Pseudo-traction Method Research Articles

Interaction of multiple parallel cracks in a pre-stressed orthotropic elastic plane

Pre-stressed structures have been widely applied in aerospace, deep mining, and civil engineering fields. This paper presents a theoretical analysis of the interaction of multiple parallel cracks in a pre-stressed orthotropic elastic material. The pseudo-traction method and complex potential method are applied to solve an associated mixed boundary-value problem. First, we derive two kinds of fundamental solutions for a pair of normal and tangential concentrated forces acting at any point on an isolated crack. Then, with these solutions, a system of Fredholm integral equations are derived by superposition. The interaction of multiple parallel cracks in a pre-stressed orthotropic elastic plane is analyzed. Numerical results show that the pre-stress has little influence on the stress intensity factors when two cracks are collinear or two cracks are far away from each other (Dh>2a or Dv>5a). For closer two parallel cracks, KI decreases with increasing pre-stress σ0 under normal loading and KII increases with increasing pre-stress σ0 under shear loading.

Bifurcation of crack pattern in two-dimensional periodic rectangular crack arrays under dynamic shear stress

Crack interaction leads to the bifurcation of crack pattern. The pseudo-traction method is used to analyze interaction among cracks. Bifurcation condition of crack growth pattern under dynamic shear stress depends upon the ratios of crack length to spacing ratio, and the crack length to the perpendicular distance between the two adjacent row ratios. Bifurcation of crack growth patters occurs when the above ratios extends critical values. Bifurcation condition for periodic rectangular cracks subjected to dynamic shear stress is analyzed theoretically. The critical parameter is dependent on crack growth velocity and the velocity of a Rayleigh wave. It is shown from numerical result that the critical parameter increases with the velocity of a Rayleigh wave, and decreases with the increase of crack growth velocity.

Effective elastic properties of the double‐periodically cracked plates

AbstractIn this paper, the interaction of double‐periodical cracks is accurately solved based on the isolating analysis procedure, superposition principle, pseudo‐traction method, Chebyshev polynomial expansion and crack‐surface collocation technique. The jump displacement crossing crack faces, the average additional strain and therefore the effective compliance of the double‐periodically cracked plate are directly determined. The numerical results for axial‐symmetrically distributed double‐periodical cracks, general double‐periodical cracks with one collinear direction as well as two sets of double‐periodical cracks with same size and square distribution are given in this paper. And the partial typical numerical results are compared with the previous works. The analysis shows that the anisotropy induced by the general double‐periodical cracks is generally not orthogonal anisotropy. Only when the double‐periodical cracks are axial‐symmetrically distributed, is the anisotropy orthogonal. In this special cases, the effective engineering constants (consist of effective elastic modulus, the effective Poisson's ratio, the effective shear modulus) of cracked plate versus crack spacing, in the plane stress and plane strain conditions, respectively, are analysed. Copyright © 2005 John Wiley & Sons, Ltd.

The interaction of doubly periodic cracks

In this paper, an extremely accurate and efficient method for computing the interaction of a set of or multiple sets of general doubly periodic cracks has been presented on the basis of superposition principle, pseudo-traction method, and isolating analysis technique. A great number of typical examples are given in this paper. The stress intensity factors (SIF), the minimum strain energy density factors (SED) of crack tip and the critical stress (CRS) of crack growth are calculated with the accuracy of six significant digits for the rectangularly distributed periodic cracks and five significant digits for the general doubly periodic cracks. The relation of the interaction effect of the double periodic cracks with the periods and the ratio of crack length to crack spacing is analyzed. Also in this paper, the key technique problems for this method are discussed.

Macrocrack–microcrack interaction problem in magnetoelectroelastic solids

Aiming at investigating the macrocrack–microcrack interactions in magnetoelectroelastic solid, the elementary solutions for a semi-infinite macrocrack and a microcrack subjected to various loading conditions are derived first. Then by using the pseudo-traction–electric-displacement–magnetic-induction method that is motivated by the well-known pseudo-traction method [J. Mech. Phys. Solids 35 (1987) 601], the present interaction problem is reduced to a system of integral equations. After solving these equations numerically by using the Chebyshev numerical integration technique, the crack tip stress intensity factors (SIFs), the electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF) as well as the mechanical strain energy release rate (MSERR) are evaluated subsequently. The J-integral for magnetoelectroelastic medium is further deduced, and it is employed to examine the correctness of the calculation results derived. Finally, take both the stress intensity factors and the mechanical strain energy release rate as measures to investigate the influences of various loading conditions, the crack location and orientation as well as the number of microcracks on the macrocrack–microcrack interactions. It is found that both the electric and magnetic loadings have a significant influence on the fracture toughness of magnetoelectroelastic solid.

J-integral analysis of the interaction between an interface crack and parallel subinterface cracks in dissimilar anisotropic materials

The J-integral analysis is presented for the interaction problem between a macro-interface crack and subinterface microcracks parallel to the former in the near-tip process zone in dissimilar anisotropic composite materials. Elementary solutions respectively considering an interface crack and a subinterface crack subjected to different loads are given from which the interaction problem is deduced to a system of integral equations with the aid of superimposing technique (i.e., the so called `Pseudo-Traction Method' abbreviated as PTM). After the integral equations are solved numerically, a consistent relation among three kinds of the J-integral values is obtained. They are induced from the macro-interface crack tip, the microcracks, and the remote field, respectively. This consistent relation of the J-integral can be used to confirm the numerical results derived by using whatever kind of technique. With the aid of J-integral analysis, the interaction behaviors between an interface crack and parallel subinterface cracks are investigated in detail, and some special physical phenomena are obtained.

Crack amplification and shielding in plane piezoelectric solids

Abstract A general solution is derived for the problem of a microcrack near the tip of a semi-infinite main crack in a two-dimensional piezoelectric medium. The interaction problem is treated by extending the method of pseudo-tractions, which was used in the study of crack-interaction problems in elastic solids, to include the pseudo-electric displacements along the microcrack. An algebraic equations system is obtained to determine the unknown pseudo-tractions and pseudo-electric displacements. The accuracy of the present formulation is confirmed by the comparison of the present solutions with those reported in the literature for the limiting cases of isotropic elastic solids. Numerical results for the field intensity factors and the mechanical strain energy release rate at the crack tips are presented to show the influence of various loading conditions, and the location and orientation of a microcrack on main crack–microcrack interaction. Amplification or shielding effects for the main crack may be introduced due to the presence of a microcrack. Electric loading is shown to play an important role in crack-interactions in piezoelectric solids. The present formulation can be easily generalized to the case of multiple microcracks near the tip of a main crack.

Interaction between an interface crack and a parallel subinterface crack in dissimilar anisotropic composite materials

In this paper, the pseudo-traction method is combined with the edge-dislocation method (i.e. PTDM) to solve the interaction problem between an interface crack and a parallel subinterface crack in dissimilar anisotropic materials. After deriving the fundamental solutions for an interface crack loaded by normal or tangential tractions on both crack surfaces and the fundamental solutions for an edge dislocation beneath the interface in the lower anisotropic material, the interaction problem is reduced to a system of a singular integral equations by adopting the well-known superposition technique. The equations are then solved numerically with the aid of the Chebyshev numerical integration and the Chebyshev polynomial expansion technique. Several typical examples are calculated and numerical results are shown in figures and tables from which a series of valuable conclusions is obtained. Since the present results should be verified and since no previous results exist to compare them with a consistency check in introduced which starts from the conservation law of the J-integral in anisotropic cases. It is shown that the check provides a powerful tool to examine the results, although it really presents a necessary condition rather than a sufficient way to the crack-tip parameters of the interface crack and the subinterface crack in the dissimilar anisotropic materials.

Further investigation of interaction between interface macrocrack and parallel microcracks in bimaterial anisotropic solids

In this paper, with the aid of superimposing technique and the Pseudo Traction Method (PTM), the interaction problem between an interface macrocrack and parallel microcracks in the process zone in bimaterial anisotropic solids is reduced to a system of integral equations. After the integral equations are solved numerically, a conservation law among three kinds ofJ-integrals is obtained which are induced from the interface macrocrack tip, the microcrack and the remote field, respectively. This conservation law reveals that the microcrack shielding effect in such materials could be considered as the redistribution of the remoteJ-integral.

Investigation of macrocrack-microcrack interaction problems in anisotropic elastic solids - Part I : General solution to the problem and application of the J-integral

A general solution is derived for the plane problem of multiple microcracks near the tip in process zone of a semi-infinite macrocrack in an anisotropic elastic solid. The pseudo-traction method, addressed thoroughly in isotropic cases, is extended to anisotropic cases. A system of Fredholm integral equations, with difficulty in evaluation of the singular integrals, is solved by invoking the asymptotic behavior of the pseudo-tractions on the macrocrack faces. The interaction effect of the release of residual stresses due to near-tip microcracking is then evaluated. The J-integral analysis is also performed to give a consistency check of the solution and some useful conclusions.

Bifurcation in Growth Patterns for Arrays of Parallel Griffith, Edge and Sliding Cracks

This paper presents the recent finding by Muhlhaus et al [1] that bifurcation of crack growth patterns exists for arrays of two-dimensional cracks. This bifurcation is a result of the nonlinear effect due to crack interaction, which is, in the present analysis, approximated by the dipole asymptotic or pseudo-traction method. The nonlinear parameter for the problem is the crack length/ spacing ratio lambda = a/h. For parallel and edge crack arrays under far field tension, uniform crack growth patterns (all cracks having same size) yield to nonuniform crack growth patterns (i.e. bifurcation) if lambda is larger than a critical value lambda(cr) (note that such bifurcation is not found for collinear crack arrays). For parallel and edge crack arrays respectively, the value of lambda(cr) decreases monotonically from (2/9)(1/2) and (2/15.096)(1/2) for arrays of 2 cracks, to (2/3)(1/2)/pi and (2/5.032)(1/2)/pi for infinite arrays of cracks. The critical parameter lambda(cr) is calculated numerically for arrays of up to 100 cracks, whilst discrete Fourier transform is used to obtain the exact solution of lambda(cr) for infinite crack arrays. For geomaterials, bifurcation can also occurs when array of sliding cracks are under compression.

Interaction between an interface crack and a parallel subinterface crack

In this paper, the pseudo-traction method addressed thoroughly in homogeneous cases is combined with the edge dislocation method to solve the interaction problem of an interface crack with a parallel subinterface crack. After deriving the fundamental solutions for a typical interface crack loaded by the normal and tangential concentrated tractions on both crack surfaces and the fundamental solutions for an edge dislocation beneath the interface, the interaction problem is reduced to a system of singular integral equations which can be solved numerically with the aid of the Chebyshev polynomial technique. Numerical results for the stress intensity factors are shown in the figures in which six kinds of material combinations presented by Hutchinson et al. [1] are considered.

On the influence of orthotropic material constants on the neutral-shielding angle in main-micro crack interaction problems

This paper presents an analysis for the influence of orthotropic material constants on the so-called neutral-shielding angle in main-micro crack interaction problems. The Fredholm integral equation and the pseudo-traction method proposed in isotropic cases are extended to treat the interaction problems in orthotropic cases. After releasing the residual stresses due to near-tip parallel micro cracking, the elastic interaction between a main crack and a parallel micro crack is evaluated. Two kinds of orthotropic materials are considered in detail. Numerical results are shown in the tables and figures. It is found that in both kinds of orthotropic cases, the influence on the angle through characteristic roots is remarkable. The trend analysis of the influence is performed and discussed.

Investigation of main-micro crack interaction problems in a unidirectional fibre reinforced composite

The elastic interaction between a main crack and an inclined micro crack in a unidirectional fibre reinforced composite is studied. The Fredholm integral method and the pseudo-traction method proposed in isotropic cases are extended to treat the main-micro crack problems in orthotropic cases. The influence of orthotropic material constants on the interaction effect, i.e. the amplification effect or the shielding effect, is discussed in detail.

Interaction between a main-crack and a parallel micro-crack in an orthotropic plane elastic solid

This paper deals with the elastic interaction between a main-crack and a parallel microcrack in an orthotropic plane elastic solid. The pseudo-traction method proposed by Chen [Engng Fract. Mech.20, 591–597 and 767–776 (1984a, b)], and Horii and Nemat-Nasser [Int. J. Solids Structures21, 731–745 (1985)] in isotropic cases is extended into orthotropic cases. After introducing two kinds of fundamental solutions, a system of Fredholm integral equations is derived from which the interaction effect of the release of residual stresses due to near-tip micro-cracking is evaluated. Numerical results are shown in figures and tables. Some useful conclusions are discussed.

Microcrack Interaction and Shear Fault Failure

The method of pseudo-tractions is used to solve the problem of interacting cracks which are arbitrarily distributed and are loaded by concentrated forces. The effect of interaction on the crack tip stress intensity factors is obtained for a set of equally spaced, parallel cracks which are loaded by both concentrated forces and uniform farfield stresses. For the special case when the cracks are collinear, the results are compared with those of the closed-form exact solution to check the accuracy of the method. The method is then applied to an array of dynamically growing wing cracks, and the growth regime of a periodic array of wing cracks of arbitrary orientation is studied. The shear fault failure mechanism is discussed in terms of such a model. The direction of the shear fault is related to the direction of a periodically distributed wing-crack array, in which tension cracks grow to a length equal to their spacing at a minimum applied load. The effect of confining pressure on the shear fault direction is discussed.

Multiple void-crack interaction

This paper presents a method for fracture analysis of a general two-dimensional system containing multiple holes, or voids, and cracks. The superposition technique is used to break the problem into a number of single-hole and single-crack problems. Each hole problem is modeled using the method of pseudo-tractions, and each crack problem is modeled by a distribution of dislocations. An integral equation approach is developed, based on two types of fundamental solutions, one due to point loads in a solid with a hole and the other due to point dislocations in an infinite elastic body. The resulting integral equations present Cauchy-type singularities only on the crack part of the multiple hole-crack problem. The results in terms of stress intensity factors (SIFs) are presented for a variety of hole-and-crack arrangements, relative sizes of cracks and holes, spacings and crack orientations. The amplification and retardation effects on SIFs are investigated. It is found that the hole-crack arrangements have significant effects on the nature of the amplification or retardation. In the fractured porous elastic medium (modeled as a crack surrounded by holes), amplification or retardation can occur, depending on the relative size of the holes and cracks and the spacing between them. Very strong retardation exists as the spacing becomes small. Some optimal retardations (void toughening) are achieved by adjusting the geometry parameters. An array of periodical crack-hole structure is examined as a numerical example.

Microslip model of strain localization in sand deformation

A microslip model of strain localization in sand deformation is proposed. The model takes account of the interaction effect among microslips which represent local inelastic deformation due to relative motion between sand particles. The formulation is based on the method of pseudo-tractions which is a general method for estimating interaction effects in elastic problems. The evolution of individual microslips is formulated. The strain localization as well as the strain softening feature is predicted by the present model. The effect of local dilatancy, which is an important mechanism in strain localization, is also discussed.

General solution to the problem of microcracks near the tip of a main crack

A general solution is obtained to the two-dimensional problem of the interaction between a main crack and multiple microcracks. Analysis is based on the method of pseudo-tractions, which is a general method for interaction problems. The stress intensity factors at the tips of both the main crack and microcracks are obtained in asymptotic forms and approximate closed-form solutions are also derived. Typical examples are presented for one collinear microcrack, one arbitrarily located microcrack and two arbitrarily located microcracks near the tip of the main crack. For one collinear microcrack the present solution is shown to coincide with the Taylor expansion of the exact solution, and inconsistencies in several previous solutions are compared. For arbitrarily located microcracks, the effects of the main crack—microcrack interaction and the microcrack—microcrack interaction are investigated. Application of the present model to the fracture process zone problem is discussed.

MECHANICAL PROPERTIES OF SOLIDS CONTAINING A DOUBLY PERIODIC RECTANGULAR ARRAY OF CRACKS

In the present study, the problem of an infinitely extended elastic solid containing a doubly periodic rectangular array of cracks is considered. The analysis is made based on the method of pseudo-tractions, and a first-order approximate but explicit solution is obtained. The study reveals the fundamental difficulty of the problems of doubly periodic cracks which gives rise to the discrepancies between the results of the previous works. The stress intensity factors of mode I and mode II and the overall compliance are derived as functions of the crack density and geometry of the crack array. The overall compliance of solids with randomly distributed, unidirectional cracks is evaluated by the self-consistent method. The validity of the self-consistent method is discussed in terms of the results of the doubly periodic cracks.