Saint-Venant Torsion of Functionally Graded Micropolar Beams

Abstract

Functionally graded materials (FGMs) are a class of composites that are increasingly used in aerospace, defence, energy, and biomedical applications because their elastic properties and microstructure can be varied continuously in space. These materials can be designed to address specific design requirements at different locations of the material. They provide a clear advantage over traditional composites in that they limit common failure modes such as delamination and de-bonding. FGMs also possess an intrinsic microstructure with non negligible characteristic length relative to the size of the component. Classical elasticity cannot account for independent micro-rotations or couple-stresses caused by the microstructure, and can therefore misrepresent the stiffness and internal stress distribution. Thus, due to the wide range of applications and advantages compared to traditional composites, it is important to study the behaviour of FGMs. This thesis investigates the torsional behaviour of functionally graded micropolar beams. The comprehensive three-dimensional Cosserat continuum framework is used to reduce the problem to a Neumann boundary value problem of anti-plane Cosserat elasticity. The problem is posed on the beam cross-section, accounting for spatially varying elastic moduli within the cross-section. Furthermore, the existence and uniqueness of weak solutions for the resulting system of partial differential equations with variable coefficients is established, using coercivity of the operator and the Lax-Milgram theorem. The general formulation is then specialized to exponentially graded circular and annular cross-section examples to demonstrate practicality. By using the inherent geometric symmetry of the circular and annular beams, the governing boundary value problem reduces to an ordinary differential equation for the radial microrotation amplitude, which is then solved numerically and compared with the corresponding homogeneous and classical cases. The results show that exponential grading on a bounded cross-section satisfies the conditions required for coercivity, so the graded micropolar torsion problem is well-posed. At a fixed level of grading, microstructure relaxes the peak shear stress, since part of the transmitted load is carried by couple-stresses. Grading in turn migrates these couple-stresses toward the stiffer region of the cross-section. Grading and micropolarity also reinforce one another: at the largest grading parameter considered, the normalized torsional rigidity of the graded micropolar bar exceeds the classical homogeneous value. It is therefore clear that grading is a controllable design parameter that can be used to stiffen a beam and redistribute stress concentrations away from critical boundaries. These results establish that functional grading and microstructure must be modelled together, since neither a classical nor a homogeneous micropolar analysis completely captures their combined effect on torsional response.