∫ ∫ ∫ ε −ε ε = δε δε σ = δε V p kl kl e ijkl ij V e kl e ijkl ij V ij ij dV D D dV dV ) ( . . . = | . | . } . { 1 3 1 ∆ + δ + δ δ ∑ ∫ ∫ = − + u tdS u tdS WQ k S i i S i i k k | . | | | . | | 1 3 1 ∆ + δ =− δ − ∑ ∫ = + − u u T dS WQ k S i i i k = （7） | . | | | . | | 1 3 1 ∆ + δ − δ δ ∑ ∫ = T dS WQ k S i i k Where （k are cohesive surfaces, ( are the displacement and traction components on the cohesive zone surfaces, kS 1,3) = , ) i i u t ( , ) i iT δ are the relative displacement and traction on the cohesive surfaces, see formulas（1）to（4）. 1∆ is the displacement of the point acted by horizontal driving force . Q Based on (7), one can develop the finite element method for scratch test problem. The incremental constitutive relation of plasticity usually is expressed as kl kl ij e kl ij jl ik ij H E E ε σ′ σ′ +ν σ + δ δ − − ν ν δ δ + +ν = & & 2 [1 (2/3)(1 ) / ] (3/2) 1 2 1 Ω σ （8） ij σ′ is deviator stress, /2 3 ij ij e σ′ σ′ σ = is effective stress; for plastic loading 1=Ω , otherwise 0 =Ω . H is plastic modulus. In uniaxial tension the film material has ε for σ ； ，for （9） / ,E =σ Y <σ N Y Y E 1/ ( / )( / ) ε = σ σ σ Y σ≥σ so that （10） 1 1/ 1 1} / ) {(1/ )( − − − σ σ = N e Y H E N Strap advance is assumed to occur in steady-state such that the stress and strain increment components can be expressed as / ) , / ( ( , ) 1 1 x x V ij ij ij ij σ ε = ∂σ ∂ ∂ε ∂ & & （11） where V is velocity of crack tip during film delamination in direction. The formula（8）is independent of V. Plastic strain components can be expressed by stress and total strain as 1x ε （12） kl e ijkl ij p ij D = ε − σ −1 A numerical method [9] which employs iteration to satisfy condition (11) is used to directly obtain the steady-state solution. Similarly, in the present analyses, adopting the fundamental relations of tensors and matrixes, (7) can be changed into the finite element relations. The steps of solving the problem can be described as follows: (1) Adopting a plastic strain distribution (in first step, take ), find displacement and strain. (2) Find stress distributions in plastic zone and unloading zone using (8), (11) and yielding condition ( 0 ε = p ij Y eσ = Y is current flow stress). (3) Find plastic strain by (12). Repeat procedures until a convergent solution is obtained. Consider that the substrate material is elastic and Young's modulus and Poisson ratio are and sE sν respectively. For further simplification, neglect the effect of mismatch of film and substrate materials, so that we take in the present analysis. During the steady-state advance of delaminated film strap, total work per unit length is ; dissipated work per unit length along the separation cohesive surface is ; and along two shear cohesive surfaces it is . Let plastic dissipation work be ( , ) ( , ) ν = ν E Es s QW W0Γ τ 0 2 Γt P WΓ . According to energy balance under steady-state advance, we have P Wt Γ Γ Γ + = + τ 0 2 0 Q （13） For elastic case, 0= PΓ . In principle, the interface separation work (interface fracture toughness) 0Γ and the material shear work (or material shear strength) could be determined by experimental measurement. The normalized horizontal driving force Q during the steady-state advance of failure strap can be τ 0Γ
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