Studies in Applied Mathematics

Berg, C. A.

doi: 10.1002/sapm1971503193pmid: N/A

The plane plastic deformation of a generally anisotropic rigid‐plastic material which possesses a yield condition dependent upon mean triaxial stress and which, through the classical associated flow rule, exhibits plastic dilatation, is considered. This model is used to represent the behavior of micro‐porous ductile metals in which the micro‐cavities may be strongly aligned due to large prior plastic strains, as for example the material surrounding the tip of an extending notch in a ductile metal. It is shown that the stress and velocity fields are hyperbolic where a line of vanishing extension rate may be found in the plane of deformation, and that the characteristics of both the stress and velocity fields coincide with the lines of vanishing extension rate. Coincidence of the characteristics of stress and velocity fields in general anisotropic plastic bodies seems not to have been expected in earlier writings, but is a natural consequence of the associated flow rule. Simple means of determining whether a given stress state at yield lies in a hyperbolic or elliptic field are discussed. The role of characteristics in providing ductile fracture nuclei is discussed.

Drew, Donald A.; Segel, Lee A.

doi: 10.1002/sapm1971503205pmid: N/A

Drew, Donald A.; Segel, Lee A.

doi: 10.1002/sapm1971503233pmid: N/A

Stanley, Richard P.

doi: 10.1002/sapm1971503259pmid: N/A

Benedek, A.; Panzone, R.

doi: 10.1002/sapm1971503281pmid: N/A

The expansion of f ∈ Lp(0, 1) Fourier series of Bessel functions of order converges to f in Lp whenever Let be the space of p‐integrable functions with respect to the measure t dt and where {sn}, n = 1, 2, …, is the set of positive zeros of Jv. Then, the expansion of in a Fourier series of functions ψn, −1 < ν < −½, converges to in whenever