2 been considered. The aim of this paper is to deal with such problems and to find a possible satisfactory solution. For this purpose it first has to deal with the existing theories systematically. The investigation has the following assumptions: continuum mechanics of observation, stationary crack, quasi-static, small deformations, isotropic and linear elastic material behavior, elastic-plastic material behavior with power-law hardening and J2 deformation theory. There are other theories and attempts to deal with and to describe such problems. These are not part of this paper, and they are not discussed here. 2. The Classical Theories of Fracture Mechanics 2.1 Griffith's Theory At first Griffith (1920) [2] published his fundamental work on the treatment of crack problems. He investigated an infinite plate with a crack-like elliptical hole under tension (Fig. 1) and gave an energy balance to this plate Fig. 1: an infinite plate with a crack-like elliptical hole under tension (Mode I) ( O) 0 a (U W O) a −Π− ≥ ∂ ∂ − − = ∂ ∂ , (2.1) where U is the work done by the external force, W the strain energy, U W ∏=− + the potential energy and O the surface energy. Under consideration of the so-called “fixed-grips“ condition and with help of the stress and displacement field of Inglis (1913) [1] and by the vanish of the short axle of the ellipse he obtained following equation E a W W 2 2 0 ′ π σ ∏= = − , (2.2) 2a λσ λσ σ σ X1 X2
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