Isogeometric analysis-based approaches for modeling solid materials and structures with defects

• Main Author: Li, Weidong
• Other Authors: Zhou Kun
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2021
• Subjects: Engineering::Mechanical engineering::Mechanics and dynamics
• Online Access: https://hdl.handle.net/10356/145737
• Institution: Nanyang Technological University

Solid materials and structures usually undergo a complex process of fracture, such as crack initiation, propagation and coalescence before experiencing final failure. The failure of materials and structures can be induced by either micro- or macro-defects, including cracks, voids, dislocations and inclusions. The fundamental understanding of fracture mechanics remains a big challenge, particularly for thin-shell structures. Researchers have mainly investigated the deformation and fracture mechanisms of solids and structures via analytical and numerical approaches. Analytical approaches show their limitations when being implemented to investigate geometrically complicated structures with various boundary conditions and to treat cracks at arbitrary positions. Numerical approaches demonstrate their efficiency and robustness on tackling these complex engineering problems over analytical ones. As a recently proposed numerical method, isogeometric analysis (IGA) exhibits significant advantages in terms of geometry exactness and higher-order approximation. Therefore, this Ph.D. research focuses on the development of IGA-based approaches and their applications to the modeling of solid materials and structures with defects. Firstly, a novel IGA-meshfree coupling approach is developed to perform both the linear and geometrically nonlinear analyses of shell structures. The analyses are based upon the assumption of the Kirchhoff Love thin-shell theory. Both parametric and physical domains are utilized for the thin-shell structures, where the former one is used to couple the IGA and meshfree methods and to obtain the latter one via mapping. The entire domain is divided into three subdomains: the subdomain described by the IGA method to ensure geometry exactness, the subdomain described by the meshfree method to achieve local refinement and the coupling subdomain described by both methods. In the coupling subdomain, the coupling formulation is obtained based on the consistency conditions to realize the smoothness between the IGA and meshfree subdomains. The coupling approach can achieve a higher convergence rate than the IGA and meshfree methods because of the realization of local refinement. The accuracy and robustness of the coupling approach are validated by solving a series of shell benchmark problems. Secondly, a strong form-based IGA-meshfree moving least-squares collocation (IMMLS-C) approach is developed for two-dimensional linear elastic fracture problems with contact loading. The IMMLS-C approach uses reproducing conditions to establish the equivalence between MLS shape functions and isogeometric basis functions, thus exhibiting advantages including the exact geometry representation, convenient crack modeling and flexible adaptive refinement. The IMMLS-C approach focuses on solving fracture problems based on strong formulations without requiring the numerical integration of Galerkin weak forms. Traction-free boundary conditions are enforced over a set of collocation points located on both sides of a crack surface. The displacement discontinuity along the crack surface is modeled by the visibility criterion, and the singularity of near-tip stress fields is captured by adaptive mesh refinement without adding tip-enrichment functions, thereby reducing the degrees of freedom. Moreover, contact constraints are enforced by introducing a penalty algorithm to the strong formulations. The numerical results demonstrate that the adaptive refinement can achieve a high convergence rate at a low computational cost. Thirdly, crack propagation in thin-shell structures is investigated via an IGA-meshfree moving least-squares (MLS) approach. The approach provides an effective strategy of adaptive mesh refinement for IGA in a straightforward meshfree manner. The adaptivity of the mesh refinement is achieved by utilizing a gradient-based error estimator to identify the meshes that need to be refined by adding linear reproducing points. The Kirchhoff Love theory is further applied in the IGA-meshfree MLS formulation to simplify the modeling of cracked thin-shell structures by neglecting the rotational degrees of freedom. In this way, the singularity of the stress fields near the crack tip and the discontinuity of the displacement fields around the crack surface can be efficiently captured by the adaptive mesh refinement to produce accurate results. A series of two-dimensional static and quasi-static crack propagation problems about thin-shell structures are investigated. The predicted propagation paths are in good agreement with the reference results. Finally, an adaptive IGA-meshfree phase-field approach is developed for the modeling of brittle fracture in three-dimensional polycrystalline materials. In this approach, the flexible local mesh refinement scheme that is inherited from a meshfree method is combined with an error estimator that includes both the phase field and its gradient to achieve adaptive refinement. In this way, propagating cracks can be dynamically tracked, and the mesh near cracks is refined in a meshfree manner without requiring a priori knowledge of crack paths. The intergranular and transgranular crack propagation patterns in polycrystalline materials can be simulated by this approach. A series of numerical examples that deal with isotropic and anisotropic fracture are investigated to demonstrate the robustness and effectiveness of the developed approach. The development of efficient and accurate IGA-based approaches provides a promising way to investigate the deformation and fracture mechanisms of solid materials and structures with defects. The understanding of these mechanisms is of significant importance to engineering structure design and failure prevention.