Numerical and Experimental Study on Curing-induced Residual Stress and Deformation of Adhesively Bonded Sandwich Structures of Dissimilar Materials

Adhesively bonded sandwich structure with dissimilar materials becomes an important means of lightweight for the next generation of autobody closure panels. However, during the baking process, the complicated change of physical properties of the adhesive can lead to structural mismatch deformation. In this paper, a multi-physics coupling numerical model of one-component hemming adhesive during the curing process is proposed, to reveal the deformation mechanism of adhesively bonded sandwich structures quantitatively. The material constitutive model of the hemming adhesive is established, considering evolution of curing properties. The curing process of typical aluminum alloy and steel bonded sandwich structure is simulated in COMSOL Multiphysics. The predicted surface deformation of the component is verified by experiment using Digital Image Correlation (DIC) technique, which captures the full-field displacement in a non-intrusive manner. Then, the development process of surface deformation of the outer panel and residual internal stress of the adhesive layer is analyzed, and the influence of temperature cycle on the maximum deformation of the component is discussed. The results show at the beginning of holding stage, the deformed component slightly rebounds, which is related to the chemical shrinkage. Mechanical strain caused by the coefficient of thermal expansion (CTE) and stiffness difference of the inner and outer panels is dominant in the adhesive layer. Reducing the curing rate and maximum holding temperature can reduce the overall deformation of the structure.


Introduction
Whether traditional fuel vehicles need to meet increasingly stringent environmental regulations, or new energy vehicles continue to improve the endurance mileage, autobody lightweight is a common and key basic technology [1][2]. The steel aluminum hybrid structure not only gives full play to the strength and manufacturing cost advantages of high-strength steel, but also takes into account the weight reduction and energy absorption characteristics of aluminum, and has become an effective way to lightweight the new generation of autobody [3]. Therefore, steel-aluminum hybrid closure panels with sandwich structure, such as doors, hoods and deck lids, gradually become the focus of lightweight research.
The sandwich structure is composed of the hemming adhesive, the inner panel and the outer panel, which are connected by the composite process of gluing, hemming and baking [4]. In general, the research and application of similar materials connection technology have been relatively mature [5]. However, when dissimilar materials are used, due to the significant differences in the stiffness and CTE, large residual stress and warping deformation of the structure are generated in the baking condition [6][7], which seriously affect the assembly accuracy and appearance perception quality of the closure panels. Therefore, in order to restrain and eliminate the structure mismatch, it is necessary to understand the deformation mechanism.
Compared with experiments, finite element simulation can effectively predict deformation and internal stress of the structure and reduce the cost of experiments, thus it has become an important research method. Due to the complex changes in the baking process, the constitutive model of the adhesive becomes the key to accurate numerical simulation. In early studies, linear model [8][9] was mainly used in the adhesive-bonded structure. As it can not describe the stress relaxation phenomenon, viscoelastic model [10] was gradually adopted. These models ignored the curing properties of the adhesive and assumed that the structure was in stress-free state before cooling. However, subsequent studies showed that many factors, such as curing temperature [11], curing rate [12], and multi-step method [13], could affect the final results. It can be seen that neglecting the mechanical properties of the adhesive in the heating and holding stage was easy to lead to large prediction deviation. Therefore, in order to achieve accurate prediction, some scholars proposed a multi-phase model covering the whole temperature cycle [14][15][16][17]. These models have been applied in coating process [15], resin mold material [16] and electronic packaging [17]. It provides a reference for the establishment of material constitutive model of the hemming adhesive.
In summary, the deformation mechanism of the adhesive-bonded sandwich structure of dissimilar materials in the baking process is not clear. It seriously limits the design and application of the new generation of material hybrid autobody closure panels.
In this study, a multi-physics coupling numerical model of one-component adhesive during the curing process is proposed. The material constitutive model of the hemming adhesive is established, considering evolution of curing properties. The curing process of typical aluminum alloy and steel bonded sandwich structure is simulated in Comsol Multiphysics. The predicted transient surface deformation is verified by DIC measurement experiment. Then, the development of the internal stress of the adhesive and the deformation of the structure are analyzed, and maximum deformation of the component under different temperature cycles is discussed.
2 Multi-physics fields coupling model

Governing equations
During the baking process, deformation process of the sandwich structure is a multi-physics fields involving heat transfer, material transformation and structural mechanics. The heat convection occurs between the outer surface of the structure and the surrounding air, while the heat transfer occurs inside. The temperature of air is heated from room temperature T0 to the highest temperature Tb at a certain heating rate K1. After holding for a period of time, the temperature drops to room temperature T0 at a certain rate K2. The differential equation of this transient heat transfer process is as follows [18][19] where ρ and c represent the density and specific heat of the material, ki is the heat conductivity coefficients (i=x, y, z), T represents the temperature, α and dα/dt represent the curing degree and instantaneous curing rate of the hemming adhesive respectively, and Hu represents the total amount of heat released by the adhesive after curing reaction.
The thermal convection boundary condition of the component is where Ts is the temperature of the component surface, Tref is the heating environment temperature, heff is the equivalent convection heat transfer coefficient of The hemming adhesive of Type Dow 1496 V is a kind of one-component thermosetting resin adhesive commonly used in the autobody closure panels.
Differential scanning calorimeter STA449C/3/G (Netzsch, Germany) is used to obtain the data in the curing process. The curing phase transition model of the hemming adhesive is an autocatalytic Kamal-sourour model [20], which is obtained by Málek where R is the universal gas constant of 8.314 Ea is reaction activation energy, and m and n are reaction order , as shown in Table 1.
In the viscoelastic structure field, the bulk modulus and shear modulus generally be used to establish the relationship between stress and strain [22]. The constitutive equation can be expressed as In the coupled physical field composed of Eq. (1)-(4), elastic and thermal strains are mainly produced in the aluminum alloy and steel panel. In addition to the elastic, thermal and chemical strains, the material stiffness of the adhesive will also change with the curing degree, temperature and time.

Constitutive model of the hemming adhesive
In order to improve the prediction accuracy and reduce the testing and calculation time of complex parameters, a four-phase curing constitutive model of the hemming adhesive is established during the baking process based on the previous studies [14][15][16][17], as shown in Fig. 1. During the phase I, as the temperature is lower than the gel temperature Tgel, the adhesive can flow and relax rapidly under the action of external force [15], thus it can offset the changes in the movement of the panels. Therefore, the effect of mechanical properties in this phase can be ignored. When the temperature exceeds Tgel, the initial shear modulus is established, and the mechanical properties mainly occur in phases II, III and IV. The total strain consists of three parts where ex ij  is mechanical strain, ch ij  is chemical strain and th ij  is thermal strain.

Phase II
Theoretically, the bulk modulus of the adhesive is related to the curing characteristics.
However, when the curing degree changes, it is found that the bulk modulus of the hemming adhesive is almost unchanged, which is similar to other polymers [23]. Therefore, this model assumes that the bulk modulus is only depends the temperature.
Based on the Tait equation, the modified bulk modulus is used in same way as in Literature [16,17] where 0 s , 1 b and 2 b are fitted constants of the adhesive material. The Tait constant C is 0.0894. The relationship between Pressure Volume Temperature (P-V-T) [23][24] is measured by high pressure dilatometer PVT6000 (Gotech, China). According to the measurement method of literature [16], the relevant parameters are shown in Table 1.
The establishment of shear modulus starts from the gel point Tgel. During the heating and holding phase, a high elastic function related to curing degree is used [10] is the shear modulus of the fully cured adhesive, gel  is the curing degree of the adhesive at Tgel.
The thermal strain can be expressed as Theoretically, the CTE of adhesive is changing with pressure, temperature and volume. As the baking process is completed under atmospheric pressure, it has nothing to do with pressure. The CTE is modeled in the same way as in literature [16][17] )]) ( tanh[ 1 ( 2 Although the chemical shrinkage and thermal expansion strains coexist, and the separation of the two is helpful to understand the process of stress generation. Many scholars have adopted linear simplified models related to curing degree [25] ch gel th ij max ) ( where ch max  is the volume chemical shrinkage between the gel point and the complete cure.

phases III and IV
During the cooling phase III and IV, the adhesive has been completely cured, and the chemical shrinkage has stopped. The model of bulk modulus is the same as the Eq.
(6)- (7), and the model of CTE is the same as Eq. (10). A linear viscoelastic relaxation function is used for the fully cured shear modulus [16] ) The deviator stress and strain can be expressed as follows where  G is the equilibrium modulus, m G and m  are the relaxation modulus and relaxation time of the m-th Maxwell unit. The stress relaxation parameters of the hemming adhesive are obtained by torsional test system with temperature chamber MTS809 (MTS, USA). According to the measurement method of literature [16], the fitted ten-order main shear relaxation modulus spectrum is shown in Table 2.
The mechanical relationship between any point and the reference temperature point can be established by using the time-temperature equivalent equation. The difference between phase III and phase IV lies in the time temperature equivalent equation. In phase III, WLF Equation [16] is used for the region above the glass transition temperature. In phase IV, Arrhenius Equation [26] is applied to the region of secondary where ref T is the reference temperature and Tg is the glass transition temperature.
The constant 1 C is 17.44 and 2 C is 51.6.

Material model of metal sheet
The material of outer panel studied is aluminum alloy AA6016-T4, which has good mechanical and bending characteristics, widely used in automobile outer panels. The inner panel material is cold-rolled steel DC04, and the main parameters are shown in Table 3. During the process of curing, as there is no plastic deformation, the linear thermoelastic constitutive model is used.

Numerical simulation
A typical flat surface-straight edge panels [4] is taken as the research object, and the aluminum alloy and steel bonded sandwich structure is shown in Fig. 2 (Table 1, 3) related to heat convection and heat transfer of different materials can be defined. In the PDE module, the general form of coefficient partial differential equation is: where u and t are the dependent variable and time respectively, f is the source form. c, a, ea, da, α, β and γ are the undetermined coefficients. By modifying the coefficients and the source term parameters (  The adhesive at the corner is divided into a triangular prism mesh with a minimum size of 0.05 mm, and the rest of the structure is divided into a hexahedral mesh as shown in Fig. 4. Along the x direction, the mesh of the structure is evenly divided, and the size of 2 mm. Along the y direction, the mesh size at the flanged part is relatively dense, which is 0.5 mm, while the mesh in other areas is relatively sparse. To improve the calculation accuracy, the quadratic Lagrangian element is used. In the temperature field, the air convection boundary condition is applied to the surface of the structure. In

Verification Experiment based on DIC
In the baking environment, the adhesive layer is sandwiched between the inner and outer panels. As the deformation is dynamic, it increases the difficulty of direct gauging.
When traditional strain gauge is used to measure the stain data of the adhesive layer in contact, the deformation accuracy of the sample is easily affected. Thus, indirect quantification of non-contact measurement becomes the best verification method [11].
To avoid the limitations of single point position measurement and record the full-field dynamic deformation of aluminum alloy outer panel, the DIC method is adopted, which is widely used in the deformation field of non-contact measurement [27].  Fig. 5a. Meanwhile, the length of the sample in the experiment is 10 mm longer than that of the simulation, and the extra 10 mm is used for clamping. Then, the sample is fixed by fixture, and placed on the refractory brick in the oven. The heating temperature cycle which is described in Section 3.1 is controlled by the oven, and the deformation results are recorded by the cameras. Finally, the data are compared and analyzed by DIC system. Each experiment is conducted three times and the average value is taken.

Comparison of methods
For the convenience of comparison, the beginning of temperature cooling is defined as initial point (t =0 min) by time equivalent conversion. Since surface quality is the focus of the sandwich structure, Fig. 6 shows the curing-induced deformation values in the z direction (Fig .4). It can be seen that the surface displacement values obtained by the experiment and the simulation are basically consistent, and the deformation trend is in good agreement. Because the width value is small, the deformation of the aluminum alloy panel changes little along the y direction. Therefore, in the case of large length-width ratio, the deformation of the sandwich structure is approximately regarded as a plane strain problem. In general, under the temperature cycle, the component first bends along the negative direction of the z-axis, then gradually stabilizes, and finally bends in the positive direction of the z-axis. The results show that the deformation process can be simulated well based on the multi-physics coupling model. As the deformation value is the largest and can reflect the degree of the structural deformation, the M-point (Fig .2) on the middle section is selected as the comparison point. Fig. 7a shows the transient variation process of M-point on the middle section obtained by using different methods. It can be seen that the curing-induced structural displacement does not start at the beginning of heating. When the molecular crosslinking reaches a certain degree, M-point begins to move in the negative direction of the z-axis. After reaching the holding temperature, the deformation begins to rebound slightly, and then enters a stable state, which is related to the curing shrinkage. Finally, with temperature cooling down, the point begins to warp in the positive direction of the z-axis. At the beginning of the cooling stage, the deformation speed of the structure is relatively slower, which is related to the faster stress relaxation of the adhesive. Fig. 7b shows the final z-axis displacement of the outer contour of the middle section. It can be seen that the traditional viscoelastic models [9,11] have a relatively large deviation from the results, which ignores the change of material properties. The multi-physics coupled model well reflect the changing process.  that the temperature almost does not change along the x direction. In the heating stage, the temperature at the corner is slightly lower than that in other regions along the y direction, which is related to the heat transfer. In the high holding stage, the temperature is about 1~2 ℃ higher at the corner than that in other regions, which is related to the exothermic of the adhesive. In the heating and holding stage, the curing field is similar to the temperature field, and the difference is relatively small in different areas. In the structural field, the von Mises stress on both sides is relatively large, especially in the cooling stage, which is mainly related to the boundary constraints.  Fig. 9 shows the development of temperature and curing degree distribution of the adhesive layer in the yz plane. It can be seen that the maximum difference in different regions of the adhesive layer is about 1~2 ℃. This is due to the high thermal conductivity of the panels and the relatively thin thickness of the adhesive layer, which results in no delaying and obvious gradient change. After heating up, the temperature continued to rise slightly due to the exothermic reaction of the adhesive. Accordingly, under the same external temperature environment, the change of the curing degree of each position has almost the same.    (Fig. 2) of adhesive layer. When the temperature exceeds Tgel, the bonding strength at the interface is gradually established between the adhesive and the panels. Since the CTE of aluminum alloy is larger than that of steel, relative dislocation movement occurs [7].
Compared with other structures, such as bimaterial [10] and bonded substructure [14], mechanical strain caused by the difference of material properties plays an important role.
At the beginning of the phase II, the influence of thermal expansion strain and mechanical strain on deformation is greater than that of chemical shrinkage. After reaching the maximum temperature 170℃, the structure produces a small amount of positive movement along the z-axis, which can be explained by continuous curing shrinkage [12], as shown in Fig. 8. In the phase III, the chemical shrinkage gradually stops and remains unchanged. And compared with thermal strain, mechanical strain began to dominate, which cause the structure bend forward along the z-axis. In the phase IV, the mechanical strain decreases gradually, due to the increase of the adhesive modulus.  since the stress relaxation is fast. With the glass transition of the adhesive, then the stress increases rapidly. In the phase IV, the stress increases slowly, which is related to the decrease of CTE and stress relaxation. Fig. 11b shows the final internal stress at the middle line of the upper adhesive layer. The tensile and peel stress are slightly larger at the edge, which is related to the boundary constraint. The shear stress direction of the left side and the right side changes, which is related to the asymmetric structure. shows the final internal stress at the middle line of the lower adhesive layer. The stress near the circular corner is relatively large, which is related to the geometric structure.  Figure 13 shows the final internal stress at the circular corner. It can be seen that the internal stress changes dramatically, and the stress concentration appears in the right angle region, due to the change of boundary conditions. The tensile and shear stresses at the circular corner are less than those at the upper and lower adhesive layers. The shear stress decreases obviously and becomes zero in the middle region. show that the final deformation of the structure can be reduced by reducing the curing rate. This is due to the heating rate affects the gel point and the bonding temperature between the adhesive and the panels [5,12]. Fig. 14b shows the curing degree curves of the adhesive under different heating rates. When the heating rate decreases, the curing curve moves to the left, which leads to lower gel temperature. On the contrary, the higher the heating rate, the higher the gel temperature. When the heating rate exceeds a certain value, as the 8 K/min, the adhesive is bonded to the panels in the heat holding stage, and the overall deformation value is the maximum [13]. Adjusting the curing kinetics characteristics of the adhesive layer can reduce the structural deformation, but it may prolong the curing time. 1). Compared with the traditional viscoelastic model, the multi-physics coupled model has higher prediction accuracy and can better reflect the deformation process of the structure, due to considering the change of adhesive material properties.

Influence of curing temperature cycles
2). At the beginning of the holding stage, the deformed component slightly rebounds, which is related to the chemical shrinkage. At the beginning of the cooling stage, the deformation rate is relatively slow, which is related to the faster stress relaxation of the adhesive.
3). The temperature difference in different areas of the adhesive layer is relatively small, about 1~2 ℃, and the difference of the curing degree is less than 0.01, due to the high thermal conductivity of the panels and the thin thickness of the adhesive layer.
4). Compared with other structures, such as bimaterial [10] and bonded substructure [14], mechanical strain caused by the CTE and stiffness difference of inner and outer panels is dominant. The direction and size of shear stress in the upper and lower adhesive layers change along the y-axis, which is related to asymmetric structure. 5). When the gel temperature is the maximum holding temperature, the overall deformation reaches the maximum. Reducing the curing rate and maximum holding temperature is helpful to reduce the overall deformation, as it reduces the gel temperature of the components.

Conflicts of interest/Competing interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Figure 1 Diagram of four-phase curing model of the adhesive under temperature cycle Figure 2 Geometric dimensions of adhesively bonded sandwich structure with aluminum alloy and steel sheet Finite element mesh of adhesively bonded sandwich structure with aluminum alloy and steel sheet Figure 5 Curing-induced distortion measurement experiment. a Non-invasive measurement schematic diagram. b

Figures
Measurement device based on DIC technology  Evolution of temperature and curing degree of the adhesive layer  Geometric dimensions of adhesively bonded sandwich structure with aluminum alloy and steel sheet Finite element mesh of adhesively bonded sandwich structure with aluminum alloy and steel sheet Figure 5 Curing-induced distortion measurement experiment. a Non-invasive measurement schematic diagram. b Measurement device based on DIC technology  Evolution of temperature and curing degree of the adhesive layer