Behavior Of Quasi-brittle Materials Research Articles

On numerical aspects of different updating schedules for tracking fracture path in strain localization modeling

A comparative study is performed that investigates numerical features of different schedules, end-of-step vs. within iterations, for updating fracture path by employing the local and global tracking strategies. Embedded strong discontinuities within an enhanced finite element framework are used to model propagating discontinuities and fracture behavior of quasi-brittle materials. It is shown that end-of-step updating, which is a standard, can cause inaccuracies in peak strength and fracture energy for large time steps. Updating within iterations rectifies the accuracy issues, but at the expense of an increased computational cost. Both schedules yield comparable performance as the step size is refined.

Constitutive behaviour of quasi-brittle materials with anisotropic friction

This paper presents an approach to a constitutive model for anisotropic quasi-brittle materials, developed in the framework of rate independent softening plasticity, involving a yield criterion in which an anisotropic friction tensor is involved. It turns out to be useful for materials characterized by ultimate behaviour which varies according to the direction, such as composite materials, anisotropic rocks, textiles, masonry. A geometrical representation of the limit domain in the case of plane stress, together with the results of laboratory tests is presented and discussed.

Modeling of unilateral effect in brittle materials by a mesoscopic scale approach

This work deals with unilateral effect of quasi-brittle materials, such as concrete. For this propose, a two-dimensional meso-scale model is presented. The material is considered as a three-phase material consisting of interface zone, matrix and inclusions - each constituent modeled by an appropriate constitutive model. The Representative Volume Element (RVE) consists of inclusions idealized as circular shapes randomly placed into the specimen. The interface zone is modeled by means of cohesive contact finite elements developed here in order to capture the effects of phase debonding and interface crack closure/opening. As an initial approximation, the inclusion is modeled as linear elastic as well as the matrix. Our main goal here is to show a computational homogenization-based approach as an alternative to complex macroscopic constitutive models for the mechanical behavior of the quasi-brittle materials using a finite element procedure within a purely kinematical multi-scale framework. A set of numerical examples, involving the microcracking processes, is provided. It illustrates the performance of the proposed model. In summary, the proposed homogenization-based model is found to be a suitable tool for the identification of macroscopic mechanical behavior of quasi-brittle materials dealing with unilateral effect.

Nonlocal damage propagation in the dynamics of masonry elements

In this work, a nonlocal damage-plasticity model for dynamic finite element analyses of cohesive structural elements is presented. The proposed cohesive model is able to reproduce the main relevant behaviors of quasi-brittle materials despite being quite simple, i.e. governed by only a few parameters which can be determined by standard laboratory tests. In particular, the model is able to reproduce the mechanisms of cohesive materials under static or dynamic loads: degradation of the mechanical properties (damage) and accumulation of irreversible strains (plasticity). Moreover, the model also simulates the cyclic macroscopic behavior of quasi-brittle materials, taking into account the loss and recovery of stiffness due to crack closure and reopening. The latter effect represents a particularly important characteristic in the case of dynamic loads. The proposed formulation is implemented as a constitutive model for two-dimensional plane stress four-node quadrilateral elements. The second order equations of motion are solved adopting the implicit Newmark time integration scheme. The proposed model is validated and its dynamic performance is numerically demonstrated through the analysis of a large-scale structural element.

A refined micromechanical damage–friction model with strength prediction for rock-like materials under compression

Inelastic deformation and damage evolution at microdefects are two essential nonlinear mechanisms that govern macroscopic mechanical behaviors of quasi-brittle solids. The present paper deals in a unified framework with two dissipative processes in microcracks: inelastic deformation due to frictional sliding and damage by crack growth, usually arising and strongly coupled in cohesive materials under compression. Contributions by this work are threefold: (i) based on the Mori–Tanaka method, the free enthalpy of the representative elementary volume composed of a matrix phase and randomly oriented and distributed penny-shaped microcracks is determined for the general case of multiple crack families. The constitutive formulations are now presented in an elegant manner by using two orientation-dependent tensorial operators; (ii) the friction criterion is formulated in terms of the local stress applied onto microcracks. This local stress contains a back stress term that allows unified modeling of material hardening/softening behavior: friction-induced hardening is attributed to the cumulation of frictional shearing while damage-related softening is induced by crack growth and coalescence; (iii) originally, strength prediction is achieved through damage–friction coupling analyses. In that process, a basic feature of the damage resistance is revealed, leading to a novel damage criterion suitable for describing and modeling nonlinear mechanical behavior of quasi-brittle materials. Moreover, trans-scale relationship between the parameters in the local criteria and experimental data from laboratory tests is set up, which is always appealing in multiscale modeling. As a first phase of validation, the refined micromechanical model is finally applied to simulate laboratory tests on a granite under triaxial compression.

Characterization and modeling of the anisotropic damage of a high-explosive composition

Modelling the behaviour of quasi-brittle materials requires dealing with the anisotropic damage evolution induced by the loading direction and the unilateral effect corresponding to the sudden opening/closure of defects. The latter was the cause deficiencies of some constitutive laws as a no continuous stress/strain response or spurious energy dissipation. In this paper, the behaviour of pressed high explosive is experimentally investigated. Evidences of a pre-existing damage, an anisotropic one developing during loading and the unilateral effect are detailed. The framework of the microplane constitutive laws previously proposed in the literature and based on the volumetric–deviatoric–tangential strain projections is used. Damage is introduced in the model and is affected by an “effectiveness” function weighting its influence with the loading direction. The calibration procedure is detailed and a discussion enables highlighting the strengths and weaknesses of the modelling approach.

Softening behavior of quasi-brittle material under full thermo-mechanical coupling condition: Theoretical formulation and finite element implementation

In this article we introduce a new thermo-damage model, which is capable of modeling the softening behavior of quasi-brittle under loading condition including temperature change. The theoretical formulation and numerical solution procedure to solve the problem in the most general case of full thermo-mechanical coupling are presented.

Structure of micro-crack population and damage evolution in quasi-brittle media

Mechanical behaviour of quasi-brittle materials, such as concrete and rock, is controlled by the generation and growth of micro-cracks. A 3D lattice model is used in this work for generating micro-crack populations. In the model, lattice sites signify solid-phase blocks and lattice bonds transmit forces and moments between adjacent sites. Micro-cracks are generated at the interfaces between solid-phase blocks, where initial defects are allocated according to given size distribution. This is represented by removal of bonds when a criterion based on local forces and defect size is met. The growing population of micro-cracks results in a non-linear stress–strain response, which can be characterised by a standard damage parameter. This population is analysed using a graph-theoretical approach, where graph nodes represent failed faces and graph edges connect neighbouring failed faces, i.e. coalesced micro-cracks. The evolving structure of the graph components is presented and linked to the emergent non-linear behaviour and damage. The results provide new insights into the relation between the topological structure of the population of micro-cracks and the material macroscopic response. The study is focused on concrete, for which defect sizes were available, but the proposed methodology is applicable to a range of quasi-brittle materials with similar dominant damage mechanisms.

Effect of stacking sequence and geometric scaling on the brittleness number of glass fiber composite laminate with stress raiser

AbstractThe fracture behavior of quasi-brittle material exhibits commonly brittle behavior with some ductility in the fracture processing zone. Therefore, a matrix of experimental works is performed to investigate the failure behaviors based on a parameter known as brittleness number, which is a measure of brittle behaviors. Glass fiber-reinforced epoxy laminates of three stacking sequences, cross-ply [0/90]2s and two quasi-isotropic [0/45/90]2s and [0/45/90/-45]s, are manufactured using hand lay-up technique. A matrix of different specimen geometric scaling with open hole of the same material is used. The fracture energy GIC and nominal strength of these composite laminates are measured experimentally using compact tension test, center crack specimen, and tension test, respectively. The results showed that the increase of homogeneity of composite laminates, which is introduced by inserting a certain angle ply for the laminate structure, enhanced the material ductility as the brittle number increased. The larger size of specimen leads to the increase of brittleness for the same stacking sequences due to the increase of stress concentration factor.

How to Enhance Efficiency and Accuracy of the Over-Deterministic Method Used for Determination of the Coefficients of the Higher-Order Terms in Williams Expansion

Using higher-order terms of the Williams expansion is necessary for assessment of fracture behavior of quasi-brittle materials. Multi-parameter fracture mechanics enables more accurate determination of the stress/displacement field even in a larger distance from the crack tip, thus the extended zone with non-elastic behavior typical for this kind of material can be well described. The so-called over-deterministic method (ODM) seems to be a suitable tool for the higher-order terms coefficients calculation, but its utilization exhibits some limitations. Therefore, extensive analyses have been performed in order to summarize recommendations regarding the mesh sensitivity, boundary conditions influence, etc. List of pieces of advice and author’s experiences presented in the end of this work should contribute to more accurate and effective utilization of the ODM.

Continuum damage model for thermo-mechanical coupling in quasi-brittle materials

In this article we introduce a thermo-mechanical damage model, which is capable of representing the behavior of quasi-brittle materials subjected simultaneously to mechanical loading and heat transfer. Special care is taken of the quasi-brittle material behavior which can be represented by the continuum damage model, along with the solution procedures for fully coupled thermo mechanical loading conditions.

A non-iterative approach for the modelling of quasi-brittle materials

Due to the softening behaviour of quasi-brittle materials, in particular the localisation of initially diffused cracking, convergence problems are often found using an iterative procedure, such as the Newton–Raphson method. This is why a new non-iterative procedure is adopted in this paper, which is inspired by the sequentially linear approach(SLA) (Rots et al. in Eng Fract Mech 75(3–4):590–614, 2008). However, several important differences between the present approach and the SLA are presented. In the present model, multi-linear material laws are adopted such that non-linearities occur only due to changes in loading/unloading states. An incremental solution is obtained until non-convergence occurs, upon which a secant approach is used in a corresponding step. The update of the stiffness in the secant approach is based on information obtained from the previous incremental solution. This method is applied to: (i) softening materials, within the scope of the discrete crack approach, and to (ii) hardening materials. As a consequence, conversely to the smeared crack approach adopted in the SLA, no mesh size sensitivity problems are obtained and there is no need to adjust material parameters. Several numerical examples are shown in order to illustrate the proposed formulation.

Fracture characterization of bone under mode II loading using the end loaded split test

Fracture energy release rate under mode II loading of bovine cortical bone is determined using a miniaturized testing device of the end loaded split test. The energy release rate is evaluated by means of a data reduction scheme based on specimen compliance, beam theory and crack equivalent concept. Experimental tests were carried out to evaluate the Resistance curve which provides a successful method to characterize fracture behavior of quasi-brittle materials like bone. A numerical analysis including a cohesive damage model was used to validate the procedure. It was demonstrated that the end loaded split test and proposed data reduction scheme provide a valuable solution for mode II fracture characterization of bone.

A comparison of two microplane constitutive models for quasi-brittle materials

This article presents a comparison of two microplane constitutive models. The basis of the microplane constitutive models are described and the adopted assumptions for the conception of these models are discussed, with regard to: decomposition of the macroscopic strains into the microplanes, definition of the microplane material laws, including the choice of variables that control the material degradation, and homogenization process to obtain the macroscopic quantities. The differences between the two models, with respect to the employed assumptions, are emphasized and expressions to calculate the macroscopic stresses are presented. The models are then used to describe the behavior of quasi-brittle materials by finite element simulations of uniaxial tension and compression and pure share stress tests. The results of the simulations permit to compare the capability of the models in describing the post critical strain-softening behavior, without numerically induced strain localization.

Discontinuous FE approach and lattice models to describe cracking behaviour in fibre-reinforced brittle materials

The numerical evaluation of the mechanical behaviour of quasi-brittle materials like rocks, concrete, ceramics etc. is difficult from the computational point of view due to complex highly nonlinear phenomena occurring in the fracture failure mechanism, and the material softening behaviour due to the cracking process usually leads to a strong mesh dependence. The use of reinforcing fibres is an efficient and economic method to enhance the mechanical behavior of such materials. For a quasi-brittle multiphase material, such as a fibre-reinforced concrete (FRC), several phenomena must be considered in the computational simulation, such as matrix cracking, fibre bridging effects, fibre debonding, fibre breaking and so on. Continuum mechanics approaches as well as micromechanical ones have been developed for the computational solution of such problems. In the present paper, the mechanical behaviour of fibre-reinforced materials is analysed by adopting both a new discontinuous-like FE approach and a lattice model. The main phenomena involved, such as crack formation and propagation, crack fibre bridging, fibre debonding, fibre breaking, etc., are taken into account and examined using the above two models. The basic assumptions and theoretical background of such approaches are outlined and, finally, some experimental data related to notched plain or fibre-reinforced concrete specimens under mixed-mode monotonic loading are analysed.

A variationally based nonlocal damage model to predict diffuse microcracking evolution

We explore a variationally based nonlocal damage model, based on a combination of a nonlocal variable and a local damage variable. The model is physically motivated by the concept of “nonlocal” effective stress. The energy functional which depends on the displacement and the damage fields is given for a one-dimensional bar problem. The higher-order boundary conditions at the boundary of the elasto-damaged zone are rigorously derived. We show that the gradient damage models can be obtained as particular cases of such a formulation (as an asymptotic case). Some new analytical solutions will be presented for a simplified formulation where the stress–strain damage law is only expressed in a local way. These Continuum Damage Mechanics models are well suited for the tension behaviour of quasi-brittle materials, such as rock or concrete materials. It is theoretically shown that the damage zone evolves with the load level. This dependence of the localization zone to the loading parameter is a basic feature, which is generally well accepted, from an experimental point of view. The computation of the nonlocal inelastic problem is based on a numerical solution obtained from a nonlinear boundary value problem. The numerical treatment of the nonlinear nonlocal damage problem is investigated, with some specific attention devoted to the damageable interface tracking. A bending cantilever beam is also studied from the new variationally based nonlocal damage model. Wood’s paradox is solved with such a nonlocal damage formulation. Finally, an anisotropic nonlocal tensorial damage model with unilateral effect is also introduced from variational arguments, and numerically characterized in simple loading situations.

Modelling fatigue in quasi-brittle materials with incomplete self-similarity concepts

In this study, a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth is proposed in order to highlight and explain the deviations from the classical power–law equations used to characterize the fatigue behaviour of quasi-brittle materials. According to this theoretical approach, the microstructural-size (related to the volumetric content of fibers in fiber-reinforced concrete), the crack-size, and the size-scale effects on the Paris’ law and on the Wohler equation are presented within a unified mathematical framework. Relevant experimental results taken from the literature are used to confirm the theoretical trends and to determine the values of the incomplete self-similarity exponents. All this information is expected to be useful for the design of experiments, since the role of the different dimensionless numbers governing the phenomenon of fatigue is herein elucidated. Finally, a numerical model based on damage mechanics and nonlinear fracture mechanics is proposed for the prediction of uniaxial S–N curves, showing how to efficiently use the information gained from dimensional analysis and how the shape of the S–N curves is influenced by the parameters of the damage model.

A two-scale model for subcritical damage propagation

The failure behaviour of quasi-brittle materials is often time-dependent. This dependence is due to physical processes taking place at the level of the micro-structure. For a rigorous modeling of the time-dependent behaviour of that kind of solids, a two-scale approach is well suited. This paper investigates time-dependent damage which microscopic origin is the subcritical micro-crack growth. We present a two-scale time-dependent damage model completely deduced from small-scale descriptions of subcritical micro-crack propagation, without any macroscopic assumptions. The passage from the micro-scale to the macro-scale is done through an asymptotic homogenization approach. At the micro-scale, the tensile failure due to the subcritical propagation of cracks is the dominant mechanism of creep observed at the macro-scale. We consider microstructures with cracks evolving in different subcritical regimes. We assume a complex propagation law that considers three characteristic regimes of subcritical crack growth, corresponding to different physical processes at the crack tip level. Numerical simulations of constant strain rate, relaxation and creep tests illustrate the ability of the developed model to reproduce different regimes of time-dependent damage response.

Gradient and fracture energy-based plasticity theory for quasi-brittle materials like concrete

In this work a thermodynamically consistent non-local gradient and fracture energy-based plasticity theory is proposed to simulate the failure behavior of concrete. The model incorporates two characteristic lengths, one due to the microcrack opening process and the other due to the non-local degradation process of the continuum in between cracks. The failure behavior of quasi-brittle materials like concrete is controlled by a decohesion mechanism expressed in terms of a combined fracture-energy and non-local gradient-based softening formulation. The two resulting characteristic lengths are functions of the stress state to describe the increasing non-locality of the degradation process as well as the reducing distance between microcracks with the increment of the confining pressure. In this way the transition from brittle to ductile post-peak response of quasi-brittle materials like concrete is realistically predicted. The thermodynamically consistent formulation covers both the hardening and softening regimes of the proposed constitutive model. The compressive meridian of the model maximum strength criterion agrees with that of Leon while constant and maximum value is adopted for the eccentricity leading to the circular forms of the failure surface deviatoric views similarly to the Drucker–Prager criterion. A volumetric non-associated flow rule is taken into account to appropriately describe the inelastic behavior of concrete in the low confinement regime. The predictive capabilities of the proposed constitutive formulation are tested against experimental results on concrete specimens in tensile and compressive regimes.

A dimensional analysis approach to fatigue in quasi-brittle materials

In this study, a generalized Barenblatt and Botvina dimensional analysis approach to fatigue crack growth is proposed in order to highlight and explain the deviations from the classical power-law equations used to characterize the fatigue behaviour of quasi-brittle materials. According to this theoretical approach, the microstructural-size (related to the volumetric content of fibres in fibre-reinforced concrete), the crack-size, and the size-scale effects on the Paris’ law and the Wöhler equation are presented within a unified mathematical framework. Relevant experimental results taken from the literature are used to confirm the theoretical trends and to determine the values of the incomplete self-similarity exponents. All these information are expected to be useful for the design of experiments, since the role of the different dimensionless numbers governing the phenomenon of fatigue is herein elucidated.