Identification of the optimal Fractional Standard Linear Solid material model from strain relaxation test measurements

An algebraic procedure for the non-parametric identification of the optimal Fractional Standard Linear Solid (FSLS) model of the material of a experimentally tested beam specimen is proposed. The standard test approach consist in estimating the material E() complex modulus by means of measuring both excitation and response in a forced excitation behaviour, but such approach generally suffers from low resolution in the frequency domain because of experimental time inherent limits. Material model identification techniques using such data as input generally suffer from this limitation, so that the numerical simulation of the dynamic behaviour of a structure made of the material under study can be generally not consistent with the expected quasi-static behaviour. It should also be taken into account that the material fractional modeling approach is typically required in order to consistently model the relaxation behavior of structures, while the standard SLS visco-elastic modeling approach can be generally consistent when a medium to high frequency dynamical behaviour has to be considered. Identification material model procedures requiring as input quasi static stress-strain relaxation response measurements in the time domain are known, but they typically suffer from low resolution in the dynamical medium to high frequency domain. Moreover, because of these limits only SLS models consisting of a single element may be typically identified. In this work, an algebraic identification technique being able to identify a FSLS material model made of N>1 elements from a set of strain relaxation J(t) creep compliance test data is proposed. E(ω) complex modulus is estimated from J(t) by means of a known numerical integration technique and a non parametric identification techinique in the frequency domain, based on the Levy’s approach, is describe in detail. A fractional Standard Linear Solid material model made up of a series arrangement of N fractional Kelvin elements, i.e., Kelvin elements adopting fractional order time derivative operators, is considered. Since the FSLS model size (N) and the fractional order associated to each element are not known in advance, a non-parametric identification procedure results. It is assumed that the fractional (num/den) order of the FSLS elements share the same (den) denominator fractional order while a different (num) value is expected to be associated to each element. The E(ω) complex modulus can be expressed as the sum of complex partial fraction terms with respect to (jω)(1/den). The associated residues and poles can be identified by means of an algebraic technique. The relationship between these parameters and the residues and poles associated to the (jω)(num/den) partial fraction form is then found. A procedure evaluating the physical and the stability properties of the solution associated to any N value is proposed in order to get the optimal N value. Some test cases numerically generated from within a known model, are used to validate the technique and some experimental test cases are also used to find the optimal equivalent FSLS material model of some composite materials obtained from polymeric matrix components mixed with different types of recycled powders. The tested composite materials are investigated for potential coating layer damping engineering applications. Results are shown and critically discussed.