expressed by the related independent parameters as follows ν β σ τ σ σ σ + = σ τ = + , , , , , ˆ , ˆ , ) ˆ ˆ )( 2 1 ( 0 0 0 N W t R t E f W t Q Y Y Y P Γ Γ Γ (14) In（14）a reference length parameter has been introduced, whose definition is , characterizing the plastic zone size in small scale yielding. ) ] /[3 (1 2 2 0 0 Y R E π −ν σ = Γ SOLUTIONS AND ANALYSES Three-dimensional elastic-plastic finite element calculation for the scratch test problem is carried out based on the concepts implemented in previous sections. The results and analyses are shown in the following. Figure 3 shows the relation of the normalized horizontal driving force (or applied work per unit area) to the maximum strength of separation and shear for two kinds of cohesive zones. The driving force changes with shear strength and separation strength are very sensitive, especially for large values of separation strength or Figure 3: Scratch work vs. material parameters Figure 4: Scratch work vs. material parameters for low hardening film for high hardening film shear strength. For the lower separation strength case, the horizontal driving force changes with the material shear strength slowly as shear strength increases, then sharply increases. For the higher separation strength, even a lower material shear strength will make the horizontal driving force increase very quickly. In the figure 3, the results of two different ratio of thin film thickness with the delaminated film width are compared. Clearly, the horizontal driving force changes sensitively with the ratios. The results correspond to the lower strain hardening exponent material, i.e., N=0.1. The separation cohesive energy along interface is taken as the normalized quantity. When the shear strength of material is zero and the elastic case is considered, the value of the normalized horizontal driving force (applied work per unit area) equals to unity. When the Figure 5: Scratch work vs. film thickness and material shear strength is not zero, the normalized indenter angles horizontal driving force will change linearly with the
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