Spline Fictitious Boundary Element Method For Analysis Of Static And Dynamic Problems Of Cracked Functionally Graded Materials

Functionally graded materials(FGMs)have been widely used in engineering structures including civil engineering structures.The appearance of crack defects on FGM has large influence on its static and dynamic characteristics,so it is important to do research on the fracture problem of FGM.Actually,due to the intrinsic uncertainties of external dynamic loadings,it is more reasonable to conduct dynamic analysis of cracked FGM using random vibration methods.Moreover,due to the complexity of manufacturing process,it has been found that the structural uncertainties have a certain influence on the dynamic responses of structures.Therefore,it is necessary to conduct compound random vibration analysis of cracked FGM with consideration of randomness for both structural parameters and external dynamic loadings.Because of the inhomogeneous property of FGM,it is difficult to capture the nonsingularity of the crack tips in dynamic and static problems,and it is more challenging to conduct random vibration analysis of fractures problems of FGM.The researches on stochastic analysis of cracked FGM are still at the starting stage focusing more on the static problem,while there are only few researches on random vibration analysis problems and no researches on compound random vibration analysis problems.Therefore,in this dissertation,spline fictitious boundary element method(SFBEM)based on Erdogan fundamental solutions is proposed for efficient static and dynamic analysis of cracked FGM plane static and dynamic problems.In addition,combined with explicit time-domain method(ETDM)for random vibration analysis,a hybrid approach is further developed for random vibration and compound random vibration analysis of cracked FGM plane problems.The work for this dissertation is described as follows:(1)SFBEM based on Erdogan fundamental solutions is proposed for static analysis of cracked FGM plane problems.The governing differential equations of the inhomogeneous cracked FGM plate are first transformed to those of the homogeneous plane problem by introducing the equivalent body forces for reflecting the inhomogeneous property of FGM.Then,Erdogan fundamental solutions for a cracked homogeneous plane are adopted in the formulation of SFBEM for static analysis of FGM crack problems,in which the stress boundary conditions on the crack surface are automatically satisfied and the singular behavior at the crack tip can be naturally reflected.Solving the above SFBEM formulation,stress intensity factors(SIFs)can be calculated directly.Therefore,the proposed method is superior in terms of accuracy and efficiency.(2)SFBEM based on Erdogan fundamental solutions is further developed for dynamic analysis of cracked FGM plane problems.Based on the SFBEM formulation for static analysis of cracked FGM plates and considering the equivalent body forces of inertia forces and damping forces,the SFBEM formulation for dynamic analysis of cracked FGM plates is derived.Then the equation of motions based on SFBEM for cracked FGM plane problems is established,which can be solved for dynamic responses of displacement,velocity,stress and SIF.Erdogan fundamental solutions for static fracture problems are used in the proposed method to obtain the dynamic SIFs of cracked FGM plates,which can avoid solving the complex fundamental solutions for dynamic fracture problems of FGM.(3)A hybrid approach with combination of SFBEM and ETDM is proposed for random vibration analysis of cracked FGM plane problems.On the basis of the SFBEM formulations,the time-domain explicit expressions of dynamic SIFs for cracked FGM plates are established by using numerical integrals.Then the explicit time-domain method(ETDM)is further used to conduct random vibration analysis of cracked FGM plates subjected to non-stationary excitations.The statistical moments of dynamic SIFs and the dynamic reliability of initial crack propagation can be obtained at high efficiency using the present approach based on the fusion of SFBEM and ETDM.(4)A hybrid approach with combination of SFBEM and ETDM is further developed for compound random vibration analysis of the cracked FGM plane problem with consideration of dynamic loading uncertainties and structural uncertainties.Based on the SFBEM formulation for dynamic analysis of cracked FGM plane problems,the time-domain explicit expressions of sensitivities for dynamic SIFs of cracked FGM plates are established.Using Taylor expansion techniques,the time-domain explicit expressions for dynamic SIFs of the cracked FGM plane with consideration of dynamic loading uncertainties and structural uncertainties can be further established.The statistical moments of dynamic SIFs and the dynamic reliability of initial crack propagation for cracked FGM plates with consideration of uncertainties for dynamic loading and structural parameters can be obtained at high efficiency using the proposed approach based on the fusion of SFBEM and ETDM.Numerical examples are investigated to show accuracy,efficiency and applicability of the present method.Then,the proposed method is further applied to solve the dynamic reliability of initial crack propagation problems for FGM components of engineering structures.The research results show that,SFBEM based on Erdogan fundamental solutions can be used to calculate static and dynamic SIFs of cracked FGM plane problems efficiently.Moreover,based on the fusion of SFBEM for cracked FGM problems and ETDM for random vibration analysis,the statistical moments of dynamic responses and the dynamic reliability of initial crack propagation for crack FGM plane problems with deterministic or random parameters can be obtained at high efficiency using the present approach.