Mechanical models, such as the maxwell model, the voigt model, and the kelvin model also called the standard linear solid 25,26 are often used to describe the viscoelastic behavior of materials. These models are often proved insufficient, however. The standard linear solid model slsm is a typical and useful model for analyzing. Is carried out and the axial force f is plotted as a function of. The transfer function of this linear system thus will be rational function, note that, as and bs are given above as polynomial of system. Formal transfer function rational transfer function iir infinite impulse response model broad class of inputoutput linear models differentiation operator formal transfer function rational function of s for a causal system m.
This will introduce the notion of the distribution of characteristic times and all the formalisms associated with the relaxationretardation spectra in the. It must be stressed in the class that the solid linear standard is a rather simple model and that almost no material could be described according to that model. The linear elastic solid has been the main material model analysed in this book thus far. Creep, relaxation and viscosity properties for basic.
The fractional order derivatives have been used in modeling the lossy inductance as well as the creep effect for movingcoil loudspeakers. Jaroslav vond rejc born on the 27th of june 1983 in opocno january 2009 czech technical university in prague. Fractional modeling on movingcoil loudspeaker linear system. The transfer function appoach for general nonlinear systems has two principle differences compared to the traditional linear techniques. If pis a pole it follows that yt ept is a solution to the 6. Creep and relaxation functions of the basic constitutive models 2. The formulation can also be quite easily extended to a generalized maxwell model. The dash pot then takes up the stress, transferring the load to the second spring as it slowly opens over time. The spring model describes an ideally elastic body dt d g s h. Transfer function of exponential signals in linear systems, exponential signals plays vital role as they come into sight in solving differential equation 1. Analysing the transfer functions of nonlinear systems in. Fractional modeling and analysis of coupled mr damping.
If a small stress is applied for a sufficiently long time, then the irreversible strains become large. Here, we revisit the generalized standard linear solid model and seek to overcome the confusion concerning the expression for the relaxation function or modulus. The constitutive equation for the linear elastic solid is then simply e 1 the creep and recovery behaviour of. In the sls configuration a maxwell element is connected in parallel with a second spring this setup is also known as the zener model. The standard linear solid sls, also known as the zener model, is a method of modeling the. The creep compliance function and the relaxation modulus have. Inaseries connection such as themaxwell model, thestresson each element is the same and equal to the imposed stress, while thetotal strain is the sumof the. Stress waves from 20 hz to 20 khz are perceived as sound above 20 khz is the ultrasonic range. Upon loading one expects the lefthand spring to stretch immediately. The standard linear solid sls model, also known as the zener model, is constructed by connecting a linear spring in parallel with a maxwell element, which consists of a linear spring and a linear dashpot in series, as shown in fig. The first is that the transfer function representation of an equivalent timedomain nonlinear system consists of a sequence of transfer functions instead of only one function in the linear case. Estimate transfer function models with prior knowledge of model structure and constraints this example shows how to estimate a transfer function model when the structure of the expected model is known and apply constraints to the numerator and denominator coefficients. Bme 315 biomechanics material properties of tendon.
Its temporal step response predicts a negative exponential function. Linear viscoelasticity numerical tours of continuum. After some manual rearrangement, the radial displacement. The linear elastic solid the linear spring the simplest way to create a model of a material is to suppose that it consists of nothing but a linear spring of stiffness e. Linear viscoelasticity an overview sciencedirect topics. The standard linear solid model sls also know as the kelvin model is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots to represent elastic and viscous components, respectively. Deriving the equations modeling these systems is beyond the scope of this course. Transfer functions the roots of the polynomial as are called poles of the system and the roots of bs are called the zeros of the system.
Cartilage has no known function other than maintaining mechanical. Viscoelastic effects are often present in loudspeaker suspensions, this can be seen in the displacement transfer function which often shows a frequency dependent value below the resonance frequency. Voigt model maxwell mdel essentially assumes a uniform distribution of stress. This constitutive equation has been put in what is known as standard form stress on. Non linear modal analysis methods for engineering structures by hugo ramon elizalde siller. Viscoelastic response is often used as a probe in polymer science, since it is. Linear elastic body, hooke body hooke model, hooke element, elastic spring, represents behavior of a perfectly elastic lossless solid material. Instead, one could, for example, use the standard linear solid sls model, which is a combination of the maxwell and kelvinvoigt models. The dashpot model describes purely viscous fluid figure 1b. Transfer functions are a frequencydomain representation of linear timeinvariant systems.
Seismic modelling, time domain, standard linear solid, constant q. Equation 7 was used to curve fit the stress response i. Strain response for a stressstep for a elastic b elastoplastic, c viscoelastic and d viscoplastic material behavior another way of representing the measurement data of the tensile experiment is by plotting the stress against the strain, resulting in the stressstrain curve. Classroom methods for demonstrating, modelling and.
Although the maxwell model did not give us a realistic result for a viscoelastic solid it is more representative of a viscoelastic fluid it did allow us to illustrate some important aspects of a viscoelastic material including the stress relaxation function gt and the creep function jt. Some basics of rheology university of nevada, reno. Often, the simpler maxwell model and the kelvinvoigt model are used. The simplest model that predicts creep and stressrelaxation is the standard linear solid sls with a parallel combination of a maxwell arrangement spring and dashpot in series with a spring fung, 1981. The relations between transfer functions and other system descriptions of dynamics is also discussed. For instance, consider a continuoustime siso dynamic system represented by the transfer function syss nsds, where s jw and ns and ds are called the numerator and denominator polynomials, respectively.
The standard linear solid sls, also known as the zener model, is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots to represent elastic and viscous components, respectively. In particular, simple analytical approximations are derived for the impact duration, t c, and for the coe cient of restitution, e. The standard linear model consider next the standard linear model, which consists of a spring in series with a kelvin unit, fig. Frequencies above 1012 hz correspond to molecular vibration and represent an upper limit for stress waves in real solids. The simulations show that the nonlinear version of the maxwell sls model can result in a time dependent. Alternative form of standard linear solid model for. Some basics of rheology standard linear solid will not relax to zero stress kelvin element maxwell element hook spring newtonian. Rational approximations of transfer functions of some. When the material is viscoelastic the strain shows time delayed response on a stress step, which indicates a. Introduction the transfer function plays an important role in the linear theory. Inaseries connection such as themaxwell model, thestresson each element is the same and equal to the imposed stress, while thetotal strain is the sumof the strainineachelement. Constitutive models of linear viscoelasticity using.
Pdf the generalized standardlinearsolid model, also called the zener model, is widely used in. Approximating constantq seismic propagation in the time. Although either can be used, for ease of calculation the 3 element generalized maxwell model gm is used for sr and epsilon dot, while the sls kelvin form of the model is often used in the case of creep. The standard linear solid model, on the other hand, considers a maxwell element which has a spring and dashpot in sequence in parallel with a. In this paper nonlinear versions of the standard linear solid model sls are investigated. In the kelvinvoigt model, membrane stress is computed as the sum of elastic and viscous stress resultants. In the linear case the transfer function fs of a control system is usually expressed as the ratio of two polynomials in the laplace operator s with real coef. Moreover, it benefits from less computational costs in computer time and memory. According to the linear theory, the viscoelastic body can be considered as a linear system with the stress or strain as the excitation function input and the strain or stress as the response function output.
A better model is the standard linear solid which is shown below. Standard linear solid model, but other models could have been used as well. The spring represents the elastic component of the models response. Mathematical modeling of linear viscoelastic impact. Constitutive models of linear viscoelasticity using laplace transform author. A novel nonlinear viscoelastic solid model sciencedirect. For the kelvin or standard linear solid model, there are two different but equivalent configurations. In this respect, the response functions to an excitation expressed by the heaviside step function. Strain in both elements of the model is the same and the total stress is the sum of the two contributions. Often more similar maxwell model and the kelvinvoigt model are used. However, the geophysical literature shows two different expressions for the relaxation function or modulus in the generalized sls model.
A variety of mathematical models were suggested to describe the stressstrain response of. In order to get analytical approximations, we consider the standard solid model as a perturbation of the kelvinvoigt section 5 or the maxwell model section 6. The dash pot then takes up the stress, transferring the load to the second spring as it. The strain as a function of time during the creep test using a unit step stress function. Pdf the generalized standardlinearsolid model and the. Show that the constitutive equations for a standard linear solid can be expressed by show that the constitutive equation for a standard linear solid model can be expressed by. Response and recovery times of elastic and viscoelastic. Modeling, control, and applications by yang fang a dissertation submitted to. Based on the fractional lossy inductance model and the fractional standard linear solid model, this work presented a fractional equivalent electrical circuit feec for modeling the linear behaviors of movingcoil loudspeakers. The constitutive equation for the linear elastic solid is then simply e 1 the creep and recovery behaviour of the material to a suddenly applied load o is. Solution of transfer function problems as stated previously, the transfer function is a common and extremely powerful method of representing a system mathematically.
Timedependent viscoelastic properties along rat small. Each element carries the same load while strains are added. The tf model object can represent siso or mimo transfer functions in continuous time or. Mathematically the transfer function is a function of complex variables.
The generalized standardlinearsolid model, also called the zener model. Transfer function representation of linear physical systems. An alternative configuration, also commonly referred to as the sls, consists of a linear spring connected in series with a kelvin element, which consists of a. Massachusetts institute of technology department of. The maxwell model does not exhibit creep since it models strain as linear function of time.
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