Adaptive Consistent Element-free Galerkin Method For Phase-field Model Of Brittle Fracture

Brittle fracture of materials and structures widely exists in various fields of national economy such as civil,mechanical,aerospace,ocean and automobile engineering.Its occurrence is usually sudden without obvious foreboding deformation and thus it severely threatens the safety in operation of engineering structures and industrial equipment.The in-depth study of brittle fracture is of great significance in revealing the mechanism of complex fracture phenomena such as crack initiation,propagation and emerging,and even in preventing accidents due to fracture of structures.Conventional analysis of brittle fracture is based on the classical Griffith theory.In numerical simulation,special treatment of the discontinuity at cracks and the stress singularity at crack tips is required.This results in the complexity in numerical simulation of multiple and 3D cracks.Moreover,the classical Griffith model of cracks is mainly used for crack propagation.It cannot directly deal with crack initiation,merging,etc.,and additional criterion has to be introduced.However,it is not easy to find an appropriate criterion.An alternative way to study cracks is the phase-field model which can be traced back to the variational principle of brittle fracture proposed in the late 1990s.This method introduces a phase field function and models cracks as the continuous transition of this function between fully broken and intact materials.Thus,the discontinuous problems of cracks are transformed into the continuous problems of the distribution of the phase field function.Tracking and handling of the discontinuity at cracks are no longer needed in numerical simulation,and this effectively simplifies the implementation of numerical simulation of multiple and 3D cracks.Moreover,the complex fracture phenomena such as crack initiation,propagation and merging can also be conveniently simulated by the phase-field model without additional fracture criteria.However,accurate description of the high gradient of the phase field in the fractured zone is required and very dense computational mesh is commonly used for spatial discretization.This leads to unaffordable computational cost and too low computational efficiency,especially for 3D fracture analysis.Aiming at this issue,the consistent element-free Galerkin(EFG)method which can accurately pass the linear and quadratic patch tests is employed in this thesis to solve the phase-field model of fracture numerically.The adaptive algorithm,which is able to automatically refine nodes in the vicinity of cracks along with their propagation,is investigated and established.Due to the developed adaptive algorithm,the number of nodes required for spatial discretization is effectively reduced and the computational efficiency of the phase-field model of fracture is improved.The work of this thesis is summarized as follows:First,the adaptive algorithm of the consistent EFG method is presented for numerical solution of problems with local high gradients.The consistent EFG method effectively improves the computational efficiency,accuracy and convergence of the standard EFG method by correcting nodal derivatives.On this basis,an adaptive algorithm for the consistent EFG method is further developed by taking full use of the merit of the EFG method,i.e.its nodal shape function does not depend on elements.In the developed algorithm,the computational nodes are refined by the local multi-level refinement of the background integration mesh.An integration scheme satisfying consistency condition is constructed for transitional background integration cells.Refinement of nodes is triggered by the criterion based on the gradient of the strain energy density.Numerical results of linear elastic examples show that the proposed algorithm is able to automatically refine the computational nodes in the region with high stress gradients and to result in reasonable distribution of computational nodes.In comparison with the adaptive analysis of the standard EFG method,the proposed method shows remarkable advantages on computational efficiency,accuracy and the smoothness of the stress fields.This provides a solid foundation for dealing with the local high gradients in the phase-field model of fracture.The establishment and numerical validation of this adaptive algorithm are presented in Chapter 4.Then,the adaptive consistent EFG method for phase-field model of brittle fracture is presented for numerical simulation and analysis of crack initiation,propagation and merging,etc.In this thesis,the phase-field model based on spectral decomposition of strain is employed to describe the mechanical behaviors of cracks and the consistent EFG method is employed to solve the phase field and the equilibrium equations numerically.In the phase-field model,the strain energy history drives the evolution of the phase field.According to this feature,a criterion based on maximum residual strain energy history and phase field is established and is used to determine the local region where refinement of computational nodes is needed.In this way,adaptive analysis of brittle fracture is achieved.Crack initiation,propagation and merging,especially the non-planar 3D crack propagation(e.g.twisting of the crack surface)are successfully simulated by the presented method with decreased number of nodes,reduced scale and improved efficiency of the computation.In addition,the proposed method presents better accuracy in comparison with the linear finite element method and the standard EFG method.The adaptive consistent EFG method for phase-field model of fracture and its numerical validation are described in Chapter 5.Finally,the gradient distribution of material parameters is further considered in the proposed adaptive consistent EFG method for phase-field model of fracture and the EFG method for the fracture analysis of functionally graded materials(FGMs)is studied.Compared with homogenous materials,the graded distribution of material parameters in FGMs leads to more complicated stress fields and thus accurate simulation of cracks becomes more difficult.In consideration of the fact that both the moving least-squares approximation possessing high smoothness and the consistent integration scheme contribute to the accurate solution of stress fields,the consistent EFG method is adopted to solve the FGMs problems and is validated by numerical examples.On this basis,the phase-field model of fracture for FGMs is introduced and is also numerically solved by the consistent EFG method.By further establishing the corresponding criterion to trigger adaptive local refinement,adaptive analysis of 2D and 3D crack propagation in FGMs is implemented.Numerical results show that the proposed method is able to accurately reflect the influence of the gradient distribution of material parameters on the final crack path and,to some extent,to reveal the fracture mechanism,i.e.crack propagation is controlled by both the strain energy history and the critical energy release rate.In Chapter 6,the element-free method and numerical results of the phase field model of fracture for FGMs are discussed in detail.For the sake of self-completeness,the basic concepts and theories of the phase field model of brittle fracture and the Galerkin meshfree methods are introduced in Chapter 2 and Chapter 3,respectively.Conclusion and future work are summarized in Chapter 7 and the design of the computer program to implement the presented method is described in the Appendix.