Equation for Linear Viscoelasticity in Simple Shear

Constitutive Equation for Linear Viscoelasticity in Simple Shear 

If the deformation is uniform (homogeneous), the stress and strain components do not vary with position and are independent of x/. There are several speciHc types of uniform deformation for which the strain and stress tensors assume a relatively simple form. One that corresponds to a commonly used experimental geometry is simple shear, where two opposite faces of the cubical element are displaced by sliding, as illustrated in Fig. 1-3. Conventionally, the 13 plane slides in the 1 direction the strains and stresses are then 

This linear constitutive equation is based on the principle that the effects of sequential changes in strain are additive -  

If the function G s) approaches zero as s approaches infinity (a condition which as we shall see corresponds to a viscoelastic liquid), there is an alternative formulationexpressed in terms of the history of the strain rather than that of the rate of strain  

An alternative constitutive equation can be written to express the strain in terms of the history of the time derivative of the stress, in the form 

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