Stress distribution around polygonal holes
Highlights
► Muskhelishvili's complex variable approach is adopted. ► The infinite isotropic plates are considered. ► The effect of hole geometry and orientation and load angle is studied. ► The arbitrary biaxial loading condition adopted for biaxial loading.
Introduction
In engineering structures, different types of holes and openings are made to satisfy some service requirements. These openings and cutouts result in strength degradation of the structure and may lead to failure. In order to predict the behaviour of the structure with such holes, it is essential to study the effect of hole geometry and loading conditions on the stress distribution around the openings.
The research on the stress distribution around holes is going on since the first solution for stress distribution around circular hole is presented by Kirsch [1] using real variables. Inglis [2] extended Kirsch's [1] work by obtaining stress field around elliptical hole. His results were exact and suitable for extreme limits of shape which an ellipse can take. However, the solution presented by Inglis [2] is difficult to apply particularly when the tip of crack is sharp. Westergaard [3] derived stress functions in terms of harmonic function to handle the stresses near crack tip. The stresses derived from Westergaard's [3] stress function satisfies equilibrium conditions, compatibility conditions and generalized Hooke's law.
Muskhelishvili [4] introduced complex variable approach to theory of elasticity. Almost independently of Muskhelishvili [4], Stevenson [5] developed an approach of two-dimensional isotropic elastic theory using complex variables. The nature of the complex potentials, the conditions for solution of the displacements and formulae for force and couple resultants are discussed.
Savin [6] and Lekhnitskii [7] used Muskhelishvili's complex variable approach [4] and found stress distribution around various shaped holes in isotropic as well as anisotropic plates. Savin [6] derived stress functions by evaluating Schwartz integral, while Lekhnitskii [7] chose series approach. Ukadgaonker and Awasare [8], [9], [10], [11] also presented solutions for stress field around circular [8], elliptical [9], triangular hole with rounded corners [10] and rectangular hole with rounded corners [11] in infinite isotropic plate, using Muskhelishvili's complex variable approach [4]. Simha and Mohapatra [12] used generalized mapping function to find the stress distribution around various shaped holes in an infinite isotropic plate. Rezaeepazhand and Jafari [13] gave the stress distribution around several non-circular cutouts in isotropic metallic plates using Lekhnitskii's solution [7]. They studied the effect of cut-out shape (triangle, square and pentagon with rounded corner), bluntness, load direction and cut-out orientation, on the theoretical stress concentration factor.
The solutions for stresses around triangular hole [14], [15], [16], [17], elliptical and triangular hole [18] in anisotropic plates with in-plane loading at infinity are also available. By choosing suitable constants of anisotropy, these solutions give very good results also for isotropic materials. These solutions are capable of handling uniaxial and biaxial in-plane loading at infinity. In order to avoid superposition of solutions of two uni-axial loading problems to solve a problem of plate with biaxial loading at infinity, a biaxial loading factor combined with an arbitrary orientation angle was used in [15], [16], [17], [18], as previously suggested by Gao [19]
The plane elasticity problem for edge notches in semi-infinite plate is addressed by Bowie [20]. The solution for stress field around discontinuities may encounter singularities. These singularities depend upon geometry of the notch and the boundary conditions. The boundary conditions decide the degree of singularity whereas the notch geometry and the applied load control the intensity of the singular stress field [21].
Lazzarin and Tovo [22], Filippi et al. [23] and Zappalorto and Lazzarin [24] presented solution for stress field near notches. Lazzarin and Tovo [22] proposed a unified approach to the analysis of linear elastic stress fields in the neighbourhood of cracks and notches. Previous solutions reported in the literature (as those due to crack [3], pointed V-notch [25], blunt crack [26], and blunt notch [27]) could be derived as special cases of that more general analytical frame. With the aim to increase the degree of accuracy in the case of blunt V-notches with large opening angles, Filippi et al. [23] gave more flexibility to the solution by enriching the potential function arrangement. Zappalorto and Lazzarin [24] have determined stress distribution for V-notches with end hole under mode I, mode II and mode III loading and calculated the relevant notch stress intensity factors. The Kolosov–Muskhelishvili approach was used to solve the in-plane elastic problem whereas the out-of-plane problem was solved by means of a specific holomorphic function.
The closed form solutions for the stress fields near a semi-elliptic circumferential notch [28], circumferential hyperbolic and parabolic notches [29], circumferential U- and blunt V-shaped notches [30], inclined notches and shoulder fillets [31] in axisymmetric shafts under torsion loading are also available in the literature.
Kolosov–Muskhelishvili's complex variable approach is useful and handy tool to study two dimensional stress analysis problem. Here, an attempt is made to find generalized solution for determining stress distribution around polygonal hole geometries like triangular, square, pentagonal, hexagonal, heptagonal and octagonal shaped holes in an infinite plate, under different types of loading conditions.
Section snippets
Mapping function
In order to find stress distribution around a hole in z-plane, the area outside the polygonal hole is mapped to a region outside unit circle in ζ-plane, which has origin at ζ=0 using Christoffel–Schwartz mapping function given below [6].
The constants R(R>0) and C1 defines size and location of the cut-out, respectively. C1 is dropped hereon as infinite plate is considered. n is number of sides of the polygon. aj are the prevertices corresponding to the vertices of
Stress functions
The basic equations of plane elasticity in complex variable form are given by Kolosov–Muskhelishvili [4] as follows:where ϕ(z),ψ(z) are the complex potentials of the complex variable z=x+iy, ϕ′(z),ψ′(z) are the first derivative of the complex potentials and ϕ″(z),ψ″(z) are the second derivative of the complex potentials.
The load is applied on the plate in such a way that the resultants are in the mid-plane XOY and the surfaces other
Results and discussion
The generalized stress functions obtained above are coded and numerical results are obtained for different loading conditions and hole geometries. The material properties taken in numerical solution are: E=200 GPa, G=80 GPa and ν=0.25.
The biaxial loading factor (λ) adopted in this work facilitate different types of loading conditions. The load angles (γ) are measured from positive x-axis.
The results discussed below include, the following type of loading conditions:
- 1.
Uniaxial tension at infinity (λ
Conclusions
The stress field around polygonal shaped cutouts in infinite isotropic plate is presented using Mushkhelishvili's complex variable approach. The effect of cutout shape, corner radius, load angle and hole orientation on stress pattern is studied for triangular, square, pentagonal, hexagonal, heptagonal and octagonal cutout shapes. The following points are concluded:
- 1.
The arbitrary biaxial loading condition adopted here facilitates implementation of any biaxial/shear loading at infinity.
- 2.
By
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