Brittle Fracture Model Research Articles

A nonlocal continuum damage model for brittle fracture

The objective of this paper is to develop a nonlocal continuum damage model for brittle fracture. A nonlocal damage conjugate force tensor is obtained from the “local” one, using an integral-type regularization technique, and the thermodynamic equations are formulated in a nonlocally generalized standard manner. A nonlocal damage model is developed based on a recently developed “local” model, which can rigorously handle damage anisotropy, distinct tensile and compressive damage behavior, and damage deactivation, so that the nonlocal model not only handles brittle fracture well but also possesses the advantageous features of the “local” model. A fully explicit integration scheme for the present model is developed and then implemented in Abaqus/Explicit via VUMAT. The present model is validated through: 1. simulating cyclic uniaxial tests on a SiC–SiC woven composite; 2. calibrating its associated material parameters via single edge notched bend (SENB) tests on concrete beams. Its versatility is demonstrated through simulating a mixed-mode fracture test on a double edge notched (DEN) specimen. It is found to be capable of producing realistic fracture paths and capturing size effects in real materials. A wide variety of mechanical behavior can be incorporated into the present model in the future.

High-accuracy phase-field models for brittle fracture based on a new family of degradation functions

Phase-field approaches to fracture based on energy minimization principles have been rapidly gaining popularity in recent years, and are particularly well-suited for simulating crack initiation and growth in complex fracture networks. In the phase-field framework, the surface energy associated with crack formation is calculated by evaluating a functional defined in terms of a scalar order parameter and its gradients. These in turn describe the fractures in a diffuse sense following a prescribed regularization length scale. Imposing stationarity of the total energy leads to a coupled system of partial differential equations that enforce stress equilibrium and govern phase-field evolution. These equations are coupled through an energy degradation function that models the loss of stiffness in the bulk material as it undergoes damage. In the present work, we introduce a new parametric family of degradation functions aimed at increasing the accuracy of phase-field models in predicting critical loads associated with crack nucleation as well as the propagation of existing fractures. An additional goal is the preservation of linear elastic response in the bulk material prior to fracture. Through the analysis of several numerical examples, we demonstrate the superiority of the proposed family of functions to the classical quadratic degradation function that is used most often in the literature.

Hybrid nonlinear surrogate models for fracture behavior of polymeric nanocomposites

We present a hybrid nonlinear surrogate model for fracture in polymeric nanocomposites. The phase field method is employed to model fracture in the polymer matrix. Since the stochastic analysis on the output of the mechanical model is prohibitively expensive, surrogate models (SM) are very attractive alternatives. In order to get an optimal and robust solution, we propose a hybrid nonlinear surrogate model (HSM) for the prediction of the fracture toughness of PNC. It is constructed with the use of the polynomial regression and the Kriging interpolation. The support data for such HSM is generated by a phase-field model for brittle fracture with six chosen input parameters. The validation of the surrogate model and by this its qualitative assessment is done based on a scanning test set algorithm. The constructed and assessed HSM is then used to present the behavior of fracture toughness of PNC with respect to various input parameters with very low computational costs and high accuracy. Within the domain of interest, the analysis shows that Young’s modulus of the matrix has no optimum value, in which, the higher input value causes higher response. On the other hand the volume fraction of clay platelets at about 5% showed stability of the response, in which, the higher input value leads to no change in the response.

A convergence study of phase-field models for brittle fracture

A crucial issue in phase-field models for brittle fracture is whether the functional that describes the distributed crack converges to the functional of the discrete crack when the internal length scale introduced in the distribution function goes to zero. Theoretical proofs exist for the original theory. However, for continuous media as well as for discretised media, significant errors have been reported in numerical solutions regarding the approximated crack surface, and hence for the dissipated energy. We show that for a practical setting, where the internal length scale and the spacing of the discretisation are small but finite, the observed discrepancy partially stems from the fact that numerical studies consider specimens of a finite length, and partially relates to the irreversibility introduced when casting the variational theory for brittle fracture in a damage-like format. While some form of irreversibility may be required in numerical implementations, the precise form significantly influences the accuracy and convergence towards the discrete crack.

Crack nucleation in variational phase-field models of brittle fracture

Phase-field models, sometimes referred to as gradient damage or smeared crack models, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith’ criterion. For a one-dimensional problem, it has been shown that they can predict nucleation upon a critical stress, provided that the regularization parameter be identified with the material’s internal or characteristic length. In this article, we draw on numerical simulations to study crack nucleation in commonly encountered geometries for which closed-form solutions are not available. We use U- and V-notches to show that the nucleation load varies smoothly from that predicted by a strength criterion to that of a toughness criterion when the strength of the stress concentration or singularity varies. We present validation and verification numerical simulations for both types of geometries. We consider the problem of an elliptic cavity in an infinite or elongated domain to show that variational phase field models properly account for structural and material size effects.Our main claim, supported by validation and verification in a broad range of materials and geometries, is that crack nucleation can be accurately predicted by minimization of a nonlinear energy in variational phase field models, and does not require the introduction of ad-hoc criteria.

A non-ordinary state-based peridynamics modeling of fractures in quasi-brittle materials

In this work, we have developed a non-ordinary state-based peridynamics model for brittle fracture in ceramics or fracture in quasi-brittle materials in general. The model is firstly validated by three numerical benchmark tests, and then it is applied to simulate the edge-on impact and drop ball test experiments. We have implemented the modified Johnson Holmquist (JH-2) constitutive damage model into the peridynamics framework at finite strain. Furthermore, the contact algorithm between the projectile and target is discussed. It is shown that the numerical results obtained from peridynamics simulations are in general agreement with those from the experiment. The comparison of experimental and numerical results indicates that the proposed peridynamics model has the ability to capture the damage propagation and other features of the brittle fracture.

A 3D finite strain model for intralayer and interlayer crack simulation coupling the phase field approach and cohesive zone model

In this study, a new 3D finite element formulation which enables simulating the interaction between brittle crack propagation and interface delamination in heterogeneous materials is presented. The Phase Field (PF) model for brittle fracture has been coupled with the Cohesive Zone Model (CZM) within the framework of the large deformation analysis. These numerical techniques have been implemented within a 8-node locking-free solid shell element, relying on the enhanced assumed strain concept, and a 8-node interface finite element, respectively. The predictive capabilities of the proposed formulation have been assessed through the simulation of cracking in flat and curved geometries under in-plane and out-of-plane loading conditions. The results show the ability of the model to predict complex crack paths where intralayer crack propagation and delamination occur simultaneously and interact. The proposed formulation provides a powerful modeling tool for the simulation of fracture phenomena in heterogeneous materials and laminate structures, which are characterized by the existence of numerous interfaces, such as in photovoltaic laminates.

Models of Long-Term Brittle Fracture of Rods in Tension and Compression Under Creep Conditions

Fracture models based on the increasing density of material and the decreasing bearing mass of material in tension and compression, respectively, are proposed. The creep strain rates modeled using the most widespread version of the flow theory. An analytical approach allows obtaining an equation identical in all cases and relating Poisson’s ratio, time function, and rheological parameter. An algorithm for determining two last parameters based on creep testing is proposed. The expected time-to-fracture values of heat-resistant alloys are in good qualitative and quantitative agreement with the experimental data.

Coupling damage and plasticity for a phase-field regularisation of brittle, cohesive and ductile fracture: One-dimensional examples

Plasticity and damage are two fundamental phenomena in nonlinear solid mechanics associated with the development of inelastic deformations and the reduction of the material stiffness. Alessi et al. [5] have recently shown, through a variational framework, that coupling a gradient-damage model with plasticity can lead to macroscopic behaviours assimilable to ductile and cohesive fracture. Here, we further expand this approach considering specific constitutive functions frequently used in phase-field models of brittle fracture. A numerical solution technique of the coupled elasto-damage-plasticity problem, based on an alternate minimisation algorithm, is proposed and tested against semi-analytical results. Considering a one-dimensional traction test, we illustrate the properties of four different regimes obtained by a suitable tuning of few key constitutive parameters. Namely, depending on the relative yield stresses and softening behaviours of the plasticity and the damage criteria, we obtain macroscopic responses assimilable to (i) brittle fracture à la Griffith, (ii) cohesive fractures of the Barenblatt or Dugdale type, and (iii) a sort of cohesive fracture including a depinning energy contribution. The comparisons between numerical and analytical results prove the accuracy of the proposed numerical approach in the considered quasi-static time-discrete setting, but they also emphasise some subtle issues occurring during time-discontinuous evolutions.

Further improvement of the Prometey model and Unified Curve method part 1. Improvement of the Prometey model

The local approach to brittle fracture is able in principle to transfer fracture data from one type of specimens to another one for different test temperatures and different material conditions. However, in practice the capacity of existing models to perform transferability need to be improved. For this reason the probabilistic model of brittle fracture known as the Prometey model has been modified. The modified model (referred to as the Prometey-M model) has been used for analysis of the transferability of the experimental results on brittle fracture for smooth and notched cylindrical tensile specimens and cracked specimens from Reactor Pressure Vessel (RPV) steel in the initial and embrittled conditions. The Prometey-M model allows the calculation of the brittle fracture probability with the same model parameters for specimens of different types tested at different temperatures. Recommendations have been given for the parameter values of the Prometey-M model that may be taken the same for RPV steels in the initial and embrittled conditions.

Phase-field model for brittle fracture. Validation with experimental results and extension to dam engineering problems

Phase-field formulations have recently emerged as promising tools to model brittle fracture. Based on the variational approach to fracture, these models aim at overcoming some of the computational challenges found in simulating complex fracture patterns and their evolution due to external or internal loads. Since most applications and validation exercises thus far have been restricted to academic benchmarks, the evaluation of phase-field fracture models against experimental results an practical engineering scenarios remains fragmented.Here we introduce a straightforward phase-field approach to simulate fluid and mechanically-driven fractures based on energy minimization and thermodynamical principles. We apply our methodology to several laboratory experiments of brittle fracture, and to fracturing processes in two full-scale concrete dams, taking into account the hydraulic forces inside the fractures. We conclude that phase-field models represent a promising computational tool that may be applied to realistic engineering scenarios.

Microbranching instability in phase-field modelling of dynamic brittle fracture

The microbranching instability occurring for rapidly propagating cracks in brittle materials has been described in various experiments as an intrinsically three-dimensional phenomenon. Using a variational phase-field model, we show that the microbranching process is, indeed, a three-dimensional instability which exhibits a strong dependence on the sample width and can be suppressed for very thin samples. We show that the phase-field internal length scale is the decisive variable governing the branching pattern, which can be either localized in the transverse direction as observed in a glass, for example, or, on the contrary, almost translational invariant with quasi-periodic structures, as observed in PMMA.

A unified phase-field theory for the mechanics of damage and quasi-brittle failure

Being one of the most promising candidates for the modeling of localized failure in solids, so far the phase-field method has been applied only to brittle fracture with very few exceptions. In this work, a unified phase-field theory for the mechanics of damage and quasi-brittle failure is proposed within the framework of thermodynamics. Specifically, the crack phase-field and its gradient are introduced to regularize the sharp crack topology in a purely geometric context. The energy dissipation functional due to crack evolution and the stored energy functional of the bulk are characterized by a crack geometric function of polynomial type and an energetic degradation function of rational type, respectively. Standard arguments of thermodynamics then yield the macroscopic balance equation coupled with an extra evolution law of gradient type for the crack phase-field, governed by the aforesaid constitutive functions. The classical phase-field models for brittle fracture are recovered as particular examples. More importantly, the constitutive functions optimal for quasi-brittle failure are determined such that the proposed phase-field theory converges to a cohesive zone model for a vanishing length scale. Those general softening laws frequently adopted for quasi-brittle failure, e.g., linear, exponential, hyperbolic and Cornelissen et al. (1986) ones, etc., can be reproduced or fit with high precision. Except for the internal length scale, all the other model parameters can be determined from standard material properties (i.e., Young’s modulus, failure strength, fracture energy and the target softening law). Some representative numerical examples are presented for the validation. It is found that both the internal length scale and the mesh size have little influences on the overall global responses, so long as the former can be well resolved by sufficiently fine mesh. In particular, for the benchmark tests of concrete the numerical results of load versus displacement curve and crack paths both agree well with the experimental data, showing validity of the proposed phase-field theory for the modeling of damage and quasi-brittle failure in solids.

Phase field modeling of brittle fracture for enhanced assumed strain shells at large deformations: formulation and finite element implementation

Fracture of technological thin-walled components can notably limit the performance of their corresponding engineering systems. With the aim of achieving reliable fracture predictions of thin structures, this work presents a new phase field model of brittle fracture for large deformation analysis of shells relying on a mixed enhanced assumed strain (EAS) formulation. The kinematic description of the shell body is constructed according to the solid shell concept. This enables the use of fully three-dimensional constitutive models for the material. The proposed phase field formulation integrates the use of the (EAS) method to alleviate locking pathologies, especially Poisson thickness and volumetric locking. This technique is further combined with the assumed natural strain method to efficiently derive a locking-free solid shell element. On the computational side, a fully coupled monolithic framework is consistently formulated. Specific details regarding the corresponding finite element formulation and the main aspects associated with its implementation in the general purpose packages FEAP and ABAQUS are addressed. Finally, the applicability of the current strategy is demonstrated through several numerical examples involving different loading conditions, and including linear and nonlinear hyperelastic constitutive models.

Experimental validation of a phase-field model for fracture

Simulations from a numerical implementation of the phase-field model for brittle fracture are compared against analytical and experimental results in order to explore the verification and validation of the method. It is found that while the intrinsic length scale associated with the phase-field model can be set arbitrarily, the scale of the fracture process zone, and the scale at which the elastic field attains the corresponding analytical brittle fracture limit could be substantially larger than this intrinsic length. It is demonstrated that with a suitable choice of this length scale, phase-field simulations can provide valid predictions of the growth of cracks in quasi-static brittle fracture.

Simulation of Laser-Induced Controlled Fracturing Utilizing a Phase Field Model

This work presents an approach to simulate laser cutting of ceramic substrates utilizing a phase field model for brittle fracture. To start with, the necessary thermoelastic extension of the original phase field model is introduced. Here, the Beer–Lambert law is used in order to model the effect of the laser on the substrate. The arising system of partial differential equations—which comprises the balance of linear momentum, the energy balance, and the evolution equation that governs crack propagation—is solved by a monolithic finite-element scheme. Finally, the influences of the laser power and the initial groove size on the manufactured work piece are analyzed numerically in simulations of a laser-cutting process.

Numerical evaluation of the phase-field model for brittle fracture with emphasis on the length scale

The phase-field model has been attracting considerable attention due to its capability of capturing complex crack propagations without mesh dependence. However, its validation studies have primaril...

Mechanism of compound fracture and removal in grinding process for low-expansion glass ceramics

The mechanism of fracture and its removal for low-expansion glass ceramics processing was studied through grinding experiments. A dynamic coefficient Kd was proposed to characterise the brittle material processing property. Using surface roughness as the evaluation index, the limitations of the empirical model and the brittle fracture model were analysed. On the basis of simple single-factor tests, two-step changes on the processing surface of brittle materials were observed for the first time, which could be described in terms of the critical depth of ductile and ductile–brittletransition (hc1) and the critical depth of ductile–brittle and brittle transition (hc2). The brittle material fracture processing and removal could be divided into three phases: ductile fracture, ductile–brittle fracture and brittle fracture. Furthermore, a compound grinding factor Q and a model of compound regime were proposed in this study, in order to comprehensively test the variation tendency of surface roughness and improve the model accuracy. A multi-factor cross-validation test also was conducted, and the results of the three models were compared with the test results. The model of compound regime (Ra3) showed the highest precision.

A Non-local Damage Model for Brittle Fracture in Metallic Structures with Stress Concentrators

This work is devoted to the development of the constitutive relations for the description of the defect evolution near the stress concentrators. A structural sensitive parameter is introduced as an averaging of the symmetrical tensor characterizing unit defect with the Boltzmann-Gibbs distribution function. Constitutive equation for structural parameter was derived under an assumption of the local thermodynamic equilibrium. The application of the developed approach is illustrated by the numerical simulation of a nonlocal fracture process occurring in a Grade-2 titanium specimen with a stress concentrator under uniaxial tension condition. Nonlocal character of the fracture near the stress concentrator was shown and physical explanation of the critical distance theory as a length of a dissipative structure growing in blow-up mode kinetics in the defect ensemble was proposed.

Applicability of Fracture Toughness Prediction by Coupled Damage Mechanics Model and Brittle Fracture Model for Ductile

Applicability of Fracture Toughness Prediction by Coupled Damage Mechanics Model and Brittle Fracture Model for Ductile