Dynamic Stress Intensity Factor Research Articles

Investigation of Mixed-Mode Crack Propagation Behaviour under Impact Loading

The fractured rock masses are often subject to impact loading, giving rise to complicated mixed-mode cracking. The propagation of the cracks may result in structural instability and project disaster and often cannot be understood by studying only the mode I fracture. Discovering the mixed-mode crack behaviour is of great interest to engineering design. Here, we report on an investigation of mixed-mode crack propagation under impact loading. A Single Cleavage Simi-Ellipse configuration was proposed with different pre-crack dip angles. Split Hopkinson pressure bar impact tests were conducted and a high-speed camera was employed to capture the cracking behaviour. With the experimental-numerical method, the dynamic initial stress intensity factors KI and KII were calculated by the finite element method. The AUTODYN was applied to study mixed-mode crack propagation numerically. The investigation demonstrates that the mixed-mode crack would adjust fracture mode continuously in the whole propagation process companied with several extending decelerations. The crack initiation mode and the dynamic initiation stress intensity factor have a relationship with the impact loading orientation. The AUTODYN can be applied in predicting rock crack propagation, offering new insights into dynamic stability analyses in engineering design.

Material point method for the propagation of multiple branched cracks based on classical fracture mechanics

The material point method (MPM) combines the advantages of both mesh-free and grid-based methods for simulating complex cracking problems without the limitations of the grid. However, the conventional MPM cannot deal with cracks directly because of the continuous nodal velocity field. Therefore, in this study, the conventional MPM is improved using three new algorithms to simulate the propagation of multiple branched cracks. The first proposed algorithm is the phantom node method using a particle cutting technique, which allows discontinuity of the velocity field around arbitrary cracks, and the second one is the multi-crack contact algorithm to deal with the contact problem between multiple objects cut by multiple cracks, avoiding crack penetration. The last one is particle interaction integral method based on classical fracture mechanics to get the dynamic stress intensity factors (DSIF) and realize the crack propagation. And all algorithms are validated by comparing with the extended finite element method. Subsequently, uniaxial compression tests of three rock samples with different pre-existing cracks are simulated using the extended MPM. Three types of crack coalescence are found, and the simulated results agree with the previous experimental results, which demonstrate the feasibility and flexibility of the proposed extended MPM for simulating the propagation of multiple branched cracks.

Time-harmonic analysis of multiple interface cracks in two dissimilar FGM half-planes: In-plane problem

In this study, the dynamic stress intensity factors (DSIFs) at the tip of several cracks located at the interface between two half-planes made of functionally graded materials (FGMs) are calculated using the Distributed Dislocation technique (DDT). In this paper, the in-plane time-harmonic loading is considered and it is assumed that the properties of the FGMs follow the law of exponential function. First, the problem of two dissimilar half-planes that are weakened by an interface Volterra type climb and glide edge dislocation is solved using the Fourier transform, and then, using the DDT, singular integral equations with a singularity of the Cauchy type are extracted. To calculate the DSIFs at the cracks tips, the resulting singular integral equations are numerically discretized and the dislocation densities on the crack surfaces are obtained. Second, the graphs are presented in the numerical results section to show the effects of change in the parameters such as non-homogeneous constant, Poisson ratio, crack length, frequency and interaction between the cracks on DSIFs.

Effect of loading rate on the mode II dynamic fracture characteristics of 40Cr steel

40Cr high strength steel is often used in aerospace and national defense fields due to its excellent mechanical properties. Therefore, the research on the mode Ⅱ dynamic fracture characteristics and failure mechanism of 40Cr high strength steel under high-speed impact has important scientific significance and engineering value. Experiments and numerical simulations were carried out to study the mode Ⅱ dynamic fracture properties of 40Cr material under high loading rates. Based on the newly designed specimen and the novel test technique for mode Ⅱ dynamic fracture, the experimental-numerical method was used to determine the dynamic stress intensity factor curve of the crack tip during the loading process. The crack initiation time of the specimen was obtained by the strain gauge pasted on the specimen, and the mode Ⅱ dynamic fracture toughness of 40Cr was finally determined. The effect of loading rate and the failure mechanism of the material were also studied. The results show that within the present loading rate range of this work (1.08−5.53 TPa·m1/2/s), the mode Ⅱ dynamic fracture toughness of 40Cr presents a positive correlation with the loading rate. Through the analysis of the fracture morphology of the recovered specimen, it is found that there exists a transition from tensile failure mode to adiabatic shear failure mode with the increase of loading rate, and the critical loading rate is about 2.92 TPa·m1/2/s. When KⅡd≤1.70 TPa·m1/2/s, the 40Cr steel mainly exhibits brittle fracture; when KⅡd is in the range of 2.12−2.92 TPa·m1/2/s, the fracture mainly presents the morphological characteristics of ductile fracture; when KⅡd ≥ 3.19 TPa·m1/2/s, the material failure is mainly caused by adiabatic shear bands. Brittle fracture is dominated by strain rate hardening mechanism, ductile fracture is dominated by strain/strain rate hardening and thermal softening mechanisms, and adiabatic shear fracture is dominated by thermal softening mechanism.

Investigation of the effect of the blast waves on the opposite propagating crack

The stress waves have a significant effect on crack propagation during blasting. In this paper, a microsecond controlled blasting case involved with the interactions between blast waves and an opposite propagating crack is studied using dynamic photoelasticity method. The results show that the stress waves influence the local stress field of dynamic crack tip , and alter the crack propagation behavior. Moreover, during crack-wave interactions, both the singular stresses and the nonsingular stresses around the crack tip changed apparently. With the incidence of dilatational wave , the non-singular stresses σ ox and σ oy around the tip of opposite propagating crack increase remarkably, inducing an increase in the crack extension resistance, and decreasing both the crack velocity and the dynamic stress intensity factor K I d . Whereas, when the shear wave impinges onto the opposite propagating crack, the crack tends to increase the crack velocity. In addition, the fractal dimension of the crack velocity decreases during the incidence of dilatational wave, and increases when the shear wave impinges upon the opposite propagating crack.

An In-depth Investigation of Bimaterial Interface Modeling Using Ordinary State-based Peridynamics

This study presents an in-depth investigation of interface modeling techniques in ordinary state-based peridynamics (OSB-PD). By performing numerical examples, the accuracy of each technique is investigated by comparison with the finite element solutions for simple bimaterial interface problems. Subsequently, the capabilities of OSB-PD in treating material interfaces with complicated geometries are demonstrated by solving various problems with a single inclusion or multiple inclusions. Finally, a cracked plate containing an inclusion is considered to study effects of the inclusion on dynamic stress intensity factor (DSIF). The OSB-PD shows its huge potential in modeling a structure with multiple defects due to the effectiveness and ease of implementation.

Dynamic crack face contact and propagation simulation based on the scaled boundary finite element method

This paper extends the static contact problem of crack faces to include dynamic contact and crack propagation based on the relevant literature (Zhang et al., 2018). The scaled boundary finite element method is used to solve dynamic crack face contact and crack propagation problems. Based on the scaled boundary finite element shape function considering side-face loads, the system dynamic equilibrium equation and related characteristic matrix are presented. The Lagrange multiplier method is used to establish the contact model between the crack faces. The meshing and remeshing algorithm is carried out by using arbitrary polygonal meshes. When a crack propagates, the remeshing operation involves only a small region of polygons around the propagating crack tip. Mesh mapping technology is used to transfer the response parameters to newly generated nodes. The proposed method is validated by analysing the impact of two identical bars and the dynamic contact of crack faces in the Koyna Dam. The results show that the dynamic contact model can effectively simulate the opening and closing of crack faces, and the longer the crack is, the greater the influence of the non-linearity of the contact along a crack face on the dynamic stress intensity factor and dynamic response. Finally, a dynamic crack propagation problem is simulated. Compared with the dynamic crack response without considering the contact constraint, when considering the contact constraint, the crack initiation angle is closer to the corresponding theoretical solution, and the crack propagation path is closer to the corresponding experimental result. Thus, the dynamic contact constraint condition of crack faces has an important influence on crack propagation.

Compound-mode crack propagation law of PMMA semicircular-arch roadway specimens under impact loading

An underground roadway usually contains defects of various types, and when the roadway is subjected to external loading, the locations of those defects influence the roadway by differing degrees. In this study, to study how the locations of defects affect crack propagation in a roadway, specimens with tunnel-type voids were made using polymethyl methacrylate, and the stress wave produced by a bullet impacting an incident rod was used as the impact load. Meanwhile, the variations in crack speed, displacement, and dynamic stress intensity factor during crack propagation were obtained using an experimental system of digital laser dynamic caustics, and the commercial software ABAQUS was used for numerical simulations. From the experiments and numerical simulations, the crack propagation path was verified and the impact fracture behavior of a semicircular-arch roadway with different defect positions was presented. The results show that when the pre-fabricated crack is on the central axis of the sample, the crack propagation is purely mode I; when the pre-fabricated crack is 5 mm from the central axis, the crack propagation alternates between mode I and a mixture of modes I and II; when the pre-fabricated crack is at the edge of the semicircular-arch roadway, the crack propagation follows the I–II mixed mode.

Strain gauge experimental study on mode I rock fracture characteristics under impact loading

Based on the analysis of the strain field of the dynamic crack tip under impact loading, a specific acute (obtuse) angle multi-point strain gauge method is proposed to study the mode I rock crack dynamic fracture. Compared with the dynamic caustics method, the feasibility of the strain gauge method for calculating the crack propagation velocity and dynamic stress intensity factor (SIF) is verified. The rock specimen mode I crack dynamic fracture behaviour under impact loading was studied by the strain gauge method and assisted by high-speed photography and scanning electron microscopy. The results showed that the mode I crack propagation behaviour calculated by two strain gauge methods under impact loading was close to the caustics experimental results. Compared with the dynamic caustics method, the multi-point strain gauge method could roughly record the crack propagation behaviour of rock under impact loading, which provides an experimental basis for studying non-transparent impact fracture. The rock section mainly experienced tensile stress failure under impact loading, manifested as mode I cracking, and the granite specimen fracture damage occurred in white spot micro-crack propagation. Under impact loading, cracks in both polymethyl methacrylate (PMMA) and granite specimens propagated stably in the middle stage of propagation, the average granite specimen crack propagation velocity was 647.9 m/s, and the average SIF value of was 5.37 MN/m3/2, which were higher than the PMMA specimen experimental results.

Research on the process of crack growth induced by explosion stress waves in rock masses with open joints

A joint in the medium has a significant impact on the propagation of explosion stress wave and the growth of explosion-induced crack. In this work, the crack growth experiment in the PMMA(polymethyl methacrylate) medium with vertical open joints under explosion loading was first carried out on the digital laser dynamic caustics experimental system (DLDC), in order to analyze the crack initiation, growth and arrest of wing cracks, and examine how the DSIF(dynamic stress intensity factor) at crack tip and the velocity of crack growth changed. The process of stress wave propagation and crack growth in this experiment was effectively inverted using the discrete lattice spring model (DLSM), the stress changes at the midpoint and endpoint of the joint were compared, and the acting characteristics of stress wave were analyzed here. Finally, the differences in velocity and acceleration of crack growth of wing cracks were compared and examined under different dip angles of joints.

Dynamic fracture behavior for functionally graded piezoelectric bi-materials with interfacial cracks near a circular hole

This work aims to develop an effective method for evaluating the stress concentration at the tip of permeable interfacial cracks near a circular hole in functionally graded piezoelectric bi-materials. The structure is loaded by anti-plane incident SH-wave and all material parameters are assumed to obey exponential variations. Fracture analysis is performed by the Green function method, which is used to solve the boundary conditions problem. The mechanical model of the cracks is constructed by crack-conjunction and crack-deviation techniques so that the crack problem is simplified as solving a series of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factors (DSIFs) at the inner and the outer crack tips can be derived. The validity of the present method is verified by comparison with references. Numerical cases reveal parametric dependence of DSIFs on the geometry of circular holes and cracks, the characteristics of the incident waves and the inhomogeneity of materials. The method proposed in this paper can circumvent the limitations of using dual integral equations to deal with asymmetric defects and opens up a new way for the research on fracture problems in functionally graded piezoelectric materials with more complex defects.

Effect of Loading Rate and Water on the Dynamic Fracture Behavior of Sandstone under Impact Loadings

The moisture content of rocks in nature often changes with the natural hydrological environment, and generally natural rock may be classified into three types, that is, dry, natural, and saturated rock. The corresponding responses to dynamic loads may have considerable difference, and therefore, impact tests in this paper were conducted on single cleavage triangle sandstone specimens with three types of conditions, respectively, under various loading rates. The ingredient of sandstone was determined by X-ray diffraction and optical microscopic image. The crack propagation parameters were recorded with the aid of a crack propagation gauge. The dynamic stress intensity factors were calculated using ABAQUS code. The results show that the fracture toughness of all the three types of specimens increase with loading rate, which could be expressed by power functions. Compared with dry and natural sandstone, the crack propagation speed of saturated sandstone specimens is more sensitive to the loading rate. Both the rate effect and the water effect, including water-weakening effect and water-enhancing effect, play a significant part in rock dynamic fracture behavior.

Experimental study on the effect of defect curvature on the impact fracture behavior of structures using the real-virtual caustics method.

Defects have significant influence on the impact fracture behavior of structures. In this paper, the real-virtual caustics method is used to study the impact fracture behavior of structures with elliptical arc defects under impact loading of the drop hammer, and the impact loading process is simultaneously analyzed. The research results show that impact loading of the drop hammer in this experiment is a multi-period dynamic loading process, and the fracture of specimens under impact loading of the drop hammer is an energy-controlled process. The running crack initiates under impact loading and propagates toward the elliptical arc defect. After reaching the end of the elliptical arc defect, the running crack arrests and accumulates energy, and then it initiates again and propagates toward the loading position. The greater the end curvature of the elliptical arc defect, the shorter the time for the running crack to stagnate and accumulate energy at the defect end, and the earlier the time for the running crack to initiate again at the defect end, the smaller the impact loading stress of the drop hammer and the dynamic stress intensity factor of running crack tip when the running crack initiates again at the defect end.

Determination of Stress Intensity Factors under Shock Loading Using a Diffraction-Based Technique

An experimental optical method has been developed for the measurement of opening and sliding notch face movements. The light passing through a thin slit is monitored by a photodiode detector. Two parts of the slit are fixed independently on the notch faces of the simulated crack. Dynamic variations of the notch face movements are recorded as an electric signal by an oscilloscope. The sensitivity of such displacement measurement is comparable with the wavelength of light. Dynamic mixed-mode stress intensity factors under shock loading were evaluated from the data obtained and subsequently compared with a numerical simulation by ANSYS software. As it was approved, the technique has shown sufficient sensitivity, good linearity, and measurement reliability. Due to its non-destructive nature and overall robustness, the arrangement is applicable even for structural component condition determination taking into consideration potentially unknown boundary conditions and the non-linear character of mechanical parameters.

Dynamic performance for piezoelectric bi-materials with an interfacial crack near an eccentric elliptical hole under anti-plane shearing

The purpose of this paper is to evaluate the stress concentration at the tip of a permeable interfacial crack near an eccentric elliptical hole in piezoelectric bi-materials under anti-plane shearing. Fracture analysis is performed by Green’s function method and the conformal mapping method, which are used to solve the boundary conditions problem. The mechanical model of the interfacial crack is constructed by interface-conjunction and crack-deviation techniques so that the crack problem is simplified as solving a series of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factors (DSIFs) at the inner and the outer crack-tips can be derived. The validity of the present method is verified by comparing with a crack emerging from the edge of a circular hole as a reference. Numerical cases reveal parametric dependence of DSIFs on the geometry of eccentric elliptical holes and interfacial cracks, the characteristics of the incident wave, the equivalent piezoelectric elastic modulus and piezoelectric parameters. The results illustrate that the eccentric distance has a great effect on the stress concentration at the crack tip, which may be harmful to the normal service of piezoelectric devices and materials. In addition, the method proposed in this paper can also deal with non-eccentric problems and has wider applicability.

Analysis of the expansion process of cracks caused by blasting stress waves in a PMMA medium with filled defects

The occurrence of defects in a medium has a significant role in the transmission of blasting stress waves and the expansion of blast-caused cracks. In this study, an experiment is first conducted on the crack expansion in a polymethyl methacrylate (PMMA) medium with vertically filled defects under a blasting load by using the digital laser dynamic caustics (DLDC) testing system. The origin, expansion, and termination of wing cracks are analyzed, and the variation of the dynamic stress intensity factor and the crack expansion velocity on the crack tip are studied. An effective inversion to the stress wave transmission and the crack expansion is achieved via a test conducted using the distinct lattice spring model (DLSM), along with a comparison between the stress variation at the center points on the wave heading side, back side, and endpoints of the defect. The characteristics of the impact due to the stress wave are also analyzed. Lastly, the expansion velocity/acceleration difference between the wing cracks are analyzed and compared with various defect characteristics (filled, open, closed, bedding).

Scattering problems for a rectangular crack in a saturated porous material: application of the Chebyshev's functions

Analytic solutions are derived for the problem of scattering of obliquely incident plane waves from a rectangular crack in a fluid-saturated porous solid. The propagation direction and polarization direction of the incident waves can be arbitrary. The solutions are based on expanding the crack opening displacements in terms of Chebyshev functions. Explicit expressions for the frequency-dependent stress intensity factors and scattering far fields are obtained. The poroelastic effects play a key role in determining the crack opening mode by changing the effective normal stress applied on the crack surfaces. Consequently, the dynamic responses of the crack can be quite different from those of conventional dry crack in an elastic solid. The effects of scattered slow-P waves as well as wave-induced crack fluid pressure on the dynamic stress intensity factors are clarified, and a corner effect is observed in the scattering patterns for moderately high frequencies. The comparisons of synthetic seismograms with the elastic model show the slow-P waves can cause significant energy loss and waveform change. This study can be used for fracture analysis of porous media and nondestructive testing for individual hydraulic fractures in reservoirs.

In-plane stress analysis of multiple parallel cracks in an orthotropic FGM medium under time-harmonic loading

This study aimed to investigate the fracture behavior of a cracked orthotropic FGM medium when exposed to an in-plane harmonic load. The mechanical feature of the medium exponentially varied along the y-axis. Using the Fourier transform method, stress was analyzed by assessing Volterra edge dislocation in orthotropic FGM media. The obtained solution was then applied to derive Cauchy singular integral equations to analyze multiple cracks. The mentioned equations were numerically solved to determine dislocation density functions on the crack face which were then used to evaluate dynamic mixed stress intensity factors. The influence of crack interaction, non-homogeneity, and load frequency on the dynamic stress intensity was numerically illustrated through examples.

Transient response of collinear Griffith cracks in a functionally graded strip bonded between dissimilar elastic strips under shear impact loading

This article analyses the interaction between a central and two symmetrically placed collinear Griffith cracks subjected to transient response under anti-plane shear impact loading. The cracks are situated in a strip constituted by functionally graded material (FGM) bonded between two dissimilar elastic strips of equal thickness. The material properties of FGM are assumed to vary exponentially as a function of thickness. Applying integral transforms, the boundary value problem reduces to a system of singular integral equations in the Laplace transformed domain. These equations are solved numerically using the Lobatto-Chebyshev collocation quadrature approach. The inverse Laplace transform is used to find the approximate expressions of dynamic stress intensity factors (DSIFs). The striking feature of the article is the study of phenomenal changes of shielding and amplification through dynamic stress magnification factors (DSMFs) at the tips of the cracks under the sudden impact loading applied at the upper material surface. The effects of impact load applied at different surfaces, positions of cracks’ axis and the thickness of the strips of the composite material on the possibilities of cracks’ arrest are depicted graphically for different particular cases.

A new method for measuring the dynamic fracture toughness under blast loads using arc-edge rectangle with edge notches

Drilling and blasting method is a common method of rock fragmentation in engineering. Meanwhile, the dynamic behaviours of rocks under blasting loads are different from those under impact loads. In this paper, a large-size arc-edge rectangle with edge notches (LAREN) specimen was posed to investigate the dynamic fracture parameters of brittle rock materials under blasting load. Three types of radial pre-cracks with different lengths were used, combining with the crack propagation gauge velocity test system, and the crack initiation time and propagation time were obtained, so as to obtain the crack propagation speed which is to be modified by the fractal theory. Also, the specimen rationality was analyzed by AUTODYN code in depth. At last, the dynamic stress intensity factors were estimated by using nonlinear finite element software ABAQUS and experimental results. It is concluded that the LAREN specimens can effectively lower the influence of the reflected tensile stress wave (by 54.18%) on the dynamic crack initiation and propagation, and have good applicability in the measurement of dynamic fracture parameters under blasting. With the development of pre-crack length, the crack time becomes longer, the average propagation speed becomes smaller, and the fluctuation of speed becomes smaller and tends to be stable. The increase of the pre-crack length also leads to the gradual decrease in the test value of the crack initiation toughness, while the test value of the propagation toughness is not constant.