Investigation of curing deformation behavior of curved fiber metal laminates

Journal Pre-proofs Investigation of curing deformation behavior of curved fiber metal laminates Lu Che, Guodong Fang, Zengwen Wu, Yunfei Ma, Jiazhen Zhang, Zhengong Zhou PII: DOI: Reference:

S0263-8223(19)30708-1 https://doi.org/10.1016/j.compstruct.2019.111570 COST 111570

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Composite Structures

Received Date: Revised Date: Accepted Date:

25 February 2019 23 September 2019 11 October 2019

Please cite this article as: Che, L., Fang, G., Wu, Z., Ma, Y., Zhang, J., Zhou, Z., Investigation of curing deformation behavior of curved fiber metal laminates, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct. 2019.111570

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Investigation of curing deformation behavior of curved fiber metal laminates Lu Che a, Guodong Fang a,, Zengwen Wu a, Yunfei Ma b, Jiazhen Zhang a, Zhengong Zhou a a

Science and Technology on Advanced Composites in Special Environments Key Laboratory, Harbin Institute of Technology, Harbin 150001, P.R. China

b

Department of Civil Engineering, Harbin Institute of Technology, Harbin 150001, P.R. China

Abstract To keep the size and shape accuracy of fiber metal laminates (FMLs) with an initial curvature during manufacturing, curing deformation behavior is investigated by using analytical, numerical and experimental methods. Based on Eckstein’s theory, an analytical model considering slippage effect between metal layer and fiber layer is developed to study the warping deformation modes of curved FMLs after curing process. The warping displacements of the curved FMLs predicted by the analytical method are in good agreement with the finite element results and experimental data. The final equilibrium configurations of the curved FMLs can be captured by using the analytical method, which can guide the size and shape control for FMLs. The effect of the side length, initial curvature radius and stacking sequence on the curing deformation characteristics of curved FMLs is studied. The curing deformation mechanisms of the curved FMLs with [Al/0/90/Al] and [Al/90/0/Al] stacking sequences are also illustrated from the perspectives of force and total strain energy. Keywords: Fiber metal laminates, interfacial slippage, residual stress, curved shape, curing deformation.



Corresponding author. Tel.: +86 451 86402396; fax: +86 451 86402386

E-mail address: [email protected] (G. Fang). 1

1. Introduction Fiber metal laminates (FMLs) have excellent damage tolerance and fatigue resistance behavior in comparison of other monolithic materials [1-3], which have been used in the fuselage panels and leading edges of tail planes of Airbus A380. There exists shape distortion phenomenon of FMLs during manufacturing due to the thermal residual stresses produced from non-symmetrical layup and the different coefficients of thermal expansion of the constituents [4, 5], especially for the thin non-symmetrical curved FMLs. It is essential to keep the size and shape accuracy for FMLs satisfying some structure assembling requirements. The FMLs are mainly used in the curved structure, such as integrated fuselage panels. The size and shape control for the curved FML structures during manufacturing is critical to the structure assembling, which is more complicated than that of flat FML structures [6]. Some analytical and numerical methods have been developed to study the warping deformation of laminates due to the thermal residual stress during manufacturing. Hyer [8, 9] proposed a non-linear analytical model to study a phenomenon that the cured shapes of thin unsymmetric cross-ply composite laminates were two approximately cylindrical shapes, rather than the saddle shape predicted by Classic Lamination Theory (CLT) [7]. The non-linear analytical model is extended from classical lamination theory including geometric nonlinearities in combination with a Rayleigh-Ritz minimization of total potential energy. To improve the accuracy of the calculated stable configurations and bifurcation behavior [10-14] and take into account more complex boundary conditions [15, 16], more complex analytical models using higher order polynomial displacement functions have been proposed further. As for hybrid laminates, several analytical models have been developed to study the bond-slip phenomenon at the metal-fiber interface by using the optimized coefficients

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of thermal expansion [17-19] and introducing the interfacial slippage strain [6]. The shape distortion and residual stresses of FMLs after the cure cycle considering the thermo-viscoelastic response of composite layer were also studied by using FE method [20]. In addition, some factors, such as tool [21, 22], chemical shrinkage [23, 24], aspect ratio [14, 25], moisture absorption [23, 26], thermal gradient [27], imperfection [28, 29], temperature dependent material properties [27, 30] and viscoelastic behavior [20, 31] etc., were taken into account to study their influences on the cured deformation behavior of unsymmetric laminates. As for cured shapes of cylindrical shell and curved plates, Ren et al. [32] investigated the shape deviations of initially cylindrical composite shells with four different stacking sequences after curing. Pirrera et al. [33] combined a Rayleigh-Ritz method with path-following algorithms to study bistable composite cylindrical shells. Eckstein et al. [34] proposed an initially curved plate model using plate von-Karman nonlinear strains instead of nonlinear shell kinematics and found that the cross-ply laminates with initial curvature exhibited multiple deformation modes within a temperature range. Based on Eckstein’s initially curved plate model, Mostafavi et al. [35] studied the thermal deformation of two or more connected curved composite plates by taking into account continuity of displacement and boundary conditions. It is noted that there is little research about stable configurations and warping deformation properties of FMLs with an initial curvature after curing. The primary purpose of the present paper is to accurately predict the cured shapes of curved FMLs under thermal stress and to expound the curing deformation mechanism of the FMLs with different stacking sequences. The requirements of structural assembly can be satisfied by controlling the size and shape for the curved FML’s structure during the fabrication process. In this paper, an analytical model

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based on Eckstein’s theory considering slippage effect is developed to study the cured shapes and deformation modes of the curved FML structure. The sixth-order out-of-plane displacement function polynomials are used to describe the initial and final shapes of FMLs, and the final shape function coefficients are solved by using Rayleigh-Ritz method. The proposed analytical method is also validated by using the out-of-plane corner displacements of six specimens with different initial curvatures and stacking sequences. To deeply understand the relatively complex deformation modes of the curved FML’s structure, the stress conditions and the changes of internal potential energy and energy barrier after curing are analyzed. In addition, the influence of side length, initial curvature radius and stacking sequence on the stable equilibrium configurations of the curved FMLs are investigated by using the analytical model. 2. Analytical model for the curved FMLs structure The initial geometry of FML before curing is a cylindrical shape as shown in Fig. 1. The geometric center of FML is set to the origin of coordinate system. The FML has a constant curvature in the x-direction and zero curvature in the y-direction. In Fig. 1, Lx and L y represent the edge lengths of the FMLs along the curved and non-curved directions, respectively. There exist initial and final deformed states for the curved FMLs during curing process. The initial state is that the FMLs are perfectly matched in curved mould before curing. The final deformed state is that the FMLs are removed from the curved mould, which may appear warping deformation due to the thermal residual stress. According to Eckstein’s theory [34], the initial FML cylindrical shell can be regarded as a curved plate. The von Karman plate kinematic equations for the curved FMLs are still satisfied. Thus, the non-linear classical lamination theory also can be used in this work. In view of the symmetry and an-

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ti-symmetry characteristics and satisfying the essential boundary conditions, the initial and final out-of-plane displacement of the curved FMLs can be expressed by using sixth-order polynomials as following expressions.

1  a10 x 2  b10 y 2  a11x 4  b11 y 4  a12 x 6  b12 y 6  e1 x 2 y 2

(1a)

2  a20 x 2  b20 y 2  a21x 4  b21 y 4  a22 x 6  b22 y 6  e2 x 2 y 2

(1b)

where a1i , b1i , e1 and a 2i , b2i , e2 represent the initial and final shape function coefficients, respectively. The initial geometry of FML as seen in Fig. 1 is a determined cylindrical shape with constant curvature along the x-direction. Thus, a10 , a11 and a12 are known constant coefficients which can be determined by the initial geometry

shape of curved FMLs. Because the initial geometry of the curved FMLs has a zero curvature in the y-direction, the coefficients b10 , b11 , b12 and e1 are equal to zero. As for the final deformed state of curved FMLs after curing, seven independent unknown shape function coefficients need to be determined. The curvatures difference between the initial and final curved plate can be defined by the second derivative of out-of-plane displacement function.

   2 2  21      2  2   x x    k 21  k11    2      2   Δk  k 22  k12      22  21   y y    k 23  k13    2     2  21     2  x  y  x  y   

(2)

The FML is in a stress-free state before curing, so only final mid-plane strain functions of curved FML can be written in the following forms

 x0  c  c1 y 2  c2 y 4  c3 y 6  c4 x 2 y 2

(3a)

 y0  d  d1 x 2  d 2 x 4  d 3 x 6  d 4 x 2 y 2

(3b)

The in-plane shear strain cannot be independent of the first two mid-plane strains, 5

and must satisfy the compatibility equation of curved surface [34].

2 2 2 0 2 0  2 x0   y   xy   22   22  22   21   21  21      2       xy  xy  y 2 x 2 x 2 y 2  xy  x y 2   

(4)

Substituting Eqs. (1) and (3) into Eq. (4), the in-plane shear strain  xy0 can be computed by integrating x and y, and the expression of mid-plane strain vector ε 0 also can be obtained. During the manufacturing process of the curved FMLs, the interface between metal layers and fiber layers exhibits the bond-slip phenomenon due to the incompatible deformation. This bond-slip phenomenon is mainly caused by the fact that the interface between metal layers and fiber layers is not perfectly bonded at the heating stage. In our previous paper [6], the prediction accuracy of the analytical model can be improved by about 10% by considering the interfacial interaction. Without considering this interfacial slippage effect, it is difficult to accurately predict the equilibrium configurations of the curved FMLs at room temperature. Therefore, the contribution of both thermal shrinkage and slippage effect to the warping deformation of FMLs should be taken into account. The thermal strain energy and slippage strain energy must be simultaneously introduced into the total potential energy function, which can be written as 

Ly



2 Ly 2

Lx 2 L  x 2



 1  ε 0 T  A B  ε 0   ε 0 T  N T   ε 0 T  N S    dxdy    2 Δk   B D  Δk  Δk   M T  Δk   M S    

(5)

where the expressions of the FML extensional, coupling and bending stiffness matrices A , B , D and the thermal force and moment vectors N T , M T are described in detail in Ref. [6]. In particular, N S and M S are the resultant force and moment vectors resulting from slippage effect, which can be given as 6

N

S



n

 

, M S   Q ij  Sj (k )

k 1

(k )



1 2 2   z k  z k 1 , 2 z k  z k 1

i, j  1,2,6 

(6)

(k )

where Q ij represents the transformed reduced stiffness matrix of the kth layer, and

 

S (k ) j

represents the interfacial slippage strain whose expression is given by

 mS 

1   E f t f

 m Em t m  a f E f t f T E f t f  Em t m E f t f  Em t m

 Sf  

1   Emtm  m Emtm   f E f t f E f t f  Emtm

E f t f  Emtm

T

(7a)

(7b)

where, the subscript m and f denote the metal layer and fiber layer (prepreg layer), respectively, and the direction of all physical quantities refers to the direction along the fiber. For instance,  mS and  Sf represent the interfacial slippage strains in the metal layer and the 0° prepreg layer in the fiber direction. Because the thermal expansion coefficient of aluminum alloy is close to that of prepreg in 90° direction and the epoxy resin matrix has a lower stiffness, the slippage strains perpendicular to fiber direction can be trivial and negligible. Similarly,  and E denote the coefficient of thermal expansion and Young’s modulus, respectively. The variable t is the individual layer thickness and T is the temperature variation ( T  100℃ ). Note that the negative sign in Eq. (7b) indicates that the direction of slippage strain in the prepreg layer is opposite to that in the metal layer. In order to describe the relative slippage of metal layer and fiber layer at the interface, a slip coefficient  depending on the material properties and fabrication process is introduced. The value of  is selected on the basis of experimental results, and its value range is  f  m    1 . More details can be found in the Ref. [6]. Substituting the expressions of curvature vector Δk and mid-plane strain vector

ε 0 into the total potential energy function  as provided in Eq. (5), the high order 7

non-linear algebraic equation has 17 unknown coefficients. In order to solve these unknown coefficients a 2i , b2i , ci , d i and e2 , the minimum total potential energy theory can be utilized. That is to say, the first order differential equation of  is zero and each item in Eq. (8) also equals to zero.

 total 

 total     a2i  total b2i  total ci  total d i  total e2  0 a2i b2i ci d i e2 fi 

(8)

 total  total  total  0 , gi   0 , pi  0 a2i b2i ci qi 

 total  total 0, o 0 d i e2

(9)

The software Mathematica [36] is used to solve the non-linear equations system as given in Eq. (9). Thus, the equilibrium configurations of the curved FMLs after curing can be determined by substituting the obtained shape function coefficients into the final out-of-plane displacement function. In addition, for a stable configuration, the Jacobian matrix of the non-linear equations system must be positive definite and vice versa. J

 f i , g i , pi , qi , o  a2i , b2i , ci , d i , e2 

(10)

3. Experiment 3.1. Curved FMLs and measurements The curved FMLs were manufactured in a curved steel mould by using the unidirectional HS4 glass fiber/epoxy prepregs and thin isotropic aluminum sheets. The basic mechanical properties of these materials are provided in Table 1. The curved moulds are made of 304 stainless steel with thickness of 3mm and are formed by roll bending process, as depicted in Fig. 2. The stiffness and thermal expansion coefficient of 304 stainless steel are 200 GPa and 17.3×10-6 /°C, respectively. In Fig. 2(a), l rep-

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resents the arc length of the mould, which is a part of the circle with a curvature radius of R. The central angle corresponding to the arc length l is denoted by  . The geometric dimensions of three steel moulds with different initial curvatures are shown in Table 2. In the present work, the influence of moulds on the cured shapes of curved FMLs is neglected due to the following reasons: Firstly, because the curved FML specimens are relatively thin, the mould does not cause a complex stress gradient inside the specimens. Secondly, the mould itself has a high stiffness, and the groove part of the mould can restrict the deformation of the curved mould during the curing process. Thirdly, the bottom aluminum sheets of FMLs are in contact with the curved mould surface. Both aluminum alloy and stainless steel are isotropic materials and their coefficients of thermal expansion are relatively close. Last but not least, there is a layer of release film placed between the steel mould and the FML specimens. Therefore, the interaction between them is extremely weak and can be assumed to be negligible. The prepregs and aluminum sheets are manually laid-up and perfectly matched with the curved steel moulds due to their relative low stiffness. The stacked laminates were vacuum bagged and cured in an autoclave. The curing temperature and pressure were 120 °C and 0.5 MPa, and molding time lasted for two hours. These laminates were taken out of the curved mould after naturally cooling to room temperature. Six curved FML specimens with the same geometrical size (450×450 mm), stacking sequence ([Al/0/90/Al], [Al/90/0/Al]) and initial curvature radius (R=400 mm, R=500 mm, R=600 mm) were fabricated. All specimens produced shape distortions and curvature changes due to the unbalanced internal stress, but still retained the cylindrical shapes are shown in Fig. 3. In order to quantitatively describe the warping deformation of FMLs, the cured

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shapes of FMLs were measured by using a handheld three-dimensional laser scanner (Handyscan700, Creaform Co., Canada). The measurement process is illustrated in Fig. 4. This laser scanner can provide the three-dimensional coordinates of data points distributed on the deformed laminate surfaces, and then fit these data points into a surface using MATLAB [37]. Thus, the out-of-plane displacements of these specimens can be used to verify the predicted results of our analytical model. 3.2. Determination of initial shape function coefficients The expression 1 as provided in Eq. (1) is the initial out-of-plane displacement function of the curved FMLs prior to curing, in which the initial shape function coefficients a10 , a11 and a12 should be determined by the initial geometry of specimens. The geometry of the mould in Cartesian coordinates is shown in Fig. 5. The shape curve of the mould is a circular arc, and can be expressed by the equation of the circle.

z  R  R2  x2

(11)

It is noted that the curvature direction of the shape curve described in Fig. 5 is opposite to the actual curvature direction of the mould. The curve equation of the mould is expanded by using Taylor’s series at x  0 , which ignoring the high order infinitesimal of x can be expressed as f ( x) 

1 2 1 1 x  3 x4  x6 5 2R 8R 16 R

(12)

Thus, the curve equation of the mould can be approximated by the polynomial function f x  . Comparing with the out-of-plane displacement function 1 , the initial shape function coefficients a10 , a11 and a12 of FMLs with different initial curvatures can be determined as shown in Table 3. 10

Fig. 6 shows the curve of 1 and the actual shape curve of mould when the curvature radius of mould is 0.5 m. If the side length Lx of the FML is 450 mm, the maximum warping displacement of the initial shape of FML before curing is 49.78 mm at the edge of the laminate, and the maximum out-of-plane displacement on curve

1 is 49.75 mm. The error between them is only -0.06%. Thus, it will not affect the predicted results of analytical model. Therefore, it is illustrated that the sixth-order out-of-plane displacement function 1 can accurately describe the initial shape of FML before curing, and the determination of the initial shape function coefficients are reasonable and accurate. 4. Results and discussion 4.1. Curing shape analysis of curved FMLs Fig. 7 shows the cured shapes of six curved FML specimens with different initial curvature radius and stacking sequence obtained by the analytical, FE and experimental methods. The analytical model can be validated by comparison with the FE results and experiment measurements. Finite element analysis (FEA) has been performed by using geometrically non-linear algorithms as described in Ref. [6]. The commercial FE program ABAQUS [38] is employed in the analysis, and the element type for the laminates is S4R. In Fig. 7, the small yellow dots represent scanned data points obtained from experimental measurements and the small red dots represent the deformed nodes extracted from ABAQUS. The upper surfaces in each subgraph represent the experimental shapes (ES), which are fitted by these scanned data points. The middle surfaces represent the finite element analysis shapes (FEAS), which are fitted by the extracted deformed nodes. The lower surfaces represent the analytical shapes (AS) based on the analytical model considering slippage effect. The z-axis represents the out-of-plane displacements of the curved FMLs in Fig. 7. The FEA 11

shape and measured shape of the curved FML are raised by 0.1 m and 0.2 m with respect to the reference plane, respectively. Thus, the central point coordinates of the analytical, FE and experimental shapes in the global coordinate system are (0, 0, 0), (0, 0, 0.1) and (0, 0, 0.2). Fig. 7(a), (b) and (c) show the initial shapes of FMLs with three curvature radii. The cured shapes of FMLs with respect to the initial shape can be divided into two states: State A and State B. State A has the same curvature direction with the initial shape, while the State B has the converse direction. The cured shape of [Al/0/90/Al] FMLs is corresponding to the State A, and the curvature in the x-direction becomes larger in comparison with the initial shape as shown in Fig. 7(d), (g) and (i). However, the cured shape of [Al/90/0/Al] FMLs changes to State B whose curvature is along the y-axis as shown in Fig. 7(f), (h) and (j). The cured shape and initial shape are always towards the same side of x-y plane. In order to find another possible stable equilibrium configuration, a force is applied on the outer surface of each FML. It is found that only specimen 4 of FMLs with [Al/90/0/Al] layup appears the snap-though phenomenon when the force is applied as shown in Fig. 7(e). The equilibrium configuration of specimen 4 after jumping is State A, and its curvature is less than that of the initial shape. It is noteworthy that the FE simulation result for this case is not shown, which is due to the significant geometric nonlinearity leading to the collapse of the FE program during the computation. The out-of-plane displacement of corner points and their errors (analytical model vs. experiment and FEA vs. experiment) are listed in Table 4. It is seen that the predicted results of these FMLs obtained by the analytical model are very close to the FE and experimental results. In Table 4, Lx’ represents the length of the curved edge of the FML projected onto the x-y plane. The value of slippage coefficient at the met-

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al-fiber interface need to be determined for the analytical model considering slippage effects. The value of  is dependent on the material properties and fabrication process, which is 0.65 in the present study. It can be found that the measured and simulated out-of-plane corner displacements are almost slightly less than the analytical model predictions, regardless of state A or state B, which can be attributed to the free edge effects [6, 28]. The higher inter-laminar stresses at the boundary result in a slight bend at the edge of the laminate. The bending deflection is more apparent near the corners. Since the free edge effects are not taken into account in the analytical model, the predicted values are slightly higher than the simulated and measured results. Meanwhile, the FE model can simulate relatively complex boundary conditions by considering the free edge effects. Thus, the results of FE simulation are closer to the experimental data than the analytical model. In order to elucidate the curing deformation mechanisms of FMLs with different stacking sequence, the deformation mode and force condition of the laminates with initial curvature radius of R=400 mm are shown in Fig. 8. The FMLs with R=400 mm have the more complicated deformation behavior as mentioned above. In the manufacturing process, the thermal stresses (thermal loads) inside the laminate are macroscopically represented as bending moments in both x and y direction. As for the specimen 1 with layup [Al/0/90/Al] as seen in Fig. 8(a), the bending moment in the x-direction Mx causes the laminate to bend deeper than the initial shape, and this equilibrium configuration is state A. The bending moment in the y-direction My is numerically equal to Mx, but the direction of bending moment is opposite. Since the laminate has a relatively high bending stiffness along y-direction, the bending moment My will not cause the snap-though phenomenon. Thus, the equilibrium configuration for state B does not exist. The curing deformation behavior of specimen 1 can also be ex-

13

plained in the view of strain energy [39]. If the initial shape is a flat plate, the total strain energy stored in state A is equal to that stored in state B. When the laminate has a certain initial curvature in the x-direction, the total strain energy of state B and energy barrier increase. With the increase of initial curvature, this energy barrier will reach the same height as the strain energy of state B. At this point, only one local minimum exists in the total potential energy curve of the laminate. Accordingly, the cured shape only has one stable equilibrium configuration of state A. As for the specimen 4 with layup [Al/90/0/Al] as shown in Fig. 8(b), the bending moment in the x-direction Mx decreases the initial curvature of the laminate. In this case, the bending moment My is sufficient to overcome the bending stiffness of laminate in the y-direction, and can cause the occurrence of snap-though phenomenon. Therefore, the state B equilibrium configuration can be observed. As for the [Al/90/0/Al] laminate, the initial curvature in the x-direction reduces the total strain energy of state B and energy barrier. However, the total potential energy still has two local minimum points. Thus, the laminate exhibits two stable equilibrium configurations which can be converted to each other. Please note that state B is more stable than state A because the minimum point of total strain energy of state B is lower than that of state A. As the initial curvature radius of laminate increases, the total strain energy of state B and energy barrier will continue to decrease. When the energy barrier is lower than the total strain energy of state A, the cured shape is only the state B equilibrium configuration, such as specimen 5, R=500 mm and specimen 6, R=600 mm. 4.2. Parametric study The influences of the edge length, initial curvature radius and stacking sequence on the deformation mode and cured shape of the FMLs are studied by using the ana-

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lytical model with slippage effect as follows. Fig. 9 shows the effect of the stacking sequences and initial curvature radius on the out-of-plane corner displacement for the 450×450 mm FMLs. The FML with layup [Al/0/90/Al] only exhibits the state A configuration as shown in Fig. 9(a), when the initial curvature radius is less than 925 mm. There exist two stable equilibrium configuration of state A and state B simultaneously, when the initial curvature radius is larger than 925 mm. The initial shapes and all the cured stable configurations obtained by the analytical model at R=400 mm and R=1500 mm are also provided in Fig. 9(a). The out-of-plane corner displacement of initial shape and stable state A gradually decreases as the initial curvature radius increases. With the increase of the initial curvature radius, the out-of-plane corner displacement of stable state B increases gradually. When the initial curvature radius increases to infinity, the initial shape of the laminate is a flat plate. The out-of-plane corner displacements of the two cured equilibrium configurations are ±39.00 mm, which has been discussed in detail in Ref. [6]. As for the layup [Al/90/0/Al] FMLs as shown in Fig. 9(b), the stable state A appears in two initial curvature radius intervals 300-400 mm and 1300-2000 mm, while the stable state B always exists. When the cured shape is closer to the flat plate, the stable state A disappears. In the initial curvature radius intervals 400-1300 mm, the energy barrier is lower than the total strain energy of state A in this range, and the total potential energy has only a local minimum point corresponding to state B. With the increase of initial curvature radius, the out-of-plane corner displacement of state A decreases for the range of 300-400 mm, and will increase for the range of 1300-2000 mm. The rate of decrease is greater than the rate of increase. The out-of-plane corner displacement of initial shape and state B gradually decreases as the initial curvature

15

radius increases, while the curve of state B changes more gently. In Fig. 9(b), two typical values of initial curvature radius (R=400 mm and R=1800 mm) are selected to exhibit the deformation modes of FMLs. The deformation law of FMLs can be clearly grasped by giving the initial shapes and all the cured stable configurations at these two points. Fig. 10 shows the predicted curves of the out-of-plane corner displacement versus the side length for the FMLs with an initial curvature radius of 400mm. As for the layup [Al/0/90/Al] FMLs as shown in Fig. 10(a), only state A configuration can be observed with the change of side length. The out-of-plane corner displacement of stable state A increases with the increase of the side length of the FML. As for [Al/90/0/Al] FMLs as shown in Fig. 10(b), the state B configuration always exists with the change of side length, and the stable configuration A appears when the side length of the FML is larger than 450 mm. The out-of-plane corner displacement of state A and state B increases with the increase of the side length of laminates. The out-of-plane corner displacement of stable state B is close to that of initial shape. It is noted that in Figs. 9 and 10, the results of FE simulation and experiment data are always slightly less than those of analytical model due to the free edge effects. When the curvature direction of cured shape changes, the cured shape and the initial shape may be on the same side or both sides of the x-y plane. Thus, the different stacking sequence and initial curvature radius cause the relatively complicated thermal stresses inside the FMLs exhibiting a variety of curing deformation modes. 5. Conclusions In the present paper, the curing deformation behavior of FMLs with initial curvature under thermal stress has been investigated, which can be helpful of guiding the design and manufacturing of FMLs. The results can be summarized as follows:

16

(1) Based on Eckstein’s initially curved plate model, the analytical model with slippage effect has been developed to accurately predict the cured shapes of curved FMLs. The predicted results of the analytical model are slightly higher than those of the FE method and the experimental data, which is attributed to the free edge effects. The moments generated by thermal residual stress may increase or decrease the initial curvature of the curved FMLs, and may also make the curvature direction of the stable configurations perpendicular to the initial shape. (2) The complicated deformation modes of the curved FMLs are explained by using the mechanism of the competition between the total strain energy stored in the stable state and the energy barrier. The initial curvature of the FMLs can lead to changes in the total strain energy of stable state and energy barrier. The local minimum points of the total strain energy correspond to the stable configurations of the curved FMLs after curing, respectively. (3) The deformation laws of the cured stable configurations varying with the edge length and the initial curvature radius under two stacking sequences are provided. At typical initial curvature radii and edge lengths, all possible curing deformation modes of the curved FMLs are provided, which will help to understand the complex deformation laws of the FMLs with initial curvature. For the stacking sequence [Al/0/90/Al] FMLs with initial curvature radius R=400 mm, only one stable equilibrium configuration exists with the increase of the edge length of the laminate. For other cases, two stable configurations can be observed simultaneously within certain ranges. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 11572101, 11672089), Natural Science Foundation of Heilongjiang

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Province, China (Grant No. A2017003), and the Fundamental Research Funds for the Central Universities (Grant No. HIT.NSRIF.2017017). References 1. Sinmazçelik T, Avcu E, Bora MÖ, Çoban O. A review: fiber metal laminates, background, bonding types and applied test methods. Mater Des 2011; 32(7): 3671-85. 2. Li H, Hu Y, Liu C, Zheng X, Liu H, Tao J. The effect of thermal fatigue on the mechanical properties of the novel fiber metal laminates based on aluminum-lithium alloy. Compos Part A 2016; 84: 36-42. 3. Zhang J, Wang Y, Zhang J, Zhou Z, Fang G, Zhao Y, et al. Characterizing the off-axis dependence of failure mechanism is notched fiber metal laminates. Compos Struct 2018; 185: 148-60. 4. Parlevliet PP, Bersee HEN, Beukers A. Residual stresses in thermoplastic composites-A study of the literature-Part I: Formation of residual stresses. Compos Part A 2006; 37(11): 1847-57. 5. Wisnom MR, Gigliotti M, Ersoy N, Campbell M, Potter KD. Mechanisms generating residual stresses and distortion during manufacture of polymer-matrix composite structures. Compos Part A 2006; 37: 522-9. 6. Che L, Zhou Z, Fang G, Ma Y, Dong W, Zhang J. Cured shape prediction of fiber metal laminates considering interfacial interaction. Compos Struct 2018; 194: 564-74. 7. Hyer MW. Some observations on the cured shape of thin unsymmetric laminates. J Compos Mater 1981; 15(2): 175-94. 8. Hyer MW. Calculations of the room-temperature shapes of unsymmetric laminates. J Compos Mater 1981; 15: 296-310. 9. Hyer MW. The room-temperature shapes of four-layer unsymmetric cross-ply laminates. J Compos Mater 1982; 16(4): 318-40. 10. Jun WJ, Hong CS. Effect of residual shear strain on the cured shape of unsymmetric cross-ply thin laminates. Compos Sci Technol 1990; 38(1): 55-67. 11. Dano ML, Hyer MW. Thermally-induced deformation behavior of unsymmetric laminates, Int J Solids Struct 1998; 35(17): 2101-20. 12. Aimmanee S, Hyer MW. Anslysis of the manufactured shape of rectangular THUNDER-type actuators. Smart Mater Struct 2004; 13: 1389-406. 18

13. Pirrera A, Avitabile D, Weaver PM. Bistable plates for morphing structures: a refined analytical approach with high-order polynomials. Int J Solids Struct 2010; 47(25-26): 3412-25. 14. Gigliotti M, Minervino M, Grandidier JC, Lafarie-Frenot MC. Predicting loss of bifurcation behaviour of 0/90 unsymmetric composite plates subjected to environmental loads. Compos Struct 2012; 94(9): 2793-808. 15. Mattioni F, Weaver PM, Friswell MI, Potter KD. Modelling and applications of thermally induced multistable composites with piecewise variation of lay-up in the planform. In: 48th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference AIAA, Honolulu, Hawaii; April, 2007. 16. Mattioni F, Weaver PM, Friswell MI. Multistable composite plates with piecewise variation of lay-up in the planform. Int J Solids Struct 2009; 46(1): 151-64. 17. Daynes S, Weaver PM. Analysis of unsymmetric CFRP-metal hybrid laminates for use in adaptive structures. Compos Part A 2010; 41: 1712-8. 18. Dai F, Li H, Du S. Cured shape and snap-through of bistable twisting hybrid [0/90/metal]T laminates. Compos Sci Technol 2013; 86: 76-81. 19. Frouzian-Nejad A, Mustapha S, Ziaei-Rad S, Ghayour M. Characterization of bi-stable pure and hybrid composite laminates-An experimental investigation of the static and dynamic responses. J Compos Mater 2019; 53(5): 653-67. 20. Abouhamzeh M, Sinke J, Jansen KMB, Benedictus R. Thermo-viscoelastic analysis of GLARE. Compos Part B 2016; 99(15): 1-8. 21. Cho M, Kim MH, Choi HS, Chung CH, Ahn KJ, Eom YS. A study on the room-temperature curvature shapes of unsymmetric laminates including slippage effects. J Compos Mater 1998; 32(5): 460-82. 22. Yuan Z, Wang Y, Wang J, Wei S, Liu T, Cai Y, et al. A model on the curved shapes of unsymmetric laminates including tool-part interaction. Sci Eng Compos Mater 2016; 25: 1-8. 23. Hufenbach W, Gude M, Kroll L. Design of multistable composites for application in adaptive structures. Compos Sci Technol 2002; 62: 2201-7. 24. Abouhamzeh M, Sinke J, Jansen KMB, Benedictus R. Closed form expression for residual stresses and warpage during cure of composite laminates. Compos Struct 2015; 133: 902-10. 25. Gigliotti M, Wisnom MR, Potter KD. Loss of bifurcation and multiple shapes of thin [0/90] unsymmetric composite plates subject to thermal stress. Compos Sci 19

Technol 2004; 64: 109-28. 26. Telford R, Katnam KB, Young TM. The effect of moisture ingress on through-thickness residual stresses in unsymmetric composite laminates: a combined experimental-numerical analysis. Compos Struct 2014; 107: 502-11. 27. Eckstein E, Pirrera A, Weaver PM. Morphing high-temperature composite plates utilizing thermal gradients. Compos Struct 2013; 100: 363-72. 28. Betts DN, Salo AIT, Bowen CR, Kim HA. Characterisation and modelling of the cured shapes of arbitrary layup bistable composite laminates. Compos struct 2010; 92: 1694-700. 29. Giddings PF, Bowen CR, Salo AIT, Kim HA, Ive A. Bistable composite laminates: effects of laminate composition on cured shape and response to thermal load. Compos Struct 2010; 92: 2220-5. 30. Moore M, Ziaei-Rad S, Salehi H. Thermal response and stability characteristics of bi-stable composite laminates by considering temperature dependent material properties and resin layers. Appl Compos Mater 2013; 20: 87-106. 31. Zhang Z, Li Y, Wu H, Chen D, Yang J, Wu H, et al. Viscoelastic bistable behaviour of antisymmetric laminated composite shells with time-temperature dependent properties. Thin Walled Struct 2018; 122: 403-15. 32. Ren L, Parvizi-Majidi A, Li Z. Cured shape of cross-ply composite thin shells. J Compos Mater 2003; 37: 1801-20. 33. Pirrera A, Avitabile D, Weaver PM. On the thermally induced bistability of composite cylindrical shells for morphing structures. Int J solids Struct 2012; 49: 685-700. 34. Eckstein E, Pirrera A, Weaver PM. Multi-mode morphing using initially curved Composite plates. Compos Struct 2014; 109: 240-5. 35. Mostafavi S, Golzar M, Alibeigloo A. On the thermally induced multistability of connected curved composite plates. Compos Struct 2016; 139: 210-9. 36. Mathematica Version 11 Reference guide. Wolfram Research, 2016. 37. Matlab Version R2012a. The MathWorks, Inc., Natick, MA, USA. 38. Abaqus analysis user’s manual, Version 6.13. Dassault Systemes; 2013. 39. Lee H, Lee J-G, Ryu J, Cho M. Twisted shape bi-stable structure of asymmetrically laminated CFRP composites. Compos Part B 2017; 108: 345-53.

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Figure Captions Fig. 1. Initial geometry of FML in the Cartesian coordinate system. Fig. 2. Three kinds of curved steel moulds with different initial curvature radius. Fig. 3. Six experimental specimens of FMLs with different stacking sequence and initial curvature radius. The geometrical size of all specimens is 450×450mm. Fig. 4. Measurement of the cured shapes using a three-dimensional laser scanner. Fig. 5. Schematic diagram of mould geometry. Fig. 6. Initial out-of-plane displacement function curve and actual shape curve of mould (The curvature radius of mould is 0.5m.). Fig. 7. Cured shapes obtained by the analytical, FE and experimental methods for these six FMLs with different initial curvatures and stacking sequences. Fig. 8. Curing deformation mode and force condition of FMLs with initial curvature radius of 400mm, and the total strain energy in the case of state A and state B. Fig. 9. Out-of-plane corner displacement as a function of the initial curvature radius for 450×450mm FMLs with layup (a) [Al/0/90Al] and (b) [Al/90/0/Al], and the deformation modes predicted by analytical model under typical initial curvature radius. Fig. 10. Out-of-plane corner displacement as a function of side length for the FMLs with initial curvature radius of R=400mm, and the deformation modes predicted by analytical model under typical side length. (a) [Al/0/90Al] and (b) [Al/90/0/Al].

21

z y

Ly

x Lx Fig. 1. Initial geometry of FML in the Cartesian coordinate system.

(a) R=400mm

h

l s

(b) R=500mm

(c) R=600mm

Fig. 2. Three kinds of curved steel moulds with different initial curvature radius.

22

Al 0° 90° Al

Specimen 1 R=400mm [Al/0/90/Al]

Specimen 4 R=400mm [Al/90/0/Al]

Specimen 5 R=500mm [Al/90/0/Al]

Specimen 3 R=600mm [Al/0/90/Al]

Specimen 2 R=500mm [Al/0/90/Al]

Specimen 6 R=600mm [Al/90/0/Al]

Al 90° 0° Al

Fig. 3. Six experimental specimens of FMLs with different stacking sequence and initial curvature radius. The geometrical size of all specimens is 450×450mm.

23

Fig. 4. Measurement of the cured shapes using a three-dimensional laser scanner.

z c(0, R)

R

Curved mould

a

b

o

x

Fig. 5. Schematic diagram of mould geometry.

0.5  0.25  x 2 x 2  x 4  2x 6

Fig. 6. Initial out-of-plane displacement function curve and actual shape curve of mould (The curvature radius of mould is 0.5m.).

24

Fig. 7. Cured shapes obtained by the analytical, FE and experimental methods for these six FMLs with different initial curvatures and stacking sequences.

25

Mx

My State A

Initial shape

Total strain energy State B

Mx

State A

My (a) R=400mm [Al/0/90/Al] Specimen 1 State A

Mx

My

Total strain energy

Initial shape

State A State B

Mx

State B

My

(b) R=400mm [Al/90/0/Al] Specimen 4 Fig. 8. Curing deformation mode and force condition of FMLs with initial curvature radius of 400mm, and the total strain energy in the case of state A and state B.

26

(a)

(b) Fig. 9. Out-of-plane corner displacement as a function of the initial curvature radius for 450×450mm FMLs with layup (a) [Al/0/90Al] and (b) [Al/90/0/Al], and the deformation modes predicted by analytical model under typical initial curvature radius.

27

(a)

(b) Fig. 10. Out-of-plane corner displacement as a function of side length for the FMLs with initial curvature radius of R=400mm, and the deformation modes predicted by analytical model under typical side length. (a) [Al/0/90Al] and (b) [Al/90/0/Al].

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Table Captions Table.1 Mechanical properties of aluminum and glass fiber reinforce plastic (GFRP). Table.2 Geometric dimensions of three different initial curvature moulds. Table.3 Initial shape function coefficients of FMLs. Table.4 Comparison of the predicted values and the measured values of out-of-plane corner displacements for six FMLs.

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Table.1 Mechanical properties of aluminum and glass fiber reinforce plastic (GFRP). Aluminum

Em =70GPa,  =0.3,  m =23.6×10-6/℃, t m =0.1mm

GFRP

E1 =54.6GPa, E2 =10.5GPa, G12 =5.5GPa,  12 =0.28, 1 =4.5×10-6/℃,  2 =32.7×10-6/℃, t f =0.15mm

Table.2 Geometric dimensions of three different initial curvature moulds. Curvature radius R (mm) Mould width s (mm) Arc length of mould l (mm) Central angle  () Mould height h (mm)

400.0 500.0 540.1 77.36 87.75

500.0 500.0 523.6 60.00 66.99

600.0 500.0 515.7 49.25 54.56

a11 1.953 1.000 0.5787

a12 6.104 2.000 0.8038

Table.3 Initial shape function coefficients of FMLs. Initial curvature radius R (m) 0.4 0.5 0.6

a10 1.250 1.000 0.8333

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Table.4 Comparison of the predicted values and the measured values of out-of-plane corner displacements for six FMLs. Specimen number Stacking sequence Initial curvature radius R (mm) Lx’ (mm) Initial State A Analytical Out-of-plane model State B displacement State A of corner FEA State B points (mm) State A Experiment State B Analytical State A model vs. State B Experiment Error (%) State A FEA vs. Experiment State B

1

2 3 [Al/0/90/Al] 400.0 500.0 600.0 426.6 435.0 439.5 61.50 49.75 41.69 98.40 88.02 80.68 91.78

81.24

73.80

92.91

85.57

77.15

5.91

2.86

4.58

-1.22

31

-5.06

-4.34

4

5 6 [Al/90/0/Al] 400.0 500.0 600.0 426.6 435.0 439.5 61.50 49.75 41.69 22.63 55.41 53.09 51.30 N/A 51.88 47.91 46.06 23.40 52.91 49.43 49.42 -3.29 4.73

7.40

3.80

N/A -1.95

-3.08

-6.80