Multi-axial Deformation Three-Dimensional Non-linear Viscoelasticity

4 Multi-axial Deformation Three-Dimensional Non-linear Viscoelasticity 

The discussion so far has been dominated by one-dimensional behaviour, reflecting the most convenient and customary materials testing methods. However, any engineering application will be for a three-dimensional body, subject to multi-axial stresses. It is now feasible to implement non-linear viscoelastic models in numerical schemes to perform analyses of structures, and this is often the motivation for generalising a viscoelastic theory to two or three dimensions. 

A very valuable technique, well known in plasticity theory, is to split the stress into its hydrostatic and deviatoric components, associated respectively with volumetric and shear strain. On the basis that creep is caused by shearing of molecules past one another, we would expect creep to be only associated with deviatoric stress. This is constructed by subtracting the hydrostatic component from the stress tensor E to give the deviatoric stress E  

In an early attempt by Pao and Marin [6] at a three-dimensional extension of their uniaxial creep law Equation (11.1), strains in the principal directions are given by terms such as 

The first term represents elasticity, and the second term represents the viscoelastic component. In the latter term, the deviatoric stress component is apparent. J2 is a scalar invariant closely associated with the deviatoric stress tensor 

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