This chapter examines the application of the computational theory for analyzing multiple discrete cracks of the mode-I type (EFCM) to study failure modes and maximum loads. It is well known that fatigue mechanisms are complicated, and it has become increasingly clear that the phenomenon is closely related to multiple-crack activities during cyclic loading. The chapter begins with numerical studies on the load-carrying capacity of a notched concrete beam subjected to various monotonic loads, focusing on the change of cracking behavior and the maximum load as the load condition changes. It is confirmed that the cracking behavior and the failure mode are very sensitive to the loading positions, which could alter significantly the load-carrying capacity of a simple beam. Also, it is found that under a given load condition, increasing the number of initial notches in a numerical model may not necessarily reduce the flexural strength of the beam. Furthermore, in the chapter, numerical studies using cyclic loads of varying loading position are carried out, aiming at clarifying the effects of changing loading positions during cyclic loading on cracking behavior and the load-carrying capacity of a notched beam.

This chapter introduces the computational theory for analyzing multiple discrete cracks of the mode-I type (EFCM). In many engineering applications involving an aging or partially damaged concrete structure, the available information on structural integrity is often limited to the crack-opening widths of cracks in the structure. What compels a seemingly ordinary surface crack to grow into a major structural crack is a complex problem involving crack interaction, and obviously its solution requires a discrete crack approach. Following the development of the fictitious crack model (FCM) by Hillerborg and his colleagues for analyzing the cracking behavior of a single crack, less progress was made in extending the method to multiple-crack problems despite extensive research efforts. The core issue here is the difficulty in determining the true cracking mode for the nonlinear crack problem, which requires the most active crack or cracks to be identified among a group of potentially active cracks during each load increment in numerical analysis. The complexity of the problem solution also stems from its computational aspect. Apparently, the numerical treatment of several varying crack surfaces requires a certain degree of flexibility and sophistication in the modeling techniques. The numerical formulation begins with a single-crack problem, and the crack equation that leads to the solution of the unknown boundary condition is established using a crack-tip-controlled modeling method. Then the crack equations for a multiple-crack problem are formulated with the crack interactions taken into account explicitly, and the validity of the numerical solutions is discussed. As illustrative examples, crack analysis are carried out on three types of structures or structural members: simple beams, tunnel linings, and concrete dams.

This chapter introduces fundamental concepts from both linear fracture mechanics (LEFM) and nonlinear fracture mechanics (NLFM) of concrete that are essential for understanding the subsequent development of the computational theories on multiple-crack analysis. The theory of the fracture mechanics of concrete, which is a branch of NLFM with its governing law for crack propagation drawn from the inelastic material behavior exhibited in an extensive fracture process zone (FPZ) ahead of an open crack, is largely developed from the theory of linear elastic fracture mechanics (LEFM). The chapter introduces the elastic theories of the crack-tip stress fields. Under the assumption of elasticity, the crack-tip stresses have an invariant form of distribution with an inverse-square-root singularity at the tip of the crack. A crack may be subjected to three different types of loading that cause displacements of the crack surfaces. In Mode I loading, the load is applied normal to the crack plane. Mode II loading refers to in-plane shear and causes the two crack surfaces to slide against each other. In Mode III loading, out-of-plane shear is applied which tends to tear the two crack surfaces apart. After defining the crack-tip fields in terms of the stress intensity factor, an energy description of the fracture process needs to be explained using Griffith's fracture theory, whose innovative concept of introducing fracture energy into the study of cracked materials laid a solid foundation for the later development of fracture mechanics. The fracture energy required for creating a unit surface of an open crack has a close relationship with the stress intensity factor. Consequently, the stress intensity factor loses its physical significance as a fracture parameter. This implies that LEFM is no longer valid, and the problem has to be analyzed based on NLFM.

This chapter focuses on a crack-analysis computer program, EFCM (CAIC-M1.FOR), for multiple-crack analysis in structural concrete. This program performs the mode-I type crack analysis in structural concrete based on the Extended Fictitious Crack Model (EFCM). The main features of this program include: numerical analysis of concrete fracture involving a number of discrete cracks of the mode-I type; two-dimensional finite element method; extended fictitious crack model theory; double precision of calculation; Gaussian elimination matrix solution method; and modified incremental stress analysis method. This program was written to carry out crack analysis in concrete strictly following the theory of the extended fictitious crack model. The usage of the program in crack analysis is illustrated by solving three sample problems including crack analysis of notched beam, crack analysis of scale model dam, and crack analysis of tunnel lining.

This chapter focuses on applications in solving real engineering problems, presenting a pseudoshell model that is developed from the extended fictitious crack model (EFCM) for crack analysis of tunnel linings. This model is used to calculate the pressure loads acting on various aging waterway tunnels of hydraulic power plants, based on crack analysis. By virtue of the extended fictitious crack model (EFCM), a different approach has emerged that focuses on the cracking behavior and lining deformation. As an application of the EFCM, a pseudoshell model for crack analysis of the tunnel lining is introduced in this chapter, which enables the crack-mouth-opening displacements (CMODs) of individual cracks to be calculated, corresponding deformations of the lining to be obtained, and pressure loads to be determined by a quasi loosening zone model. Through the case studies, the chapter demonstrates that for tunnels with cracked linings, the pseudoshell model and the quasi loosening zone model can work together to provide an effective means for calculating the ground pressure based on the CMOD.

This chapter introduces a book that focuses on the fracture mechanics of structural concrete. The book aims to discuss the latest developments in computational theories on multiple-crack analysis and mixed-mode fracture in structural concrete and the application of these theories to solve important engineering problems. Crack analysis is indispensable to clarify the various mechanisms by which cracks occur and evaluate the damaging effects that these cracks inflict on concrete structures. The peculiar nonlinearity that occurs in the fracture process zone (FPZ) ahead of an open crack makes the theory quite different from the whole range of problems in classical continuum mechanics. In addition, the frequent encounter of multiple cracks and mixed-mode fracture in real situation problems that require creative approaches highlights the challenging feature of crack analysis, which compels one to advance fracture mechanics of concrete further and to expand its applications wider.

This chapter focuses on a crack-analysis computer program, CAIC-M12.FOR, for mixed-mode crack analysis. The program CAIC-M12.FOR has been developed by extending the mode-I crack analysis program CAIC-M1.FOR, and except for the fracture mode involved, the main features, structure, and flow of the program are basically the same as those of CAIC-M1.FOR. It performs crack analysis of the mixed-mode type (mode I + mode II) in structural concrete. However, solving a mixed-mode crack problem requires additional operations of setting pairs of unit cohesive forces in the direction tangential to the crack surface, calculating sliding displacement and other coupled influence coefficients, and formulating the crack equation for mixed-mode fracture. Two types of shear-transfer model are employed in the program—the bilinear type and the trilinear type—and the variables are used to define these models. The chapter explains the formulation of the crack equation for mixed-mode fracture in matrix form, how the direction of shear force is determined based on the direction of crack surface sliding, and subroutines that have been added in CAIC-M12.FOR for mixed-mode crack analysis. Furthermore, the usage of the program in crack analysis is illustrated by solving a problem.

This chapter examines the fundamental concept of the fictitious crack model (FCM) and related issues in its numerical implementation. The FCM proposed by Hillerborg, Modeer, and Petersson (1976) for studying crack formation and crack growth in concrete brought two new concepts to the study of concrete fracture: the physical presence of an FPZ (“microcracked zone” in their original words) in front of an open crack due to strain localization and a constitutive law for crack propagation stipulating the relations between the cohesive forces and the crack-opening displacement (COD) inside the FPZ—in other words, the tension-softening relation. These are the fundamental concepts of the FCM that have become the basis of fracture mechanics of concrete. Furthermore, the chapter addresses the issue of stress singularity underlying the FCM formulation. Some effective modeling techniques are introduced, including the dual-nodes method for crack-path modeling, the path-shifting method for remeshing of curvilinear crack trajectories, and incremental stress analysis.

This chapter examines the application of the computational theory for analyzing multiple discrete cracks of the mode-I type (EFCM) to study crack interaction and localization in concrete. Crack interaction is an important issue in the crack analysis of concrete. Because the local stress field and the crack driving force for a given flaw can be significantly affected by the presence of one or more neighboring cracks, clarifying the effect of crack interaction is the key to a clear understanding of various cracking behaviors, including crack localization. In linear elastic fracture mechanics (LEFM), it is known that depending on the relative orientation of the neighboring cracks, the crack interaction can either magnify or diminish the stress intensity factor. Many studies have examined the interaction effects of multiple cracks in the fracturing process of concrete. In the discrete approach that allows the interaction of multiple cracks to be studied most straightforwardly, an explicit mathematical formulation of the crack interaction is possible. Such an approach enables crack interaction to be quantified and various cracking behaviors (such as why some cracks are active, while others are not and why crack localization begins early in some cases and is delayed in others) to be studied based on the nature and the intensity of the crack interactions involved. In order to derive the coefficient of interaction, a numerical formulation of three discrete cracks can be carried out based on the extended fictitious crack model (EFCM).

This chapter discusses the issue of mixed-mode fracture. In establishing the fictitious crack model (FCM) for analyzing the mode-I type of cracks in concrete, Hillerborg et al. (1976) had also indicated the possibility of applying the concept to model other types of fracture, such as the shear fracture of the mode-II or mode-III type. Much effort has since been made to extend the fictitious crack model to mixed-mode fracture because most practical fracture problems in concrete are mixed-mode, involving modes I and II. In modeling the shear transfer mechanism in the fracture process zone (FPZ), numerical studies in this category often rely on interface elements: The stiffnesses of interface connections in the normal and tangential directions are assumed to be functions of the crack surface deformation. However, this approach to introducing shear to the crack surface is inexplicit and approximate. An accurate specification of tangential tractions based on a given shear transfer law can hardly be achieved through regulating the stiffness of tangential connections that lack clear physical meaning. Since the shear transfer law defines the mode-II fracture energy, the lack of accuracy in implementing the law in numerical analysis could lead to erroneous results and misleading conclusions on the role and influence of the mode-II fracture parameters. Hence, a straightforward extension of the FCM is needed. To apply the mixed-mode FCM and EFCM to engineering problems, the fracture tests on scale models of a gravity dam are remodeled as mixed-mode fracture, focusing on the influence of shear on the crack path in dams.