## Extended higher order theory for sandwich plates of arbitrary aspect ratio

##### Abstract

In recent years advances in technology have allowed the transition of composite structures from secondary to primary structural components. Consequently, a lot of applications demand development of thicker composite structures to sustain heavier loads. Typical sandwich panels consist of two thin metallic or composite face sheets separated by a honeycomb or foam core. This configuration gives the sandwich panel high stiffness and strength and enables excellent energy absorption capabilities with little resultant weight penalty. This makes sandwich structures a preferred design for a lot of applications including aerospace, naval, wind turbines and civil industries. Most aerospace structures can be analyzed using shell and plate models and many such structures are modeled as composite sandwich plates and shells. Accurate theoretical formulations that minimize the CPU time without penalties on the quality of the results are thus of fundamental importance.
The classical plate theory (CPT) and the first order shear deformation theory (FSDT) are the simplest equivalent single-layer models, and they adequately describe the kinematic behavior of most laminates where the difference between the stiffnesses of the respective phases is not huge. However, in the case of sandwich structures where the core is a much more compliant and softer material as compared to the face sheets the results from CPT and FSDT becomes highly inaccurate. Higher order theories in such cases can represent the kinematics better, may not require shear correction factors, and can yield much more accurate results.
An advanced Extended Higher-order Sandwich Panel Theory (EHSAPT) which is a two-dimensional extension of the EHSAPT beam model that Phan presented is developed. Phan had extended the HSAPT theory for beams that allows for the transverse shear distribution in the core to acquire the proper distribution as the core stiffness increases as a result of non-negligible in-plane stresses. The HSAPT model is incapable of capturing the in-plane stresses and assumes negligible in-plane rigidity. The current research extends that concept and applies it to two-dimensional plate structures with variable aspect ratios. The theory assumes a transverse displacement in the core that varies as a second order equation in z and the in-plane displacements that are of third order in z, the transverse coordinate. This approach allows for five generalized coordinates in the core (the in-plane and transverse displacements and two rotations about the x and y-axes respectively).
The major assumptions of the theory are as follows:
1) The face sheets satisfy the Euler-Bernoulli assumptions, and their thicknesses are small compared to the overall thickness of the sandwich section; they undergo small strains with moderate rotations.
2) The core is compressible in the transverse and axial directions; it has in-plane, transverse and shear rigidities.
3) The bonding between the face sheets and the core is assumed to be perfect.
The kinematic model is developed by assuming a displacement field for the soft core and then enforcing continuity of the displacement field across the interface between the core and facesheets. The constitutive relations are then defined, and variational and energy techniques are employed to develop the governing equations and associated boundary conditions.
A static loading case for a simply supported sandwich plate is first considered, and the results are compared to existing solutions from Elasticity theory, Classical Plate Theory (CPT) and First-Order Shear Deformation Plate Theory (FSDT).
Subsequently, the governing equations for a dynamic analysis are developed for a laminated sandwich plate. A free vibration problem is analyzed for a simply supported laminated sandwich plate, and the results for the fundamental natural frequency are compared to benchmark elasticity solutions provided by Noor. After validation of the new Extended Higher Order Sandwich Panel Theory (EHSAPT), a parametric study is carried out to analyze the effect of variation of various geometric and material properties on the fundamental natural frequency of the structure.
After the necessary verification and validation of the theory by comparing static and free vibration results to elasticity solutions, a nonlinear static analysis for square and rectangular plates is carried out under various sets of boundary conditions. The analysis was carried out using variational techniques, and the Ritz method was used to find an approximate solution. The kinematics were developed for a sandwich plate undergoing small strain and moderate rotations and nonlinear strain displacement relations were evaluated.
Approximate and assumed solutions satisfying the geometric boundary conditions were developed and substituted in the total potential energy relations. After carrying out the spatial integrations, the total potential energy was then minimized with respect to the unknown coefficients in the assumed solution resulting in nonlinear simultaneous algebraic equations for the unknown coefficients. The simultaneous nonlinear equations were then solved using the Newton-Raphson method.
A convergence study was carried out to study the effect of varying the number of terms in the approximate solution on the overall result and rapid convergence was observed. The rapid convergence can be attributed to the fact that the assumed approximate solution not only satisfies the geometric boundary conditions of the problem but also the natural boundary conditions.
During calculations four cases of boundary conditions were considered
1) Simply Supported with moveable edges.
2) Simply Supported with fixed edges.
3) Clamped with moveable edges.
4) Clamped with fixed edges.
For movable boundary conditions, in-plane displacements along the normal direction to the supported edges are allowed whereas the out-of-plane displacement is fixed. For the immovable boundary condition cases, the plate is prevented from both in-plane and out-of-plane displacements along the edges. For the simply supported cases rotations about the tangential direction are allowed, and for the clamped cases no rotations are allowed.In recent years advances in technology have allowed the transition of composite structures from secondary to primary structural components. Consequently, a lot of applications demand development of thicker composite structures to sustain heavier loads. Typical sandwich panels consist of two thin metallic or composite face sheets separated by a honeycomb or foam core. This configuration gives the sandwich panel high stiffness and strength and enables excellent energy absorption capabilities with little resultant weight penalty. This makes sandwich structures a preferred design for a lot of applications including aerospace, naval, wind turbines and civil industries. Most aerospace structures can be analyzed using shell and plate models and many such structures are modeled as composite sandwich plates and shells. Accurate theoretical formulations that minimize the CPU time without penalties on the quality of the results are thus of fundamental importance.
The classical plate theory (CPT) and the first order shear deformation theory (FSDT) are the simplest equivalent single-layer models, and they adequately describe the kinematic behavior of most laminates where the difference between the stiffnesses of the respective phases is not huge. However, in the case of sandwich structures where the core is a much more compliant and softer material as compared to the face sheets the results from CPT and FSDT becomes highly inaccurate. Higher order theories in such cases can represent the kinematics better, may not require shear correction factors, and can yield much more accurate results.
An advanced Extended Higher-order Sandwich Panel Theory (EHSAPT) which is a two-dimensional extension of the EHSAPT beam model that Phan presented is developed. Phan had extended the HSAPT theory for beams that allows for the transverse shear distribution in the core to acquire the proper distribution as the core stiffness increases as a result of non-negligible in-plane stresses. The HSAPT model is incapable of capturing the in-plane stresses and assumes negligible in-plane rigidity. The current research extends that concept and applies it to two-dimensional plate structures with variable aspect ratios. The theory assumes a transverse displacement in the core that varies as a second order equation in z and the in-plane displacements that are of third order in z, the transverse coordinate. This approach allows for five generalized coordinates in the core (the in-plane and transverse displacements and two rotations about the x and y-axes respectively).
The major assumptions of the theory are as follows:
1) The face sheets satisfy the Euler-Bernoulli assumptions, and their thicknesses are small compared to the overall thickness of the sandwich section; they undergo small strains with moderate rotations.
2) The core is compressible in the transverse and axial directions; it has in-plane, transverse and shear rigidities.
3) The bonding between the face sheets and the core is assumed to be perfect.
The kinematic model is developed by assuming a displacement field for the soft core and then enforcing continuity of the displacement field across the interface between the core and facesheets. The constitutive relations are then defined, and variational and energy techniques are employed to develop the governing equations and associated boundary conditions.
A static loading case for a simply supported sandwich plate is first considered, and the results are compared to existing solutions from Elasticity theory, Classical Plate Theory (CPT) and First-Order Shear Deformation Plate Theory (FSDT).
Subsequently, the governing equations for a dynamic analysis are developed for a laminated sandwich plate. A free vibration problem is analyzed for a simply supported laminated sandwich plate, and the results for the fundamental natural frequency are compared to benchmark elasticity solutions provided by Noor. After validation of the new Extended Higher Order Sandwich Panel Theory (EHSAPT), a parametric study is carried out to analyze the effect of variation of various geometric and material properties on the fundamental natural frequency of the structure.
After the necessary verification and validation of the theory by comparing static and free vibration results to elasticity solutions, a nonlinear static analysis for square and rectangular plates is carried out under various sets of boundary conditions. The analysis was carried out using variational techniques, and the Ritz method was used to find an approximate solution. The kinematics were developed for a sandwich plate undergoing small strain and moderate rotations and nonlinear strain displacement relations were evaluated.
Approximate and assumed solutions satisfying the geometric boundary conditions were developed and substituted in the total potential energy relations. After carrying out the spatial integrations, the total potential energy was then minimized with respect to the unknown coefficients in the assumed solution resulting in nonlinear simultaneous algebraic equations for the unknown coefficients. The simultaneous nonlinear equations were then solved using the Newton-Raphson method.
A convergence study was carried out to study the effect of varying the number of terms in the approximate solution on the overall result and rapid convergence was observed. The rapid convergence can be attributed to the fact that the assumed approximate solution not only satisfies the geometric boundary conditions of the problem but also the natural boundary conditions.
During calculations four cases of boundary conditions were considered
1) Simply Supported with moveable edges.
2) Simply Supported with fixed edges.
3) Clamped with moveable edges.
4) Clamped with fixed edges.
For movable boundary conditions, in-plane displacements along the normal direction to the supported edges are allowed whereas the out-of-plane displacement is fixed. For the immovable boundary condition cases, the plate is prevented from both in-plane and out-of-plane displacements along the edges. For the simply supported cases rotations about the tangential direction are allowed, and for the clamped cases no rotations are allowed.