Communications in Nonlinear Science and Numerical Simulation
Identification of constitutive parameters for fractional viscoelasticity
Introduction
The fractional derivative has become an extremely adequate tool to model mechanical properties of viscoelastic materials [1], because it is such an intimate descriptor of viscoelastic materials behavior that only a small number of parameters are enough to accurately represent a particular material [1], [2], [3], [4], [5]. An important problem connected with the fractional rheological models, as mentioned by Lewandowski [6], is the estimation of the model parameters from experimental data. As a matter of fact, there are a number of literatures related to this issue. However, these work seem to be mainly driven by determining parameters of simple fractional viscoelastic devices (such as viscoelastic dampers, dynamic vibration isolators etc. [6], [7], [8], [9], [10], [11], [12], [13]), or by identifying fractional viscoelastic parameters of simple and homogeneous structures (such as beams [14]), instead of the parameters identification of a homogeneous/inhomogeneous fractional viscoelastic field. On the other hand, these work were mostly carried out in the frequency domain. When measurement data are limited and polluted, an error may arise in the integral transform from time domain to the frequency domain, as shown in the Section 4.
With the above consideration, this paper attempts to tackle with two issues, i.e.
- 1.
The identification of constitutive parameters for fractional viscoelastic fields, instead of for simple devices or simple homogeneous structures.
- 2.
The identification of fractional constitutive parameters in the time domain, instead of in the frequency domain.
In the Section 1, a Finite Element (FE) equation is derived for the direct problem, which can be solved by the Laplace transform or FD method for 2-D homogenous and regionally inhomogeneous fractional viscoelastic fields.
In Sections 2 The governing equations of 2-D direct fractional viscoelastic problem, 3 The inverse problem for identification of fractional viscoelastic constitutive parameters, the identification of fractional viscoelastic constitutive parameters is treated as an optimization problem that is solved by an ACO algorithm [15].
The ACO algorithm needs the solution provided in the Section 1, but does not need derivatives of solutions with respect to constitutive parameters. These derivatives are demanded for the gradient based algorithm, and seem unease to calculate accurately due to a difficulty caused by the Gamma function. Although the ACO algorithm has successfully been applied to solve various kinds of inverse problems [16], [17], [18], [19], it seems first time to be used to solve inverse fractional viscoelastic problems.
In the Section 4, an identification problem of fractional viscoelastic constitutive parameters for a plate with a rectangular opening is investigated. The impacts of regional inhomogeneity, noisy data, and spatial arrangement of measurement points etc. on the solution are taken into account. The solutions obtained by the proposed approach are less sensitive to the noisy data given in this paper, and different spatial arrangements of measurement points seem no significant impact on the solution. Numerical results indicate that the proposed approach is available for the identification of constitutive parameters of homogeneous/regionally inhomogeneous fractional viscoelastic fields in the time domain.
Section snippets
The governing equations of 2-D direct fractional viscoelastic problem
The equilibrium equation of two-dimensional quasi-static problems is given by [20]The relationship between the strain and displacement is described bywhere and stand for the vectors of stress and strain, respectively, x = {x1, x2} represents the vector of coordinates, b refers to the vector of the body force, and represents the vector of displacements.
The boundary conditions
The inverse problem for identification of fractional viscoelastic constitutive parameters
The inverse problem in this paper mainly involves the combined identification of ακ, υκ, and , the solution can be obtained by minimizing a cost functional defined bywhere , {ακ}T = {α1, α2, …, ακ, ..., αMT}, {υκ}T = {υ1, υ2, …, υκ, …, υMT}, , . and refer to measured and estimated values of displacement
Implementation of ant colony algorithm
The major computing steps of the Gridding Partition based continuous ACO algorithm [25] include
Step 1
Set error bound
Set initial lower and upper bounds is the number of variables to be identified
Set number of the ants K
Set number of the grids Nop
Set pheromone value ϑ0
Set evaporation rate of the pheromone
Set maximum number of iteration NLS
Set number of iteration kstep = 0
Step 2
Divide into Nop parts with the size
Set
Numerical verification and discussion
For simplicity, all the computing parameters are assumed dimensionless.
Let us consider a plate with a rectangular opening as shown in Fig. 1 that also gives a description of FE mesh with 128 8-node elements. The size of computing domain is 10 × 10.
At first, let us examine a Laplace transformation with limited and polluted measurement data in the time domain at x1 = 3, x2 = 3, the computing parameters are f1 = 1 × 106, f2 = 2 × 105, α = 0.3217, , , υ = 0.15 is given by [26]
Conclusions
The major contribution of this paper includes
- 1.
A numerical model in the time domain is presented to identifying unknown constitutive parameters for homogeneous/regionally inhomogeneous fractional viscoelastic fields, instead of for simple fractional viscoelastic devices or simple homogeneous fractional viscoelastic structures.
- 2.
An explicit semi-analytical numerical solution is derived for the direct homogenous fractional viscoelastic problem, which provides a quick solution of the direct problem.
Acknowledgments
The research leading to this paper is funded by NSF (10421002, 10772035, 10721062, 11072043), NKBRSF (2010CB832703).
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