9 we have tacitly assumed, that the volume fraction of ΔV is like in Figure 2, without analyzing it. To check whether this assumption for the deformation of the actual crack front surface is true, the deformation of the crack front surface must be again examined carefully. Let us now consider the body with cracks (Fig. 1) again. Under loading, the crack will extend and at the same time it will open, as shown below in Figure 3. As already known, only a small crack front surface will help for crack growth. Thus, the volume fraction ΔV must be included by surfaces A0, A1 and Aβ. Especially the surface Aβ describes the crack opening and this has been neglected until now. By using the Gaussian theorem the basic equation (3.3) can be written in the new volume fraction ΔV (Fig. 3) as follows ∫ ∫ ∫ β β Δ→ Δ→ Δ Δ→ • σ Δ + σ Δ σ ε = Δ ∏ = A j 0 i 1 ij t 0 A 0 j 0 i 1 ij t 0 V 0 ij 1 ij t 0 u m dA 2 1 t 1 u m dA lim 2 1 t 1 dv lim 2 1 t 1 lim 0 . (4.11) The first term of equation (4.10) has already been dealt above. The second term is new and we will study it more precisely now. By replacing the time t with the crack length a and by use of the Stockes's theorem, the second term can be reformulated as follows Fig. 3: real deformation of the crack front surface ∫ ∫ =− ε ε ∂ σ σ Δ β β Δ→ 0A qmlljk m ij i q k A j 0 i 1 ij t 0 u n dA 2 1 l u m dA 2 1 t 1 lim , (4.12) where kl is the direction vector of the crack extension, ijk ε is the alternative tensor and i ∂ is the differential operator. Now we receive a second new quantity Tk ∫ = ε ε ∂ σ 0A qmlljk m ij i q k u n dA 2 1 T , (4.13) which describes the distortion of the crack front surface. It can easily be proved that the surface integral Tk above is path-independent by using the Gaussian theorem 0 u )dV u ( 2 1 ( u n u n )dA 2 1 u n dA 2 1 T V k j ij i j k ij i B j ij i k k ij i j B qmlljk m ij i q k = = ∂ ∂ σ −∂ ∂ σ = ∂ σ −∂ σ = ε ε ∂ σ ∫ ∫ ∫ where B is an arbitrary closed area. It can be seen above, that ΔV A0 A1 Aβ
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