Experimental Identification of the Constitutive Model of Viscoelastic Non-Standard Materials

Abstract

There is an increasing interest towards the use of non-conventional material such as Functionally Graded Materials (FGM) for aerospace and automotive mechanical applications. Classical material models, e.g. Kelvin or Zener, can show some limitations in describing the viscoelastic behavior of these materials. A numerical and experimental approach to identify the optimal model order and the parameters of the constitutive material relationship in the frequency domain is proposed. The constitutive equation is modeled by means of a generalized Kelvin model and expressed in the form of a rational function. To describe the complex material behavior, high order polynomials are needed for the rational function and the problem of finding the function coefficients can be ill-conditioned. Different approaches for the rational function parameters identification are compared. A least square error identification technique adopting Forsythe orthogonal polynomials is proposed. The selected procedure is first applied on numerically estimated measurements with noise, and then on real measurement data obtained by forced vibration testing of Polytetrafluoroethylene specimens.

Experimental identification of the material standard linear solid model parameters by means of dynamical measurements

“…The material D(ω) stress(σ^)–strain(ε^) relationship can be modelled by means of a SLS N order generalized Kelvin model (Amadori and Catania, 2016, 2017; Findley et al, 1989), resulting from series connecting N K-blocks, the i -th block being a parallel connection of an elastic Hooke ( E i parameter) and a viscous Newton ( β i parameter) component, i = 1,…, N . Since for the i -th block, the following equation holds…”

“…In this work, a single-cantilever experimental setup (Amadori and Catania, 2016, 2017) is considered. A sinusoidal excitation, whose frequency belongs to a Δ fixed limited range, is applied to a beam specimen of known geometry, and both force and displacement ( ν ) response are measured in correspondence of the same experimental dof.…”

“…The standard linear solid (SLS) material model is commonly used to model the σ stress ε strain relationship of many conventional and unconventional materials in the frequency domain, and it consists of a generalized Kelvin (K) model made up of N K-blocks connected in series (Amadori and Catania, 2016, 2017; Findley et al 1989). Nevertheless, starting from dynamic mechanical measurements on specimens made of the material under study, the identification of the SLS material model order and parameters is not a trivial task since the material D(ω)=1/E0⋅trueσ^(ω)/trueε^(ω), where true( )^=ℱ() is the Fourier transform operator and E0=trueσ^(jω=0)/trueε^(jω=0), has to be first identified in a useful Δ = [ ω min , ω ma x ] frequency range.…”

“…Nevertheless, starting from dynamic mechanical measurements on specimens made of the material under study, the identification of the SLS material model order and parameters is not a trivial task since the material D(ω)=1/E0⋅trueσ^(ω)/trueε^(ω), where true( )^=ℱ() is the Fourier transform operator and E0=trueσ^(jω=0)/trueε^(jω=0), has to be first identified in a useful Δ = [ ω min , ω ma x ] frequency range. Since the D(ω=ωs), s=1..nf dependency on the SLS material model parameters can be analytically defined in closed form for every ω s frequency measured value, a set of n f highly non-linear equations in the model parameter unknowns result (Amadori and Catania, 2016, 2017). Moreover, a non-parametric identification problem results since the N model order is unknown.…”

Identification of the Extended Standard Linear Solid Material Model by Means of Experimental Dynamical Measurements

“…The material uniaxial σ stress v/s ε strain unknown frequency function, σ/ε = E(j⋅ω), where () refers to the Fourier transform operator, j = ̅̅̅̅̅̅ ̅ − 1 √ , and ω is the circular frequency, can be experimentally estimated by means of input-output measurements made on a beam specimen in stationary forced vibration conditions and in low strain response conditions. Dynamic Mechanical Analysis (DMA) test systems are generally used for this task [1], and homogeneous, uniform beam specimens made with the material under study, excited in a flexural, tension or compression experimental set-up under known boundary conditions (BC) at the beam ends, are generally taken into account. The beam is harmonically excited at a fixed ω frequency value by means of a f force in correspondence of a fixed axial position.…”