The wedge subjected to tractions: a paradox resolved

✍ Scribed by J. P. Dempsey

• Publisher: Springer Netherlands
• Year: 1981
• Tongue: English
• Weight: 473 KB
• Volume: 11
• Category: Article
• ISSN: 0374-3535
• DOI: 10.1007/bf00042478

✦ Synopsis

The classical two-dimensional solution provided by L6vy for the stress distribution in an elastic wedge, loaded by a uniform pressure on one face, becomes infinite when the opening angle 2c~ of the wedge satisfies the equation tan 2c~, = 2~,. Such pathological behavior prompted the investigation in this paper of the stresses and displacements that are induced by tractions of O(r -°J) as r-~ 0. The key point is to choose an Airy stress function which generates stresses capable of accommodating unrestricted loading. Fortunately conditions can be derived which pre-determine the form of the necessary Airy stress function.

The results show that inhomogeneous boundary conditions can induce stresses of O(r-°~), O(r -'~ In r), or O(r -~ In 2 r) as r--~ 0, depending on which conditions are satisfied. The stress function used by Williams is sufficient only if the induced stress and displacement behavior is of the power type. The wedge loaded by uniform antisymmetric shear tractions is shown in this paper to exhibit stresses of O(ln r) as r -~ 0 for the half-plane or crack geometry. At the critical opening angle 2c%, uniform antisymmetric normal and symmetric shear tractions also induce the above type of stress singularity. Not anticipating such stresses, L6vy used an insufficiently general Airy stress function that led to the observed pathological behavior at 2a,.