5 3. Uniform Formulations of the Energy Release Rate 3.1 General Formulations Let us first consider the change of the potential energy in a cracked elastic body shown in Figure 2(a), where the deformation of the crack front surface is adopted as shown in Figure 2(b), Fig. 2: (a) a cracked body, (b) the crack front field ∫ ∫ − ∏= − = • • v s i i t t u dA dt d wdV dt d (W U) . (3.1) In the above equation Π is the potential energy, W is the strain energy, w is the specific strain energy, U is the work done by the external force, ij σ is the stress tensor, ij ε is the strain tensor, iu is the displacement vector, it the force on the surface ts and V the volume, ( ) d/dt = • is the material time derivative and t the time-like parameter. It follows ∫ ∫ ∫ Δ Δ→ • • • +Δ Δ + − ∏= V t 0 V s i i w(t t)dV t 1 wdV t u dA lim t . (3.2) By looking at the first and the second term in the equation (3.2) and by the use of the Gaussian theorem, the terms cancel each other, so we get ∫ Δ Δ→ • +Δ Δ ∏ = V t 0 w(t t)dV t 1 lim . (3.3) This equation (3.3) is the basic equation from which we can derive different quantities. x1 x2 u 0 i = it auf ts V A0 A1 ΔV (a) (b)
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