Modelling Of Quasi-brittle Materials Research Articles

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.

Energy degradation mode in nonlocal Macro-Meso-Scale damage consistent model for quasi-brittle materials

The accurate simulation of crack initiation and propagation plays an important role in exploring the failure mechanism and safety evaluation of engineering structures. The proposal of the fracture phase-field theory has made a breakthrough in crack simulation. Recently, the nonlocal macro-meso-scale consistent damage (NMMD) models are more efficient in predicting crack initiation and propagation. However, the energy degradation mechanism caused by the cracking phase, which plays a critical role in these models, is still more or less empirically determined. In this paper, assuming that the geometric damage in the NMMD model follows an exponential decay distribution and advocates the equivalence principle of energy dissipation, the intrinsic correlation between the energy degradation mode and the influence domain size is explored. It is of physical interest to find that the faster the energy degradation in the initial stage, the slower the energy degradation in the later stage, requiring a larger range of influence domain. The example of a three-point bending of the notched beam with different energy degradation modes validates this discovery. Moreover, the detailed implementation for the secondary development of the NMMD model in the Abaqus platform is provided. Several two- and three-dimensional fracture benchmarks including the dynamic cracking problem and thermal–mechanical coupling problem are verified with the well-established solver in Abaqus.

Introducing a novel LLD-CMOD equivalence fracture model for quasi-brittle materials exposed to three-point bending

Effective crack models have been preferred in the fracture of concrete thanks to their analytical solution for equivalent effective crack length. Nevertheless, the main hypothesis in the effective crack models, the Load-LLD or Load-CMOD equivalence between a real cohesive material and a corresponding virtual specimen, is controversial. The present study aims to offer a distinct perspective on the effective crack approach. To this end, the Load-LLD or the Load-CMOD equivalence was replaced by the LLD-CMOD equivalence in the proposed fracture model for the notched beams under three-point bending. In the proposed LLD-CMOD equivalence fracture model, three fracture toughness parameters are used, two of them depend on the evolution of the fracture process zone (FPZ) and the other is for the peak load measured. That is, KICin and KICde define the limit of the nonexistent FPZ and the fully developed FPZ, respectively, but KICun corresponds to the peak load only. The test results of the notched beams revealed that KICin and KICun show size independence, but this was not the case for KICde. The proposed fracture model was also compared with the Double-K model through the experimental results from the literature. Accordingly, a good agreement between both models was achieved in terms of the effective crack length and fracture toughness at the peak load.

Anisotropic damage state modeling based on harmonic decomposition and discrete simulation of fracture

This study proposes an anisotropic damage state modeling of the effective (damaged) elasticity tensor E˜ as a function of damage based on (i) a discrete element model for quasi-brittle material and (ii) a decomposition of the elasticity tensor in covariants. A procedure is proposed to measure the evolution of effective elasticity tensors computed by a beam-particle model. Various multiaxial damaging loadings allow us to constitute a dataset of around 76 000 effective elasticity tensors (our virtual testing reference set). We then cross-identify the anisotropic/tensorial damage state for the whole dataset. A detailed analysis of the dataset, using the distance to orthotropy as a guideline, justifies representing the induced micro-cracking by a single second-order damage variable D, even in the final stages with strong micro-cracks interaction. To formulate the damage state coupling, we use a reconstruction formula of orthotropic elasticity tensors in terms of invariants and (tensor) covariants. Thanks to this formula, some parts of the effective elasticity tensors E˜ (such as the dilatation part) are modeled exactly from the single damage variable. Constitutive equations are proposed for the remaining parts of E˜ (such as its generalized shear modulus and fourth-order harmonic part) using physical assumptions from micro-mechanics and a sparse data driven approach. The proposed anisotropic damage state coupling E˜(D) accurately models the damaged elasticity tensors in multiaxial loading, proportional or non-proportional, up to high damages. The present study firstly highlights the need for an anisotropic damage model for quasi-brittle materials and, secondly, offers a methodology to formulate the damage state coupling by explicit formulas introducing at most two dedicated parameters: the (optional) nonlinear shear-damage coupling parameter m and the harmonic prefactor h.

Vector-based damage-driven computational homogenization with localized gradient enhanced boundary conditions for multi-scale modelling of quasi-brittle materials

Although a lot of progress has been made in computational homogenization, it is still a difficult challenge to emerge and evolve localization bands robustly and to tackle the size dependency issue for brittle and quasi-brittle materials. In this paper, a vector-based damage-driven computational homogenization with localized gradient enhanced boundary conditions is proposed. In the homogenization, the crack band theory is used to up-scale the damage at the micro level to the macro level based on the unit cell size and element size. The vector-based homogenization introduces damage in the principal direction based on the smeared crack model. The updated localized displacement gradients are introduced in the boundary conditions of unit cells to localize crack propagation between adjacent unit cells. The localized displacement gradients are updated iteratively to consider crack open and closure as well as the influences of surrounding elements. Incremental sequentially linear analysis is used to guarantee computational robustness. Size objectivity and localization have been demonstrated for different unit cell sizes and element sizes compared with the direct simulation method (DSM). In addition, an array of voids is considered in the notched beam test for further validation.

Regularized local damage model for quasi-brittle material

This paper presents an improved local damage model for quasi-brittle materials. A parameter, namely damage, in the range [0,1] is used to characterize the material from intact state to complete failure. Classical local damage model is known for low computational cost but it suffers from numerical issues such as difficult convergence and mesh-density-dependent results. Various non-local models have therefore been proposed, however the computational cost is increased, which hinders the applicability in practical problems. Furthermore, the width of the damage zone predicted by the non-local models is usually non-physically large. Here, the improvement is the introduction of the fracture energy and element-size into the calculation of damage parameters to mitigate the weakness of the local damage model, while keeping low computational cost. The employment of polygonal element is also proposed to utilize the advantage of high-accuracy, thus less number of elements (than the usual 3-node triangular or 4-node quadrilateral elements) can be used. The bi-energy norm-based equivalent strain is for the first time considered in a local damage model. This quantity is based on the maximum strain criterion, but with a modification to take into account the property of quasi-brittle materials such as concrete, limestone, etc., that load capacity in compression is higher than in tension. Accuracy and efficiency of the proposed model is investigated through comparison with reference results from experiments and other numerical methods.

A new quasi-brittle damage model implemented under quasi-static condition using bond-based peridynamics theory for progressive failure

A novel quasi-brittle damage model implemented under quasistatic loading condition using bond-based peridynamics theory for progressive failure is proposed to better predict damage initiation and propagation in solid materials. Since peridynamics equation of motion was invented in dynamic configuration, this paper applies the adaptive dynamic relaxation equation to achieve steady-state in peridynamics formulation. To accurately characterise the progressive failure process in cohesive materials, we incorporate the dynamic equation with the novel damage model for quasi-brittle materials. Computational examples of 2D compressive and tensile problems using the proposed model are presented. This paper presents advancement by incorporating the adaptive dynamic equation approach into a new damage model for quasi-brittle materials. This amalgamation allows for a more accurate representation of the behavior of damaged materials, particularly in static or quasi-static loading situations, bringing the framework closer to reality. This research paves the way for the peridynamics formulation to be employed for a far broader class of loading condition behaviour than it is now able to.

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 one dimensional ubiquitiform damage model for quasi-brittle materials

Considering the self-similar characteristic of irregular pore network of materials during the damage evolution process, a pore ubiquitiform model (PUM) is developed to characterize the multiscale pore or microcrack network of quasi-brittle materials, and then based on the PUM, the ubiquitiform damage model (UDM) is proposed to describe the damage evolution process of quasi-brittle materials under quasi-static uniaxial tensile loading. The values of porosity and pore specific surface area estimated from the PUM and the stress-deformation curves including the softening curve calculated from the UDM are in good agreement with previous experimental data, respectively. It can be found that, the PUM can be adopted to characterize the self-similar multiscale pore network and estimate the porosity and pore specific surface area, and the UDM can describe the damage evolution process of quasi-brittle materials under quasi-static uniaxial tensile loading. Meanwhile, the UDM can characterize the self-similar characteristic of pore network during the damage evolution process of quasi-brittle materials through the evolving ubiquitiform complexity.

A simple implementation of localizing gradient damage model in Abaqus

With the localizing gradient enhancement, a damage model for quasi-brittle materials is able to achieve regularized softening responses, with localized damage profiles corresponding to the development of macroscopic cracks, to resolve the numerical spurious effects induced by the conventional gradient enhancement. The typical implementation strategy for a gradient enhanced model is to solve the system of governing equations simultaneously. Focusing on the finite element (FE) package Abaqus, a user element subroutine is required to define the finite elements with additional degrees of freedom for the nonlocal field. Moreover, with user elements, additional effort is required to visualize the numerical results. To an inexperienced engineer/researcher, these requirements can be challenging. In this paper, a simple implementation of the localizing gradient damage model is elaborated. By utilizing the in-built coupled thermo-mechanical elements in Abaqus, the user only needs to define the material constitutive laws, as well as the sensitivity terms with respect to the field variables. Post-processing of results can be done directly in Abaqus. The applicability and ease of implementation are demonstrated via several examples, including those that utilize the Abaqus features of element deletion, contact between surfaces, as well as the incorporation of cohesive elements. Sample files can be downloaded from https://github.com/leonghien/Localizing-Gradient-Damage-with-UMAT-UMATHT

A damage-driven integration scheme in physically non-linear transient analysis for quasi-brittle materials

The standard Newton–Raphson scheme suffers from divergence problems due to softening, negative tangent stiffness, bifurcations or snap-back when modelling fracture of quasi-brittle materials, such as masonry and concrete. Convergence can be achieved by applying sequentially linear analysis (SLA) in some cases. However, it has difficulties in dealing with cases in which the displacement history matters such as dynamic damage modelling due to its total approach. Recently, incremental sequentially linear analysis (ISLA) which combines the merits of the N–R method and SLA, has been proposed. The solution search path follows damage cycles sequentially with secant stiffness corresponding to local damage increments, which traces both damage history (explicit) and displacement history (implicit). This work focuses on dynamic damage modelling of quasi-brittle materials accounting for physical non-linearity as well to address the transient effects in the fracture process. The standard ISLA, similar to the large time increment (LATIN) method, the extended finite element method (X-FEM) and strong discontinuity approaches (SDA) which are proposed to capture the crack propagations for quasi-brittle materials, is not suitable for dynamic damage modelling. The objective of this work is therefore to propose a damage integration scheme (damage-driven instead of time integration scheme) based on ISLA for non-linear transient analysis of structures under dynamic loading while retaining the merits of the previous ISLA. The proposed damage integration scheme has been compared with the standard time integration scheme in a dynamically loaded bar example, showing good agreements and sufficient accuracy. The proposed method has been validated further for a notched beam under tension and four-point bending. Since all physical non-linearity is linearized in damage cycles with the explicit secant stiffness of the reduced elastic material model, the algorithm possesses high robustness.

A phase-field regularized cohesive zone model for quasi-brittle materials with spatially varying fracture properties

This study develops a numerical method combining the phase-field regularized cohesive zone model (PF-CZM) and random fields for modeling of complicated mesoscopic fracture in quasi-brittle materials. In this method, the material’s fracture properties such as tensile strength and fracture energy are assumed as spatial random variables and represented by Weibull random fields (RFs), which are mapped to finite element meshes. The PF-CZM with cohesive softening laws are used to model quasi-brittle multi-crack initiation and propagation without remeshing. The new method is first validated by Monte Carlo simulations of a mesoscale concrete example under uniaxial tension, with the effects of correlation length and variance in RFs investigated. Two concrete beam examples, under mode-I and mixed-mode fracture respectively, are then successfully modelled. It is found that the developed method can predict realistic, complicated fracture processes and load-carrying capacity of little mesh-dependence. It provides an effective tool for calculation of structural reliability caused by materials’ random heterogeneity.

Fracture modelling of plain concrete using non-local fracture mechanics and a graph-based computational framework

In this article, we developed a thermodynamically consistent non-local microcracking model for quasi-brittle materials with application to concrete. The model is implemented using a novel graph-based finite element analysis (GraFEA) approach that allows for (i) the probabilistic modeling of the growth and coalescence of microcracks, (ii) the modeling of crack closure using a kinematics-based approach, and (iii) the modeling of rate effects on microcracking. The developed theoretical model and its computational framework is also implemented into the dynamics-based Abaqus/Explicit finite element program through a vectorized user-material subroutine interface. We further demonstrate the procedure for obtaining the parameters (including the non-local intrinsic material length scale, which governs the fracture process) and consequently validate the simulations with independent experimental results.

Two-scale random field model for quasi-brittle materials

A new idea of two-scale random field is presented by taking the mechanical behaviors of quasi-brittle materials as an example. In this way, the random fluctuation of material damage at the micro-meso-scale and the spatial correlation of mechanical properties at the macro-scale are both well described by the random distribution of fracture strain. Moreover, it is proved that by introducing a transform matrix, the mesoscopic random fields can be described by a two-scale random field. Based on this, a two-step numerical implementation of the two-scale random field is proposed, and the mechanical behaviors of quasi-brittle materials can be analyzed by incorporating the probability density evolution theory. Finally, an experimental example is presented to demonstrate the capability of the proposed model. Ideal agreements are achieved against the experimental results. Particularly, the randomness of material properties at both scales can be well reproduced.

A pragmatic orthotropic elasticity-based damage model with spatially distributed properties applied to short glass-fibre reinforced polymers

This article presents a simple progressive damage model for quasi-brittle materials, combining orthotropic elasticity with a scalar damage model including spatial variation of the damage initiation strain and the crack band method for softening regularization. The model’s performance is first analyzed from a numerical point of view and then demonstrated for tensile tests (0°, 45° and 90°), open-hole tensile tests (0°) and three-point bending (0° and 90°) tests of short fibre-reinforced polypropylene with 15 wt.% and 30 wt.% glass fibres. Despite its simplicity, the model captures the anisotropic elastic and inelastic behaviour observed in experiments. The model is applicable for orthotropic brittle or quasi-brittle materials, where the variability in elastic properties is negligible and the orientation dependency of the fracture strain is small or not relevant for the application.

Phase-field simulation of crack propagation in quasi-brittle materials: COMSOL implementation and parameter sensitivity analysis

Recently, the use of the phase-field method (PFM) to simulate the fracture process of brittle materials has attracted increasing attention. The PFM describes the fracture process through a series of differential equations, thus avoiding tedious crack surface tracking and offering advantages in simulating crack initiation, propagation, and bifurcation. The essence of the PFM is a multifield coupling problem, so it is supposed that the COMSOL Multiphysics commercial finite element software, which is particularly suitable for solving multifield coupling problems, should be more efficient and simpler to implement for the PFM. In this paper, a crack propagation model for quasi-brittle materials based on PFM is implemented in COMSOL Multiphysics by means of the solid mechanics module and secondary development interfaces of the partial differential equation (PDE), domain ordinary differential equation (ODE) and differential algebraic equation (DAE). Combined with the collected tensile and shear numerical simulation data, validation studies are carried out both qualitatively and quantitatively. In addition, considering that the PFM involves many parameters would create a significant amount of work for model calibration.Therefore, multifactor sensitivity analysis based on the orthogonal test method is used to identify the parameter' sensitivity. The results show that the use of the solid mechanics module and interfaces of the PDE and domain ODE and DAE are effective for phase-field modelling, and the proposed method could reasonably characterize the whole fracture process of quasi-brittle materials. The sensitivity analysis results revealed that Young's modulus (E) and critical energy release rate (G c) are the main factors affecting the output results of the model.

A Novel Fractional Plastic Damage Model for Quasi-brittle Materials

A Novel Fractional Plastic Damage Model for Quasi-brittle Materials

A viscoelastic micropolar peridynamic model for quasi-brittle materials incorporating loading-rate effects

A viscoelastic micropolar peridynamic (VMPD) model is proposed to describe the nonlinear deformation and fracture behaviors of quasi-brittle materials considering loading-rate effects. The governing equations of bonds connecting material points are reformulated by introducing the definition of bond deformation rates. Two peridynamic parameters, corresponding to the normal stiffness and the tangential stiffness of the bond, respectively, are introduced to ensure that the strain energy obtained from the proposed peridynamic model is consistent to that from the continuum viscoelastic mechanics. A novel failure criterion is developed to describe the dynamic progressive fracture of quasi-brittle materials. Damage functions and dynamic strengths of the bond are introduced to capture loading-rate effects for solid materials. The proposed model is verified by comparing its predictions with those from experimental observations. Numerical examples demonstrate that the nonlinear viscoelastic behaviors of quasi-brittle materials such as relaxation, softening–hardening behaviors as well as mix-mode fractures under dynamic loads with different loading rates are well captured by the proposed model.

Correction to: Phase field fracture in elasto-plastic solids: a length-scale insensitive model for quasi-brittle materials

Correction to: Phase field fracture in elasto-plastic solids: a length-scale insensitive model for quasi-brittle materials