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Department of Technology Fundamentals. This article focuses on fractional Maxwell model of viscoelastic materials, which are a generalization of classic Maxwell model to non-integer order derivatives. To build a fractional Maxwell model when only the noise-corrupted discrete-time measurements of the relaxation modulus are accessible for identification is a basic concern. For fitting the original measurement data an approach is suggested, which is based on approximate Scott Blair fundamental fractional non-integer models, and which means that the data are fitted by solving two dependent but simple linear least-squares problems in two separable time intervals. A complete identification algorithm is presented.

Maxwell model site top

Maxwell model site top

Maxwell model site top

Maxwell model site top

Maxwell model site top

If both the above conditions are satisfied, stop the procedure taking as the FMM parameters. It is shown that FMM can be used to describe the viscoelastic mechanical properties of biological materials. Note, that the curves of FMM relaxation modulus shown in Figures 4 and Maxwell model site top has the characteristic shape of the relaxation modulus of viscoelastic materials obtained in experiment. I am blessed beyond measure to live this life, and I would not have had it Olsen twins sex movies the efforts of those both past and present. Machado, V. For these model parameters the accurate fit to experiment data is achieved, as shown in the Figure 8. A historical review of applications can Maxwell model site top found in [ 8 ]. Rop, X. CRC Press.

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Department of Technology Fundamentals. This article focuses on fractional Maxwell model of viscoelastic materials, which are a generalization of classic Maxwell model to non-integer order derivatives.

To build a fractional Maxwell model when only the noise-corrupted discrete-time measurements of the relaxation modulus are accessible for identification is a basic concern.

For fitting the original measurement data an approach is suggested, which is based on approximate Scott Blair fundamental fractional non-integer models, and which means that the data are fitted by solving two dependent but simple linear least-squares problems in two separable time intervals. A complete identification algorithm is presented. The usability of the method to find the fractional Maxwell model of real biological material is shown.

The parameters of the fractional Maxwell model of carrot root that approximate the experimental stress relaxation data closely are given. Fractional calculus is a branch of mathematical analysis that generalizes the derivative and integral of a function to non-integer order [ 1 ]. Application of fractional calculus in classical and modern physics greatly contributed to the analysis and our understanding of physico-chemical and bio-physical complex dynamical systems, since it provides excellent instruments for the description of memory and properties of various materials and processes.

During the last two decades fractional calculus has been increasingly applied to mathematical modelling in physics [ 2 , 3 ], engineering [ 4 , 5 ], and especially to rheology [ 6 , 7 ], where fractional calculus constitute a valuable mathematical tool to handle viscoelastic aspects of systems and materials mechanics. Models involving fractional derivatives and operators have been found to better describe some real phenomena than integer-order differential equations [ 1 - 5 ], whence there are many new exciting areas of fractional models and fractional calculus applications, such as the automatic control, the modeling of biological, medical and environmental.

A historical review of applications can be found in [ 8 ]. The survey on fractional models from biology and biomedicine is presented in [ 9 ], see also other papers cited therein. In recent decades fractional derivatives were found quite flexible, especially in the description of viscoelastic polymer materials [ 10 ]. Viscoelastic materials present a behaviour that implies dissipation and storage of mechanical energy.

Research studies conducted during the past few decades proved that these models are also an important tool for studying the behaviour of biological materials [ 11 ]: wood, fruits, vegetables, animals tissues, etc. Viscoelasticity of the materials manifests itself in different ways, such as gradual deformation of a sample of the material under constant stress creep behaviour , and stress relaxation in the sample when it is subjected to a constant strain.

In an attempt to describe some of the above effects mathematically several constitutive laws have been proposed which describe the stress—strain relations in terms of quantities like creep compliance, relaxation modulus, the storage and loss moduli and dynamic viscosity.

Some of these constitutive laws have been developed with the aid of mechanical models consisting of combinations of springs and viscous dashpots. For over five decades classical exponential behaviour models have been widely applied to describe the viscoelastic properties of biological materials. Maxwell, Kel- vin-Voight and Zener models are used to mathematical modelling of stress relaxation and creep processes [ 11 - 13 ].

For these models the relationship between the stress and deformation of the material is approximated though an ordinary differential or integral equations. However, relaxation or creep processes deviating from the exponential Debye decay behaviour are often encountered in the dynamics of biological complex materials [ 6 , 13 ]. For such materials a stretched exponential decay KWW model Kohlrausch-Williams-Watts [ 12 ], hyperbolic type decay Peleg model [ 13 ] or power type behaviour models [ 14 , 15 ] are used to approximate experimentally obtained relaxation modulus or creep compliance data.

To this end, fractional rheological models, originally pioneered by Nutting [ 17 ] and Scott Blair [ 18 ], have proven to be a concise and elegant framework for predicting the response of complex fluids such as liquid foods using a small number of parameters [ 16 ].

By replacing the springs and dashpots of the classical viscoelastic models by the Scott Blair elements, several fractional models, including the fractional Maxwell, fractional Voigt and fractional Kelvin models, have been proposed [ 6 , 19 ]. In this paper fractional Maxwell model FMM model is considered, which relates the stress to the strain in the material by means of using differential fractional equation [ 3 , 6 , 7 ].

While fitting data to exponential sum models, like classic Maxwell model, is a very old problem, which has been studied for a long time, until now there are only a few papers concerned mainly to finding fractional Maxwell model. Common choice of the model quality measure is the mean square approximation error, leading to a least-squares identification problem. Thus, in [ 7 , 21 ] an approach is considered, which consists on the determination of the FMM parameters by two approximate Scott Blair models identification in two separate time intervals, for small and large times respectively.

The aim of the paper is to develop a complete procedure for FMM identification using relaxation modulus data from ramp stress relaxation test. We consider a linear viscoelastic material subjected to small deformations for which the uniaxial, nonaging and izotropic stress-strain equation can be represented by a Boltzmann superposition integral [ 22 ]:.

By assumption, the exact mathematical description of the relaxation modulus G t is completely unknown, but the value of G t can be measured with a certain accuracy for any given value of the time t.

Fractional Scott Blair model [ 7 ] is described by the fractional differential equation. To illustrate the structure of fractional models a fractional element, in addition to the standard purely elastic and purely viscous element, must be introduced - see Figure 1c.

This approach makes it possible to include a whole range of dissipation mechanisms in a single three parameter rheological element. Classic viscoelastic Maxwell model is the arrangement of ideal spring in series with a dashpot see Figure 2a described by the first order differential equation.

For details of the fractional model construction see [ 6 , 7 ]. The above result is obtained in [ 6 ] by applying Fourier and Mellin transforms, while in [ 23 ] the same result is obtained by an approach involving the Laplace-Mellin transform technique. A deep insight into the complex properties of FMM gives the infinite hierarchical structures of elementary fractional model 2 presented in Figure 3 composed of ideal spring and dashpot elements, which compose the classic Maxwell model.

Note, that the curves of FMM relaxation modulus shown in Figures 4 and 5 has the characteristic shape of the relaxation modulus of viscoelastic materials obtained in experiment. Sequential ladder realization on the fractional Scott Blair element [ 6 ; Fig. Based on 8 , the relaxation modulus G t 7 can be rewritten as. From the following asymptotic property [ 24 ]. The same asymptotic approximate models are considered in [ 7 ] and [ 21 ], but in the last paper stringent applicability conditions, derived above, are imposed.

The approximate models 10 , 12 and the exact FMM are summarized in Figure 6 , where logarithmic scale is used both for relaxation modulus and time scales. In result better fit to experimental data is obtained. In this paper a complete two-interval identification scheme based on approximate models 10 and 12 and using the linear least-squares is presented and applied to FMM of biological material identification. The scheme is inspired by [ 7 , 21 ] papers.

However, precise applicability conditions are formulated here for the first time. For computational methods of relaxation modulus determination see, for example [ 12 , 25 ]. Fitting data to the original FMM 7 is a very difficult problem of nonlinear optimization, numerically difficult and often ill- conditioned.

Here, the linear least-squares identification routine will be used to estimate FMM parameters based on the logarithmic transformation of the experimental data and equations 10 and 12 which, respectively, yields.

Now, classical linear least squares method can be applied to find optimal approximate models. Therefore, the least-squares identification of the log- linearized models consists of determining the model parameters minimizing the indices 18 , 19 by solving the following standard optimization problems. The model parameters optimal in the sense of 21 are given by the known formulas:. The formulas for optimal and are analogous. According to definitions 13 and 14 the positivity constraint can be neglected for k 2 ,k 2 in 20 , 21 optimization tasks.

It can be shown, based on the Czebyszev equality, that the optimal parameters and are positive, if and only if, the following conditions are satisfied. In view of 14 and 13 the pairs of parameters are not independent. Therefore, from 14 and 13 we have. Taking into account the above, the calculation of the approximate values of FMM parameters involves the following steps. Determine the set of log-transform measurement data Compute the estimates and according to formulas 22 , 23 and, next, compute the estimates of the relaxation time and elastic modulus based on 24 and In order to ascertain if the models with parameters are a satisfactory approximation of measurement data in time intervals compute the optimal identification indices and examine if and for S, a preselected small positive error.

Otherwise, go to step 6. In order to ascertain if the Scott Blair models 10 , 12 with parameters and are a satisfactory approximation of the original FMM examine if 26 If both the above conditions are satisfied, stop the procedure taking as the FMM parameters.

The stopping rules from Step 5 guarantee the good quality of the log-linearized models in the chosen time intervals and correspond with those commonly used in the optimal identification techniques. The conditions 26 , 27 are imposed to guarantee the applicability of Scott Blair models 10 , 12 to approximate the original FMM.

Both the pairs of conditions must be satisfied simultaneously in order to guarantee good quality of the resulted model. Note, that 26 and 27 are a posteriori conditions, since the applicability of the identification procedure cannot be checked, earlier than after the experiment is performed. The next example shows, how the identification scheme can be used for identification of FMM of real material. A cylindrical sample of During the two-phase stress relaxation test performed by Bohdziewicz [ 14 ], in the first loading phase of the time period 0, Next, during the second phase of the time period Next, the relaxation modulus measurements were computed for the time interval 0, The inaccurate fit of the resulting FMM to the experimental data is illustrated in Figure 7.

Next, the recurrent scheme of the algorithm have been realized and successive approximations of FMM parameters have been determined. Repeated composition of the scheme operations has led to parameters listed in the last row of Table 1. For these model parameters the accurate fit to experiment data is achieved, as shown in the Figure 8. The model errors are not big see Table 1 , but for a few first FMM approximations the accuracy is insufficient, especially in long time region.

Next, the generalized discrete Maxwell model, which presents a relaxation of exponential type given by a finite Dirichlet-Prony series [ 11 , 12 ]. Least-squares approximations are used, and two, three and four-parameter optimal models were determined. Figure 9 presents experiment data and the optimal Maxwell models. It is easy to observe, that the Maxwell model is inappropriate for description of this relaxation process.

Thus, the classic Maxwell models do not fully characterize the true viscoelastic behaviour of the biological material. A better fit to experimental data can be obtained, if the FMM is used. Note, that also the range of the relaxation time estimates is lower for FMM than for classic Maxwell models, even if the conditions 26 , 27 of the scheme applicability are not sharply fulfilled - compare Tables 1 and 2. In this work, a complete simple procedure is proposed for the fractional Maxwell model identification based on the discrete-time stress relaxation experimental data.

The asymptotic approximate log-linearized Scott Blair models, composed of only two parameters, are used in separate time intervals. The linear least squares are applied to find the best Scott Blair models in two separable appropriately chosen time intervals. An applicability conditions of the approximate identification scheme are derived. It is shown that FMM can be used to describe the viscoelastic mechanical properties of biological materials. The effectiveness is compared with the classical two, three and four parameter Maxwell model approximations.

While the idea of using the approximate Scott Blair models to find the FMM parameters is known and already appeared in [ 7 , 21 ], the complete identification scheme is presented in this paper for the first time. Data correspond to usage on the plateform after The current usage metrics is available hours after online publication and is updated daily on week days.

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Maxwell and kelvin voight models of viscoelasticity presentation

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