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.

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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.”