Rapid crack propagation and arrest in the dcb specimen—an improved numerical analysis

A two-dimensional linear elastodynamic analysis of crack initiation, fast crack propagation and crack arrest in the DCB specimen is presented. The analysis is performed using the previously developed SMF2D code in its generation mode. The experimentally measured crack tip motion, as well as the specimen's geometry and its material characteristics, serve as input to the simulation. The dynamic stress intensity factor, the dynamic energy release rate and the various distributions of energies are subsequently evaluated. The numerical results are found to be in very good agreement with analytical and experimental findings.     

Elastodynamic response of a penny-shaped crack in a cylinder of finite radius

The elastodynamic response of a penny-shaped crack in a cylinder of finite radius is investigated in this study. A step stress is applied to the crack surface resulting in transient behavior. The stress field near the crack front and the dynamic stress intensity factor are determined. Numerical resifits on the dynamic stress intensity factor are obtained to show the influence of inertia, geometry and their interactions on the load transfer to the crack.     

A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

An experimental investigation of dynamic crack propagation

The dynamic fracture behavior of polymethylmethacrylate (bdPMMA) has been investigated. The specimens were in the form of rectangular sheets with sharp notches. The elastodynamic crack tip stress field and the crack velocity were determined by the use of resistance strain gauges. An analytic expression for the dynamic crack tip stress field was used to evaluate the dynamic stress intensity factors, and the dynamic arrest toughness was also determined.The dynamic response of the stresses at the notch tip at varying loading rates was considered and some “hysteresis” fracture phenomena were observed.     

Analysis of the optical method of caustics for dynamic crack propagation

In the interpretation of experimental data on dynamic crack propagation in solids obtained by means of the optical method of caustics, it has been customary to neglect the effect of material inertia on the stress distribution in the vicinity of the crack tip. In this paper, the elastodynamic crack tip stress field is used to establish the exact equations of the caustic envelope formed by the reflection of light rays from the surface of a planar solid near the tip of a propagating crack. These equations involve the instantaneous crack tip speed, the material parameters and the instantaneous dynamic stress intensity factor, and they can be used to determine the stress intensity factor for given material parameters and crack tip speed. The influence of inertial effects on stress intensity factor measurements for system parameters typical of experiments with PMMA specimens is considered. It is found that the stress intensity factor values inferred through a dynamic analysis may differ by as much as 30–40% from values based on a quasi-static analysis.     

A FEM analysis of crack arrest experiments

Crack arrest experiments are performed on edge-notched specimens of a high strength steel. The crack velocity and the displacements at the boundaries are continuously measured during the experiments. This information is then used in a subsequent FEM analysis to evaluate the dynamic stress intensity factor before and after crack arrest. Dynamic effects are seen to influence the process a considerable time after the arrest. A discussion of the validity of the arrest condition in a linear theory is made based upon experimental results. The inverse problem is also considered, i.e. to predict the crack growth event, the arrest length and the corresponding stress intensity factors when the crack propagation toughness is known for the material. The predicted values are shown to exhibit good agreement with the experimental results.     

An elastodynamic fracture analysis in a centre-cracked plate

A two-dimensional linear elastodynamic analysis of crack initiation and fast crack propagation in a centre-cracked plate, subjected to constant tension is presented. The analysis is performed using the previously developed SMF2D code in its generation mode. The experimentally measured crack tip motion, as well as the specimen's geometry and its material characteristics serve as input to the simulation. The dynamic stress intensity factor, the dynamic energy release rate, and the various energy distributions are subsequently evaluated. Special attention is given to the influence of the energy supplied to the body during the fracture process due to the work done by the external tractions.     

Application of the weight function method to two-dimensional elastodynamic fracture mechanics

In this paper, the weight function method is used for two-dimensional mixed-mode crack analyses of clastostatic and elastodynamic problems. By the use of the Laplace transformation method and an indirect boundary element method, the dynamic stress intensity factors for a finite sheet containing a central or an edge crack are evaluated. A Green's function method is introduced which depends on the weight function for an impulsive applied load. The Green's function can be used to determine stress intensity factors for arbitrary time dependence of the boundary conditions. The stress intensity factors obtained by the weight function method are compared where possible, with existing solutions.     

Study of mode I crack dynamic propagation behaviour and rock dynamic fracture toughness by using SCT specimens

To study crack dynamic propagation behaviour and rock dynamic fracture toughness, a single cleavage triangle (SCT) specimen was proposed in this paper. By using these specimens and a drop‐weight test system, impact experiments were conducted, and the crack propagation velocity and the fracture time were measured by using crack propagation gauges. To examine the effectiveness of the SCT specimen and to predict the test results, finite difference numerical models were established by using AUTODYN code, and the simulation results showed that the crack propagation path agrees with the test results, and crack arrest phenomena could happen. Meanwhile, by using these numerical models, the crack dynamic propagation mechanism was investigated. Finite element code ABAQUS was applied in the calculation of crack dynamic stress intensity factors (SIFs) based on specimen dimension and the loading curves measured, and the curves of crack dynamic SIFs versus time were obtained. The fracture toughness (including initiation toughness and propagation toughness) was determined according to the fracture time and crack speeds measured by crack propagation gauges. The results show that the SCT specimen is applicable to the study of crack dynamic propagation behaviour and fracture toughness, and in the process of crack propagation, the propagation toughness decreases with crack propagation velocity, and the crack arrest phenomena could happen. The critical SIF of an arrest crack (or arrest toughness) was higher than the crack propagation toughness but was lower than the initiation toughness.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

Three-dimensional dynamic fracture analysis with the dual reciprocity method in Laplace domain

In this paper, the dual reciprocity boundary element method in the Laplace domain has been developed for the analysis of three-dimensional elastodynamic fracture mechanics mixed-mode problems. The boundary element method is used to calculate the unknowns of transformed boundary displacement and traction and the domain integrals in the elastodynamic equation are transformed into boundary integrals by the use of the dual reciprocity method. The transformed dynamic stress intensity factors are determined by the crack opening displacement (COD) directly in the Laplace domain. By using Durbin's inversion technique, the dynamic stress intensity factors in the time domain are obtained. Several numerical examples are presented to demonstrate the good agreement with existing solutions.     

Elastodynamic analysis of the Finite punch and finite crack problems in orthotropic materials

The transient elastodynamic response of the finite punch and finite crack problems in orthotropic materials is examined. Solution for the stress intensity factor history around the punch corner and crack tip is found. Laplace and Fourier transforms together with the Wiener–Hopf technique are employed to solve the equations of motion in terms of displacements. A detailed analysis is made in the simplified case when a flat rigid punch indents an elastic orthotropic half-plane, the punch approaches with a constant velocity normally to the boundary of the half-plane. An asymptotic expression for the singular stress near the punch corner is analyzed leading to an explicit expression for the dynamic stress intensity factor which is valid for the time the dilatational wave takes to travel twice the punch width. In the crack problem, a finite crack is considered in an infinite orthotropic plane. The crack faces are loaded by impact uniform pressure in mode I. An expression for the dynamic stress intensity factor is found which is valid while the dilatational wave travels the crack length twice. Results for orthotropic materials are shown to converge to known solutions for isotropic materials derived independently.     

Green's functions for the stress intensity factor evolution in finite cracks in orthotropic materials

The elastodynamic response of an infinite orthotropic material with finite crack under concentrated loads is examined. Solution for the stress intensity factor history around the crack tips is found. Laplace and Fourier transforms are employed to solve the equations of motion leading to a Fredholm integral equation on the Laplace transform domain. The dynamic stress intensity factor history can be computed by numerical Laplace transform inversion of the solution of the Fredholm equation. Numerical values of the dynamic stress intensity factor history for some example materials are obtained. This solution can be used as a Green's function to solve dynamic problems involving fini te cracks.     

An analytical method for mixed-mode propagation of pressurized fractures in remotely compressed rocks

An analytical method for mixed-mode (mode I and mode II) propagation of pressurized fractures in remotely compressed rocks is presented in this paper. Stress intensity factors for such fractured rocks subjected to two-dimensional stress system are formulated approximately. A sequential crack tip propagation algorithm is developed in conjunction with the maximum tensile stress criterion for crack extension. For updating stress intensity factors during crack tip propagation, a dynamic fictitious fracture plane is used. Based on the displacement correlation technique, which is usually used in boundary element/finite element analyses, for computing stress intensity factors in terms of nodal displacements, further simplification in the estimation of crack opening and sliding displacements is suggested. The proposed method is verified comparing results (stress intensity factors, propagation paths and crack opening and sliding displacements) with that obtained from a boundary element based program and available in literatures. Results are found in good agreements for all the verification examples, while the proposed method requires a trivial computing time.     

裂纹止裂技术及裂纹动态扩展规律研究

为了对动态荷载作用下水泥粉煤灰砂浆的裂缝动态扩展行为进行研究,提出了一种大尺寸带V型底边的半圆边裂纹(SECVB)试件,其V形底部具有止裂功能。SECVB试件的V形底部设计为180°,150°和120°三个角度。采用落锤冲击装置进行了冲击试验,并使用裂纹扩展计(CPG)用于测量裂纹扩展的相关参数。利用有限差分程序AUTODYN对裂纹扩展行为进行了数值模拟,并用有限元程序ABAQUS计算了裂纹的动态应力强度因子(DSIF);根据CPG测量的裂纹萌生时间和扩展时间来确定临界应力强度因子。试验和数值模拟结果表明,SECVB试件适合于研究动态荷载作用下水泥粉煤灰砂浆的裂纹扩展行为和止裂行为。在裂纹扩展过程中,裂纹可能在一段时间内止裂,并且裂纹在起始时刻的断裂韧度高于裂纹扩展时的断裂韧度。     

Axisymmetric dynamic response of a penny-shaped crack to a pair of transient concentrated forces

The three-dimensional axisymmetric elastodynamic response of a penny-shaped crack embedded in an infinite elastic solid subjected to a pair of transient concentrated forces is investigated. The forces are applied on the symmetry axis perpendicular and symmetric to the crack surfaces, including the special case when the forces act precisely on the crack surfaces. A time-domain boundary integral equation method is applied for computing the crack-opening displacement and subsequently the time dependence of the dynamic stress intensity factors. Numerical calculations are carried out for various geometry parameters and the results are discussed. It is found that the location of the applied concentrated forces inducing the highest dynamic stress intensity factors differs from that producing the highest static values.     

Elastic-plastic finite element analysis of dynamic fracture

Recent evidence has pointed to the possible inadequacy of elastodynamic treatments of rapid crack propagation and crack arrest. This paper describes the development of a dynamic elastic-plastic finite element capability designed to address this concern by taking direct account of crack tip plasticity. Comparisons with known dynamic fracture mechanics solutions and with experimental data are made to demonstrate the fidelity of the approach. A comparison with an elastodynamic solution in an impact loaded 4340 steel bend specimen is also made. This result reveals that a significant effect of crack tip plasticity may exist even for high strength materials.     

Elastodynamic stress-intensity factors for a semi-infinite crack under 3-D combined mode loading

The dynamic stress intensity factor histories for a half plane crack in an otherwise unbounded elastic body are analyzed. The crack is subjected to a traction distribution consisting of two pairs of suddenly-applied shear point loads, at a distance L away from the crack tip. The exact expression for the combined mode stress intensity factors as the function of time and position along the crack edge is obtained. The method of solution is based on the direct application of integral transforms together with the Wiener-Hopf technique and the Cagniard-de Hoop method, which were previously believed to be inappropriate. Some features of solutions are discussed and the results are displayed in several figures.     

混凝土/花岗岩界面Ⅰ-Ⅱ复合型动态断裂试验研究

为研究混凝土/岩石界面在复合型应力条件下的动态断裂性能,考虑四种应变率(10-5~10-2 s-1)及三种模态(21.8°,41.7°和60.3°)工况,对混凝土/花岗岩复合试件进行了四点剪切试验,获得了荷载与裂缝张开位移及裂缝剪切位移的关系曲线;结合界面力学理论和结构动力分析得到了界面Ⅰ型和Ⅱ型动态应力强度因子,据此得到并分析了断裂韧度、应变能释放率的率相关性及模态比相关性。结果表明:在所研究的应变率和模态角范围内,同一时刻的裂缝张开位移均大于裂缝剪切位移;Ⅰ型和Ⅱ型断裂韧度均随应变率的提高而增加,Ⅰ型断裂韧度随模态角的增大而减小,Ⅱ型断裂韧度随模态角的增大而增加;应变能释放率随应变率和模态角的增加均呈现出增长趋势。     

Path-independent integrals in fracture dynamics using auxiliary fields

“Path-independent” integrals are presented for use in fracture dynamics. One class of integrals is based on a reciprocal theorem and the other is related to the energy flow into a moving crack tip. These integrals may be used to numerically calculate the dynamic stress intensity factor in elastodynamic crack propagation problems. Several examples are presented using the release node technique to model crack propagation.