Behavior Of Quasi-brittle Materials Research Articles

A staggered local damage model for fracture analysis in bi-material structures

This article is devoted to extension of the recently developed enhanced local damage model for failure prediction in bi-material structures. Compared to non-local models, the enhanced local model offers lower computational cost while the inherent mesh-dependency issue is treated. By defining equivalent strain based on the bi-energy norm concept and Mazars’s criterion, which considers both tensile and compressive strain components, the model aligns with the behavior of quasi-brittle materials. The state of material point is indicated by a damage parameter, ranging from 0 to 1, to represent the evolution from being fully intact to complete failure. An efficient staggered scheme is introduced, in which the equilibrium equation and the update of damage parameter are decoupled. The proposed model is validated with a series of three-point bending experimental tests on PMMA/Al6061 specimens reported by Lee and Krishnaswamy (2000). Good agreement is observed between the proposed model and experimental data, as well as numerical results from other authors, in crack path prediction.

Mesh objective characteristic element length for higher-order finite beam elements

The use of fracture energy regularization techniques can effectively mitigate the mesh dependency of numerical solutions caused by the strain softening behavior of quasi-brittle materials. However, the successful regularization depends on the correct estimation of the crack bandwidth in Finite Element solutions. This paper aims to present an enhanced crack band formulation to overcome the strain localization instability especially for the higher-order elements developed in the framework of Carrera Unified Formulation (CUF). Besides, a modified Mazars damage method incorporating fracture energy regularization is employed to describe the nonlinear damage behavior of the concrete. To evaluate the efficiency of the proposed crack band formulation, three experimental concrete benchmarks are selected for the numerical damage analysis. By comparing numerical and experimental results, the proposed method can guarantee mesh objectivity despite varying finite element numbers and orders, indicating perseved fracture energy consumption within proposed higher-order beam models.

A novel non-linear peridynamic model for quasi-brittle materials under cyclic loading

In the present paper, a novel non-linear peridynamic (PD) model is proposed to numerically simulate the fatigue behaviour of quasi-brittle materials. The model implements a non-linear degradation law for the bonds, function of the number of loading cycles. More precisely, a calibration function and a calibration parameter are included in such a law. The novel PD model is applied to simulate cyclic three-point bending experimental tests performed on concrete notched beams (available in the literature). The results are presented in terms of fatigue lifetime, Paris law parameters and crack path.

Physically consistent nonlocal macro–meso-scale damage model for quasi-brittle materials: A unified multiscale perspective

Despite great efforts in capturing the damage evolution law in multiscale approaches in damage mechanics, much less attention has been paid to the conversion from geometric damage to energy dissipation. In the present paper, the newly proposed nonlocal macro–meso-scale damage (NMMD) model is physically consolidated and a multiscale point of view is consistently adopted in both the damage evolution and energy dissipation. In this model, for each macroscopic material point a mesoscopic structure is attached such that all the point pairs composed of this point and the points in its influence domain are connected to it when the material is intact. The deformation field under loading will result in deformation of point pairs, and thus mesoscopic damage related to a point pair will occur if the deformation index of this point pair exceeds the prescribed threshold. Such mesoscopic change among point pairs connected to a material point within its influence domain has two-fold consequences: geometrically the accumulation of mesoscopic damage will result in macroscopic discontinuity, and simultaneously the free energy will be dissipated, leading to degradation of mechanical behavior of quasi-brittle materials. In other words, the damage as a metric of the geometric discontinuity will lead to the damage in the sense of energy dissipation and degradation of mechanical behavior for quasi-brittle materials, which can be physically captured by the ratio of summation of all the dissipated mesoscopic free energy in point pairs to the free energy of intact material within the influence domain. This energy-based macroscopic damage could then be inserted into the framework of continuum damage mechanics. Therefore, the conversion from geometric damage to energetic damage is implemented on the mesoscale instead of the macroscale, and thus a macroscopic energetic degradation function is not needed. In addition, the structured strain which is suitable for quasi-brittle materials can be adopted to determine the deformation index of point pairs. Correspondingly, the macroscopic free energy can be split into dissipative and non-dissipative parts. In this way, several inconsistencies in the previous NMMD models are remedied. The influence of mesoscopic model parameters is investigated in depth, and several numerical examples including mode-I, mixed-mode and compressive splitting fracture problems of quasi-brittle materials are carried out. Numerical results indicate that the proposed model can not only well capture the cracking process with or without initial cracks but also quantitatively predict the load–deformation curves without mesh size sensitivity. Moreover, due to the inherent physical consistency, in mixed-mode cracking problems the proposed model performs better than the previous NMMD model. Problems to be further studied are also discussed.

A thermodynamically consistent strain difference-based nonlocal damage mechanics approach for failure analysis of quasi-brittle materials

The nonlocal integral approaches are established methods known for their effectiveness in dealing with the challenges of ill-posed behaviour and mesh sensitivity during the strain softening process, especially within classical continuum theories. The present work extends the strain difference-based nonlocal approach to continuum damage mechanics (CDM) theories for predicting the failure behaviour of quasi-brittle materials. The strain difference approach is adopted for two reasons: (i) it maintains physical realism and indirectly incorporates nonlocal effects while computing both the damage variable and the damage energy release rate, and (ii) the constitutive model maintains a symmetry nature in constructing the nonlocal part of the stiffness matrix, reducing the computational cost in strain-softening problems. The state and damage evolution equations for the present nonlocal approach are formulated within the thermodynamic framework. Numerical examples involving 2D and 3D cases are solved: (i) to study the influence of nonlocal parameter ‘α’ on the damage distribution while capturing the post-peak behaviours, (ii) to demonstrate the effectiveness of the present methodology by obtaining mesh-independent results and accurately predicting damage propagation paths, and (iii) to compare the experimental results and numerical structural response for coarse mesh, highlighting both computational efficiency and reduction of mesh-bias dependency. The results obtained by the strain difference-based nonlocal damage model agree well with the experimental results in the literature.

Application of a Closed-Form Model in Analyzing the Fracture of Quasi-Brittle Materials.

Fracture failure in quasi-brittle materials poses a persistent challenge in materials science and engineering. This study presents a thorough investigation of the Boundary Effect Model (BEM), offering a nuanced understanding of the size effect on fracture properties. The conceptual framework, evolutionary process, and applicability scope of BEM are elucidated, highlighting its accuracy and reliability in calculating fracture properties across various quasi-brittle materials. Through the integration of BEM with diverse fracture tests-such as three-point bending, four-point bending, and wedge-splitting-a linear correlation between maximum failure loads and material fracture properties is established. Notably, the study demonstrates that fracture properties, determined by BEM, can be regarded as consistent material constants across specimens of varying sizes, initial notch lengths, geometries, and microstructures. Validation of the BEM's reliability encompasses the analysis of 140 fracture test results involving concrete, hard rocks, and bamboo scrimber. The synergy of non-linear and linear BEM analyses emerges as a robust approach for accurately predicting the fracture behavior of quasi-brittle materials. This comprehensive exploration sheds light on the potential of the Boundary Effect Model as a valuable tool for predicting and understanding fracture mechanics in diverse materials and scenarios. This research serves as an effective approach to accurately evaluating the fracture properties of quasi-brittle materials, which is of great practical significance for material design, engineering construction, and various industrial applications.

Predicting fracture behavior of quasi-Brittle materials using quantitatively varied geometric specimens: A design method

Fracture toughness (KIC) and tensile strength (ft) are two important parameters for describing the fracture performance of quasi-brittle materials. The accurate determination of both parameters is crucial for the safety design and stability evaluation of real engineering structures. A model and method are developed in this paper for simultaneously determining the KIC and ft for quasi-brittle materials without size effects. And using the geometric structural parameter ae as a key design parameter. This method can get KIC and ft to be obtained by fitting curves to any two sets of quantitative specimens with different geometric structures with ae ratio equal to 3. Additionally, fracture full curves and peak load prediction lines with 95% reliability are established for different fracture modes using quantitative design specimens with an ae ratio of 3. These curves and lines can accurately predict fracture behavior and peak loads of large specimens or engineering structures. The proposed design method is verified by combining experimental studies conducted by other scholars on quasi-brittle materials and the simulation results of ABAQUS software. This paper achieves the objective of accurately predicting the fracture properties of quasi-brittle material structures in the field, using quantitative laboratory specimens of various geometries.

Modeling tensile failure of concrete considering multivariate correlated random fields of material parameters

This paper presents a novel approach for modeling the tensile failure of quasi-brittle materials by incorporating a multivariate random field to represent material parameters in the phase field model. The aim is to characterize and propagate uncertainties in the behaviors of materials, taking into account the spatial variability and probabilistic dependence. Copula theory is employed to investigate the probability distributions and dependence configuration of the parameters. The proposed framework combines the generation approach for the multivariate random field with the phase field model, resulting in a complete numerical analysis methodology. The governing equations of the deterministic phase field model for quasi-brittle solids are outlined, followed by a description of the multivariate random field for quasi-brittle media and its uncertainty characterization procedure using copula theory. A numerical approach for generating samples of the multivariate random field is introduced. The numerical analysis procedure for the tensile fracture of concrete is presented, and the effects of material uncertainties on failure patterns and macroscopic responses are discussed. The results demonstrate that the developed methodology effectively captures the random damage and fracture processes in concrete specimens. The interaction between the correlation length of the random field and the characteristic length scale of the phase field significantly influences the probabilistic characteristics of the random responses. The research findings emphasize the importance of determining the correlation length values accurately. This study contributes to the field of structural reliability analysis by providing a comprehensive framework for modeling the stochastic behavior of quasi-brittle materials. The integration of a multivariate random field and copula theory allows for the realistic representation of material uncertainties and their probabilistic dependence. The proposed methodology, combined with the probability density evolution method (PDEM), offers a powerful tool for analyzing and predicting the probabilistic evolution of material behaviors under tensile loading conditions.

Phase-field modeling of stochastic fracture in heterogeneous quasi-brittle solids

Owing to the random nature of heterogeneity, damage and fracture behavior of quasi-brittle materials exhibits a considerable degree of uncertainty. Computational modeling of stochastic fracture in quasi-brittle materials has become an indispensable tool for analysis and design of engineering structures. To this end, we present in this paper a computational framework to capture probabilistic fracture in heterogeneous quasi-brittle solids by combining the random field theory and the phase-field cohesive zone model (PF-CZM). The spatial variation of the material strength and fracture energy is represented by a cross-correlated bivariate random field generated by the Karhunen–Loève expansion. The recently proposed PF-CZM is employed to simulate the stochastic crack nucleation and propagation in quasi-brittle solids. The objectivity of the Monte-Carlo simulation is achieved by imposing a specific condition on the phase-field length scale parameter and the correlation length of the random field. In particular, upon this condition the width of the fracture process zone (FPZ) is considerably smaller than the correlation length of the random field such that the material inside the FPZ does not exhibit significant spatial variations of mechanical properties. As the fracture energy is intrinsically incorporated in the PF-CZM, it is unnecessary in this case to explicitly consider the FPZ width. The resulting probabilistic PF-CZM together is applied to the Monte-Carlo simulations of fracture in concrete structures of different geometries. It is shown that the stochastic simulation results are insensitive to both the phase-field length scale parameter and the finite element mesh discretization as in the previous deterministic analyses. Enhanced with the specific condition on the involved characteristic lengths, the PF-CZM provides a viable tool for stochastic simulations of damage and fracture in quasi-brittle structures.

Relationship between probabilistic characteristics of microscopic and macroscopic strength within the stochastic phase-field model

Stochastic phase-field models can reproduce the nonlinearity and randomness of the mechanical behavior of quasi-brittle materials within a unified macroscopic continuum framework. To study the probabilistic characteristics of microscopic and macroscopic strength within the stochastic phase-field model and their relationship, a series of numerical results of a specimen subjected to uniaxial tension using a stochastic phase-field model is presented. The probabilistic characteristics of the macroscopic responses and the underlying mechanisms are discussed. Based on the numerical results, a probabilistic model is developed, which can effectively determine the probability information of the macroscopic structural strength of quasi-brittle uniaxial tensile specimens. In particular, the interaction of several characteristic scales involved in the stochastic phase-field model can be demonstrated.

Numerical study and experimental validation of the size effect of smooth and mode I cracked semi-circular bend specimens

The edge-cracked semi-circular bend (SCB) specimen subjected to three-point bending loading is used in many applications to measure the fracture behavior of quasi-brittle materials. The main objective of the present work was to study the effect of the crack length to SCB specimen radius ratio (a/R), span to specimen diameter ratio (S/D), and specimen size on its flexural and mode I crack growth behavior. The contour integral method was implemented using the 3-D finite element method to determine the mode I stress intensity factor. In addition, high-strength concrete specimens were experimentally studied to validate the numerical results. The results show that the maximum compression stress is not sensitive to the S/D value, while the tensile stress is very sensitive. The value of S/D is the main parameter controlling the crack driving force (i.e., the crack mouth opening displacement (CMOD) and the normalized stress intensity factor, YI). For the same S/D, the SCB specimen diameter value change has a marginal effect on CMOD and YI. The specimen with S/D = 0.8 showed that it is the most compatible specimen with three-point bending test conditions, regardless of the SCB specimen size. A good agreement between the numerical and experimental results was achieved.

Size effect on fracture behavior of quasi-brittle materials during uniaxial compression tests

Size effect on fracture behavior of quasi-brittle materials during uniaxial compression tests

Determining the fracture toughness of quasi-brittle materials with notched four-point bending tests

Fracture is common failure mode of quasi-brittle materials when suffering tensile loads, the accurate determination of the fracture toughness becomes extremely vital. Notched four-point bending (4-p-b) test is one of main fracture test methods, but is rarely applied in evaluating the fracture behavior of quasi-brittle materials. In this study, on the basis of relevant standards, the specific requirement of 4-p-b fracture test is proposed. Then the closed-form solution is used to establish the relation between the four-point bending test results and fracture toughness on the basis of the Boundary Effect Model. The influence of test conditions, boundary effect and distribution characteristics of fracture properties are discussed in detail. After that, two types of hard rock and concrete are utilized to verify the reliability of proposed methods. The results show that four-point bending test and corresponding analytical solution are suitable for determining the fracture toughness of quasi-brittle materials, regardless of the different initial notch length and material categories.

A non-linear cohesive zone model for low-cycle fatigue of quasi-brittle materials

The low-cycle fatigue behaviour of quasi-brittle materials (e.g., concrete and rock) that is characterized by fatigue induced inelastic deformation significantly affects the integrity and serviceability of engineering structures. However, the low-cycle fatigue mechanism and fatigue-controlled fracture process of quasi-brittle materials is not clear. This paper develops a new cyclic cohesive zone model (CZM) for low-cycle fatigue of quasi-brittle materials. Based on in-situ stress and damage state, a nonlinear fatigue damage model is proposed and implemented into the cyclic CZM. The fatigue parameters are determined based on S–N curve. A worked example for monotonic and cyclic loading of concrete beam under three-point bending is presented to demonstrate the application of the developed numerical model. After validation against experimental data, the fatigue crack mechanisms are discussed and a comprehensive parametric study is carried out to investigate the effects of fatigue parameters, stress levels and loading sequences on the fatigue failure. It has been found that there are three stages for the development of crack mouth of displacement, i.e., crack initiation, stable growth and rapid fracture which are caused by combined static and fatigue damage, fatigue damage, and combined static and fatigue damage dominated by static damage, respectively. The developed cyclic CZM is practically significant and its parameters are easy to be determined based on S–N curve. It provides a new and useful tool for low-cycle fatigue crack modelling of quasi-brittle materials.

A unified non-linear energy dissipation-based plastic-damage model for cyclic loading

A new energy-dissipation-based rate-independent constitutive law within the framework of elastoplasticity coupled with damage is proposed. With this methodology, the inelastic strains and the stiffness degradation exhibited by quasi-brittle materials under monotonic or cyclic loading conditions are taken into account. The proposed constitutive model is able to capture micro-cracks closure-reopening effects due to load reversal. A wide variety of hardening/softening laws on the stress–strain relationship are described and considered for the novel normalized plastic-damage energy dissipation internal variable. This normalized internal variable allows the model to be independent on the sign of the load and dissipate different fracture energies (tensile, compressive and potentially shear) in a natural way. Several numerical examples are presented in order to ensure the efficiency and validity of the proposed model for simulating the non-linear behaviour of quasi-brittle materials under monotonic and cyclic loading. Some numerical aspects of the implemented algorithm and the return mapping procedure are also described in detail and discussed.

A novel peridynamic approach for fracture analysis of quasi-brittle materials

In the present paper, a novel peridynamic approach is proposed to simulate the fracture behaviour of quasi-brittle materials. It consists in: (i) an improved bond force-stretch relationship to implement into peridynamic models, in order to simulate the fracture behaviour of quasi-brittle materials, and (ii) the proposal to consider, into the above models, the material energy release rate as a 3D random field, to take into account the stochastic nature of quasi-brittle material properties. Such a novel peridynamic approach is verified by analysing four case studies related to quasi-brittle materials, that is: rock, expanded polystyrene, and concrete.

Construction of Discrete Element Constitutive Relationship and Simulation of Fracture Performance of Quasi-Brittle Materials.

In order to solve the problem that the built-in parallel bond model in the discrete element software cannot adequately simulate the post-peak fracture behavior of quasi-brittle materials, a linear cohesive model was established. First, two particles are used to simulate the interface constitutive behavior in different modes. The results show that the new model can better simulate the behavior of Mode-I fracture, Mode-II fracture, and Mixed-mode fracture. Then, the influence of micro-parameters on the newly constructed constitutive model is analyzed, which provides a basis for the determination of micro-parameter values. Finally, the proposed softening model is applied to a three-point bending test of mortar, and the fracture behavior obtained is compared to the acoustic emission results. The simulation results also show that the constitutive model we built can be used to simulate the fracture behavior of quasi-brittle materials such as mortar and concrete.

Mechanical fracture parameters of concrete drill-core specimens supported by a slenderness ratio study

The detailed analytical and experimental investigation of the fracture behaviour of quasi-brittle materials is an endeavour which has been ongoing worldwide for many years. Such materials are usually characterized in terms of their mechanical fracture parameters, which are determined based on the evaluation of quasi-static fracture experiments. One of the most commonly used building materials with a quasi-brittle response is concrete, which is most often based on a cement matrix. It is sometimes also necessary to characterize concrete included in existing structures. In this case, test specimens are obtained by core drilling, and the investigation is conducted with the requirement to maximize the number of parameters obtained while minimizing the number of test specimens drilled from the structure. This paper focuses on the mechanical fracture parameters of core-drilled specimens taken from a selected concrete structure. Tests were performed on cylindrical specimens with a chevron-notched stress concentrator in the three-point bending configuration in order to determine modulus of elasticity, fracture toughness and fracture energy. Subsequently, theoretical compressive strength was estimated and tests for the determination of compressive strength values were performed focusing on dependence on the slenderness ratio, i.e. the relationship between the compressive strength and the length to diameter ratio of the cylindrical specimens. In relation to the obtained mechanical fracture parameters, selected specimens were analysed and three-dimensionally characterized via high-resolution X-ray computed tomography.

A subcritical microcrack informed creep-damage model for quasi-brittle materials

A subcritical microcrack informed creep-damage model is proposed to describe the creep-failure behavior of quasi-brittle materials where the subcritical microcracks propagation is characterized by the subcritical damage. Total free energy is decomposed into the time-dependent component and time-independent component where the time-dependent Helmholtz free energy is associated with time-dependent behavior. Driven by the time-dependent free energy the subcritical damage growth is originated from the nano-scale fracture kinetics. Within the continuum damage framework, a modified stochastic damage model is introduced to account for instantaneous behavior and subcritical damage evolution. The viscoelastic behavior is characterized by the Burgers model. The proposed model is validated through several numerical simulations including uniaxial loading tests, uniaxial creep tests and bending creep test. The results demonstrate that the present model is promising for the long-term analysis of underground structures.

A new brittle-elastoviscoplastic fluid based on the Drucker–Prager plasticity

A new brittle-elastoviscoplastic (BEVP) fluid model is presented in this paper. This model is relatively simple to use, as it contains few material parameters and a simple fixed-point algorithm is effective for solving the coupled system of equations. The model combines some existing fundamental features such as elasticity, plasticity and brittle damage. The combination of them is based on thermodynamics that ensures the positivity of the dissipation and the Onsager symmetry. Moreover, thermodynamics allows to point out the link between thixotropy and damage in the context of elastoviscoplastic (EVP) fluids. Theoretical results on the Drucker–Prager plasticity criterion are completed in order to use it here. Preliminary results with the proposed BEVP model are very encouraging: it is able to represent the pre-failure, failure and post-failure behavior of quasi-brittle materials. • A new brittle-elastoviscoplastic (BEVP) fluid model. • A design process based on thermodynamics. • Application to brittle complex materials. • Numerical simulations for the uniaxial compression benchmark.