Time-dependent Stress Intensity Factor Research Articles

Transient in-plane analysis of cracked general orthotropic planes

An edge dislocation with a time-dependent Burgers vector in an orthotropic plane is analyzed. The stress field caused by a dynamic point force in the intact plane is determined. The displacement discontinuity of dislocation and direction of point load are at arbitrary angles with the Material Principal Axes (MPA), the general orthotropic plane. The density of dislocations is employed to construct integral equations for orthotropic planes weakened by several interacting cracks, making arbitrary angles with the MPA. The numerical solution of integral equations results in the density of dislocations on a crack surface. The results are employed to determine time-dependent Stress Intensity Factors (SIFs) at the tips of a crack and Crack Opening Displacement (COD). In the problem of general orthotropic planes containing cracks subjected to dynamic loads, crack orientations may cause their partial/total closing. Under the hypothesis of frictionless contact, we devise an iterative procedure to handle crack closing, as well. By contrast to cracks in the directions of MPA, expressions for the two modes of SIFs contain the density functions of both the climb and glide of the edge dislocation. Therefore, mode I SIF at an open tip of a crack may not be positive. Conversely, positive mode I SIF may occur at a closed crack tip. Numerical results of the SIFs and COD for several examples of orthotropic planes under dynamic point loads, while weakened by single and double parallel cracks, are presented. Moreover, in double cracks cases, the interaction between cracks is studied.

A Precise Time-Domain Expanding Crack-Tip Element for 3D Planar and Axial Dynamic Cracks

Unlike two-dimensional (2D) crack problems of which the analytical symplectic eigensolutions are available, the analytical symplectic solution of three-dimensional (3D) planar and axial dynamic cracks has not been derived yet. We thus propose a trial displacement field for the 3D planar and axial crack problem using the existing solutions of the 2D crack problems and construct a new crack-tip element on that basis. The recently developed precise integration scheme for 3D planar and axial cracks under dynamic loading (Hu, X. F., Chen, W. H., Zhang, P. and Yao, W. A. [2021] “An explicit crack-tip element for stationary dynamic cracks,” Theor. Appl. Fract. Mech. 112, 102886) is further enhanced. A precise time-domain expanding algorithm is employed for the temporal discretization. All the field variables are expanded using a series expansion scheme, and the dynamic problem is then transformed into a series of quasi-static elastic problems. All the sub-elastic problems are related in a recursive manner. Taking advantage of a smart automatic truncating approach, the number of expanding terms can be determined automatically subjected to a prescribed error tolerance. A few numerical examples are investigated to demonstrate the developed method, numerical solutions indicate that the time dependent stress intensity factors (SIFs) can be calculated in a simple way and the present method has higher efficiency than the recently developed time integration scheme while having comparable accuracy.

Mechanics of a pressurized penny-shaped crack in a poroelastic halfspace

This paper examines the axisymmetric problem of a penny-shaped crack located in a poroelastic halfspace that is modelled by Biot poroelasticity. The paper first examines the axisymmetric problem of the mechanics of fluid injection over a circular area within a halfspace that would create the skeletal stress state necessary to initiate fracture. The triggering of the fracture is assumed to create an axisymmetric penny-shaped crack whose surfaces will be subjected to fluid pressure. The mechanics of the penny-shaped crack in terms of its potential to extend in an axisymmetric fashion is examined by formulating the poroelastic mixed boundary value problem and solving the coupled Fredholm integral equations of the second-kind obtained through successive applications of integral transform techniques. The analysis of the poroelasticity problem of fluid-injection into the crack gives rise to time-dependent skeletal stress intensity factors (SIFs) at the crack tip; these are combined with a mixed-mode brittle skeletal fracture criterion to establish the injection pressures that can lead to the extension of the penny-shaped crack in a self-similar fashion.

Mixed-mode dynamic stress intensity factors by variation technique with finite block method

The variational technique has been extended to calculate the dynamic stress intensity factors with the meshless finite block method in this paper. The Lagrange series are utilized to formulate the meshless approach in the normalized domain with coordinate mapping technique for practical problems. The stress intensity factors are evaluated by a variation of crack length for the mixed-mode fracture problems. Based on the static reference, the stress intensity factors in the Laplace space are determined with the boundary integral and domain integral. The Durbin inversion method is utilized to determine the time-dependent stress intensity factors. The applications of the proposed technique to two-dimensional fracture mechanics are demonstrated with three examples. Comparisons are made with the boundary element method and the meshless method in order to demonstrate the efficiency and accuracy of the proposed method.

The foundations of fracture impedance spectroscopy

The foundations of Fracture Impedance Spectroscopy (FIS) as a theoretical framework for analyzing and predicting fatigue and corrosion fatigue crack growth rate in real structures under complex loading waveforms from laboratory data obtained under simple, “pristine” loading conditions is presented. FIS is based upon an analogy between current flow (crack growth rate) in an electrical circuit in response to an applied, time-dependent voltage (time-dependent stress intensity factor). In this paper, the framework of the theory and its application for two different aerospace alloys are presented.

Study of Creep–Fatigue Crack Growth Behavior in a Gas Turbine Casing

A creep–fatigue crack growth on the outer surfaces of a turbine casing was assessed, and the turbine casing’s overall lifetime was predicted. The crack location, size, and direction were determined using nondestructive tests. Temperature distribution, measured using thermography method, was applied as boundary conditions in FEM simulation. ABAQUS software was utilized for calculating the stress distribution of the casing in accordance with the real cycles of the turbine. Both the creep and the fatigue crack growths were predicted using the ZENCRACK code. The Paris equation was applied in order to estimate the fatigue life, and the time-dependent Paris equation was used to predict the creep life. It was shown that after a certain amount of time has passed, the crack stress intensity factor (K) and time-dependent stress intensity factor (Ct) decrease and stop in reaching the fatigue threshold stress intensity factor (Kth) and creep fracture mechanics parameter (C t * ) values, respectively.

An analytical model for the fracture behavior of the flexible lithium-ion batteries under bending deformation

To understand the influence of the bending deformation on the stress evolution and crack propagation in nano flexible electrode during electrochemical cycling, an analytical model is developed based on core-shell structure in a cylindrical electrode. In the model, the analytical solution of stress specialized for the cylindrical electrode in the process of bending deformation and phase transformation is clarified. Further, the weight function is utilized to calculate the time-dependent stress intensity factor by combining the stress profiles as found in the analytical work. Thus, the behavior of the preexisting center and edge cracks in the electrode is discussed to investigate the effect of the initiation position on crack propagation. It is found that cracks tend to spread more easily in the early stages of discharging process due to the larger slope of the SIF curve within small edge cracks. Further, an analytical fracture mechanics study is presented and the formula of critical size for the flexible electrode is derived based on Griffith criterion, below which the crack does not spread under the superposition of the diffusion stress and the bending stress. In the light of the fracture mechanics study of the flexible electrode, the present work sheds some light on stress engineering and structural design of durable flexible lithium-ion batteries.

Integral equation analysis for cracked strip of orthotropic functionally graded material

In this paper, the integral equations are formulated and solved numerically for a cracked orthotropic strip with Functionally Graded Materials (FGMs) subjected to both static and dynamic loads. The Green's function of displacement discontinuity in an orthotropic FGM strip is derived by using the Fourier transform and the Laplace transform techniques with the shear modulus and the mass density varying exponentially, and constant Poisson's ratio for functionally graded materials. It has been shown that the transformed Green's solutions have the same order of singularities as that in the static case. The normalized time-dependent stress intensity factors are obtained with the Durbin’s inversion method for the Laplace transform. The static stress intensity factors are compared with the results given by finite element analysis and excellent agreements are achieved.

A fracture mechanics study of the phase separating planar electrodes: Phase field modeling and analytical results

Abstract It is well known that phase separation could severely intensify mechanical degradation and expedite capacity fading in lithium-ion battery electrodes during electrochemical cycling. Experiments have frequently revealed that such degradation effects could be substantially mitigated via reducing the electrode feature size to the nanoscale. The purpose of this work is to present a fracture mechanics study of the phase separating planar electrodes. To this end, a phase field model is utilized to predict how phase separation affects evolution of the solute distribution and stress profile in a planar electrode. Behavior of the preexisting flaws in the electrode in response to the diffusion induced stresses is then examined via computing the time dependent stress intensity factor arising at the tip of flaws during both the insertion and extraction half-cycles. Further, adopting a sharp-interphase approximation of the system, a critical electrode thickness is derived below which the phase separating electrode becomes flaw tolerant. Numerical results of the phase field model are also compared against analytical predictions of the sharp-interphase model. The results are further discussed with reference to the available experiments in the literature. Finally, some of the limitations of the model are cautioned.

Anti-plane interfacial crack with functionally graded coating: static and dynamic

The anti-plane displacement discontinuity method is applied to establish the Fredholm integral equation of the first kind for the orthotropic Functionally Graded Material (FGM) coatings subjected to static/dynamic shears. The shear modulus and mass density are assumed to vary exponentially through the thickness. The static and dynamic fundamental solutions with anti-plane displacement discontinuity are derived for orthotropic FGM coating by using Fourier transform method and Laplace transform method. It has been shown that the transformed fundamental solution with orthotropic coatings has the same order of hyper-singularity as in the static case, i.e. O(1/r2), and the Chebyshev polynomials of the second kind are used to solve the integral equations numerically. The time dependent stress intensity factors are obtained directly from the coefficients of the Chebyshev polynomials with the aid of Durbin’s Laplace transform inversion method. A comparative study of FGM versus homogeneous coating is conducted, and the dependence of the stress intensity factors in the coating/substrate system on the material property (orthotropic) and thickness of coating is examined. Two examples including the static/dynamic loads are given as benchmarks for the numerical methods and application in composite engineering.

Transient electromechanical cracking of finite piezoelectric bodies

This article studies the problem of a finite crack in a finite piezoelectric medium subjected to transient electric and mechanical loads. Unlike past studies that mainly focused on the infinitesimal cracks, the current article considers the effect of the medium boundaries on the transient fracture behavior of piezoelectric materials. A system of singular integral equations is formulated in terms of the crack surface displacement and electric potential. Effects of crack length and medium size on the crack-tip field intensity factors are investigated in detail. Numerical results for the time-dependent stress and electric displacement intensity factors are obtained. Electrically impermeable crack assumption, electrically permeable crack assumption, and notch of finite height are investigated. Some new conclusions are drawn.

Some basic problems in anisotropic bimaterials with a viscoelastic interface

This research is devoted to the study of anisotropic bimaterials with Kelvin-type viscoelastic interface under antiplane deformations. First we derive the Green’s function for a bimaterial with a Kelvin-type viscoelastic interface subjected to an antiplane force and a screw dislocation by means of the complex variable method. Explicit expressions are derived for the time-dependent stress field induced by the antiplane force and screw dislocation. Also presented is the time-dependent image force acting on the screw dislocation due to its interaction with the Kelvin-type viscoelastic interface. Second we investigate a rectangular inclusion with uniform antiplane eigenstrains embedded in one of the two bonded anisotropic half-planes by virtue of the derived Green’s function for a line force. The explicit expressions for the time-dependent stress field induced by the rectangular inclusion are obtained in terms of the simple logarithmic and exponential integral functions. It is observed that in general the stresses exhibit the logarithmic singularity at the four corners of the rectangular inclusion. Our results also show that when one side of the rectangular inclusion lies on the viscoelastic interface, the interfacial tractions are still regular at the two corners of the inclusion which are located on the interface. Last we address a finite Griffith crack normal to the viscoelastic interface by means of the obtained Green’s function for a screw dislocation. The crack problem is formulated in terms of a resulting singular integral equation which is solved numerically. The time-dependent stress intensity factors at the two crack tips are obtained and some interesting features are discussed.

On a semi-infinite crack penetrating a piezoelectric circular inhomogeneity with a viscous interface

We investigate a semi-infinite crack penetrating a piezoelectric circular inhomogeneity bonded to an infinite piezoelectric matrix through a linear viscous interface. The tip of the crack is at the center of the circular inhomogeneity. By means of the complex variable and conformal mapping methods, exact closed-form solutions in terms of elementary functions are derived for the following three loading cases: (i) nominal Mode-III stress and electric displacement intensity factors at infinity; (ii) a piezoelectric screw dislocation located in the unbounded matrix; and (iii) a piezoelectric screw dislocation located in the inhomogeneity. The time-dependent electroelastic field in the cracked composite system is obtained. Particularly the time-dependent stress and electric displacement intensity factors at the crack tip, jumps in the displacement and electric potential across the crack surfaces, displacement jump across the viscous interface, and image force acting on the piezoelectric screw dislocation are all derived. It is found that the value of the relaxation (or characteristic) time for this cracked composite system is just twice as that for the same fibrous composite system without crack. Finally, we extend the methods to the more general scenario where a semi-infinite wedge crack is within the inhomogeneity/matrix composite system with a viscous interface.

Multiple surface cracks in a piezoelectric layer bonded to an elastic substrate under transient electromechanical loads

Piezoelectric materials are usually bonded to elastic substrates to form so-called smart material structures that are capable of sensing and actuating in a controlled manner in response to a stimulus. Such smart structures are often operated at transient electromechanical environments. In this paper, the problem of a periodic array of surface cracks in a piezoelectric layer bonded to an elastic substrate subjected to transient electromechanical loads is studied. A system of singular integral equations is formulated in terms of the crack surface displacement and electric potential. Numerical results include the time-dependent stress and electric displacement intensity factors. Effects of crack spacing and electromechanical coupling on the crack tip field intensity factors are investigated in details. Numerical examples are given for a single piezoelectric layer and a piezoelectric layer bonded to an elastic substrate, subjected to independent mechanical and electrical impact loads. Some useful conclusions are given.

Transient response of a piezoelectric layer with a penny-shaped crack under electromechanical impacts

In this paper, the dynamic response of a piezoelectric layer with a penny-shaped crack is investigated. The piezoelectric layer is subjected to an axisymmetrical action of both mechanical and electrical impacts. Two kinds of crack surface conditions, i.e., electrically impermeable and electrically permeable, are adopted. Based upon integral transform technique, the crack boundary value problem is reduced to a system of Fredholm integral equations in the Laplace transform domain. By making use of numerical Laplace inversion the time-dependent dynamic stress and electric displacement intensity factors are obtained, and the dynamic energy release rate is further derived. Numerical results are plotted to show the effects of both the piezoelectric layer thickness and the electrical impact loadings on the dynamic fracture behaviors of the crack tips.

Time-Dependent Effect of Viscoelastic Substrate on a Cracked Body under Inplane Load

Theoretical and experimental methods dealing with the effect of a viscoelastic substrate on a cracked body under inplane load are presented in this article. A generalized trimaterial solution is solved as a convergent series in terms of complex potentials via the successive iterations of the alternating technique in order to satisfy the continuity condition along the interfaces between dissimilar media. This trimaterial solution is then applied to the problem of a finite thickness layer bonded to a half-plane substrate. Using a standard solid model to formulate the viscoelastic constitutive equation, the real time stress intensity factors can be directly obtained from the Laplace domain. In the experiment, an aluminum cracked plate bonded to a polymer substrate is tested in a tension machine, where the displacement of the specimen is recorded by a high precision digital camera. The time-dependent stress intensity factor is determined by an inverse calculation through the crack opening displacement. The comparisons between the theoretical and experimental results are discussed in the final.

Branching instabilities in rapid fracture: Dynamics and geometry

We propose a theoretical model for branching instabilities in 2-dimensional fracture, offering predictions for when crack branching occurs, how multiple cracks develop, and what is the geometry of multiple branches. The model is based on equations of motion for crack tips which depend only on the time dependent stress intensity factors. The latter are obtained by invoking an approximate relation between static and dynamic stress intensity factors, together with an essentially exact calculation of the static ones. The results of this model are in qualitative agreement with a number of experiments in the literature.

Interfacial steady‐state and transient thermal fracture of dissimilar media using the boundary element contact analysis

AbstractA boundary element procedure is formulated to treat frictional contact problems of thermally stressed structures. The aim is the computation of steady‐state and time‐dependent thermal stress intensity factors of interfacial cracks in two‐dimensional bimaterial structures when crack closure conditions are present. The effect of friction between the crack faces is taken into consideration. The problems considered are formulated in an incremental and iterative fashion because of their non‐linear nature. Near crack tip singularities of temperature and displacement fields are modelled through appropriate quarter‐point singular elements. Fracture parameters are evaluated from nodal displacements of singular elements utilizing proper formulas. Numerical results are compared with available solutions from the literature, where possible. Good agreement between them can be found. Copyright © 2005 John Wiley & Sons, Ltd.

The time-dependent interaction between inclusions and a cracked matrix subjected to antiplane shear

The time-dependent interaction between multiple circular inclusions and a cracked matrix in the antiplane viscoelastic problem is discussed in this paper. The fundamental elastic solution is obtained as a rapidly convergent series in terms of complex potentials via successive iterations of Mobius transformation in order to satisfy continuity conditions on multiple interfaces. Based on the correspondence principle, the Laplace transformed viscoelastic solution is then directly determined from the corresponding elastic one. In association with the singular integral technique, the time-dependent mode-III stress intensity factor of the crack tip can be solved numerically in a straightforward manner. Finally, some typical examples of an arbitrary crack lying in a matrix with various material properties under various loading types are also discussed. The results show that, depending on the relative locations and material properties of inclusions, the evolution of the stress intensity factor (SIF) may increase or decrease with time.

Does the continuum theory of dynamic fracture work?

We investigate the validity of the linear elastic fracture mechanics approach to dynamic fracture. We first test the predictions in a lattice simulation, using a formula of Eshelby for the time-dependent stress intensity factor. Excellent agreement with the theory is found. We then use the same method to analyze the experiment of Sharon and Fineberg. The data here are not consistent with the theoretical expectation.