3 where W0 is the strain energy without crack, E E ' = for plane stress, ) E E/(1 2 ' = −ν for plane strain, E is the elastic modulus and ν the Poisson’s Ratio. With Eq. (2.2) he got the energy release rate E a (2a) W (2a) G 2 I ′ σ π = ∂ ∂ =− ∂ ∂Π =− . (2.3) The index I denotes the mode I and the factor 2 means the whole elliptical crack length. For the surface energy O he assumed the form = γ O 4a , (2.4) where γ is the specific surface energy and should be a material constant. By substituting Eq. (2.3) and (2.4) into Eq. (2.1), the Eq. (2.1) becomes ≥ γ G 2 I . (2.5) With this, he predicted that the unstable crack growth occurs when the equation (2.5) is fulfilled. From the Griffith's investigation it can be summarized: His study is based on a two-dimensional, infinite plate with an elliptic crack-like hole under mode I loading case. He established the global energy balance for the whole body and recognized that under the fixed grips condition the strain energy change could only be considered and a residual amount of the strain energy change has to be given in order to proceed with the crack. This residual amount of energy divided by the crack change must have the same size or is larger than the surface energy, which then can create the new surface. From this energy, he has introduced a well-known fracture mechanics quantity which is referred to as "energy release rate". It was unclear whether the Griffith's theory is applicable for general or complicated crack problems. 2.2 Irwin's Work Irwin (1957, 1964) [3, 4] has attempted to answer the questions above. He extended Griffith's theory for mode I to mode II and III for linear elastic materials. Under consideration that the energy to close the crack (2.6) ∫ σ +σ +σ Δ = Δ → 1A 1 2 1 3 0 23 1 2 0 22 1 1 0 12 a 0 u u )n dA ( u 2 1 a 1 I lim (2.6) has to be equal to the energy to extend the crack, he obtained a well-known equation )ν (1+ E K + E′ K + E′ K G= I= 2 III 2 II 2 I , (2.7) where ij σ is the stress tensor, iu the displacement vector and in the normal on the crack front surface. 0 () refers to the state of the time t and 1 () the state of the time t t +Δ , A1 is the crack front surface to the state of time t t +Δ and III II IK ,K ,K are the stress intensity factors for mode I, mode II and mode III.
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