Experimental and mathematical modeling for fracture of rock joint with regular asperities

This paper is aimed at developing a mathematical model for the deformation behavior of a rock joint that explicitly accounts for the effects of joint surface topography. The present work is focused on rock joints with triangle-shaped regular asperities. Specimens of artificial rock joint with triangle-shaped asperities were made of simulated rock material and tested in the laboratory. Experimental results are examined to identify three mechanisms that influence the deformation of a rock joint: sliding, separation of asperity contact-faces, and shear fracture of asperities. A modeling methodology is then described and the behaviors of an asperity contact-face, including separation, sliding and shear fracture are discussed. The stress-deformation relationship of a rock joint is subsequently derived and the model performance is evaluated by comparing the predicted results from the derived model and the measured results from experiments.     

岩石爆破损伤模型及其研究进展

损伤模型明确反映了岩石应变率效应及岩石动载破坏的“脆断”和“灾变”特性,并为岩石爆破破碎区的确定和爆破块度预测提供了直接而可行的判据。本文系统介绍了岩石爆破损伤模型的研究进展及成果,指出了其工程应用前景。     

Microstructure-based damage and fracture modelling of alumina coatings

Effect of microstructure on damage and fracture evolution due to elastic anisotropy in alumina coatings and a mismatch in thermal expansion of the coating and substrate are examined by means of numerical simulations. Random microstructures of alumina coatings with different porosity levels are generated by the Monte Carlo method based on the Poisson distribution of the number of microdefects (voids/cracks) and their stereological size-shape distributions in plasma-sprayed ceramic coatings. The damage evolution law extended to complex stress states is applied to damage analysis. Failure mechanisms for coatings under uniform thermal loading are investigated with an explicit account for various microstructures.     

Multiple scale meshfree methods for damage fracture and localization

The Reproducing Kernel Particle Method (RKPM), which utilizes the fundamental notions of the convolution theorem, multiresolution analysis and meshfree properties, is reviewed. The multiple-scale RKPMs are then proposed as an alternative to commonly used numerical methods such as the finite element method. The elimination of a mesh, combined with the filtering properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and localization problems. This class of methods has been applied to shear band problems, and large deformation fracture and damage problems.     

Micromechanical modeling of damage and fracture of unidirectional fiber reinforced composites: A review

An overview of methods of the mathematical modeling of deformation, damage and fracture in fiber reinforced composites is presented. The models are classified into five main groups: shear lag-based, analytical models, fiber bundle model and its generalizations, fracture mechanics based and continuum damage mechanics based models and numerical continuum mechanical models. Advantages, limitations and perspectives of different approaches to the simulation of deformation, damage and fracture of fiber reinforced composites are analyzed.     

Correlation between fracture and damage for quasi-brittle bi-material interface cracks

Fracture at a bi-material interface is essentially mixed-mode, even when the geometry is symmetric with respect to the crack and loading is of pure Mode I, due to the differences in the elastic properties across an interface which disrupts the symmetry. The linear elastic solutions of the crack tip stress and displacement fields show an oscillatory type of singularity. This poses numerical difficulties while modeling discrete interface cracks. Alternatively, the discrete cracks may be modeled using a distributed band of micro-cracks or damage such that energy equivalence is maintained between the two systems. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in quasi-brittle bi-material interface beams. The study is aimed at large size structures made of quasi-brittle materials failing at concrete-concrete interfaces. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. It is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.     

Energy-based equivalence between damage and fracture in concrete under fatigue

Damage in concrete members, occur in a distributed manner due to the formation and coalescence of micro-cracks, and this can easily be described through a local damage approach. During subsequent loading cycles, this distributed zone of micro-cracks get transformed into a major crack, introducing a discrete discontinuity in the member. At this stage, concepts of fracture mechanics could be used to describe the behavior of the structural member. In this work, an approach is developed to correlate fracture and damage mechanics through energy equivalence concepts and to predict the damage scenario in concrete under fatigue loading. The objective is to smoothly move from fracture mechanics theory to damage mechanics theory or vice versa in order to characterize damage. The analytical methods developed here have been exemplified with some already available data in the literature. The strength and stiffness reduction due to progressive cracking or increase in damage distribution, has been characterized using the available indices such as the strength reduction and stiffness reduction factors. It is seen through numerical examples, that the strength and stiffness drop indices using fracture and damage mechanics theory agree well with each other. Hence, it is concluded, that through the energy approach a discrete crack may be modeled as an equivalent damage zone, wherein both correspond to the same energy loss. Finally, it is shown that by knowing the critical damage zone dimension, the critical fracture property such as the fracture energy can be obtained.     

Fracture mechanics and damage mechanics based fatigue lifetime prediction of adhesively bonded joints subjected to variable amplitude fatigue

The first part of the paper describes an investigation into the behaviour of adhesively bonded single lap joints (SLJs) subjected to various types of variable amplitude fatigue (VAF) loading. It was seen that a small proportion of fatigue cycles at higher fatigue loads could result in a significant reduction in the fatigue life. Palmgren-Miner’s damage sum tended to be less than 1, indicating damage accelerative load interaction effects. In the second part of the paper, fracture mechanics (FM) and damage mechanics (DM) approaches are used to predict the fatigue lifetime for these joints. Two FM based approaches were investigated, which differed with respect to the cycle counting procedure, however, both approaches were found to under-predict the fatigue lifetime for all the types of spectra used. This was attributed to the inability of the FM based models to simulate the crack initiation phase. A DM based approach was then used with a power law relationship between equivalent plastic strain and the damage rate. Good predictions were found for most of the spectra, with a tendency to over-predict the fatigue life.     

Damage zone around crack tip and fracture toughness of rubber-modified epoxy resin under mixed-mode conditions

Damage zones that form around crack tips before the onset of fracture provide significant data for evaluating the fracture behavior of polymeric materials. The size of the damage zone correlates closely with the fracture toughness of the resin. In this study, we investigate the relationship between the fracture toughness and damage zone size around crack tips of a rubber-modified epoxy resin under mixed-mode conditions. The fracture toughness, G_C, based on the energy release rate, is measured using an end-notched circle type (ENC) specimen. The deformation of rubber particles in the damage zones is also observed using an optical microscope. The results show that the fracture toughness, G_C, of the rubber-modified epoxy resin is closely related to the area of the damage zone. In the specimen with a loading angle of 30°, the rubber particles were deformed ellipsoidally due to the difference between the first and second principal stresses.     

Fracture and damage mechanics modelling of thin-walled structures - An overview

This paper reviews the most important current approaches for residual strength prediction of thin-walled structures. Crack driving force parameters such the linear elastic stress intensity factor and its plastic zone corrected extension for contained yielding conditions, the crack tip opening displacement δ₅, the crack tip opening angle CTOA, the cohesive zone model parameters, separation energy, critical tensile stress and critical separation and the parameters of the damage models of Gurson-Tvergaard-Needleman type are introduced and discussed with respect to their benefits and limitations for the simulation of plane and stiffened panels. In addition, specific aspects of modern non-integral and integral structures which pose a challenge are addressed. These comprise multi-site damage, crack deviation and branching, welding residual stresses, strength mismatch in material compounds and problems in bonded structures, such as delamination. A number of examples are provided to illustrate the potential of the various approaches.     

Determination of normal and shear residual stresses from fracture surface mismatch

The contour method of residual stress measurement has recently been adapted to measure fractured, rather than cut specimens. The fracture contour method was capable of determining normal residual stresses acting prior to the plane-strain failure of a large aluminium alloy forging, but shear residual stresses could not be measured (Prime et al., 2014).We demonstrate that the application of digital image correlation to topographic measurements of a fracture surface pair allows the determination of shear residual stresses in addition to the normal stress component. Miniature compact tension samples were extracted at an angle from a bent beam to give a known variation in normal and shear residual stress on the fracture plane. The material used was a metal matrix composite, which could be deformed plastically to introduce a known distribution of stresses and also present limited plasticity upon fracture, allowing plane-strain condition in a small specimen. The samples were fractured at cryogenic temperatures to further restrict plasticity. Although the fracture surface was non-planar and evidence suggested the occurrence of plasticity near the edges, experimental results correlated fairly well with the calculated normal and shear residual stress profiles.     

Evaluation of directional mesh bias in concrete fracture simulations using continuum damage models

In the present comparative study, we investigate the influence of directional mesh bias on the results of failure simulations performed with isotropic and anisotropic damage models. Several fracture tests leading to curved crack trajectories are simulated on different meshes. The isotropic damage model with a realistic biaxial strength envelope for concrete is highly sensitive to the mesh orientation, even for fine meshes. The sensitivity is reduced if the definition of the damage-driving variable (equivalent strain) is based on the modified von Mises criterion, but the corresponding biaxial strength envelope is not realistic for concrete. The anisotropic damage models used in this study capture reasonably well arbitrary crack trajectories even if the biaxial strength envelope remains close to typical experimental data. Their superior performance can be at least partially attributed to their ability to capture dilatancy under shear, which is revealed by a comparative analysis of the behavior of individual models under shear with restricted or free volume expansion.     

Delamination damage analyses of adhesively bonded lap shear joints in laminated FRP composites

Three-dimensional non-linear finite ele- ment analyses have been carried out to evaluate the out-of-plane stresses in the adhesive layer existing between the lap and the strap adherends of the Lap Shear Joint (LSJ) in laminated FRP composites for varied delamination lengths. The delaminations are presumed to be pre-embedded in the thin resin rich layer existing between the first and second plies of the strap adherend. Sublaminate technique has been used to model the LSJ with the delamination. Contact finite element analyses have been performed in order to avoid interpenetration of delaminated surfaces. The effects of varied delamination lengths on the peel and interlaminar shear stresses and the individual modes of Energy Release Rate (ERR) in the delamination zones are highlighted in this paper. It is seen that three-dimensional effects exist near the free edges of the overlap end of the joint. The delamination propagation significantly affects the stress distributions in the adhesive layer existing between the lap and the strap adherends of the LSJ. The variations of interlaminar stresses and ERRs on both the delamination fronts are found to be significantly different and thus, indicate that the propagation of delamination does not occur at same rate at the two delamination fronts. This may throw some light to the evaluation of structural integrity of the LSJ in the presence of pre-embedded delaminations.     

A micro-kink theory for determination of shear modulus of a unidirectional composite lamina

A hitherto unavailable concept of fiber micro-kink responsible for microscopic shear damage is introduced. Fiber micro-kinking is caused by crystallite disorientations, as detected by the Raman and X-ray measurements, inside a carbon fiber. In addition, the fibers themselves, constituting a unidirectional composite lamina, may contain micro-flaws such as fiber misalignment. The matrix material is assumed to be linear elastic. A mechanistic (as opposed to empirical) approach for derivation of the constitutive relation in shear, based on the concept of micro-kinking (caused by crystallite disorientations) and fiber misalignment, is developed. This is the shear counterpart of the micro-cracking theory for damage assessment of composites under tension. An expression for the nonlinear shear modulus is derived based on the assumption of uniform distribution of micro-kinks and fiber misalignment defects.     

R-curve behavior in fracture of notched porous ceramics

Notched specimens of porous silicon carbide (SiC) with porosity 37% were fractured under four-point bending. A single edge notch with six depths ranging from 0.1 to 2.8 mm was introduced to the specimen with a height of 7 mm. The fracture of specimens with a notch depth of 0.1 mm did not start from the notch, but from the intrinsic defect. The size of the non-damaging notch is about 0.1-0.2 mm and roughly equal to the size of SiC particles. When the notch depth was larger than 0.4 mm, the fracture started from the notch for all specimens. The record of the strain gage glued on the compression surface of the specimen as a function of the load showed nonlinearity before reaching the maximum load. The critical stress intensity factor was nearly constant for crack initiation from the notch. The resistance curve was constructed by estimating the crack length from the compliance change of the specimen, and was used for determining the maximum load point in bending tests. Fractographic observations showed the fracture path along the binder phase between silicon particles.     

Influence of notch severity on the impact fracture behavior of aluminum alloy 7055

In this paper, the conjoint influence of notch severity and test temperature on the impact behavior of an Al-Zn-Mg-Cu alloy 7055 in the T7751 microstructural condition is presented and discussed. Notch angles of 45°, 75° and 90° were chosen for a standard charpy impact test specimen containing two notches. For a given angle of the notch the increase in dynamic fracture toughness, with test temperature, is most significant for the least severe of the notches, i.e. 45°. At a given test temperature, the impact toughness of the T7751 microstructure decreased with an increase in notch severity. An increase in notch severity resulted in essentially Mode I dominated fracture at all test temperatures. The influence of localized mixed-mode loading is minimal for the alloy has low dynamic toughness. The impact fracture behavior of the alloy is discussed in light of alloy microstructure, mechanisms governing fracture and the deformation field ahead of a propagating crack.     

Analysing and modelling the 3D shear damage behaviour of hybrid yarn textile-reinforced thermoplastic composites

The presented work focuses on the examination of the 3D shear damage behaviour and its phenomenological failure process of a thermoplastic composite made of E-glass/polypropylene hybrid yarn with a woven reinforcement. Experimental shear characterisation is performed by means of the Iosipescu testing approach for both in-plane and through-thickness directions. A procedure for the manufacturing of through-thickness shear specimens is presented in this study. The characterisation of the chronological failure process and deformation analysis is supported by high speed camera system and Digital Image Correlation. Based on the experimental observations, material modelling strategies are derived and performed within the finite element environment Ls-Dyna.     

Triggering of dry snow slab avalanches: stress versus fracture mechanical approach

This paper analyses the conditions for triggering of dry snow slab avalanches. As suggested by several Authors, we assume as a basic mechanism for avalanche triggering the mode II fracture of the weak layer lying beneath the stiff snow slab, i.e. we assume the presence of super-weak zones in the basal layer. By means of a linear elastic analysis, the shear stresses in the weak layer as well as the strain energy release rate caused by an increment of the super-weak zone are evaluated. Hence we introduce a stress failure criterion as well as an energy one. It is shown that the latter criterion can be seen as an extension of the criterion firstly proposed by McClung [McClung, D.M., 1979. Shear fracture precipitated by strain softening as a mechanism of dry slab avalanche release. Journal of Geophysical Research, 84(B7), 3519-3526.] for dry snow slab avalanche release. Finally we couple the two criteria, showing that the weak layer can fail only in a min–max band of thickness.     

Early fatigue crack growth as the damage accumulation process

A probabilistic approach to modeling of the initial stage of fatigue crack growth is suggested based on the concepts of continuum damage mechanics. The material is presented as a set of microstructural elements with randomly distributed properties. Both the grains and intergranular boundaries are considered as the elements of microstructure. The parameters of resistance of each element to damage accumulation are considered as random variables. These parameters are distributed among the elements independently that allows to model the damage process in polycrystalline materials. The damage measure depends on the characteristic normal and tangential stresses in order to take into account the tensile and shear fracture modes for each element of microstructure. It is assumed that a nucleus of a crack is initially present near the body surface as a single completely ruptured element. The final damage of an element is considered as the crack advancement. The crack is modelled as a sequence of ruptured grains for the transgranular fracture, and as a sequence of couples of neighboring ruptured grains when the intergranular rupture is considered. Numerical simulation is performed to illustrate feasibility of the proposed model. In particular, non-planar crack propagation, blunting, kinking and branching of cracks at the early stage is demonstrated. The non-monotonous pattern of the short crack growth process is observed. Statistical scattering of the current crack size and the crack growth rate as functions of the cycle number and the crack depth is studied.