Fracture Process In Quasi-brittle Materials Research Articles

Phase field modeling of crack propagation in three-dimensional quasi-brittle materials under thermal shock

Quasi-brittle materials create intricate crack patterns on their surface and internal three-dimensional crack networks when exposed to thermal shock conditions. Their cracking behavior is notably impacted by their quenching temperature and size. In this paper, a coupled three-dimensional phase field-cohesive zone model that accounts for thermoelastic fracture is developed to reproduce the quenching and cracking experiments of ceramic balls. The accuracy of the phase field model is demonstrated by simulating the phase field of a single-element cooling process and comparing it with the analytical solution. The quench fracture process of the 3D ceramic ball is then simulated. The phase field simulations show that the surface crack morphology during the quenching process is somewhat random. Still, the crack distribution density is determined by the quenching temperature difference and ceramic ball size. Furthermore, a threshold size is present during the quenching process. When the ceramic ball's radius falls below this threshold, the surface crack pattern no longer appears during quenching. Consequently, the phase-field-cohesive zone model represents an effective means of predicting three-dimensional fracture processes in quasi-brittle materials subjected to thermodynamically coupled loading.

A modified cohesive damage-plasticity model for distinct lattice spring model on rock fracturing

In this study, a modified cohesive damage-plasticity model (m-CDPM) is developed for a distinct lattice spring model (DLSM) to simulate the damage evolution and fracture process in quasi-brittle materials. A hyper-elastic segment is introduced into the original CDPM to enable the DLSM to handle special cases associated with negative or vanishing shear stiffness in the DLSM framework when attempting to link micro- and macro-constitutive parameters. Relationships between the macro experimental parameters and their micro counterparts are theoretically derived following the tradition of the DLSM with consideration of the particle size effect. The correctness and feasibility of the proposed model are verified through numerical examples and a comparison with available experimental or numerical solutions.

Coupled continuous–discrete formulation based on microplane and strong discontinuity models for representing non-orthogonal intersecting cracks

Fracture processes in quasi-brittle materials are governed by the strain localization phenomenon, which involves the formation of localized damage zones and cohesive cracks. In this work, we present numerical tools to model strain localization from the onset of localized damage to the formation and propagation of multiple intersecting cracks. Two main ingredients are used for this purpose: (i) a microplane model to describe the initial anisotropic damage phase; (ii) the strong discontinuity method to introduce cracks as strong discontinuities in the damaged continuum using the Embedded Finite Element Method (E-FEM). Here, we formulate the microplane microdamage model within a thermodynamic framework by means of simple constitutive laws on each microplane. In order to describe intersecting cracks, we extend the standard E-FEM to accommodate two strong discontinuities. The coupling between the microplane microdamage model with the strong discontinuity model is achieved using a transition method based on the energy equivalence between both models. Exploiting the anisotropic description provided by the microplane model, transition criteria are formulated based on the quantities defined on each microplane. The proposed methodologies are illustrated using several elementary test cases involving both simple and complex stress-strain states.

Size effect in heterogeneous materials analyzed through a lattice discrete element method approach

In the Lattice Discrete Element Method (LDEM), different types of mass are considered to be lumped at nodal points and linked by means of one-dimensional elements with arbitrary constitutive relations. In previous studies on the tensile fracture behavior of rock samples, it was verified that numerical predictions of fracture of non-homogeneous materials using LDEM models are feasible and yield results that are consistent with the experimental evidence available so far. In the present paper, a discussion of the results obtained with the LDEM is presented. A set of rock specimens of different sizes, subjected to monotonically increasing simple tensions, are simulated with LDEM. The results were analyzed from the perspective of the brittleness number, proposed by Alberto Carpinteri, to measure the brittleness level of the structure under study. The satisfactory correlation between the experimental results and LDEM results confirms the robustness of this method as a numerical tool to model fracture processes in quasi-brittle materials.

A boundary element method formulation for quasi-brittle material fracture analysis using the continuum strong discontinuity approach

The implicit formulation of the boundary element method is applied to bi-dimensional problems of material failure involving, sequentially, inelastic dissipation with softening in continuous media, bifurcation and transition between weak and strong discontinuities. The bifurcation condition is defined by the singularity of the localization tensor. Weak discontinuities are related to strain localization bands of finite width, which become increasingly narrow until to collapse in a surface with discontinuous displacement field, called strong discontinuity surface. To associate such steps to the fracture process in quasi-brittle materials, an isotropic damage constitutive model is used to represent the behaviour in all of them, considering the adaptations that come from the strong discontinuity analysis for the post-bifurcation steps. The crack propagation across the domain is done by an automatic cells generation algorithm and, in this context, the fracture process zone in the crack tip became totally represented.

Virtual elements and zero thickness interface-based approach for fracture analysis of heterogeneous materials

A novel procedure for analyzing fracture processes in quasi-brittle materials and consisting in combining the discrete crack approach by means of interface elements with the Virtual Element Method is proposed and developed in this work. In particular, the proposed procedure is used in the simulation of non-linear mechanical response and cracking of cement-based composites at the mesoscopic level of observation. Thereby three components are recognized: mortar, coarse aggregates and mortar–aggregate interfaces. In this regard VEM constitutes a powerful and efficient tool to represent the complex geometries of the inclusions in composite materials, such as coarse aggregates in concrete. Actually, patches with any number of edges (not necessarily convex), hanging nodes, flat angles, collapsing nodes, etc., can be easily handled in the VEM framework while retaining the same approximation properties of FEM. On the other hand, classical zero-thickness interface elements (IEs) are employed for modeling stress-crack opening processes. A series of numerical results, not only at the mesoscopic but also at the macroscopic level of observation, are presented to demonstrate the soundness and capabilities of the proposed approach based on combinations of VEM and IEs.

Coupled domain decomposition–proper orthogonal decomposition methods for the simulation of quasi-brittle fracture processes

In this paper, we discuss a strategy to reduce the computational costs of the simulation of dynamic fracture processes in quasi-brittle materials, based on a combination of domain decomposition (DD) and model order reduction (MOR) techniques. Fracture processes are simulated by means of three-dimensional finite element models in which use is made of cohesive elements, introduced on-the-fly wherever a cracking criterion is attained. The body is initially subdivided into sub-domains; for each sub-domain MOR is obtained through a proper orthogonal decomposition (POD) of the equations governing its evolution, until when it starts getting cracked. After crack inception within a sub-domain, the solution is switched back to the original full-order model for that sub-domain only. The computational gain attained through the coupled use of DD and POD thus depends on the geometry of the body, on the topology of sub-domains and, on top of all, on the spreading of cracking induced by load conditions. Numerical examples concerning well-established fracture tests are used for validation, and the attainable reduction of the computing time is discussed at varying decomposition into sub-domains, even in the absence of a full exploitation of parallel computing potentialities.

A higher-order stress-based gradient-enhanced damage model based on isogeometric analysis

The micro-damage associated with diffuse fracture processes in quasi-brittle materials can be described by continuum damage mechanics. In order to overcome the mesh dependence of local damage formulations, non-local and gradient-enhanced approaches are often employed. In this manuscript, a higher-order stress-based gradient-enhanced formulation is proposed, which exploits the higher-order continuity of B-spline functions in isogeometric analysis (IGA). The proposed formulation does not require the decomposition of the fourth-order model into two second-order models. Two numerical examples are presented to demonstrate the performance of the formulation and to compare the obtained solutions with results from conventional gradient-enhanced damage formulation.

An embedded crack in a constant strain triangle utilizing extended finite element concepts

This paper revisits the formulation of the CST element with an embedded discrete crack taking advantage of the direct formulations developed within the framework of the extended finite element method, XFEM. The result is a simple element for modeling cohesive fracture processes in quasi-brittle materials. The element is easily fitted a standard FEM code, and as such it is an alternative to more cumbersome XFEM elements which require special d.o.f.’s and extra administration. The crack description is embedded, in the sense that extra d.o.f.’s controlling the crack opening are eliminated at the element level. The cracked element is stress-compatible in the sense that stresses are continuous across the crack. A special shape function is introduced to allow for the discontinuous displacements without eradicating the stress compatibility. The simplicity of the element comes at the cost of inter-element discontinuity of displacements. The formulation is based on a variational principle of virtual work involving only the interpolation of displacements. The good performance of the element is demonstrated through the comparison with three benchmark tests in which a single crack is propagated: The center cracked sheet in uni-axial tension, the three-point bending test and the four-point shear beam test.

Fully-developed FPZ length in quasi-brittle materials

This paper studies the development of fracture processes in quasi-brittle materials. We propose to use the length of the fracture process zone (FPZ) once it is fully developed as a material parameter. This assumption allows us to build an analytical formulation that reproduces the mechanical behavior of any specimen as a cohesive crack advances. Extensive comparisons with experimental results lead us to define a new characteristic length that commensurates with the fully-developed FPZ and that together with the new analytical model, is used to provide a complete and consistent study on the fracture process. In particular, the size-effect deriving from our formulation coincides with the statistical size-effect law of Bažant for small and medium sizes, whereas it smoothly converges to size-independent results as size increases. The analytical cohesive formulation developed here is validated against experimental results on various types of normal and high-strength concretes as well as construction ceramics for several experimental set-ups and test scales. Because of its simplicity as compared with numerical models for fracture, this analytical formulation constitutes a powerful tool for studying fracture processes in a wide variety of mechanical configurations. Meanwhile, analytical expression for a fully-developed FPZ length is given for a general type of cohesive law.

A continuous–discontinuous approach to simulate fracture processes in quasi-brittle materials

A macroscopic framework for the simulation of failure processes in quasi-brittle materials is proposed. The framework employs the partition of unity (PU) concept and introduces a new cohesive zone model, capturing the transition between the initial continuum state and the final discrete state. The model is generic in a sense that it allows extension of most continuum models to a discontinuous framework in an efficient and robust way, thereby adding the effect of macro-crack formation by the growth and coalescence of micro-defects. Both material failure and interface failure can be studied with this formulation.

Towards fully automatic modelling of the fracture process in quasi-brittle and ductile materials: a unified crack growth criterion

Fully automatic finite element (FE) modelling of the fracture process in quasi-brittle materials such as concrete and rocks and ductile materials such as metals and alloys, is of great significance in assessing structural integrity and presents tremendous challenges to the engineering community. One challenge lies in the adoption of an objective and effective crack propagation criterion. This paper proposes a crack propagation criterion based on the principle of energy conservation and the cohesive zone model (CZM). The virtual crack extension technique is used to calculate the differential terms in the criterion. A fully-automatic discrete crack modelling methodology, integrating the developed criterion, the CZM to model the crack, a simple remeshing procedure to accommodate crack propagation, the J2 flow theory implemented within the incremental plasticity framework to model the ductile materials, and a local arc-length solver to the nonlinear equation system, is developed and implemented in an in-house program. Three examples, i.e., a plain concrete beam with a single shear crack, a reinforced concrete (RC) beam with multiple cracks and a compact-tension steel specimen, are simulated. Good agreement between numerical predictions and experimental data is found, which demonstrates the applicability of the criterion to both quasi-brittle and ductile materials.

Modelling of fracture process in concrete using a novel lattice model

Papers deals with simulations of fracture process in quasi-brittle materials like concrete with a novel lattice model. Concrete was described mainly as a three-phase material composed of aggregate, cement matrix and interfacial transition zones. The calculations were carried out for concrete specimens subject to uniaxial extension, shear and extension and three-point bending. Two-dimensional and three-dimensional simulations were performed. The advantages and disadvantages of the proposed model were outlined.

Mode I Fracture Surface of Granite: Measurements and Correlations with Mechanical Properties

This paper deals with the experimental characterization of the fracture surface of granite, obtained from direct tensile tests. The objective is to address the dependence of Mode I fracture energy on the fracture surface's roughness, as well as to analyze the fracture propagation process based on the surface features. In addition, the influence of microstructural aspects on the fracture surface is also discussed. Different types of granites were studied, similar to the ones used in the fabric of ancient buildings, with distinct petrographical characteristics. The fracture surfaces were inspected with a novel 3D laser topographical inspection system aiming at acquiring the texture of fracture surface profiles at various locations on the surface. The relief of the fracture surfaces was characterized by means of classical parameters, namely, mean roughness and root-mean-square roughness. A deeper analysis was carried out by the separation of micro- and macroroughness regimes. This study reveals a clear linear trend between fracture energy and mean microroughness. Additionally, it was found that microstructural aspects, like planar anisotropy and grain size, as well as weathering state, influence the fracture surface. The results allow further understanding of the fracture process in quasi-brittle materials and contribute to the assessment of the mechanical characterization of granite under tension.

Determination of double-Determination of double- K criterion for crack propagation in quasi-brittle fracture Part I: experimental investigation of crack propagation

The results of experimental investigations using laser speckle interferometry on small size three-point bending notched beams and using photoelastic coating and the strain gauges on very large size compact tension specimens of concrete are presented in detail. The investigations showed that there exists a stage of stable crack propagation before unstable fracture occurs. The results are in agreement with other researchers' investigations using moire interferometry, holographic interferometry, dye-impregnation method and microscope. Further detailed study shows that the three different states, i.e., crack initiation, stable crack propagation and unstable fracture can be distinguished in the fracture process in concrete structures. In order to predict the crack propagation during the fracture process in quasi-brittle materials a double-K criterion is proposed. The double-K criterion consists of two size-independent parameters. Both of them are expressed in terms of the stress intensity factors. One of them reflects the initial cracking toughness, denoted with Kini, which can be directly evaluated by the initial cracking load, Pini, and the precast crack length, a0, using a formula of LEFM. The other one refers to the unstable fracture toughness, denoted with Kun, which can be obtained inserting the maximum load, Pmax, and the effective crack length, a, into the same formula of LEFM. The values of the two parameters, KIcini and KIcun, obtained from the small size three-point bending notched beams and the large size compact tension specimens show that KIcini and KIcun are size-independent. Evaluating with the K-resistance curves obtained from the same test data, it is found that the proposed double-K criterion is equivalent to it in basic principle, but, the double-K criterion can be applied more easily than the K-resistance curve. Finally, as a practical example, the application of the double-K criterion to the prediction of the crack propagation in a concrete dam is discussed.