Abstract
We consider the elastic stress near a hole with corners in an infinite plate under biaxial stress. The elasticity problem is formulated using complex Goursat functions, resulting in a set of singular integro-differential equations on the boundary. The resulting boundary integral equations are solved numerically using a Chebyshev collocation method which is augmented by a fractional power term, derived by asymptotic analysis of the corner region, to resolve stress singularities at corners of the hole. We apply our numerical method to the test case of the hole formed by two partially overlap** circles, which can include either a corner pointing into the solid or a corner pointing out of the solid. Our numerical results recover the exact stress on the boundary to within relative error \(10^{-3}\) for modest computational effort.
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References
Muskhelishvili NI (1953) Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen
Kolosoff G (1914) On some properties of problems in the plane theory of elasticity. Z Math Physik 62:384–409
Savin GN (1970) Stress distribution around holes. NASA, Washington, D.C
Pan Z, Cheng Y, Liu J (2013) Stress analysis of a finite plate with a rectangular hole subjected to uniaxial tension using modified stress functions. Int J Mech Sci 75:265–277
Motok MD (1997) Stress concentration on the contour of a plate opening of an arbitrary corner radius of curvature. Mar Struct 10:1–12
Ling CB (1948) The stresses in a plate containing an overlapped circular hole. J Appl Phys 19:405–411
Driscoll TA, Trefethen LN (2002) Schwarz–Christoffel map**. Cambridge University Press, Cambridge
Di Carlo A, Gurtin ME, Podio-Guidugli P (1992) A regularized equation for anisotropic motion-by-curvature. SIAM J Appl Math 52:1111–1119
Golovin AA, Davis SH, Nepomnyashchy AA (1998) A convective Cahn–Hilliard model for the formation of facets and corners in crystal growth. Physica D 122:202–230
Gurtin ME (1993) Thermomechanics of evolving phase boundaries in the plane. Oxford University Press, New York
Siegel M, Miksis MJ, Voorhees PW (2004) Evolution of material voids for highly anisotropic surface energy. J Mech Phys Solids 52:1319–1353
Williams ML (1952) Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J Appl Mech 19:526–528
Chiu CH (2020) The model of eye-shaped voids: Elasticity solution and its applications in material failures via morphological transformation. J Mech Phys Solids 137:103822
Wu CH (1982) Unconventional internal cracks, part 1: symmetric variations of a straight crack. J Appl Mech 49:62–68
Burton WK, Cabrera N, Frank FC (1951) The growth of crystals and the equilibrium structure of their surfaces. Philos Trans R Soc A 243:299–358
Cabrera N (1964) The equilibrium of crystal surfaces. Surf Sci 2:320–345
Srolovitz DJ, Davis SH (2001) Do stresses modify wetting angles? Acta Mater 49:1005–1007
Mikhlin SG (1957) Integral Equations. Pergamon, London
Gonzalez O, Stuart AM (2008) A first course in continuum mechanics. Cambridge University Press, Cambridge
Soutas-Little RW (1999) Elasticity. Dover, New York
Herring C (1951) Some theorems on the free energies of crystal surfaces. Phys Rev 82:87–93
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in Fortran 77: the art of scientific computing, vol 2. Cambridge University Press, Cambridge
Boyd JP (2001) Chebyshev and Fourier spectral methods. Dover, New York
Golub GH, Welsch JH (1969) Calculation of Gauss quadrature rules. Math Comput 23:221–230
Bremer J, Gimbutas Z, Rokhlin V (2010) A nonlinear optimization procedure for generalized Gaussian quadratures. SIAM J Sci Comput 32:1761–1788
Hoskins JG, Rokhlin V, Serkh K (2019) On the numerical solution of elliptic partial differential equations on polygonal domains. SIAM J Sci Comput 41:A2552–A2578
Clenshaw CW (1955) A note on the summation of Chebyshev series. Math Comput 9:118–120
Knops RJ, Payne LE (1971) Uniqueness theorems in linear elasticity. Springer, New York
Shampine LF (2008) Vectorized adaptive quadrature in MATLAB. J Comput Appl Math 211:131–140
Epperson JF (1987) On the Runge example. Am Math Mon 94:329–341
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We thank Jeremy Hoskins for his helpful discussion on numerical aspects of this work.
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Appendix A: Asymptotic analysis near the corner for overlap** circles case.
Appendix A: Asymptotic analysis near the corner for overlap** circles case.
Here we derive the asymptotic behavior of the stress near the corner for the overlap** circle case of Sect. 4.5. The trace of the stress tensor for the overlap** circles case is given by Eq. (63) which we write as
where
with
and
Here, the polar angle from the center of the circle \(\gamma \), polar angle \(\theta \), amount of overlap between the circles \(\alpha \), tensions \(N_{1}\) and \(N_{2}\) are defined as in Sect. 4.4. We use \(\alpha =2\pi /3\) as an example in this Appendix. The result can be generalized to other \(\alpha \) in \(\pi /2<\alpha <\pi \) using a similar approach as described here.
The corner location is at \(\theta =\pi /2\). As \(\theta \rightarrow \pi /2\), \(\gamma \rightarrow \pi /2+\arcsin (\cos \alpha )\). Thus,
It follows from (70) that
which gives \(\xi \rightarrow \infty \) as \(\theta \rightarrow \pi /2\). Thus the \(\cos (s\xi )\) term in integral (67) is highly oscillatory near the corner which may cause inaccuracy of the numerical integration.
We determine the behavior of the integral near the corner by applying asymptotic analysis. Let \(\theta =\pi /2-\varepsilon \) where \(0<\varepsilon \ll 1\), then \(\cos \theta =\sin \varepsilon =\varepsilon -\varepsilon ^{3}/6+{\mathcal {O}}(\varepsilon ^{5})\) and \(\sin \theta =\cos \varepsilon =1-\varepsilon ^{2}/2+{\mathcal {O}}(\varepsilon ^{2})\). By Eq. (71), we have
Thus we have
Substitute Eq. (75) into Eq. (70), the asymptotic approximation of \(\cosh \xi \) is
Thus,
Now consider the integral in (67). F has properties \(F(-s)=F(s)\), \(F(s)\rightarrow 0\) as \(s\rightarrow \pm \infty \) and \(F(s,\alpha )\) is bounded for all s (\(s=0\) is a removable singularity), so we can rewrite the integral on \(-\infty<s<+\infty \) as
We evaluate I by considering the contour integral on complex plane \(z=s+\text {i}t\) as follows:
then
where C is the line along s axis from \(-R\) to \(-R\). Note \(F(z,\alpha )\) is analytic in upper half-plane \(\text{ Im }(z)>0\) except at zeros of the denominator located by the roots of
There are infinitely many singularities of \(F(z,\alpha )\) on the upper half-plane (see Fig. 21). We claim that since \({\tilde{I}}\) contains \(\text {e}^{\text {i}\xi z}=\text {e}^{-\xi t}\text {e}^{\text {i}\xi s}\) and \(\xi \gg 1\), the integral \({\tilde{I}}\) can be approximated by the contribution from the residual of the singularity that occurs at the location with smallest imaginary part t in the upper half-plane. For \(\alpha =2\pi /3\), as shown in Fig. 21, this first singularity lies on the imaginary axis, and so satisfies \(s=0\) and
For \(\alpha =2\pi /3\) the solution of Eq. (82) is \(t_1^*\approx 0.6157\) and the first singularity is at \(z=\text {i}t_1^*\). We construct a rectangular contour on the complex plane with the first singularity inside as in Fig. 22 to evaluate the integral (79), where C is the line segment from \(z=-R\) to \(z=R\), \(\varGamma _1\) is the line segment from \(z=R\) to \(z=R+\text {i}t_2\), \(\varGamma _2\) is the line segment from \(z=R+\text {i}t_2\) to \(z=-R+\text {i}t_2\), and \(\varGamma _3\) is the line segment from \(z=-R+\text {i}t_2\) to \(z=-R\). The height of the rectangle is defined by \(t_2=\pi /(2\alpha )>t_1^*\) such that only the first singularity is inside the contour. By the residue theorem and taking the limit \(R\rightarrow \infty \),
Now consider the integrals on \(\varGamma _1\), \(\varGamma _2\), and \(\varGamma _3\). First, on \(\varGamma _1\)
We can bound |F| in (84) using following inequalities
Then
Thus,
For the same reason,
For the \(\varGamma _2\) contour
All three integrals in the square brackets of Eq. (92) are finite, so the bound on the integral on contour \(\varGamma _2\) is of order \(\text {e}^{-\xi t_2}\) for \(\xi \gg 1\).
Finally, consider the residue from the singularity:
From Eq. (77), \(\xi \gg 1\) near the corner. Then the integral on \(\varGamma _2\) is asymptotically smaller than the residue at \(z=\text {i}t_1^*\) because \(\text {e}^{-\xi t_2}\ll \text {e}^{-\xi t_1^*}\). Thus the dominant asymptotic contribution to the integral is
Using this result in Eq. (67) we obtain
where \(t_{1}^{*}\) is the smallest non-zero root of Eq. (82). Recalling that \(\varepsilon =\pi /2-\theta \) is the proximity to the corner, since \(t_{1}^{*}<1\), \(\sigma _{x}+\sigma _{y}\) has an integrable singularity at the corner. Note the exponent \(1-t_1^*\) matches the exponent for the singular solutions for an infinite wedge geometry [12], as Eq. (82) is equivalent to Eq. (30). More generally, for \(\pi /2<\alpha <\pi \) (cases like Fig. 13) there is an integrable singularity with \(1<t_1^*<3/2\), and for \(0<\alpha <\pi /2\) (cases like Fig. 9) there is no singularity because \(0<t_1^*<1\).
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Wang, W., Spencer, B.J. Numerical solution for the stress near a hole with corners in an infinite plate under biaxial loading. J Eng Math 127, 13 (2021). https://doi.org/10.1007/s10665-021-10104-8
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DOI: https://doi.org/10.1007/s10665-021-10104-8