Plane Strain in Linear Viscoelasticity

Considerable simplification is obtained if the number of dimensions in a problem reduces to two, just as in the elastic case. One specific way that this happens, which will be of interest in later chapters, is when plane strain conditions prevail. We will discuss briefly the characteristics of plane strain, omitting mention of the closely related topic of plane stress. 

Under plane strain conditions parallel to the xy plane, the displacement u (r, 0 is constrained to be zero, while the other components are independent of z. This would occur for example if all the bodies in the problem were uniform and infinitely extensive in the z direction, and are acted upon by external stresses independent of z. It follows that 

We confine the discussion to the case of an isotropic material. The constitutive equations (1.9.6) give that 

This normal stress must operate in order to maintain plain strain conditions. Problem 2.8.1 Show that 

Consider the Papkovich-Neuber form (2.7.1) in the two-dimensional case. All dependence on z = must drop out, so we put y/ = 0. There are two equations. It is our object to combine them into one complex equation. Let us for a moment assume that [Gladwell (1980)] 

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