Numerical investigation of dynamic crack branching under biaxial loading

Dynamic crack growth and branching of a running crack under various biaxial loading conditions in homogeneous and heterogeneous brittle or quasi-brittle materials is investigated numerically using RFPA2D (two-dimensional rock failure process analysis)-Dynamic program which is fully parallelized with OpenMP directives on Windows. Six 2D models were set up to examine the effect of biaxial dynamic loading and heterogeneity on crack growth. The numerical simulation vividly depicts the whole evolution of crack and captured the crack path and the angles between branches. The path of crack propagation for homogenous materials is straight trajectory while for heterogeneous materials is curved. Increasing the ratio of the loading stress in x-direction to the stress in y-direction, the macroscopic angles between branches become larger. Some parasitic small cracks are also observed in simulation. For heterogeneous brittle and quasi-brittle materials coalescence of the microcracks is the mechanism of dynamic crack growth and branching. The crack tip propagation velocity is determined by material properties and independent of loading conditions.     

An experimental investigation into dynamic fracture: III. On steady-state crack propagation and crack branching

This is the third in a series of four papers in which problems of dynamic crack propagation are examined experimentally in large, thin sheets of Homalite-100 such that crack growth in an unbounded plate is simulated. In the first paper crack initiation resulting from stress wave loading to the crack tip as well as crack arrest were reported. It was found that for increasing rates of loading in the microsecond range the stress intensity required for initiation rises markedly. Crack arrest occurs abruptly without any deceleration phase at a stress intensity lower than that which causes initiation under quasi-static loading.In the second paper we analyze the occurrence of micro cracks at the front of the running main crack which control the rate of crack growth. The micro cracks are recorded by real time photography. By the same means it is shown that these micro cracks grow and turn away smoothly from the direction of the main crack in the process of branching.In the present paper we report results on crack propagation and branching. It is found that crack propagation occurs at a constant velocity although the stress intensity factor changes markedly. Furthermore, the velocity is determined by the stress wave induced intensity factor at initiation. The terminal velocity in Homalite-100 was found to be about half the Rayleigh wave speed (0.45 C _r ). These observations are analyzed in terms of a microcrack model alluded to in the second paper of this series. A mechanism for crack branching is proposed which considers branching to be a natural evolution from a cloud of microcracks that accompany and lead the main crack. These results are believed to apply to quasi-brittle materials other than Homalite-100 and the reasons for this belief are discussed briefly in the first paper of this series.In the final paper of the series the effect of stress waves impinging on the tip of a rapidly moving crack is examined. Waves affect the velocity and the direction of propagation as well as the process of crack branching.     

Path and kinetics of branching from defects under uniaxial and biaxial compressive loading

In order to understand the physical significance of branching features, especially path and kinetics, formed from defects in confined geological or mining conditions, a set of branch fractures obtained under uniaxial and biaxial loading at the tip of an isolated pre-existing oblique open slot was studied in PMMA (polymethylmethacrylate) plates. It was found that the branching always initiated perpendicular to the local plane tangent to the slot edge at the branch crack root while the branch crack root-crack tip distance increased with the slot-loading axis angle; the influence of biaxial loading is also discussed. The study of the stress field linked to an elliptical slot under shear conditions confirms these experimental observations and predicts the influence of the radius of curvature at the crack tip on the branching distance. The conditions of propagation are studied in terms of strain energy release rate along the stress trajectory starting from the point of maximum tensile stress at the slot edge, taking into account the presence of microcracks stemming from the slot tip induced by the sawing method. This allows one to describe the three successive régimes of propagation of the branch crack: spontaneous, catastrophic and controlled ruptures, according to the intensity of the applied compressive uniaxial stress and the size of pre-existing microcracks at the edge of the open slot. The computed variation of the branch crack length versus time agrees well with our experimental observations.     

Studies of dynamic crack propagation and crack branching with peridynamics

In this paper we discuss the peridynamic analysis of dynamic crack branching in brittle materials and show results of convergence studies under uniform grid refinement (m-convergence) and under decreasing the peridynamic horizon (δ-convergence). Comparisons with experimentally obtained values are made for the crack-tip propagation speed with three different peridynamic horizons. We also analyze the influence of the particular shape of the micro-modulus function and of different materials (Duran 50 glass and soda-lime glass) on the crack propagation behavior. We show that the peridynamic solution for this problem captures all the main features, observed experimentally, of dynamic crack propagation and branching, as well as it obtains crack propagation speeds that compare well, qualitatively and quantitatively, with experimental results published in the literature. The branching patterns also correlate remarkably well with tests published in the literature that show several branching levels at higher stress levels reached when the initial notch starts propagating. We notice the strong influence reflecting stress waves from the boundaries have on the shape and structure of the crack paths in dynamic fracture. All these computational solutions are obtained by using the minimum amount of input information: density, elastic stiffness, and constant fracture energy. No special criteria for crack propagation, crack curving, or crack branching are used: dynamic crack propagation is obtained here as part of the solution. We conclude that peridynamics is a reliable formulation for modeling dynamic crack propagation.     

Why do cracks branch? A peridynamic investigation of dynamic brittle fracture

In this paper we review the peridynamic model for brittle fracture and use it to investigate crack branching in brittle homogeneous and isotropic materials. The peridynamic simulations offer a possible explanation for the generation of dynamic instabilities in dynamic brittle crack growth and crack branching. We focus on two systems, glass and homalite, often used in crack branching experiments. After a brief review of theoretical and computational models on crack branching, we discuss the peridynamic model for dynamic fracture in linear elastic–brittle materials. Three loading types are used to investigate the role of stress waves interactions on crack propagation and branching. We analyze the influence of sample geometry on branching. Simulation results are compared with experimental ones in terms of crack patterns, propagation speed at branching and branching angles. The peridynamic results indicate that as stress intensity around the crack tip increases, stress waves pile-up against the material directly in front of the crack tip that moves against the advancing crack; this process “deflects” the strain energy away from the symmetry line and into the crack surfaces creating damage away from the crack line. This damage “migration”, seen as roughness on the crack surface in experiments, modifies, in turn, the strain energy landscape around the crack tip and leads to preferential crack growth directions that branch from the original crack line. We argue that nonlocality of damage growth is one key feature in modeling of the crack branching phenomenon in brittle fracture. The results show that, at least to first order, no ingredients beyond linear elasticity and a capable damage model are necessary to explain/predict crack branching in brittle homogeneous and isotropic materials.     

Crack branching: A “twin-crack” model based on macroscopic energy fracture criteria

An improved approach for the prediction of the branching angle of running cracks is presented. It is based on the assumption that from the early steps of propagation, ahead of the running main crack-tip a tuft of microcracks is formed, from which two symmetric microcracks prevail over the others at the final steps before macroscopic branching. Further expansion of these two microcracks, which can be predicted by existing macroscopic fracture criteria, results in macroscopic branching. The unknown initial orientation of the two dominant microcracks represents the stochastic part of the whole branching phenomenon, while the main crack geometry, velocity and external load refer to the deterministic part. The results obtained by applying the T-criterion for the expected branching angle show a satisfactory agreement with existing experimental data.     

Dynamic effects of inclusions and microcracks on a main crack

The shielding and amplification effects of multiple inclusions and microcracks on the tip or the growth path of a main crack under dynamic loading are investigated using numerical simulations. The simulations employ a combined numerical tool based on the time-domain boundary element method together with the sub-region technique for multi-regions, the maximum circumferential stress criterion for crack-growth direction and discrete modelling for the crack propagation. New elements of constant length are added to the moving crack-tip to simulate the growth at an adaptable time-step according to the crack propagation criterion. Of particular interest is the study of the effects of the sizes and positions of inclusions and microcracks and the material combinations on the dynamic stress intensity factors and the crack growth path. The numerical results demonstrate the crack-tip shielding and amplification effects of inclusions and micro-cracks.     

Crack growth mechanism in precracked torsional fatigue specimens

The mechanism of crack growth of precracked materials was investigated under cyclic torsion loading with and without axial static stress using 4340 steel. During the crack growth, branching of the crack was observed. The length of the crack between the first branching points was dependent on loading conditions. This length was longer when the applied shear amplitude and the static axial stress level were higher. When the crack tips were opened by the tension loads, the crack had the tendency to grow in a shear mode during cyclic torsion. It was found that friction of the crack surfaces prevented shear mode crack growth. Furthermore, at higher stresses the initiation of new microcracks was observed in front of the main crack and their density was dependent on loading conditions. This helped the crack grow in a shear mode before and after branching.     

Rupture crack branching in solids with structural hierarchy

Both growth and branching of sharp cracks in perfect single crystals are studied. Strength and deformation criteria of sharp crack branching are proposed. These criteria describe brittle, quasi-brittle, and ductile material behavior under fracture propagation. For inner cracks, simple relations have been obtained that describe crack branching when curves of Kolumb-Mohr type theoretical strength for the generalized deformation mode are known. Multiple crack branching, associated with a multiplicity of eigenvalues when the system is buckled, is discussed. It has been ascertained that the principle of local symmetry in the vicinity of a crack tip is valid for perfect single crystals if the symmetry axis of a crystal is in line with the crack axis. When asymmetric perturbations of an atomic lattice occur in the vicinity of a crack tip or the symmetry axis of a single crystal is inconsistent with the crack axis, the principle of local symmetry fails. Solids with a hierarchy of regular structures that are typical for Macro-, Micro- and Nano-scales are studied in the same way. The curves of theoretical strength (Kolumb-Mohr type) of every structural level of material are considered to be known for the generalized stress state.     

Rupture crack branching in solids with structural hierarchy

Both growth and branching of sharp cracks in perfect single crystals are studied. Strength and deformation criteria of sharp crack branching are proposed. These criteria describe brittle, quasi-brittle, and ductile material behavior under fracture propagation. For inner cracks, simple relations have been obtained that describe crack branching when curves of Kolumb-Mohr type theoretical strength for the generalized deformation mode are known. Multiple crack branching, associated with a multiplicity of eigenvalues when the system is buckled, is discussed. It has been ascertained that the principle of local symmetry in the vicinity of a crack tip is valid for perfect single crystals if the symmetry axis of a crystal is in line with the crack axis. When asymmetric perturbations of an atomic lattice occur in the vicinity of a crack tip or the symmetry axis of a single crystal is inconsistent with the crack axis, the principle of local symmetry fails. Solids with a hierarchy of regular structures that are typical for Macro-, Micro- and Nano-scales are studied in the same way. The curves of theoretical strength (Kolumb-Mohr type) of every structural level of material are considered to be known for the generalized stress state.     

Instability during cohesive zone growth

Tensile microcracking of quasi-brittle materials is studied by means of micromechanics, based on (i) an elasto-damaging cohesive zone model accounting for cohesive softening and (ii) a dilute distribution of non-interacting microcracks of uniform orientation and size. Considering virgin microcracks (initially without cohesive zones), macroscopic tensile load increase results in growth of cohesive zones ahead of stationary (non-propagating) cracks and, subsequently, in crack propagation which, notably, will be encountered before the cohesive zones are fully developed i.e. onset of instable cohesive zone growth will be encountered at a load level (i) at which tractions are still transmitted across the inner edges of the cohesive zones and (ii) at which the separation at the inner edges of the cohesive zones is smaller than its critical value. Focusing on onset of instable cohesive zone growth, the chosen approach allows for accessing quantities characterizing the stability limit (e.g., the intensity of the macroscopic loading and the opening at the inner edges of the cohesive zones), without raising the need for non-linear Finite Element analyses. It is shown that the tensile macrostrength of materials containing virgin microcracks is larger than the one related to cracks with already initially fully developed cohesive zones, and related strength differences are quantified for a wide class of cohesive softening behavior. The proposed model is validated by comparing model predictions with an exact solution (available for the special case of constant cohesive tractions) and with results from reliable Finite Element analyses. The paper will be of interest for engineers involved in testing and/or in modeling of quasi-brittle media including cementitious materials and rock.     

The characteristics of an inclined crack in anisotropic paperboard under biaxial loading

In this paper, a fracture mechanics method was applied for the evaluation of crack behaviour in anisotropic paperboard subjected to biaxial uniform loading. The experiment was performed to determine the crack propagation angle and the fracture strength of paperboard under biaxial loading with the cruciform specimen optimized by FEM simulation. The effects of biaxial loads on the critical stress ratio and crack propagation angle for various inclination angles were investigated. The experimental results were compared with theoretical results, which were calculated by using the Normal Stress Ratio Criteria. The experimental results for crack propagation angle and critical stress show good agreement with theoretical results.     

Near‐threshold behaviour of shear‐mode fatigue cracks in metallic materials

This review paper presents a brief state‐of‐the art of the research on long fatigue shear‐mode cracks and describes some recent results on effective crack growth thresholds and mode I branching conditions achieved by the authors for ARMCO iron, titanium with two different microstructures, nickel and stainless steel. A special technique for preparation of fatigue precracks enabled us to substantially suppress the crack closure (friction) effects at the beginning of the experiment, and the measured threshold values could be considered to be very close to the effective ones. In all investigated materials, the effective thresholds under the remote mode II loading were found to be about 1.7 times lower than those under the remote mode III loading. Effective thresholds under mode II loading of investigated materials were found to follow a simple formula assembled by the shear modulus G, the magnitude of Burgers vector b and a goniometrical function n_α of the mean deflection angle that depends on the number of available crystallographic slip systems. These quantities determine the intrinsic material resistance to mode II crack propagation at the threshold. A simple criterion for mode I branching in terms of effective threshold values well reflects a transition from the shear‐mode to the opening‐mode controlled crack propagation at the threshold. The associated transition deflection angle of 40° is a material independent constant.     

Numerical analysis of interaction and coalescence of numerous microcracks

The deformation and failure behaviors of brittle or quasi-brittle solids are closely related to interaction and propagation of stochastically distributed microcracks. The influence of microcrack interaction and evolution on the mechanical properties of materials presents a problem of considerable interest, which has been extensively argued but has not been resolved as yet. In the present paper, a novel numerical method is used to calculate the effective elastic moduli and the tensile strength, and to simulate the failure process of brittle specimens containing numerous microcracks. The influences of some crack distribution parameters reflecting the non-uniform spatial concentration, size and orientation distributions are examined. The effective elastic moduli and the tensile strength of brittle materials exhibit different dependences on microcrack interaction. For example, two microcrack distributions that lead to the identical effective elastic moduli may cause a pronounced difference in the tensile strengths and failure behaviors of materials. By introducing two criteria for microcrack growth and coalescence in terms of Griffith’s energy release rate, the above numerical method is extended to simulate the coalescence process of microcracks that results in a fatal crack and the final rupture of a specimen.     

Fatigue crack growth in a coarse-grained iron–silicon alloy

The fatigue crack growth in a coarse-grained Fe-2 wt% Si alloy is investigated in considerable detail. At room temperature, the material is quasi-brittle and the failure mode is mixed cleavage. Under a mode-I cyclic loading, the crack can overcome the barrier effect of grain boundaries, which dominates the overall damage evolution. Associated with the geometrically necessary crack front branching, there are two possible break-through modes for a fatigue crack to propagate across a high-angle grain boundary.     

Quasi-brittle fracture diagrams under low-cycle fatigue (variable amplitude loadings)

Stepwise propagation of the internal mode I crack under cyclic loading is considered. Variable amplitude loads under cyclic loading conditions are studied for elastoplastic material. For analysis of this process, it is proposed to use diagrams of quasi-brittle fracture of solids under variable amplitude cyclic loading conditions. One of curves of the proposed diagram bears resemblance to the Kitagawa-Takahashi diagram. Estimates of average dimensionless velocity of stepwise crack propagation per loading cycle have been obtained in an explicit form for plain specimens of finite width. The relations derived for the average crack growth rate can be considered as structural expressions for plotting Paris’ curves.     

Cyclic loadings and crystallization of natural rubber: An explanation of fatigue crack propagation reinforcement under a positive loading ratio

Natural rubber is known to have excellent fatigue properties. Fatigue crack propagation studies show that, under uniaxial tension loading, fatigue crack growth resistance increases with the loading ratio, even if the peak stress increases. Studies dealing with crack initiation confirm this trend. If strain induced crystallization is believed to play a major role in this reinforcement process, it is not clear yet by which mechanism this reinforcement takes place. Using SEM investigation, it is shown here that the reinforcement process is associated with strong crack branching in the crack tip region. From experimental results it is shown that under particular reinforcing loading condition a cyclic strain hardening process can be observed on the natural rubber which is able to overcome classically observed softening effects. A cumulative strain induced crystallization process is proposed to explain the stress ratio effect on fatigue crack initiation and propagation properties of natural rubber.     

Stability of dynamically propagating cracks in brittle materials

Summary Dynamic crack propagation and bifurcation phenomena are investigated analytically by utilizing the strain energy density fracture criterion in the framework of catastrophe theory. The effect of biaxial stress, loading imperfections (mixed-mode loading), Poisson's ratio, state of stress as well as crack tip propagation speed on the crack path directional stability is analyzed. Special crack path stability charts for (un)stably propagating cracks are obtained, and their connection with the experimentally recorded crack tip stress field is addressed. It is shown that a slight change of the normal stress acting parallel to a crack at its tip (crack-parallel stress) may be able to affect the crack surface roughening and/or branching velocity considerably. It is also indicated that under small tensile crack-parallel stress, the crack propagation is stable only when the crack propagation speed is less than about 30% of the relevant shear wave speed. The crack becomes unstable, and its surfaces roughen severely at a higher speed, and the crack bifurcates at the highest propagation speed, some 45% of the shear wave speed. It is suggested that superimposing mode-II (shear) loading will enhance the dynamic crack path stability while increasing crack propagation speed will reduce the stability of crack propagation. It is expected that under compressive crack-parallel stress no crack surface roughening will occur before the crack stably bifurcates.     

Characterization of Loading Rate Effects on the Interactions between Crack Growth and Inclusions in Cementitious Material

The microcapsule-enabled cementitious material is an appealing building material and it has been attracting increasing research interest. By considering microcapsules as dissimilar inclusions in the material, this paper employs the discrete element method (DEM) to study the effects of loading rates on the fracturing behavior of cementitious specimens containing the inclusion and the crack. The numerical model was first developed and validated based on experimental results. It is then used to systematically study the initiation, the propagation and the coalescence of cracks in inclusion-enabled cementitious materials. The study reveals that the crack propagation speed, the first crack initiation stress, the coalescence stress, the compressive strength and the ultimate strain increase with the loading rate. The initiation position, the propagation direction, the cracking length and the type of the initiated cracks are influenced by the loading rates. Two new crack coalescence patterns are observed. It is easier to cause the coalescence between the circular void and a propagating crack at a slow loading rate than at a fast loading rate.