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The repeatedly applied low-intensity loads would lead to the damage and fatigue crack growth of mechanical structures made of quasi-brittle materials. In numerical modelling, these two mechanisms are normally treated differently and separately; the damage is usually associated with nonlocal approaches, while the fatigue crack growth is related to the local stress intensity range at the crack tip. In this study, a discrete element model for damage and fatigue crack growth of quasi-brittle materials is proposed, which is able to model the damage and fatigue crack growth simultaneously in one single model. The proposed model achieves the implementation of a continuum damage model in a discrete element code, which is a helpful enrichment of this numerical method. The evaluation method of the stress intensity range during the damage evolution provides a way to couple both failure mechanisms. This feature allows crack initiation to be induced by localized damage and a progressive transition to a fracture behaviour with the crack propagation. Independent parameters for the fatigue damage model and fatigue crack growth model are admitted without any previous calibration. The numerical results are in good agreement with the theoretical predictions of damage and fracture mechanics, and intact and precracked samples are analysed under fatigue loading to show the consistent coexistence of fractured and damaged zones in a single model.

The size and boundary effects of quasi-brittle fractures have been studied systematically in the past decades [

The fatigue behaviour can be studied experimentally and numerically. The experimental study is commonly expensive and time-consuming and sometimes impossible in the case of huge structures, while the numerical study is time- and cost-efficient and can effectively enable researchers to optimize the experimental effort required [

Both continuous and discrete numerical methods have their own advantages and shortcomings in the modelling of fracture and damage phenomena. The finite element method (FEM) is cumbersome because the mesh requires to be updated to match the geometry of the discontinuity [

In this study, a discrete element approach is proposed based on a local description of damage and fracture. The continuum damage model is implemented in a discrete element code and verifies the theoretical prediction. In Section

CDMs are capable of predicting the fatigue life of the specimen without large cracks. During the fatigue tests, as the material is deformed, the initiation, growth, and coalescence of microdefects decrease the stiffness (degradation of material properties), which is represented by the growth of the damage variable

Several stress- and strain-based continuum damage models have been established by the researchers, including the models established by Castro and Sanchez [

Bodin et al. proposed an elastic isotropic continuum damage model for fatigue, which characterizes the decrease in stiffness with cyclic loading. The damage growth criterion is based on a modified Rankine criterion with zero threshold damage growth. Evolution of damage (local version of the model) is controlled by the strain state of the material by a scalar equivalent strain, which can be written as follows:

In the nonlocal version of Bodin’s model, the “local” equivalent strain

The rate of damage growth is defined as a function of the local equivalent strain rate:

The function of damage

The existence of cracks can significantly reduce the fatigue life of a component or the whole structure. Some physical evidence is quite intuitive: at higher stresses, a crack tends to propagate “faster” (with respect of the number of cycles), and at similar stress levels, a bigger crack tends also to propagate “faster.” The propagation of localized cracks depends on many parameters and is not well characterized by damage models. Fatigue crack growth models (FCGMs) take into account the variation of the stress intensity factor during the cycles to describe the crack propagation [

Fatigue crack growth in a wide variety of brittle and quasi-brittle materials [

The discrete element method (DEM) was originally developed by Cundall for modelling granular and particulate systems [

Challenges related to calibration between the macro- and micromaterial parameters can be avoided if a close-packed assembly (regular hexagonal packing) is adopted, as shown in Figure

Equivalent (a) continuous and (b) discrete media (close-packed assembly).

DEM discretizes a material using elements of simple shapes (circles, spheres, or blocks) that interact with neighbouring elements according to laws of interaction that are applied at points of contact. At each time step, the computation of all contact forces is followed by the application of Newton’s second law to the particles. Each contact force has normal and tangential components,

(a) Unit cell and (b) contact law (modified from [

Young’s modulus

The mean values of the components of the tensors of stress and strain are based on the behaviour of one pair of contacts (

(a) Contact displacements and (b) contact forces of respective adjacent particles. (c) Mean stresses and the orientation of their principal values [

The normal and tangential components of each contact (

The stress tensor (in two dimensions) can be defined by the values of the principal stresses

The value of

Most of the fatigue laboratory tests of a high number of cycles consist of applying a sinusoidal displacement (or force) with constant amplitude at the boundary of the sample. During testing, the variation of global stiffness is monitored, which is defined as the ratio between the amplitudes of the force and the displacement. In order to numerically model these time-consuming laboratory tests, Bodin et al. [

In this section, a local version of Bodin’s L2R damage model is implemented in a discrete element code. A close-packed assembly (regular hexagonal packing) is adopted due to the direct relationship between the macroscopic parameters (Young’s modulus

The damage of all the contacts in the assembly is calculated in the same way and updated at the same time. The evaluation of the damage per contact can be summarized by the following operations:

DEM elastic analysis and identification of the stress and strain fields (Section

Evaluation of the principal stresses: the stress tensor

Calculation of the equivalent strain (Equation (

Local equivalent strain of the contact: the equivalent strains of the contact pairs are averaged to contact points. Since damage is associated with the single contact, the mean equivalent strain of the contact pairs around a certain contact (calculated during the previous operation) is adopted as the local equivalent strain for the contact. Only the existing contact pairs of the scheme shown in Figure

Evaluation of the damage growth: the damage growth rate is defined as a function of the local equivalent strain rate

Evaluation of the damage and update: the value of Young’s modulus (

Contact pairs associated with the average of the equivalent strain of the contact

An uncracked plate with dimensions

An uncracked plate subjected to imposed sinusoidal strain.

The discrete element model is shown in Figure

The discrete element model of an uncracked plate subjected to imposed sinusoidal strain.

The numerical results are compared with Bodin’s L2R damage model predictions, as shown in Figures

The theoretical and numerical predictions of the increase of damage for an uncracked plate subjected to fatigue loading.

The theoretical and numerical predictions of the decrease of global stiffness for an uncracked plate subjected to fatigue loading.

The calculation speed of the discrete element modelling depends on the discretization level, the discrete element code, computing power, etc. This calculation was performed on an Intel Core i7-6700K

In view of the crack initiation and propagation, the failure modes of quasi-brittle materials subjected to fatigue loading can be described by four stages, including crack nucleation (Stage I), short crack growth (Stage II), large crack growth (Stage III), and ultimate failure (Stage IV) (Figure

The four stages of the fatigue failure process [

The crack nucleation stage and most of the short crack growth stage were shown to be well described by the continuum damage model. However, as the crack length increases, the decrease in the global stiffness becomes dominated by crack propagation. At this point, the continuum damage model failed, resulting in fast propagation due to the stress singularity at the crack tip. Hence, it is necessary to adopt a fatigue crack growth model (e.g., Paris’ law) to estimate better the fracture behaviour during the end of stage II and stage III.

Damage models (Section

In the present work, a damage approach (Section

The propagation of a crack can be analysed as the creation or extension of the boundaries of a given geometry. In fracture mechanics, this transformation is usually controlled by the energy release during the process. Despite the different existing criteria of crack propagation, roughly a crack may be created or propagated where the stress (and/or strain) is maximized (Figure

Localization of the local maxima of the stress/strain for (a) a cracked plate and (b) a simply supported beam.

In an elastic system, a simple verification of the local maximum value of the principal stress may be enough to identify potential localization of crack tips. However, during fatigue, the value of the stress tends to decrease due to the degradation of elastic properties of the material where the stress is concentrated. A better indicator, in this case, is shown to be the damage increment per cycle

Damage increment per cycle of a center cracked plate.

In the present model, the possibility of crack propagation will only be considered on contacts, which locally maximizes the damage increment per cycle, as shown in Figure

A damage value

In crack growth models, the energy release is considered to be localized exclusively at the crack tip. An overdamaged zone near a crack tip leads automatically to inconsistent evaluations of the energy release rates at the crack tip. In order to avoid this disturbance due to unphysical damage values, if damage reaches

(a) Scale effect on the damage value and (b) contact points in the potential crack path.

The principal components of the contact forces

Principal components of force and displacement for a certain contact being (a) the first contact (defined clockwise) in a pair and (b) the second contact in a pair.

Evaluation of the energy release at the crack tip (contact

The second contact

Evaluation of the energy release at the second contact close to the crack tip during a fatigue test.

Based on the relation between the energy release rate and the stress intensity factor in plane stress [

The evolution of the damage variable

A contact which presents

Figure

Center cracked plate subjected to imposed sinusoidal stress.

In Figure

Comparison of the range of stress intensity factor

The effect of the particle size

For the condition of imposed stress with constant amplitude, the amplitude of the displacement at the edge of the sample tends to increase during the fatigue test. This behaviour is due to the decrease of the global stiffness of the sample caused by damage of the material and the propagation of the crack. The process of the stiffness degradation can be quantified by the ratio of the initial displacement amplitude and its instantaneous value at a time during the test, which is a function of the number of cycles

Figure

Stiffness degradation as a function of the number of cycles

Figure

Damage distribution and crack propagation of a precracked plate with the initial crack length

The increase of the number of cycles causes the evolution of the high damage zone due to the extension of the fatigue crack, as shown in Figure

The damage and fatigue crack growth are normally modelled separately by different approaches. The damage is usually modelled by nonlocal approaches implemented in a finite element code, which may produce reasonable global sample behaviour, based on unrealistic material behaviour. The fatigue crack growth model is related to the local stress intensity range at the crack tip, which can produce reasonable rupture patterns but fails to predict the correct fatigue life and global behaviour when there is no crack or only small cracks in the sample. In order to reduce these limitations of two different approaches, a simple numerical scheme coupling damage and fracture mechanics in a discrete element environment was proposed. The association of these different mechanical formulations allows the reproduction of experimental evidence: before material rupture by damage models and during crack propagation by crack growth models. In parallel, important drawbacks of each approach are avoided.

In this study, the local version of the continuum damage model was successfully implemented in the discrete element code and compared to the theoretical prediction showing good agreement under homogeneous stress conditions. This discrete element fatigue model would be a helpful enrichment for the discrete element method and shares as well the advantages of this numerical method in terms of the construction of the models with voids, imperfections, or heterogeneities and the simulation of crack initiation and propagation. The evaluation method of the stress intensity range during the damage evolution provides a way to couple both failure mechanisms. This feature allows crack initiation to be induced by localized damage and a progressive transition to a fracture behaviour with the crack propagation. Independent parameters for the fatigue damage model and fatigue crack growth model are admitted without any previous calibration. The numerical results are in good agreement with the theoretical predictions of damage and fracture mechanics, and intact and precracked samples are analysed under fatigue loading to show the consistent coexistence of fractured and damaged zones in a single model. The proposed model can be further extended to study the crack nucleation and short crack growth problems for varied microstructures. Such a numerical model can be constructed by representing different materials (aggregates, mortar, etc.) as clusters of discrete elements with different parameters, including the mechanical properties of the contacts and model parameters for the continuum damage model and fatigue crack growth model.

The practical applications and precisely comparison with the experimental results will be performed in the future study, and the implementations of the aggregate or grain size in the analytical equations and discrete element code would make this model more realistic and robust.

The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.

The authors declare no conflicts of interest.

This manuscript is based on Xiaofeng Gao’s doctoral thesis. This research was supported by the program of the China Scholarship Council and China Postdoctoral Science Foundation (Grant no. 2018M631478).