Some Aspects of Model Equations Development for Viscous Materials

Mechanics, Materials Science & Engineering, May 2017

ISSN 2412-5954 25

Irina Viktorova1, a, Sofya Alekseeva1, b, Muhammed Kose2, c 1

Department of Mathematical Sciences, Clemson University, Clemson South Carolina

2

Department of Mechanical Engineering, Clemson University, Clemson South Carolina

a

iviktor@clemson.edu

b

salekse@clemson.edu

c

muhammk@clemson.edu DOI 10.2412/mmse.81.48.85 provided by Seo4U.link

Keywords: viscoelasticity, creep, stress, strain, relaxation, viscous, elastic.

ABSTRACT. Materials with viscous properties, polymers and composites with polymer matrix, have found increasingly wider applications in modern industry. It is known that temperature and moisture are very important for correct estimation of the environmental response of these materials. Therefore, the construction of the constitutive equations for the material elements under real conditions of exploitation, takes on special significance in modern mechanical engineering.

Viscoelastic models. The term viscoelasticity is derived from two terms: elastic (spring) and viscous (an oilis strain and is stress.

odels can describe, in some cases, the rheological behavior of materials.

Investigations such as this were alluded to during the middle of the 19th century in the works of Maxwell, Voigt, Kelvin and many others. The elements can be connected in series or parallel. In the

(1)

For the parallel case, the Voigt model is valid (2) blished by Magnolithe GmbH. This is an open access article under the CC BY-NC-ND license http://creativecommons.org/licenses/by-nc-nd/4.0/

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Later on, it was shown that the simplest models constructed from the two elements could only be used to describe either creep behavior through the Voigt model or relaxation processes through the Maxwell model.

Of course, one can construct a complex system, which will contain more elements. This is a natural way to develop the generalization of the models (1) and (2). In this way, one finds a relationship between -hand sides

(3)

This differential relationship can be transformed into an integral equation. It was shown [1] that if n = m, equation (3) is equivalent to the integral equation

Here

Parameters and are connected with coefficients of equation (3) as shown in [1-2]. Because of the large number of parameters mentioned, the above equations are rather difficult.

Some empirical, or semi-empirical, approaches are popular for the description of creep and relaxation processes. These theories are called the technical creep theories, they can be thought of as the maximum limitation of a number of variables and stating some propositions on a functional relationship between these variables. It is then necessary to connect them in an analytical relation, the best of which will be a theory that demonstrates the best correlation with tests.

Creep theory must be able to describe the general material behavior, using the simplest tests, with time dependent stress and strains, given the determination of a strain changing law using a stress changing law and vice versa. In a particular case, the theory must allow one to construct the relaxation curves using a series of creep curves, which is a cornerstone for any time-dependent theory.

Three basic creep theories to note: 1. Time-hardening theory

for example

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2. Flow theory

for example 3. Strain-hardening theory

for example Power functions are the most popular choice for , though exponential or other functions that demonstrate strong correlation with experiments can be chosen.

The approaches mentioned above were developed originally for the description of creep (or relaxation) of metals a deformation without the elastic component. The application of such approaches is often limited because, if the parameters of an equation for description of creep curves at constant stress can be successfully estimated, then one can find only qualitative correlation for the description of relaxation curves.

Other approaches were being developed when almost only metals were used as construction materials. Creep in metals only becomes apparent at high temperatures and under certain loading conditions.

The situation changed when polymers and composites with a polymer matrix found a wider application in industry. Time-dependent properties of these materials are clearly evident at and below room temperature. Therefore, it was necessary to conduct a complex investigation of the behavior of these materials. Moreover, models must be established, which would use the same set of parameters for creep, relaxation, loading with various regimes, cyclic loading, and unloading, etc.

Hereditary-type theories The only approach from those currently known that answers all the above demands is an introduction of the hereditary principle in mechanics of materials and a construction of the corresponding constitutive equation. This approach aroused a revolution in the field of viscoelasticity, which began with the work of the German scientist Boltzmann [3]. This work was published in 1878 and shows a way by which the viscous behavior of materials can be explained using the conception of heredity. mechanical process is defined by some function , A reaction response of a solid or system of solids is determined by some function u(t). In general, the value of the function u(t) in the present time t is affected not only by the value of an action at specific time t, but also by the whole history of the function v changing. Therefore, the function u can be written as a function of v

In the dependence of this type of function, one will find various types of constitutive equations. The stress-strain state of a material can be considered: Let us consider that some stress was applied at the instant of time and was perpetuated for a time . The material retains a memory of the action of the stress in the form of a small deformation , which is proportional to the stress and the MMSE Journal. Open Access www.mmse.xyz

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time and depends on the difference between the present time t and the instant , also known as (time-lag) It gives . With the addition of an elastic term, the following integral equation becomes the constitutive model

(4)

Here E is the elastic modulus and K

is the kernel of the integral equation, which describes the

ns in biology The Italian scientist Volterra chose the other approach for the construction of the hereditary equation [4The basic exponential model of population development belongs to Malthus [6]. The main assumption was that the birth and death rates were equal. It was supposed that the rate of change for the number of individuals , is proportional to the common number (N) of the individuals at that time

(5)

The solution of the above equation shows that if time rises in the arithmetic progression, then the value N rises in the geometric progression. Malthus concluded that wars and other conscious destructions of the population on Earth had to be considered since a population grows at a considerable rate. In the 19th century, these works were analyzed and criticized. For example, it was proposed that the coefficient was not a constant but a decreasing function of N connected with a competition inside the species population

(6)

where and are constants. Solving the equation gives

This shows that as , , which is a constant value. The equations can be modified by adding some other terms, such as periodic terms, which can appear because of a seasonal prevalence of weather. n kinds. or n differential equations must be written. The simplest example is the association consisting of two kinds: predator and victim. For this case, we have

Victim:

(7)

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Predator: Here, the coefficient of increase, 1, for victims has a positive value because, in the absence of predators, the quantity of victims will increase. The coefficient, 2, is negative because the quantity of predators will naturally decrease in the absence of victims. The terms, which are concerned with competition, depend on the number of kinds meeting and are proportional to N1N2. The terms are positive for predators and negative for victims.

Analysis of these equations led to interesting conclusions. For example, the periodicity of fluctuations in the biological associations was proven. This property of periodicity was known previously, but ecologists proposed that the phenomenon should be explained by other factors, for example by the seasonal change of weather, or destruction from the human side. he International Mathematical Congress in 1913. He stated that even destruction of both kinds of fish (predators and victims) leads to a decrease in the number of predators and to an increase in the number of victims. On the other hand, stopping fishing leads to an increase of predators. This effect was confirmed via statistical analysis of fishing in the Adriatic Sea during WWI by biologist Later on, Volterra developed a general theory for n of species and introduced the concepts of conservative and dissipative associations. An association of the type seen in equation (5) is conservative, but an association of the type described in equation (6) is dissipative. The equations (7) also represent a conservative association. The equations for the dissipative association can be obtained from (7) with account of inside competition

Now, the rates of change of N1 and N2 will be damped so their amplitudes will decrease over time and have a tendency to occupy an equilibrium. Volterra showed that this property is universal for the associations, which he named as dissipative. This caused him to think about an analogy with mechanics. Conservative associations are ideal but do not exist in nature, while dissipative associations are more common. The analogy with mechanics is seen in the fact that to the conservative mechanical system with friction, the mechanical energy of the system will decrease if oscillations are damped.

Having analyzed the differential equations, which describe the oscillations of mechanical systems and biological fluctuations, Volterra proved the identity of the equations.

The similarity between biology and mechanics even more pronounced when considering the phenomenon of after-effect, or delay. After-effect can be easily explained in biological content The number of predators, N2(t), in the present depends not only on the quantity of victims in the present time, t, but also on the quantity of victims in some previous instant, , and thus depends on the time elapsed from the instant to the present time t, and thus on difference t- . From the equations (7), we can now obtain the system with the equation that contains an after-effect integral

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The above is the particular case of a more symmetric system, where the first equation also contains an after-effect integral.

Pekar introduced the phenomenon of after-effect or delay, in physics in his studies of elasticity, magnetism, electricity and other phenomena. The same principle can be seen in mechanics. Therefore, in the inorganic world, a memory of the past exists. For example, to determine the stress-strain state of some deformed structural element, it is necessary to know the previous stress states.

General principle of heredity Influence of past on future is observed in areas other than biology, physics and mechanics. It clearly exists in various fields of human activity such as economics, politics, psychology and the general development of civilization. Therefore, after-effect is one of the basic laws of nature and human development.

To take into account a continuous sequence of previous states, it is necessary to use the integral and integro-differential equations, with the functions, which depend on some period of time that proceeds the present. For mechanical systems, we assume that the past history acts as a force that can be described by the functional

This additional force is a resultant of the elementary actions from the previous intervals ( , + ). Functions must be decreasing because of the assumption that the aftereffect is weaker with an increasing difference between the present time and the time of the force application. If we know the displacement during a period of time equal to the after-effect duration and the outside forces in the next interval of time, then it is possible to calculate the displacements, which will develop in the next moment.

The phenomenon of after-effect, its mathematical formulation, analyses of dissipative processes and fluctuations have been considered in detail in the works of Volterra.

Generally speaking, both the organic and inorganic natural worlds are subject to the same laws, which are described by the same equations whose analysis allows one to demonstrate the general concept of the laws of nature.

Kernels of the integral equation Functions

, which are is introduced under the integral, are the kernel of the integral equation.

analysis of the equation. Therefore, in mechanics, the first condition of kernel choice is strictly correlated with the test results, with creep tests as the most popular for viscoelastic materials [1].

Boltzmann was the first to suggest the singular form of a kernel:

. It gives an infinitely large

rate of deformation at an initial moment, which is verified by experiments. However, not only the rate of deformation becomes infinitely large but also the deformation, itself. Later, Duffing [7] suggested to use the function , where 0< <1, as a kernel. This kernel was named after Abel. Its shortcoming is in the fact that at the limiting creep conditions the strain, as time, , which contradicts with the experimental data. However, it was experimentally proven [8-9] that such type of kernel might be used, especially for nonlinear viscoelasticity, because it allows to accurately MMSE Journal. Open Access www.mmse.xyz

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describe the behavior of materials for the interval of at least seven-eight times orders. This result means that if the kernel parameters are determined from the short-term creep tests, the behavior of the material in construction at much longer exploitation periods can be predicted.

Many authors use a kernel in the form of the exponential function , which is convenient for calculations. However the drawback is for short periods of loading, the estimates do not correspond to experimental results. Nonetheless extrapolation over large periods of time for some cases can be sufficiently close to the tests. Attempts to combine the properties of exponents and weak singularity kernels lead to the construction of many known types: Bronsky, Slonimsky, Rzhanitsyn, Koltunov, etc. The analysis of some of them can be found in [10]. Their weakness is in the difficulty of finding the resolvent solution, which restricts the application of the constitutive equations with such kernels.

The simplicity of mathematical solutions is the second crucial factor for the choice of an integral equation kernel. Such transforms into a simple algebraic relationship is rare in practice. Instead more complex processes and arbitrary types of loading, with given strains but not stresses, occur more often in engineering kernels.

It must be emphasized that hereditary-type models have significant advantages in comparison with differential-type models, which are often empirical in nature. The latter models are used for the description of some specific type of loading, such as creep, relaxation, cyclic loading, etc. Therefore, these models are restricted by narrow limits of application. It should be noted that the integral equations for each of the above-mentioned types of loading can be transformed into the simple algebraic equations via the integral operators and successfully used for the cases when the type of loading changes during universal and satisfies all the constraints that a kernel must satisfy. It was named an exponential kernel of arbitrary order [1,11-12]

the series becomes an exponential function.

The great input of Rabotnov consists in the construction of the class of the new resolvent functions and algebraic operators. The advantage of these functions is in the procedure of finding the resolvent, which is the same type of function but with some different parameters, which can be easily calculated.

This makes it po which solutions to hereditary-elastic problems must be such that one must solve an ordinary elastic problem by treating the operators as elastic constants. In the finite solution, these elastic constants ought to be replaced by the operators, and the resulting solution must be interpreted through the algebraic inverse operators.

Thus, the introduction of the new class of exponential functions with arbitrary order together c solid mechanics based on classical theory of elasticity.

However the use of the linear theory is rather limited since the linear region cannot be found for some materials such as polymers or composites with polymer matrix and gives good results only at moderately high stresses.

Nonlinear equations MMSE Journal. Open Access www.mmse.xyz

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The most general nonlinear equation was written by Volterra and represents an infinite series of the multiple integrals

If we choose a large number of terms in the above equation and hypothetically determine the parameters of all the kernels , we can describe any process of deformation with any precision. That was the approach taken in [13-15] and many others in construction of constitutive or governing equation. Certain assumptions or restrictions in material behavior such as in compressibility, same behavior in tension and compression and others can simplify the above infinite series and kernel functions involved. But it is clear that use of the multiple integrals and determination of a large number of hereditary kernels is a very difficult task. One of the practical applications of such an approach through the quasising some prepositions. The new equation contains three terms of the series of ordinary integrals rather than multiple integrals.

(8)

The left hand side of the equation, the nonlinear function strain ( ) is called a curve of instantaneous sixties, when a large number of experiments were carried out on the short and long-term creep of glass-reinforced plastics. It was shown that the equation with the exponential kernel of arbitrary order allowed the prediction of long-term creep with the highest accuracy. In recent years Rabotnov's type kernel had been successfully used in combination with optimization techniques for the creep modeling of new type of viscous materials nanocomposites [17].

Application to dynamic problems The constitutive equation (8) can be applied both: the quasistatic and dynamic processes of loading.

With an increase of strain rate, the strain diagrams are being shifted vertically and condensed, aspiring in a limit to the curve of the instantaneous deformation ( ).

For dynamic problems, two basic approaches that have been subject to various modifications should the existence of two curves (static and dynamic) of deformation. The dynamic curve is located above the static one and determined from any dynamic experiment. The other approach, Sokolovskyon stress-strain diagram on the rate of loading and is determined experimentally. This process is not which is bounded by the two diagrams found at and . For example, in the works [1819], it is shown that this approach, based on the principle of heredity, allows one to describe processes of creep and quasistatic loading with different rates and can also be used to find the solution to the

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waves propagation problems. It must be emphasized that the same parameters of a kernel and the same curve of the instantaneous deformation can be chosen.

The equation (8) can be used both for the analysis of polymers, composite materials and metals [20].

Temperature and moisture effects Environmental factors such as temperature (high and low) and moisture are known to affect the viscous properties of polymers and composites thus leading to a numerous research efforts of introduction temperature and moisture level functions into the constitutive equation. Many introduce both temperature and moisture in a purely formal way with the parameters of the kernel and the instantaneous deformation curve, the modulus of elasticity in the elastic case, are dependent on both temperature and moisture in the most general way. Evidently, to determine the whole set of parameters requires an extensive experimental program. The combined introduction of temperature and moisture effects into the constitutive equations is quite rare.

The principle of the time-temperature analogy was introduced by Ferry [21]. It is based on the concept that time, dependent on temperature, should be introduced in an equation

Here aT is the factor of temperature shift and is determined from the formula:

where C1 and C2 are empirical constants and T0 is the reduced temperature.

The type of the constitutive equation is unchanged, the value of time in the constitutive equation depends on temperature. The solutions to problems remain the same as before but demand an additional interpretation. Such an approach allows one to consider the thermo-rheological identical processes, which is convenient for engineering practices since it is possible to carry out short-term experiments at higher temperatures and to predict the behavior of a material at lower temperatures and longer times of loading. The principle of the time-temperature analogy was thoroughly developed

by Latvian school of mechanics [22]. The principle of the time-moisture analogy was formulated as well. It was based on the same concepts.

The analogy principles used in engineering allow to estimate the behavior of various materials.

However, there are considerable difficulties connected with the determination of the complete set of parameters, the number of which is too large for nonlinear case and it leads to the lack of uniqueness in the determination of the parameters. Moreover, considerable difficulties arise when solving some specific nonlinear problems of mechanics.

The fundamental idea of a completely different approach [23] is based on assumption that the instantaneous deformation curve, ( ) is considered as a curve of absolute zero temperature. The constitutive equation can be presented in the form

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where ( ) is bounding from above the deformation process. It has been shown that a function of temperature influence can be chosen as a power function , where T is the temperature in Kelvin. For convenience of calculation, the function can be taken as (9)

The influence of moisture can be accounted for in a similar manner [24-25] and function f2(W) can be introduced under the integral, where W is determined from the weight increase in percent at moisture saturation level

(10)

Here, W0 is some empirical constant of the percent weight decrease of an absolutely dry material compared to the weight of the material under the room moisture conditions.

The constitutive equation becomes

(11)

Summary. Substantial experimental data verifies the validity of such an approach. With increasing moisture-saturation, the effect of creep becomes increasingly more pronounced and the stress-strain diagrams shift increasingly further down. On the other hand, viscous effects will become considerably instantaneous deformation curve. This makes one think that the application of the moisture influence function should be the same as the temperature function. However, the experiments on moisture influence are labor-intensive and demand carefully taken measurements with special equipment, thus very few have been recorded in literature.

The most important problem is the construction of a moisture-saturation scale with at least two basic points. In terms of temperature in degrees of Celsius, the points are the melting temperature of ice, the SI temperature system, the points used are at absolute zero, 0K (function f1(T) can be built. But since there is no sufficiently reliable scale available to measure the moisture saturation, it is a difficult task to build the moisture influence function f2(W). However, in (10), the value W0 has a definite physical sense. If W% becomes equal to W0 during a drying process, then f2(W)=0 and as shown in (11), the equation of the instantaneous deformation curve ( ) can be written in a form independent of any other loading conditions. The value W0 must be one of the basic points on the scale of moisture saturation with the other point being the moisture content at room conditions.

The validity of such approach had been confirmed in [26], where the influence of various effects, including moisture, on the strength properties of polymethylmetacrilate was considered. It was shown that at a low loading rate of 10-3 sec-1, the presence of moisture leads to a considerable decrease of strength and increase of viscous effects. Another possible application of the approach in connection with experimental data on polymers and composites is given in [27-28].

However, the combined influence of temperature and moisture on the behavior of materials can be more complex than described above by equation (11). Some secondary effects might appear.

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