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.

Many practical cases of fracture can be considered as one-dimensional, that is, propagating in one dimension and characterised by opening (mode I) and shearing (mode II) action only with no tearing (mode III) action. A double cantilever beam (DCB) represents the most fundamental one-dimensional fracture problem. There has however been considerable confusion in calculating its mixed-mode energy release rate (ERR) partition. In this work, new and completely analytical mixed-mode partition theories are developed for one-dimensional fractures in isotropic homogeneous and laminated composite DCBs, based on linear elastic fracture mechanics (LEFM) and using the Euler and Timoshenko beam theories. They are extended to isotropic homogeneous and laminated composite straight beam structures and isotropic homogeneous plates based on the Kirchhoff-Love and Mindlin-Reissner plate theories. They are also extended to non-rigid elastic interfaces for isotropic homogeneous DCBs. A new approach is used, based on orthogonal pure fracture modes. Two sets of orthogonal pairs of pure modes are found. They are distinct from each other in the present Euler beam and Kirchhoff-Love plate partition theories and coincide on the first set in the present Timoshenko beam and Mindlin-Reissner plate partition theories. After the two sets of pure modes are shown to be unique and orthogonal, they are used to partition mixed modes. Interaction is found between the mode I and mode II modes of the first set in the present Euler beam and Kirchhoff-Love plate partition theories. This alters the ERR partition but does not affect the total ERR. There is no interaction in the present Timoshenko beam or Mindlin-Reissner plate partition theories. The theories distinguish between local and global ERR partitions. Local pureness is defined with respect to the crack tip. Global pureness is defined with respect to the entire region mechanically affected by the crack. It is shown that the global ERR partition using any of the present partition theories or two-dimensional elasticity is given by the present Euler beam or Kirchhoff-Love plate partition theories. The present partition theories are extensively validated using the finite element method (FEM). The present beam and plate partition theories are in excellent agreement with results from the corresponding FEM simulations. Approximate 'averaged partition rules' are also established, based on the average of the two present beam or plate partition theories. They give close approximations to the partitions from two-dimensional elasticity. The propagation of mixed-mode interlaminar fractures in laminated composite beams is investigated using experimental results from the literature and various partition theories. The present Euler beam partition theory offers the best and most simple explanation for all the experimental observations. It is in excellent agreement with the linear failure locus and is significantly closer than other partition theories. It is concluded that its excellent performance is either due to the failure of materials generally being based on global partitions or due to the through-thickness shear effect being negligibly small for the specimens tested. The present partition theories provide an excellent tool for studying interfacial fracture and delamination. They are readily applicable to a wide-range of engineering structures and will be a valuable analytical tool for many practical applications.

Advances in the design and manufacturing technologies have greatly enhanced the utility of fiber reinforced composite materials in aircraft, helicopter and space- craft structural components. The special characteristics of composites such as high strength and stiffness, light-weight corrosion resistance make them suitable sub- stitute for metals/metallic alloys. However, composites are very sensitive to the anomalies induced during their fabrication and service life. Also, they are suscepti- ble to the impact and high frequency loading conditions because the epoxy matrix is at-least an order of magnitude weaker than the embedded reinforced carbon fibers. On the other hand, the carbon based matrix posses high electrical conductivity which is often undesirable. Subsequently, the metal matrix produces high brittleness. Var- ious forms of damage in composite laminates can be identified as indentation, fiber breakage, matrix cracking, fiber-matrix debonding and interply disbonding (delam- ination). Among all the damage modes mentioned above, delamination has been found to be serious for all cases of loading. They are caused by excessive interlaminar shear and normal stresses. The interlaminar stresses that arise in the case of composite materials due to the mismatch in the elastic constants across the plies. Delamination in composites reduce it’s tensile and compressive strengths by consid- erable margins. Hence the knowledge of these stresses is the most important aspect to be looked into. Basic theories like the Euler-Bernoulli’s theory and Timoshenko beam theory are based on many assumptions which poses limitation to determine these stresses accurately. Hence the determination of these interlaminar stresses accurately requires higher order theories to be considered. Most of the conventional methods of determination of the stresses are through the solutions, involving the trigonometric series, which are available only to small and simple problems. The most common method of solution is by Finite Element (FE) Method. There are only few elements existing in the literature and very few in the commercially available finite element software to determine the interlaminar stresses accurately in the composite laminates. Accuracy of finite element solution depends on the choice of functions to be used as interpolating polynomials for the field variable. In-appropriate choice will manifest in the form of delayed convergence. This delayed convergence and accuracy in predicting these stresses necessiates a formulation of elements with a completely new concept. The delayed convergence is sometimes attributed to the shear locking phenomena, which exist in most finite element formulation based on shear deformation theories. The present work aims in developing finite elements based on higher order theories, that alleviates the slow convergence and achieves the solutions at a faster rate without compromising on the accuracy. The accuracy primarily depends on the theory used to model the problem. Thus the basic theories (such as Elementary Beam theory and Timoshenko Beam theory) does not suffice the condition to accuratley determine the interlaminar stresses through the thickness, which is the primary cause for delamination in composites. Two different elements developed on the principle of super-convergence has been presented in this work. These elements are subjected to several numerical experiments and their performance is assessed by comparing the solutions with those available in literature. Spacecraft and aircraft structures are light in weight and are also lightly damped because of low internal damping of the material of construction. This increased exibility may allow large amplitude vibration, which might cause structural instability. In addition, they are susceptible to impact loads of very short duration, which excites many structural modes. Hence, structural dynamics and wave propagation study becomes a necessity. The wave based techniques have found appreciation in many real world problems such as in Structural Health Monitoring (SHM). Wave propagation problems are characterized by high frequency loads, that sets up stress waves to propagate through the medium. At high frequency, the wave lengths are small and from the finite element point of view, the element sizes should be of the same order as the wave lengths to prevent free edges of the element to act as a free boundary and start reflecting the stress waves. Also longer element size makes the mass distribution approximate. Hence for wave propagation problems, very large finite element mesh is an absolute necessity. However, the finite element problems size can be drastically reduced if we characterize the stiffness of the structure accurately. This can accelerate the convergence of the dynamic solution significantly. This can be acheived by the super-convergent formulation. Numerical results are presented to illustrate the efficiency of the new approach in both the cases of dynamic studies viz., the free vibration study and the wave propagation study. The thesis is organised into five chapters. A brief organization of the thesis is presented below, Chapter-1 gives the introduction on composite material and its constitutive law. The details of shear locking phenomena and the interlaminar stress distribution across the thickness is brought out and the present methods to avoid shear locking has been presented. Chapter-2 presents the different displacement based higher order shear deformation theories existing in the literature their advantages and limitations. Chapter-3 presents the formulation of a super-convergent finite element formulation, where the effect of lateral contraction is neglected. For this element static and free vibration studies are performed and the results are validated with the solution available in the open literature. Chapter-4 presents yet another super-convergent finite element formulation, wherein the higher order effects due to lateral contraction is included in the model. In addition to static and free vibration studies, wave propagation problems are solved to demonstrate its effectiveness. In all numerical examples, the super-convergent property is emphasized. Chapter-5 gives a brief summary of the total research work performed and presents further scope of research based on the current research.