Boltzmann superposition principle 285 alternating

The Boltzmann Superposition Principle Alternate form of the LVE Equation... 

In this chapter we describe the common forms of viscoelastic behaviour and discuss the phenomena in terms of the deformation characteristics of elastic solids and viscous fluids. The discussion is confined to linear viscoelasticity, for which the Boltzmann superposition principle enables the response to multistep loading processes to be determined from simpler creep and relaxation experiments. Phenomenological mechanical models are considered and used to derive retardation and relaxation spectra, which describe the time-scale of the response to an applied deformation. Finally we show that in alternating strain experiments the presence of the viscous component leads to a phase difference between stress and strain. 

The pseudoelastic design method may be used for components submitted to intermittent loading, provided that the intervals during which the material is unloaded are sufficient to allow virtually complete recovery. Some manufacturers provide recovery data that enable the validity of this assumption to be tested. Alternatively, the Boltzmann superposition principle may be used to determine whether the assumption gives a satisfactory approximation (see... 

Equations 7 and 9 are two of many possible expressions of the Boltzmann superposition principle that the effects of mechanical history are linearly additive, where the stress is described as a function of rate of strain history or alternatively the strain is described as a function of the history of rate of change of stress. Many early treatises have elaborated on the effects of different sequences of stressing or straining according to this principle, and it has been tested by a variety of experiments. The procedure may be illustrated by some characteristic sequences of stress history. 

This is the alternative form of the Boltzmann superposition principle and it is of importance in particular in rheological treatments. 

It is not convenient to represent the result of a creep experiment in terms of the relaxation modulus, since it is the stress that is controlled rather than the strain. For this purpose we can use an alternative expression of the Boltzmann superposition principle in which stress is the independent variable. For the case of simple shear, this is shown by Eq. 4.20. 

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