Interaction of a parabolic notch with a generalized singularity
Introduction
A notch is a common feature necessarily figuring in many mechanical components, possibly included for practical requirements such as component assembly (the retention of clips, for example) or power transmission, as in the spline. A knowledge of the stress field ahead of notches is essential during design and fatigue assessment. To understand the stress distribution near notches, great efforts have been put into studying this topic. There are two fundamental classes: ‘sharp’ and ‘smooth’ or ‘blunt’.
Considering, first, the sharp notch, Williams (1952) developed a method for expanding the Airy stress function as an eigenfunction series, presenting an asymptotic solution for the distribution of stress around a sharp V-notch. The notch stress intensity factors are related only to the first terms in the Williams’ solution. The factors can be thought of as the extension of the conventional stress intensity factors used for cracked components. But, the most important property of the solution found is that the eigensolutions provide very precise information about stress gradients and the spatial distribution of stresses; they therefore provide a lot of information about the neighbourhood in which cracks may potentially start, and it is straightforward to represent the environment present in a complicated prototype in a laboratory experiment to a very high degree of fidelity.
Turning, now, to blunt notches, Neuber (1958) dealt with the stress concentration for a blunt notch using the first term in a series expansion of Airy biharmonic potential functions. The simple stress concentration factor was adopted to directly characterize the stress field of blunt notches at a point (see, e.g., Creager & Paris, 1967; Savruk & Kazberuk 2006; Kim & Cho, 2012; Radaj, 2014; Bahrami et al., 2019). It seems that this approach is perceived to be simpler in practical engineering so that it has been widely employed (e.g., Kullmer & Richard, 2006; Gómez et al., 2007; (Livieri and Segala, 2007); Zappalorto & Lazzarin, 2014; Wang & Schiavone, 2021). It is attractive because of its simplicity, but a lot of information is lost by this approach. In particular, there is no information about the gradients of stress, nor their spatial distribution.
Approaching the problem from the opposite direction, by taking advantage of the Kolosov-Muskhelishvili complex variable formulae (Muskhelishvili, 1953), the sharp V-notch was investigated with finite representative complex terms by Carpenter (Carpenter, 1984a, 1984b, 1994, 1995). And, following Carpenter's efforts, by combining the Kolosov-Muskhelishvili approach with Neuber's conformal mapping (Muskhelishvili, 1977; Neuber, 1958), a substantial attempt to provide a unified approach to the analysis of cracks, sharp and blunt V-notches was proposed by Lazzarin and Tovo (1996). In order to improve the accuracy of the approach and its application, much further work has been done (see, e.g., Filippi et al., 2002; Lazzarin et al., 2011; Mirzaei et al., 2020). In these studies notch problems are treated by a stress function series expansion, or by employing a series to find asymptotic or approximate solutions.
The work described above represents very considerable progress in our understanding of the problems and, in particular, the progress made by the Italian research group is spectacular, but there are some basic challenges still remaining. Firstly, strictly speaking, the aforementioned analysis has been conducted in an asymptotic or approximate sense. It is hard to cover various load conditions, including the far-field loads which are always represented by generalized stress intensity factors, and the near-field loads such as force loads and displacement loads (dislocations, eigenstrain distribution) acting in the neighborhood of notches. Secondly, apart from some finite element simulations, damage such as crack initiation, crack propagation, or plastic deformation, occurring near the root of notches, due to the high stress concentration, is hard to model in terms of an explicit approach at present, but may reveal an important damage mechanism. These basic challenges demand a more rigorous and thorough analysis of blunt notch problems.
Once the geometry of a notch is specified, its external load can be always theoretically represented by force loads and displacement loads. Force loads can be expressed in terms of concentrated body forces through the superposition principle, while displacement loads can be represented by dislocations, which can also be used to simulate all kinds of damage problems. These indicate that a fundamental notch-singularity interaction model is needed to be established. Subsequently, by virtue of Green's function methods, a systematic and generic study of the stress field of the notch and its damage problems would be feasible. This is the basic motivation for the present work. As a first step, the parabolic notch is studied. The parabola is chosen here with two intentions: (i) the vicinity of a rounded notch tip can always be approximated by a parabola of appropriate curvature, and (ii) as an illustration, as the parabola and the approach developed in this study can be generalized to other smooth shapes.
To this end, in this study we develop a fundamental solution for a parabolic notch interacting with a generalized singularity. The singularity may represent either a concentrated force or an edge dislocation, or their various combinations which may lead to moment load, residual strain load and so on. The intended application of the results is two-fold; first, as a means of generating a wide range of equivalent remote loads in a simple and versatile way, and secondly, to provide the possibility of analyzing local notch root plasticity in a way which correctly tracks out the plastic flow rules.
Following this strategy, we use complex variable methods to derive closed-form solutions to the problem of a parabolic notch interacting with the generalized singularity. The basic geometry we will consider is shown in Fig. 1, in which, the generalized singularity is arbitrarily located within the infinite notched solid.
The procedure followed is outlined as follows: first, the Kolosov-Muskhelishvili formula and the generalized singularity are presented in §2, and the basic model is formulated in §3. The solution for the interaction of a parabolic notch and a generalized singularity is derived in § 4. Subsequently, in §5, as an example, the analytical solution for the parabolic notch interacting with an edge dislocation is analyzed, and the force present on the dislocation is derived. Finally conclusions are drawn in §6.
Section snippets
The Kolosov-Muskhelishvili complex framework and its form for a generalized singularity
In this section, the Kolosov-Muskhelishvili complex potential and boundary condition are introduced. The general singularity model is introduced which provides basic concept for next analysis, and then complex variable formula for in-plane problems is presented.
problem
Consider a parabolic notch with a surface Γ which extends to infinity, as shown in Fig. 1, where a generalized singularity is located in the substrate material. The parabolic surface can be described by the following function in a polar coordinate set as set out by (Zappalorto et al., 2008). where h and H are constant numbers, and H has a length dimension. Its corresponding expression in rectangular coordinate system is
This implies that any parabolic notch
Solution
In this section, some integral formulae will be manipulated, in a form which is of general application to notch problems. By virtue of these identities, Eq. (3.10) will be solved.
Example: Analytical solution for the parabolic notch interacting with an edge dislocation
In §4, we have obtained the general solution. In this section, we take the singularity to be an edge dislocation, as shown in Fig. 5 as an example to demonstrate explicitly the stress field. It might be an actual edge dislocation, so that the solution gives information about the state of stress induced. On the other hand, it might be used as the kernel to model, for example, a crack emanating from the stress raiser.
For simplicity, but without loss of the generality, we here set which would
Concluding remarks
In this study, through the complex conformal mapping technique, the analytical solution for a parabolic notch in a plane infinite solid, interacting with a generalized singularity has been derived. The generalized singularity may be thought of as representing either an edge dislocation or a concentrated force. To demonstrate its application and verify its validation, the influence of a parabolic notch on a nearby edge dislocation has been investigated in detail. The driving force on the
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that cdocould have appeared to influence the work reported in this paper.
Acknowledgements
The support of the Sir Joseph Pope Fellowship from Nottingham University is much appreciated. This work was also partially supported by National Natural Science Foundation of China (grant no. 12072254).
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