Array Of Coplanar Cracks Research Articles

The beneficial effect of full or partial autofrettage on the combined 3-D stress intensity factors for inner coplanar crack arrays and ring cracks in a spherical pressure vessel

The distributions of the combined 3-D Stress Intensity Factor (SIF), KIN = KIP + KIA, due to both internal pressure and autofrettage along the front of coplanar crack arrays as well as ring cracks emanating from the bore of an overstrained spherical pressure vessel are evaluated. The 3-D analysis is performed using the finite element (FE) method employing singular elements along the crack front. A novel realistic autofrettage residual stress field incorporating the Bauschinger effect is applied to the vessel. The residual stress field is simulated using an equivalent temperature field in the FE analysis. Numerous coplanar crack array configurations are analyzed as well as ring cracks of various depths. SIFs distributions are evaluated for coplanar crack arrays of densities δ = 0–0.95, consisting of cracks of depth to wall thickness ratios of a/t = 0.05–0.4, and ellipticities of a/c = 0.2–1.0, prevailing in fully or partially autofrettaged spherical vessels of different geometries R0/Ri = 1.1, 1.2, and 1.7, bearing three levels of autofrettage (ε = 50%, 75%, and 100%). Furthermore, SIFs for inner ring cracks of various crack depth to wall thickness ratios of a/t = 0.025–0.6 prevailing in the same fully or partially autofrettaged spherical pressure vessels are also evaluated. In total, about two hundred different crack configurations are analyzed. A detailed study of the influence of the above parameters on the prevailing SIF is conducted. The results clearly demonstrate the favorable effect of autofrettage which may considerably reduce the prevailing effective stress intensity factor, thus delaying crack initiation and slowing down crack growth rate, and hence, substantially prolonging the total fatigue life of the vessel. Furthermore, the results emphasize the importance of properly accounting for the Bauschinger effect including re-yielding, as well as the significance of the three-dimensional analysis herein performed. Furthermore, it is shown that in some cases the commonly accepted approach that the SIF for a ring crack of any given depth is the upper bound to the maximum SIF occurring in an array of coplanar cracks of the same depth is not universal.

3-D Stress Intensity Factors due to Full Autofrettage for Inner Radial or Coplanar Crack Arrays and Ring Cracks in a Spherical Pressure Vessel

Three dimensional, Mode I, Stress Intensity Factor (SIF) distributions for radial or coplanar crack arrays as well as ring cracks emanating from the inner surface of an autofrettaged spherical pressure vessel are evaluated. The 3-D analysis is performed via the finite element (FE) method employing singular elements along the crack front. A novel realistic autofrettage residual stress field incorporating the Bauschinger effect is applied to the vessel. The residual stress field is simulated in the FE analysis using an equivalent temperature field. Numerous radial and coplanar crack array configurations are analyzed as well as ring cracks of various depths. SIFs distributions are evaluated for arrays of radial or coplanar cracks consisting of cracks of depth to wall thickness ratios of a/t=0.1-0.6, and ellipticities of a/c=0.2-1.0 prevailing in a fully autofrettaged spherical vessels, ε=100%, of different geometries R0/Ri=1.1, 1.2, and 1.7. SIFs are evaluated for radial arrays containing n=1-20 cracks, and for arrays of coplanar cracks of δ=0-0.95 densities. Furthermore, SIFs for inner ring cracks of various crack depth to wall thickness ratios of a/t=0.025-0.6 are also evaluated. In total, about three hundred different crack configurations are analyzed. A detailed study of the influence of the above parameters on the prevailing SIF is conducted. The results clearly demonstrate the favorable effect of autofrettage which may considerably reduce the prevailing effective stress intensity factor, thus delaying crack initiation and slowing down crack growth rate, and hence, substantially prolonging the total fatigue life of the vessel. Furthermore, the results emphasize the importance of properly accounting for the Bauschinger effect including re-yielding, as well as the significance of the three dimensional analysis herein performed. Furthermore, it is shown that in some cases the commonly accepted approach that the SIF for a ring crack of any given depth is the upper bound to the maximum SIF occurring in an array of coplanar cracks of the same depth is not universal.

3-D stress intensity factors due to full autofrettage for inner radial or coplanar crack arrays and ring cracks in a spherical pressure vessel

Three dimensional, stress intensity factor for inner radial or coplanar crack arrays and ring cracks prevailing in an autofrettaged spherical pressure vessel are evaluated. The 3-D analysis is performed via the finite element method implementing a novel realistic autofrettage residual stress field incorporating the Bauschinger effect. SIFs are evaluated for arrays of radial or coplanar cracks and ring cracks of depth to wall thickness ratios of a/t=0.1–0.6, and ellipticities of a/c=0.2–1.0 in fully autofrettaged spherical vessels, ε=100%, of Ro/Ri=1.1,1.2, and 1.7. SIFs are evaluated for radial arrays containing n=1–20 cracks, and for arrays of coplanar cracks of δ=0–0.95 densities.

Elastic waves at periodically-structured surfaces and interfaces of solids

This paper presents a simple treatment of elastic wave scattering at periodically structured surfaces and interfaces of solids, and the existence and nature of surface acoustic waves (SAW) and interfacial (IW) waves at such structures. Our treatment is embodied in phenomenological models in which the periodicity resides in the boundary conditions. These yield zone folding and band gaps at the boundary of, and within the Brillouin zone. Above the transverse bulk wave threshold, there occur leaky or pseudo-SAW and pseudo-IW, which are attenuated via radiation into the bulk wave continuum. These have a pronounced effect on the transmission and reflection of bulk waves. We provide examples of pseudo-SAW and pseudo-IW for which the coupling to the bulk wave continuum vanishes at isloated points in the dispersion relation. These supersonic guided waves correspond to embedded discrete eigenvalues within a radiation continuum. We stress the generality of the phenomena that are exhibited at widely different scales of length and frequency, and their relevance to situations as diverse as the guiding of seismic waves in mine stopes, the metrology of periodic metal interconnect structures in the semiconductor industry, and elastic wave scattering by an array of coplanar cracks in a solid.

Three-dimensional stress intensity factors for ring cracks and arrays of coplanar cracks emanating from the inner surface of a spherical pressure vessel

Certain spherical pressure vessels are composed of two hemispheres joined together by a girth weld. These vessels are susceptible to multiple cracking along the weld resulting in one or more cracks developing from the inner surface of the vessel and creating either a ring (circumferential) crack, or an array of coplanar cracks on the equatorial-weld plane. In order to assess the fracture endurance and the fatigue life of such vessels it is necessary to evaluate the Stress Intensity Factors (SIFs) distribution along the fronts of these cracks. However, to date, only two solutions for the SIF for an internal ring crack as well as two 3-D solutions for a single internal semi-elliptical crack prevailing in various spherical pressure vessels are available. In the present analysis, mode I SIF distributions for a wide range of ring, lunular, and crescentic cracks are evaluated. The 3-D analysis is performed, via the FE method employing singular elements along the crack front. SIFs for numerous ring cracks of different depths prevailing in thin, moderately thick, and thick spherical vessels are evaluated first. Subsequently, three-dimensional mode I SIF distributions along the crack fronts of a variety of lunular and crescentic crack array configurations are calculated for three spherical vessel geometries, with outer to inner radii ratios of R0/Ri=1.01, 1.1, and 1.7. SIFs are evaluated for arrays of density δ=0–0.99; for a wide range of crack-depth to wall-thickness ratios, a/t, from 0.025 to 0.95; and for various lunular and crescentic cracks with ellipticities, i.e., the ratio of crack-depth to semi-length, a/c, from 0.2 to 1.5. The obtained results clearly indicate that the SIFs are considerably affected by the three-dimensionality of the problem and by the following parameters: the crack density of the array-δ, the relative crack depth-a/t, crack ellipticity-a/c, and the geometry of the spherical vessel-η. Furthermore, it is shown that in some cases the commonly accepted approach that the SIF for a ring crack of any given depth is the upper bound to the maximum SIF occurring in an array of coplanar cracks, of the same depth, is not universal.

Guided interfacial waves at a periodic array of coplanar disbonds in a solid

This paper is concerned with guided elastic waves at a periodic array of coplanar disbonds in a solid, and the role they play in the scattering of bulk waves. It is approached as a mixed boundary condition problem pertaining to two elastic half spaces which are joined together within infinitely long regularly spaced strips, and unattached in between. It provides an approach to modeling wave interactions at an array of coplanar cracks, solids with slightly uneven surfaces which are in contact, stoping in mine operations where a narrow seam of a mineral is removed leaving regularly positioned pillars for roof support, and other analogous physical situations. In the analysis, a truncated Fourier series representation of the wave field is invoked, and the boundary conditions are applied at a discrete set of points within the repeat interval at the interface to determine the Fourier coefficients. The dispersion relation for interfacial waves is obtained within a one dimensional folded Brillouin zone scheme, and it is shown that there are branches associated with leaky interfacial waves, which at certain isolated points in wavevector space uncouple from the bulk wave continuum to exist as secluded supersonic interfacial waves. These observations are able to explain striking resonant features in the scattering of bulk waves at the interface.

Guided elastic waves at a periodic array of thin coplanar cavities in a solid

This paper is concerned with guided elastic waves at a periodic array of thin coplanar cavities in a solid and the role they play in the scattering of incident longitudinal and transverse waves in the plane perpendicular to the cavities. It is treated as a mixed boundary-condition plane strain problem pertaining to two elastic half spaces which are joined together within infinitely long regularly spaced strips and unattached in between. It provides an approach to modeling wave interactions at an array of coplanar cracks, partially bonded surfaces, mining stopes, and other analogous physical situations, and it is of relevance in the topical field of phononic crystals. The method of analysis brought to bear on this problem involves smoothing out the discontinuities in the boundary conditions, invoking a truncated Fourier series representation of the wave field and applying the boundary conditions at a discrete set of points within the repeat interval at the interface to determine the Fourier coefficients. The dispersion relation for interfacial waves is obtained, and it is shown how (in the supersonic domain with respect to bulk transverse waves) there are branches associated with leaky interfacial waves, which at certain isolated points in $k$ space uncouple from the bulk wave continuum to exist as secluded supersonic interfacial waves. These observations are able to explain striking resonant features in the scattering of bulk waves at the interface.

Influence of the Interface Periodic Array of Coplanar Cracks on the Free-Surface Oscillations

Let the boundary surface between two elastic media be a periodic array of coplanar cracks. The lower medium is a half space

Three-Dimensional Interaction Effects in an Internally Multicracked Pressurized Thick-Walled Cylinder—Part II: Longitudinal Coplanar Crack Arrays

In the first part of this paper, the interaction effects among many radial, internal, semi-circular, and semi-elliptical cracks in a pressurized, thick-walled vessel were quantified. In the present paper, the mode I stress intensity factor (SIF) distribution for numerous longitudinal coplanar, internal, semi-circular, and semi-elliptical arrays of surface cracks in an infinite, pressurized, thick-walled cylinder are evaluated. The 3-D analysis is performed by the finite element (FE) method and the submodeling technique, employing singular elements along the crack front. The effects of dense and sparse interacting longitudinal coplanar crack arrays on the SIFs are studied for a wide range of crack depth to wall thickness ratios, a/t, from 0.05 to 0.6; and, for various ellipticities of the crack, i.e., the ratio of the crack depth to semi-crack length, a/c, from 0.2 to 2.0. An analysis is performed to determine the influence of the three major parameters—crack density, crack ellipticity, and crack depth—on the interaction effects between adjacent cracks. The results clearly indicate that crack density, and, in some cases, ellipticity have opposing effects on the SIF of longitudinal crack arrays as compared to radial crack arrays. As a result of these contrasting behaviors, thick-walled cylinders having combined longitudinal and radial crack arrays would need further study.

SH-Waves in a Medium Containing a Disordered Periodic Array of Cracks

Reflection and transmission of an SH-wave by a disordered periodic array of coplanar cracks is investigated, and subsequently its application to the dispersion and attenuation of an SH-wave in a disorderedly cracked medium is also treated. This is a stochastic boundary value problem. The formulation largely follows Mikata and Achenbach (1988b). The problem is formulated for an averaged scattered field, and the governing singular integral equation is derived for a conditionally averaged crack-opening displacement using a quasi-crystalline-like approximation. Unlike our previous study (Mikata and Achenbach, 1988b) where a point scatterer approximation was used for the regular part of the integral kernel, however, no further approximation is introduced. The singular integral equation is solved by an eigenfunction expansion involving Chebyschev polynomials. Numerical results are presented for the averaged reflection and transmission coefficients of zeroth order as a function of the wave number for normal incidence, a completely disordered crack spacing, and various values of the ratio of crack length and average crack spacing. Numerical results are also presented for the dispersion and attenuation of an SH-wave in a disorderedly cracked medium.

Reflection and Transmission by a Periodic Array of Coplanar Cracks: Normal and Oblique Incidence

Scattering of in-plane elastic waves by a periodic array of coplanar cracks is investigated. This problem was first treated by Angel and Achenbach. The method of their solution was based on the Fourier series expansion of the Green-Lame potentials. In this paper, we have extended van der Hijden and Neerhoff’s treatment of scattering of in-plane elastic waves by a single crack to the periodic case. Major advantages of our treatment over Angel and Achenbach are: (1) the formulation is relatively simple, straightforward, and unified for both normal and oblique incidence, and (2) more importantly, there is no need for numerical integrations, because all the integrations involved in the formulation can be analytically evaluated thanks to the periodicity of the problem and the coplanarity of the cracks. Numerical results are presented and compared with those of Angel and Achenbach. Some new numerical results are also presented.

Role of residual stress field interaction in strengthening of particulate-reinforced composites

A quantitative analysis was conducted on the effect of residual thermoelastic stress concentrations on the strength of particle-reinforced brittle matrix systems. The analysis is derived from the stress intensity factor for a periodic array of coplanar cracks emanating from the matrix-particle interface. It is shown that the major drop in strength occurs at smaller volume fractions of second phase where the residual stress field interaction effects are minimal. The effect of volume fraction on strength becomes important at larger volume fractions (normally above 10–15%). The theory is compared with experimental measurements of strength for glass and alumina matrix composites as a function of the particle volume fraction, its size, and thermal mismatch Δα.