Fractional Viscoelastic Models

Chapter 3 Fractional Viscoelastic Models Linear viscoelasticity is certainly the field of the most extensive appli- cations of fractional calculus, in view of its ability to model hereditary phenomena with long memory. Our analysis, based on the classical linear theory of viscoelstic- ity recalled in Chapter 2, will start from the power law creep to justify the introduction of the operators of fractional calculus into the stress-strain relationship. So doing, we will arrive at the fractional generalization of the classical mechanical models through a correspondence principle. We will devote particular attention to the generalization of the Zener model (Standard Linear Solid) of which we will provide a physical interpretation. We will also consider the effects of the initial conditions in prop- erly choosing the mathematical definition for the fractional derivatives that are expected to replace the ordinary derivatives in the classical models. 3.1 The fractional calculus in the mechanical models 3.1.1 Power-Law creep and the Scott-Blair model Let us consider the viscoelastic solid with creep compliance, J (t)= a Γ(1 + ν ) t ν , a> 0 , 0 <ν< 1 , (3.1) where the coefficient in front of the power-law function has been in- troduced for later convenience. Such creep behaviour is found to 57

Fractional Viscoelastic Models

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