On the determination of higher order terms of singular elastic stress fields near corners

Summary In two-dimensional elasticity stresses at reentrant corners exhibit singular behavior. The stress field is of the form, where (r, ) are polar coordinates centered at the tip of the corner, andf_i(;_i are smooth functions. For practical use of this series the eigenvalues_i (which are generally complex numbers) are required in order of ascending real part. The problem then is to find the roots of a transcendental equation (eigenequation) in the complex plane and arranged in order of ascending real part.A theorem is proved on the number, location and nature of the roots of this equation with the real part in fixed intervals of length . Excellent initial estimates of the real part of the complex roots become available, and so are bounds, within which single real roots exist. This enables the determination of any number of roots in ascending order of real part. The critical angles at which the eigenvalues change nature are also determined. It is shown that for certain cases and for the symmetric mode of deformation, the eigenvalue =1 does not represent a rigid body rotation, therefore it has to be included in the series representation of the stresses. The coefficientsK_i can be determined by recently developed extraction techniques, thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture, crack initiation, plastic zone estimation etc.Dedicated to Professor Ivo Babuka on the occasion of his 60th birthdayPresented at the Conference: The Impact of Mathematical Analysis on the Solution of Engineering Problems, 17–19 September 1986, University of Maryland, College Park, Maryland, USA