Development of isogeometric finite element method to analyze and control the responses of the laminated plate structures Development of isogeometric finite element method to analyze and control the responses of the laminated plate structures Development of isogeometric finite element method to analyze and control the responses of the laminated plate structures
Introduction
This dissertation aims to create an isogeometric finite element method for analyzing and controlling the responses of laminated plate structures This chapter provides an overview of isogeometric analysis and highlights the latest advancements in laminated plate structures relevant to the dissertation Additionally, it outlines the key drivers and innovative aspects of the research to offer readers a clearer understanding of the dissertation's content.
An overview of isogeometric analysis
Over time, the advantages of numerical procedures have become widely acknowledged, leading to their extensive development for computing, analyzing, and simulating the response and dynamic characteristics of laminated plates Popular numerical techniques include the boundary element method (BEM), finite element method (FEM), finite difference method (FDM), mesh-free method, and finite volume method (FVM) These numerical methods are generally categorized into two main groups.
- Group 1: methods that require meshing; e.g FEM, FDM, BEM and FVM
- Group 2: methods that do not require meshing; e.g mesh-free methods
Finite Element Method (FEM) is widely recognized for its effectiveness in solving diverse technical problems, making it a universal tool for addressing boundary and initial value challenges across engineering and scientific research fields Despite its versatility and power, FEM does have limitations, including issues like overstated stiffness, inaccuracies in stress results for linear elements, and meshing complications To mitigate these disadvantages, three potential solutions are proposed.
- Improve the finite element spaces
- Improve both the variational method and the finite element spaces
Therefore, the isogeometric analysis (IGA) is proposed in order to implement the aforementioned solutions At first, it is necessary to know a brief history of IGA and what it is
Since its introduction in the early 1940s, computers have played a crucial role in mathematical computations and solving engineering challenges, leading to the development of Computer-Aided Engineering (CAE) The advancement of computer hardware and algorithms since the 1960s has given rise to the Finite Element Method (FEM), which has become the predominant numerical tool for addressing partial differential equations in physical problems This method has gained global recognition, resulting in numerous articles and books dedicated to its study Today, FEM is a well-established technique widely utilized across various industries.
The rapid advancements in modern technology have led to the creation of increasingly complex structures, exemplified by the fact that a typical personal automobile consists of around 3,000 parts, whereas a Boeing 777 is made up of approximately 100,000 parts This multitude of components results in a more intricate process for modeling, analysis, and construction, presenting a significant challenge for traditional Finite Element Method (FEM) approaches.
In 1966, French engineers Pierre Bézier from Renault and Paul de Faget de Casteljau from Citroën pioneered geometry modeling, utilizing Bernstein polynomials to create curves and surfaces Riesenfield's contemporary advancements in 1972 and Versprille's generalization to NURBS in 1975 further enhanced these developments, laying the foundation for Computer Aided Design (CAD), which is now a standard tool for geometry representation in industry Despite this, CAD systems have evolved independently alongside Computer Aided Engineering (CAE) for several decades, primarily due to various influencing factors.
Designers focus on creating systems for effective visualization, while analysts prioritize simplicity for rapid computation, reflecting the computing power limitations of the past As technology has advanced, the demand for analyzing increasingly complex structures has grown However, a significant challenge arises from the differences in geometric descriptions between Computer-Aided Design (CAD) and Computer-Aided Engineering (CAE), necessitating the simplification and conversion of CAD models into compatible formats for finite element analysis This results in a considerable amount of redundant work.
Despite significant efforts to automate the conversion process from CAD to CAE, existing techniques remain unsuitable for industrial application due to their unreliability in replacing manual corrections The challenges arise from the complexity of creating accurate meshes for intricate geometries, which often leads to substantial loss of geometric information This issue stems from the necessity of refining the mesh to capture detailed features of the computational domain, requiring interaction with the design model However, the analysis-suitable model is merely an approximation of the design model, making direct interaction impractical and preventing the retention of the original geometry's precise details.
Figure 1.1: Analysis procedure in FEA Due to the meshing, the computational domain is only an approximation of the CAD object
Figure 1.2: Analysis procedure in IGA No meshing involved, the computational domain is thus kept exactly
In 2005, Hughes, Cottrell, and Bazilievs introduced Isogeometric Analysis (IGA), a technique that eliminates the need for complex conversions between systems by using the same basis functions from CAD (B-splines/NURBS) for analysis This approach results in exact meshes and higher continuity, significantly reducing computational costs as meshes are generated directly within CAD, fostering better collaboration between Finite Element Analysis (FEA) and CAD Since its inception, extensive research has demonstrated IGA's successful application across various fields, including structural analysis, fluid-structure interaction, electromagnetics, and higher-order partial differential equations IGA's ability to deliver higher accuracy due to its smoothness and continuity between elements motivates the exploration of a new numerical method beyond standard finite elements, which will be presented in this dissertation through an alternative approach based on Bézier extraction.
Literature review about materials used in this dissertation
This dissertation explores four distinct material types: laminated composite plates, piezoelectric laminated composite plates, piezoelectric functionally graded porous plates reinforced with graphene platelets, and functionally graded piezoelectric material porous plates.
Laminated composite plates are crucial structural elements widely utilized in various engineering fields, including civil, aerospace, and automotive engineering, due to their exceptional mechanical properties such as high strength-to-weight and stiffness-to-weight ratios, as well as wear resistance The design flexibility offered by the stacking sequence and layer thickness of these composites enhances their application in engineering Consequently, researchers are increasingly focused on understanding the bending behavior, stress distribution, and natural vibrations of laminated composites, making the study of their static and dynamic responses essential This thesis introduces a novel approach to analyzing laminated composite plates using Isogeometric Analysis (IGA), building on existing applications like Kirchhoff–Love plates and Reissner–Mindlin plates Recent studies, such as those by Lezgy-Nazargah et al and Valizadeh et al., have explored static and transient analyses using various plate theories, yet the investigation of laminated composite plates through IGA with Bézier extraction and a generalized unconstrained higher-order shear deformation theory (UHSDT) remains underexplored Thus, this research aims to fill that gap by providing a comprehensive study of laminated composite plates.
Piezoelectric materials are innovative substances that exhibit a unique coupling of electrical and mechanical properties, enabling the conversion between electrical and mechanical energy When subjected to mechanical loads, these materials generate electricity, while applying an electric field can alter their shape This dual functionality has led to widespread applications in smart structures across various sectors, including aerospace, automotive, military, and medical fields Numerous numerical methods have been developed to analyze the behavior of plates integrated with piezoelectric layers For instance, Mitchell and Reddy introduced the classical plate theory (CPT) using third-order shear deformation theory (TSDT) to derive the Navier solution for composite laminates with piezoelectric components Similarly, Suleman and Venkayya utilized classical laminate theory (CLT) combined with four-node finite elements to conduct static and vibration analyses of laminated composites with piezoelectric layers, optimizing numerical integration and addressing hourglass issues.
Recent advancements in the mechanics of adaptive piezoelectric actuators and sensors have been made through the development of higher-order finite formulations and analytical closed-form solutions [26] Liew et al [27] explored the post-buckling behavior of piezoelectric functionally graded materials (FGM) plates under thermo-electro-mechanical loadings, employing a semi-analytical solution with a Galerkin-differential quadrature integration algorithm based on higher-order shear deformation theory (HSDT) Liu et al [28-29] introduced the radial point interpolation method (RPIM) utilizing first-order shear deformation theory (FSDT) and classical plate theory (CPT) with a four-node non-conforming rectangular plate bending element to analyze the static deformation and dynamic responses of plates integrated with sensors and actuators Furthermore, Hwang and Park [30] investigated plates equipped with piezoelectric sensors and actuators through the discrete Kirchhoff quadrilateral (DKQ) element, applying the Newmark method for direct time response analysis under negative velocity feedback control.
7 layerwise generalized finite element formulation [31] and the layerwise based on analytical formulation [32] were investigated for piezoelectric composite plates And
FE formulations based on HSDT for analysis of smart laminated plates were studied in ref [33]
For vibration control, Bailey et al [34] and Shen et al [35] investigated smart beams integrated with piezoelectric layers using analytical solutions Tzou and Tseng
Recent advancements in piezoelectric materials have led to the development of a piezoelectric thin hexahedron solid element for analyzing and controlling plates and shells equipped with dispersed piezoelectric sensors and actuators Liew et al combined a meshfree model based on First-Order Shear Deformation Theory (FSDT) to explore shape control of piezo laminated composite plates under various boundary conditions Wang et al employed the finite element method for dynamic stability analysis of piezoelectric composite plates, utilizing Lyapunov’s energy functional derived from general governing equations of motion with active damping He et al investigated shape and vibration control of functionally graded materials (FGM) plates integrated with sensors and actuators based on Classical Plate Theory (CPT) Selim et al examined active vibration control of FGM plates with piezoelectric layers using Higher-Order Shear Deformation Theory (HSDT) and the element-free IMLS-Ritz method Furthermore, Phung-Van et al conducted nonlinear transient analysis of piezoelectric FGM plates under thermo-electro-mechanical loads using the generalized shear deformation theory with Isogeometric Analysis (IGA).
1.3.3 Piezoelectric functionally graded porous plates reinforced by graphene platelets (PFGP-GPLs)
Porous materials are widely used in engineering fields such as aerospace, automotive, and biomedical applications due to their lightweight nature, excellent energy absorption, and heat resistance However, the presence of internal pores significantly reduces their structural stiffness To address this limitation, reinforcing these materials with carbonaceous nanofillers, including carbon nanotubes (CNTs), has emerged as a promising solution.
8 graphene platelets (GPLs) [51-52] into the porous materials is an excellent and practical choice to strengthen their mechanical properties
In recent years, researchers have focused on porous materials reinforced by GPLs due to their lightweight nature and exceptional properties, including energy absorption and thermal management Artificial porous materials like metal foams combine desirable physical and mechanical characteristics, making them widely used in lightweight structural and biomaterial applications The incorporation of GPLs enhances material performance while reducing structural weight through increased porosity This synergy significantly improves mechanical properties while preserving the lightweight potential of the structures By adjusting pore sizes, densities in various directions, and GPL dispersion patterns, the performance of these materials can be further optimized.
FG porous plates reinforced by GPLs (FGP-GPLs) have been introduced to obtain the required mechanical characteristics [61–63]
Recent studies have focused on the effects of graded porous layers (GPLs) and porosities on structural behavior under various conditions Researchers such as Kitipornchai et al and Chen et al have applied the Ritz method and Timoshenko beam theory to analyze free vibration, elastic buckling, and nonlinear post-buckling performance of functionally graded (FG) porous beams Yang et al explored the uniaxial, biaxial, shear buckling, and free vibration responses of FG porous-GPLs using the First-Order Shear Deformation Theory (FSDT) and Chebyshev-Ritz method Furthermore, Li et al employed Isogeometric Analysis (IGA) to assess static, free vibration, and buckling behaviors of FG porous-GPLs Additionally, the geometrically nonlinear responses of piezoelectric functionally graded porous-GPL plates were examined, with D Nguyen-Dinh et al investigating the nonlinear thermo-electro-mechanical dynamics of shear deformable piezoelectric sigmoid functionally graded sandwich cylindrical shells on elastic foundations.
D Nguyen-Dinh et al introduced a novel method for analyzing the nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube-reinforced composite double curved shallow shells Additionally, Li et al explored the nonlinear vibration and dynamic buckling of sandwich functionally graded plates and graphene platelet-reinforced laminates supported by a Winkler-Pasternak elastic foundation using classical plate theory (CPT).
Most studies have primarily concentrated on piezoelectric plates that incorporate layers specifically addressing the core made of Functionally Graded Materials (FGM) or Functionally Graded Carbon Nanotube Reinforced Composites (FG-CNTRC) Additionally, there remains a limited scope of research on the geometrically nonlinear static and dynamic analyses of piezoelectric FG plates subjected to various loading conditions.
1.3.4 Functionally graded piezoelectric material porous plates (FGPMP)
Traditional piezoelectric devices, typically made from multiple layers of different materials, face issues such as cracking, delamination, and stress concentrations at their interfaces To address these challenges, functionally graded materials (FGMs) have been introduced, which feature a continuous variation of material properties throughout their thickness by blending two distinct materials This innovation aims to mitigate the disadvantages associated with piezoelectric laminated composites Furthermore, the development of functionally graded piezoelectric materials (FGPMs) combines two types of piezoelectric materials in a single direction, resulting in enhanced properties compared to conventional piezoelectric materials Consequently, FGPMs have garnered significant interest from researchers focused on the analysis and design of advanced smart devices in recent years.
Nowadays, there are many modern techniques to fabricate FGMs such as centrifugal solid-particle method, centrifugal mixed-powder method, plasma
Functionally Graded Piezoelectric Materials (FGPMs) often exhibit inherent porosities due to manufacturing imperfections, akin to the natural structures of wood and stone, which also contain porous phases While these porous materials are widely utilized in lightweight structures and biomaterials, they significantly compromise structural strength The mechanical and electrical behaviors of imperfect FGPM plates differ markedly from those of perfect FGPM plates, highlighting the necessity of investigating the effects of porosity on FGPM technology Understanding these implications is crucial for advancing the performance and application of FGPMs.
Recent studies have focused on the electro-mechanical behavior of functionally graded (FG) piezoelectric structures Notably, Zhong and Shang developed an exact three-dimensional solution for a fully simply-supported FGPM rectangular plate grounded along its edges Additionally, Jadhav and Bajoria investigated both free and forced vibration control of FG piezoelectric plates under electro-mechanical loading Kiani et al also contributed to the field by examining the buckling behavior of these structures.
Recent studies have focused on the analysis of functionally graded (FG) piezoelectric materials, particularly Timoshenko beams under thermo-electrical loading Sharma and Parashar employed Mindlin plate theory and the generalized differential quadrature method to investigate the natural frequencies of annular FGPM plates Li and Pan conducted static and free vibration analysis of FG piezoelectric microplates using modified couple-stress theory Additionally, Behjat and Khoshravan explored geometrical nonlinearity in the bending and free vibration analysis of FG piezoelectric plates through finite element methods (FEM) Most recently, Zhu et al presented an analytical approach for free and transient vibration analyses of FGPM plates, accommodating various boundary conditions.
Recent literature highlights significant advancements in the study of functionally graded porous (FGPMP) plates Barati et al explored the free vibration of these plates using an improved four-variable theory, considering both even and uneven porosity distributions They also examined the buckling behavior of higher-order graded piezoelectric plates with porosity on elastic foundations Ebrahimi et al investigated the free vibration properties of smart shear deformable plates made from porous magneto-electro-elastic (MEE-FG) materials Additionally, Wang studied the electro-mechanical vibration behavior of FGP plates with porosities, while Wang and Zu focused on the porosity-dependent nonlinear forced vibrations of functionally graded piezoelectric material plates Notably, these studies primarily utilized analytical approaches suitable for simple rectangular geometries, whereas practical applications often involve more complex shapes Therefore, the development of a robust numerical method is essential, as higher-order approximate methods significantly enhance the accuracy of numerical solutions for FGPMP plates The numerical results obtained could serve as valuable references for future research.
Goal of the dissertation
The dissertation explores the advancement of isogeometric finite element methods to analyze and control the behavior of laminated plate structures It aims to introduce a novel isogeometric formulation utilizing Bézier extraction for the analysis of laminated composite plates The study encompasses three analytical approaches: static analysis, free vibration analysis, and dynamic transient analysis, applied to various laminated structures, including laminated plates, piezoelectric laminated composite plates, piezoelectric functionally graded porous plates reinforced with graphene platelets, and functionally graded piezoelectric material porous plates Additionally, the research focuses on implementing active control mechanisms for these structures.
12 algorithm is applied to control static and transient responses of laminated plates embedded in piezoelectric layers in both linear and nonlinear cases.
The novelty of dissertation
This dissertation contributes several novelty points coined in the following points:
The generalized unconstrained higher-order shear deformation theory (UHSDT) presents a significant advancement by eliminating the requirement for shear correction factors and allowing for zero-shear stresses at the top and bottom surfaces of plates This innovative approach enhances the accuracy and efficiency of structural analysis in engineering applications.
It is written in general form of distributed functions Two distributed functions which supply better solutions than reference ones are suggested
The proposed method leverages Isogeometric Analysis (IGA) to seamlessly integrate Finite Element Analysis (FEA) with traditional NURBS-based Computer-Aided Design (CAD) tools, a numerical approach introduced by Hughes et al in 2005 Despite its advancements, there remain compelling areas for further research in this field.
• IGA has surpassed the standard finite elements in terms of effectiveness and reliability for various engineering problems, especially for ones with complex geometry
In this article, we explore the innovative use of Isogeometric Analysis (IGA) based on Bézier extraction, which replaces conventional IGA methods This approach utilizes Bernstein shape functions, enabling the use of a consistent set of shape functions across all elements, similar to standard Finite Element Method (FEM) This significant advancement facilitates seamless integration into existing finite element codes with minimal modifications, distinguishing it from previous research conducted in Vietnam.
Current research still reveals a gap in the study of porous plates reinforced with graphene platelets integrated into piezoelectric layers, utilizing Isogeometric Analysis (IGA) based on Bézier extraction for both linear and nonlinear assessments.
13 active control technique for control of the static and dynamic responses of this plate type is also addressed
• In this dissertation, the problems with complex geometries using multipatched approach are also given This contribution seems different from the previous dissertations which studied IGA in Viet Nam.
Outline
The dissertation contains seven chapters and is structured as follows:
Chapter 1 provides an overview of the historical development of IGA, detailing the state-of-the-art advancements in four material types discussed in this dissertation It outlines the motivation and novelty behind the research, while also guiding the reader through the organization of the thesis for a comprehensive understanding of its content.
Chapter 2 presents an in-depth exploration of isogeometric analysis (IGA), focusing on key concepts such as B-spline basis functions, non-uniform rational B-splines (NURBS), and their applications in curves and surfaces It also covers B-spline geometries and refinement techniques Additionally, the chapter highlights Bézier extraction and evaluates the advantages and disadvantages of IGA in comparison to the finite element method.
Chapter 3 offers a comprehensive overview of plate theories and material properties essential for subsequent chapters It begins by detailing various plate theories, including those that will be utilized later in the text Additionally, the chapter introduces four key material types: laminated composite plates, piezoelectric laminated composite plates, functionally porous plates reinforced with graphene platelets embedded in piezoelectric layers, and functionally graded piezoelectric material porous plates.
Chapter 4 presents the results of static, free vibration, and transient analyses of laminated composite plates featuring diverse geometries, reinforcement orientations, and boundary conditions The analysis employs Isogeometric Analysis (IGA) utilizing Bézier extraction throughout the chapter Additionally, the study incorporates two piezoelectric layers adhered to the top and bottom surfaces of the laminated composite plates.
Composite plates are analyzed for static, free vibration, and dynamic responses, utilizing a displacement and velocity feedback control algorithm for active control The numerical examples presented demonstrate the accuracy and reliability of the proposed method.
Chapter 5 explores an isogeometric Bézier finite element analysis for the bending and transient behavior of functionally graded porous (FGP) plates reinforced with graphene platelets (GPLs) embedded in piezoelectric layers, termed PFGP-GPLs The study investigates the influence of GPL weight fractions, dispersion patterns, porosity distribution types, and external electric voltages on the structural performance, providing novel numerical results that serve as reference solutions for future research Additionally, the chapter expands on the analysis of nonlinear static and transient responses of PFGP-GPLs, incorporating constant displacement and velocity feedback control methods to actively manage the geometrically nonlinear static and dynamic behaviors of the plates, while accounting for structural damping through a closed-loop control system.
Chapter 6 explores the benefits of functionally graded piezoelectric material porous plates (FGPMP), characterized by their continuously varying material properties in the thickness direction, modeled using a modified power-law formulation The study employs two porosity models—uniform and non-uniform distributions—and incorporates an electric potential field represented by a combination of cosine and linear variations to satisfy Maxwell’s equations under quasi-static conditions Additionally, the research investigates several FGPMP plates with curved geometries, for which analytical solutions remain unknown This study aims to provide a reference solution for future research in the field.
• Finally, chapter 7 closes the concluding remarks and opens some recommendations for future work
Concluding remarks
This chapter provides an overview of Isogeometric Analysis (IGA) and the materials used, highlighting the key drivers and novel aspects of this dissertation It also outlines the organization of the dissertation, which consists of nine chapters The subsequent chapter will present a detailed exploration of the isogeometric analysis framework.
Introduction
This chapter provides a comprehensive overview of the benefits of Isogeometric Analysis (IGA) in comparison to traditional methods such as Finite Element Method (FEM), B-splines, and Non-Uniform Rational B-Splines (NURBS) It also includes a concise discussion on refinement techniques, numerical integration, and a summary of the IGA procedure, highlighting its efficiency and effectiveness in computational analysis.
Advantages of IGA compared to FEM
IGA offers several advantages over traditional FEM, particularly in maintaining the computation domain regardless of the level of discretization This stability simplifies contact detection between two surfaces, especially in large deformation scenarios where their relative positions change significantly Furthermore, IGA allows for precise and accurate modeling of sliding contact, which is crucial for applications sensitive to geometric imperfections, such as shell buckling analysis and boundary layer phenomena in fluid dynamics.
NURBS-based CAD models streamline the mesh generation process by eliminating the need for geometry clean-up or feature removal, significantly reducing the time spent on these tasks, which typically account for around 80% of total analysis time Additionally, mesh refinement becomes quick and efficient, as it does not require communication with CAD geometry This efficiency arises from the shared basis functions used in both modeling and analysis, allowing for straightforward geometry partitioning and automated knot insertion algorithms that create precise new elements in the mesh.
The method of interelement higher regularity with a maximum of C p − 1, particularly in the absence of repeated knots, is ideally suited for mechanics problems involving higher-order derivatives, such as the Kirchhoff-Love shell, gradient elasticity, and the Cahn-Hilliard equation of phase separation This suitability arises from the direct use of B-spline/NURBS bases for analysis Unlike FEM, where basis functions are defined locally within an element and exhibit C 0 continuity across boundaries, IGA’s basis functions span multiple contiguous elements, ensuring greater regularity and interconnectivity, resulting in highly continuous approximations Additionally, this enhanced smoothness leads to improved convergence rates compared to traditional methods, especially when paired with a new k-refinement technique Importantly, the extended support of the basis does not increase the bandwidth of the numerical approximation, maintaining the sparse matrix bandwidth akin to classical FEM functions.
Some disadvantages of IGA
This method, however, presents some challenges that require some special treatments
A key challenge in utilizing B-splines and NURBS in Isogeometric Analysis (IGA) is their tensor product structure, which restricts true local refinement Consequently, any insertion of knots results in global changes throughout the computational domain.
The absence of the Kronecker delta property complicates the implementation of inhomogeneous Dirichlet boundary conditions and the exchange of forces or physical data in coupled analyses.
The IGA's enhanced foundational support leads to the generation of denser system matrices, featuring a greater number of nonzero entries, in contrast to the Finite Element Method (FEM), which results in the loss of the tri-diagonal band structure.
B-spline geometries
Performing summation over each univariate B-spline basis function multiplied with its associated control point, P i a B-spline curve of order p in d is obtained as
C P N P (2 1) where n is the number of univariate B-spline basis function in direction and the univariate B-spline basis N i p , ( ) is then defined recursively over p starting with piecewise constants (p = 0) as
In this context, it is assumed that when the denominator of an arbitrary coefficient term equals zero, that term is defined to have a value of zero Each control point, denoted as P_i, is associated with a specific basis function that determines its influence on the curve While these are analogous to nodal coordinates in Finite Element Analysis (FEA), the non-interpolatory nature of B-splines alters the interpretation of control point values For instance, Figure 2.1 depicts a quartic B-spline curve based on the knot vector Ξ={0, 0, 0, 0, 0, 1/3, 1/3, 1/3, 2/3, 1, 1, 1, 1, 1}, where red squares represent physical knots or element boundaries The open knot vector results in the interpolation of the two end control points, and due to the multiplicity of four at the knot 1/3, the curve interpolates one control point, ensuring C^0 continuity at this knot.
A B-spline curve possesses the following properties:
• If p= −n 1 (i.e., the order of a B-spline curve is equal to the number of control points minus 1), this B-spline curve reduces to a Bézier curve
• B-spline curve is a piecewise polynomial curve
• Clamped B-spline curve interpolates the two end control points P 1 and P n+ 1
• Strong convex hull property: an arbitrary B-spline curve is kept inside the convex hull of its control polygon
• Invariance with respect to affine transformations
• Local modification: modifying the position of P i affects C ( ) only in the interval i , i p + + 1 )
• C ( ) is C p k − continuous at a knot of multiplicity k
Each Bézier segment can connect to another with a desired level of continuity, up to a maximum of C p − 1 This means that an arbitrary curve can be represented through two distinct methods.
Figure 2 2: The B-spline curve in Figure 2 1 can be described by three concatenated Bézier curves Due to interelement C 0 continuity, this representation produces more control points than the B-spline one
A tensor-product B-spline surface of order (p, q) for an arbitrary patch is constructed parametrically by a sum over B-spline functions multiplied with the associated control points as for B-spline curve
In the context of bivariate B-splines, S P (2 3) represents a surface defined by m and n univariate B-spline basis functions in two parametric directions The control net, denoted as P i j, is created by connecting m × n control points The bivariate B-spline basis function, N i j p q ( , ), is defined accordingly to facilitate the construction of the surface.
Properties of a B-spline surface inherit directly from their univariate counterparts as the result of its tensor product nature.
Refinement technique
B-splines offer a flexible approach to geometry description and analysis, allowing for the enhancement of its polynomial bases without altering the fundamental geometry or requiring interaction with CAD software.
B-splines utilize two primary refinement strategies: knot insertion and degree elevation, which correspond to h-refinement (subdivision) and p-refinement (order elevation) in finite element analysis (FEA) The integration of these strategies leads to hp-refinement in FEA However, in isogeometric analysis (IGA), the sequence of h- and p-refinement affects the refined polynomial basis, allowing for greater control over element size, polynomial order, and basis continuity, resulting in additional degrees of freedom This introduces a new refinement strategy known as k-refinement The subsequent sections will briefly illustrate these three refinement methods for univariate B-splines, while for multivariate B-splines, the tensor product structure enables the independent application of univariate algorithms across each parameter direction and corresponding control mesh.
Knot insertion is a technique used for refinement in approximation spaces, similar to mesh refinement, where the mesh size (h) is decreased while maintaining global continuity Unlike mesh refinement in Finite Element Analysis (FEA), which results in C0 continuity at element boundaries, knot insertion offers enhanced flexibility in controlling both element size and continuity between elements This method is grounded in a specific mathematical relation.
In the context of C P P C (2.5), the knot vectors for the original and refined spaces are defined as Ξ = {ξ1, ξ2, , ξn, ξp+1} and Ξ' = {ξ1, ξ2, , ξn, ξm+p+1}, respectively, where m represents the number of newly inserted knots The process of knot insertion necessitates the reevaluation of both basis functions and the associated control points For clarity, we will focus on the insertion of a single knot, ξ, into the knot span [ξj, ξj+1), with j denoting the specific position within the knot vector.
22 vector The n + 1 new basis functions are thus derived from the newly extended knot vector Ξ and the new control points are computed as a linear combination of the old ones as [90]
The insertion of a single knot requires the evaluation of p new control points, while all other control points remain aligned with the original layout Existing knots can be repeated to maintain the continuity and support of B-splines by inserting their values up to the multiplicity of p This process does not alter the polynomial order or create new elements As illustrated in Figure 2.3, the original curve on the left is refined by dividing its second element into two, resulting in a new control polygon P on the right after inserting a knot at ξ = 0.83 Although the refined curve retains the same geometry as the original, it now features an additional control point and basis function, increasing the total count by one.
Figure 2 3: An illustration of h-refinement for a B-spline curve
The process of refinement in B-spline curves involves systematically increasing the order while maintaining fixed interelement regularity A B-spline curve of order p is characterized as a piecewise polynomial curve, allowing for the elevation of all spline segments by a factor of t ≥ 1 without altering the overall shape of the curve The p-refinement algorithm is designed to determine a new knot vector Ξ = {ξ₀, , ξₙ₊ₚ₊ₜ} and updated control points {Pᵢ} based on the specified incremental order of elevation t.
The C C P (2.8) equation involves n control points of C(ξ), with N i p t,(ξ) representing the B-spline basis functions of order k + m defined on the knot vector Ξ The knot vector Ξ and the number of control points n can be directly determined through a specific procedure.
In the context of spline functions, the positive integers m1, , ms represent the multiplicities of internal knots Given that C(ξ) exhibits smoothness up to C^(p m - i) continuity at the knot ξi, it is essential for C(ξ) to maintain the same level of regularity at this knot To adhere to this requirement, each unique knot in the original knot vector must be replicated accordingly.
In this dissertation, the new control points \( P_i \) of the B-spline curve of order \( p + t \) are computed using the algorithm proposed in [92], which involves three key steps: decomposing the B-spline curve into Bézier segments, raising the order of each segment, and rejoining the elevated Bézier segments to form the new B-spline curve This process is illustrated in Figure 2.4, where the original quartic curve is shown on the left and the refined curve, which has an increased order, is displayed on the right, along with their corresponding basis functions The original curve is defined by the knot vector.
= and the control polygon P Rising the order of the curve by one resulting a quintic curve with the new knot vector
= and the new control polygon P It can be seen that the geometry of the curve is preserved, but each element is now influenced by one additional control point
Figure 2 4: An illustration of p-refinement for a B-spline curve
This refinement strategy involves first elevating the order of the coarse mesh, followed by knot insertion By performing knot insertion after order elevation, the refined basis attains high order and maximum continuity, specifically C p − 1 [95] This innovative refinement method offers a unique approach that differs from traditional finite element analysis (FEA) techniques, such as h- and p-refinements.
Figure 2 5: An illustration of k-refinement
K-refinement, like p-refinement, increases the number of control points during the refinement process However, unlike p-refinement, which introduces excessive degrees of freedom, k-refinement achieves the same level of approximation error with considerably fewer degrees of freedom.
Figure 2 illustrates the concept of k-refinement, which differs from classical p-refinement in the refinement of linear elements In the classical p-refinement process, knot insertion is followed by order elevation, resulting in nine quadratic basis functions that maintain C0 continuity at the element boundaries Conversely, the k-refinement approach first elevates the order and then inserts knots, leading to five quadratic basis functions with C1 interelement continuity For a more comprehensive discussion on k-refinement, please refer to source [95].
NURBS basis function
In CAD and isogeometric analysis, geometry is commonly represented using Non-Uniform Rational B-splines (NURBS), which are derived from conic projective transformations of B-splines in higher dimensions NURBS offer a significant advantage as rational basis functions, enabling the precise representation of complex shapes, including conic sections, that polynomial basis functions cannot accurately depict.
A NURBS curve is created by multiplying each control point's component from the control mesh \( P_i \) with a corresponding positive scalar weight \( w_i \) This process is followed by dividing the result by the weighting function \( W(\xi) \).
(2 12) where R i p ( ) is the univariate piecewise NURBS basis function defined by
Figure 2 6 demonstrates two circles that are represented by both NURBS and B- spline in the corresponding solid and dotted curves Their control points are depicted
27 by black balls with the associated weights also given for the NURBS case It is clear that only the NURBS curve is able to represent the circle exactly
Figure 2 6: Two representations of the circle The solid curve is created by
NURBS which describes exactly the circle while the dotted curve is created by B- splines which is unable to produce an exact circle
NURBS share many properties with B-Splines, and when all weights are equal, NURBS simplify to B-Splines However, the derivatives of NURBS are more complex than those of B-Splines, as discussed in Subsection 2.5.2 Key properties of NURBS include their flexibility in representing both standard geometric shapes and freeform curves, making them essential in computer-aided design and graphics.
• For open knot vectors, NURBS basis functions constitute a partition of unity
• The continuity and support of NURBS basis functions are the same as for B- splines
• NURBS are pointwise non-negative
NURBS are capable of accurately representing a broad range of curves, including conic sections, thanks to their strong convex hull property The derivation of NURBS surfaces involves applying a projective transformation to the established equations used for NURBS curves Consequently, NURBS surfaces are defined through this transformation process.
S P (2 14) where NURBS basis functions in parameter space of two dimensions are defined by
(2 15) in which the bivariate weighting function in the denominator is given by
In modeling, the circular plate is a common conic section that can be accurately represented by a NURBS surface, as shown in Figure 2.7 There are two primary methods for parameterizing a circular NURBS surface at a coarse mesh level The first method utilizes eighteen control points, resulting in four elements, while the second method requires only nine control points to create a single element Each parameterization approach has its own singularities; the first method has one singularity at the center where nine control points overlap, whereas the second method has four singularities at specific locations of control points P1, P3, P7, and P9 Typically, the second approach is favored in analysis due to its superior parameterization Another frequently encountered conic section in design is the annular plate, also illustrated in Figure 2.7 This construction features an internal interface, marked by a red line, where the first and last control points in the circumferential direction converge It is crucial to address this interface in analysis and to develop an effective strategy for managing the control variables associated with these control points.
Figure 2 7: Two representations of the same circular plate
Figure 2 8: A annular plate represented by NURBS surface.
Isogeometric discretization
In isogeometric analysis, the solution space is approximated using geometry functions such as B-splines or NURBS, ensuring compatibility between the geometry design and analysis phases through isoparametric discretization This dissertation specifically addresses composite plates, focusing on a two-dimensional geometry represented by a bivariate tensor-product single patch geometry map \( F(\xi, \eta): \Omega \rightarrow \Omega \), where the parameter domain \( \Omega \) is defined as a unit square in \( \mathbb{R}^2 \), \( \Omega = [0, 1]^2 \) The geometric mapping \( F(\xi, \eta) \) effectively translates points from the parameter domain to the physical domain.
= = x F P (2 17) where P i j , are the control point locations in physical space, R i j p q , , ( , ) are the basis functions defined in Eq.(2 15)
Through the isoparametric concept, B-spline or NURBS basis functions are employed to describe the displacement field and the test functions in spatial coordinates x and y as
= = u x u u (2 18) where u i j , is the (i, j)-th component of the vector of control variables u, obtained from the solution of a discretized PDEs system.
Numerical integration
Isogeometric analysis utilizes B-spline/NURBS basis functions to divide the computational domain into elements, with grid size determined by the spans of knots in parameter space This parameter space, defined by knot vectors, can be a line, rectangle, or cube, depending on its dimension, and is accurately mapped onto the corresponding physical geometry The basis functions maintain C∞ continuity within any non-zero knot span, allowing for the numerical integration of key components—such as the mass matrix, stiffness matrix, and load vector—on each non-zero knot span This integration is performed similarly to standard finite element analysis (FEA) using Gaussian quadrature.
Unlike traditional Finite Element Analysis (FEA), which assigns individual parameterizations to each element, the B-spline/NURBS approach utilizes a local parametric space that is specific to patches In FEA, the mapping from parent to physical elements is performed separately for each element Conversely, B-spline/NURBS geometrical mapping encompasses multiple elements within a patch, necessitating two mappings to numerically evaluate the elemental integrals that define the entries of characteristic matrices or vectors.
The process involves two key steps: first, integrals are mapped back to parametric space through a geometric transformation, and second, they are further transformed to the parent space using standard transformation rules, where numerical integration is performed.
To enhance clarity in this presentation, we denote coordinates in physical, parametric, and parent spaces as x, ξ, and ξ, respectively Elements are similarly named in the order of e, e, e Utilizing this notation, we can express the coordinate transformation from parent space to parametric space.
(2 19) and the Jacobian of this transformation is hence given as follows
Figure 2 9: The numerical integration procedure performed in Isogeometric
To demonstrate this integration procedure, let’s take example of the integral of elemental mass matrix which is given by
(2 21) where N is the matrix of basis functions, F ξ ( ) is geometric mapping defined in Eq
The Jacobian matrix \( J_{\xi} \) is defined in Eq (2.20), and the final integral can be computed using standard Gaussian quadrature, utilizing Gauss points where \( p \) and \( q \) represent the orders of B-spline/NURBS basis functions in each parametric direction However, it is important to note that Gaussian numerical quadrature is often sub-optimal, and there are more efficient numerical integration schemes available for Isogeometric Analysis (IGA) Figure 2.9 illustrates these integration mappings.
10 summarizes essential steps to analyze a problem using the isogeometric framework
Figure 2 10: Summary of IGA procedure.
Bézier extraction
The primary aim of IGA using Bézier extraction is to substitute the globally defined B-spline/NURBS basis functions with Bernstein shape functions, utilizing a consistent set of shape functions across each element, similar to traditional finite element methods (FEM) This approach facilitates straightforward integration of conventional finite element processes for smooth, higher-degree basis functions, highlighting the advantages of B-spline/NURBS basis functions in computational applications.
In the realm of structural analysis, traditional finite element methods (FEM) face challenges due to the absence of local domains in global structures, complicating stiffness formulation and numerical computations These computations often require transformation into parent elements, with basis functions defined in parametric space However, by utilizing Bernstein polynomials as basis functions in Bézier extraction, isogeometric analysis (IGA) simplifies computations, mirroring the efficiency of FEM Notably, Bernstein basis functions maintain C0-continuity, similar to Lagrangian shape functions in FEM, enhancing their applicability in structural analysis.
The native implementation of Isogeometric Analysis (IGA) codes presents challenges that complicate their integration into the existing finite element framework A key issue is that this approach requires different B-spline basis functions for each element, unlike traditional Finite Element Analysis (FEA), which utilizes uniform basis functions across all elements However, as noted in Section 2.3.2, B-spline curves can be represented as concatenated C^0 Bézier curves This observation suggests that it is feasible to convert a B-spline patch into a series of piecewise C^0 Bézier elements, thereby allowing for a finite element representation of B-splines or NURBS.
2.9.2 Bézier decomposition and Bézier extraction [97-98]
It follows that the same curve can be described by two equivalent formulas as
The C N P B P (2 22) equation involves vectors of B-spline (N T) and Bézier (B T) basis functions, with their associated control points stored in vectors P and P, respectively The process of extracting individual Bézier curves from a B-spline curve is known as Bézier decomposition This decomposition is typically achieved through knot insertion, where existing knots are added until their multiplicities match the polynomial order, ensuring that the continuity between them is C 0.
Given a knot vector Ξ= 1, 2, , n p + + 1 and a collection of control points
P P which determine a B-spline curve By applying the knot insertion
35 procedure in Eq (2 6) to a set of knots 1, 2, , j , , m that needs to be replicated to produce the Bézier decomposition from a B-spline curve, one can write
P C P (2 23) where P 1 =P Eq (2 23) is the matrix form of Eq (2 6) obtained when inserting a single knot j ,j =1, 2, ,m to the original knot vector which the matrix C j is defined as
By applying the transformation outlined in Eq (2.24) for each inserted knot \( \xi_j \), we obtain the final collection of control points \( P_{m+1} \), which defines the Bézier segments of the decomposition Subsequently, by setting \( P_b = P_{m+1} \) and defining \( C_T = (C_m^T, C_{m-1}^T, C_1^T) \), we can derive the desired results.
The B-spline curve can be expressed as convex linear combinations of control points, represented by the matrix C, known as the Bézier extraction operator, which ensures that the rows sum to unity Notably, the construction of matrix C relies exclusively on a knot vector, making the operator applicable to both B-splines and NURBS By integrating equations (2.22) and (2.25), we can establish a relationship between B-spline basis functions and Bernstein basis functions.
The B-Spline basis functions are derived by multiplying the matrix C with the Bézier basis functions, also known as the Bernstein basis This method simplifies the integration of Isogeometric Analysis (IGA) into existing Finite Element Analysis (FEA) codes by allowing the implementation of an element that uses the Bernstein basis and includes a Bézier extraction matrix C For Non-Uniform Rational B-Splines (NURBS), the application of the extraction operator follows a specific procedure.
The formula of weighting functions defined in Eq.(2 11) can be rewritten in matrix form as
= = N w = CB w = B C w = B w = (2 27) where w b =C w T are the corresponding weights of the Bernstein basis functions Now, one rewrites Eq (2 13) in matrix form as follows:
R WN (2 28) in which Wis the diagonal matrix of control points’ weights defined as
Replacing matrix Nin Eq (2 28) by the relation in Eq (2 26) yields the formula that expresses NURBS basis in terms of Bernstein basis as
The relationship between the NURBS control points, P, and the Bézier control points, P b is defined as
P W C WP (2 31) where W b is the diagonal matrix form of the Bézier weights recast from the vector form w b as
For higher-dimension bases, the extraction operators are straightforwardly defined as the tensor product of the univariate ones.
Concluding remarks
In this chapter, the fundamental developments of Isogeometric Analysis have been addressed which are summarized as follows
IGA aims to eliminate the need for converting CAD files into CAE codes by utilizing the same basis functions of CAD for analysis, streamlining the process and enhancing efficiency.
B-spline basis functions can be efficiently calculated using the Cox-de Boor algorithm, which operates on a specified knot vector Additionally, the derivatives of these functions can be expressed as linear combinations of lower-order basis functions.
A B-spline curve is constructed through a linear combination of basis functions and associated control points, while B-spline surfaces and volumes are similarly defined, utilizing the tensor product structure inherent in B-splines.
B-splines provide three types of mesh refinement: h-refinement, p-refinement, and k-refinement The first two methods are similar to element subdivision and order elevation in finite element analysis (FEA) However, k-refinement is unique to B-splines, enhancing interelement continuity significantly.
NURBS in d-dimensional space are characterized by conic projecting B-splines in d+1 dimensions, where the coordinates of the additional dimension represent strictly positive weights This transformation enables the precise representation of conic sections.
• NURBS geometry therefore is defined similarly as B-spline one
Numerical integration in NURBS-based Isogeometric Analysis (IGA) involves a two-step mapping process The initial mapping transitions from the natural or parent space to the parametric space, followed by a subsequent mapping that converts the parametric space into the physical space.
• Since the same B-spline/NURBS curve can be represented by concatenated Bézier curves, one can decompose the B-spline/NURBS curve into several C 0
Bézier elements for using in the analysis This procedure makes the IGA approach backward compatible with conventional FEM codes
Overview
This chapter provides a comprehensive overview of plate theories and outlines the specific theories applicable to subsequent chapters It details the material properties of key subjects, including laminated composite plates, piezoelectric laminated composite plates, functionally graded porous plates reinforced with graphene platelets embedded in piezoelectric layers, and functionally graded piezoelectric material porous plates.
An overview of plate theories
Various plate theories are utilized to analyze the behavior of laminated composite plates, with three-dimensional (3D) elasticity theory initially proposed to accurately predict static problems and assess transverse shear stresses However, this approach incurs high computational costs, particularly for thin structures, and struggles with complex geometries and boundary conditions To address these limitations, equivalent single layer (ESL) plate theories have been developed, transforming 3D problems into quasi-2D scenarios by examining stress and deformation kinematics through the plate thickness This thesis employs ESL plate theories for calculating and simulating laminated composite plate behavior due to their simplicity and lower computational demands Notable ESL models include classical plate theory (CPT), first-order shear deformation theory (FSDT), and higher-order shear deformation theories (HSDTs).
3.2.1 The higher-order shear deformation theory
3.2.1.1 The third-order shear deformation theory
Higher-order shear deformation theories (HSDTs) have been developed to avoid the necessity of a shear correction factor (SCF) and to facilitate the smooth distribution of shear stress in structural analysis Among these, the third-order shear deformation theory (TSDT) introduced by Reddy is the most widely utilized, particularly for plates and shells with varying thicknesses TSDT achieves a parabolic distribution of shear stresses across the plate thickness, eliminating the need for SCF An illustration comparing the undeformed and deformed configurations of transverse normal using classical plate theory (CLPT), first-order shear deformation theory (FSDT), and TSDT is presented in Figure 3.1.
The displacement field can be expressed as follows
The TSDT satisfies the disappearing of transverse shear stresses in the top and bottom surfaces of plates and provides more accurate inter-laminar stress distributions than FSDT
Figure 3 1 Deformation of transverse normal using CLPT, FSDT and TSDT
3.2.1.2 The generalized higher-order shear deformation theory
Numerous higher-order shear deformation theories have been developed to effectively analyze and simulate the behavior of laminated composite plates Among these, the most widely recognized theory, as referenced in works [18-19, 115-117], is derived from classical plate theory.
(3 2) where f(z) is a shape function which determines the distribution of the transverse shear strains and stresses across the thickness of the plates and it is chosen so that the
The shear stress-free surface conditions on the top and bottom surfaces of the plate are met, indicating that f z'( )=0 at z= ±h/2 Various distributed functions f(z) that satisfy this condition are listed in Table 3.1 The current theory can be readily adapted to other theories presented in Table 3.1 by selecting the appropriate distribution function f(z) along the thickness of the plate.
Table 3 1: The various forms of shape function
Figure 3.2 illustrates the shapes of the function f(z) and its derivative f’(z) in relation to the thickness of the plates, demonstrating that both functions satisfy the free stress conditions at the top and bottom of the plates.
Figure 3 2 Distribution function f(z) and its derivation g(z) versus the thickness of the plates
In Eq (3 2), if distribution function f(z) is equal to zero, the higher-order shear deformation theory will take the form of classical plate theory (see in Eq.(3 3) )
By defining f(z) = z and substituting x = − +w , x x for Eq (3 4), the first- order shear deformation theory (FSDT) is obtained as
3.2.2 The generalized unconstrained higher-order shear deformation theory (UHSDT)
The Third-Order Shear Deformation Theory (TSDT) incorporates cubic variations of in-plane displacements, constrained by transverse displacements and rotations, while inaccurately assuming that transverse shear stresses are negligible at the plate's top and bottom surfaces To address the issue of shear traction parallel to plate surfaces, Leung introduced the Unconstrained Third-Order Shear Deformation Theory (UTSDT), which offers a more accurate approach Moreover, UTSDT is applicable in various scenarios, enhancing its utility in structural analysis.
The Unconstrained Third-Order Shear Deformation Theory (UTSDT) addresses 44 problems related to contact friction and flow fields, differing from Reddy's Traction-Free Shear Deformation Theory (TSDT) by permitting finite transverse shear strain on both the upper and lower plate surfaces While the governing differential equations of UTSDT share a similar complexity with those of TSDT, UTSDT provides more accurate solutions when compared to the three-dimensional exact solution This advanced theory incorporates seven displacement components, consisting of six in-plane displacements and one transverse displacement, enhancing its applicability in engineering analyses.
This thesis introduces a novel unconstrained higher-order shear deformation theory (UHSDT) utilized for calculations in Chapter 4 While UHSDT incorporates seven displacement components akin to those in the unified theory of shear deformation theory (UTSDT), its higher-order rotations are influenced by an arbitrary function f(z) throughout the plate thickness.
In the context of unconstrained higher-order shear deformation theory (UHSDT), the third-order function \( f(z) = z^3 \) is utilized, illustrating that the shear stress distribution across plate thickness is influenced by factors such as the number of layers, layer thickness, and material properties This leads to the development of a generalized UHSDT that effectively captures nonlinear behavior throughout the plate thickness, offering improved solutions compared to traditional third-order shear deformation theory (UTSDT) Consequently, this prompts a deeper exploration of UHSDT for enhanced analytical outcomes.
The unconstrained theory based on HSDT can be rewritten in a general form using an arbitrary function f(z) as follows:
( ) and w x y t, , are seven displacement variables which must be determined
This article introduces two newly proposed shape functions, along with the shape functions of the Uniformly Transverse Shear Deformation Theory (UTSDT), emphasizing the distributed function f(z) across the plate thickness These shape functions effectively capture the nonlinear behavior throughout the plate thickness, leading to improved analytical results.
In the context of UTSDT, f(z) is defined as a continuous function with a nonlinear first derivative across the plate thickness An optimal function, denoted as f_op(z), exists that provides the most accurate solutions, yet identifying f_op(z) continues to be an unresolved issue Two specific functions f(z) are presented for further illustration in Table 3.2.
Table 3 2: Three used forms of distributed functions and their derivatives
3.2.3 The C 0 -type higher-order shear deformation theory (C 0 -type HSDT)
The discussed theories necessitate C 0 -continuity and C 1 -continuity in the approximate or generalized displacement fields Both the Higher-Order Shear Deformation Theory (HSDT) and the Classical Plate Theory (CPT) are linked to the derivation of transverse displacement, also known as slope components.
In some numerical methods, it is often difficult to enforce boundary conditions for slope components due to the unification of the approximation variables Therefore, a
C 0 -type HSDT is rather recommended [119-120]
In this thesis, the authors promote a C 0 -type HSDT for PFGP-GPLs and FGPMP plates shown in chapters 5 and 6 This theory contributes to increase the novelty of the dissertation
According to the generalized higher-order shear deformation theory in Eq.(3
2), the displacement field of any points in the plate has five unknowns and can be rewritten by:
In the equation (3.7), the variables u0, v0, and w0 represent the in-plane and transverse displacements, while θx and θy indicate the rotation components in the y-z and x-z planes, respectively The notation ',x' and ',y' signifies the derivative of any function with respect to the x and y directions.
To avoid the order of high-order derivation in approximate formulations and easily apply boundary conditions similar to the standard finite element procedure, additional assumptions are made as follows:
Substituting Eq (3 8) to Eq.(3 7), it can be written:
0 0 0 u = u v w T ; u = − x y 0 T ; u = x y 0 T (3 9) From Eq (3 9), it can be seen that the compatible strain fields only request
C 0 -continuity This theory is named as the C 0 -type higher-order shear deformation theory
Based on the C 0 -type higher-order shear deformation theory, the bending and shear strains are expressed by:
(3 11) in which f z ( )is the derivation of the function f(z) which is chosen later.
Laminated composite plate
3.3.1 Definition of laminated composite plate
A composite material, often referred to as a composite, is made up of two or more distinct materials, including a matrix and reinforcing fibers This unique combination results in a new material that exhibits superior mechanical properties compared to its individual components, such as enhanced strength-to-weight and stiffness-to-weight ratios, as well as improved wear resistance and reduced weight.
Laminated composites consist of multiple layers, or lamina, made from various materials, which are combined to achieve desired mechanical properties The process of creating a laminated composite typically involves assembling three distinct lamina layers, each incorporating different types of fibers.
Figure 3 3 Configuration of a lamina and laminated composite plate
To set the basic equations of the building block of a composite laminate, there are some assumptions as:
• a lamina is a continuum; i.e., no empty spaces or holes exist
• a lamina behaves as a linear elastic material; i.e., Hooke's law is used
3.3.2 Constitutive equations of laminated composite plate
The generalized Hooke's law for an anisotropic material is expressed by: i Q ij j
48 where i are the stress components, j are the strain components and Q ij are the
In the context of a 2D problem, the "reduced" material coefficients, denoted as Q_ij, correspond to the components of an orthogonal Cartesian coordinate system (x1, x2, x3) Typically, there are 21 independent elastic constants for these coefficients However, for orthotropic materials in three-dimensional scenarios, the number of material parameters is simplified to 9.
Figure 3.4 depicts the material coordinate system (x₁, x₂, x₃), where the x₁ axis aligns with the fiber direction, the x₂ axis is oriented transversely within the lamina plane, and the x₃ axis extends perpendicular to the lamina plane.
Figure 3 4 Configuration of a lamina and laminated composite plate
Using rule of mixture, the lamina constants are defined as follows
In laminated composites, the parameters E_f, E_m, ν_f, ν_m, φ_f, φ_m, and G_f, G_m represent the Young's moduli, Poisson's ratios, volume fractions, and shear moduli of the fiber and matrix materials, respectively The shear moduli G_f and G_m are derived from these fundamental properties, facilitating a comprehensive understanding of the composite's mechanical behavior.
By neglecting z for each orthotropic layer, the constitutive equation of k th layer in the local coordinate system derived from Hooke’s law for a plane stress is given by
(3 15) in which reduced stiffness components, Q ij k , are expressed by
The stress - strain relationship in the global reference system (x,y,z) is computed by
0 0 0 k k k xx xx k k yy yy k k xy xy k k xz xz k k yz yz
Figure 3 5 Material and global coordinates of the composite plate. where Q ij k is the transformed material constant matrix and is written in detail as:
4 sin cos sin cos sin 2 2 sin cos cos
2 2 sin cos sin cos cos sin sin cos cos sin
Local and global coordinates of the laminated composite is shown in Figure 3 5.
Piezoelectric material
Piezoelectric materials, particularly Lead Zirconate Titanate (PZT) and Polyvinylidene Fluoride (PVDF), are among the most recognized smart materials, playing a crucial role in intelligent structures equipped with sensors and actuators Their applications span structural health monitoring, vibration and noise suppression, and precision positioning, highlighting their significance in various engineering fields The piezoelectric effect facilitates the conversion between electrical and mechanical energy, enabling mechanical deformation when an electric field is applied PZT, with a recoverable strain of 0.1%, is widely utilized as both an actuator and sensor across a broad frequency range, while PVDF, recognized since the late 1960s and commercially available since the early 1980s, is primarily employed as a sensor The study of these piezoelectric materials has gained considerable traction over the past three decades, underscoring their importance in advancing technology.
Piezoelectric materials, initially developed as sea-level transducers using ultrasound waves by Paul Langevin and his team in France, have since found diverse applications These include ultrasonic transducers for sonar and medical imaging, compact piezoelectric motors, structural monitoring systems, active damping elements, and ignition systems.
3.4.2 The basic equation of piezoelectric material
This section introduces the electro-mechanical equations governing piezoelectric materials, grounded in IEEE standards These equations effectively describe the properties of piezoelectric materials, which are assumed to exhibit linear behavior In this dissertation, both mechanical stress and electric field are considered linear, aligning with the established IEEE guidelines.
The linear piezoelectric constitutive equations can be expressed as follow [125-
(3 19) where and σ are the strain vector and the stress vector, respectively; c denotes the elastic constant matrix The electric field vector E, can be defined as grad
In this study, we examine specific piezoelectric materials, focusing on the formulation of stress piezoelectric constant matrices (e), strain piezoelectric constant matrices (d), and dielectric constant matrices (g), as outlined in reference [41].
Piezoelectric functionally graded porous plates reinforced by graphene
This study presents a sandwich plate model characterized by a length \( a \), width \( b \), and a total thickness \( h = h_c + 2h_p \), where \( h_c \) represents the thickness of the porous core layer and \( h_p \) denotes the thickness of the piezoelectric face layers, as illustrated in Figure 3.6.
Figure 3 6 Configuration of a piezoelectric FG porous plate reinforced by
The article presents three distinct types of porosity distribution along the thickness of plates, which include two non-uniformly symmetric types and one uniform type, as depicted in Figure 3.7 Furthermore, it explores three different GPL dispersion patterns, illustrated in Figure 3.8, for each porosity distribution, focusing on the GPL volume fraction in each pattern.
In the analysis of non-uniformly distributed porous materials without graded porous layers (GPLs), the Young's moduli are represented by E1' for the maximum and E2' for the minimum values In contrast, E' indicates the Young's modulus for a uniform porosity distribution This variation in Young's moduli reflects the smooth gradation of properties along the thickness direction.
(a) Non-uniform porosity distribution 1 (b) Non-uniform porosity distribution 2
Figure 3 8 Three dispersion patterns 𝐴, 𝐵 and 𝐶 of the GPLs for each porosity distribution type [127]
The material properties including Young’s moduli 𝐸(𝑧), shear modulus 𝐺(𝑧) and mass density 𝜌(𝑧) which alter along the thickness direction for different porosity distribution types can be expressed as
Non uniform porosity distribution 1 Non uniform porosity distribution 2 Uniform porosity distribution cos( / ), ( ) cos( / 2 / 4),
In the study of porosity distribution, it is established that for non-uniform and uniform types, the relationships 𝐸 1 = 𝐸 1 ′ and 𝐸 1 = 𝐸 ′ hold true The maximum mass density of the porous core is represented by 𝜌 1, while the coefficient of porosity, denoted as 𝑒 0, can be accurately determined through specific calculations.
Through Gaussian Random Field (GRF) scheme [40], the mechanical characteristic of closed‐ cell cellular solids is given as
Then, the coefficient of mass density 𝑒 𝑚 in Eq (3 22) is possibly stated as
Also according to the closed‐cell GRF scheme [128], Poisson’s ratio 𝜈(𝑧) is derived as
( ) 0.221 1(0.342 1.21 1), v z = p +v p − p + (3 27) in which 𝜈 1 represents the Poisson’s ratio of the metal matrix without internal pores and 𝑝 ′ is given as
To ensure a meaningful and fair comparison of FG porous plates with varying porosity distributions, it is essential to equate the mass per unit surface area (𝑀) of these plates This equivalence can be calculated accurately, allowing for a consistent evaluation across different designs.
Then, the coefficient in Eq (3 23) for uniform porosity distribution can be defined as
The volume fraction of GPLs alters along the thickness of the plate for three dispersion patterns depicted in Figure 3 8 can be given as
(3 31) where 𝑆 𝑖1 , 𝑆 𝑖2 and 𝑆 𝑖3 are the maximum values of GPL volume fraction and 𝑖 = 1,2,3 corresponds to two non‐uniform porosity distributions 1, 2 and the uniform distribution, respectively
The relationship between the volume fraction 𝑉 𝐺𝑃𝐿 and weight fractions 𝛬 𝐺𝑃𝐿 is given by
GPL m GPL GPL GPL e z dz V e z dz
By the Halpin‐Tsai micromechanical model [129-131], Young’s modulus 𝐸 1 is determined as w w
The average dimensions of Graphene Platelet Reinforcements (GPLs) are represented by their width (𝑤 𝐺𝑃𝐿), length (𝑙 𝐺𝑃𝐿), and thickness (𝑡 𝐺𝑃𝐿) The Young’s moduli of the GPLs (𝐸 𝐺𝑃𝐿) and the metal matrix (𝐸 𝑚) are crucial for understanding their mechanical properties Additionally, the mass density (𝜌 1) of the GPLs can be calculated, and Poisson’s ratio (𝜈 1) for the GPLs reinforced within a porous metal matrix can be derived using the rule of mixtures.
= + (3 36) where 𝜌 𝐺𝑃𝐿 , 𝜈 𝐺𝑃𝐿 and 𝑉 𝐺𝑃𝐿 are the mass density, Poisson’s ratio and volume fraction of GPLs, respectively; while 𝜌 𝑚 , 𝜈 𝑚 and 𝑉 𝑚 = 1 − 𝑉 𝐺𝑃𝐿 represent the mass density, Poisson’s ratio and volume fraction of metal matrix, respectively.
Functionally graded piezoelectric material porous plates (FGPMP)
Consider a FGPMP plate with the length a, the width b and the thickness h
The study focuses on a plate composed of a mixture of two piezoelectric materials, PZT-4 and PZT-5H, subjected to an electric potential, as illustrated in Figure 3.9 The materials are positioned with material 1 at the top surface (z=h/2) and material 2 at the bottom surface (z=-h/2) Two types of functionally graded piezoelectric porous plates, FGPMP-I and FGPMP-II, are analyzed For the evenly distributed FGPMP-I, the effective material properties of the piezoelectric porous plates are determined using a modified power-law model.
, 1,1 , 1, 2 , 1,3 , 3,3 , 5,5 , 6,6 g u l l u l ij ij ij ij ij ij c z c c z c c c h i j
2 2 g u l l u l ij ij ij ij ij ij e z e e z e e e i j h
2 2 g u l l u l ij ij ij ij ij ij k z k k z k k k i j h
(3.37) where c ij , e ij and k ij are defined as above, g is the power index that represents the material distribution across the plate thickness, is the material density; the symbols
57 u and l denote the material properties of the upper (material 1) and lower surfaces
(material 2), respectively, and is the porosity volume fraction
In the FGPMP-II model, the uneven distribution of porosity is characterized by a concentration around the middle surface of the cross-section, with diminished porosity observed at both the top and bottom To determine the effective material properties, specific computational methods are applied.
, 1,1 , 1, 2 , 1,3 , 3,3 , 5,5 , 6, 6 g u l l u l ij ij ij ij ij ij z z c z c c c c c h h i j
2 2 g u l l u l ij ij ij ij ij ij z z e z e e e e e i j h h
2 2 g u l l u l ij ij ij ij ij ij z z k z k k k k k i j h h
Figure 3.9 Geometry and cross sections of a FGPMP plate made of PZT-4/PZT-5H
The study examines the impact of porosity volume fraction on the elastic coefficient \( c_{11} \) of porous functionally graded piezoelectric materials (FGPM) plates made from PZT-4 and PZT-5H, highlighting variations with thickness and power index values For a perfect FGPM with \( \alpha=0 \), the elastic coefficient remains continuous from the PZT-4 rich top surface to the PZT-5H rich bottom surface In contrast, for the porous FGPMP-I type, the elastic coefficient profile mirrors that of the perfect FGPM, but with reduced stiffness due to porosity The FGPMP-II type exhibits maximum elastic coefficients at the top and bottom surfaces, decreasing toward the mid-zone, indicating a unique distribution of porosity Ultimately, the elastic coefficient amplitude of the FGPMP-II plate matches that of the perfect FGPM at the surfaces and aligns with the FGPMP-I plate at the mid-surface.
Concluding remarks
This chapter provides a comprehensive overview of plate theories that will be referenced in subsequent chapters, along with essential information on various materials These include laminated composite plates, piezoelectric laminated composite plates, piezoelectric functionally graded porous plates reinforced with graphene platelets, and functionally graded piezoelectric material porous plates The discussion covers different types of functionally graded porous materials, specifically Perfect FGPM, FGPMP-I, FGPMP-II, and FGPMP with a grading factor of g=0.1.
Figure 3.10 Variation of elastic coefficient c 11 of FGPMP plate made of PZT-