A continuum damage mechanics method for fracture simulation of quasi‐brittle materials

This paper addresses a novel continuum damage‐based method for simulating failure process of quasi‐brittle materials starting from local damage initiation to final fracture. In the developed method, the preset characteristic length field is used to evaluate damage instead of element, which is used to reduce the spurious sensitivity. In addition, damage is only updated in the most dangerous location at a time for considering stress redistribution due to damage evolution, which is used to simulate competitive fracture process. As cases study, representative numerical simulations of two benchmark tests are given to verify the performance of the developed continuum damage‐based method together with a used damage model. The simulation results of the crack paths for two concrete specimens obtained from the developed method matched well with the corresponding experimental results. The results show that the developed continuum damage‐based method is effective and can be used to simulate damage and fracture process of brittle or quasi‐brittle materials. And the simulation results based on the developed method depend only the preset characteristic length field and not grid mesh.     

Multiscale domain decomposition analysis of quasi‐brittle heterogeneous materials

A hybrid multiscale framework is presented, which processes the material scales in a concurrent manner, borrowing features from hierarchical multiscale methods. The framework is used for the analysis of non‐linear heterogeneous materials and is capable of tackling strain localization and failure phenomena. Domain decomposition techniques, such as the ?nite element tearing and interconnecting method, are used to partition the material in a number of non‐overlapping domains and adaptive re?nement is performed at those domains that are affected by damage processes. This re?nement is performed in terms of material scale and ?nite element size. It is veri?ed that the results are independent of the chosen domain decomposition. Moreover, the multiscale analyses are validated with reference solutions obtained with a full ?ne‐scale solution procedure. Copyright © 2011 John Wiley & Sons, Ltd.     

Incremental‐secant modulus iteration scheme and stress recovery for simulating cracking process in quasi‐brittle materials using XFEM

In this paper, an incremental‐secant modulus iteration scheme using the extended/generalized finite element method (XFEM) is proposed for the simulation of cracking process in quasi‐brittle materials described by cohesive crack models whose softening law is composed of linear segments. The leading term of the displacement asymptotic field at the tip of a cohesive crack (which ensures a displacement discontinuity normal to the cohesive crack face) is used as the enrichment function in the XFEM. The opening component of the same field is also used as the initial guess opening profile of a newly extended cohesive segment in the simulation of cohesive crack propagation. A statically admissible stress recovery (SAR) technique is extended to cohesive cracks with special treatment of non‐homogeneous boundary tractions. The application of locally normalized co‐ordinates to eliminate possible ill‐conditioning of SAR, and the influence of different weight functions on SAR are also studied. Several mode I cracking problems in quasi‐brittle materials with linear and bilinear softening laws are analysed to demonstrate the usefulness of the proposed scheme, as well as the characteristics of global responses and local fields obtained numerically by the XFEM. Copyright © 2006 John Wiley & Sons, Ltd.     

Combined continuum damage‐embedded discontinuity model for explicit dynamic fracture analyses of quasi‐brittle materials

In this paper, a novel constitutive model combining continuum damage with embedded discontinuity is developed for explicit dynamic analyses of quasi‐brittle failure phenomena. The model is capable of describing the rate‐dependent behavior in dynamics and the three phases in failure of quasi‐brittle materials. The first phase is always linear elastic, followed by the second phase corresponding to fracture‐process zone creation, represented with rate‐dependent continuum damage with isotropic hardening formulated by utilizing consistency approach. The third and final phase, involving nonlinear softening, is formulated by using an embedded displacement discontinuity model with constant displacement jumps both in normal and tangential directions. The proposed model is capable of describing the rate‐dependent ductile to brittle transition typical of cohesive materials (e.g., rocks and ice). The model is implemented in the finite element setting by using the CST elements. The displacement jump vector is solved for implicitly at the local (finite element) level along with a viscoplastic return mapping algorithm, whereas the global equations of motion are solved with explicit time‐stepping scheme. The model performance is illustrated by several numerical simulations, including both material point and structural tests. The final validation example concerns the dynamic Brazilian disc test on rock material under plane stress assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

Comparison of a phase‐field model and of a thick level set model for brittle and quasi‐brittle fracture

This paper provides a comparison between one particular phase‐field damage model and a thick level set (TLS) damage model for the simulation of brittle and quasi‐brittle fractures. The TLS model is recasted in a variational framework, which allows comparison with the phase‐field model. Using this framework, both the equilibrium equations and the damage evolution laws are guided by the initial choice of the potential energy. The potentials of the phase‐field model and of the TLS model are quite different. TLS potential enforces a priori a bound on damage gradient whereas the phase‐field potential does not. The TLS damage model is defined such that the damage profile fits to the one of the phase‐field model for a beam of infinite length. The model parameters are calibrated to obtain the same surface fracture energy. Numerical results are provided for unidimensional and bidimensional tests for both models. Qualitatively, similar results are observed, although TLS model is observed to be less sensible to boundary conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

A probabilistic nonlocal model for crack initiation and propagation in heterogeneous brittle materials

A probabilistic damage model is developed to study crack initiation and growth in quasi‐brittle materials. Two different thresholds are considered to describe these mechanisms. A Weibull model is used to account for the randomness of crack initiation(s) and then a fracture mechanics based threshold is considered to model crack propagation. The model is integrated in a finite element code via a nonlocal damage approach. A regularization operator based on a stress regularization is introduced. Both damage thresholds are checked using the ‘regularized’ stress field to avoid mesh dependence. The interaction between propagating cracks and potential initiation sites is accounted for. Copyright © 2012 John Wiley & Sons, Ltd.     

Local/global model order reduction strategy for the simulation of quasi‐brittle fracture

This paper proposes a novel technique to reduce the computational burden associated with the simulation of localized failure. The proposed methodology affords the simulation of damage initiation and propagation while concentrating the computational effort where it is most needed, that is, in the localization zones. To do so, a local/global technique is devised where the global (slave) problem (far from the zones undergoing severe damage and cracking) is solved for in a reduced space computed by the classical proper orthogonal decomposition while the local (master) degrees of freedom (associated with the part of the structure where most of the damage is taking place) are fully resolved. Both domains are coupled through a local/global technique. This method circumvents the difficulties associated with model order reduction for the simulation of highly nonlinear mechanical failure and offers an alternative or complementary approach to the development of multiscale fracture simulators. Copyright © 2011 John Wiley & Sons, Ltd.     

Phase‐field material point method for brittle fracture

The material point method for the analysis of deformable bodies is revisited and originally upgraded to simulate crack propagation in brittle media. In this setting, phase‐field modelling is introduced to resolve the crack path geometry. Following a particle in cell approach, the coupled continuum/phase‐field governing equations are defined at a set of material points and interpolated at the nodal points of an Eulerian, ie, non‐evolving, mesh. The accuracy of the simulated crack path is thus decoupled from the quality of the underlying finite element mesh and relieved from corresponding mesh‐distortion errors. A staggered incremental procedure is implemented for the solution of the discrete coupled governing equations of the phase‐field brittle fracture problem. The proposed method is verified through a series of benchmark tests while comparisons are made between the proposed scheme, the corresponding finite element implementation, and experimental results.     

A fracture mechanics interpretation of the DNV brittle fracture criteria for ships and mobile offshore units

ABSTRACT Design procedures against brittle fracture are either based on empirical experience based methods or theoretical fracture mechanics based methods. Frequently, the fracture mechanics based assessment procedures are suspected to be tools by which to get acceptance for poor quality. In reality, these suspicions are somewhat unfounded, since the manufacturing standards, implicitly, also assume some specific defect size. The standards contain specific inspection criteria, which in themselves lead to some specific possible accepted defect size. The problem has been that the manufacturing standards have not been evaluated based on an advanced fracture mechanics analysis. Here the Det Norske Veritas (DNV) rules for ships and mobile offshore units are analysed, applying the state‐of‐the‐art fracture mechanics assessment method SINTAP. Based on the assessment, the consistency of the DNV rules, with respect to strength level and inspection criteria is revealed. The assessment, also creates the possibility to extend the rules to thinner and higher strength steels.     

The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading

The purpose of this paper is to revisit the maximum tensile stress (MTS) criterion to predict brittle fracture for mixed mode conditions. Earlier experimental results for brittle fracture of polymethylmethacrylate (PMMA) using angled cracked plates are also re-examined. The role of the T -stress in brittle fracture for linear elastic materials is emphasized. The generalized MTS criterion is described in terms of mode I and II stress intensity factors, K _I and K _II and the T- stress (the stress parallel to the crack), and a fracture process zone, r _c . The generalized MTS criterion is then compared with the earlier experimental results for PMMA subjected to mixed mode conditions. It is shown that brittle fracture can be controlled by a combination of singular stresses (characterized by K ) or non-singular stress ( T -stress). The T -stress is also shown to have an influence on brittle fracture when the singular stress field is a result of mode II loading.     

基于定长裂缝试件的脆性材料尺寸效应实验方法

由于脆性或准脆性材料内各类微缺陷的影响,材料的力学性能,如名义破坏应力,　刚度以及断裂韧性等随试件的大小而改变,具有明显的尺寸效应。通常情况下,描述材料尺寸效应的Ｂａｚａｎｔ尺寸效应律是建立在一系列相似试件的基础上通过实验方法确定的。　本文提出了一种新的用含固定长度裂缝试件测定断裂韧性和有效断裂过程区大小的实验方法和计算公式。与相似试件测定方法相比,实验结果吻合很好。根据本文提出的定长裂缝试件实验方法,在保证与相似试件相同脆性指数范围的前提下,可以用小试件进行测量。     

Analyse the role of the non‐singular stress in brittle fracture by BEM coupled with eigen‐analysis

A coupled model resulting from the boundary element method and eigen‐analysis is proposed in this paper to analyse the stress field at crack tip. This new combine method can yield several terms of the non‐singular stress in the Williams asymptotic expansion. Then the maximum circumferential stress (MCS) criterion taken the non‐singular stress into account is introduced to predict the brittle fracture of cracked structures. Two earlier experiments are re‐examined by the present numerical method and the role of the non‐singular stress in the brittle fracture is investigated. Results show that if more terms of non‐singular stress are taken into account, the predicted crack propagation direction and the critical loading by MCS criterion are much closer to the existing experimental results, especially for dominating mode II loading conditions. Moreover, numerical results manifest that Williams series expansion can describe the stress field further from the crack tip if more non‐singular stress terms are adopted.     

Aspects of ductile fracture and adaptive mesh refinement in damaged elasto‐plastic materials

Numerical simulation of elasto‐plastic problems involving multi‐fracturing materials requires a reliable failure prediction technique and a robust solution algorithm. This work approaches ductile fracture by means of continuum damage mechanics, from which two new failure criteria based on coupled and uncoupled damage analysis are derived. A two‐parameter stress update algorithm for damaged materials based on a Newton–Raphson iterative procedure is presented. A posteriori error estimators using ductile failure concepts are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

A generalized finite‐strain damage model for quasi‐incompressible hyperelasticity using hybrid formulation

A new generalized damage model for quasi‐incompressible hyperelasticity in a total Lagrangian finite‐strain framework is presented. A Kachanov‐like reduction factor (1 ? D) is applied on the deviatoric part of the hyperelastic constitutive model. Linear and exponential softening are defined as damage evolution laws, both describable in terms of only two material parameters. The model is formulated following continuum damage mechanics theory such that it can be particularized for any hyperelastic model based on the volumetric–isochoric split of the Helmholtz free energy. However, in the present work, it has been implemented in an in‐house finite element code for neo‐Hooke and Ogden hyperelasticity. The details of the hybrid formulation used are also described. A couple of three‐dimensional examples are presented to illustrate the main characteristics of the damage model. The results obtained reproduce a wide range of softening behaviors, highlighting the versatility of the formulation proposed. The damage formulation has been developed to be used in conjunction with mixing theory in order to model the behavior of fibered biological tissues. As an example, the markedly different behaviors of the fundamental components of the rectus sheath were reproduced using the damage model, obtaining excellent correlation with the experimental results from literature. Copyright © 2015 John Wiley & Sons, Ltd.     

A quasi‐linear reproducing kernel particle method

Reproducing kernel particle method (RKPM) has been applied to many large deformation problems. RKPM relies on polynomial reproducing conditions to yield desired accuracy and convergence properties but requires appropriate kernel support coverage of neighboring nodes to ensure kernel stability. This kernel stability condition is difficult to achieve for problems with large particle motion such as the fragment‐impact processes that commonly exist in extreme events. A new reproducing kernel formulation with ‘quasi‐linear’ reproducing conditions is introduced. In this approach, the first‐order polynomial reproducing conditions are approximately enforced to yield a nonsingular moment matrix. With proper error control of the first‐order completeness, nearly second‐order convergence rate in L2 norm can be achieved while maintaining kernel stability. The effectiveness of this quasi‐linear RKPM formulation is demonstrated by modeling several extremely large deformation and fragment‐impact problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Linear and nonlinear solvers for variational phase‐field models of brittle fracture

The variational approach to fracture is effective for simulating the nucleation and propagation of complex crack patterns but is computationally demanding. The model is a strongly nonlinear non‐convex variational inequality that demands the resolution of small length scales. The current standard algorithm for its solution, alternate minimization, is robust but converges slowly and demands the solution of large, ill‐conditioned linear subproblems. In this paper, we propose several advances in the numerical solution of this model that improve its computational efficiency. We reformulate alternate minimization as a nonlinear Gauss–Seidel iteration and employ over‐relaxation to accelerate its convergence; we compose this accelerated alternate minimization with Newton's method, to further reduce the time to solution, and we formulate efficient preconditioners for the solution of the linear subproblems arising in both alternate minimization and in Newton's method. We investigate the improvements in efficiency on several examples from the literature; the new solver is five to six times faster on a majority of the test cases considered. © 2016 The Authors International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd.     

Level-set topology optimization for maximizing fracture resistance of brittle materials using phase-field fracture model

Fracture is one of the most common failure modes in brittle materials. It can drastically decrease material integrity and structural strength. To address this issue, we propose a level-set (LS) based topology optimization procedure to optimize the distribution of reinforced inclusions within matrix materials subject to the volume constraint for maximizing structural resistance to fracture. A phase-field fracture model is formulated herein to simulate crack initiation and propagation, in which a staggered algorithm is developed to solve such time-dependent crack propagation problems. In line with diffusive damage of the phase-field approach for fracture; topological derivatives, which provide gradient information for the topology optimization in a LS framework, are derived for fracture mechanics problems. A reaction-diffusion equation is adopted to update the LS function within a finite element framework. This avoids the reinitialization by overcoming the limitation to time step with the Courant-Friedrichs-Lewy condition. In this article, three numerical examples, namely, a L-shaped section, a rectangular slab with predefined cracks, and an all-ceramic onlay dental bridge (namely, fixed partial denture), are presented to demonstrate the effectiveness of the proposed LS based topology optimization for enhancing fracture resistance of multimaterial composite structures in a phase-field fracture context.     

Extension of the modified Bai‐Wierzbicki model for predicting ductile fracture under complex loading conditions

The ductile fracture behaviour of metallic materials is strongly dependent on the material's stress state and loading history. This paper presents a concept of damage initiation and failure indicators and corresponding evolution laws to enhance the modified Bai‐Wierzbicki model for predicting ductile damage under complex loading conditions. The proposed model considers the influence of stress triaxiality and the Lode angle parameter on both damage initiation and the subsequent damage propagation. The model parameters are calibrated for C45E + N steel using a series of mechanical tests and numerical simulations. The enhanced approach is applied to the modelling of various mechanical tests under proportional and non‐proportional loading conditions and successfully predicts the ductile damage behaviour in these tests.