Dynamic Crack Model Research Articles

Variability of Spectral Estimates of Stress Drop Reconciled by Radiated Energy

ABSTRACTA review of a collection of theoretical source spectral models revealed: (1) Despite the well-known variation in predicting static stress drop Δσs from the seismic moment and corner frequency, all models, especially the three conventional models, suggest that earthquakes radiate about half of the available strain energy into the surrounding medium. This similarity justifies a less model-dependent approach to estimate Δσs, though estimates for natural earthquakes rely on apparent seismic radiation efficiency (=2σa/Δσs; σa is apparent stress of an earthquake). (2) When one attempts to use Δσs and spectral models to make predictions, such as apparent stress σa, there is a model-dependent discrepancy between the σa inferred from theoretical energy partitioning and the σa predicted using spherical mean corner frequency. Their ratio cp varies significantly from 1.0 for the Brune (1970, 1971) model to 6.38 for the Madariaga (1976) model. If one uses spectral models to predict the ground motion, cp must be considered. (3) We infer that the constancy of the “stress parameter” (Δσ˜) found in engineering seismology (e.g., Boore, 1983; Atkinson and Beresnev, 1998) is similar to having constant apparent stress, σa (e.g., Ide and Beroza, 2001). The observation that Δσ˜ is generally larger than the average static stress drop Δσs for global M >5.5 shallow crustal earthquakes in active tectonic regions implies that these earthquakes radiate, on average, more seismic energy than predicted from the conventional dynamic crack models.

Prediction of Fracture Pattern in Defective Single Walled Carbon Nanotubes Using Molecular Structural Mechanics

An atomistic-based dynamic crack model for recreating execution of carbon nanotubes by considering vacancy defects is projected. The idea of the fracture model is taking into account the supposition that carbon nanotubes, when stacked, carry on like space-casing structures. The finite element method is utilized to analyse the nanotube structure and the molecular structural mechanics approach with beam element is used to simulate the non-linear force field of the C–C bonds. The model has been applied to defected single walled chiral, arm chair and zigzag nanotubes subjected to critical tension. The defect presented here (model) is one absent atom at the centre of the nanotube. The anticipated crack advancement, failure stresses of the nanotubes correspond exceptionally well with sub-atomic mechanics model from the literature.

Analytical Solutions of Dynamic Crack Models of Bridging Fiber Pull-Out in Unidirectional Composite Materials

An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane was analyzed, and the crack extension should occur in the format of self-similarity. When the fiber strength is over its maximum tensile stress, the fiber breaks. By means of complex variable functions, the problem considered can be easily translated into Reimann-Hilbert mixed boundary value problem. Utilizing the built dynamic model of bridging fiber pull-out in unidirectional composite materials, analytical solutions of the displacements, stresses and stress intensity factors under the action of increasing loads Pt5/x5, Px5/t4 are obtained, respectively. After those analytical solutions were used by superposition theorem, the solutions to arbitrary complex problems were acquired.

Analytical Solutions of a Symmetrical Dynamic Crack Model of Bridging Fibers in Unidirectional Composites

When composite materials occur crack, their fibrous locations will produce bridging fibers. A symmetrical dynamic crack model of bridging fibers in unidirectional composite materials are not probed as deeply by virtue of the complexity, cockamamie and difficulty in mathematical operations. In the light of the theory of complex variable functions, the problems discussed can be facilely translated into Remann-Hilbert problems. Analytical solutions of the displacements, stresses and stress intensity factors under the action of variable loads Pt6/x6, Px6/t5 are attained, respectively. After those analytical solutions were used by superposition theorem, the solutions of arbitrary complex problems were acquired.

Dynamic Crack Propagation Problems of Fibre-Reinforced Concrete

Applying the built dynamic crack model of fibre concrete, bridging fiber segment is substituted by loads. When a crack propagates its fiber continues to break. By the approaches of the theory of complex functions, the problems dealt with can be translated into Riemann-Hilbert problems. Analytical solutions of the displacements, stresses, and dynamic stress intensity factors under the action of of moving variable loads and , respectively, are attained by the ways of self-similar measures. After those analytical solutions are utilized by superposition theorem, the solutions of arbitrary complex problems can be obtained.

A dynamic asymmetrical crack model of bridging fiber pull-out in unidirectional composite materials

As a rule, when a crack happens in composite materials, the fibrous system will generate bridging fibers resulted in the asymmetrical extending of the crack. In this paper, a dynamic asymmetrical crack model of bridging fiber pull-out in unidirectional composite materials is built for analyzing the distributions stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is determined by the maximum tensile stress, the fiber ruptures, and hence the crack propagation should also occur in self-similar modality. The formulation involves the development of a Riemann-Hilbert problem. The analytical solution of an asymmetrical extension crack in unidirectional composite materials under the conditions of moving increasing loads Pt2/x2 and Px2/t is concluded, respectively. Based on relative material properties, the variable law of dynamic stress intensity factors was depicted perfectly. After the conclusion of analytical solutions with the superposition theorem, the solutions of arbitrary complex problems could be acquired.

An asymmetrical dynamic crack model of bridging fiber pull-out of composite materials

When a crack occurs in composite materials, the fibrous system will form bridges, and the crack extends asymmetrically as a rule. In this paper an asymmetrical dynamic crack model of bridging fiber pull-out of composite materials is presented for analyzing the distributions stress and displacement under the loading conditions of an applied nonhomogenous stress and the traction forces. Thus the fiber failure is ascertained by maximum tensile stress, the fiber ruptures and hence the crack propagation should also appear in the modality of self-similarity. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of composite materials under the conditions of an increasing force Px2/t2, Pt2/x is obtained respectively. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be gained.

A Simplified Method Study of Aluminium Alloy Fatigue Crack Tip Parameters Computed by Elastic-Plastic Finite Element Model

Fatigue crack propagation of aluminium alloy 7049-OA has been studied by non-linear finite element business-oriented software ABAQUS, and elastic-plastic finite element models of static fatigue crack and dynamic fatigue crack of center crack panel (CCP) specimens are also built. Based on the finite element computation results, the differences of stress and crack opening displacement around crack tip of static crack model have been compared with those of dynamic crack model. The compared results showed that the finite element computation results of dynamic crack model can be replaced by the results calculated by the static crack model. Fatigue crack tip parameters of aluminium alloy CCP specimens can be calculated by elastic-plastic finite element model of static crack. This is an effective method to cut down the computation expense and promote the computational efficiency.

Analytical and numerical treatment of a dynamic crack model

We discuss the propagation of a running crack in a bounded linear elastic body under shear waves for a simplified 2D-model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked, bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u(y, t) and the one-dimensional crack tip trajectory h = h(t). We assume that the crack grows straight. Based on a paper of Nicaise-Sandig, we derive an improved formula for the ordinary differential equation of motion for the crack tip, where the dynamical stress intensity factor occurs. The numerical simulation is an iterative procedure starting from the wave field at time t = ti. The dynamic stress intensity factor will be extracted at t = ti. Its knowledge allows us to compute the crack-tip motion h(ti+1) with corresponding nonuniform crack speed assuming (ti+1 − ti) is small. Now, we start from the cracked configuration at time t = ti+1 and repeat the steps. The wave displacements are computed with the FEM-package PDE2D. Some numerical examples demonstrate the proposed method. The influence of finite length of the crack and finite size of the sample on the dynamic stress intensity factor will be discussed in detail.

An Asymmetrical Self-Similar Dynamic Crack Model of Bridging Fiber Pull-Out in Unidirectional Composite Materials

When composite materials produce a crack, their fibrous position will form bridges, moreover the crack usually propagates in the modality of asymmetrical self-similarity. In this paper an asymmetrical self-similar dynamic crack model of bridging fiber pull-out in unidirectional composite materials is built. Analytical solutions of the stresses, displacements, stress intensity factors and bridging fiber fracture speeds for this model under the action of moving loads and homogeneous loads, respectively, are acquired by self-similar measures. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be gained.

Dynamic crack models on problem of bridging fiber pull-out of composite materials

An elastic analysis of an internal central crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane was performed. A dynamic model of bridging fiber pull-out of composite materials was presented. Resultingly the fiber failure is governed by maximum tensile stress, the fiber breaks and hence the crack extension should occur in self-similar fashion. By the methods of complex functions, the problem studied can be transformed into the dynamic model to the Reimann-Hilbert mixed boundary value problem, and a straightforward and easy analytical solution is presented. Analytical study on the crack propagation subjected to a ladder load and an instantaneous pulse loading is obtained respectively for orthotropic anisotropic body. By utilizing the solution, the concrete solutions of this model are attained by ways of superposition.

On the Expected Relationships among Apparent Stress, Static Stress Drop, Effective Shear Fracture Energy, and Efficiency

We consider expected relationships between apparent stress τ a and static stress drop Δ τ s using a standard energy balance and find τ a = Δ τ s(0.5 – ξ ), where ξ is stress overshoot. A simple implementation of this balance is to assume overshoot is constant; then apparent stress should vary linearly with stress drop, consistent with spectral theories (Brune, 1970) and dynamic crack models (Madariaga, 1976). Normalizing this expression by the static stress drop defines an efficiency η sw = τ a/Δ τ s as follows from Savage and Wood (1971). We use this measure of efficiency to analyze data from one of a number of observational studies that find apparent stress to increase with seismic moment, namely earthquakes recorded in the Cajon Pass borehole by Abercrombie (1995). Increases in apparent stress with event size could reflect an increase in seismic efficiency; however, η sw for the Cajon earthquakes shows no such increase and is approximately constant over the entire moment range. Thus, apparent stress and stress drop co-vary, as expected from the energy balance at constant overshoot. The median value of η sw for the Cajon earthquakes is four times lower than η sw for laboratory events. Thus, these Cajon-recorded earthquakes have relatively low and approximately constant efficiency. As the energy balance requires η sw = 0.5 – ξ , overshoot can be estimated directly from the Savage–Wood efficiency; overshoot is positive for Cajon Pass earthquakes. Variations in apparent stress with seismic moment for these earthquakes result primarily from systematic variations in static stress drop with seismic moment and do not require a relative decrease in sliding resistance with increasing event size (dynamic weakening). Based on the comparison of field and lab determinations of the Savage–Wood efficiency, we suggest the criterion η sw > 0.3 as a test for dynamic weakening in excess of that seen in the lab.

Heterogeneous distribution of dynamic stress drop and relative fault strength recovered from the results of waveform inversion: the 1984 Morgan Hill, California, earthquake

Abstract The dynamic rupture process of the 1984 Morgan Hill, California, earthquake (M = 6.2) has been investigated on the basis of a three-dimensional (3D) dynamic shear crack model, from previous waveform inversion results. For this purpose, a locking fracture criterion is introduced for rupture propagation, from which a lower bound of the peak shear stress just before rupturing, and hence of the relative fault strength, has been estimated at each fault segment. The distribution of static and dynamic stress drops has also been independently evaluated from that of fault slips by linear and nonlinear inversion procedures, respectively. The results show that large slips located from the hypocenter to about 10 km and 14 to 17 km along the strike at depths between 8 and 12 km result from local dynamic stress drops exceeding 40 and 140 bars, respectively. Negative stress drops down to −15 bars are required to explain very low slips over a shallow fault section. If a non-negative stress-drop condition is imposed, somewhat larger slips are obtained there than those from the waveform inversion. The above results suggest that there could be a zone of velocity-strengthening frictional behavior in the shallow crust, which may have arrested slip motion during rupture propagation. The strength excess is found to be generally small but somewhat larger at a small zone that has delayed rupture propagation. The dynamic rupture initiated from a small nucleus zone with a low stress drop, propagated southeastward, breaking the deeper fault section with high stress drop, and then broke a relatively high strength zone after a short time of arrest, with highest stress drop.

Dynamic rupture analysis: Inversion for the source process of the 1990 Izu‐Oshima, Japan, earthquake (M = 6.5)

A waveform inversion has been applied to strong motion data using a dynamic shear crack model. We studied the 1990 Izu‐Oshima earthquake (MJMA = 6.5), which has vertical strike‐slip faulting with unilateral rupture propagation. The inversion has two steps, a waveform inversion and a crack inversion, that are applied iteratively. A waveform inversion is used to determine the distribution of rupture starting times and slip dislocations using the slip functions calculated by the initial crack model, or by previous crack inversion. A crack inversion is used to calculate dynamic crack propagation that explains the results of the above inversion. In this step, we use the estimated rupture times as a locking fracture criterion; the maximum shear stress attained before a fault segment breaks gives a lower bound estimate of the peak shear strength at each fault segment. Then the dynamic stress drop distribution is estimated from the slip distribution obtained from waveform inversion assuming a dynamic crack model. From the results, we determine the rise time distribution and the distribution of a dimensionless stress ratio S defined as (strength excess)/(stress drop). Our analysis gives the following picture of the rupture process of the 1990 Izu‐Oshima earthquake: (1) An asperity‐type faulting having large slip and high stress drop was detected in the region around the initiation point of rupture. (2) South of the asperity zone, barrier‐type faulting characterized by incoherent propagating rupture, small slip, long rise time, and high strength excess was detected. This zone corresponds to the intersection of the fault with the 1978 earthquake (MJMA = 7.0).

Evidence for and implications of self-healing pulses of slip in earthquake rupture

Dislocation time histories of models derived from waveforms of seven earthquakes are discussed. In each model, dislocation rise times (the duration of slip for a given point on the fault) are found to be short compared to the overall duration of the earthquake (∼ 10%). However, in many crack-like numerical models of dynamic rupture, the slip duration at a given point is comparable to the overall duration of the rupture; i.e. slip at a given point continues until information is received that the rupture has stopped propagating. Alternative explanations for the discrepancy between the short slip durations used to model waveforms and the long slip durations inferred from dynamic crack models are: (1) the dislocation models are unable to resolve the relatively slow parts of earthquake slip and have seriously underestimated the dislocations for these earthquakes; (2) earthquakes are composed of a sequence of small-dimension (short duration) events that are separated by locked regions (barriers); (3) rupture occurs in a narrow self-healing pulse of slip that travels along the fault surface. Evidence is discussed that suggests that slip durations are indeed short and that the self-healing slip-pulse model is the most appropriate explanation. A qualitative model is presented that produces self-healing slip pulses. The key feature of the model is the assumption that friction on the fault surface is inversely related to the local slip velocity. The model has the following features: high static strength of materials (kilobar range), low static stress drops (in the range of tens of bars), and relatively low frictional stress during slip (less than several hundreds of bars). It is suggested that the reason that the average dislocation scales with fault length is because large-amplitude slip pulses are difficult to stop and hence tend to propagate large distances. This model may explain why seismicity and ambient stress are low along fault segments that have experienced large earthquakes. It also qualitatively explains why the recurrence time for large earthquakes may be irregular.

A dynamic crack model with viscous resistance to crack opening: Antiplane shear mode

There are several phenomena which have been observed in experiments involving brittle materials subjected to dynamic loading which do not seem to follow the usual concepts of linear elastic fracture mechanics. There are many loading situations that occur in practice are such that inertia as well as material rate sensitivity must be taken into account. In order to go a step further to model more realistic material response under dynamic loading without introducing undue complexity, a viscous strip model is introduced in antiplane shear mode. It is considered as a transient problem and the solutions are exact. Some of the information obtained is quite different from the well known models based on purely elastic response. A fracture criterion based on a dynamic loading condition will be proposed and discussed.

Dynamical fault rupture processes in heterogeneous media

The fault rupture processes in a horizontally layered medium and also in a three-dimensionally heterogeneous structure are investigated on three-dimensional, spontaneous dynamic shear crack models. The wave equations for three-dimensional space are solved numerically by a finite difference scheme under the appropriate boundary conditions and the finite stress fracture criterion. The heterogeneous properties of the elastic medium, particularly the existence of low-velocity zones, give remarkable effects on the rupture process, yielding appreciably decelerated rupture velocities and large fault displacements in and around the zones and strong motions in the near-field. The large fault displacements are enhanced when the rupture breaks the ground surface. The static seismic moment in the case with a low-velocity zone is essentially the same as in a homogeneous half-space. An attempt is made to simulate the rupture process of a moderate-size earthquake, by applying the above shear crack model in a heterogeneous medium with depth-dependent and laterally heterogeneous stress drop. The model appears to explain the observed features to a satisfactory degree.

Application of dynamic shear crack models to the study of the earthquake faulting process

The earthquake faulting process can be modelled as propagating two or three dimensional shear cracks. Since analytical methods cannot be used to study finite cracks, several numerical approaches to this problem have been taken, of which the numerical boundary integral method is here described in detail. The process of crack growth can be determined by the use of a fracture criterion. Since the material properties such as fracture strength and frictional properties vary over the surfaces of real faults, two types of complex faulting models-the “barrier” model and the “asperity” model-are discussed in the context of seismology.

A dynamic source model for the San Fernando earthquake

abstractWe model the San Fernando earthquake as a propagating rupture in a half-space, using for the slip-time-history on the fault plane analytical expressions which approximate the slip functions of dynamic crack models obtained by Das and Aki (1977a, b). We synthesize the strong ground motions and accelerations at the Pacoima Dam site and compute the teleseismic signals for different models of cracks. Three major featuras of the data–the strong pulse associated with the beginning of the rupture, the high acceleration phase on the Pacoima Dam records, and the presence of ripples on the teleseismic seismograms–which are not compatible with a smooth rupture process, are well explained by a crack with barriers model where the rupture encounters, along the fault plane, barriers or obstacles of high strength materials which may remain unbroken after the passage of the rupture front. A high-stress drop (400 to 500 bars) is required in the hypocentral area to explain the high-amplitude short-duration first pulse of the teleseismic records. This indicates a high level of tectonic stress in the area. A study of the earthquake series following the main shock shows that the aftershocks which took place in the region where major slipping occurred during the earthquake may represent the release of some of the barriers.