Unlike published shell theories, the main two parts of this thesis are devoted to the asymptotic construction of a refined theory for composite laminated shells valid over a wide range of frequencies and wavelengths. The resulting theory is applicable to shells each layer of which is made of materials with monoclinic symmetry. It enables one to analyze shell dynamic responses within both long-wavelength, low- and high-frequency vibration regimes. It also leads to energy functionals that are both positive definiteness and sufficient simplicity for all wavelengths. This whole procedure was first performed analytically. From the insight gained from the procedure, a finite element version of the analysis was then developed; and a corresponding computer program, DVAPAS, was developed. DVAPAS can obtain the generalized 2-D constitutive law and recover accurately the 3-D results for stress and strain in composite shells. Some independent works will be needed to develop the corresponding 2-D surface analysis associated with the present theory and to continue towards full verification and validation of the present process by comparison with available published works.

A series of composite beam models are constructed for efficient high-fidelity beam analysis based on the variational-asymptotic method (VAM). Without invoking any a priori kinematic assumptions, the original three-dimensional, geometrically nonlinear beam problem is rigorously split into a two-dimensional cross-sectional analysis and a one-dimensional global beam analysis, taking advantage of the geometric small parameter that is an inherent property of the structure. The thermal problem of composite beams is studied first. According to the quasisteady theory of thermoelasticity, two beam models are proposed: one for heat conduction analysis and the other for thermoelastic analysis. For heat conduction analysis, two different types of thermal loads are modeled: with and without prescribed temperatures over the crosssections. Then a thermoelastic beam model is constructed under the previously solved thermal field. This model is also extended for composite materials, which removed the restriction on temperature variations and added the dependence of material properties with respect to temperature based on Kovalenoko’s small-strain thermoelasticity theory. Next the VAM is applied to model the multiphysics behavior of beam structure. A multiphysics beam model is proposed to capture the piezoelectric, piezomagnetic, pyroelectric, pyromagnetic, and hygrothermal effects. For the zeroth-order approximation, the classical models are in the form of Euler-Bernoulli beam theory. In the refined theory, generalized Timoshenko models have been developed, including two transverse shear strain measures. In order to avoid ill-conditioned matrices, a scaling method for multiphysics modeling is also presented. Three-dimensional field quantities are recovered from the one-dimensional variables obtained from the global beam analysis. A number of numerical examples of different beams are given to demonstrate the application and accuracy of the present theory. Excellent agreements between the results obtained by the current models and those obtained by three-dimensional finite element analysis, analytical solutions, and those available in the literature can be observed for all the cross-sectional variables. The present beam theory has been implemented into the computer program VABS (Variational Asymptotic Beam Sectional Analysis).

The issue of accurately determining the effective properties of composite materials has received the attention of numerous researchers in the last few decades and continues to be in the forefront of material research. Micromechanics models have been proven to be very useful tools for design and analysis of composite materials. In the present work, a versatile micromechanics modeling framework, namely, the Variational Asymptotic Method for Unit Cell Homogenization (VAMUCH), has been invented and various micromechancis models have been constructed in light of this novel framework. Considering the periodicity as a small parameter, we can formulate the variational statements of the unit cell through an asymptotic expansion of the energy functional. It is shown that the governing differential equations and periodic boundary conditions of mathematical homogenization theories (MHT) can be reproduced from this variational statement. Finally, we employed the finite element method to solve the numerical solution of the constrained minimization problem. If the local fields within the unit cell are of interest, the proposed models can also accurately recover those fields based on the global behavior. In comparison to other existing models, the advantages of VAMUCH are: (1) it invokes only two essential assumptions within the concept of micromechanics for heterogeneous material with identifiable unit cells; (2) it has an inherent variational nature and its numerical implementation is shown to be straightforward; (3) it calculates the different material properties in different directions simultaneously, which is more efficient than those approaches requiring multiple runs under different loading conditions; and (4) it calculates the effective properties and the local fields directly with the same accuracy as the fluctuation functions. No postprocessing calculations such as stress averaging and strain averaging are needed. The present theory is implemented in the computer program VAMUCH, a versatile engineering code for the homogenization of heterogeneous materials. This new micromechanics modeling approach has been successfully applied to predict the effective properties of composite materials including elastic properties, coefficients of thermal expansion, and specific heat and the effective properties of piezoelectric and electro-magneto-elastic composites. This approach has also been extended to the prediction of the nonlinear response of multiphase composites. Numerous examples have been utilized to clearly demonstrate its application and accuracy as a general-purpose micromechanical analysis tool.

Rotor (helicopter/wind turbine) blades are typically slender structures that can be modeled as beams. Beam modeling, however, involves a substantial mathematical formulation that ultimately helps save computational costs. A beam theory for rotor blades must account for (i) initial twist and/or curvature, (ii) inclusion of composite materials, (iii) large displacements and rotations; and be capable of offering significant computational savings compared to a non-linear 3D FEA (Finite Element Analysis). The mathematical foundation of the current effort is the Variational Asymptotic Method (VAM), which is used to rigorously reduce the 3D problem into a 1D or beam problem, i.e., perform a cross-sectional analysis, without any ad hoc assumptions regarding the deformation. Since its inception, the VAM based cross-sectional analysis problem has been in a constant state of flux to expand its horizons and increase its potency; and this is precisely the target at which the objectives of this work are aimed. The problems addressed are the stress-strain-displacement recovery for spanwise non-uniform beams, analytical verification studies for the initial curvature effect, higher fidelity stress-strain-displacement recovery, oblique cross-sectional analysis, modeling of thin-walled beams considering the interaction of small parameters and the analysis of plates of variable thickness. The following are the chief conclusions that can be drawn from this work: 1. In accurately determining the stress, strain and displacement of a spanwise non-uniform beam, an analysis which accounts for the tilting of the normal and the subsequent modification of the stress-traction boundary conditions is required. 2. Asymptotic expansion of the metric tensor of the undeformed state and its powers are needed to capture the stiffnesses of curved beams in tune with elasticity theory. Further improvements in the stiffness matrix can be achieved by a partial transformation to the Generalized Timoshenko theory. 3. For the planar deformation of curved laminated strip-beams, closed-form analytical expressions can be generated for the stiffness matrix and recovery; further certain beam stiffnesses can be extracted not only by a direct 3D to 1D dimensional reduction, but a sequential dimensional reduction, the intermediate being a plate theory. 4. Evaluation of the second-order warping allows for a higher fidelity extraction of stress, strain and displacement with negligible additional computational costs. 5. The definition of a cross section has been expanded to include surfaces which need not be perpendicular to the reference line. 6. Analysis of thin-walled rotor blade segments using asymptotic methods should consider a small parameter associated with the wall thickness; further the analysis procedure can be initiated from a laminated shell theory instead of 3D. 7. Structural analysis of plates of variable thickness involves an 8×8 plate stiffness matrix and 3D recovery which explicitly depend on the parameters describing the thickness, in contrast to the simplistic and erroneous approach of replacing the thickness by its variation.

An asymptotically-exact methodology is presented for obtaining the cross-sectional stiffness matrix of a pre-twisted, moderately-thick beam having rectangular cross sections and made of transversely isotropic material. The beam is modelled with-out assumptions from 3-D elasticity. The strain energy of the beam is computed making use of the constitutive law and the kinematical relations derived with the inclusion of geometrical nonlinearities and initial twist. Large displacements and rotations are allowed, but small strain is assumed. The Variational Asymptotic Method (VAM) is used to minimize the energy functional, thereby reducing the cross section to a point on the reference line with appropriate properties, yielding a 1-D constitutive law. In this method as applied herein, the 2-D cross-sectional analysis is performed asymptotically by taking advantage of a material small parameter and two geometric small parameters. 3-D strain components are derived using kinematics and arranged as orders of the small parameters. Warping functions are obtained by the minimisation of strain energy subject to certain set of constraints that renders the 1-D strain measures well-defined. Closed-form expressions are derived for the 3-D non-linear warping and stress fields. The model is capable of predicting interlaminar and transverse shear stresses accurately up to first order.

Composites are materials which cater to the present and future needs of many demanding industries, such as aerospace, as they are weight-sensitive for a given requirement of strength and stiff ness, corrosion resistant, potentially multi-functional and can be tailored according to the application. However, they are in particular difficult to join as they cannot be easily machined, without introducing damages which can eventually grow. Any structure is as strong as its weakest joint. Most of the joints belong to the category of mechanically-fastened joints and they pose enormous challenges in modeling due to contact phenomena, nonlinearity and stress concentration factors. It is therefore a necessity to construct an efficient model that would include all the relevant contact phenomena in the joints, as it has been pointed out in literature that damage typically initiates near the joint holes. The focus of this work is to describe the construction of an asymptotically-correct model using the Variational Asymptotic Method (VAM). Amongst its many potential applications, VAM is a well-established analytical tool for obtaining the stress and strain fields for beams and shells. The methodology takes advantage of the small parameter that is inherent in the problem, such as the ratio of certain characteristic dimensions of the structure. In shells and beams, VAM takes advantage of the dimension-based small parameter(s), thereby splitting the problem into 2-D + 1-D (for beams) and 1-D + 2-D (for shells), in turn offering very high computational efficiency with very little loss of accuracy compared to dimensionally unreduced 3-D models. In this work, the applicability of VAM is extended to two-dimensional (2-D) and three-dimensional (3-D) frictionless contact problems. Since a generalised VAM model for contact has not been pursued before, the `phantom0 step is adopted for both 2-D and 3-D models. The development of the present work starts with the construction of a 2-D model involving a large rectangular plate being pressed against a rigid frictionless pin. The differential equations governing the problem and the associated boundary conditions are obtained by minimizing the reduced strain energy, augmented with the appropriate gap function, by using a penalty method. The model is developed for both isotropic and orthotropic cases. The boundary value problem is solved numerically and the displacement field obtained is compared with the one obtained using commercial software (ABAQUSr) for validation at critical regions such as the contact surfaces. Banking on the validation of the 2-D model, a 3-D model with a pin and a finite annular cylinder was constructed. The strain energy for the finite cylinder was derived using geometrically exact 3-D kinematics and VAM was applied leading to the reduction in the strain energy for isotropic and orthotropic materials in rectangular and cylindrical co-ordinates. As in the 2-D case, the reduced strain energy, subject to the inequality constraint of the gap function, is minimized with respect to the displacement field and the corresponding boundary value problem is solved numerically. The displacements of the contact surface and the top surface of the annular cylinder are compared with those from ABAQUS and thus validated. The displacement fields obtained using the current 2-D and 3-D models show very good agreement with those from commercial finite element software packages. The model could be re ned further by using the gap function derived in this work and applying it to a plate model based on VAM, which could be explored in the future.

No description available.

In aerospace applications, use of laminates made of composite materials as face sheets in sandwich panels are on the rise. These composite laminates have low transverse shear and transverse normal moduli compared to the in-plane moduli. It is also seen that the corresponding transverse strength values are very low compared to the in-plane strength leading to delaminations. Further, in sandwich structures, the core is subjected to significant transverse shear stresses. Therefore the interlaminar stresses (i.e., transverse shear and normal) can govern the design of sandwich structures. As a consequence, the first step in achieving efficient designs is to develop the ability to reliably estimate interlaminar stresses. Stress analysis of the composite sandwich structures can be carried out using 3-D finite elements for each layer. Owing to the enormous computational time and resource requirements for such a model, this process of analysis is rendered inefficient. On the other hand, existing plate/shell finite elements, when appropriately chosen, can help quickly predict the 2-D displacements with reasonable accuracy. However, their ability to calculate the thickness-wise distributions of interlaminar shear and normal stresses and 3-D displacements remains as a research goal. Frequently, incremental refinements are offered over existing solutions. In this scenario, an asymptotically correct dimensional reduction from 3-D to 2-D, if possible, would serve to benchmark any ongoing research. The employment of a mathematical technique called the Variational Asymptotic Method (VAM) ensures the asymptotical correctness for this purpose. In plates and sandwich structures, it is typically possible to identify (purely from the defined material distributions and geometry) certain parameters as small compared to others. These characteristics are invoked by VAM to derive an asymptotically correct theory. Hence, the 3-D problem of plates is automatically decomposed into two separate problems (namely 1-D+2-D), which then exchange relevant information between each other in both ways. The through-the-thickness analysis of the plate, which is a 1-D analysis, provides asymptotic closed form solutions for the 2-D stiffness as well as the recovery relations (3-D warping field and displacements in terms of standard plate variables). This is followed by a 2-D plate analysis using the results of the 1-D analysis. Finally, the recovery relations regenerate all the required 3-D results. Thus, this method of developing reduced models involves neither ad hoc kinematic assumptions nor any need for shear correction factors as post-processing or curve-fitting measures. The results are most general and can be made as accurate as desired, while the procedure is computationally efficient. In the present work, an asymptotically correct plate theory is formulated for composite sandwich structures. In developing this theory, in addition to the small parameters (such as small strains, small thickness-to-wavelength ratios etc.,) pertaining to the general plate theory, additional small parameters characterizing (and specific to) sandwich structures (viz., smallness of the thickness of facial layers com-pared to that of the core and smallness of elastic material stiffness of the core in relation to that of the facesheets) are used in the formulation. The present approach also satisfies the interlaminar displacement continuity and transverse equilibrium requirements as demanded by the exact 3-D formulation. Based on the derived theory, numerical codes are developed in-house. The results are obtained for a typical sandwich panel subjected to mechanical loading. The 3-D displacements, inter-laminar normal and shear stress distributions are obtained. The results are compared with 3-D elasticity solutions as well as with the results obtained using 3-D finite elements in MSC NASTRAN®. The results show good agreement in spite of the major reduction in computational effort. The formulation is then extended for thermo-elastic deformations of a sandwich panel. This thesis is organized chronologically in terms of the objectives accomplished during the current research. The thesis is organized into six chapters. A brief organization of the thesis is presented below. Chapter-1 briefly reviews the motivation for the stress analysis of sandwich structures with composite facesheets. It provides a literature survey on the stress analysis of composite laminates and sandwich plate structures. The drawbacks of the existing anlaytical approaches as opposed to that of the VAM are brought out. Finally, it concludes by listing the main contributions of this research. Chapter-2 is dedicated to an overview of the 3-D elasticity formulation of composite sandwich structures. It starts with the 3-D description of a material point on a structural plate in the undeformed and deformed configurations. Further, the development of the associated 3-D strain field is also described. It ends with the formulation of the potential energy of the sandwich plate structure. Chapter-3 develops the asymptotically correct theory for composite sandwich plate structure. The mathematical description of VAM and the procedure involved in developing the dimensionally reduciable structural models from 3-D elasticity functional is first described. The 1-D through-the-thickness analysis procedure followed in developing the 2-D plate model of the composite sandwich structure is then presented. Finally, the recovery relations (which are one of the important results from 1-D through-the-thickness analysis) to extract 3-D responses of the structure are obtained. The developed formulation is applied to various problems listed in chapter 4. The first section of this chapter presents the validation study of the present formulation with available 3-D elasticity solutions. Here, composite sandwich plates for various length to depth ratios are correlated with available 3-D elasticity solutions as given in [23]. Lastly, the distributions of 3-D strains, stresses and displacements along the thickness for various loadings of a typical sandwich plate structure are correlated with corresponding solutions using well established 3-D finite elements of MSC NASTRAN® commerical FE software. The developed and validated formulation of composite sandwich structure for mechanical loading is extended for thermo-elastic deformations. The first sections of this chapter describes the seamless inclusion of thermo-elastic strains into the 3-D elasticity formulation. This is followed by the 1-D through-the-thickness analysis in developing the 2-D plate model. Finally, it concludes with the validation of the present formulation for a very general thermal loading (having variation in all the three co-ordinate axes) by correlating the results from the present theory with that of the corresponding solutions of 3-D finite elements of MSC NASTRAN® FE commercial software. Chapter-6 summarises the conclusions of this thesis and recommendations for future work.

Composite materials show heterogeneity at different length scales. hence concurrent multiscale analysis is the only reliable method to analyze them. But unfortunately there is no concurrent multi-scale strategy that is efficient, and accurate while addressing all kinds of problems. This lack of reliability is partly because there is no micro-mechanical model which inherently keeps all relevent global information with it. This thesis tries to fill this gap. The presented micro-mechanical model not only homogenizes the micro-structure but also keeps the global information with it. Most of the micro-mechanical models in the literature extract the Representative Volume Element (RVE) from the continuum for analysis which results in loss of information and accuracy. In the present approach also, the RVE has been extracted from the continuum but with the major difference that all the macro/meso-scopic parameters are accounted for. Five macro/meso-scopic one dimensional parameters have been defined which completely define the effect of continuum. 11 for one dimensional stretch, _1 for torsion, __ (_ = 2, 3) for bending and _33 for uniform pressurization due to the presence of the continuum. Further, the above macro/meso-scopic parameters are proven, by the asymptotic, theory to be constant at a cross section but vary, in general, over the length of the fiber. Hence, the analysis is valid for any location and is not restricted to any local domain. Three major problems have been addressed: • Homogenization and analysis of RVE without any defects • Homogenization and analysis of RVE with fiber-matrix de-bonding • Homogenization and analysis of RVE with radial matrix cracking. Variational Asymptotic Method (VAM) has been used to solve the above mentioned problems analytically. The results have been compared against standard results in the literature and against 3D FEA. At the end, results for “Radial deformation due to torsion” problem will be presented which was solved “accidentally.”

Quick and accurate predictions of equivalent properties for thin-walled composite structures are required in the preliminary design process. Existing literature provides analytical solutions to some structures but is limited to particular cases. No unified approach exists to tackle homogenization of thin-walled structures such as beams, plates, or three-dimensional structures using the thin-walled approximation. In this work, a unified approach is proposed to obtain equivalent properties for beams, plates, and three-dimensional structures for thin-walled composite structures using mechanics of structure genome. The adopted homogenization technique interprets the unit cell associated with the composite structures as an assembly of plates, and the overall strain energy density of the unit cell as a summation of the plate strain energies of these individual plates. The variational asymptotic method is then applied to drop all higher-order terms and the remaining energy is minimized with respect to the unknown fluctuating functions. This has been done by discretizing the two-dimensional unit cell into one-dimensional frame elements in a finite element description. This allows the handling of structures with different levels of complexities and internal geometry within a general framework. Comparisons have been made with other works to show the advantages which the proposed model offers over other methods.