Weight function for an elliptic crack in an infinite medium. I. Normal loading
01 Jun 2000-International Journal of Fracture (Kluwer Academic Publishers)-Vol. 103, Iss: 3, pp 227-241
TL;DR: In this paper, an integral equation method has been used to derive the crack opening displacement of an elliptic crack in an infinite elastic medium subjected to a concentrated pair of point force loading at an arbitrary location on the crack faces.
Abstract: A recently developed integral equation method has been used to derive the crack opening displacement of an elliptic crack in an infinite elastic medium subjected to a concentrated pair of point force loading at an arbitrary location on the crack faces. These results have been used to obtain the stress intensity factor along the elliptic crack front which corresponds to the weight function for an elliptic crack under normal loading. Analytical expression of the weight function can be used to derive the stress intensity factor for both polynomial loading as well as non-polynomial loading.
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TL;DR: The authors investigated 90 elementary teach- ers' ability to identify two systematic error patterns in subtraction and then prescribe an instructional focus and found that more than half of the teachers chose to address basic subtraction facts first during instruction regardless of error type.
Abstract: The present study investigated 90 elementary teach- ers' ability to identify two systematic error patterns in subtraction and then prescribe an instructional focus. Presented with two sets of 20 completed subtraction problems comprised of basic facts, computation, and word problems representative of two students' math performance, participants were asked to examine each incorrect subtraction problem and describe the errors. Participants were subsequently asked which type of error they would address first during math instruction to correct students' misconceptions. An analysis of the data indicated teachers were able to describe specific error patterns. However, they did not base their instruc- tional focus on the error patterns identified, and more than half of the teachers chose to address basic subtraction facts first during instruction regardless of error type. Limitations of the study and implications for practice are discussed.
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TL;DR: In this paper, a coupled finite-volume (FV)/finite-area (FA) model was proposed to simulate the propagation of multiple hydraulically driven fractures in two and three dimensions at the wellbore and pad scale.
Abstract:
This paper presents the formulation and results from a coupled finite-volume (FV)/finite-area (FA) model for simulating the propagation of multiple hydraulically driven fractures in two and three dimensions at the wellbore and pad scale. The proposed method captures realistic representations of local heterogeneities, layering, fracture turning, poroelasticity, interactions with other fractures, and proppant transport. We account for competitive fluid and proppant distribution between multiple fractures from the wellbore. Details of the model formulation and its efficient numerical implementation are provided, along with numerical studies comparing the model with both analytical solutions and field results. The results demonstrate the effectiveness of the proposed method for the comprehensive modeling of hydraulically driven fractures in three dimensions at a pad scale.
40 citations
TL;DR: In this paper, a general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks for the calculation of stress intensity factors in embedded elliptical cracks.
35 citations
Cites background from "Weight function for an elliptic cra..."
...Several attempts have been made to derive the weight functions for embedded elliptical crack by solving the problem analytically, see recent development in [13,14]....
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TL;DR: DDSim as mentioned in this paper is a next-generation damage and durability simulator with a hierarchical, multiscale, "search and simulate" strategy, which consists of three levels: initial, reduced order, conservative screening, based on a linear finite element analysis of the uncracked component, to determine the most life-limiting locations for intrinsic material flaws.
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References
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TL;DR: In this paper, the surface and the deepest point of a semi-elliptical crack in a finite thickness plate were derived from a general weight function and two references stress intensity factors.
162 citations
"Weight function for an elliptic cra..." refers background in this paper
...Shen and Glinka (1991) derived weight functions for the surface and the deepest point of a surface semi-elliptical crack in a finite thickness 2 T.K....
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TL;DR: In this article, the relevant weight functions of all three modes are represented in closed form with the aid of elementary transcendentals, and their derivation as displacements of fundamental fields is simple and without recourse to integral transforms of any kind.
155 citations
"Weight function for an elliptic cra..." refers methods in this paper
...Bueckner (1987) applied an ingenious method to obtain the analytical form of the weight function for a circular crack under normal and shear loading....
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01 Jan 1989
144 citations
133 citations
"Weight function for an elliptic cra..." refers methods in this paper
...present problem. The method based on solution of Laplace’s equation in terms of spheroidal coordinates as used extensively by Vijayakumar and Atluri (1981) and Kassir and Sih (1975) for elliptical crack problem, too cannot be used for the present problem....
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TL;DR: In this article, the authors apply the method developed by Rice [1] to solve the elasticity problems of somewhat circular planar tensile cracks under arbitrary load distributions, based on a known solution for the stress intensity factor along a circular crack due to a pair of wedgeopening point forces on its surfaces.
Abstract: In this paper we apply the method developed by Rice [1], of solving for the elastic field of a crack with a front perturbed from some reference shape, to solve the elasticity problems of somewhat circular planar tensile cracks under arbitrary load distributions. The method is based on a known solution for the stress intensity factor along a circular crack due to a pair of wedge-opening point forces on its surfaces. A full solution, accurate to first order in the deviation from a circular shape, is derived for the stress intensity factor and the crack opening displacement distributions. The results of a perturbation in a harmonic wave form suggest that a circular crack, under axially symmetric loading, can be configurationally unstable (not grow as a circle) for loadings that increase in intensity with distance from the center. Circular cracks with harmonic shape perturbations are found to have the same form of variation of the stress intensity factor with arc length along the crack edge (to first order accuracy) as found in previous work for a half plane crack. As a test case for the perturbation solution, an elliptical planar tensile crack under uniform tension is viewed as being perturbed from a circular crack. Results derived from the perturbation formulae through numerical evaluation are compared with the exact solutions existing in the literature. The perturbation results show a very good match with the exact solutions even when the semi-axis lengths of the elliptical crack differ by a factor of two (and by as much as a factor of three when special choices of the reference circular crack location are made). This suggests that the perturbation procedure presented here, while theoretically exact only to first order, can be used to produce acceptable results for some planar cracks whose shapes deviate appreciably from a circle.
128 citations
"Weight function for an elliptic cra..." refers methods in this paper
...Gao and Rice (1987) obtained the weight function for a circular crack under normal loading satisfying various boundary conditions at infinity using the results of Rice (1985)....
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