ISSN:
0945-3245
Keywords:
AMS(MOS): 65N30
;
CR: G1.8
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary In two-dimensional elasticity stresses at reentrant corners exhibit singular behavior. The stress field is of the form $$\sigma = \Sigma {\rm K}_\iota r^{\lambda _2 - 1} f_i (\theta ;\lambda _i )$$ , where (r, θ) are polar coordinates centered at the tip of the corner, andf i (θ;λ i are smooth functions. For practical use of this series the eigenvaluesλ 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 π. 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 λ=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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01395878
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