Fracture Of Quasi-brittle Materials Research Articles

Discrete modeling of elastic heterogeneous media

Discrete models provide advantages in simulating fracture in quasi-brittle materials due, in part, to their simplicity in representing cracking and other forms of displacement discontinuity. However, the stress analyses that form the basis for fracture simulation are complicated by difficulties in modeling the Poisson effect and other aspects of elastic behavior. The capabilities of Voronoi-cell lattice models, which are a form of particle-based lattice model, for elastic stress analysis are evaluated. It is found that the conventional means for representing the Poisson effect in particle-based lattice models result in spatially correlated stress oscillations that, at first glance, mimic the effects of material heterogeneity. The correlation length is dependent on discretization size. Alternatively, material heterogeneity can be introduced into elastically uniform lattice models via random assignments of material properties, independent of mesh size and geometry.

A C0 scalar field approach for describing sprain variables in finite element analysis of fracture

A new approach is proposed to account for the impact of sprain variables for computational simulation of fracture in quasi-brittle materials. For an isotropic solid, these variables are equal to the Laplacians of the components of the displacement field, and they were introduced in a recent study by Zhang and Bazant to resolve spurious mesh effects for cases involving softening material response. This is possible by establishing localization-resisting forces which are conjugate to the displacements with respect to a sprain energy functional, defined as a function of the sprain variables. To enable the evaluation of the sprain variables in a conventional, C0 continuum finite element approximation, this paper introduces a set of scalar fields, each being weakly equal to the Laplacian of one of the components of the displacement vector field. This leads to a straightforward implementation with standard continuum (solid) elements, eliminating the need for approximations satisfying higher continuity requirements or for ad-hoc, finite-difference algorithms which were adopted in the original pertinent work for the calculation of the sprain variables.

An optimized unibond dual-parameter peridynamic model for deformation and fracture simulation of quasi-brittle materials

Peridynamic is a nonlocal theory, which is an effective tool for predicting fracture in solids. The unibond dual-parameter peridynamic (UDPD) model can break through the Poisson's ratio limitation compared with the bond-based peridynamic model. However, the traditional algorithm for the integrated volume is designed according to the analytical value of partial length in the one-dimensional model, which will produce non-negligible errors for the two- and three-dimensional models. Besides, the classical UDPD model is restricted to ideal brittle materials rather than quasi-brittles, and the long-range force effect is generally omitted for simplified analysis. In this work, an optimized UDPD model is developed to overcome the above drawbacks in simulating the deformation and fracture of quasi-brittle materials. First, the novel algorithm to obtain a more accurate integrated volume is introduced into the UDPD model. Second, considering the long-range force effect and the deformation characteristics, the nonlinear constitutive force function coupled with the cosine type influence function is developed for the UDPD model. Then, the corresponding bond stiffnesses are derived. The proposed model is verified through several representative numerical applications, which suggests that the predicted results match well with the experimental observations and former simulations. It is indicated that the proposed model reveals a significant improvement in the convergence and precision of results, which can not only retain the stability of the classical UDPD model, but also precisely analyze the deformation behavior and damage evolution process of quasi-brittle materials.

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.

Notch depth and root radius effects on quasi-brittle fracture of materials related to grain size

Fracture of quasi-brittle materials is sensitive to geometry of defect (e.g. notch). In this study, an analytical model predicting fracture stress of specimen is proposed to consider notch depth and root radius effects which are related to average grain size G by analyzing facture measurements on ten ceramics (G: 1–500 μm) and four rocks (G: 300–810 μm). In addition, normal distribution theory is incorporated to describe the inevitable scatter of fracture measurements due to material heterogeneity and/or machining/testing errors. The results show that the proposed model can reliably predict fracture behavior of notched specimens with 96% reliability.

ESTUDIO DE LA INFLUENCIA DE LA ADICIÓN DE FIBRAS EN LA ROTURA EN MODO II DE MATERIALES CUASIFRÁGILES

Fracture of quasi-brittle materials, such as concrete or gypsum, has been studied for decades. These studies have given light on the fracture mechanisms as well as on the parameters that drive these processes, allowing producing numerical models that reproduce fracture using tools such as the Finite Element Analysis. On another note, the three possible Modes of fracture that can be identified in the field of Fracture Mechanics, Mode I is very well known, experimentally reproduced and used as the base for most numerical models. Nevertheless, in some cases fracture takes place as a combination of a stress state that induces a fracture mechanism where Modes I and II are involved. In this sense, Mode II fracture has been studied less often, partly because Mode I fracture is more usual, but also because reproducing Mode II experimentally is quite more complex. This study is focused on inducing Mode II fracture on fibre-reinforced gypsum specimens by using two different experimental tests. More specifically, the experimental test proposed by the Japanese standards JSCE and the push-off test are employed. The displacement field is obtained by means of the digital image correlation technique (DIC), which helps to understand how fibres modify the fracture mechanisms of the gypsum matrix when this takes place under Mode II conditions.

On simulating impact fracture in high-strength concrete using GraFEA

In this work, we use a recently-formulated Graph-based Finite-Element Analysis (GraFEA) framework developed for modeling deformation and non-local fracture in quasi-brittle materials to simulate the fracture response of high-strength concrete (HSC) samples under impact loads. By fitting the material parameters in the theory to match a macroscopic force–displacement response obtained from a three-point bending experiment conducted under quasi-static conditions, we show that the GraFEA computational method is able to independently validate the experimentally-determined impulse response, obtained from impact experiments conducted under various initial impact speeds imposed by a drop-weight on HSC samples that are 6 to 7 orders of magnitude higher than the quasistatic tests. Furthermore, the GraFEA-based FEM simulations are also able to reproduce experimental crack propagation speeds in a HSC sample under different initial impact speeds to good accord.

Simulation of dynamic fracture in quasi-brittle materials using a finite element modeling approach enhanced by moving mesh technique and interaction integral method

This work presents an efficient modeling approach for predicting dynamic crack propagation mechanisms in quasi-brittle materials. The proposed method consists of a standard Finite Element code enhanced by Moving Mesh (MM) technique consistent with the Arbitrary Lagrangian-Eulerian (ALE) formulation. The MM is used to reproduce the geometry evolution caused by dynamically growing cracks. More precisely, the nodes of the computational mesh around the crack tip change position during the simulation consistently with the growth of the crack front. In such a context, the ALE formulation ensures an effective re-location of mesh nodes inside the computational domain, reducing the overall amount of remeshing events. The motion of the computational mesh takes place according to previsions dictated by classic fracture criteria, which define crack initiation conditions, the direction of propagation, and the velocity of the advancing crack front. Such conditions depend on Dynamic Stress Intensity Factors at the crack front. Hence, accurately predicting these fracture variables is mandatory to ensure consistent mesh motions. To this end, the proposed approach implements the ALE formulation of the M-integral, which enables computing fracture variables on moving elements without losing accuracy. The validity of the proposed modeling strategy has been assessed through comparisons with numerical data and analytical formulations reported in the literature.

Enhancement of the smoothing gradient damage model with alternative equivalent strain estimation for localization failure

This paper aims at enhancing the capability of the smoothing gradient-enhanced damage method in modeling localization failure, especially mixed-mode fracture in quasi-brittle materials, e.g., double edge notch — DEN by Nooru-Mohamed test. As a novelty, we integrate an alternative technique for estimation of the equivalent strain based on Ottosen’s failure criterion, the four-parameter strain equivalent method that characterizes the sensitivity of deformation stages and loading path conditions, into our damage theory to form a new approach for better capturing the mixed-mode fracture behavior. Our approach allows to use the same low-order finite elements for approximation of both displacements and equivalent strain fields without any stress oscillation, highly suitable for practice. The new approach owns several desirable features of an advanced method including mesh insensitivity, and numerical examples in two-dimension indicate the applicability of the developed model in capturing the crack paths observed in experiments. Interpreting principal stresses in the failure envelopes further sheds light on insight into the damage behavior of the developed model.

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.

Mechanism-based estimations of the relationship between the critical distance and the grain size for quasi-brittle fracture

It is well known that the grain size has significant influence on the fracture of quasi-brittle materials. In order to give a quantitative description of this influence, many researchers have tried to find empirical relationships between the critical distance and the grain size, connecting a macroscopic parameter with the material microstructures. In this paper, a more rational model of this relationship is proposed based on the energy balance in the fracture process zone. This model naturally considers the size effect and can be implemented to both a nearly plain specimen and a specimen containing a long crack. It is not only helpful to explain the existing empirical estimations, but also reveals more underlying mechanisms of the quasi-brittle fracture.

Three-dimensional modeling of fracture in quasi-brittle materials using plasticity and cohesive finite elements

This paper presents a three-dimensional discrete approach based on the use of interface cohesive elements with zero thickness to model the crack formation and propagation in quasi-brittle materials. The interface constitutive model incorporates concepts of nonlinear fracture mechanics and plasticity. A Coulomb-based surface is used as yield function, and an exponential traction–separation curve as softening law. The model features a non-associated flow rule with an implicit integration scheme and is able to simulate the propagation of single and multiple cracks in mode I and mixed-mode. The proposed model was tested through extension and shear paths and applied in the simulation of experimental tests in concrete specimens available in the literature. Also, a study on the effect of the use of different bulk elements was performed. Numerical results, including peak loads and load-CMOD curves showing softening behavior, are in good agreement with the experimental data.

Multiple Pseudo-Plastic Appearance of the Dynamic Fracture in Quasi-Brittle Materials.

Understanding and simulating the dynamic response of quasi-brittle materials still remains as one of the most challenging issues in structural engineering. This paper presents the damage propagation material model (DAMP) developed in order to obtain reliable results for use in structural engineering practice. A brief overview focuses on the differences between fracture mechanics studies, and engineering material modelling is presented to highlight possible guideline improvements. An experimental dynamic test performed on ultra-high-performance concrete specimens was used to obtain evidence of the physical behaviour of brittle materials with respect to specimen size variations and, consequently, to verify the reliability of the material equations proposed. Two widely used material models (RHT and M72R3), as representatives of the classical brittle material models for structural purposes, and the proposed material model are compared. Here, we show how: (i) the multiple structural strength of brittle materials arises from the damage propagation process, (ii) there is no contradiction between fracture mechanics and the engineering approach once the velocity of damage propagation is chosen as fundamental material property and (iii) classical dynamic material models are intrinsically not objective with related loss of predictive capability. Finally, the original material model equation and the experimental strategy, dedicated to its extended verification, will be shown in order to increase the design predictiveness in the dynamic range considering structure and specimen size variations. The dynamic stress increasing factor (DIF), experimentally observed and widely recognised in literature as a fundamental concept for quasi-brittle material modelling, has been reviewed and decomposed in its geometrical and material dependencies. The new material model defines its DIF starting from the physical quantities of the damage propagation velocity applied to the test case boundary conditions. The resultant material model predictiveness results improved greatly, demonstrating its ability to model several dynamic events considering size and dynamic load variations with a unique material property set without showing contradictions between numerical and experimental approaches.

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

Phase-field methods for fracture have been integrated with plasticity for better describing constitutive behaviours. In most of the previous phase-field models, however, the length-scale parameter must be interpreted as a material property in order to match the material strength in experiments. This study presents a phase-field model for fracture coupled with plasticity for quasi-brittle materials with emphasis on insensitivity of the length-scale parameter. The proposed model is formulated using variational principles and implemented numerically in the finite element framework. The effective yield stress is calibrated to vary with the length-scale parameter such that the tensile strength remains the same. Moreover, semi-analytical solutions are derived to demonstrate that the length-scale parameter has a negligible effect on the stress–displacement curve. Five representative examples are considered here to validate the phase-field model for fracture in quasi-brittle materials. The simulated force–displacement curves and crack paths agree well with the corresponding experimental results. Importantly, it is found that the global structural response is insensitive to the length scale though it may influence the size of the failure zone. In most cases, a large length-scale parameter can be used for saving the computational cost by allowing the use of a coarse mesh. On the other hand, a sufficiently small length-scale parameter can be selected to prevent overly diffusive damage, making it possible for the proposed phase-field model to simulate the fracture behaviour with $$ \varGamma $$ -convergence.

Phase field modeling of fracture in Quasi-Brittle materials using natural neighbor Galerkin method

Recently developed phase field models of fracture require a diffusive crack representation based on an introduction of a crack phase field. We outline a thermodynamically consistent framework for phase field models of crack propagation in elastic solids, develop incremental variational principles. In this work, the numerical implementation of the phase field model for brittle fracture using natural neighbor Galerkin method is presented. Phase field method diffuses the sharp discontinuity by introducing a regularized crack functional. The notion of regularized crack functional provides a basis for defining suitable convex dissipation functions which govern the evolution of the crack phase field. The use of natural neighbors to dynamically decide the compact support at a nodal point makes it truly mesh free and nonlocal. The method uses smooth non-polynomial type Sibson interpolants, which are C0 at a given node and C∞ everywhere else. This allows to capture the interface region in a phase field model more accurately with only few nodes. For a given accuracy, there is an improvement in the computational time with the use of natural neighbor interpolants. Several benchmark problems were solved to investigate the efficiency of the numerical implementation of phase field model using natural neighbor Galerkin method.

Vectorial stress-separation laws for cohesive cracking: in concrete and other quasibrittle materials

In the past 25 years, a considerable number of works on fracture of quasibrittle materials or structures has been published. A large fraction of such works makes use of cohesive crack models and, therefore, deals with cohesive traction–separation laws. However, in most instances, this appears to be a secondary aspect in the publications, the principal one being the variant of the numerical algorithm presented in the research. This makes it difficult to compare the various available formulations for the, necessarily vectorial, traction–separation laws ($$\mathbf{t}$$–$$\mathbf{w}$$ laws). The present work aims at initiating a systematic comparison between the $$\mathbf{t}$$–$$\mathbf{w}$$ laws, and this is done by comparing a subset of the possible formulations using exactly the same numerical algorithms. A damage-based model is presented which includes as particular cases most of the already published laws and which allows an independent specification of: (a) the softening function; (b) the lack of coaxiality of $$\mathbf{t}$$ and $$\mathbf{w}$$; (c) the shape of the fracture criterion in the traction space; and (d) the ratio of fracture energies in pure modes I and II. The performance of the models is numerically investigated for a simple, but crucial, case to reveal possible spurious influences of the foregoing issues (b)–(d).

Damage and fracture mechanics approaches to mixed-mode discrete fracture with dilatancy

In this manuscript mixed-mode fracture in quasi-brittle materials is studied. A damage model is first introduced in the framework of the discrete crack approach, in which the limit surface is defined in the traction space. Dilatancy is included in this model as an additional compliance. Some experimental results are simulated, which give rise to further refinement of the model due to the effect of dilatancy. One new model is introduced, based on a nonlinear fracture mechanics approach. This is a purely incremental hypoelastic model, which cannot be derived in the framework of the damage mechanics theory. The numerical results of the two models are compared. In particular, it is found that the fracture mechanics model can better meet the requirements needed for the simulation of the experimental results.

Phase-field modeling of fracture for quasi-brittle materials

This paper addresses the modeling of fracture in quasi-brittle materials using a phase-field approach to the description of crack topology. Within the computational mechanics community, several studies have treated the issue of modeling fracture using phase fields. Most of these studies have used an approach that implies the lack of a damage threshold. We herein explore an alternative model that includes a damage threshold and study how it compares with the most popular approach. The formulation is systematically explained within a rigorous variational framework. Subsequently, we present the corresponding three-dimensional finite element discretization that leads to a straightforward numerical implementation. Benchmark simulations in two dimensions and three dimensions are then presented. The results show that while an elastic stage and a damage threshold are ensured by the present model, good agreement with the results reported in the literature can be obtained, where such features are generally absent.

Meshfree modeling of the dynamic mixed-mode fracture in FRC through an eigensoftening approach

This work is concerned with the numeric study of dynamic mixed-mode fracture in fiber reinforced concrete (FRC). A recently developed eigensoftening algorithm to deal with the fracture of quasi-brittle materials is employed in a meshfree framework. Three point bending tests on notched beams reinforced with steel fibers carried out through a drop weight device at two loading velocities are modelled. Since the notch was placed with an offset from the middle section, mixed-mode crack formation was facilitated. Assuming a linear softening stress-equivalent crack opening relation, the numerical methodology is first validated against experimental results on plain concrete. Subsequently, it is applied to study the dynamic fracture of fiber reinforced concrete with a newly-proposed bilinear relation. The studies reveal that the total fracture energy dissipation plays an important role on the peak reaction load, whereas the transitional point between the two branches has a significant influence on the crack patterns. Parametric studies on the notch position are also carried out. The results show that the notch position has a significant influence on the crack patterns.

Micro-crack damage in strip of fracture process zone

Damage and fracture of quasi-brittle materials are non-reversible processes of micro-defects and micro-cracks, often accompanied by a strip of micro-crack damage along the end of the macro-crack or gap. In this work, the micro-crack damage parameters in the strip are calculated, and a mechanics model is proposed to analyze the relation between deformation and force. The mechanics parameters are also determined using meso-mechanics homogenization of the damage zone of the micro-crack strip. Then, calculations of the deformation and stress in the damage strip are introduced, and the damage factor is estimated using a cohesive crack model. Finally, the relationship between the micro-crack and macro-fracture zone is correlated to electron microscopy data of a cast iron specimen and numerical results. For the damage factor, damage density, and geometrical factors of micro-cracks, the meso-mechanics scale is correlated to macro-mechanics in the damage fracture process zone by combining the homogenization method and cohesive crack model of nonlinear fracture.