13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- than the material resistance in the sense of plastic dissipation: ( ) 0> E σ . Furthermore, stability of a structure whose volume is V can be deduced as p p 1 : : d 2 Δ = Δ Δ ∫ V E V σ σ C . (5) This equation shows that ΔE is also the norm of plastic-stress increment field p Δ σ . If ΔE = 0, then p Δ σ is always zero and the structure is stable. If ΔE > 0, the structure is unstable. 2.2. Expression in FE analysis In this section, DRT is deduced and implemented in elasto-plastic FE analysis. For simplicity, the problem is restricted in displacement method, which means the kinematic admissibility is naturally satisfied. Since 1σ is a equilibrium stress-field, it satisfies equilibrium condition: T 1d =∑∫ Ve e V σ F B . (6) F is equivalent nodal force vector of external loads. B denotes the displacement gradient matrix. Applied with Eq. (3) and after simple manipulations, Eq. (6) can be recast into the following expression: T p T T 1 d ( )d d Δ = Δ = − = − ∑ ∑ ∑ ∫ ∫ ∫ U B B F B σ σ σ σ Ve Ve Ve e e e V V V . (7) ΔU is the driving force of the deformation process that can be termed the unbalanced force. It’s also referred to as the residual force in FEM, which is a set of equivalent nodal forces of the difference between the two stress fields 1σ and σ. 2.3. Fracture analysis based on DRT Liu Y. R. et al. proved both theoretically and experimentally that the unbalanced force can be used as the prediction and measurement of failure mechanism [8]. Failure occurs where there is unbalanced force subjected to prescribed loads, and the structure is unstable in the sense of PCE. So the unbalanced force can be used to evaluate fracturing of structure. According to DRT, unbalanced forces are the driving force of structural failure, and fracture is part of the failure mechanism. Thus, unbalanced forces could be the determination of fracture location. The area where unbalanced forces occur is the location where the fracture initiates. Furthermore, the amount of unbalanced force incurred by external load represents the extent of fracture propagation. The development of unbalanced forces is the process of propagation of the fracture. 3. Application in high arch dam In this section, unbalanced forces are applied to indicate initiation of dam heel cracking, and to verify the dominating cracks from multi-crack arch dam. 3.1. Dam heel cracking analysis of Baihetan arch dam Baihetan arch dam, located in an asymmetrical “V”-shaped valley, is 289 m high. Both 3-D finite element numerical and geo-mechanical experiments are performed. The finite element mesh model is shown in Fig. 2(a). Various rock materials and faults are simulated. The size of FE model is as follows: Upper stream: 1.5 times of the height of dam (500 m);
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