Crack formation under compression

This paper presents a theoretical analysis of the formation of a tensile stress zone (separation zone) in a crack of finite width under uniaxial transverse compression, assuming the existence of prefracture zones that satisfy a linear one-parameter relationship between stress and displacement vectors. A crack formation criterion is stated, and practical fracture criteria are analyzed from the point of view of locality of the crack formation process.Simferopol' University. Translated from Dinamicheskie Sistemy, No. 10, pp. 3–8, 1992.     

A criterion for the growth of cracks with bonds in the end zone

A fracture criterion which takes account of the work done in the deformation of bonds in the end zone of a crack is proposed for analysing the quasistatic growth of a crack with bonds in the end zone. The energy condition that the deformation energy release rate at the crack tip is equal to the rate of deformation energy consumption by the bonds in the end zone of the crack (the first fracture condition) corresponds to the state of limit equilibrium of the crack tip. The rupture of bonds at the trailing edge of the end zone is determined by the condition for their limiting traction (the second fracture condition). Starting from these two conditions, the processes of subcritical and quasistatic crack growth are considered for the case of a rectilinear crack at interface of materials and the two basic fracture parameters, the critical external load and the size of the end zone of the crack in the state of limit equilibrium, are determined. Analytical expressions are obtained for the deformation energy release rate at the crack tip and the rate of deformation energy consumption by the bonds and, also, the dependences of the critical external load and size of the end zone of the crack on the crack length in the case of a rectilinear crack in a homogeneous body with bond tractions which are constant and independent of the external load. The limit cases of a crack which is filled with bonds and a crack with a short end zone are considered.     

Scattering of antiplane shear wave by a propagating crack at the interface of two dissimilar elastic media

An analysis of the scattering of horizontally polarized shear wave by a semi-infinite crack running with uniform velocity along the interface of two dissimilar semi-infinite elastic media has been carried out. The mixed boundary value problem has been solved completely by the Wiener-Hopf technique. The effect of different values of the material parameter, the angle of incidence of incident wave and the crack propagation velocity on the stress intensity factor have been illustrated graphically.     

On the integrable case of a Riemann-Hilbert boundary value problem for two functions and the solutions of certain mixed problems for a composite elastic plane

A method for solving the Riemann-Hilbert boundary value problem with piecewise-constant coefficients is generalized /1/. It is shown that the following static problems of a composite elastic plane with three kinds of connection conditions allow of exact solutions: 1) the splicing line is weakened by a system of loaded slots and a transverse shear crack or the edges of one of the slots are partially contacting, or one of the slots is cleaved by a rigid insert; 2) the splicing line is reinforced by a system of thin rigid inclusions and there is one arbitrarily located delamination zone; 3) the elastic half-planes are contacting (with slip) on a certain section of their boundaries, and mixed boundary conditions in the displacements and stresses are given on the rest of the boundaries.

In the general case the Riemann-Hilbert boundary value problem for many functions reduces to the problem of a linear conjugation, and then to Fredholm integral Eqs./2/. Closed solutions are obtained in certain special cases /3–5/. For applications we mention the papers /6, 7/, where problems are considered concerning slits at the interface of two elastic media with two kinds of physical boundary conditions taken into account simultaneously.     

Singularities at the tip of a crack on the interface of piezoelectric bodies

Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezoelectric media under various boundary conditions on the crack surfaces. The singularity exponents form the spectrum of a certain polynomial pencil, and although explicit formulas are not available, this spectrum is described completely though. The mathematical results apply to problems in fracture mechanics. In this way the Griffith formulas are obtained for increments of energy functionals due to the growth of the crack, and the notion of energy release matrix is introduced. Normalization conditions for bases of singular solutions are proposed to adapt them to energy, stress, and deformation fracture criteria. Connections between these bases are determined, and additional properties of the deformation basis related to the notion of electric surface enthalpy are established. Bibliography: 44 titles. Dedicated to Vsevolod Alekseevich Solonnikov Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 241–271.     

On the assessment of cracks within the interface of thin layers on a substrate

Although bimaterial wedge or notch configurations are identified as potential weak locations, the assessment of the degree of criticality of cracks in such regions is still a demanding problem. The singular character of the stress field at cracks or at bimaterial notches can be calculated analytically or numerically. The angle of the direction of potential crack initiation may also be determined, but the decisive question is whether a hypothetical crack will be initiated or not. An essential question in the context of crack assessment is to find a criterion for crack nucleation. For that aim, the hypothesis of Leguillon is modified. Herein, the crack is assumed to be critical when and only when both the released energy and the local stress reach critical values along a hypothetical crack of finite length. This concept can be transferred to a bimaterial interface configuration of a thin layer on a substrate. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Problem of thermoelectromagnetoelasticity for a piezoelectric/piezomagnetic bimaterial with interface crack

We consider a plane strain problem for a piezoelectric/piezomagnetic bimaterial space with a crack in the region of the interface of the materials. At infinity, tensile and shear stresses and heat, electric, and magnetic flows are set. Using representations for all mechanical, thermal, and electromagnetic factors in terms of piecewise analytic functions, we formulate problems of linear conjugation that correspond to a model of an open crack and models taking into account the contact zone in the vicinity of a crack tip. Exact analytic solutions of the indicated problems are constructed. Expressions for stresses, the electric and magnetic inductions, jumps of derivatives of displacements, and electric and magnetic potentials on the interface are written. The coefficients of intensities of the indicated factors are presented. We derive a transcendental equation for the determination of the real length of the contact zone. The dependences of this length and the coefficients of intensity on the set external influences are investigated.     

压电基体中部分脱开的刚性导体椭圆夹杂分析

通过利用八维Stroh公式以及共形映射、解析延拓和奇点分析技术,获得了对一压电基体中已部分脱开的刚性导体椭圆夹杂二维问题的闭合形式全场解答。也推导了一些新的恒等式和求和式,通过这些恒等式及求和式可获得沿界面应力和电位移分布以及刚性夹杂转动的实形式表示。正如所预料的,在脱开界面的端部应力及电位移显现出与在压电材料Griffith界面裂纹的研究中所发现的相似的奇异行为。最后也给出了几个算例以展示所得到解答的一般性以及各种载荷条件、几何参数和机电常数等对界面处应力及电位移分布的影响。     

Scattering of a SH-wave by an elastic fiber of nonclassical cross section with an interface crack

The problem of interaction of a plane time-harmonic SH-wave with an elastic fiber of quasi-square or quasi-triangular cross section, when an interface crack is present between an infinite elastic matrix and the fiber, is considered. The modified null-field method taking into account the asymptotic behavior of the solution at crack tips is exploited for obtaining numerical results. The effects of fiber shape, fiber/matrix material combination, debonding (crack size), and direction of wave incidence on the scattering amplitude in the far zone are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 245–254, March–April, 2008.     

The effect of the interface between media on stress concentration in the vicinity of a crack

The problem of determining the effect of the media interface in a piecewise-homogeneous body on the stress concentration in the vicinity of cracks located in one of the half-spaces is reduced to a system of two-dimensional singular equations of Newtonian potential type. We study the effect of the relations of the elastic constants of the materials of a body composed of two half-spaces on the stress intensity factors in the vicinity of a crack. We study different variants of the crack location relative to the interface. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 64–69.     

Interlaminar Crack Propagation Analysis of ENF Specimens Based on the Cohesive Zone Model北大核心CSCD

Based on the classical laminated plate theory and the cohesive zone model, a theoretical model for general delamination cracked laminates was established for crack propagation of pure mode Ⅱ ENF specimens. Compared with the conventional beam theory, the proposed model fully considered the softening process of the cohesive zone and introduced the nonlinear behavior of ENF specimens before failure. The predicted failure load is smaller than that under the beam theory and closer to the experimental data in literatures. Compared with the beam theory with only fracture toughness considered, the proposed model can simultaneously analyze the influences of the interface strength, the fracture toughness and the initial interface stiffness on the load-displacement curves in ENF tests. The results show that, the interface strength mainly affects the mechanical behavior of specimens before failure, but has no influence on crack propagation. The fracture toughness is the main parameter affecting crack propagation, and the initial interface stiffness only affects the linear elastic loading stage. The cohesive zone length increases with the fracture toughness and decreases with the interface strength. The effect of the interface strength on the cohesive zone length is more obvious than that of the fracture toughness. When the adhesive zone tip reaches the half length of the specimen, the adhesive zone length will decrease to a certain extent. Copyright ©2022 Applied Mathematics and Mechanics. All rights reserved.     

Delamination of Composites along the Interface as Buckling Failure of the Stressed Layer

A refined model for free vibrations of a Griffith crack in layered composites is investigated. The delamination of the composites along an interface is studied in the context of growth of instability of the stressed layer. This is analogous to the bending of a beam on an elastic foundation. The second-order gradient terms are taken into account in the power-series expansion of deflections. These terms are responsible for nonlinear phenomena in the interaction zone. The zero-order frequency of free vibrations of the crack along the interface and the time of instability growth are determined.     

Subcritical and critical states of a crack with failure zones

The problem on the stress–strain state near a mode I crack in an infinite plate is solved in the frame of a cohesive zone model. The complex variable method of Muskhelishvili is used to obtain the crack opening displacements caused by the cohesive traction, which models the failure zone at the crack tip, as well as by the external load. The finite stress condition and logarithmic singularity of the derivative of the separation with respect to the coordinate at the tip of a physical crack are taken into account.The cohesive traction distribution is sought in a piecewise linear form, nodal values of which are being numerically chosen to satisfy the traction-separation law. According to this law, the cohesive traction is coupled with the corresponding separation and fracture toughness. The tips of the physical crack and cohesive zone (geometric variables) along with the discrete cohesive traction are used as the problem parameters determining the stress-strain state. If the crack length is included in the set, then the critical crack size can be found for the given loading intensity.The obtained determining system of equations is solved numerically. To find the initial point for a standard numerical algorithm, the asymptotic determining system is derived. In this system, the geometric variables can be easily eliminated, which make it possible to linearize the system.In the numerical examples, the one-parameter traction-separation laws are used. Influence of the shape parameters of the law on the critical crack size and the corresponding cohesive length is studied. The possibility of using asymptotic solutions for determining the critical parameters is analysed. It is established that the critical crack length slightly depends on the shape parameter, while the cohesive length shows a strong dependence on the shape of cohesive laws.     

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.     

A CRACK PROBLEM WITH A BROKEN LINE INTERFACE

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Scattering of a plane wave on a wavy surface

Numerical methods are proposed for solving problems of diffraction of a plane electromagnetic wave on a nonhomogeneous interface of two media. Strictly periodic cylindrical surfaces are considered, as well as surfaces that are nonplanar on a finite section. Algorithms are developed for computing the characteristics of the reflected field in the far zone. Numerical results are reported for various electrodynamic models, and their limits of applicability are investigated.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 86–111.     

Implementation of a cohesive zone model for the investigation of the dynamic behavior of a rotating shaft with a transverse crack

The influence of a transverse crack on the vibration of a rotating shaft has been at the focus of attention of many researchers. The knowledge of the dynamic behavior of cracked shaft has helped in predicting the presence of a crack in a rotor. Here, the changing stiffness of the cracked shaft is investigated based on a cohesive zone model. This model is developed for mode-I plane strain and accounts for triaxiality of the stress state explicitly by using basic elastic-plastic constitutive relations. Then, the proposed numerical solution is compared to the switching crack model, which is based on linear elastic fracture mechanics. The cohesive zone model is implemented in finite element techniques to predict and to analyse the dynamic behavior of cracked rotor system. Timoshenko beam theory is used to model the discrete shaft under the effect of gravity, unbalance force and gyroscopic effect. The analysis includes the cohesive function for describing the breathing crack and the reduction of the second moment of area of the element at the location of the crack. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Micromechanics of polymer fracture

Small-angle x-ray scattering has been used to investigate the formation of embryonic submicroscopic cracks in polymers under a load. The main characteristics of crack formation in various loading regimes are analyzed. It is shown that there is a relation between the submicrocrack concentration and the deformation of the loaded polymer. The principal parameters of crack formation determining the strength properties of the polymer are found to be the transverse dimension of the initial submicrocracks relative to the loading axis, which is determined by the structural heterogeneity of the material, and the submicrocrack concentration in the prefracture state. The principles of the micromechanics of polymer fracture are formulated on the basis of the results of an analysis of the quantitative relationship between these parameters. The dominant role of the surface in the fracture process is demonstrated by comparing the parameters of crack formation in the interior and at the surface of the loaded polymer.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 5, pp. 792–801, September–October, 1974.