Multiple crack fatigue growth modeling by displacement discontinuity method with crack-tip elements

This paper presents a numerical approach for modeling multiple crack fatigue growth in a plane elastic infinite plate. It involves a generation of Bueckner’s principle, a displacement discontinuity method with crack-tip elements (a boundary element method) proposed recently by the author and an extension of Paris’ law to a multiple crack problem under mixed-mode loading. Because of an intrinsic feature of the boundary element method, a general multiple crack growth problem can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is conveniently modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. Fatigue growth modeling of an inclined crack in an infinite plate under biaxial cyclic loads is taken into account to illustrate the effectiveness of the present numerical approach. As an example, the present numerical approach is used to study the fatigue growth of three parallel cracks with same length under uniaxial cyclic load. Many numerical results are given.     

Static and extending interface cracks with contact zones in anisotropic as well as in piezoelectric bimaterials

An interface crack with an electrically permeable and mechanically frictionless contact zone in a piezoelectric bimaterial under the action of a remote mixed mode mechanical loading as well as thermal and electrical fields is considered in the first part of this paper. By use of the matrix‐vector representations of thermal, mechanical and electrical fields via sectionally‐holomorphic functions the problems of linear relationships are formulated and solved exactly both for an electrically permeable and an electrically impermeable interface crack. For these cases the transcendental equations and clear analytical formulas are derived for the determination of the contact zone lengths and the associated fracture mechanical parameters. A plane strain problem for a crack with a frictionless contact zone at the leading crack tip extending stationary along an interface of two semi‐infinite anisotropic spaces with a subsonic speed under the action of various loading is considered in the second part of this paper. By introducing of a moving coordinate system connected with the crack tip and by using the formal similarity of static and propagating crack problems the combined Dirichlet‐Riemann boundary value problem is formulated and solved exactly for this case as well and a transcendental equation is obtained for the determination of the real contact zone length. It is found that the increase of the crack speed leads to an increase of the real contact zone length and the correspondent stress intensity factors which increase significantly for a quasi‐Rayleigh wave speed.     

理想弹塑性Ⅲ型扩展裂纹的全新和精确分析

本文采用线场分析方法对理想弹塑性Ⅲ型准静态扩展裂纹进行了分析.本文的意义在于突破了小范围屈服理论的限制.通过求得裂纹线附近塑性区应力和位移率的通解,并将此通解(而不是过去一直采用的特解)与弹性场的精确解(而不是线弹性奇异K场)在裂纹线附近的弹塑性边界上匹配,本文得出了裂纹线附近塑性区的应力变形场、塑性区的长度及弹塑性边界的单位法向量的全新和精确解答.本文的分析放弃了小范屈服理论的所有近似假定并且不再附加任何其它的近似假定,本文的结果在裂纹线附近是足够精确的.本文的结果表明:对理想弹塑性Ⅲ型准静态扩展裂纹,不存在“定常扩展状态”,且裂纹线附近塑性应变不存在奇异性.本文还对裂纹稳定扩展过程讨论了两种重要情形.     

Interaction between three moving Griffith cracks at the interface of two dissimilar elastic media

The paper deals with the interaction between three Griffith cracks propagating under antiplane shear stress at the interface of two dissimilar infinite elastic half-spaces. The Fourier transform technique is used to reduce the elastodynamic problem to the solution of a set of integral equations which has been solved by using the finite Hilbert transform technique and Cooke’s result. The analytical expressions for the stress intensity factors at the crack tips are obtained. Numerical values of the interaction effect have been computed for and results show that interaction effects are either shielding or amplification depending on the location of each crack with respect to other and crack tip spacing.     

理想塑性固体中逐步扩展裂纹的弹塑性场

本文从三维的塑性流动理论出发,导出了关于理想塑性固体平面应变问题的基本方程。利用这些方程,分析了不可压缩理想塑性固体的逐步扩展裂纹顶端的弹塑性场。得到了关于应力和速度的一阶渐近场。分析了弹性卸载区的演变过程和中心扇形区的发展过程。预示了出现二次塑性区的可能性。最后给出了关于应力场二阶渐近分析。     

Application of configurational forces in the context of a kinked crack

The application of the configurational force approach in crack problems is often used in order to establish fracture criteria that are adapted to a specific material behaviour. The tangential component of the calculated vectorial quantity that acts at the crack tip is a generalisation of the conventional J-integral and can be interpreted as the energy release rate when the crack extends in this direction. However, the interpretation of nontangential components in the same way, and hence the interpretation of this vectorial quantity as the crack driving force, is not consistent with established kink criteria in the special case of linear elastic fracture mechanics. As a classical example, an in-plane loaded crack in a homogeneous isotropic linear elastic material is considered under the small strain assumption. Using the expansion of stress intensity factors at the extended crack tip, nontangential components of the configurational force can be interpreted as sensitivities to crack deflection. This perspective has the potential of generalisation which can be applied to more complex situations in order to study the interplay between mechanical fields in the vicinity of the crack tip and the microstructural influence within the process zone. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一维正方准晶中半无限裂纹问题的解析解

运用广义复变函数方法,通过构造适当的广义保角映射,研究了含有沿准周期方向穿透的半无限裂纹的一维正方准晶的反平面弹性问题,给出了在部分裂纹面上受均匀面外剪切时应力场和裂纹尖端应力强度因子的解析解.将此方法进一步推广到半无限裂纹垂直于一维正方准晶的准周期方向穿透的情形中,得到了相应的平面弹性问题的解析解.当准晶体的对称性增加时,还可以得出一维四方准晶相应问题的解析解.     

Perturbation method for solving the nonlinear eigenvalue problem arising from fatigue crack growth problem in a damaged medium

An analytical solution of the nonlinear eigenvalue problem arising from the fatigue crack growth problem in a damaged medium in coupled formulation is obtained. The perturbation technique for solving the nonlinear eigenvalue problem is used. The method allows to find the analytical formula expressing the eigenvalue as the function of parameters of the damage evolution law. It is shown that the eigenvalues of the nonlinear eigenvalue problem are fully determined by the exponents of the damage evolution law. In the paper the third-order (four-term) asymptotic expansions of the angular functions determining the stress and continuity fields in the neighborhood of the crack tip are given. The asymptotic expansions of the angular functions permit to find the closed-form solution for the problem considered.     

An extended thermal-medium crack model

It is important to investigate the effects of heat conduction of crack interior on thermoelastic fields of a cracked material. In this paper, an extended thermal-medium crack model is proposed to address the influences of the thermal conductivity inside an opening crack on the induced thermoelastic fields. Then the problem of a penny-shaped crack in a transversely isotropic material is investigated under applied mechanical and uniform heat flow loadings. Based on the Hankel transform technique, the governing partial differential equations are transformed to ordinary differential equations, then to a system of coupled dual integral equations. The thermoelastic fields around the penny-shaped crack are obtained explicitly by solving the derived dual integral equations. Numerical results are reported to show the influences of the thermal conductivity of crack interior on partial insulation coefficient, temperature change across crack and thermal stress intensity factor. As compared to the known thermal-medium crack model, the proposed one exhibits more applicability.     

三角形弹性夹杂对裂纹的影响

研究了三角形弹性夹杂和裂纹之间的相互影响问题。应用Chau和Wang导出的面力边值问题的边界积分方程为基本方程,用夹杂和基体交界面上的面力和位移的连续性条件为补充方程,从而得到了一组能够解决夹杂和裂纹相互影响问题的方程,最后的方程组用一种新的边界单元法求解。计算了各种不同的夹杂和基体的材料常数以及夹杂和基体之间不同距离情况下裂纹尖端的应力强度因子。文中结果对研究新型复合材料有一定的应用价值。     

Simulation of Autofrettage in Curvilinear Coordinates

Pressure vessels subjected to high mechanical loadings can fail due to microscopic crack growth, i.e., stress corrosion cracking. A popular way to increase the crack resistance of a structure is by autofrettage. The standard approach to obtain a numerical solution for the involved fields is to discretize w.r.t. Cartesian coordinates as implemented in common commercial FE programs. However, this approach is computationally costly in comparison to pure elastic simulations—often rendering this method practically unfeasible. The current work shows that autofrettage can be simulated more efficiently in extensive bodies that can be described by a bijective map where certain symmetries are available in load and form. The discretization and the mathematical formalism of this approach in curvilinear coordinates are described. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

带裂纹十次对称二维准晶平面弹性的无摩擦接触问题

借助经典平面弹性复变函数方法，研究了单个刚性凸基底压头作用下，带任意形状裂纹十次对称二维准晶半平面弹性的无摩擦接触问题.利用十次对称二维准晶位移、应力的复变函数表达式, 带任意形状裂纹的准晶半平面弹性无摩擦接触问题被转换为可解的解析函数复合边值问题，进而简化成一类可解的Riemann边值问题.通过求解Riemann边值问题，得到了应力函数的封闭解, 并给出了裂纹端点处应力强度因子和压头下方准晶体表面任意点处接触应力的显式表达式.从压头下方接触应力的表达式可以看出, 接触应力在压头边缘和裂纹端点处具有奇异性.当忽略相位子场影响时, 该文所得结论与弹性材料对应结果一致.数值算例分别给出了单个平底刚性压头无摩擦压入带单个垂直裂纹和水平裂纹的十次对称二维准晶下半平面的结果.该文所得结论为准晶材料的应用提供了理论参考.     

纯扭正交异性复合材料板的断裂分析

对受纯扭载荷作用的线弹性正交异性复合材料板裂纹尖端附近的断裂性态进行探讨。利用复变函数方法，通过求解偏微分方程的边值问题，推出了裂纹尖端附近的弯矩、扭矩、应力和位移的表达式，最后给出了数值算例。     

Fatigue life investigation using non-destructive testing methods

Experimental investigations of the fatigue life prediction of two different specimen forms made from steel 42CrMo4 are described. The fatigue load is defined as a cyclic loading with constant and non constant amplitudes in tension range and with different stress ratios. This loading range leads to a high cyclic fatigue behaviour of both specimens. During the test, the growth of the fatigue crack is monitored using two different non-destructive methods, namely acoustic emission and the electric resistance. The acoustic emission is used for detecting sound waves as a result of dissipation of elastic energy during microcracks and cracks growth, as well as for the detection of the place of crack initiation. The electric resistance is used for the monitoring and quantitative investigation of the crack growth. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A singular perturbation approach to integral equations occurring in poroelasticity

Singular perturbation theory is used to solve the integral equationswhich occur when treating finite-length crack problems in porouselastic materials. The method provides the stress intensityfactors which characterize the near crack tip stress and displacementfields for small times. The method also gives the stress andpore pressure fields on the fracture plane for small times relativeto the diffusive time scale. In this paper, the authors treatcrack problems which are unmixed in the pore pressure boundarycondition on the fracture plane. The Abelian result that smalltimes correspond, in Laplace transform space, to large valuesof the transform variable is used to formulate the problemsin terms of a small parameter. Rescaling on this small parameterleads to inner problems which are eigensolutions of the semi-infiniteproblems treated earlier by the authors. The outer solutionsare given by elastic eigensolutions together with appropriatefluid dipole responses. These outer solutions give the completestress and pore pressure fields except in the neighbourhoodof the crack tips; in this region the outer solutions are asymptoticallymatched with inner solutions. The full outer solutions are givenhere as an asymptotic expansion for small times and enable thedevelopment of the outer fields to be followed in real time.A reciprocal theorem in Laplace transform space is used to checkthe small-time solutions. The inner problem is rescaled to asemi-infinite crack problem, so eigensolutions of this semi-infiniteproblem are used together with the known asymptotic behaviourof the real solution to identify the stress intensity factor.The stress intensity factor is then related to an integral involvingthe inner limit of the outer solution together with the eigensolutionof the semi-infinite problem. Using this integral, we recoverthe result for the stress intensity factor found using singularperturbation theory. A ‘nearly’ invariant integralanalogous to the invariant M integral used in elastostaticsis derived. Unfortunately, the poroelastic analogue is not invariant,although it is used to verify the small-time results.     

Growth of arterial cracks in polymers subject to static and fatigue loading

The growth kinetics of artificial and natural cracks (the former arising from notches) are studied in viscose, di- and triacetyl cellulose, nitrocellulose, polymethylmethacrylate, polystyrene, polypropylene, and carpon films by a micro-motion-picture method over a wide temperature range under both static and cyclic (fatigue) loading. In all the cellulose materials studied, the time required to form the nuclei of visible cracks is much shorter than the total life of the sample. The initial rate of crack growth depends exponentially on the applied stress and test temperature, so that the equation for the growth rate of an arterial crack is analogous to the general life equation. The different ways in which stress and test temperature affect the parameters of the equation describing the kinetics of crack growth enable us to distinguish the effect of local heating and that of the relaxation processes in the fatigue problem.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 2, pp. 253–265, March–April, 1970.     

纯扭各向异性复合材料板断裂分析的复变函数方法

对受纯扭载荷作用的线弹性各向异性纤维复合材料板裂纹尖端附近的应力场进行探讨.选取带复参数的挠度函数,利用复变函数方法和待定系数法,借助边界条件,确定复参数,从而推出了裂纹尖端附近的弯矩、扭矩、应力和位移计算公式.所得到的公式在有关的断裂分析中有一定的实用价值和参考作用,最后给出了数值算例.     

A boundary element analysis for stress intensity factors of multiple circular arc cracks in a plane elasticity plate

In this paper, a numerical approach for analyzing interacting multiple cracks in infinite linear elastic media is presented. By extending Bueckner’s principle suited for a crack to a general system containing multiple interacting cracks, the original problem is divided into a homogeneous problem (the one without cracks) subjected to remote loads and a multiple crack problem in an unloaded body with applied tractions on the crack surfaces. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by considering the latter problem, which is analyzed easily by means of the displacement discontinuity method with crack-tip elements proposed recently by the author. Test examples are given to illustrate that the numerical approach is very accurate for analyzing interacting multiple cracks in an infinite linear elastic media under remote uniform stresses. In addition, the displacement discontinuity method with crack-tip elements is used to analyze a multiple crack problem in a finite plate. It is found that the boundary element method is also very accurate for investigating interacting multiple cracks in a finite plate. Specially, a generalization of Bueckner’s principle and the displacement discontinuity method with crack-tip elements are used to analyze multiple circular arc crack problems in infinite plate in tension (including: Two Collinear Circular Arc Cracks, Three Collinear Circular Arc Cracks, Two Parallel Circular Arc Cracks, Three Parallel Circular Arc Cracks and Two Circular Arc Cracks) in a plane elasticity plate. Many results are given.     

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals is considered. The Fourier transform technique is used to solve a moving crack problem under the action of antiplane shear and inplane electric field. Full elastic stresses of phonon and phason fields and electric fields are derived for a crack running with constant speed in the periodic plane. Obtained results show that the coupled elastic fields inside piezoelectric quasicrystals depend on the speed of crack propagation, and exhibit the usual square-root singularity at the moving crack tip. Electric field and phason stresses do not have singularity and electric displacement and phonon stresses have the inverse square-root singularity at the crack tip for a permeable crack. The field intensity factors and energy release rates are obtained in closed form. The crack velocity does not affect the field intensity factors, but alters the dynamic energy release rate. Bifurcation angle of a moving crack in a 1D hexagonal piezoelectric quasicrystal is evaluated from the viewpoint of energy balance. Obtained results are helpful to better understanding crack advance in piezoelectric quasicrystals.     

Fracture of a piezoelectric fiber/elastic matrix composite

A piezoelectric fiber/elastic matrix system subjected to axially symmetric mechanical and electric loads is considered. The fiber contains a penny-shaped crack located at its center perpendicularly to the fiber. By using the Fourier and Hankel transforms, the problem is reduced to the solution of an integral equation. Numerical solutions for the crack tip fields are obtained for various crack sizes and different fiber volume fractions. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 301–318, May–June, 2006.