Time-domain boundary element analysis of dynamic near-tip fields for impact-loaded collinear cracks

A time-domain boundary integral equation method has been developed to calculate elastodynamic fields generated by the incidence of stress (or displacement) pulses on single cracks and systems of two collinear cracks. The system of boundary integral equations has been cast in a form which is amenable to solution by the boundary element method in conjunction with a time-stepping technique. Particular attention has been devoted to dynamic overshoots of the stress intensity factors. Elastodynamic stress intensity factors for pulse incidence on a single crack have been computed as function of time, and they have been compared with results of other authors. For collinear macrocrack-microcrack configurations the stress intensity factors at both tips of the macrocrack have been computed as functions of time for various values of the crack spacing and the relative size of the microcrack.     

Perturbation solution for near-tip fields of cracks growing in elastic perfectly-plastic compressible materials

With = 1/2 – ( - Poisson's ratio) as a small parameter, perturbation expansion is made, based upon the generally accepted solution for near-tip fields of cracks growing in elastic perfectly-plastic incompressible materials under plane strain. Asymptotic solutions up to the order ² are obtained for near-tip fields of cracks growing quasi-statically and steadily in elastic perfectly-plastic compressible materials. The near-tip field has a 5-sector structure. As 1/2 the 5-sector solution degenerates to the 4-sector solution for the case of incompressible materials. The perturbation expansion provides deeper insight into the process of degeneration of the 5-sector solution to the 4-sector one, and also gives approximate analytical solutions for the case of compressible materials.

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Résumé On réalise l'expansion d'une perturbation en faisant légèrement varier le paramètre = 1/2 – ( = Module de Poisson) et en se basant sur la solution g'enéralement admise pour des champs au voisinage de l'extrémité de fissures en croissance dans des matériaux incompressibles parfaitement élastiques-plastiques soumis á déformation plane.On obtient des solutions asymptotiques jusqu'à l'ordre ² pour des champs au voisinage de l'extrémité de fissures qui croissent de manière quasi-statique et stable dans des matériaux compressibles élastiques-parfaitement plastiques. Ce type de champ a une structure comportant cinq secteurs. Lorsque tend vers 1/2, la solution à cinq secteurs dégénère en la solution à quatre secteurs relative au cas des matériaux imcompressibles.L'expansion de la perturbation procure une vision plus profonde de ce processus de dégénérescence, et donne des solutions analytiques approchèes pour le cas des matériaux compressibles.

     

Higher-order solution for near-tip fields of cracks growing in elastic perfectly-plastic compressible materials

The higher-order asymptotic solution of a quasi-static steadily propagating mode-I crack under the plane strain condition in an elastic perfectly-plastic compressible material is studied. In order to statisfy the higher-order compatibility equation for the rate of deformation in the centered fan sector, the stress near the crack tip is expanded asymptotically as an irregular logarithmic power series. The higher order terms near the crack tip were successfully derived. These higher order solutions are distinctly different from those for a stationary crack. The present solution for a growing crack is a one-parameter near-tip field based on a characteristic length A, through which the influence of loading and crack geometry enter into the near-tip field. This feature is substantiated by the numerical solution obtained by A.G. Varias and C.F. Shih. Comparisons between the analytic solution and the numerical results are presented.Presented at the Far East Fracture Group (FEFG) International Symposium of Fracture and Strength of Solids, 4–7 July 1994 in Xi'an, China.     

Elastodynamic near-tip fields for a rapidly propagating interface crack

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar isotropic elastic solids is investigated. Both anti-plane and in-plane motions are considered. The anti-plane displacements and the in-plane displacement potentials are sought in the separated forms $\text{r}{}^{\mathrm{q}}\text{F(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. The mathematical statement of the problem reduces to a second-order linear ordinary differential equation in θ, which can be solved analytically. Formulation of the boundary and interface conditions leads to an eigenvalue problem for the singularity exponent $\text{q}$. For the in-plane problem, root $\text{q}$ is found to be complex. Thus, the stresses exhibit violent oscillations within a small region around the crack tip, and the solutions have physical significance only outside this region. The angular stress distributions are plotted for various crack speeds, and it is found that at a high enough speeds the direction θ of maximum stress moves out of the interface. This result indicates that a running interface crack may move into one of the adjoining materials.     

T-stress of cracks loaded by near-tip tractions

T-stress solutions were derived for tractions acting on the crack-faces near a crack tip. Such solutions are of interest for the determination of the leading term of a weight function representation of T-stresses and the computation of an “intrinsic” T-stress for cracks growing in a material with a rising crack growth resistance. First, the type of the Green’s function for T-stresses is theoretically established. Then, results of finite element computations are reported for edge-cracked bars, DCB and CT specimens, which are suited for the determination of the first series term. As an application of the Green’s functions, the T-stresses caused by bridging interactions very close to the crack tip are computed.     

Condition for stable growth of branched cracks

In order to clarify the reason why the stable growth of branched cracks occurs in delayed failure, while not in other subcritical crack propagation process such as fatigue, the stress intensity factor after crack branching in delayed failure was dropped to various values, and the propagation behavior of both cracks was investigated.The well balanced growth of branched cracks in delayed failure occurs only when the crack propagation velocity after crack branching belongs to the region II where the crack propagation velocity is constant independently of $\text{K}$. The fatigue cracks at the tips of artificially branched cracks, on the other hand, can not propagate stably, and only either crack propagates preferentially.The exponent in the crack propagation law ($\text{d}\text{a/}\text{d}\text{t =}\text{c}{}_1\text{K}{}^{\mathrm{<mtext>m</mtext>}}\text{or}\text{d}\text{a/}\text{d}\text{N =}\text{c}{}_2\text{(ΔK)}{}^{\mathrm{<mtext>m</mtext>}}$) expresses the degree of unbalance growth of branched cracks. The stable growth of branched cracks occurs only when the crack propagation velocity is constant independently of $\text{K}$ or $\text{ΔK}$, i.e. $\text{m}\text{= 0}$.     

Plane strain stress intensity factors for branched cracks

A method of calculating stress intensity factors for branched and bent cracks embedded in an infinite body has been developed. The branches are always assumed to be sharp cracks and are modelled by dislocation distributions. The original crack may be either sharp or of elliptical cross-section with finite root radius. Hence, the method which has a precision ±2%, is also applicable to the study of crack branches emanating from elliptical holes and, approximately, also from notches. The following detailed calculations have been made assuming mode I loading: branched sharp crack with branches of equal and different length, bent sharp crack, and one and two crack branches emanating from the crack with a finite root radius. Bending of a sharp crack under mixed mode loading has also been studied. The criteria of maximum tensile stress and maximum energy release rate used in the study of direction of crack propagation are discussed.     

Continuous near-tip fields for a dynamic crack propagating in a power-law elastic-plastic material

By use of the J ₂ flow theory and the rectangular components of field quantities, the near-tip asymptotic fields are studied for a dynamic mode-I crack propagating in an incompressible power-law elastic-plastic material under the plan strain conditions. Through assuming that the stress and strain components near a dynamic crack tip are of the same singularity, the present paper constructs with success the fully continuous dominant stress and strain fields. The angular variations of these fields are identical with those corresponding to the dynamic crack propagation in an elastic-perfectly plastic material (Leighton et al., 1987). The dynamic asymptotic field does not reduce to the quasi-static asymptotic field in the limit as the crack speed goes to zero. This revised version was published online in July 2006 with corrections to the Cover Date.     

Two-parameter characterization of the near-tip stress fields for a bi-material elastic-plastic interface crack

A particular case of interface cracks is considered. The materials at each side of the interface are assumed to have different yield strength and plastic strain hardening exponent, while elastic properties are identical. The problem is considered to be a relevant idealization of a crack at the fusion line in a weldment. A systematic investigation of the mismatch effect in this bi-material plane strain mode I dominating interface crack has been performed by finite strain finite element analyses. Results for loading causing small scale yielding at the crack tip are described. It is concluded that the near-tip stress field in the forward sector can be separated, at least approximately, into two parts. The first part is characterized by the homogeneous small scale yielding field controlled by J for one of the interface materials, the reference material. The second part which influences the absolute value of stresses at the crack tip and measures the deviation of the fields from the first part can be characterized by a mismatch constraint parameter M. Results have indicated that the second part is a very weak function of distance from the crack tip in the forward sector, and the angular distribution of the second part is only a function of the plastic hardening property of the reference material.     

Surface effects on the near-tip stress fields of a mode-II crack

On the physical nature, most crack tips are not ideally sharp but have a small curvature radius. Both surface energy and crack-root curvature affect the stress and displacement fields in the vicinity of the crack tip. In the present paper, a numerical method, which incorporates the effect of surface elasticity into the finite element method, is employed to study the surface effects on the mode-II crack tip fields. It is found that when the curvature radius of the crack root decreases to micro-/nanometers, surface elasticity has a significant influence on the stresses near the crack tip. For a mode-II crack, surface effects alter both the magnitude and position of the maximum stresses, as is different from a mode-I crack, in which case only the stress magnitude is influence by surface stresses.     

Crack retardation equations for the propagation of branched fatigue cracks

The stress intensity factors (SIF) associated with branched fatigue cracks can be considerably smaller than that of a straight crack with the same projected length, causing crack growth retardation or even arrest. This crack branching mechanism can quantitatively explain retardation effects even when plasticity induced crack closure cannot be applied, e.g. in high R-ratio or in some plane strain controlled fatigue crack growth problems. Analytical solutions have been obtained for the SIF of branched cracks, however, numerical methods such as Finite Elements (FE) or Boundary Elements (BE) are the only means to predict the subsequent curved propagation behavior. In this work, a FE program is developed to calculate the path and associated SIF of branched cracks, validated through experiments on 4340 steel ESE(T) specimens. From these results, semi-empirical crack retardation equations are proposed to model the retardation factor along the crack path. The model also considers the possible interaction between crack branching and other retardation mechanisms.     

Elastodynamic near-tip fields for a crack propagating along the interface of two orthotropic solids

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar orthotropic elastic solids is solved numerically, for in-plane motion. The cartesian displacements are sought in the separated forms, $\text{r}{}^{\mathrm{p}}\text{U(θ)}$ and $\text{r}{}^{\mathrm{p}}\text{V(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. This reduces the mathematical statement of the problem to two complex second-order linear ordinary differential equations for complex functions $\text{U(θ)}$ and $\text{V(θ)}$. By means of the finite difference method, a matrix eigenvalue problem of the type $\text{ΣA}{}_{\mathrm{ij}}\text{(p)X}{}_{\mathrm{j}}\text{= 0}$, is obtained where $\text{A}{}_{\mathrm{ij}}\text{(p)}$ are polynomials of the complex variable $\text{p}$ and $\text{X}{}_{\mathrm{j}}$, are complex unknowns. An iterative numerical scheme for determining $\text{Im(p)}$ is developed and the roots $\text{p}$ as well as angular stress and displacement distributions are calculated and plotted for various material combinations. Comparison with exact solutions for the case of dissimilar Isotropie solids indicates good accuracy of the numerical solution. The orthotropic nature of the materials is shown to have a significant effect on stress maximums.     

Determination of the near-tip stress fields for the scattering of P-waves by a crack

Summary A conservation law implied by the field equations of linear elastodynamics is derived, and a procedure based on this conservation law is given for the direct determination of the near-tip stress fields arising from the scattering of normally incident P-waves by a crack in a homogeneous, isotropic elastic medium.     

The analytic solution of near-tip stress fields for perfectly plastic pressure-sensitive material under plane stress condition

Different from dense metals, many engineering materials exhibit pressure-sensitive yielding and plastic volumetric deformation. Adopting a yield criterion that contains a linear combination of the Mises stress and the hydrostatic stress, the analytic solutions of plane-stress mode I perfectly-plastic near-tip stress fields for pressuresensitive materials are derived. Also, the relevant characteristic fields are presented. This perfectly plastic solution, containing a pressure sensitivity parameter , is shown to correspond to the limit of low-hardening solutions, and when =0 it reduces to the perfectly plastic solution of near-tip fields for the Mises material given by Hutchinson [1]. The effects of material pressure sensitivity on the near-tip fields are discussed.     

Plane-strain mixed-mode near-tip fields in elastic perfectly plastic solids under small-scale yielding conditions

Within the context of the small-strain approach, plane-strain mixed-mode near-tip fields of a stationary crack in an elastic perfectly plastic Mises solid under small-scale yielding conditions are examined by finite element methods. Steady-state stress fields in the immediate vicinity of the crack tip are obtained as the remote loading of the elastic K-field increases. Asymptotic crack-tip solutions consisting of constant stress sectors, centered fan sectors, and an elastic sector are then constructed accordingly. The asymptotic crack-tip stress solutions agree well with the numerical results for a whole spectrum of mixed-mode loadings. Our mixed-mode near-tip solution with an elastic sector differs from that of Saka et al. by one (plastic) constant stress sector situated between the elastic sector and the neighbouring fan sector. The effect of the existence of the elastic sector on the near-tip fields is discussed in the light of the computational results. The plastic mixity factor of the near-tip field is given as a function of the elastic mixity factor of the prescribed K-field. This function is well bounded by that of the perfectly plastic limit of the corresponding solutions for power-law hardening materials given by Shih. Some issues pertaining to the numerical procedures such as the implementation of the small-scale yielding assumption are also addressed.     

Derivation of the near-tip stress and displacement fields for a moving crack without using complex functions

This note presents a mathematically simple approach to deriving the equations for the mixed-mode stress and displacement fields near the tip of a moving crack. Coordinate transformations allow the results to be derived without the use of complex functions.     

Influences of non-singular stresses on plane-stress near-tip fields for pressure-sensitive materials and applications to transformation toughened ceramics

In this paper, we investigate the effects of the non-singular stress (T stress) on the mode I near-tip fields for elastic perfectly plastic pressure-sensitive materials under plane-stress and small-scale yielding conditions. The T stress is the normal stress parallel to the crack faces. The yield criterion for pressure-sensitive materials is described by a linear combination of the effective stress and the hydrostatic stress. Plastic dilatancy is introduced by the normality flow rule. The results of our finite element computations based on a two-parameter boundary layer formulation show that the total angular span of the plastic sectors of the near-tip fields increases with increasing T stress for materials with moderately large pressure sensitivity. The T stress also has significant effects on the sizes and shapes of the plastic zones. The height of the plastic zone increases substantially as the T stress increases, especially for materials with large pressure sensitivity. When the plastic strains are considered to be finite as for transformation toughened ceramics, the results of our finite element computations indicate that the phase transformation zones for strong transformation ceramics with large pressure sensitivity can be approximated by those for elastic-plastic materials with no limit on plastic strains. When the T stress and the stress intensity factor K are prescribed in the two-parameter boundary layer formulation to simulate the crack-tip constraint condition for a single-edge notch bend specimen of zirconia ceramics, our finite element computation shows a spear shape of the phase transformation zone which agrees well with the corresponding experimental observation.     

Determination of the growth of branched cracks by numerical methods

In order to determine the growth of branched cracks in brittle materials, a static stress analysis for a branched crack model is performed by the finite element method. Assuming vanishing Mode II stress intensity factor as the governing criterion the growth is followed in several steps. Secondary branching is also analysed. In a quasi-dynamic analysis energy balance considerations are used to study growth of two branches of unequal lengths. It is shown that at low velocities the shorter branch will be rapidly arrested, while at very high velocities the two branches will continue to grow with nearly the same velocity.