 MateriaIp roperties of each part

Because the discrete non-uniformity of the honeycomb structure bringsmany difficulties to ordinarymechanical analysis，it is necessary to perform equivalent calculations on the mechanical properties of the honeycomb structure，the honeycomb structure is regarded as an orthotropic entity［20］，as shown in Fig.4.It is also necessary to make an accurate analysis of the thermal performance inside the honeycomb.According to Timoshenko beam theory［21］and sandwich panel theory［22］，the equivalent parameters of honeycomb structure elastic modulus，Poisson’s ratio and shear modulus are established.Then the heat transfer coefficient of the honeycomb structure is calculated accord-ing to the SP method［23］and Fatemi［24］method，and the specific heat capacity of the honeycomb structure is obtained by the volume mixing principle.Finally，the equivalent thermal expansion coefficient of the honeycomb material is obtained according to the ANSYS finite elementanalysis.The equivalentmaterial properties are shown in Table 2：

Fig.4 Equiva Ient honeycom b structure

Tab Ie 2 Equiva Ientmateria Ip roperties

4 Therm al coup ling of sandw ich

4.1 Resin equivalentmodeling

The thermal expansion of the resin also has a great influence on the deformation of the sandwich structure.At this stage，most of the research is aimed at establishing a curing kinetic model for resin-based composite materials，but the calculation cost is too high due to the complexity of the resin constitutive model.In addition，mostof them only consider the influence of the thermal expansion coefficientmismatch between themold and the sandwich and rarely consider the influence of the thermal expansion of the resin on the deformation of the sandwich.Taking into account the change of the state of the resin during the heating process，an equivalentmodel of the resin is established based on the curing kinetic model of the resin.

The curing reaction of resin is an extremely complicated process，which is generally represented by the followingmathematicalmodel［25］：

Where，f（T，a）is a function of temperatureTand curing degreea.And the curing kineticsmodel of3501-6 resin is：

In Eq.（2）and Eq.（3），K1，K2andK3are respectively the reaction rate constants of the 3501-6.Ai（i=1，2，3）is the frequency factor，ΔEi（i=1，2，3）is the activation energy.

According to the curing kineticsmodel of 3501-6，the relationship between the glass transition temperature and curing degree of3501-6 resin system is established［26］as：

According to Eq.（4），the relationship between the glass transition temperature and curing degree of the 3501-6 resin system is shown in Fig.5.

It can be seen from the Fig.5 that as the degree of curing increases，the glass transition temperature also increases.The state of the resin changes from liquid to glass and finally cooled to solid with changes in temperature.

Fig.5 The re Iationship cu rve betw een g Iass transition tem perature and curing degree

The equivalent model of the resin is composed of two parts；one is the influence of the state change of the resin itself on the sandwich structure；the other is the equivalent treatment of the overlap between the substrates.This fully considers the effect of resin on the sandwich during the thermal deformation of the sandwich.Since the resin is liquid during the heating process and the influence of the resin on the composite material can be ignored in the simulation.As the temperature rises，the resin gradually solidifies.When the resin curing degree reaches 0.95，the temperature is 180℃and the resin began to withstand shear stress.For the change of the resin state，it is achieved by changing the contact state and setting the life and death element in ANSYS，as shown in Fig.6.

Fig.6 Schem atic diagram of contact state

Since there is no internal stress in the resin when the temperature is raised，the state of the resin at this time is set as a dead element，but at this time the contact state of the panel and the honeycomb is set to no separation due to the presence of liquid resin between them.When the temperature rises to 180℃，the panel and aluminum honeycomb are bonded together due to the solidification of the resin.Meanwhile，the resin can withstand the shearing force.

Therefore，the resin state is set as a live element.Through the establishment of a simple model，the set of resin states are compared，including the effects of no resin，with resin，with life and death element，and setting the reference temperature on deformation，as shown in Fig.7.It is found by comparison that when there are resin and life and death units，it has a greater impact on the deformation of the structure in Fig.8.

Fig.7 Com parison of resin sim u Iation cond itions

Fig.8 Deform ation after setting the Iife and death eIem ent

Since the semi-cylindrical sandwich in this paper is made up of sixteen substrates overlapped，the overlap between the substrate and the substrate is bonded with resin.Due to the shape of the mold is round；the overlap of the substrate creates certain difficulties for finite element simulation.Therefore，the substrate overlap is needed to dealwith as shown in Fig.9.The contact between the substrates is set to be permeable.In other words，the two substrates will expand toward each other after the substrate is heated and expanded.This solution solves the problem of mutual slip between the substrates caused by the change of the resin state during the thermoforming of the sandwich.

Fig.9 Equiva Ient form of overIapping structure

4.2 Boundary conditions

The load consists of two parts，one part is the thermal load caused by temperature changes，the other part is the force load caused by vacuuming and applying pressure in the autoclave.According to the structure forming process，the structure is first vacuumed and then the structure is put into the autoclave as the temperature rises.The pressure is applied and the maximum pressure is 0.2 MPa when the temperature rises to100℃.When the temperature drops to22℃，the structure is cooled under pressure.Finally，the vacuum state of the structure is broken and the pressure is zero as shown in Fig.10.

Fig.10 Boundary condition setting

4.3 Grid independence verification

In order to enable the 3Dmodel to be simulated and analyzed in the ANSYS workbench，the current simplified model needs to be meshed.The model analyzed in this paper is a cylindricalmodelwith a simple shape and it is divided by a hexahedralmesh.

After verification of grid independence in Fig.11，the total number of grids in the overall structure of the final analysis plan is determined to be 40 000.Using the Hex Dominantmethod of division，the final number ofmesh elements is40 000 and the total number of nodes is 130 000.

Fig.11 Grid independence verification

5 Sim u lation resu lt analysis

5.1 Deformation analysis of sandwich

According to the results of finite element analysis，we observed the deformation of the model before and after heating，before and after demolding.It can be seen from the Fig.12 that when the temperature rises to 180℃，themaximum deformation of themold and the sandwich structure appears at the top.At this time，the sandwich structure deforms following the deformation of themold and the resin solidifies.Due to the difference in thermal expansion coefficientbetween the sandwich and themold，internal stress is generated inside.The internal stress inside the sandwich cannot be released due to the action of the resin when the heat preservation ends and the temperature drops.The separation of the sandwich structure and themold causes the internal stress release of the sandwich and the deformation and rebound of the sandwich structure are caused when the entire process is completed.It can be seen from Fig.12 that the deformation value of the top of the sandwich after demolding is 3.74 mm，while the deformation value of the sandwich structure during actual processing and forming is3～4mm.The simulation result is consistentwith the actual deformation.

Fig.12 Com parison of deform ation of sandw ich at different times

5.2 Stress analysis of sandwich

Due to the inconsistency of the thermal expansion coefficient of the sandwich and themold aswell as the solidification of the resin，the sandwich has a stress relief phenomenon before and after demolding（Fig.13）.According to Fig.13，it can be seen that the maximum internal stress of the sandwich structure before demolding is 67.56 MPa and themaximum internal stress after demolding is 15.55 MPa.The release of internal stress caused the sandwich to shrink.

Fig.13 Com parison of stress distribution before and after dem o Iding of the Iower pane Iof the sandw ich

5.3 Calculation of roundness error ofmold

In order to systematically study the structural performance of the sandwich structure after curing and optimize its performance，it is necessary to determine the indicators tomeasure the performance of the sandwich structure.The shrinkage of the sandwich structure and the roundness error of themold inner surface are used as indicators to measure the performance of the sandwich structure.The roundness error of the inner surface of the mold is calculated by the least squares circle method［27］，which can be derived from the geometric relationship and formula shown in Fig.14：

Where，θis the angle between the center of the least squares circle and the coordinate origin，is the number of derived nodes，andri（i=1，2，…，n）is the distance from the node to the coordinate origin.Then calculate the center and radius of the least squares fitting circle and calculate the distance from each point of the measured actual contour to the center of the least square circle：

The algebraic difference between themaximum value and theminimum value is the roundness error：

According to the calculation of the least squares circlemethod，the roundness error of themold is0.155 7 mm.

Global sensitivity analysis（GSA）and design optimization are becoming more and more important in practical engineering applications.Surrogate model techniques are used to explicitly approximate the out-

6 Param eter sensitivity analysis

put.In the polynomial regressionmodel，krigingmodel（Kriging），radial basis function（RBF）and other surrogatemodels，we use the RBF model to combine the output（objective function and constraint function）with the input（design variables and parameters）［28］link up.

Fig.14 Least squares circ Ie method

6.1 Design of experiment

The purpose of this paper is to calculate the influence ofmold and panel thermal expansion coefficient，panel elasticmodulus，honeycomb side length，honeycomb wall thickness，panel Poisson’s ratio，and resin Poisson’s ratio on the shrinkage of sandwich structure.The value range of each variable is shown in Table 3.

Tab Ie 3 Va Iue range of each variab Ie

For these nine variables，we adopt the Latin Hypercube experimental design method take ninety sample points as training points and teen sample points as test points in Fig.15 to test the accuracy of the constructed surrogatemodel.

Fig.15 Taking points in Latin hypercube（the p revious two variab Ies are exam p Ies）

6.2 RBF model

Radial Basis Function neural network is a neural network proposed by J.Moody and C.Darken in the 1980s［29］.According to the sampling planX=｛x（1），x（2），…，x（n）｝Tand the responseY=｛y（1），y（2），…，y（n）｝T，we can get a fixed radial basis functionfto represent the relationship between the sample point and the response value：

Where，c（i）represents the center of the i-thmirror basis function，φrepresents the radial basis function used，wirepresents the weight of each basis function.In this paper，the radial basis function is Gaussian function：

In order to verify the feasibility of the RBFmodel，we take the determination coefficientR2as the criterion，and ninety sets of data as training points and teen sets of data as test points respectively，and test the accuracy of the constructed RBFmodel.The calculatedR2is as follows Table.4 shows：

Tab Ie 4 Accuracy of surrogate m ode I

6.3 Global sensitivity analysis

The Sensitivity analysis method is a Monte Carlo（MC）algorithm based on variance，which can calculate the sensitivity index of a single parameter，Si，and the interactions between these parameters through the ratio of each sensitivity index to the corresponding total sensitivity index.Specifically，considering an integrable function，f（x）in the following form［30］：

Where，1≤i1＜… ＜is≤n，fi1…is（xi1，…，xis）is a function of a unique subset of variables from x.Eq.（11）is also called the ANOVA representation off（x）if

Square the Eq.（12）and integrate it on the n-dimensional unithypercube to obtain the following equation：

Di1…isis the partial variance caused by the simultaneous changes of factorsi1toisin the model response，andDis the total variance off（x）.Then the overall sensitivity of each variable can be calculated as：

Through comparative analysis of the various parameters that affect the shrinkage of the sandwich，it is clearly that the thermal expansion coefficient of the mold and the panel directly affects the shrinkage of the sandwich.The elastic modulus of the panel，the thermal expansion coefficient and the elastic modulus of the resin have a small affect on the shrinkage of the sandwich.Through the comparative analysis of the various parameters that affect the roundness error of the sandwich，it is concluded that the thermal expansion coefficient of the resin is themost important factor affecting the roundness error.The elasticmodulus of the panel，the thermal expansion coefficient of the mold and the panel have a obviously affecton the roundness error of the sandwich.

Fig.16 Sensitivity ana Iysis of various param eters

7 Conclusion

（1）In this paper，a finite element analysis is carried out on the forming of the first-stage tail section of a sandwich composite with a diameter of 5 meters.The first-stage tail section is simplified into a 1/32 model and boundary conditions are applied.The honeycomb is equivalently processed，and the resin equivalentmodel is established using life and death element technology in curing process.We explored and determined themesh size to ensure the calculation accuracy and calculation speed.The analysis results show that there is residual stress inside the sandwich when it is not released，which will cause the sandwich structure to shrink；According to the initial design plan，the average shrinkage of the top of the sandwich structure is 3.312 mm.The roundness error of the mold is calculated by the least square method to be 0.155 7 mm.

（2）This paper studies that the influence of factors such asmold and panel thermal expansion coefficient，panel and resin thermal expansion coefficient，elastic modulus，Poisson’s ratio，honeycomb side length，honeycomb wall thickness on the shrinkage and roundness error of the sandwich structure.Nine parameters that have a greater impacton the shrinkage and roundness errors of the sandwich structure are selected to construct a surrogate model and the accuracy is tested.

（3）The sensitivity analysis of various parameters is carried out by using the sobol’smethod.The results show that the thermal expansion coefficient of themold and panel are themain factors affecting the shrinkage of the sandwich.The thermal expansion coefficient of the resin is the most important factor affecting the roundness error.