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Statistical theory of stresses in rubber-like materials.

The general theoretical equation representing principal stresses gives them in the form

[sigma = [[sigma].sub.m ] + [sigms.sub.a] tan[phi]

where [[sigma].sub.m] represents the mean stress, [[sigma].sub.a] represents the intensity of shear stresses and [phi] is an angle quantifying the distinction between the current state of stress and a reference one. It turns out that the [phi] plays an essential role in the constitutive modeling of rubber-like materials. In physically justifying this role, we provide the physical interpretation of [phi] and describe its role in the energetic components of the material. Based on this, we then mechanically characterize the principal values of the stress tensor: They represent averages of the pressures created by the distributions of intermolecular forces inside material. These ensembles of forces are then discussed in detail and are related to such phenomena as rubber friction, rubber fatigue and fracture, energy dissipation and heat build-up in rubber cycling. The theory is illustrated by practical examples.

It is hardly necessary to elaborate upon the importance of an energy function when it comes to characterization of rubbers for industry applications. Suffice it to say that, at the modern level of industrial operations, the necessary calculations regarding performance of rubber parts depend heavily on our knowledge of the energy function for rubber. Now, the basis of this energy function is, on one hand, the experimental data on rubber samples and, on the other hand, our capability of insight into the intimate molecular structure of rubber. This last part of the problem seems to be the responsibility of statistics, unilaterally assisted by experiments of a non-mechanical nature (chemical, x ray, neutron scattering, etc.). The statistics concentrate mainly upon chain molecular properties, whereby the macromolecule is supposed to have a random shape in space, thus relating to stochastic concepts like random walk or Brownian motion.

In the engineering of rubber parts, one hardly has the time and means to dedicate oneself to a detailed study, along the lines just sketched, of the rubber itself. Rather, the only thing at one's disposal at this level of research is the simple mechanical experiment of loading specimens of rubber, which are close to the condition of application envisaged. This fact raises a legitimate question: Is there a possibility to infer upon the internal macromolecular properties of robbers based on just simple mechanical experiments only, without relying on sophisticated experiments of any other nature? We dedicated some study to this problem, and the present work is one of the reports on results. Partial results showing to what extent this approach, in its mechanical theoretical form is helping in modeling, have been reported elsewhere (ref. 1). In the present work, we only concentrate upon the details of a statistical theory.

Two are basically the experimental observations to start with in developing the theory. The first one is of a kinematical nature: The experimental loading data for rubber are well represented by a linear function of trigonometric tangent of a linear function of time, something of the form

f = A + B * tan (a * t + b)

Here, f is the experimental load, t is the time of experiment and A, B, a and b are constants. Evidently, if the experiments are conducted under constant strain rate, as is usually the case, then the time in the above equation can be replaced by the measure of strain. The equation fits the data fairly well in any kind of simple experiments. We have two examples in figures 1 and 2, referring to a slender (uniaxial) specimen and a strap 6.5 cm long, 13 cm wide and 1.5 mm thick. The general conclusion drawn from such examples is that the tangent fit goes well in the high strain region of data, the goodness-to-fit deteriorating significantly in the low strain region. This deterioration seems to be more marked as the experiment departs from the condition of uniaxiality, for instance in the case of compression or extension of thicker specimens. This fact compelled us to look into another kind of curve fitting of the data, which occasioned a second experimental observation necessary in developing the theory. Namely, no matter the type of simple loading experiment we consider and the condition of the specimens, the data are exquisitely represented over the whole range of strain by a simple ratio of two polynomials, one of third degree and the other of second degree, something of the form

f = x * A[x.sup.2] + Bx + C a[x.sup.2] + bx + x

Here, x is a measure of strain (does not matter what measure), f is the load, and A, B, C, a, b and c are constants. Examples of such rational function fit are shown in figures 1 and 2 by comparison with the tangent fit, and in figure 3, where a comparison of uniaxial data and planar data is presented. Moreover, it turns out that, if in the above rational function we switch the places of x and f, the curve fit of the experimental data remains valid and is of the same quality (see figure 4).


Given these experimental observations, our general task above simplified, so to speak, came down to unraveling a relationship between the two types of curve fit just described. This relationship, coming simply from the fact that the two relations are the expression of the same statistics, is reported in what follows.

Stresses in an isotropic continuum

Let us start with a brief digression on what would be an experimentally measured stress according to established theories of deforming continuum. This is important in that the analysis produces a macroscopic quantity to be targeted by the microscopic statistical physics of molecular chains. As a first guess, we can say that what is physically perceived in experiments is a sort of average, more specifically a space average of all interparticle forces inside the material. So our task would have to be that of relating the experimental stress to whatever we can discover as a rational average of the intermolecular forces in a solid.

On the other hand, it is widely accepted in engineering problems that the stress in a piece of solid is characterized by a positively defined symmetric tensor. Experimentally, the philosophy is that we are supposed to measure its eigenvalues. As far as tensors are employed to characterize stresses, and we are using a continuum theory, there are two averages we might suspect as emerging into measurement: One of them is the average volumetric stress (pressure), the other is the average shear stress in a point. One naturally understands that we refer the stress in a certain point in a solid to both directions and planes through that point. A part of the task is then to evaluate the experimental stress as functions of the above mentioned averages, and then check to see if this description blends within the experimental constitutive behavior as based on them.

In order to do this, we have to bring on stage specific assumptions of isotropy and homogeneity: A body is assumed isotropic in the sense that it has normal and shear components of stress on any plane through a point of that solid, and homogeneous in the sense that this situation is valid for any point of the solid. This characterization of isotropy is akin a continuum assumption. For, a real body has a physical structure manifested especially by the fact that the internal forces act on, and normal to, privileged planes in privileged points. This is most obvious in crystalline solids, but we can confidently extend it as a general characteristic of the solid bodies. The fact that a solid experiences internal forces normal and tangential with respect to each conceivable plane in any point means not only that it is a continuum, but a homogeneous continuum. Let us put this assumption into equation. This was first done, as far as we know, by V.V. Novozhilov (ref. 2). There are more recent approaches to the problem; all of them are based on the same geometrical idea (see a discussion by Beatty, ref. 3). We will follow, however, the original work of Novozhilov, first for the sake of simplicity and, secondly, because it applies to any tensor. Thus, whatever we say about stresses is valid also for any physical entity that happens to be quantified by a tensor.

Call n the unit normal vector to a plane through a point in a continuum solid. Then the normal component of the Cartesian tensor [sigma] on that plane is, using the summation convention over repeated indices:

[[sigma].sub.n = [[sigma].sub.ij][n.sub.j][n.sub.j] (1)

The equation (1) allows us to write the square of shear component of the tensor on that very plane as


These equations are valid if the vector of stress with respect to the plane (the traction vector) is defined as

[[sigma].sub.k] = [[sigma].sub.kl][n.sub.l] (3)

In other words, the stress tensor defines a linear application from the unit sphere to stress vectors. To simplify the things as much as possible, we use the fact that the stress tensor, being symmetric, has orthogonal principal directions in every point of the solid. Thus, we refer the calculations to a local frame of reference as given by the principal directions of the tensor. The quantities from equations (1) and (2) can then be written as


Now we take, in our reference frame,

[n.sub.1] = sin[theta] cos[phi] [n.sub.2] = sin[theta] sin[phi] [n.sub.3] = cos[theta] (5)

where [theta] and [phi] are spherical polar angles and define the space average of a certain quantity Q(n) in a point by the formula


with [OMEGA] the solid angle around the point and d[[OMEGA].sub.n] the elementary solid angle in the direction n:

[OMEGA] = 4[pi], d[[OMEGA].sub.n] = sin[theta]d[theta]d [phi] (7)

Applying the definition from equation (6) to the quantities in equation (4), and performing the necessary integrations, gives

[[sigma].sub.n = 1/3 ([[sigma].sub.1] + [[sigma].sub.2] + [[sigma].sub.3])

[[sigma].sup.2.sub.t] = 1/15 [[([[sigma].sub.1] - [[sigma].sub.2]).sup.2] + [([[sigma].sub.3] - [[sigma].sub.1].sup.2] (8)

We are thus to accept that these quantities, or quantities related to them, are somehow perceived experimentally in measurements, at least if we are working with an isotropic material, which is the case of rubber. Consequently, an experimental measure of the stress in solids, at any level of analysis, must be defined as a function of the two quantities from equation (8).

Experimental values in terms of space averages

Here we show that the experimental specification of the stresses is always accompanied by an inherent indeterminacy of the principal stresses, calling for supplementary statistics over those based on directions in an isotropic continuum, as pre-sented in the previous section. The indeterminacy in question occurs, essentially, because our claim that, at the macroscopic level, we cannot perceive but averages, depending on the quantities from equation (8). Now, there are two levels of organization in a solid (ref. 4). The first one is the one just presented, where the molecules are simply taken as individuals. This level of organization complies well with the tensor concept of stresses in the solid. However, there is another level of organization, which in the case of filled rubbers seems to be critical, that of macromolecules. The representation of stresses has further to cope with this reality, and the theory provides indeed the means for this. In order to see how and where, recall that the principal values of a 3 x 3 matrix are the roots of a cubic polynomial (the characteristic equation of that matrix). In solving a cubic equation (ref. 5), it helps to know two more distinguished polynomials specifically related (ref. 5) to a certain cubic one: One of them is second degree and is called Hessian (H) of the given cubic, the other is third degree and is called Jacobian (T) of the given cubic. We then have the following identity (ref. 5) between cubic itself (f), its Hessian (H) and its Jacobian (T):

4[H.sup.3] = [theta] * [f.sup.2] - [T.sup.2] (9)

Here, [DELTA] denotes the so-called discriminant of the given cubic. The expression in the right hand side of this equation can be decomposed into two factors, each of the third degree, because a cubic and its Jacobian are prime with respect to each other. On the other hand, the left hand side is a product of two perfect cubes, for the Hessian is a quadratic polynomial. The identity (9) then shows that each factor of the right hand side is proportional to a factor of the expression from the left hand side, and this proportionality can be taken in two ways at will. However, for a fixed choice between those two ways, the proportionality factors are reciprocal to one another. Indeed, the Hessian can be factorized in infinitely many ways as

H = mU * 1/m V (10)

where U and V are first degree reciprocally prime binomials and m is any nonzero number. Thus, the identity (9) can be written as the system

[square root of ([DELTA])] f + T = 2[m.sup.3][U.sup.3], [square root of (DELTA)] T = 2[m.sup.-3][V.sup.3] (11)

Adding these expressions together gives:

[square root of ([DELTA]) f = [m.sup.3][U.sup.3] + [m.sup.-3][V.sup.3] (12)

One can further decompose the right hand side in linear factors, to the effect that (12) becomes

[square root of ([DELTA]) f = (mU + [m.sup.-1]V) * ([omega]mU + [[omega].sup.2][m.sup.-1]V) * ([[omega].sup.2]mU + [omega][m.sup.-1]V) (13)

where [omega] [equivalent to] -1/2 + i[square root of (3/2)] is the cubic root of unity. This form allows us to find the roots of the cubic equation f = 0. For the sake of completeness, we mention that the Jacobian of a cubic can also be obtained from (11) as a difference of cubes:

T = [m.sup.3][U.sup.3] - [m.sup.-3][V.sup.3]

= (mU - [m.sup.-1] V) * ([omega]mU - [[omega].sup.2][m.sup.-1]V) * ([[omega].sup.2]mU + [omega][m.sup.-1]V) (14)

This shows that the roots of Jacobian are of the same nature as the roots of the cubic itself: In the formula giving the roots of our cubic, we do not have to change but a sign in both the denominator and numerator in order to get the roots of the corresponding Jacobian. Let us write the roots of the cubic equation as they come out from equation (13).

For a cubic with real roots (the case of characteristic equation of the stress matrix), the roots of its Hessian are the complex, h, and its complex conjugate, h say. Likewise, m is complex of unit modulus. Equation (13) then gives the roots as


with k [equivalent to] [m.sup.-2]. These equations offer the physical meaning of the complex numbers h, [bar.h] and k in terms of principal values. First, by simply solving them we have


where we used the equation (8) and [phi] is the arbitrary phase of k. Thus, h is a linear combination of the two averages from equation (8), but this is not sufficient to give physical meaning to this complex number.

For a known tensor, which is usually supposed to be the case in continuum mechanics, h is perfectly defined by the principal values of the tensor. However, in experimental measurements, this is hardly the case. Rather, inside the material, there is an addition of forces induced by experimental constraints or constraints of another nature, and what we measure is a number depending not only on the quantities from equation (8). Here, the phase [phi] is completely arbitrary, and we show now that it is dictated by the orientation of the shear stresses in a certain octahedral plane. In order to do this, notice that the shear stresses giving the second one of the quantities from equation (8) can be taken as a vector on the octahedral plane, in a reference frame represented by the principal directions of the stress. The components of this vector are given by the principal values of the deviator of stress. Specifically, the vector is given by the following column matrix:


where we used the equations (15) for the principal stresses. Now, take as reference in our octahedral plane the vector (17) for [phi] = 0 (k = 1), when the root of the Hessian is strictly defined by the principal values, i.e.,


The angle between vectors (17) and (18) is given by

cos[theta] [equivalent to] 1/2 ([square root of (k)] + 1/[square root of (k)] (19)

Thus, the phase of k is in fact the angle made by the octahedral shear stress vector with its correspondent from the case where this vector is completely determined by the principal values. This last case is the ideal case, always supposed to happen in the tensor theory of stresses.

Equation (16) above gives, therefore, the principal stresses in the form

[sigma] = a + b * tan [phi] (20)

where a and b are average properties, obtained on account of tensor character of stresses, and [phi] is an angle, having a part determined by stresses and a part completely arbitrary. "Tan" denotes the trigonometric tangent function. The experimental values relate to simple states of stress. For instance, in the uniaxial case, [phi] from equation (20) is identical with [phi] from equation (16), so that the experimental values of stresses in the uniaxial case are dependent on a completely arbitrary parameter. The same happens for the biaxial state and, in general, any state of stress satisfying the linear relation

2[[sigma].sub.1] - (1 + B [square root of (3)])[[sigma].sub.2] - (1 - B [square root of (3)])[[sigma].sub.3] = 0 (21)

where B is a constant number, is in the same situation: Its principal stresses do not depend but on the averages from equation (8) and an arbitrary parameter. This parameter is practically the phase angle from equation (16). We can guess that it is somehow related to the progress of deformation. By studying the uniaxial data, for instance, we can prove that this is the case. Indeed, the uniaxial data from constant rate experiments can be closely represented by a function like (20) with [phi] = m * [epsilon] + n, where [epsilon] is the displacement and m and n are some constants (figures 1 and 2).

The associated statistical theory

Up to this point, we have discussed only a type of statistics related to the tensor character of stresses. The presence of a still arbitrary parameter in the expression of experimental stresses shows that the job is not completely done when considering the stress as a tensor. This observation has been made earlier by Fried (ref. 4). This author's contention seems to be that we need two kinds of averages; one as above, dictated by the tensor character of stresses, the other over the physical ensemble as represented by the macromolecular chains. We will exploit the idea in some other direction, by starting with the observation that, due to the presence of our arbitrary angle, the fact that the left hand side of equation (20) qualifies as a mean suffices in order to give us the possibility to characterize the statistics beyond the directional considerations presented above. Intuitively, the left hand side of equation (20) must be a mean over the equilibrium forces between molecules, and the formula (20) reflects only partially its structure, while showing something more: This equilibrium depends upon the stage of deformation process as represented by [phi], and this stage is controlled by the microscopic association of the molecules into macromolecules. This is, again, a natural conclusion, coping with our intuitive image of deformation process and with the definition of stresses as equilibrium quantities. We wonder what is the statistical nature of this ensemble, i.e., basically what is the distribution function characterizing it. In order to show this, we simply differentiate equation (20) with respect to [phi] and then eliminate this parameter. The result is the following differential equation of Riccati type:

d[sigma] / d[sigma] = A [[sigma].sup.2] + 2B[sigma] + C (22)

where A, B and C are some constants depending on the Novozhilov's directional means from equation (8).

We have seen before what is the physical meaning of [phi] for a certain state of stress. If it is in relation with the progress of deformation, this means it is measurable. To obtain a further insight into the nature of the statistics we are after, let us notice that the most non-committal probability distribution that we can obtain from experimental data having at our disposal only measured values of the mean is, according to the principle of maximum information entropy (ref. 6), an exponential distribution: If X is the physical quantity to be statistically characterized, and [theta] is the measured value or a certain function of the measured value, then the exponential distribution of the values [xi] of X over a certain ensemble of values is given by

[P.sub.[theta]] (x) = [theta] * [e.sup.-[theta][xi]] (23)

Now, in real cases, it may happen that X is not allowed to run on the whole real positive axis. This may be due to the fact that the real axis is already endowed with an apriori measure or to the fact that X has a limited range, there is a partition of real axis, which, from a mathematical point of view, comes to the same model. The model describes the following family of elementary probabilities depending on one experimental parameter [theta]:

[F.sub.[theta]] (d[xi]) = ([N.sub.m] [([theta])).sup.-1] [e.sup.-[theta][xi]] (24)

where m (d[xi]) is the apriori measure of the real line, and the normalization factor is defined by equation


These probability distributions, used in all kinds of physical applications, have the remarkable property, independent of the apriori measure m (m(d[xi])) of the real numbers, that their variance (VAR) can be related to their mean (x) by


A particular subclass of exponentials, with large applications, in experiments as well as in theory, is the family of distributions with quadratic variance function, for which the variance as a function of mean is a quadratic polynomial. For these exponentials, the mean satisfies the differential equation

dx([theta]) / d[theta] = [r.sub.1] [x.sup.2] + 2[r.sub.2]x + [r.sub.3] (27)

where [r.sub.1], [r.sub.2] and [r.sub.3] are three real constants characterizing the distribution and accessible to measurement. Comparing our differential equation (22) for experimental stresses with the equation here obtained for the mean of a quadratic variance distribution function, one can say that the experimental stress is the mean of a quadratic variance distribution, provided we interpret the angle [phi] as the parameter of a family of these distributions characterizing our rubber. Then, under condition [r.sub.1][r.sub.3] - [r.sup.2.sub.2] > 0


This formula is of the form given in equation (20). If now the parameter [theta] is taken as the angle of equation (16), the equation (28) can be interpreted as the mean stress on a certain plane. This very plane is not so important by itself and, for experimental purposes, can be identified with a cross-sectional plane of the experimental specimen. Then the numbers [r.sub.1], [r.sub.2] and [r.sub.3] or their counterparts in experimental records: x = A + B tan ([alpha][theta] + [beta])


are certainly related to the internal molecular properties of the specimen. In the specific case of rubbers, these properties are those of the macromolecular chains and filler dispersions.

The above considerations leave one important subject uncovered; the relationship between the parameter [theta] of the family of quadratic variance distribution functions and the experimental recording. All we can say is what already has been said before, namely, that the engineering experience to date indicates that the angle [phi] of representation of the stresses is in direct connection with the progress of deformation; this fact is reinforced by the close representation of experimental data for uniaxial tension (or compression) by equation (20) with the phase depending linearly on the recorded deformation (figures 1 and 2). If, in a general situation, the deformation is quantified by the experimental stretch [lambda], then a relationship between [theta] and [lambda] is to be expected. We are now in no position of finding directly this relationship, but there is an indirect way to it, namely, the accordance with experimental data. Here enters the second stage of our experimental observations, namely about representing the data with a rational function. In other words, if, according to our philosophy, the experimental stress must be a mean, how do we account for a mean that is a rational function of stretch? Notice that everything in exponential statistics depends on the measure m (d[xi]) we use for the characterization of the ideal continuum approximating the real body. Thus, we must find the measures that best suit our experimental needs. These measures may not always be positive, but the fact remains that no matter which feature, they must always give results in finite terms. In order to exemplify this fact, we choose the polynomial measures as given in table 1. The table offers the normalization factors as functions of [theta] for different polynomial measures of the real line. By the simple transformation

= 1 / [lambda] (30)

we can bring these normalization factors to experimental terms and see if we can have a real situation. In short, we take the situation represented by the last row of table 1 in terms of the physical parameter [lambda]:


For this family of distributions, the mean is

[bar.x] = [lambda] 6a [[lambda].sup.2] + 4b[lambda] + c / 2a [[lambda].sup.2] + 2b[lambda] + c (32)

This mean can be cast into form

[bar.x] = A + B [lambda] + C / [lambda] + [alpha] + D / [lambda] + [beta] (33)

more suitable for practical purposes. The experimental force must be proportional with the quantity (32) [or (33), does not really matter], which thus replaces the equation (20) from the ideal uniaxial case. Notice that what is really changing here is not the nature of the statistic to be considered, but the experimental measure of the stage of deformation characterizing the ensembles of equilibrium intermolecular forces inside rubber. Mention should be made that, based on these results, the stretches do not seem to be tensors in their algebraical nature, as usually taken for granted. Figures 1-4 show the quality of representing different kinds of experimental data by function (32).

Consequences regarding the energy function of rubbers

In view of equation (33), the nominal stress, S say, will then be represented in a particularly convenient form by


The advantage of using equation (34) for the nominal stress becomes more obvious if we integrate it in order to obtain the energy function for the model. The result is


Now, equation (35) can be advantageously used if we accept that the stretch is indeed a tensor, and what we measure in the experiment is just one of its eigenvalues. Assuming then, on one hand, isotropy to the effect that equation (34) is the same no matter of the eigenvalue of the stretch tensor, and on the other hand, the Valanis-Landel hypothesis (ref. 7) stating that the total energy function is the sum of the three components of the energy along the principal directions of stretch tensor, we have


where we have used the notations


Notice that this energy is not a pure first invariant theory, and this fact algebraically complicates the matter. However, the complication is no more than algebraic and it can be easily overcome. Notice also that we have here both a generalization of the neohookean model and of the Varga type constitutive model (ref. 8). This result is particularly encouraging in our endeavor for constitutive modeling. Partial results based on this energy function have already been reported (ref. 1). However, these results are not entirely satisfactory, and we relegate this tact to the oversimplifying assumption that the stretch should be a tensor.

Summary and conclusions

Summarizing this work, we can say that it is indeed possible to characterize statistically the internal structure of rubbers just starting from simple mechanical experiments. In other words, the internal statistic of rubbers is intimately imprinted into the experimental loading curve. This kind of data pleads for a non-Gaussian statistic of both intermolecular forces and chain lengths, of which we give here no further details than the basics. From a technical point of view, one can say that the statistical ensembles in question are, obviously, those of intermolecular forces in equilibrium. However, we are able to say something more: This equilibrium is described by a parameter depending approximately linearly on the degree of advancement of deformation, and representing the angle of orientation of octahedral shear stress in the octahedral plane as referred to the principal directions of the stress tensor. The two kinds of curve fits we found as representing experimental data for rubbers reflect the same statistical nature of the problem. The difference between them is due to the simple fact that there are apriori different measures of the set of real numbers to be employed in the calculations of the statistics involved. Namely, in all cases, the rational representation should prevail anyway. However, it is the tangent representation that gives physical interpretation of the statistical theory. Because this tangent representation seems to work better for the ideal uniaxial case than for others, it appears that the apriori measure of the real numbers is an ingredient that becomes more and more important with the difference between experimental setup and the state of stress induced inside the specimen. This seems to be a natural thing again, inasmuch as the statistical description of different ensembles asks for different partitions of the real numbers, and these partitions are indeed represented by the apriori measures used to construct the probabilities.

In explicitly considering the two kinds of structural averages as illustrated in this work, the current theories of rubber deformation rely upon the tensor character of stretches (ref. 4), in order to use the space averages even for the characterization of macromolecules. This fact does not seem to be in order, once the stretches are modeled by simple strings in a finite number in order to obtain limited calculable results. The reason for this is, in our opinion, the fact that the stretch is not a tensor, but a parameter characterizing a dynamical equilibrium, like absolute temperature in the case of ideal gases, for instance. In any case, should the algebraical nature of stretch change, e.g., it should be considered as a vector or a matrix, this fact would be dictated only by the nature of the equilibrium distribution it describes. That changes first: Still a quadratic variance distribution function, it becomes a multidimensional distribution, having a matrix parameter characterizing the internal mechanical equilibrium necessary in the definition of stresses.

The presence of a disposable parameter in the theory is instrumental in the theory of contact stresses. The description of friction phenomena is particularly manageable, for the disposable parameter of our theory is basically kinematical in nature. The form of the coefficient of friction can then be easily explained in terms of the evolution of the stresses at the contact zone: The work of external force is, in the first stage of friction, used to create states of stress in the two materials in contact, such that their octahedral planes are parallel. In this situation, the external force has a maximum. Once this is done, the external force continues to work until the two shear vectors are brought to parallelism. The expenditure of force is gradually smaller, reaching finally a steady value. One can easily understand then that roughness works against the ideal alignment of both the octahedral planes and the two shear vectors characterizing the materials in contact.


(1.) N. Mazilu, J.L. Turner and J.D. Ulmer, report to "ABAQUS Users Conference," Boston, MA, May 2004.

(2.) V.V. Novozhilov, Prikl. Mat. MekhH., 15, p. 2 (1951), "Foundations of nonlinear theory of elasticity, "fourth printing, Graylock, Rochester, NY., 1971.

(3.) M.F. Beatty, J. Elasticity, 70, p. 65 (2003).

(4.) E. Fried, Journ. Mech. Phys. Solids, 50, p. 571 (2002).

(5.) W.S. Burnside and A. W. Panton, "The theory of equations," Dover Publications, New York, 1960.

(6.) E.T. Jaynes, Phys. Rev., 106, p. 620 (1957).

(7.) K.C. Valanis and R.F. Landel, J. Appl. Phys., 38, p. 2,997, (1967).

(8.) O.H. Varga, "Stress-strain behaviour of elastic materials," Interscience, New York, 1966.

by Nicolae Mazilu, John L. Turner and James D. Ulmer, Bridgestone Firestone North American Tire
Table 1--list of polynomial measures and
correcponding normalization factors

Apriori measure Normalization factor

m(d[xi]) = d[xi] [N.sub.m][theta] = 1 / [theta]
m(d[xi]) = [xi]d[xi] [N.sub.m][theta] = 1 / [[theta].sup.2]
m(d[xi]) = [[xi].sup.2]d[xi] [N.sub.m][theta] = 2 / [[theta].sup.3]
m(d[xi]) = [[xi].sup.3]d[xi] [N.sub.m][theta] = 6 / [[theta].sup.4]
m(d[xi]) = [[xi].sup.4]d[xi] [N.sub.m][theta] = 24 / [[theta].sup.5]
m(d[xi]) = (a[xi] + b)d[xi] [N.sub.m][theta] = a + b[theta] /
m(d[xi]) = (a[xi] + 2b[xi] + [N.sub.m][theta] = c[[theta].sup.2] +
 c)d[xi] 2b[theta] + 2a / [[theta].sup.3]
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Author:Ulmer, James D.
Publication:Rubber World
Date:Sep 1, 2006
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