Interacting mechanism and initiation prediction of multiple cracks

The maximum Mode Ⅰ and Mode Ⅱ stress intensity factors(SIFs), KI,kmax(θ) and KII,kmax(θ)(0°<θ<360°), of inclined parallel multi-crack varying with relative positions(including horizontal and vertical spacings) are calculated by the complex function and integration method to analyze their interacting mechanism and determine the strengthening and weakening zone of SIFs. The multi-crack initiation criterion is established based on the ratio of maximum tension-shear SIF to predict crack initiation angle, load, and mechanism. The results show that multi-crack always initiates in Mode Ⅰ and the vertical spacing is better not to be times of half crack-length for crack-arrest, which is in good agreement with test results of the red-sandstone cube specimens with three parallel cracks under uniaxial compression. This can prove the validity of the multi-crack initiation criterion.

Solution of stress intensity factors of multiple cracks in plane elasticity with eigen COD formulation of boundary integral equation

The concept of eigen crack opening displacement （COD） can be defined as the COD of a crack in infinite plate under the tractions acting on the crack surface. By introducing this concept, the eigen COD formulation of boundary integral equation is proposed in this paper, together with the solution procedures for multiple crack problems in plane elasticity. With the proposed approach, the multiple crack problems can be solved with the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix as that in the numerical Green’s function （NGF） approach but without the trouble to determine the complementary solutions since the standard boundary element discretization on the crack surface is no longer required with the proposed approach. Some numerical examples computing the stress intensity factors are presented and compared with those in literature to show the accuracy and the effectiveness of the proposed approach.

Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane

The Schmidt method is adopted to investigate the fracture problem of multiple parallel symmetric and permeable finite length mode-III cracks in a functionally graded piezoelectric/piezomagnetic material plane. This problem is formulated into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. In order to obtain the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The results show that the stress, the electric displacement, and the magnetic flux intensity factors of cracks depend on the crack length, the functionally graded parameter, and the distance among the multiple parallel cracks. The crack shielding effect is also obviously presented in a functionally graded piezoelectric/piezomagnetic material plane with mul- tiple parallel symmetric mode-III cracks.

A numerical method for multiple cracks in an infinite elastic plate

This article examines the interaction of multiple cracks in an infinite plate by using a numerical method. The numerical method consists of the non-singular displacement discontinuity element presented by Crouch and Startled and the crack tip displacement discontinuity elements proposed by the author. In the numerical method implementation, the left or the right crack tip element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The numerical method is called a hybrid displacement discontinuity method. The following test examples of crack problems in an infinite plate under tension are included： “ center-inclined cracked plate”, “interaction of two collinear cracks with equal length”, “interaction of three collinear cracks with equal length”, “interaction of two parallel cracks with equal length”, and “interaction of one horizontal crack and one inclined crack”. The present numerical results show that the numerical method is simple yet very accurate for analyzing the interaction of multiple cracks in an infinite plate.

Peridynamic Study on Fracture Mode and Crack Propagation Path of a Plate with Multiple Cracks Subjected to Uniaxial Tension

How to simulate fracture mode and crack propagation path in a plate with multiple cracks is an attractive but difficult issue in fracture mechanics.Peridynamics is a recently developed nonlocal continuum formulation that can spontaneously predict the crack nucleation,branch and propagation in materials and structures through a meshfree discrete technique.In this paper,the peridynamic motion equation with boundary traction is improved by simplifying the boundary transfer functions.We calculate the critical cracking load and the fracture angles of the plate with multiple cracks under uniaxial tension.The results are consistent with those predicted by classical fracture mechanics.The fracture mode and crack propagation path are also determined.The calculation shows that the brittle fracture process of the plate with multiple cracks can be conveniently and correctly simulated by the peridynamic motion equation with boundary conditions.

Semi-analytical solution for internal forces of tunnel lining with multiple longitudinal cracks

Longitudinal cracks on the tunnel lining significantly influence the performance of tunnels in operation.In this study,we propose a semi-analytical method that provides a simple and effective way to calculate the internal forces of tunnel linings with multiple cracks.The semi-analytical solution is obtained using structural analysis considering the flexural rigidity for the cracked longitudinal section of the tunnel lining.Then the proposed solution is verified numerically.Using the proposed method,the influences of the crack depth and the number of cracks on the bending moment and modified crack tip stress are investigated.With the increase in crack depth,the bending moment of lining scetion adjacent to the crack decreases,while the bending moment of lining scetion far away from the crack increases slightly.The more the number of cracks in a tunnel lining,the easier the new cracks initiated.

Estimation of structure crack propagation based on multiple factors correction

By deriving the stress concentration factor of theestimation approach for residual fatigue life’ an estimationapproach for structure crack propagation based on multiplefactors correction is proposed. Then’ the quantitativeexpression among the structure factor’ stress ratio’ loadingtype’ the manufacture processing factor and the crackpropagation is achieved. The proposed approach iimplemented in a case study for an instance structure’ and theinfluences of correction factors on the crack propagation areanalyzed. Meanwhile’ the probabilistic method based onWeibull distribution probability density function is selected toevaluate the precision of the corrected estimation approach’and the probability density of results is calculated by theprobability density function. It is shown that the resultsestimated by the corrected approach is more precise than thoseestimated by the fracture mechanics, and they are closer to thetest data.

A linearly-independent higher-order extended numerical manifold method and its application to multiple crack growth simulation

The numerical manifold method(NMM)can be viewed as an inherent continuous-discontinuous numerical method,which is based on two cover systems including mathematical and physical covers.Higher-order NMM that adopts higher-order polynomials as its local approximations generally shows higher precision than zero-order NMM whose local approximations are constants.Therefore,higherorder NMM will be an excellent choice for crack propagation problem which requires higher stress accuracy.In addition,it is crucial to improve the stress accuracy around the crack tip for determining the direction of crack growth according to the maximum circumferential stress criterion in fracture mechanics.Thus,some other enriched local approximations are introduced to model the stress singularity at the crack tip.Generally,higher-order NMM,especially first-order NMM wherein local approximations are first-order polynomials,has the linear dependence problems as other partition of unit(PUM)based numerical methods does.To overcome this problem,an extended NMM is developed based on a new local approximation derived from the triangular plate element in the finite element method(FEM),which has no linear dependence issue.Meanwhile,the stresses at the nodes of mathematical mesh(the nodal stresses in FEM)are continuous and the degrees of freedom defined on the physical patches are physically meaningful.Next,the extended NMM is employed to solve multiple crack propagation problems.It shows that the fracture mechanics requirement and mechanical equilibrium can be satisfied by the trial-and-error method and the adjustment of the load multiplier in the process of crack propagation.Four numerical examples are illustrated to verify the feasibility of the proposed extended NMM.The numerical examples indicate that the crack growths simulated by the extended NMM are in good accordance with the reference solutions.Thus the effectiveness and correctness of the developed NMM have been validated.

Three-Steps-Meshing Based Multiple Crack Identification for Structures and Its Experimental Studies

Multiple crack identification plays an important role in vibration-based crack identification of structures. Traditional crack detection method of single crack is difficult to be used in multiple crack diagnosis. A three-step-meshing method for the multiple cracks identification in structures is presented. Firstly, the changes in natural frequency of a structure with various crack locations and depth are accurately obtained by means of wavelet finite element method, and then the damage coefficient method is used to determine the number and the region of cracks. Secondly, different regions in the cracked structure are divided into meshes with different scales, and then the small unit containing cracks in the damaged area is gradually located by iterative computation. Lastly, by finding the points of intersection of three frequency contour lines in the small unit, the crack location and depth are identified. In order to verify the effectiveness of the presented method, a multiple cracks identification experiment is carried out. The diagnostic tests on a cantilever beam under two working conditions show the accuracy of the proposed method： with a maximum error of crack location identification 2.7% and of depth identification 5.2%. The method is able to detect multiple crack of beam with less subdivision and higher precision, and can be developed as a multiple crack detection approach for complicated structures.

Equivalent elastic compliance tensor for rock mass with multiple persistent joint sets:Exact derivation via modified crack tensor

Discontinuities constitute an integral part of rock mass and inherently affect its anisotropic deformation behavior.This work focuses on the equivalent elastic deformation of rock mass with multiple persistent joint sets.A new method based on the space geometric and mechanical properties of the modified crack tensor is proposed,providing an analytical solution for the equivalent elastic compliance tensor of rock mass.A series of experiments validate the capability of the compliance tensor to accurately represent the deformation of rock mass with multiple persistent joint sets,based on conditions set by the basic hypothesis.The spatially varying rules of the equivalent elastic parameters of rock mass with a single joint set are analyzed to reveal the universal law of the stratified rock mass.

A generalized cover renewal strategy for multiple crack propagation in two-dimensional numerical manifold method

Partition of unity based numerical manifold method can solve continuous and discontinuous problems in a unified framework with a two-cover system,i.e.,the mathematical cover and physical cover.However,renewal of the topology of the two-cover system poses a challenge for multiple crack propagation problems and there are few references.In this study,a robust and efficient strategy is proposed to update the cover system of the numerical manifold method in simulation of multiple crack propagation problems.The proposed algorithm updates the cover system with a bottom-up process:1)identification of fractured manifold elements according to the previous and latest crack tip position;and 2)local topological update of the manifold elements,physical patches,block boundary loops,and non-persistent joint loops according to the scenario classification of the propagating crack.The proposed crack tracking strategy and classification of the renewal cases promote a robust and efficient cover renewal algorithm for multiple crack propagation analysis.Three crack propagation examples show that the proposed algorithm performs well in updating the cover system.This cover renewal methodology can be extended for numerical manifold method with polygonal mathematical covers.

Numerical Comparison Research on the Solution of Stress Intensity Factors of Multiple Crack Problems

A newly developed approach without crack surface discretization for modeling 2D solids with large number of cracks in linear elastic fracture mechanics is proposed with the eigen crack opening displacement (COD) boundary integral equations in this paper. The eigen COD is defined as a crack in an infinite domain under fictitious traction acting on the crack surface. Respect to the computational accuracies and efficiencies, the multiple crack problems in finite and infinite plates are solved and compared numerically using three different kinds of boundary integral equations (BIEs): 1) the dual BIEs require crack surface discretization;2) the BIEs with numerical Green’s functions (NGF) without crack surface discretization, but have to solve a complementary matrix;3) the eigen crack opening displacement (COD) BIEs in the present paper. With the concept of eigen COD, the multiple crack problems can be solved by using a conventional displacement discontinuity boundary integral equation in an iterative fashion with a small size of system matrix as that in the NGF approach, but without troubles to determine the complementary matrix. Solution of the stress intensity factors of multiple crack problems is solved and compared in some numerical examples using the above three computational algorithms. Numerical results clearly demonstrate the numerical models of eigen COD BIEs have much higher efficiency, providing a newly numerical technique for multiple crack problems. Not only the accuracy and efficiency of computation can be guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks.

Fatigue crack growth investigation on offshore pipelines with three-dimensional interacting cracks

Due to high cost of full-scale experimental setup, this study presents a numerical model on fatigue behaviours of offshore pipeline with multiple coplanar cracks under cyclic tensile loadings. The validation on numerical results is made by other researchers' experimental results, and significant parameters affecting fatigue crack growth are studied.

A continuous-discontinuous cellular automaton method for cracks growth and coalescence in brittle material

A method of continuous-discontinuous cellular automaton for modeling the growth and coalescence of multiple cracks in brittle material is presented. The method uses the level set to track arbitrary discontinuities, and calculation grids are independent of the discontinuities and no remeshing are required with the crack growing. Based on Grif- fith fracture theory and Mohr-Coulumb criterion, a mixed fracture criterion for multiple cracks growth in brittle mate- rial is proposed. The method treats the junction and coales- cence of multiple cracks, and junction criterion and coales- cence criterion for brittle material are given, too. Besides, in order to overcome the tracking error in the level set ap- proximation for crack junction and coalescence, a dichotomy searching algorithm is proposed. Introduced the above the- ories into continuous-discontinuous cellular automaton, the present method can be applied to solving multiple crack growth in brittle material, and only cell stiffness is needed and no assembled global stiffness is needed. Some numerical examples are given to shown that the present method is efficient and accurate for crack junction, coalescence and percolation problems.

Data Fusion of Multiple Curvature Mode Shapes for Structural Damage Diagnosis

This study aims to develop a reliable method for locating multiple cracks in a beam using data fusion of multiple curvature mode shapes.The identification of multiple damage in beams based on curvature mode shape has become a research focus in recent decades.However,the curvature mode shape method using a single mode carries limited damage information,and fails to reveal all cracks when multiple damage detection is required.To address this limitation,the curvature mode shape method is further enhanced by a data fusion algorithm to fuse damage information in multiple modes.The effectiveness of the method is proved through a numerical case of a beam with multiple cracks.The applicability of the method is experimentally validated by detecting multiple cracks in a composite laminated beam with mode shapes acquired by a scanning laser vibrometer.The results show that the method of fusing multiple curvature mode shapes carries more complete damage information than the traditional one using a single mode,and can successfully locate all cracks in a damaged beam.

Compensation of stress intensity factors in hollow cylinders containing several cracks under torsion by electro-elastic coating

In this article, a formulation for a hollow cylinder reinforced with an electroelastic layer is investigated. The hollow cylinder and its electro-elastic coating are under the Saint-Venant torsional loading. First, the solution to the problem containing a Volterra-type screw dislocation is obtained by using the Fourier transform. The problem is then reduced to a set of Cauchy singular integral equations by the distributed dislocation method. Finally, several examples are presented to show the effect of the electro-elastic coating on the reduction of the stress intensity factors at the crack tips.

Numerical solutions for cracks in an elastic half-plane

The behavior of the stress intensity factor at the tips of cracks subjected to uniaxial tension σχ^∞= p with traction-free boundary condition in half-plane elasticity is investigated. The problem is formulated into singular integral equations with the distribution dislocation function as unknown. In the formulation, we make used of a modified complex potential. Based on the appropriate quadrature formulas together with a suitable choice of collocation points, the singular integral equations are reduced to a system of linear equations for the unknown coefficients. Numerical examples show that the values of the stress intensity factor are influenced by the distance from the cracks to the boundary of the half-plane and the configuration of the cracks.

Fracture behaviors of AZ80 magnesium alloy during multiple forging processes

The initiation, propagation and the accompanied dislocation structures of the cracks in AZ80 magnesium alloy during multiple forging processes were investigated. The results show that the cracks firstly initiate at the Mg/Mg17Al12 interface under the hoop tensile stress on equatorial free surface. On further deformation, the cracks in the Mg17Al12 particles tend to propagate along the grain boundaries(GBs) in a zigzag pattern and link with adjacent cracks in other Mg17Al12 particles to form one whole crack, leading to the fracture surface. Low deformation temperature and too many forging passes during the deformation will promote the nucleation of interfacial microcrack inside the specimens due to the strong plastic strain incompatibility and the high internal stresses near the GBs. The loading axis rotating during the process can change the stress field at the tip of cracks, leading to the change of the crack propagating path and assisting in inhabiting microcracking development.

Research on the Plane Multiple Cracks Stress Intensity Factors Based on Stochastic Finite Element Method

Taylor stochastic finite element method (SFEM) is applied to analyze the uncertainty of plane multiple cracks stress intensity factors (SIFs) considering the uncertainties of material properties, crack length, and load. The stochastic finite element model of plane multiple cracks are presented. In this model, crack tips are meshed with six-node triangular quarter-point elements; and other area is meshed with six-node triangular elements. The partial derivatives of displacement and stiffness matrix with resp...