104 4.10 The first ten non-dimensional bending frequencies (¯ωm) of smart composite plate. f) for different sets of boundary conditions. ITOT in terms of 3D EKM for a square hybrid sandwich plate (b) under five different boundary conditions for S=5, 10 and 20.
PREFACE
In such smart hybrid plates, the edge effect is more complex due to the presence of electromechanical coupling and may lead to the loss of the actuation and sensing authority of the piezoelectric layers. It is also proposed to develop 3D piezoelectric solutions for the dynamic response of the same.
HISTORICAL BACKGROUND OF EKM
In this method, the initially chosen function is not required to satisfy the boundary conditions of the problem and can therefore be chosen arbitrarily. Besides the above-mentioned advantage of EKM, the convergence in solution is achieved very quickly, which is investigated in this work in successive chapters.
LITERATURE REVIEW
Pioneered Researchers in the Field of Piezoelasticity
The vibration behavior of smart plates is presented by Chandra and Agarwal [27] and Batra et al. Tzou and Fu [35] Distributed vibration sensing and control of continua using segmented distributed piezoelectric sensors and actuators.
Piezoelasticity Solution for Static Analysis of Smart Hybrid Plates
92] presented a free vibration analysis of composite laminates for simply supported and clamped boundary conditions using 3D. To the best of the author's knowledge, there is no 3D piezoelasticity solution for free vibration analysis of hybrid and sandwich composite plates for arbitrary boundary conditions using multi-term EKM.
OBJECTIVES OF THE PRESENT WORK
ORGANISATION OF THE THESIS
In Chapter 3, 3D EKM is used to present the free vibration analysis of elastic composite and sandwich plates together with the effect of adhesive characteristics on the free vibration behavior of composite plates are investigated. The accuracy of the present solution is verified by comparison with the results of other theories and with ABAQUS 3D FE solutions.
GOVERNING EQUATIONS
It is perceived that, in this method, convergence in results is achieved very quickly in only two to three terms of the Fourier expansion in the iteration process. The interfaces of the piezoelectric layers with the adjacent elastic layers are taken as grounded (φ = 0) for effective actuation/sensing.
FOURIER SERIES-GENERALIZED EKM SOLUTION
First Iteration Step
In the first step, the functions fli(ξ1) are treated as known, while gil(ζ) is determined for each layer. Since fli are known analytic functions consisting of exponential and trigonometric functions, the integrations are ⟨. but in the above elements they were evaluated in a tight form.
Second Iteration Step
The general solution of the system of first-order ODEs with constant coecients given by Eq. 2.30) is obtained by the procedure described in Ref. Similar to the thickness direction, fli(ξ1) is divided into two groups: (i) F¯ containing 8n primary variables appearing in the boundary conditions (2.15), and (ii) Fˆ containing the remaining 5nvariables:. 2.19), and this time it is integrated over the thickness directionζ.
NUMERICAL RESULTS AND DISCUSSIONS
Stress Field at the Interface of Long Bi-material Strip
The distribution of the normal stress (¯σz) along the interface (z = 0) obtained from the solution is plotted in Fig. The order of singularities in the stress distribution is calculated by fitting a power law curve over the plot up to r/a =0.05, and these are compared in table 2.2 with their theoretical values calculated from Eq.
Edge Effects in Piezoelectric and Smart Hybrid Laminated Plates
Having established the accuracy of the mixed-field multiterm EKM for edge stresses, we now present results for edge effects in piezoelectric and smart hybrid laminated plates. The response of the piezoelectric plate is obtained for the compressive load case 1, while the smart hybrid sandwich plate is analyzed for both compressive and potential load cases.
Single-Layer Piezoelectric Plate
The changes of the voltage ¯σx,¯σy,τ¯zx through the thickness and the electric potential ¯ϕ near the clamped edge of the C-C PZT plate are presented in the figure. The nature of the in-plane stress distribution shows a significant change with the aspect ratio (b/a).
Smart Hybrid Sandwich Plate
The nature of the distribution of the interlaminar shear stress ¯τzx at the actuator interface is similar for simply supported and free edges, and differs from the clamped edge where a stress singularity occurs at the support, which is similar to the pin-force model that is commonly used. used for the load transfer mechanism [158]. Here too, the predicted response is in excellent agreement with the FE solution, except at the boundary.
Effect of x-Location, Aspect Ratio and Thickness Ratio on the Response
The effect of b/a on the longitudinal distribution of interlaminar shear stress ¯τzx at the piezoelectric layer-host laminate interface is shown in Fig. The effect of the actuator thickness on the nature of longitudinal profile of ¯τzx is also investigated in Fig.
Accurate Estimation of Interlaminar Stresses for the Adhesive Bonded
It is clear from the graph that when the elastic stiffness of the adhesive increases, counterintuitively the stress concentration at the layer interface increases, which is due to the fact that the increased stiffness has a negative effect on the bonding property of the adhesive.
CONCLUSIONS
But for the potential load case, the interphase and center values of the interlaminar stresses increase slightly with the thickness of the adhesive layer. The lower modulus of elasticity allows a significant reduction in interlaminar stresses for both pressure and potential load cases.
INTRODUCTION
Benchmark natural frequencies are tabulated for composite and sandwich laminates (first time) for various boundary conditions. The effect of span to thickness (S) and in-plane modulus ratios on the natural frequencies are also studied.
THEORETICAL FORMULATIONS
The accuracy of the 3D EKM is established by extensive comparison study with results from other theories and with the 3D FE solution of ABAQUS. The plate is subjected to arbitrary support conditions (simply supported, clamped and free) at edges ξ1 = 0 and 1.
FOURIER SERIES-GENERALIZED EKM SOLUTION
Solution Along Thickness Direction (z)
Its complementary solution has the form G¯c(ζ) =eλζY, which upon substitution into the homogeneous part of Eq. For a non-trivial solution, the determinant is zero and ω can be obtained by finding the roots of the equation|det(Kd)|=0 using the bisection method.
Solution Along In-plane Direction (x)
Since the functions gl(ζ) are known in narrow form, the above integrations are calculated in narrow form. The same procedure is used to obtain the natural frequency˜ ω01=ωnas in Sec. 3.16) and is similarly solved by satisfying the boundary conditions at ξ1 = 0.1.
RESULTS AND DISCUSSION
Free Vibration Analysis of Cross-ply Laminated Plates
The dimensionless natural frequencies are compared with Boscolo and Banerjee [82] for various boundary conditions in Table 3.2 for the thick composite slab (S=5) laying [0◦/90◦] with in-plane modulus ratio Y1/Y2= 30, except for the SS case where the additional results for Y1/Y2=3 are presented. The effect of the spacing-to-thickness ratio (S) on the fundamental frequency parameters (¯ωn) for arbitrary boundary conditions is shown in Table 3.3 for symmetric lamination.
Free Vibration Analysis of Soft Core Sandwich Plate
The percentage decrease variation in natural frequencies of sandwich plate (b) for F-F boundary conditions compared to S-S and C-F boundary conditions is shown in figure. The first three bending mode shapes with frequencies of thick sandwich plate (S = 5) for different boundary conditions are presented in Fig.
Accurate Estimation of the Influence of Adhesive Bonding on the Free
CONCLUSIONS
The generalized mixed-field multi-term extended Kantorovich method (EKM) for piezoelasticity solution for static cases developed in Chapter 2 is further extended to 3D piezoelasticity solution for free vibration [164] of Levy-type laminated plate integrated with piezoelectric actuation. ators and sensors in this chapter. In this chapter, multi-term 3D EKM has been used for free vibration analysis of single layer piezoelectric and bimorph plates and also for multilayer composite and sandwich plates with integrated top and bottom piezoelectric layers.
THEORETICAL FORMULATION
If the piezoelectric layer is used as an actuator, ϕ is prescribed (closed circuit condition) and if it is used as a sensor, Dz is prescribed (open circuit condition) at the outer surfaces. The mechanical boundary conditions at the edges ξ1 = 0 and 1 can be prescribed for a given support type, e.g.
FOURIER SERIES-GENERALIZED EKM SOLUTION
First Iteration Step
As mentioned earlier, the initial trial functions in EKM are not necessary to meet the prescribed boundary conditions. Since the variations δgli are arbitrary, the coefficients of δgil in the resulting expression must vanish individually.
Second Iteration Step
After obtaining the desired natural frequency, Eq. 4.30) is solved for determining the unknown constants by solving linear algebraic equations and then by solving Eq. 4.29) modal displacements, electrical variables and stresses are obtained.
RESULTS AND DISCUSSION
Validation with 3D Exact Results for Simply Supported (S-S) Case
The natural frequencies are compared with the 3D exact [167] results for S=5, 10, 20 and also with the additional results for S=1000 which are validated using the computer program of Kapuria and Achary [101]. It is observed that the present EKM results are in exact agreement with the exact 3D results for thick to thin plates.
Validation for Other Boundary Conditions
From the table it is clear that the current calculated frequencies are in good agreement with the analytical solution of Farsangi et al. The deviation is more significant for thick plates, as observed for CC cases, thanks to the 2D approximation (FSDT theorem) of Farsangi et al .
Some New Benchmark Results
The bending natural frequencies for the first ten modes determined by the present 3D EKM are produced in Table 4.9 for the F-F case. The effect of adhesive density on the natural frequency of the plate is investigated and presented in Table 4.15.
CONCLUSIONS
The increase in adhesive thickness reduces the natural frequencies for all types of boundary conditions. This is due to the fact that glue, being a material with a lower modulus of elasticity, acts as a vibration suppressor when increasing the layer thickness. The denser glues help increase the natural frequencies, as seen with simply supported and clear-free enclosures.
MATHEMATICAL MODELLING
- Geometry
- The Strain-Displacement and Constitutive Relations
- Governing Differential Equations and Boundary Conditions
- Levy-type Solution
The generalized stress resultants F1, F2, F3 and F4 can be expressed in terms of displacement and potential variables by substituting the expressions for σ,τ,D and Dz from Eq. hybrid sheet dielectric matrices. The displacements, stress resultants, electric potential, stresses, and electric displacement at any point on the plate can be calculated using Eq.
FREE VIBRATION OF ELASTIC LAMINATED PLATES
Validation
The accuracy of the current ZIGT result was verified by direct comparison with the 3D exact [101] and 3D EKM [159] results for all circular simply supported cases (S-S) in Table 6.1 and with the FSDT [170] and third order shear deformation theory (TSDT) results ) [171] for other types of boundary conditions in Table 6.2. The bending fundamental frequency of a Levy-type single-layer isotropic plate is presented in Table 6.2 and compared with TSDT [171] and FSDT [170] results for different types of boundary conditions.
Assessment
The percent error of the lowest eight bending frequencies for the three-layer composite laminated plate is presented in Table 6.6 for the S-S and C-C cases with S = 10 and 20 together with the reference EKM 3D results. The percentage errors of ZIGT and TOT frequency for the lowest five bending frequencies are presented in Table 6.7 according to the 3D EKM results for a five-layer sandwich plate [90◦/0◦/Core/0◦/90◦] under five boundary conditions (S–S, C –C, C–S, C–F and F–F) for S = 5, 10 and 20.
STATIC ANALYSIS OF RECTANGULAR HYBRID PLATES
Longitudinal variations of deflection, stresses and electrical displacement under potential loading for moderately thick sandwich plate with C-F and C-S boundary conditions are presented in Fig. However, ITOT results are erroneous even in the inner part of the plate specific for ¯σx.
FREE VIBRATION ANALYSIS OF RECTANGULAR HYBRID PLATES
Validation
Assessment
CONCLUSIONS
Three-dimensional free vibration analysis of cross-layer laminated rectangular plates via 2D and exact models. Three-dimensional free vibration analysis of Levy-type laminated plates using the multinomial extended Kantorovich method.
General configuration of a rectangular hybrid plate
Geometry and coordinate system of a L-layer smart hybrid plate
Geometry of bi-material strip
Stress distribution along the interface of bi-material strip
Configurations of (a) piezoelectric plate and (b) smart hybrid sandwich plate
Longitudinal variations of displacements, stresses, electric potential and electric
Longitudinal variations of displacements, stresses, electric potential and electric
Through-thickness distributions of stresses and electric potential for square piezo-
Effect of b/a on through-thickness variations of stresses and electric potential for
Longitudinal variations of displacements, stresses and electric displacement for
Authors pioneered in smart structure research
Material properties
Comparison of order of stress singularity (λ − 1)
Material constants
Comparison of the fundamental frequency parameter, ¯ ω for simply-supported (S-
