Fracturing Section Research Articles

Energy Evolution Characteristics and Hydraulic Fracturing Roof Cutting Technology for Hard Roof Working Face during Initial Mining: A Case Study

In the process of mining, a large area of hard roof will be exposed above a goaf and may suddenly break. This can easily induce rock burst and has a significant impact on production safety. In this study, based on the engineering background of the hard roof of the 2102 working face in the Balasu coal mine, the spatial and temporal characteristics of the strain energy of the roof during the initial mining process were explored in depth. Based on a theoretical calculation, it is proposed that hydraulic fracturing should be carried out in the medium-grained sandstone layer that is 4.8–22.43 m above the roof, and that the effective fracturing section in the horizontal direction should be within 30.8 m of the cutting hole of the working face. The elastic strain energy fish model was established in FLAC3D to analyze the strain energy accumulation of the roof during the initial mining process. The simulation and elastic strain energy results show that, as the working face advances to 70–80 m, the hard roof undergoes significant bending deformation. The energy gradient increases with the rapid accumulation of strain energy to a peak value of 140.54 kJ/m3. If the first weighting occurs at this moment in time, the sudden fracture of the roof will be accompanied by the release of elastic energy, which will induce rock burst. Therefore, it is necessary to implement roof cutting and pressure relief before reaching the critical step of 77 m. To this end, the comprehensive hydraulic fracturing technology of ‘conventional short drilling + directional long drilling’ is proposed. A field test shows that the hydraulic fracturing technology effectively weakens the integrity of the rock layer. The first weighting interval is 55 m, and it continues until the end of the pressure at the 70 m position. The roof collapses well, and the mining safety is improved. This study provides an important reference for hard roof control.

Optimizing fracturing techniques for enhanced hydrate dissociation in low-permeability reservoirs: Insights from numerical simulation

Natural gas hydrate represents an environmentally friendly alternative energy. However, the low-permeability hydrate reservoirs pose significant challenges to commercial production. To optimize hydrate production, reservoir stimulation methods like hydraulic fracturing, gas fracturing, and explosive fracturing are potential. In the investigation of the fracturing effect on hydrate dissociation behavior, we developed a three-dimensional model that comprehensively incorporates the arrangement of fracture planes, encompassing fracture permeability, spacing, and height. The model was implemented using a self-developed hydrate simulator which was validated at multiple scales. We firstly conducted a comparative analysis, contrasting the conventional depressurization method in horizontal wells with a combined depressurization and fracturing method. The comparison suggested that fracturing brings significant promotion (113.5%) on hydrate dissociation. Sensitivity analysis revealed fracture permeability and fracture height evidently exhibit an optimal value that facilitates the dissociation behavior. The optimal value of fracture permeability is dependent on the geological background. The optimal value of fracture height occurs just before penetrating the hydrate-bearing sediment layer and the overlying/underlying formations, i.e., near the thickness of the hydrate-bearing sediment layer. The impact of spacing on gas volume is limited while decreasing the spacing leads to more evident localized dissociation and differential subsidence of hydrate reservoir. It is necessary to balance production and geological risks when selecting the fracture spacing. The conclusions suggest that the fracturing operation need to be optimized by controlling the injection rate and the fracturing interval. It is not advisable to blindly pursue the effectiveness of reservoir stimulation but rather to consider the geological conditions. Additionally, we have found that it is preferable to establish barriers between the reservoir and overlying/underlying formations near the hydraulic fracturing section before reservoir stimulation, to prevent excessive early water production. In conclusion, this study would provide valuable insights for the optimization of fracturing techniques in stimulation of hydrate reservoir.

Microseismic activity characteristics and range evaluation of hydraulic fracturing in coal seam

Accurately evaluating hydraulic fracturing (HF) activity characteristics and impact range is critical to optimizing the HF scheme design. This study proposes an adaptive threshold peak detection method and a wavelet transform-support vector machine (WT-SVM) polarity analysis method based on microseismic (MS) monitoring data from a coal mine in Guizhou, China. The results show that the adaptive threshold peak detection method effectively improves the accuracy and stability of the P-wave initial arrival pickup for low signal-to-noise ratio (SNR) signals. The main frequency band of the MS signal of HF is focused on 150–300 Hz. The improved particle swarm optimization algorithm was used by introducing a random-weight strategy to determine the high-density impact areas of the HF within approximately 20 m on both sides of the fracturing borehole. Density cloud map analysis revealed a small area of the blank zone at the junction of the fracturing section, which provided a basis for fracturing parameter design and scheme optimization. The directional characteristics of the MS waveform were quantitatively described by polarity analysis, where the dip angle distribution ranged from 39° to 49°, the strike angle ranged from 61° to 74°, the slip angle was concentrated near 44°, and the stretch angle ranged from 61° to 74°. The WT-SVM proposed in this study overcame the effects of low SNR signals and the limitations of a small-area station network; 62% of the shear ruptures and 38% of the tensile ruptures of the HF were obtained. Shear cracks were the primary expression of the HF rupture type.

New approach of evaluating fracturing interference based on wellhead pressure monitoring data: a case study from the well group-A of Fuling shale gas field

Well pattern infilling has become an effective means for improving the development effect of gas reservoirs in unconventional gas reservoirs. The hydraulic fracturing of infill wells causes widespread fracturing interference between new and old wells. Because fracturing interference has a significant influence on the production of old wells, it is urgent to evaluate the degree of fracturing interference. This paper proposes a new approach to evaluating fracturing interference between new and old wells, which is based on a systematic analysis of the variation pattern of old well wellhead fracturing during the fracturing process of new wells. This new approach not only provides a semi-quantitative evaluation for the degree of fracturing interference between fracture sections of new and old wells but also achieves inter-well connectivity evaluation between new and old wells. This new approach is applied in well group A of the Fuling gas field to demonstrate its analysis process. The results show the different types of fracturing interference result in different levels of pressure response between each fracturing section and the old wells. For example, The pressure rise of old well A7-1 is more obvious in the fracturing process of the 2nd, 14th, and 13th sections of new well A68-5, and the old well A7-2 has significant fracturing interference with the 1st, 2nd, 3rd, 4th, and 6th sections. This achieves a semi-quantitative characterization of fracturing interference between new and old wells. The degree of fracturing interference between the old well A7-2 and the new well A68-5 is the strongest in well group A, which is the effect of compression fracture interference. The old wells A7-3 and A15-3 are the least impacted by fracturing interference, and follow the old wells A15-2 and A7-1. This result has implications for assessing the degree of fracturing interference and inter-well connectivity in unconventional gas reservoirs.

Factors affecting casing equivalent stress in multi-cluster fracturing of horizontal shale gas wells: finite element study on Weirong Block, southern Sichuan Basin, China

Multi-cluster fracturing technology was often used in horizontal well reservoir reconstruction to achieve production increase, which also affected casing equivalent stress distribution. This paper focuses on multi-cluster fracturing and establishes a fracturing model in line with the reality. The three-dimensional finite element model of multi-cluster fracture-formation-cement sheath-casing was proposed, the influence of cluster spacing and fracturing cluster number on casing equivalent stress was studied. On this basis, a single segment 8-cluster three-dimensional finite element model was developed. The influence of rock elastic modulus, casing inner wall pressure, geostress change and elastic modulus of cement sheath on casing equivalent stress was simulated from two aspects of uniform and non-uniform extrusion of wellbore. Actual data was used and analyzed for the fracturing section of a well in Weirong Block, southern Sichuan Basin, China. The results showed that the casing equivalent stress decreased with the increase of fracture dip angle. The casing equivalent stress increased with the increase of cluster spacing; however, it decreased with the increase of rock elastic modulus. The casing equivalent stress increased with the increase of casing wall pressure. Also, the cracks extrude the casing evenly did not affect the change on casing equivalent stress. It was also found that, when casing was uniformly squeezed by multiple fractures, the difference of ground stress had little effect on casing equivalent stress, while non-uniform extrusion had greater effect on casing equivalent stress. Further, when there was no wellhead pumping pressure, the casing equivalent stress increased with the increase of the elastic modulus of the cement sheath, and decreased on the contrary. The elastic modulus of rock was lower than that of cement sheath, and the casing equivalent stress increased with the increase of the elastic modulus of cement sheath, and decreased on the contrary. The research results had certain guiding significance for the prevention and control of casing damage in fracturing section.

A novel framework combining production evaluation and quantification of development parameters for shale gas wells

Accurate prediction of estimated ultimate recovery (EUR) and quantification of important engineering and geological parameters play a decisive role in production parameter optimization and investment decision of shale gas wells. This study proposes an integrated learning framework for EUR prediction and production parameter quantification in shale gas wells using CatBoost algorithm. CatBoost is designed to identify complex relationships efficiently and accurately between geological and engineering parameters and target values EUR. Furthermore, the EUR prediction model based on CatBoost was used to identify important features. Various machine learning interpretation methods are used to visualize the marginal effects of important and interesting features in an attempt to optimize production parameters by quantifying the features. Experiments are conducted to evaluate the performance of the proposed the integrated learning framework in the deep shale gas well data set of Luzhou block in Sichuan Basin, China. Results demonstrate that compared with other mainstream ensemble learning algorithms, the proposed EUR prediction model based on CatBoost algorithm performs well on small data sets, with a prediction error of only 12.31%. The post-interpretation tool of the model was used to find out four important characteristics affecting EUR in Luzhou block, namely, contents of brittle minerals, fracturing length, porosity and fracturing section. The optimal combination of geological and engineering parameters for the block to maximize EUR is identified by using the established efficient and accurate EUR prediction model as input and the visualization of the marginal effects of the features of interest as output. This work can effectively provide guidance for EUR prediction and production parameter optimization of shale gas wells.

Distribution Characteristics and Cause Analysis of Casing Deformation during Hydraulic Fracturing of Shale Gas Wells

The distribution characteristics and causes of casing deformation during hydraulic fracturing were investigated. Finite element models of a casing–cement sheath-formation assembly, considering a multi-stage fracturing process, were established. The axial displacement law of formation during multi-stage fracturing was analyzed. Furthermore, considering the existence of faults, the mechanism of casing deformation caused by fault slippage due to axial displacement was analyzed. Finally, the influence of fault parameters, as well as the casing’s geometry and mechanical parameters, on casing deformation were clarified. The results show that the casing deformation point is densely located near landing point A and sparsely located near toe point C. The entry of high-pressure fracturing fluid into the formation would reduce the effective stress of the formation and could even change it from compression to tension, which would mean that the formation in the fractured section would be stretched in the axial direction, squeezing the formation in the non-fractured area and causing axial displacement; the closer the formation is to point A, the greater the displacement. The change in casing deformation can be divided into three stages. In the first, the casing deformation is zero when the fracturing section is far from the fault. In the second, the casing deformation increases significantly when the fracturing section is close to the fault. In the third, the casing deformation increases slowly when the fracturing section deviates from the fault. The closer the fault is to landing point A, the greater the length, the smaller the dip angle, and the larger the casing deformation, and vice versa. In contrast, increasing casing wall thickness and steel grade has little effect on relieving the casing deformation.

A new insight into casing shear failure induced by natural fracture and artificial fracture slip

Casing deformation is often encountered due to hydraulic fracturing in the unconventional reservoir. Analysis of 3D seismic, microseismic and logging data shows that the casing shear failure is the primary casing failure mode during hydraulic fracturing. The casing deformation is closely related to the fracture system near the wellbore, but the mechanism and law behind this phenomenon are not completely clear. The slip mechanism of the fracture system and the characteristics of casing deformation were investigated to prevent casing deformation and identify the fracture with large slip distance. Based on the Mohr-Coulomb and maximum circumferential stress theory, the slip mode of natural fracture and the slip mechanism of artificial fracture were analyzed, respectively. Then, two numerical models were established to calculate the slip distance with finite fracture height and corresponding casing deformation. Furthermore, a monitoring model of casing deformation was presented based on microseismic data and real-time injection parameters. The result shows three slip modes in the fracture system crossing wellbore under different fluid pressures, including shear-slip under low fluid pressure, open-slip under medium fluid pressure and propagation-slip under high fluid pressure. The slip distance is associated with fracture length, fracture height, fluid pressure in the fracture and fracture angle. Hydraulic fracturing not only activates natural fractures and faults but also causes the slip of artificial fractures created by hydraulic fracturing, which is responsible for the casing shear deformation after fracturing and during production. The greater the stress deflection angle, the longer and higher the artificial fracture, the more likely the casing deformation in the fracturing section. Therefore, the scale of artificial fracture should be controlled during hydraulic fracturing. This study will provide guidance for predicting fracture slip distance and casing deformation prevention.

Simulation Study on Interactive Propagation of Hydraulic Fractures and Natural Fractures in Shale Oil Reservoir

Natural fractures are developed in the shale oil reservoir, and the hydraulic fracture (HF) morphology is complex. The fracture shape can be inverted by using fracture propagation numerical simulation technology, which provides guidance for fracturing parameter design and fracturing process optimization. However, the existing models still have many deficiencies in the interactive propagation mechanism of HF and natural fracture (NF). Based on the three interactive modes of HF and NF (HF propagation without NF, HF propagation with full NF, and HF propagation with half NF), this work establishes the fracture propagation model and puts forward the simulation calculation method. The Brinkman equation is used to modify the leakage model based on Darcy’s law, and G1701H well is taken as an example to simulate the fracture propagation law under different interaction modes. The research shows that there is a transition region between the HF wall and rock matrix. The greater the porosity and permeability of the rock matrix, the more significant the influence of the transition region on leakage. The NF zone will change the propagation direction of the main fracture. When there are multiple clusters of fractures in the same fracturing section, only some HFs meet with natural fractures, and it is easy to form a “T”-shaped fracture network. The results improve the existing hydraulic fracturing model and provide help for fracturing parameter design and fracture parameter inversion of the shale oil reservoir.

Investigation on the influence of multiple fracture interference on hydraulic fracture propagation in tight reservoirs

The combination of horizontal well drilling and staged fracturing technology is one of the most effective ways to increase unconventional oil and gas production. The orientation of the horizontal wellbore is mostly considered to be consistent with the minimum horizontal principal stress in the numerical simulation. However, horizontal wellbores are usually at an angle to the minimum principal stress in field operations. In this case, the dynamic growth of the fractures will be limited during subsequent perforation and fracturing operations. Therefore, the objective of this paper is to build an effective numerical model to investigate the propagation of hydraulic fractures in the case of a horizontal wellbore that is not co-linear with the minimum horizontal principal stress. The extended finite element method is applied for multi-stage fracturing flow-solid coupling in horizontal wells. In addition, the effect of sequential and alternate fracturing on the propagation of fractures in horizontal wells was compared. The simulation results show that fracture growth on one side of the horizontal wellbore is severely restricted as the pinch angle increases. It is most evident in sequential fracturing mode while increasing the fracture spacing is one of the effective ways to reduce inter-fracture interference. Meanwhile, compared to sequential fracturing, alternate fracturing is more effective in reducing inter-fracture interference, allowing for effective fracturing at closer fracture spacing. Finally, the density of the fracturing section has been increased, and a higher reservoir reconstruction area can be obtained. This work, for the first time, evaluates the dynamic propagation of fractures when the horizontal wellbore is not collinear with the minimum horizontal principal stress orientation and helps operators to optimize the fracturing process.

Dynamic Stress Field Evolution of the Shale Oil Reservoirs: Development and Optimization of Well Patterns

Research on shale oil has gained considerable attention with regard to global unconventional oil exploration, which is of the strategic significance in the field of energy. Considering the characteristics of continental shale reservoirs, a three-dimensional fluid–solid coupling model of a multistage fracturing horizontal well was established based on the poroelasticity theory. The difference in stress sensitivity among different layers was also considered in this model. The evolution of pore pressure and stress field during the production process was analyzed through numerical simulation. A comparison of the planar well pattern and stereo well pattern revealed that the latter has less interwell-flow interference and smaller alteration angle of the minimum principal stress; thus, in the latter pattern, intersection with existing hydrofractures during refracturing can be avoided. It is more advantageous to refracturing in the middle of the outside horizontal section, but with increase in production time, the optimal fracturing section shrinks. Considering the maximum recovery factor and interwell interference, well-spacing optimization is recommended from 200 m to 300 m.

Production data analysis of shale gas using fractal model and fuzzy theory: Evaluating fracturing heterogeneity

The development of shale-gas reservoirs has been dramatically accelerated with the technical breakthroughs made in the fields of horizontal well drilling and multistage hydraulic fracturing. After being restructured by hydraulic fracturing, the shale reservoir would have a complicated fracture network featuring great randomness. The accurate evaluation of the fracture network structure and the quantitative calculation of the fracturing fluid flowback are keys to the efficient development of shale-gas fields. In this paper, a new fractal seepage model of single fracturing section was established based on the conversion relationship of gas-liquid flow fractal dimension. Then, the sensitivity of model parameters and the flow characteristics of single fracture section were analyzed to determine the reason that gas production “rises first, and then drops down” at the initial production phase. Finally, the mathematical description of the heterogeneity of fracture networks was further proposed based on Fuzzy theory. Case studies in Fuling shale-gas field were performed by means of the Extended Monte Carlo Simulation and the optimal fracturing heterogeneity factors of typical wells are 0.25, 0.10 and 0.15. Compared with the homogeneous model, the accuracies of gas production and water production calculated by heterogeneous model are increased by up to 15% and 20%, respectively.

The impact of coal macrolithotype on hydraulic fracture initiation and propagation in coal seams

Macrolithotypes control the pore-fracture distribution heterogeneity in coal impacting stimulation via hydrofracturing and the coalbed methane (CBM) production. Given that it is affected by the discontinuities, hydraulic fracture geometry is complex in the vertical plane and is different from a simple fracture in a homogeneous reservoir. However, the initiation and propagation mechanism in the vertical plane is unclear. To clarify this, the cohesive zone finite element approach, with macrolithotype contributions included, was used to simulate and analyze the hydraulic fracture propagation. The experimental tests showed that, the bright and semi-bright coal usually have higher microfracture (cleat) density accompanied by the lower mechanical properties than that of the semi-dull and dull coals. The behavioral differences are likely to impact the geometry evolution of hydraulic fractures and which appears to vary when fracturing the different coal macrolithtypes. Thus, the cohesive zone finite element approach was used with two models to capture macrolithotype impacts. The result show that, when fracturing the dull coal (model 2), the overall propagation region rapidly displayed a simple plane in shape because of the less development of natural fractures. With the influence of the larger elastic modulus, the high-stress zone would be easy formed and suddenly release to generate pressure pulse when the hydraulic fracture penetrated the interface. As the hydraulic fracture initiates from the bright coal (model 1), the presence of the existing diverse cleat network contribute greatly to the increase of cracks number to form complex fractures. However, the opening of natural fractures will lead to the diversion of fracturing fluid, and the larger elastic modulus of the interlayer also plays a limiting role in the height of the hydraulic fracture. In addition, the monitoring of hydraulic fracture was carried out and shown that the height of the major fracture in model 1 was restricted and limited by the bright coal; and the height in model 2 is usually larger than the dull coal thickness, indicating that the hydraulic fracture has cut through the fracturing section (dull coal) and embedded into the upper and lower layers.

Effect of inter-cluster interference on fracture morphology in multi-cluster staged fracturing for shale reservoir

Multi-cluster staged fracturing technology is an effective measure to stimulate the reservoir properties. However, the inter-cluster interference effect is obvious when the cluster spacing is very narrow, which seriously affects the effect of fracturing. In order to understand the interference among fracturing clusters within the single fracturing section of the shale horizontal wells during multi-cluster staged fracturing, a finite element model is developed by using ABAQUS finite element simulation software. On this basis, the influences of factors on the fracture morphology are studied. The simulation results have shown that the cluster spacing is the most important factor affecting inter-cluster interference. With the increase in the distance between adjacent clusters, the interference among the fracturing fractures decreases and the propagation of different fractures become homogeneous or similar. Moreover, the increase in the elastic modulus of the shale formation promotes the propagation of the fractures longitudinally, but it hinders the crack opening of the fracture laterally. In addition, properly increasing the injection rate of fracturing fluid during fracturing is more advantageous for obtaining long and wide fractures. Besides, the effect of the fracturing fluid viscosity on fracture width is greater than that on the fracture half-length. The simulation results show the existence of inter-cluster interference comprehensively, which can provide a reference for the design and optimization of multi-cluster staged fracturing to some extent.

Numerical simulation of the conveyance characteristics of fracturing ball in the horizontal section

Ball drop-type fracturing tools that can improve permeability and production efficiency have played important roles in well stimulation during the exploitation of unconventional hydrocarbon reservoirs. In this study, computational fluid dynamics (CFD) with a standard k-ε turbulence model and a dynamic fluid body interaction model was applied to describe the coupling interaction between fracturing ball and the carrying fluid to understand the conveyance characteristics of fracturing ball in the horizontal section. The motion behavior of fracturing ball moving across the horizontal section with different flow rates and ball sizes is investigated. Simulation results indicate that the fracturing ball exhibits collision and rebound during the conveyance in the horizontal section. Minimum flow rates are required for the fracturing balls with different sizes to pass through the ball seat; otherwise, they will be trapped in the horizontal section and hardly reach the target fracturing section. On-site fracturing experiments are also conducted to validate the CFD study results. A general agreement between the numerical simulation and the on-site fracturing experiment is obtained. Therefore, these analyses have significant importance for the design of fracturing ball and fracturing parameters.

Variation rules of fracture initiation pressure and fracture starting point of hydraulic fracture in radial well

Abstract Radial well technology in combination with hydraulic fracturing technology has gained encouraging achievements as a new method of increasing production in oil fields. Compared with conventional perforation fracturing, radial well fracturing has an obvious advantage in breaking through polluted borehole areas, and it is superior to horizontal well fracturing because of the shorter construction period, less consumption of fracturing fluid and lower damage to the reservoir. Currently, in China, the study on the position of fracture starting point and fracture propagation of radial well is still at the preliminary stage and the fracture initiation pressure and position of fracture starting point remain unclear. Consequently, it's difficult to design radial well completion parameters (length of radial well, diameter of radial well and fracturing truck units) precisely and implement the technology more efficiently. Based on fluid-solid coupling effect and the maximum tensile-stress criterion, this paper follows the concept of dynamic analysis and analyzes the influence rule of the length, diameter and azimuth of radial well, horizontal in-situ stress and natural fracture on fracture initiation pressure and fracture starting point under the stress of strike-slip fault by using ABAQUS to simulate and study local stress accumulation situation caused by drilling vertical well section, radial well section and fracturing section through finite element method. The result shows that initiation pressure and distance between well and fracture starting point increases as the length, diameter and azimuth of radial well section rise. Azimuth is most influenced, followed by length, and lastly diameter of radial well section. When the horizontal in-situ stress ratio (σH:σh) is decreased from 1.9 to 1.1, if the azimuth is 0°, the initiation pressure increases by 41.35%; if the azimuth is 90°, the initiation pressure declines 0.8%. However, the positions of these two fracture starting point remain unchanged. The permeability increases by four orders of magnitudes and the fracture initiation pressure goes up 34.5%, with no influence on the fracture starting point. When there existed natural fractures in the reservoir, the intersection between fracture section and radial well section firstly shows fracturing. Besides, the fracture initiation pressure and the hydrostatic fluid column pressure in vertical wellbore are equivalent. Reduced length and diameter of radial well section as well as reservoir permeability and properly increased azimuth of radial well are conducive to fracture at the toe end of the radial well section. On the contrary, fracture at shaft linings of vertical well section is easily occurred. The research result can be used to predict the direction of fracture propagation to some extent and is favorable for designing parameters of radial well completion and fracturing operation.

Energy dissipation due to fracture of a beam at low temperature

We consider the energy dissipation associated with fracture of a beam at low temperature. A two-dimensional finite element analysis is used to compute the stored elastic energy in the test beam assembly just prior to fracture and to interpret the experimentally determined heat output. By utilizing the Laplace transform technique, the dynamic fracture response of the beam subjected to pure bending is also studied, and instantaneous crack length, crack tip speed and fracturing section bending moment are obtained. The amount of energy dissipated during cryogenic fracture of the beam is calculated using a dynamic strain energy release rate, and the conversion rate of the dissipated energy into the interpreted heat output is estimated.

Effect of shear and rotary inertia on dynamic fracture of a beam or plate in tensile loading

The dynamic fracture response of a long beam of brittle material subjected to tensile loading is studied. If the magnitude of the applied tensile loading is increased to a critical value, a crack will propagate from one of the longitudinal surfaces of the beam. As an extension of previous work, the effect of shear and of rotary inertia on the tensile loading and the induced bending moment at the fracturing section is included in the analysis. Thus an improved formulation is presented by means of which the crack length, crack tip velocity, bending moment and axial force at the fracture section are determined as functions of time after crack initiation. It is found that the rotary effect diminishes the bending moment effect and retards total fracture time whereas the shear has an opposite effect. Thus by combining the two effects (to simulate to first order the Timoshenko beam) overall fracture is retarded. The results also apply for plane strain fracture of a plate in tensile loading provided the value of the elastic modulus is appropriately modified.

On the symmetric dynamic fracture of a beam under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied by means of two different one-dimensional models. If the magnitude of the applied loading is increased quasi-statically to a critical value, two coplanar edge cracks are assumed to propagate across the beam's cross section. The first model parallels that of [6] with the crack length, crack speed and the loading on the fracturing section being determined as functions of time after fracture initiation. The second model is derived by means of energy considerations in the vicinity of the fracturing section. The results obtained from both models are similar except during the final phase of the fracture process.

Effect of bending moment on the dynamic fracture of a beam or plate under tensile loading

The dynamic fracture response of a long beam of brittle elastic material under tensile loading is studied. If the magnitude of the applied loading is increased to a critical value, a crack is assumed to propagate across the beam cross section. As an extension of previous work, an induced bending moment generated during fracture is incorporated into the analysis and this improved formulation is presented. The crack length, crack tip speed, axial force and bending moment on the fracturing section are determined as functions of time after crack initiation. It is found that the bending moment has a significant effect on the fracture process in that it tends to retard fracture and causes a drastic change in the slope of the loading curve for large crack depths. Finally, by appropriate change of the elastic modulus, the results may be applied to plane strain fracture of a plate in pure tensile loading.