A study of anisotropic and viscoelastic ductile fracture

The problem of ductile fracture in anisotropic and viscoelastic solids is an important engineering problem. In this paper the Dugdale model is assumed, that is, the yielded zone is replaced by a constant yield stress. Solutions are presented for anisotropic static, orthotropic dynamic and anisotropic viscoelastic solids.

The anisotropic static and orthotropic dynamic solutions were obtained by the complex variable approach. Stress functions which must satisfy a generalized bi-harmonic equation, are represented in terms of two analytic functions of two different complex variables. In this way boundary value problems can be reduced to problems of complex function theory. The anisotropic and dynamic solutions are thus seen to be completely analogous and thus similar relations are obtained for both cases.

The yield stress is assumed to follow a Von Mises' yield criterion which was adopted to the anisotropic-dynamic case. For the static anisotropic and the orthotropic dynamic cases the following results were obtained;

1. The plastic zone is given by the same relation as in the isotropic - static case,

(𝒍/a) = cos [ (πT)/ (2Y) ]

1. Any anisotropic or corresponding dynamic function may.be obtained from the corresponding isotropic or static function by simply multiplying the isotropic or static function by a coefficient.

2. A limit on yielding along the line of the crack, and therefore a limit on the anisotropy and the velocity are derived.

The anisotropic viscoelastic solution is obtained from the static solution from Graham's extension of the correspondence principle after it is shown that the problem fits the restriction set by this technique. The effects of anisotropy in the material are handled by the inclusion of the generalized creep compliance and relaxation modulus. Once these terms are evaluated for each material an approximate inversion method for the Laplace transform may have to be used.

For the case of constant external load it is shown that the a xx stress may be separated into a time variation and a space variation. Several different viscoelastic materials are assumed and the stresses solved for.

In the limit all results reduce to the isotropic and static solutions. Finally, a large bibliography is included to serve those who wish to investigate the area further.