A model on the curved shapes of unsymmetric laminates including tool-part interaction

1 Introduction

Some cross-ply laminates can display a bistable property under certain geometric conditions, whereby the laminate can lead to two stable configurations at room temperature (RT). Figure 1 shows a square [0₂/90₂]_T laminate with two stable shapes of equal curvature in opposite directions. One cylindrical shape has a large curvature in the x direction and an imperceptible curvature in the y direction. The other cylindrical shape has a large curvature in the y direction and an imperceptible curvature in the x-direction. One stable shape can change to the other by applying a small amount of energy. This is known as the snap-through phenomenon [1], making bistable laminates of interest for a wide range of engineering applications. This mechanism behind the bistable behavior of cross-ply laminates can be explained by the existence of residual stress developed during cooling-down process due to the mismatches of the elastic modulus and thermal expansion coefficients (CTE) between the direction along and perpendicular to the fiber direction. During the cooling-down process from high curing temperature to RT, the mismatch of CTE in the two adjacent plies which are stacked with different orientations, leads to inter-ply residual stresses.

Figure 1:

Two stable shapes of a square [0₂/90₂]_T laminate at room-temperature.

The importance of predicting the room-temperature shape of asymmetric laminates has received a great deal of attention. The earliest work can be traced back almost 25 years when Hyer [2] reported that shape of unsymmetric laminates at RT do not conform the predictions of the Classical Lamination Theory (CLT) which is a geometrically linear theory. Then Hyer proposed a model incorporating the non-linear strain of von Kármán and analyzed the curved shaped of laminates using the Rayleigh-Ritz energy method and the minimum total potential energy principle. Following Hyer’s study, many investigators have studied the room-temperature shape of unsymmetric composite laminates. Cho et al. [3], [4] proposed a higher-order plate model by taking the slippage effect induced between the tool plate and laminate into consideration to analysis the curved shape of unsymmetric laminate. Dano and Hyer introduced a new model, hitherto considered state-of-the art, in which they used directly for the first time approximations for the laminate mid-plane strains. Initially they used a set of complete polynomials with 28 unknown coefficients, but then they realized that the use of 14 coefficients was enough [1], [5], [6]. This approach is named Extended Classical Lamination Theory (ECLT). Cantera et al. [7] predicted the curved shapes of bistable laminates by incorporated the mechanical curvatures and through-the-thickness strain to the ECLT and found that the length associated with bifurcation point increases when considering a uniform value of through-the-thickness strain. The curved shapes of asymmetric laminates with an initial curvature were also investigated using a modified shell model [8], [9], [10]. It was found that for some shells with small radius, only one stable configuration existed. Brampton et al. [11] investigated the influence of material, geometric and environmental uncertainties on laminate curvatures.

As mentioned above, most studies have been focused on the thermal stresses caused by the mismatch of coefficients of thermal expansion longitudinal and transverse to the fiber direction during the cooling-down process. However, fewer studies have been seen about the effect of tool-part interaction during the heating-up process on the deformation of cross-ply laminates. Whereas, many researchers have shown that the tool-part interaction during the heating-up process could induce the composite part to deform after curing process [12], [13], [14], [15], [16], [17], [18], [19], [20]. Twigg et al. [12], [13] found that even thin, balanced laminates fabricated on flat tooling usually can exhibit a concave down deformation after curing process and they attributed this phenomenon to the effect of tool-part interaction. Even at low degree of cure, prior to resin gelation, considerable shear stress between tool and part could be observed [16]. Usually, the tool material has much higher CTE than that of composite. When the tool and composite part are forced together due to autoclave pressure and subjected to a temperature ramp, a shear interaction between the tool and the part along their interface will arise and place the laminate in tension. As a result of low shear modulus of composite part during curing, a stress gradient through the thickness of the laminate emerges and then locks in as the resin cures, which causes the composite part to warp on removal of the tool [18], [21], [22], [23]. Deformations due to tool-part interaction in laminates have been investigated in many factors, including tooling material [24], tool surface properties [25], release agent [12], [26], [27], pressure [12], [16], [27], stacking sequence [26], [28] and geometry [12], [26], [29], but there is still relatively little quantitative data about the induced stress through the laminate thickness generated by tool-part interaction which requires extensive experimental and material characterization.

In the present study, the model proposed by Dano and Hyer in Ref. [5] is considered as a basic approach. It has been modified by the incorporation of through-the-thickness stress according to the tool-part interaction during heating-up process to predict the room-temperature curvature shapes of cross-ply laminates. The process-induced stresses through the thickness of the laminate due to tool-part interaction were introduced by assuming an interfacial shear stress between tool and laminate. With the use of experimental data, two interfacial shear stresses which are introduced to describe the tool-part interaction can be calibrated to an appropriate value and reasonable predictions could be achieved.

2 Analytical model

2.1 Interaction between tool and laminate during heating-up process

The tool-part interaction during heating-up process which induces the composite part to be stretched is one of the most important sources of residual stress in composite structures. Cho et al. [3] used a slippage model to predict the room-temperature curvature shapes. In their model, two dimensionless coefficients β₁ and β₂ were introduced to indicate the degree of tool-part interaction, which produced linear induced stress across the thickness of the laminate. However, this model does not take the material characterization during curing process into account where many researchers have showed that the through-thickness stress distribution strongly depends on the material properties of the part during heating-up process [12], [13], [14]. Arafath et al. [14] used the closed-form solution to calculate the residual stress due to the effect of tool-part interaction and found that the material properties at the initial stages of curing are crucial in predicting the processed deformations accurately. In the present study, a model was developed to take the effect of tool-part interaction into account without requiring more complicated material characterization and rein cure kinetics. The driving force at the tool-part interface due to the CTE mismatch between tool and composite laminate during the heating-up is represented by an interfacial shear stress. The model is formulated using a Cartesian coordinate system as shown in Figure 1. A composite part stretched due to the expansion of the tool can be simplified to a classical beam undergoing tractions at the top and bottom surfaces, as shown in Figure 2. The induced stress and displacement can be written in the following form [14].

Figure 2:

Schematic of a beam under applied shear tractions on the top and bottom surfaces.

(1)uxx=∑n = 1N{sin(knx)(Aneβnz+Bne−βnz)}+εtherxσxx=∑n = 1N{Exxkncos(knx)(Aneβnz+Bne−βnz)}τxz=∑n = 1N{Gxzβnsin(knx)(Aneβnz−Bne−βnz)}

where β_n=ck_n, c=Exx/Gxz,k_n=(2n− 1)π/L, n=1, 2, 3, …, u_xx and σ_xx are the beam axial displacement and stress, respectively, and τ_xz is the shear stress, E_xx and G_xz are the beam axial elastic modulus and transverse shear modulus, respectively, ε_ther is the axial free thermal strain and L is the length of the laminate in the x direction. A_n and B_n are unknowns and can be calculated by applying the boundary conditions at the bottom and top surfaces (τ_xz=τ_b at z=0 and τ_xz=τ_t at z=t), seen in Figure 2.

The composite laminate is made of multi-layers with different fiber orientations and the boundary conditions at the top and bottom surfaces of laminate are shown in Figure 3. From Equation (1), axial induced stress and strain variation in the thickness direction can be given by:

Figure 3:

Model of the laminate.

(2)uxxi=∑n = 1N{sin(knx)(Anieβniz+Bnie−βniz)}+εtherixσxxi=∑n = 1N{Exxikncos(knx)(Anieβniz+Bnie−βniz)}Δεxxi=∑n = 1N{kncos(knx)(Anieβniz+Bnie−βniz)}τxzi=∑n = 1N{Gxziβnisin(knx)(Anieβniz−Bnie−βniz)}

where a superscript of i denotes the coefficients connected with the i-th layer; Ani and Bni are unknowns associated with the i-th layer; Δεxxi is the residual strain in the x direction. For m layers, there are 2m unknowns which can be determined from the equilibrium and continuity conditions on the interface between each layer.

An interface layer, which is very thin and the only stress that it transfers is a shear stress across its thickness, is introduced with shear modulus G_s and thickness t_s in Ref. [14]. The shear stress transmitted by the interface layer due to tool-part interaction is given by:

(3)τs=Gsu10−utool0ts

where u10 and utool0 are the longitudinal displacements of the laminate and tool at the position z=0. The shear modulus of the elastic interface layer is calibrated with the experimental result. According to the numerical results and the closed-form solution in Ref. [14], the shear stress varies linearly along the length of the laminate in the interface between the tool and laminate. The maximum shear stress value in the interface layer varies with the shear layer modulus and part length. However, it was reported in experiments [13], [21] that during the heating-up portion of the cure cycle, the interfacial shear stress should be constant along the laminate length. As a result, an interfacial shear stress τ_bottom between the tool and laminate is introduced in this paper and assumed to be constant along the laminate length. Thus, the continuity condition on the interface between the tool and laminate (at z=0) is as follows:

(4)τbottom=∑n = 1N{Gxz1βn1sin(knx)(An1−Bn1)}

The layer axial displacement and shear stress should be equal on the interface between the two neighbor layers, which means that at a generic i-th interface (z=z_i), these are:

(5)uxxi=uxxi + 1   and   τxzi=τxzi + 1

which lead to the following two equations:

(6)∑n = 1N{sin(knx)(Anieβnizi+Bnie−βnizi)}+εtherix =∑n = 1N{sin(knx)(Ani + 1eβni + 1zi+Bni + 1e−βni + 1zi)}+εtheri + 1x

(7)∑n = 1N{Gxziβnisin(knx)(Anieβnizi−Bnie−βnizi)}=∑n = 1N{Gxzi + 1βni + 1sin(knx)(Ani + 1eβni + 1zi−Bni + 1e−βni + 1zi)}

Equations (6) and (7) constitute 2(m−1) equations for (m−1) interfaces. The shear stress τ_top at the top surface due to the free boundary condition can be expressed as:

(8)τtop=∑n = 1N{Gxzmβnmsin(knx)(Anmeβnmzm−Bnme−βnmzm)}=0

From Equations (5–8), the unknowns Ani and Bni of each layer can be determined and the residual stress and strain of each layer can be calculated.

3 Results and discussion

Cho et al. [3] investigated the deformation of composite parts made from laminates DMS-2224 with an average cured ply thickness of 0.14 mm. Flat parts were made on aluminum tool or rubber tool, which are widely used in manufacturing process. Analytical results are computed under the assumption that temperature is uniform across the laminate. This is a reliable hypothesis for thin and small parts (the laminate is 1.12 mm thick and 300 mm long).

In this article, a two-step model including anisotropy in the thermal expansion and tool-part interaction was developed to predict the room-temperature shape of asymmetric laminate. The two steps of the model represent the heating-up process and cooling-down process. Arafath et al. [14] found that the resin modulus during curing process was crucial in predicting the process-induced stress and they finally calibrated the initial resin modulus by correlating the experimental results with their analytical predictions. The results showed that the initial resin modulus Er0 is 4.71 kPa for the T800H/3900-2 CFRP material which was six orders of magnitude less than the fully cured resin modulus. The properties used in the proposed model during heating-up process, are assumed from the optimization analysis in Ref. [14]. A reasonable assumption for the material used in this study is also set six orders of magnitude less than the fully cured modulus. However, the modulus parallel to fiber primarily reflects the properties of fiber and is essentially constant with time. After heating-up process, the properties of the material are switched to glassy properties as shown in Ref. [3]. The mechanical properties are assumed to be constant in each step, with a jump in the magnitude of the properties between the two states, as shown in Table 1.

In the heating-up process, the laminate is assumed not to sustain any mechanical stress in the transverse direction, whereas it can sustain some fiber stresses due to tool-part interaction in the longitudinal direction. The residual strains induced due to tool-part interaction are calculated by selecting appropriate values of the interfacial shear stress τbottomx and τbottomy. In the cooling-down process, the resin vitrifies and transforms to the glassy state. The induced stress and strain developed in the heating-up process will be incorporated in Equation (12) using Dano and Hyer’s model and final deformation can be predicted. It should be noted that the laminate shows no any deformation before the cooling-down process due to the autoclave pressure and the encapsulation by vacuum bag.

The shape characteristics of [0₂/90₂]_T laminate as a function of side length with different tool material are depicted and compared with the analytical solutions obtained by Dano and Hyer’s theory in Figure 4. Since the Dano and Hyer’s model does not take the tool-part interaction into consideration, there will be no any difference in the curvature prediction between the smooth Al tool and rubber tool for [0₂/90₂]_T laminate. However, different curvatures were observed by different tool material used in experiments [3]. The interaction between tool and laminate should be the primary reason, as shown in Figure 5. In theory, one [0₂/90₂]_T laminate cylindrical shape due to the effect of anisotropic properties in cooling-down process has a large curvature in the x direction and an imperceptible curvature in the y direction at RT, as shown in Figure 5C. However, the tool-part interaction due to the expansion of the tool during heating-up will place the laminate in tension and residual stress will develop in fiber direction. The residual stress will lead to a bending moment, which results in an opposite curvature in the x direction, as shown in Figure 5B. That is an important reason why the curvature predicted by Hyer’s model is larger than the curvature measured in experiment, as shown in Figure 5D.

Figure 4:

Theoretical and experimental results for [0₂/90₂]_T laminate.

Figure 5:

Effect of tool-part interaction during heating-up process for [0₂/90₂]_T laminate.

τbottomx and τbottomy are the interfacial shear stress of tool-part interaction in the x and y direction, respectively, which are unknowns and can be derived from the correlation of theoretical solutions with the experimental data. It has to be noted that the magnitude of τbottomx and τbottomy are usually not equal and always affected by some other parameters such as the magnitude of autoclave pressure and tool surface condition. The change of curvature as a function of side length of the laminate is depicted and compared with experimental results and analytical solutions both obtained by Dano and Hyer’s theory and this paper’s model. The interfacial shear stress τbottomx and τbottomy were determined by correlating theoretical solutions with experimental one point data (side length of laminate=0.3 m).

As we know, the room-temperature shape of asymmetric laminate depends on its size. When the sidelength of a square laminate is smaller than a critical value, only a unique saddle shaped is predicted. When the sidelength of a square laminate is larger than this critical value, two cylindrical shapes or a saddle shape will be predicted. This critical value is called bifurcation point. In Figure 6, smooth aluminum tool is used. The combination of τbottomx and τbottomy which provided the best agreement with the experimental results were −6.9 kPa and 12.6 kPa, respectively. It was reported that during the heat-up portion of the cure cycle, the maximum shear stress ranged from 0.01 MPa to 0.16 MPa [20], [21], [28]. It can also be seen that the bifurcation point moved toward the direction of increasing sidelength. This phenomenon can be also observed in Figures 7–9 . The bifurcation point obtained by Hyer’s model (=39 mm) is smaller than that predicted by our model (=43 mm). As shown in Figure 6, for laminates >43 mm, the cylinder predicted by our model will be shallower than the results obtained through the Hyer’s basic model which demonstrates that the effect of tool-part interaction cannot be negligible. It can be seen that the results predicted by our model considering tool-part interaction during the heating-up agree fairly well with the experimental results in most cases expect the results for the 25 mm part in the first experimental set.

Figure 6:

Theoretical and experimental results for [0₂/90₂]_T laminate for the Al tool.

Figure 7:

Change of curvature as a function of the side length for [0₄/90₄]_T laminate for the Al tool.

Figure 8:

Change of curvature as a function of the side length for [0₂/90₂]_T laminate for the rubber tool.

Figure 9:

Change of curvature as a function of the side length for [0₄/90₄]_T laminate for the rubber tool.

Next the effect of laminate thickness is discussed. In Figure 7, the curvature of the laminate with [0₄/90₄]_T was investigated. The combination of τbottomx and τbottomy which provided the best agreement with the experimental results were −18.8 kPa and 140 kPa, respectively. As the thickness of laminate increases for [0_n/90_n]_T cross-ply layup, the predicted values of coefficients τbottomx and τbottomy are increased. However, the magnitude of τbottomy/τbottomx has changed. For [0₂/90₂]_T and [0₄/90₄]_T cross-ply laminate, the ratio of τbottomy to τbottomx is −1.826 and −7.447, respectively. It seems that the thickness of the laminate has effect on the magnitude of τbottomy/τbottomx. The results of the analytical model and experimental data are in agreement, however the present model underestimates the magnitudes near the bifurcation point. The results also illustrate that the bifurcation point moved toward the direction of increasing side length.

The theoretical and experimental results for the rubber tool are shown in Figures 8 and 9. The results show that the predicted values of coefficient τbottomx and τbottomy are not equal for the smooth Al tool which is due to the different tool material and tool surface properties. The combination of τbottomx and τbottomy were −15.5 kPa and 28.2 kPa, respectively, for [0₂/90₂]_T cross-ply laminate with rubber tool. For [0₄/90₄]_T cross-ply laminate with rubber tool, the combination of τbottomx and τbottomy changed to −15 kPa and 112 kPa, respectively. However, the magnitude of τbottomy/τbottomx has almost kept unchanged for the same thickness laminate. For smooth Al tool and rubber tool of [0₂/90₂]_T cross-ply laminate, the ratio of τbottomy/τbottomx is −1.826 and −1.819, respectively. For smooth Al tool and rubber tool of [0₄/90₄]_T cross-ply layup, the ratio of τbottomy/τbottomx is −7.447 and −7.466, respectively. Good agreement with the experimental results is also generally observed and supported the assumption of the part-tool interaction condition. The results also illustrate that the bifurcation point moved toward the direction of increasing side length. Cantera et al. [7] found that the length associated with bifurcation point increases when considering a uniform value of through-the-thickness strain. In our model, taking the tool-part interaction into consideration has the same influence on the shift of bifurcation point. More measurement method or analytical model should be developed to investigate the reason of the shift of bifurcation point.

4 Conclusions

In this article, the influence of tool-part interaction on the curved shapes of unsymmetric laminates was investigated. Based on the assumption that slip occurs between tool and composite part during the heating-up phase, an analytical model was developed to predict the shape of unsymmetric laminates after cooling-down process. By introducing two interfacial shear stresses which could be calibrated with experimental data, the complex tool-part interaction was investigated. It was shown that unsymmetric laminates could have considerable residual stresses due to tool-part interaction after the heating-up process. The curvature values predicted by the proposed model were compared with the experimental results by Cho (2003) and good agreements were obtained.