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Journal ArticleDOI

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
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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Citations
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Journal ArticleDOI
Paul J. Riccomini1Institutions (1)
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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77 citations


Journal ArticleDOI
Xin Wang1, Grzegorz Glinka2Institutions (2)
Abstract: This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new 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. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).

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30 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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Journal ArticleDOI
John M Emery1, Jacob D. Hochhalter2, Paul A. Wawrzynek3, Gerd Heber2  +1 moreInstitutions (3)
Abstract: Current tools for fatigue life prediction of metallic structural components are limited in some or all of the following capabilities: geometry of, and boundary conditions on, the affected structural component, automation of the simulation process, randomness of the primary variables, and physics of the damage evolution processes. DDSim, a next-generation damage and durability simulator, addresses each of these limitations with a hierarchical, multiscale, “search and simulate” strategy. This hierarchical strategy consists of three levels. Level I, described in this paper, performs an 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. Initial flaw size can be specified deterministically, or generated randomly from statistical descriptions of the microstructure and used in Monte Carlo simulation. The result is a scalar field of predicted life over the entire domain of the structure. The benefits of the Level I analysis include a high degree of automation, solution speed, and easy implementation of high performance parallel computing resources. A validation case study of Level I is described. Levels II and III are outlined herein, but will be described in further detail in subsequent papers. The Level II analysis uses FRANC3D to accurately predict the number of cycles consumed by microstructurally large crack growth processes. Level III performs multiscale analyses to accurately predict the cycles consumed in microstructurally small crack growth processes.

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28 citations


Journal ArticleDOI
01 Apr 2020-Spe Journal
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.

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25 citations



References
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Journal ArticleDOI
James R. Rice1Institutions (1)
Abstract: It is shown that if the displacement field and stress intensity factor are known as functions of crack length for any symmetrical load system acting on a linear elastic body in plane strain, then the stress intensity factor for any other symmetrical load system whatsoever on the same body may be directly determined. The result is closely related to Bueckner's (1970) weight function, through which the stress intensity factor is expressed as a sum of work-like products between applied forces and values of the weight function at their points of application. An example of the method is given wherein the solution for a crack in a remotely uniform stress field is used to generate the expression for the stress intensity factor due to an arbitrary traction distribution on the faces of a crack. A corresponding theory is developed in an appendix for three-dimensional crack problems, although this appears to be directly useful chiefly for problems in which there is axial symmetry.

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873 citations


1


"Weight function for an elliptic cra..." refers background or methods in this paper

  • ...Rice (1972) independently showed that weight functions could be determined by differentiating known elastic displacement field solutions with respect to crack length....

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  • ...Foundations of the three dimensional weight function theory were given independently by Rice (1972) in the Appendix of that publication and by Bueckner (1973) in a review....

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Journal Article
Abstract: A state of plane strain in a notched or cracked elastic domain under the action of boundary tractions is considered. It is shown that the stress intensity factor K at a root of a notch can be re presented in the form of a weighted average of the tractions, and that the weight functions involved can be derived from the boundary displacements of two special stress fields, each of which is characterized by a "fundamental singularity" at the root and by the absence of externally impressed forces.

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834 citations


"Weight function for an elliptic cra..." refers background in this paper

  • ...The concept was first introduced by Bueckner (1970) for two-dimensional cracked bodies....

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Book
01 Jan 1980-

829 citations


"Weight function for an elliptic cra..." refers background or methods in this paper

  • ...Now using the following relations (see Gradshteyn and Ryzhik, 1980 )...

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  • ...Here it may be noted that the only contributing term in the weight function is W 0 0 . All other terms contribute zero either due to the orthogonal property of sine and cosine functions, or due to orthogonality of Jacobi’s polynomials or due to the relation ( Gradshteyn and Ryzhik, 1980 )...

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  • ...respect to between 0 and 1, one obtains after making use of the orthogonality property (see Gradshteyn and Ryzhik, 1980 ):...

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Book ChapterDOI
J. F. Knott1Institutions (1)
01 Jan 1983-
Abstract: Calculations of the theoretical strengths of crystalline solids are usually based on idealised forms of atomic force-displacement curves, in which the force is defined as the differential with respect to distance of the inter-atomic or inter-ionic energy. The energy curve is similar to that for a diatomic molecule in that it represents the resultant of inter-atomic repulsions and attractions; the nature and strength of the attractive forces depending on the bond type: ionic, covalent, metallic, or Van der Waals. Differences in character between a lattice and a molecule occur at separations of order one lattice spacing, when Friedel oscillations in the screening charge cause the long range component of the interaction potential in the lattice to undergo a damped oscillation about zero. For small displacements, the atomic force/displacement curve is linear, having a slope equivalent to Young’s modulus, E. A lattice also has shear stiffness, denoted by the shear modulus μ. Both E and μ are defined macroscopically, usually for randomly-oriented polycrystals which are assumed to be isotropic. In single crystals, both the tensile stiffness and the shear stiffness vary with the orientation of the crystal with respect to the tensile axis and these variations can be substantial: in iron, for example, the minimum value of E is in the [100]direction (132 GPa at room temperature) and the maximum value is in the [111] direction (260 GPa).

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671 citations


Journal ArticleDOI
M. K. Kassir, G.C. Sih, James R. Rice1Institutions (1)

371 citations


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No. of citations received by the Paper in previous years
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20191
20161
20154
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20132