13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- in a three dimensional nonlinear FEM program, and successfully applied in dam heel cracking and multi-crack analysis of arch dam. Both FEM analysis and geo-mechanical experiments are performed on Baihetan and Xiaowan arch dams. Results show that unbalanced forces can be used as the indication of fracture initiation and propagation. 2. Deformation Reinforcement Theory 2.1. Definition of structural stability Geotechnical structure are characterized by magnificent scale and complicated configurations and working conditions. The classical elasto-plastic theory aims at solving the displacement and stress fields that simultaneously satisfy all the basic equations in boundary value problem, including kinematic admissibility, equilibrium condition and constitutive equations. However, the existence of such solution requires that the structure is stable, i.e., a state where no failure occurs [9]. Structural instability occurs when action is greater than resistance, and the difference between action and resistance defines the unbalanced force. Considering the arbitrary kinematical and equilibrium stress-field, 1σ, which is termed the trial elastic stress: e 1 0 0 : = +Δ = + Δ σ σ σ σ ε D . (1) The kinematical and stable stress-field, σ, which is the real stress response, could be identified by the following minimization problem: ( ) ( ) ( ) ( ) ( ) 1 1 min , 0, 1 : : . 2 ∀ ≤ = yc yc yc yc yc E f E σ σ σ σ −σ σ −σ C (2) Eq. (2) is known as the closest-point projection method (CPPM) [10], as shown in Fig. 1. Figure 1. Diagram of elastic-plastic stress adjustment The difference between 1σ and σ is the plastic-stress increment field p Δ σ : p 1 Δ = − σ σ σ. (3) The plastic-stress increment field p Δ σ leads to the plastic-strain increment field p p : Δ = Δ ε σ C , while C is the fourth-order compliance tensors. Clearly, the minimization variable σ restricted by the yield criterion can be viewed as the material resistance while the minimization objective E in Eq. (2), termed the volume density of the plastic complementary energy (PCE), represents the difference between the plastic dissipations of the stress action and the material resistance, ( ) ( ) ( ) 1 1 1 : : . 2 = E σ σ −σ σ −σ C (4) Thus, stability of a material point can be interpreted as the condition that the stress action is greater
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