Free vibration characteristics and impact response of functionally graded conical shell

Abstract

Functionally graded materials (FGMs) categorized as an advanced class of composite materials, consist of novel inhomogeneous mixture of materials like ceramic and metals with smooth changes of its constituents’ volume fraction along the thickness direction. These materials do not contain well distinguished boundaries or interfaces between their different regions as in the case of conventional composite materials but have numerous advantages that make them appropriate in potential applications due to reduction of in-plane and through-the thickness transverse stresses, improved thermal properties, high toughness, etc. FGMs possess good chances of reducing mechanical and thermal stress concentration in many structural elements because of smooth transition between the properties of the components and thereby cracking or delamination, which are often observed in conventional multi-layer systems are avoided. FGMs consisting of metallic and ceramic components are well-known to enhance the properties of high temperature thermal-barrier application where the ceramic part has good thermal resistance and metallic part has superior fracture toughness. Thus FGMs have great potential in applications where the operating conditions are severe, including spacecraft heat shields, nuclear reactors, biomedical implants, etc. A functionally graded shallow conical shell is a structural element of considerable technical significance and can be idealized as turbo machinery blades under rotation that can be extensively used in the aviation, energy, nuclear and mechanical industries. In a weight-sensitive and high thermal gradient application, FGM materials are advantageous because of their light weight, high strength, stiffness and thermal barrier ceramics components. In addition, FGM materials can be tailored to cater the design requirements of strength, stiffness thermal barrier application. The prior knowledge of free vibration characteristics of such turbomachinery blades is utmost important to avoid resonance effect ensuring longer life of such components, preventing unscheduled shutdown of the machineries. The composition of FGM constituents’ such as ceramics and metals can be used with help of prior knowledge of natural frequencies. Moreover, the initial stress system in a rotating shell due to centrifugal body forces has the cascading effect on the natural frequency appreciably. Thus, the free vibration characteristics have crucial influence on safe performance of such FGM shell structures. On the other hand outside/inside debris or small torn out objects from the turbo machines can have impact of such conical shell blade with low velocity. Therefore, the ABSTRACT viii susceptibility to damage due to low velocity impact caused by foreign objects can accelerate the degradation of strength and can promote the structural instability. Hence the low velocity impact performances are crucial for designing of an impact mitigating system In realistic situations, pretwisted conical shell structures have geometrical complexities arising due to their specific applications in various service environments. A typical dynamic parameters need to be used considering the rotation effect of these structural elements. Therefore a good understanding of the dynamic behaviour of FGM pretwisted conical shells requires close attention in order to confirm the operational safety. Accordingly, the present study is intended to investigate two key aspects of the dynamic behaviour of pretwisted FGM rotating conical shells, namely, free vibration characteristics and dynamic low velocity impact response. Being a proficient analysis tool to the design engineer the finite element method is employed to address the present problems. An eight-noded isoparametric shell element is used for the finite element formulation considering rotary inertia effect and transverse shear deformation based on Mindlin’s theory. The dynamic equilibrium equation is derived from Lagrange’s equation neglecting the Coriolis effect for moderate rotational speeds. A modified Hertzian contact law considering permanent indentation is used to calculate the contact force along with other impact response parameters. Using the Newmark’s time integration scheme the time dependent equations of the shell and the impactor are solved. The static equilibrium equations and the standard eigen value problem are solved by Gauss elimination technique and QR iteration algorithm, respectively. Finite element codes are developed and validated with those published results in the open literature after performing a suitable convergence study and verification of the results. The results are primarily obtained for FGM pretwisted and untwisted conical shells for the triggering parameters like different FGM power law index, rotational speeds, twist angles and porosity factors on the natural frequencies. The mode shapes for the FGM conical shells are also presented. Numerical solutions are also obtained for time dependent impact response of FGM conical shells subjected to low velocity impact. Parametric studies are conducted to investigate the effects of prime parameters like different FGM power law index, angle of twist, velocity of impactor, location of the impactor, shell thickness, mass of the impactor and porosity factors on impact performance. The results are discussed in detail with graphs and tables and the conclusions are laid down concentrating on the significant findings. The future scopes of the present work are also projected to carry out the further investigations.