Example Thin film undergoing plane strain extension

3 Example Thin film undergoing plane strain extension 

For the case of thin films being deformed plastically under the action of equi-biaxial states of stress, the stress history follows a straight line path in stress space. In some of the experimental methods for study of thin film plasticity that have been described in this section, the in-plane stress components are not equal, in general, and the trajectory in stress space during deformation is not a straight line. Consider a film material being deformed under plane strain extension in the a —direction the plane strain constraint is enforced in the. j—direction. The elastic modulus is E and the Poisson ratio is Pf = 0.3. The initial yield locus for the material in terms of tensile yield stress stress (Tyf is the Mises condition. The material 

Initially, the response is within the elastic range of behavior. In this range, the plane strain constraint e z = 0 is sufficient to determine the stress trajectory. From Hooke s law. 

Once plastic ffow begins, the total strain rate is the sum of the elastic and plastic strain rates. The plastic strain rate is given in terms of stress by (7.60), and the expressions for total strain rate are 

The plane strain constraint still requires that tzz = 0 and the imposed extensional strain requires that ixx = rryJEi. Therefore, the left-hand sides of both equations are known. The equations (7.88) and (7.89) then provide a pair of first-order ordinary differential equations to be integrated for (Txx and Tzz as functions of 

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