Generating line spectra from experimental responses. Part IV ...

Rheologica Acta

Rheol Acta 33:60-70 (1994)

Generating line spectra from experimental responses. Part IV: Application to experimental data I. Emri and N.W. Tschoegl University of Ljubljana, Ljubljana, Slovenia California Institute of Technology, Pasadena, California, USA

Abstract: The previously reported algorithms for deriving line spectra (respondance time distributions) from synthetic or smoothed experimental responses is here extended to experimental data. The earlier algorithm was modified to improve performance in the presence of experimental errors. The effect of smoothing the data with the aid of the cubic spline function was examined. The performance of the modified algorithm was studied comprehensively. Auto-predictions and cross-predictions of storage and loss compliances from the generated line spectra were in excellent agreement. In equally good agreement were the line spectra obtained from compliance data and from stress relaxation data obtained on the same material. Key words: Line spectrum - relaxation spectrum - retardation spectrum spline function

Introduction

In three previous papers (Emri and Tschoegl, 1993; Tschoegl and Emri, 1993; Tschoegl and Emri, 1992) we reported three forms of an iterative computer algorithm that determine respondance time (relaxation or retardation time) distributions f r o m simulated (synthetic) or smoothed experimental data. In the first paper we described the f o r m of the algorithm in which it applies to relaxation modulus or creep compliance data. In the second paper we examined the algorithm as applied to storage and loss modulus or compliance data. Finally, in the third paper we described the application of the algorithm to the interconversion of relaxation and retardation line spectra. In all three papers we examined synthetic data that were obtained f r o m suitable mathematical generating functions (Tschoegl, 1989, p. 314ff). These data were free f r o m r a n d o m experimental error, and allowed us to demonstrate the nature and the power of the algorithm without the need to take into account possible complications arising f r o m superposed r a n d o m experimental error. The present paper is the fourth in this series, and deals with experimental data, i.e., data that are not free of experimental error. In contrast to all other methods of which we are aware; our algorithms compute each spectrum line

utilizing only well-defined subsets (windows) of the complete set of experimental data. The subsets are chosen according to the f o r m of the kernel functions in the canonical (i.e., integral) representations (Tschoegl, 1989, p. 160) of the experimental response functions. Each spectrum line is obtained f r o m the data within its window using least squares minimization. The minimization effectively smoothes the data within each window that contains more than one datum point. The procedures which we have applied in the earlier papers to synthetic data work well even with experimental data in the presence of moderate experimental error. They do so particularly when the experimental data are subjected to additional smoothing. We used a cubic spline function (de Boer, 1978) to smooth the data on the logarithmic time (or frequency) scale, and - as shown later - have obtained excellent results with it. I f the data set is reasonably dense, the self-smoothing property of the algorithms m a y still cope quite well with normal data scatter even in the absence of additional smoothing. I f there is appreciable data scatter over relatively sparse data, it is advisable to make use of a modification designed to improve the performance of the algorithms. The modification consists in minimizing the relative instead of the absolute error between the

I. Emri and N..W. Tschoegl, Generating line spectra from experimental responses

61

data and the calculated approximation. This modification increases the computation time but does so only slightly. We compare typical results obtained with and without the modification in the experimental section.

the experimentally obtained material function be given by the set {(~(zj);j = 1,2 . . . . . M} of M datum points. Each of these points can then be modeled by

ModifiCation to the algorithms

6(zj)=qGg-(Gg-{G~})

When we applied the algorithms to computer generated (synthetic) response functions, the error term for the j t h spectrum line consisted only of the (normalized) error of approximation, A~, because any (normalized) experimental error, Aft, could only have resulted from limitations on the resolving power of the computer and was, therefore, effectively zero. This allowed us to find an explicit, closed form expression for evaluating the jth spectrum line based on the minimization of the absolute error. When working with experimental data this procedure - as mentioned above - may or may not serve depending on the magnitude of the experimental error and on the number of datum points available in each window. Non-negligible experimental errors on relatively sparse data sets may significantly affect the evaluation. To illustrate: let the overall accuracy of the experimental determination be, say, 10°70. In the transition region there may well arise a difference of the order of one decade in the value of the modulus or the compliance at the two ends of the window. Thus, an error of 10070 at the higher end is of the same order of magnitude as the modulus or compliance value at the lower one. In this case the absolute value Of the :error at the higher end is itself larger than the magnitude of any datum point at the lower end. These datum points therefore do not contribute to the evaluation of the spectrum line. T o overcome this problem it is necessary to base the' minimization on the relative rather than the absolute error. This effectively equalizes the influence of the errors associated with all the datum points within the window. Unfortunately, minimization of the relative error leads to an expression for the evaluation of each line that cannot be solved in closed form. We outline the required modification in what follows. We remark that this modification does not simply replace the calculations involving the absolute error which we have introduced earlier (Emri and Tschoegl, 1983; Tschoegl and Emri, 1993; Tschoegl and Emri, 1992) but is used in addition to them. In introducing the relative error modification we now first consider data obtained in strain response, i.e., as relaxation, storage, or loss modulus data. Let

{Ge} + (Gg -{

Ge}) .= 1h (7:i)K(z"i, Zj) + Oj

~lh(zi)K(zi, ZJ)+OJ (1)

(Gg-{Ge}) i~=l h(gi)K(Ti, zj)-4- Oj . Here, z} is either t} or o~j, 6 ( z j ) is either G(tj), G'(co}), or G"(o)j), h(z'i) is the normal&ed i th line of the relaxation spectrum, H ( r ) , K(7:i,zj) is the appropriate kernel function, i.e., r

K(7:i, Zj) =

tj/ri) , z} = tj

exp ( 1

22'

I +oJj Ti

~ojri 1 "]-O)j2 2T' i

zJ=~°s

(2)

Zj = O)j ,

and Oj is the term for the absolute error as defined earlier (Emri and Tschoegl, 1993). N i s the preselected number of spectrum lines to be calculated. The brackets, {}, indicate that Ge is present when the material is arrheodictic (i.e., does not show steady-state flow) and absent otherwise. Computation of spectra from the respondances (Tschoegl and Emri, 1992), i.e., from the relaxance, Q(s), and the retardance, U(s), requires the kernel function 1/(1 +risj). Let us now turn to data obtained in stress response, i.e., as creep compliance, or storage or loss compliance data. If the response function is given by the set [~(zj);j = 1 , 2 , . . . , M } , then we may write

,~(zj) = N

r

l (zi)K (ri, Zj) + {(bfZj} + Oj

J~- (J~- 4) i=1 N

Jg+(J[e°}-Jg) ~ l(zi)K(ri, Zg)+Oj

(3)

i=1 N

(j~l_ 4) E l(rOK(r~,z;) + {¢~i/z;] + O ; . i=1

Here, ,~(zj) is either J(tj),J'(o~j), or J"(%), l(ri) is the normalized i th line o f the retardation spectrum,

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Rheologica Acta, Vol. 33, No. 1 (1994)

L(7), and Of is the steady-state fluidity. The brackets, {}, indicate that the flow term is present when the material is rheodictic (i.e., exhibits steadystate flow) but is absent otherwise. Similarly, J~} represents the steady-state compliance, j0, in a rheodictic, and the equilibrium compliance, Je, in an arrheodictic material. We now introduce the dimensionless functions

r G(Zj)-[Ge}

~ (zj) =

a g -- B (Zj)

-{Gel

Gg

(~(zj) ,_ Og-[Gel

or

or

relative error of approximation ,,~ (Zj)

-

-

]g

(4)

J[e°1- Jg or

@(ZJ)--[Of//Z;} j~l_ jg

=

(5)

Z%)+A s ,

where

E

fi2,

where the skd and Sk, u are the first and the last discrete points in the window belonging to the k th spectrum line. Minimizing the error according to OQe _ 0

(11)

,

Ofk where fk is the k th spectrum line, we find

8Qg_2 E 5j =--0 , 0fk J=sk,~ 8fe

denotes the theoretical (error-free) value of the normalized experimental datum point, ~ (zj), in which f('ri) stands for either h(l:i) or l(Ti). Aj is the normalized absolute error given by

and obtain

[Z(zj)l s

(13)

Singling out the k th relaxation time, Eq. (6) can then be recast as k-1

Z % ) = ~ f(ri)K(ri, z)) +f('cg)K(7:k,Z)) i=I N

Aj-j{eOOlJ_jg

(7)

and can now be expressed as

Aj = 5[ (zj) - Z ( z 2) .

(12)

(6)

i=1

Gg-{G.}Oior

(10)

J = Sk, l

j = sk,l

Z % ) = ~ f(ri)K(ri, zfl

(9)

J = Sk, u

QR=

J=~k,. f (z]) -Z(zj) ~ (zj)K(rg, Zj) = 0 . i=N

Aj-

~j= AJ = ¢2(zj)-Z(zj) z%) z%)

The sum of squares of the Oj within Window 2 then becomes

These functions are obtained through normalization using the viscoelastic c o n s t a n t s Gg and {Ge}, or Jle°l and Jg, respectively (Tschoegl, 1989, p. 229). In principle, nomalization is a matter of convenience (Emri and Tschoegl, 1993). Thus, if the normalization factors, Gg-[Ge} or fle°l-Jg, are unknown, they may be replaced by the difference between the highest and the lowest members of the data set. The same normalization factors are then used in denormalizing the calculated normalized spectra when this is desired. When making comparisons between spectra that were normalized with different normalization factors, both spectra must, of course, be denormalized with the same factor. With the help of Eqs. (4), we can write

s%)

The algorithms calculate each spectrum line from certain portions of the data set which we have called Boundary Window or Window 1, and Modeling Window or Window 2, respectively. These windows contain those subsets of a given set of data from which datum points are selected for the determination of each spectrum line as explained in detail previously (Emri and Tschoegl, 1993; Tschoegl and Emri, 1993; Tschoegl and Emri, 1992). For the reasons explained at the beginning of this section we now introduce the

(8)

+ i = k~+ l f(ri)K(ri, z j ) ,

(14)

Those values of the KQ:i, zj) that are located sufficiently far to the left of rk effectively vanish. We can, therefore, rewrite Eq. (14) as

I. Emri and N.W. Tschoegl, Generating line spectra from experimental responses k-1

Z(zj) = ~ f(ri)K(ri, Zj) + f(rk)K(rk, Zj) i=m N

+

~

(15)

f(ri)K(zi, Zj) ,

i=k+l

where m depends on the particular form of K(7:i, zj) and on the number of spectrum lines per decade as explained in detail in our earlier papers (Emri and Tschoegl, 1993; Tschoegl and Emri, 1993; Emri and Tschoegl, 1992). Using Eq. (11) and the abbreviation k-1

N

Y( Zj) = ~ f ( r i l K ( r i , z j ) + i=m

~

f(zilK(zi, Zj)

i=k+l

(16)

to simplify the notation, we obtain the equation

j : se,. S. (zj) - [Y (zj) + f(vk)K(rk, gj)l

E

J = sk, l

[ Y (Zj) +f(rk)K(rk, Zj)] 3

× ~ (zj)K(rk, zj) = 0 .

(17)

This equation must be satisfied to minimize the sum of squares of the relative quadratic errors, Qk, in the k th window. It must be solved numerically by iteration for it cannot be made explicit for f(zk) as is possible when minimizing the absolute error. We therefore now recast Eq. (17) in a form in which it becomes suitable for recursion. We let

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in a first sweep through the data using the algorithms based on minimization of the absolute error (Emri and Tschoegl, 1993; Tschoegl and Emri, 1993; Tschoegl and Emri, 1992). Such a sweep is quite fast since it involves no iteration. We found that repeated sweeping tends to speed up the final iterations based on the relative error. In the calculations reported here, we swept three times. Even the iterations are generally quite fast. Other things being equal, the computation time depends on the nature of the kernel function (cf. Eqs. (2)). On a PC equipped with a 33 MHz Intel 384 CPU chip and a math coprocessor a line spectrum can generally be obtained from relaxation modulus or creep compliance data in under a minute. Storage modulus and compliance data may take 2 to 3 minutes, while loss modulus and compliance data may require 5 to 6 minutes. Obtaining a spectrum from a respondance may take longer. The respondances are the operational forms of the impulse responses, the relaxance and the retardance (Tschoegl, 1989, p. 40). Since they are one another's reciprocal, they are the basis for converting from relaxation to retardation behavior and vice versa (Tschoegl and Emri, 1992). A computer program for carrying out these and other rheological operations is expected to become available shortly. 2)

Experimental

(20)

We extensively examined two sets of experimental data obtained on an uncrosslinked polyisobutylene. Samples of this material had been distributed by the (then) U.S. National Bureau of Standards to a number of experimenters in a round-robin exercise. The exercise was designed to test the validity of different experimental determinations of the linear viscoelastic properties of a typical polymer. The first set of the polyisobutylene data was obtained in tensile stress relaxation experiments by Catsift and Tobolsky (1955) and will be referred to under the acronym CT. The second set represents storage and loss compliance data in shear obtained and discussed by Fitzgerald, Grandine, and Ferry (1953), and by Ferry, Grandine, and Fitzgerald (1953). These data will be referred to under the acronym FGF. We will use these acronyms further on wherever they are required for the sake of clarity. A comparison of the two sets allows us to assess the use of the algorithm in comparing experimental data

The starting set of spectrum lines for the iteration is {f(l) (rk); k = 1,2 . . . . . N}. This starting set is obtained

~) Interested readers should write to Minerva International, 566 California Blvd, Pasadena, CA 91105, USA.

gi+l)(Tk) -- • []e(i)(2"k)] [/{°(rk)]

(18)

where f(i)(rk) is the value of f(rk) obtained in the ith iteration, while

d ~ o (~)1 = J =