Figure 1: Scratch test geometry and ductile failure Figure 2: Double cohesive zone model Let ( ）be relative displacements at each cohesive surface along direction ( ), respectively, and define a normalized displacement quantity 1 2 3 , , δ δ δ 1 2 3 , , x x x 2 3 2 2 2 1 1 λ =δ δ +δ +δ − c , （1） The critical condition for the cohesive zone is, 1 λ = . For the separation-dominated cohesive zone case, , while for the shear-dominated cohesive zone case, c cn δ =δ ct cδ =δ . The traction relations, ( ) σ λ and ( ), on the cohesive zone surfaces are sketched in figure 2. τ λ The traction component expressions can be formulated in detail as follows. Define a potential function δδ δ =δ σλ′λ′ ∫λ d cn ( ) ( , , ) 0 1 2 3 Π （2） then, one will derive out the traction expressions easily ( , , ) ( ) ) , , , , ) ( 1 2 3 1 2 3 1 2 3 δ δ δ λδ ( σ λ = ∂δ ∂ ∂δ ∂ ∂δ ∂ = cn T T T Π Π Π （3） Similarly, for the shear-dominated cohesive surface, one reads ( , , ) ( ) , , ) 1 2 3 1 2 3 δ δ δ λδ ( τ λ = ct T T T （4） Adhesion work per unit area along the cohesive surface can be written as ) ˆ (1 2 1 2 1 0 = σδ +λ −λ cn Γ （5） for the separation zone, and ) ˆ (1 2 1 2 1 0 = τδ +λ −λ τ ct Γ （6） for the shear cohesive zone. Earlier work has shown that the shape parameters (λ and ) of cohesive zone model have the secondary influence on the analytical results. In the present analysis, we take . Moreover, for reducing the number of governing parameters, we take 1 2λ ( , ) (0.15, 0.5) 1 2 λ λ = c ct cn δ =δ =δ , then from (5) and (6), one have . = τ σ τ ˆ / ˆ / 0 0 Γ Γ ENERGY BANLANCE AND ELASTIC-PLASTIC MECHANICS METHOD The double cohesive zone model has been sketched by figure 2. The variation equation for the total system can be written as
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