Analytical, numerical, and experimental modeling methods are presented to predict deformation after the cure process of thin unsymmetric laminates for piezoelectric actuation. During fabrication, laminates deform to several post-cure room temperature shapes. Thin cross-ply laminates deform to a circular cylindrical post-cure shape while thicker laminates deform to a saddle shape. Post-cure shapes are dependent on ply orientation, thickness, and material properties. Because, CLT alone does not always predict the correct post-cure room temperature shape of the thin composite laminates, an extension of CLT with the Rayleigh-Ritz technique and potential energies are used to better predict these shapes. Finite element models are used to predict the post-cure room temperature shapes. Thin composite laminates are modeled coupling heat transfer and structural mechanics, which are necessary for modeling the cure process. Modeling the fabrication process captured important data such as residual stresses from the cure process, room temperature shapes, and bi-stability of the composite laminates. To validate these analytical and numerical results, experiments were con- ducted using macro-fiber composite (MFC) patches for morphing the laminates. The experimental piezoelectric morph- ing results relate well to analytical and numerical results.

Smart material systems are vastly becoming an integral part in engineering applications. One of the phenomena used in smart material systems is piezoelectricity. Using unsymmetric bi-stable composites, piezoelectric effects can be implemented to achieve a snap through to the other stable shape of the composite [

After curing, a once flat laminate will deform into one of multiple shapes based on the laminate layup and the suggested cure temperature. Research and analysis are being done to predict the deformations and deflections of laminates. Dano and Hyer (2001) predicted the forces and moments that cause cure deformation using common theories based on displacements and strain fields [

Hyer and Jilani (1998) presented a method for predicting the deformation of rectangular laminates cooled from the cure temperature to that of the room temperature [

The Classical lamination theory suggests that a laminate cooled from the cure temperature will achieve a saddle shape, the shape between the two stable shapes. This suggestion is not always correct. When cooled from the cure temperature, laminates can also attain one of the stable cylindrical shapes (concave up or down). M. Schlecht, K. Schulte, and Hyer (1995) extended the Classical lamination theory to approximate these shapes then compared these results to finite element analysis [

Hyer also discusses the subject matter of stress analysis of fiber composites [

M. Gigliotti and M. R. Wisnom studied the curvature due to the cure process of an AS4/8552 [0/90] laminate [

The snap through phenomenon for piezoelectric systems are of interest and will be investigated in further research; the present paper focuses on the curing process and approximating the stresses, strains, and curvatures of the unsymmetric laminate during this process. As Schlecht and Schulte studied the thermal effects of the cure process on room-temperature laminates, this paper establishes analytical and numerical modeling for stress, strain, and curvature prediction after the cure process [

M. W. Hyer also used strain energy theories and virtual work formulation to predict the post cure displacements and curvatures of the laminate [3,5]. These methods were evaluated using various types of lay-ups and cure temperatures. M. W. Hyer formulated the necessary equations to predict the variables then compared these values to experimental values of the same laminate. The most common lay-up was that of [0/90] T and it was used due to the simplicity of the governing equations. Libo Ren and Azar Parvizi-Majidi investigated the CLT and how the predictions of the post cure shape were not always correct [

As seen in the literature, bi-stability of the composite laminate is achieved during fabrication. After the cure process, multiple deformation shapes can be observed based on the ply orientation and material [

The fabrication and cure process of unsymmetric laminates are essential elements of the smart material morphing structure. Models were developed to achieve a better understanding of these processes for piezoelectric actuation. An analytical model is presented first to predict the post-cure shape and deformation. The analytical model was based on both Hyer’s and Vizzini’s composite material work [3,19,20]. Numerical analysis was done to compare to the analytical results, and is presented following the analytical models. Multi-physics Finite Element Models were created for numerical modeling of both the fabrication process and piezoelectric actuation [

AS4 carbon fiber pre-impregnated laminae are used for the composite laminates of this study. Due to the thickness of the carbon fiber pre-pregs, the laminates in this study are classified as thin composite laminates where the thickness to length ratio is less than $\frac{1}{20}$. This ratio is important for post-cure deformation shapes, thin laminates will deform to a circular cylindrical shape whereas a thicker laminate will deform to that of the saddle shape. The first model was done with the CLT for composite laminates. For CLT analysis, the following key assumptions of the material are made:

• assume smeared properties through thickness;

• assume perfectly bonded laminae with no defects;

• assume zero bond line thickness between plies (no epoxy between plies);

• assume Kirchhoffs hypothesis is valid; plane sections remain planar and perpendicular sections remain perpendicular.

The first assumption states that the individual fibers of the laminae are homogeneous, not varying with direction. The second and third assumptions are made to eliminate any complications with bonding the laminate. The models of the laminate are assumed to be perfectly bonded with no epoxy layer between the plies. The fourth assumption is made in relation to Kirchhoffs hypothesis. This assumptions states that the strain is linear through the thickness of the laminate. The laminate is then treated as an orthotropic material with three planes x, y, and z (also known as 1, 2, and 3). Treating the laminate as an orthotropic material, nine engineering constants of the material are required including Youngs modulus, Poisson ratio, and shear modulus of the material one in each directional plane. These constants are then used to determine the reduced material stiffnesses of the laminate [

where Q_{ij} are the reduced material stiffnesses [

where θ is the arbitrary ply orientation. The next step in the analysis is to calculate the forces and moments that occur during the cure process. In order to determine these resultant forces and moments the extensional, bending-extension coupling, and the bending stiffness tensors also named the ABD matrix, must be calculated. Using the rotated stiffness tensor equations and the middle surface lamina thickness, the ABD matrix can be determined for each direction of the laminate [_{ij} component of the ABD matrix which is the bending stiffness of the laminate. If D_{16} =D_{26} = 0, then this assumption can be made. Unsymmetric laminates contain this type of coupling stiffness while symmetric laminates do not. The force and moment constitutive equations are given by

where N and M are the forces and moments. To calculate these strains and curvatures values, the ABD matrix can be inverted to solve for the desired variables. If the forces and moments were to be determined and the strains and curvatures were known, the simple calculation can be done with the matrices. Continuing the CLT, the laminate is treated as orthotropic material to equate the thermal strains to the coefficients of thermal expansion multiplied by the thermal gradient. The residual stresses and strains arising from the cure process can then be found [

The CLT predicts a saddle shape will occur each time a thin unsymmetric laminate cools from the elevated cure temperature to room temperature. This is due to the failure to capture reference strain due to the large out-ofplane deformations as the laminate is cooled. The Rayleigh-Ritz technique accounts for the large out-of-plane deformations arising from the room temperature cooling. The maximum states of the potential energy are the unstable equilibrium conditions of the laminate. The minimum states of the potential energy are the conditions needed to characterize the deformations of the laminate cooling to room temperature. Using the same engineering constants, mentioned earlier, the reduced stiffnesses of the laminate are found. The coefficients of thermal expansion are also transformed to the ply orientation not aligned in the principal axis. The process is continued to using the method of invariants mentioned in the earlier section to determine the reduced stiffnesses of the laminate. The thermal strains and stresses are then evaluated using the following equations:

where the thermal strains, ε^{T}, are given by:

where αi are the coefficients of thermal expansion. The middle surface strains and curvatures can then be approximated. Where u^{o}, v^{o}, and w^{o} are the displacement fields:

The stresses and strains are then approximated using

The post-cure room temperature shape can then be modeled. This step does not include bonding the MFC actuator, therefore only thermal effects are accounted for. The strain energy or total potential energy is given by:

Carrying out the integrations in equation 27, the potential energy is then reduced to an algebraic equation in terms of the coefficients a, b, c, and d. Once the integrations are computed, Equation (27) will show that the potential energy is also in terms of material properties and geometries of the laminate. These coefficients are then determined by solving the nonlinear algebraic equations that is reduced from equating to zero the first variation of the potential energy with respect to the coefficients,

where C is a, b, c, and d. The undetermined coefficients were found using Newton’s iterative method. These solutions relate to the equilibrium shapes of the cooled room temperature laminate. To check stability of the solution for the laminate, the second variation of the potential energy must be positive definite.

The following is an example of the results found for the analytical methods mentioned for the post-cure room temperature shapes of the laminates.

The laminates studied were [0/90], [0_{2}/90_{2}], [0/45], and [45/90] square and rectangular laminates. When cooling to room temperature, the cross-ply laminates will deform to one of three possible equilibrium shapes. Two being stable cylindrical shapes and one being the unstable saddle shape, the shape between the two cylindrical shapes. After analytical formulation, these equilibrium shapes can be predicted and are shown in the meshes of

As seen in plot two of

Next, predictions can be made for the [0_{2}/90_{2}] laminate using the same techniques as described earlier. With more plies than the [0/90] laminate, the curvature is expected to be different due to the increase of plies. The transverse curvature for the ﬁrst cylindrical shape (expected shape after curing) of the [0_{2}/90_{2}] laminate is expected to be along the length of side $L_y$. When compared with the [0/90] laminate this curvature is in opposite the direction along the other axis. The equilibrium shapes, shown as meshes, of the [0_{2}/90_{2}] laminate in

The first plot (a) in _{2}/90_{2}] laminate. Plot b represents the shape between the two cylindrical shapes, the saddle shape. Plot c shows the expected actuated displacement shape of the [0_{2}/90_{2}] laminate.

The [0/45] laminate was modeled using the RayleighRitz technique as described above. With a ply orientation at 45^{˚}, it is expected that the laminate will not deform to one of the equilibrium cylindrical stable shapes. Instead, the [0/45] laminate should deform to the saddle shape. As research shows, this shape is unstable compared to the simple cross-ply laminates. The stability is determined by the second variation of the total potential energy, or the second derivative of the Jacobian matrix. If this matrix is positive definite, which confirms that the equilibrium solution is stable. As shown in

The curvature of the [0/45] laminate is along both the x and y axis. This is a characteristic of the saddle shape. After fabrication, the [0/45] laminate should deform to this saddle shape. For piezoelectric actuation the [0/45] laminate is still going to snap through to another saddle shape (shown in later chapters), but unlike the simple

cross-ply laminates, due to the instability of the [0/45] laminate removal of the snap through force will cause the laminate to snap back to its original shape shown in

Multi-physics numerical FEA models were used to compare analytical models and experimental results. The present models required implementing the Thermal-Structural Interaction mode, which couples both, the structural mechanics and the heat transfer modules. The analysis type of the stress-strain and the heat transfer modules were chosen as static and transient types respectively. Due to the small thicknesses of the laminates studied, models were done using linear and quadratic mesh element types.

The Thermal-Structural interaction mode allowed for including thermal expansion due to the coefficients of thermal expansion(CTE) during the cure process. The CTE’s along with geometry and other material parameters, play key roles during the heating and cooling of the laminate. In numerical model 3-D space, new coordinate systems are needed to specify certain fiber orientations for plies not oriented in the global or principal material axes. This feature is done by rotating the global coordinate system from the original x, y, z axis to the angle needed for the specific ply. Allowing $0^o$ to represent the default or global coordinate system (no need to change coordinate system if ply is oriented at $0^o$), the remaining plies coordinate systems can be changed using the consecutive rotation axes option. After the laminate is drawn with the specified dimensions, the material properties and conditions can be specified in the structural mechanics subdomain settings of the model. The heat transfer sub-domain settings controls the heat source and the thermal properties of the laminate. The structural domain settings control the material properties of the laminate, including the fiber orientation.

The laminate model was treated as an orthotropic material as stated earlier. In these settings, the necessary engineering constants along with the selection of the global coordinate system and specified created coordinate systems can be entered. The initial pressure of the laminate can be selected. An initial pressure was used to model the mold pressing the laminate flat during the heating of the laminate. Similar to the sub-domain settings, there are separate boundary condition settings for the structural application and the heat transfer application. Modeling free thermal expansion, there were no constraints on the boundaries or edges of the laminate. Compared with the analytical model, this assumption can be made due to the absence of external loads during the cooling of the laminate to room temperature. When solving the model, the solver parameters are important for accuracy and solver convergence. For the structural application mode, the solver was chosen based on the static analysis nature of the problem. This analysis type differs from the heat transfer application mode, however. For the heat transfer application mode, the analysis was chosen to be transient based on the steps of the curing process. Therefore, the analysis type needed to couple both the static and transient nature of the curing process.

The numerical analysis allowed for more complex lamina behavior than the CLT assumptions used in the analytical modeling. For the structural mechanic analysis mode of the numerical modeling, a few important equations were used to define the domain of the laminate. Using the solid, stress-strain application for 3-D modeling, the straindisplacement relationship equations were specified as follows:

The stress-strain relation of the numerical analysis is given by the generalized Hooke’s Law formulation:

where D is the stiffness matrix of the material. The Structural Mechanics mode is based on the weak formulation of equilibrium equations in the global stress components. For 3-D models, the equilibrium equation is represented by:

where is the stress tensor and F is the volume or body forces [

The sub-domain setting for the heat transfer mode was specified using Fourier’s law:

where T is the temperature being measured at that instant, Q is the heat source, and k is the thermal conductivity. Specifying the heat transfer boundary equation using the heat flux equation,

where q_{o} is the inward heat flux of the composite being modeled, h is the heat transfer coefficient, T_{inf} is the external temperature, Const is the numerical constant for heat transfer relations, and T_{amb} is the ambient temperature. The heat transfer boundary equation is a form of the heat convection equation. The second half of the heat transfer boundary equation is the heat transfer due to radiation. Although shown, it was not used in the models. The equations in this section are all derived or are a form of the same analytical equations used in the numerical analysis. Using the same material properties as in

Applying an initial pressure of 5.861 × 10^{5} Pa to the laminate models the press applied from the mold during the heating of the laminate. The laminate is then heated to 449.817 K then cooled to room temperature, 296.483 K.

The [0/90] laminate was modeled with a total thickness of 0.36002 mm with square length of 76.2 mm and rectangular lengths of L_{x} = 61 mm and L_{y} = 120 mm which classified the laminates as thin laminates. In addition to being thin laminates, they were simple cross-ply laminates therefore the saddle shape is not present here. Increasing the number of plies to a [0_{2}/90_{2}] laminate, different cylindrical room temperature shapes are observed. _{2}/90_{2}] laminate. The [0_{2}/90_{2}] laminate was modeled at 0.72004 mm total thickness. The lengths were the same as the [0/90] laminate. Therefore the saddle shape is not seen here. For the [0_{2}/90_{2}] laminate, the symmetric circular transverse curvature is in the positive z-direction. This curvature is seen in the first and second plot of

For laminates that are not simple cross-ply laminates, a saddle shape as the room temperature post-cure shape is expected. Experimental and analytical work [

done to better understand the saddle shape and the instability of the laminate. The simple cross-ply laminates deform while cooling to room temperature to a symmetric circular shape. This is seen in the numerical solutions above. More complex laminates will deform unsymmetrically with a twist curvature similar to the saddle shape. This twist curvature develops as the laminate cools to room temperature. Applying a force to this saddle shape will cause deformation, but unlike the simple cross-ply laminates, removal of the snap through force will cause the laminate to return to its original shape. This classiﬁes the laminate in the saddle shape as unstable. For this study, [0/45] and [45/90] were used for experimentation. Using the same parameters as the simple cross-ply laminates, the [0/45] laminates were modeled.

^{˚} ply. The twist curvature which characterizes the saddle shape is seen in all three plots. For the rectangular [0/45] laminate in plot 1, two edges deform upward in the positive z-direction while the other two edges deform downward in the negative z-direction.