Spherical Cavity in an Infinite Solid under Uniaxial Tension

Stress concentration at a spherical cavity (a pore, void or casting defect) in an infinite elastic solid under remote uniaxial tension, from the closed-form solution of Southwell and Gough (1926) and Goodier (1933): the peak stress on the equator, the compressive stress at the poles, their dependence on Poisson's ratio, and the decay of the axial stress along the equatorial plane; compared with the plate hole (Kt = 3).

  • Closed-form 3D solution: Kt = (27 − 15ν)/(14 − 10ν) at the equator, about 2.05 for ν = 0.3

  • Compressive stress at the poles and the stress along the equatorial radius

  • Comparison with the two-dimensional round hole

Cavity, Material and Load

Unit System: /

Convention

Spherical cavity of radius a in a solid under remote tension S along z; point P on the equatorial plane

Inputs





Spherical Cavity
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Spherical Cavity in an Infinite Solid

Spherical Cavity in an Infinite Solid under Uniaxial Tension

Problem Statement

An infinite isotropic elastic solid contains a spherical cavity of radius a (a pore, a gas void, a casting defect) and carries a remote uniaxial tension S along z. Wanted are the stress concentration on the cavity surface and the decay of the stress away from it.

Spherical cavity under uniaxial tension

Key Assumptions

Infinite solid, linear elastic isotropic, traction-free cavity, uniaxial remote stress. Biaxial or triaxial remote states superpose (hydrostatic tension gives 1.5S all over the surface); an ellipsoidal cavity, a cavity near a free surface or a cluster of pores are different problems.

Solution

Southwell and Gough (1926) and Goodier (1933) solved the problem in spherical coordinates with the Boussinesq–Papkovich displacement potentials. On the cavity surface the axial stress is largest on the equator, the circle in the plane normal to the load,

Kt = (27 − 15ν) / [2(7 − 5ν)]    (1.93 for ν = 0, 2.05 for ν = 0.3, 2.17 for ν = 0.5),

and the poles on the load axis are in compression, σ = −(3 + 15ν)S/[2(7 − 5ν)] (−0.68S at ν = 0.3). Along the equatorial radius

σz(r) = S[1 + (4 − 5ν)a³/(2(7 − 5ν)r³) + 9a5/(2(7 − 5ν)r5)],

which gives the equatorial factor at r = a (the tool's built-in check) and decays as (a/r)³: the disturbance is confined to about two radii, against three or four for the plate hole whose field decays as (a/r)².

Why Less than 3

In a plate the load diverted by a hole must pass on either side of it in the plane. Around a sphere it can pass on all sides, so the equatorial stress rises by only about 2 instead of 3, and the result depends on Poisson's ratio because the lateral contraction of the surrounding material contributes to the equatorial strain. The same effect makes a cylindrical hole in a thick body (plane strain) keep the factor 3 while a spherical pore of the same size is far less severe, which is why porosity is less damaging than a through hole of the same diameter, though pores near a surface or in clusters interact.

Limits

  • Only the equatorial-plane distribution is given; the full field is available in Timoshenko and Goodier.
  • Fatigue: a pore acts as a crack initiator; the elastic factor is the starting point, and the small-defect size effect (Murakami's √area approach) is a separate method.


References

Southwell, R. V., and Gough, H. J. (1926). "On the Concentration of Stress in the Neighbourhood of a Small Spherical Flaw; and on the Propagation of Fatigue Fractures in Statistically Isotropic Materials". Philosophical Magazine, Series 7, Vol. 1, No. 1, pp. 71–97. First solution of the spherical cavity in uniaxial tension and the factor (27 − 15ν)/(14 − 10ν).

Goodier, J. N. (1933). "Concentration of Stress Around Spherical and Cylindrical Inclusions and Flaws". Transactions of the ASME, Journal of Applied Mechanics, Vol. 55, APM-55-7, pp. 39–44. The spherical cavity and inclusion by the Boussinesq–Papkovich method; the equatorial and polar stresses used here.

Timoshenko, S. P., and Goodier, J. N. (1970). Theory of Elasticity, 3rd ed. McGraw-Hill, New York. Chapter 12, article "Stress concentration at a spherical cavity": the equatorial factor, the polar stress and the distribution σz(r) along the equatorial plane in the form implemented.

Pilkey, W. D., and Pilkey, D. F. (2008). Peterson's Stress Concentration Factors, 3rd ed. John Wiley & Sons, Hoboken. Chapter 4, "Holes": spherical cavities under uniaxial and hydrostatic stress and ellipsoidal cavities.

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