Kolosov-Muskhelishvili equations

The stresses may be calculated by differentiating (2.8.5). One finally obtains the viscoelastic generalization of the Kolosov-Muskhelishvili equations which we collect together as follows ... 

Kolosov-Muskhelishvili Equations Adapted to the Half-Plane... 

I. Method of Solution. The method of solution is based on the viscoelastic Kolosov-Muskhelishvili equations, adapted to a half-space. Explicit solutions to the first and second boundary value problems are presented in detail. In these cases no restrictions on material behaviour are necessary. In the case of mixed boundary value problems where surface friction is present, it is necessary to make the proportionality assumption. Limiting frictional contact problems are... 

Our approach is based on the viscoelastic Kolosov-Muskhelishvili equations, as in the case of contact problems. Consider the general form, given by (2.8.9). These relations must hold over the entire complex plane, where 0(z, /) V ( > 0 are analytic everywhere, except on some or all of the real axis, and go to zero at infinity as 1/z, according to (2.8.12). 

As in the previous chapter, we rely upon an adaptation of the Kolosov-Muskhelishvili equations. Problems for which the normal and shear forces are prescribed on the surfaces of open, and possibly growing, cracks may be solved without any restriction on viscoelastic material behaviour. The case of a single crack is studied in detail and explicit formulae are derived for stress intensity factors and displacements across the crack surface. 

Equations (2.8.9) extend to viscoelasticity the Kolosov-Muskhelishvili solution of plane elasticity, which is widely used in Chaps. 3, 4. 

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