- Email: [email protected]
Application of mathematical statistics to a study of long-term polymer strength
970
A . A . VALISHINand E. M. KARTASHOV
REFERENCES 1. R. Y. TING, J. Appl. Polym, Sci. 20: 3017, 1976 2. G. V. NESYN, V. N. MANZHAI and V. P. SHIBAYEV, Vysokomol. soyed. B28: 714, 1986 (Not translated in Polymer Sci. U.S.S.R.)
PolymerScienceU.S.S.R. Vol. 31, No. 4, pp. 970-975, 1989 Printedin Poland
0032-3950/89 $10.00+.00 © 1990PergamonPress pl¢
APPLICATION OF MATHEMATICAL STATISTICS TO A STUDY OF LONG-TERM POLYMER STRENC;TH* A. /tk. VALISHIN a n d E. M. KARTASHOV M. V. Lomonosov Institute of Light Chemical Technology, Moscow ( R e c e i v e d 28 June 1988)
A methodology based on multivariate regression analysis and statistical modelling is proposed for experimental investigation of dependence of long-term strength on stress, temperature, and other factors. A complete statistical analysis of the Zhurkov and Bartenev models has been carried out. The minimum size of experiment required for an unequivocal determination of parameters in a model has been found. THE level of statistical methods used in experimental investigation of strength is by no means satisfactory. Thus, from the whole arsenal of regression methods only the simplest type of one-dimensional least-squares analysis is used and even then without the appropriate statistical analysis. Multivariate regression ~nalysis, which forms the basis of statistical models described by functions of several variables, is almost completely ignored. In this paper we want to explain the modern approach to the establishment of dependences and to demonstrate it on the example of polymer strength considered as a function of time and temperature. Application of digital computers enables one to dispense with graphical methods and extrapolations that invariably lead to a loss of accuracy. Two formulae were proposed to describe the stress and temperature dependences of long-term strength of uniaxially stretched polymers: the Zhurkov law [1] for solid polymers r = ro exp (( Up - y t r ) / k T )
(1)
and the Bartenev formula for elastomers [2] r = B a - m exp (Uo/kT)
(2)
The formalism used in these equations is generally known; ~o, Uo, ~', B, and m are constants which must be found from experimental data. An accurate determination of these parameters is prerequisito for reaching valid physical conclusions. Long-term strength is usually measured separately as a function of stress and temperature; the latter dependence is often found by the so-called method of intersects [1]. The parameters in formulae (1) and (2) are found graphically by means of far extrapolations [1, 2]. Such methods are highly * Vysokomol. soyed. A31: No. 4, 877-882, 1989.
Study of long-term polymer strength
971
unstable: small errors in slopes of the straight lines are amplified by the extrapolation and result in low accuracy of the final characteristics of long-term strength. From this point of view the large scatter of values to be found in standard reference textbooks is understandable. A more promising approach to the evaluation of constants in the above equations consists in that the strength is considered as a function of at least two variables, which is then processed by methods of mathematical statistics. Let us rewrite formula (1) in the form
r/='go + fl2 y + flj2 x y , where x=tr, y = l / k T , r/=log r, flo=log to, f l 2 = M U o ; Formula (2) can be written as
(3)
ill2 = - M y ; with M = l o g e.
,I = 'go + 'g, x + 'g2 y
(4)
where x = l o g a, y = 1/kT, r/=log r, ,go-log B, 'gl = - m , and 'gz=MUo. In the terminology of,regression analysis the variables x and y are called predictors (or factors), and the variable r/is the response. The Bartenev formula (4) represents a model of first order since it is linear in the predictors, while the Zhurkov formula (3) is a second-degree model as it contains a quadratic cross-term. The modern regression analysis includes a check whether models such as (3) and (4) adequately describe existing experimental data. If one of them is found to be adequate, its coefficients are determined along with their accuracy. If none of the models tested is found to be adequate, a more general model is conceived and the analysis starts from the beginning. Such "tuning" then leads to the optimum model. F r o m the geometrical point of view the set of all predictor values forms what is called the factorial space, in our case two-dimensional (a surface). The response (strength) is measured at several values of stress and temperature, i.e., on an "experimental subset" of points (cells) in the factorial space. Let the number of such points be N; they can be distributed in the factorial space either regularly, i.e., lie on some curves, or chaotically. The first case corresponds to some sort of experimental design, in the second instant the measurements are performed with randomly selected values of factors. References [3, 4] present a strategy which uses two levels of temperature, i.e., the measuring points lie on two parallel lines in the factorial plane. In the extended factorial space which includes a third axis coresponding to the response, the models (3) and (4) represent a response surface. It is easy to show that the response surface is a hyperbolic paraboloid in the case of the Zhurkov model (3); the so-called pole, whereat the straight lines of the stress and tempelature dependences converge, is the saddle point of the surface which contains all the above straight lines. The response surface of the Bartenev model is a simple plane. Assume that in the j-th experimental cell the response (i.e., log r) has been measured p:times,
j=I,--N. The total number of observations then is N o = ~
pj, where obviously No>~N. Let w:j
J=~
be the l-th replica (i= 1, pj) in th.e j-th experimental cell ( j = 1, N); the quantity
PJ ~=z
designates the mean in the j-th cell. The observed responses wtj can be taken as consisting of a regular (r/j) and a random part (etj); thus,
wo=~lj+el~,
i = 1 , pj,
j=I,N
(6)
Neither of them is known, only their sum is accessible experimentally. Useful information is contained only in the regular components, whilst the random part reflects "noise" due to effects beyond the control of the experimenter. The regular part is a parameter of the model; the components e~j are assumed to be independent random variables with zero mean and variance D(e~j) given by
D(ect)= trZ/toj,
i= I, p j,
j= L-,IV,
(7)
972
A. A. VALISHINand E.'M. KARTASHOV
where o~j is the statistical weight of the j-th cell respecting the possible variation of variance between individual cells. The viariance (7) is usually unknown and only the quantities a 2 and roj can be estimated from the experimental data. Z h u r k o v model. This model can be represented by
~j=Bo+F~y~+/h~zj+~j,
i=l, ~,
(8)
where z = x y , and "~j is the mean experimental error in t h e / t h cell. The measured quantities, i.e., the individual factors and the response, usually have quite different physical meaning and different dimensions; this brings about some inconvenient computational consequences, as the necessity to operate simultaneously with very small and very large numbers leads to rounding errors. To avoid this complication the factors and the response are usually coded; for the model (8) the coded variables have the form x~ = xj. x , sx
y~= yj -- y , s~
,
~j . . . .
zj -- z , s~
wj -- w s~
w ~ = ~ ,
(9a)
where N
N
N
.,
W'~"
.,
N
J-I
S2~ -
~. eojpj(xj--~)~.
(9b)
J:l
The remaining quantities which appear in formula (ga) are defined similarly. As a result, the Zhurkov model assumes the simple form k
k
k
--
/=1, N
wj=ct2yj+ct3zj+ej,
(10)
The actual physical parameters of the Zhurkov model are related to the coded coefficients in equation (10) through Uo = M - ~ ( s w / s , ) ~2 = - M-
log~o =
~-
ICsw/s,)
(sw/s,) ~
-
~3
(11)
(s~/s,)-~3
In the following we shall assume all quantities to be coded and drop the superscripts in formulae (9) and (10). Let us now introduce the following matrix notation: w r = (~z . . . . , WN),the vector of observations whose components are the coded experimental response values (the sperscript T designates matrix transpose); the error vector T = ( ~ .-., eN); the empirical regression vector w =(w~ . . . . w~) with components given by the values of response predicted by the assumed model. Let us further introduce the design matrix F of type N x 2, consisting of components given by coded values of regressors in all experimental cells,
the vector of regression coefficients r = (e2, e3), and the vector of their estimates Q-- (e ~, e 3). The sums of elements in all columns of the coded design matrix (12) are zero, whilst their squares add to unity. The coded Zhurkov model in ~he matrix notation assumes the form w= Fa+~
(13)
The statistical estimate ~ of the vector a is constructed so as to extract maximum useful information from the measured values of the response, in other words, by minimizing the random corn-
Study of long-term polymer strength
973
ponent. To do that one defines the so-called loss function, equal to the weighted sum of squared random components for all observations, i.e., j=~rGe,
(14)
where G, a diagonal matrix of type N × N, is the weight matrix with elements ¢ojpj, j--1, N. The requirement that the loss function be minimized leads to a system of equations B a = F r Gw ,
(15)
B= : a F
(16)
where is a symmetric matrix of type 2 × 2, called the information matrix. It is then easy to find the minimum extent of experiment required for unambiguous estimation of parameters in the Zhurkov model by solving the system of equations (15): both the stress and temperature must be varied at least on two levels with replication. Assuming these conditions to be satisfied, we obtain from (15) ~t = B - 1FrGw (17) This formula defines the statistical estimates ~2 and ~3 of the sought parameters in model (10): they are usually called the least squares (LS) estimates. It can be shown [5] that, provided the structure of the model is correct, the LS estimates are unbiassed, effective, and substantial. Statistical estimates of the physical parameters in the Zhurkov model can be derived from (11) in the form Uo = M
(sw/sy) ~t2
~x
-1
y=M
A
(18)
(sw/sz)cta
logro = ( N - I l r - ~ r B - I F r G )
w,
where ~r=(~vsw/s~, -zsw/s,) and l r = (I, 1. . . . ,1). The variances of the estimates (18) are "~
D(Uo)=a
2
M
--2
2
--1
(s~/s,) b,1
D (~) = a 2 M - 2(sw/s=)2b~
(19)
D (log to) = a 2 ( N - 2 Sp G- 1+ ~r B- iO, where b~ x are elements of the matrix B-~ (the inverse of the information matrix B). The estimates Uo, Y, log ro are in general correlated. The coded values of the response predicted by the model, . wl, J = 1, N, i.e., the vector of empirical regression, can be derived from the matrix equation txk
- Fa
(20)
V ( ~ ) = a: F B ~ ~F r
(21)
The variance matrix of the vector ~ is
k . m mdiwdual . . The diagonal elements are the variances of the predicted values of the response Awj experimental cells; differences in the variances reflect the differences in the accuracy of prediction. When the off-diagonal elements are non-zero, the predicted values of the response ~,j are correlated, in contrast to the directly observed values ~,j. The variance trz which appears in the above formulae can be estimated from replicate determination of response inside the experimental cells of the factorial plane: in particular, the statistic N
$2 "-
serves as an unbiassed estimate of o 2.
J=l
pj
I=l No-N
(22)
974
A. A. VALISFIINand E, M. KARTASHOV
The LS estimates of the regression coefficients are unbiassed estimates of the actual coefficients, provided the model adequately represents the experimental data. The postulated model is said to be adequate if it yields unbiassed estimates [5]. The adequacy is checked by comparing the statistic j 2 _ equation ( 2 2 ) - with another unbiassed estimate of the variance o 5. The latter is calculated from the residual sum of squares as
s:=
(w- ~)~G (w - ~) N-2
(23)
The statistic s,2 represents art unbiassed estimate of o 2 if the model structure has been correctly selected , i.e., the model is adequate. Hence, by comparing the actual values of the statistics s 2 and s2 by means of the Fisher criterion, the adequacy of the model can. be verified. The model is adequate if the ratio of ttte two statistics is insignificant; in this case the two stat~sttcs . . s,2 and s,2 are pooled into a new unbiassed estimate of the variance as: N
pj
:=
(24) Jft
1=1
Inserting the final value of a 2 into (19) and (21), we can estimate all variances and covariances. The accuracy of the estimates (18) is expressed by the confidence interval which includes the actual value of the parameter at a significance level 6 (with confidence probability 1 - 6). The 100(1-6) ~ confidence limits for the physical parameters that enter formula (1) then are Up= U o + t ( 1 - 6 / 2 , v) tD(O'o)] t/2
r=? +_t(1-6/2, v) [b(~)l '/~
(25)
log re =logz'o + t ( 1 - - 6 / 2 , v) [D(log~'o)] U2 , where t(l -tr/2, v) is the quantil of Student distribution, corresponding to the significance level 6, v = No - 2 is the number of degrees of freedom. The confidence intervals defined by (25) are useful when the design matrix is orthogonal and, consequently, the estimates (18) are independent. The orthogonality can be assured by selecting a suitable experimental design. When the columns of the design matrix are not orthogonal, in particular in a passive experiment, the common confidence interval which includes (with a probability 1 - 6) the true values of the parameters is much more accurate. The 100(l - J ) confidence intervals for the activation energy Up and for the structure-sensitive coefficient 7 are defined by the inequalities
b~,(sr/s,)(Uo- Up) 2 - 2 b a 2 ( U o - Uo)(~-7)+b22(s=/sy)(~-7)2<~2s2M-2(sw/s,s=)F(6,
2, v)
(26)
where btj are elements of the information matrix B and F(6, 2, v) is the quantil of Fisher distribution. They define the interior of an ellipse in the plane with coordinates Up, 7, centered at (Up, ~). As an example consider the data collected in the Table, which refer to long term strength of oriented polyamide [1]. By means of formulae (16) to (18) we obtained the following estimates: Up = 189.10 kJ/mole, ~ = 2.08 x 10-as m 3, log re = - 12"03. In this case, where no replicate determinations are available, the adequacy of the model can be checked by means of the multiple correlation coefficient [6] R* =.
(~,j - ~,)2
(wj - ~')'
(27)
J=l
The resulting value of R= is 0.9998. The Fisher criterion proves it to be highly significant, so that there is no reason to reject the Zhurkov model, which explains the scatter of experimental points from the overall mean to within 99"98~o. As no replicate experiments were performed, the variance o 2 was estimated from the residual variance (23) as s2= ~ 2 = 2.94 × 10-5. The variances and the confidence limits can be estimated from (19) and (25), respectively. At the confidence level of 95~/o the actual
Study of long-term polymer strength
975
values of the parameters in formula (1) lie within the limits Uo=189.10_+2.54 kJ/mole ~,=(2.08+0.07)x 10 -2s m "s log ro = - 12.03 +__0"30
(28)
The 955/0 confidence ellipse for Uo and ?, constructed according to formula (26), is very asymmetric: its eccentricity e---0-88 shows that the estimates Uo and ~ are strongly correlated. EXPERIMENTAL DATA ON LONG-TERM STRENGTH OF ORIENTED POLYAMIDE [I] AND NON-VULCANIZED
ELASTOMERSKS-30 [7] tr, MPa 294 323.4 343 362.6 392 274"4 303.8 343 362-6 372.4
T, K
i ! : ! [
i
Polyamide 301 301 301 301 301 353 353 353 353 353
log x [see]
log a [MPa]
14.4 13.8 13.3 12.9 12"3 10.9 10'3 9"6 9'2 9"1
0"03 0"26 0'44 0"58 0.67 0'03 0,26 0"46 0.55 0.67
T, K SKS-30 333 333 333 333 333 393 393 393 393 393
I
log r [sec] 5-0 4.0 3.6 3.0 2-8 3.6 2.7 2.0 1.6 1.2
Bartenev model The data were coded exactly as in the Zhurkov model. The design matrix of the coded model has the form k
F=
k
. .
(29)
x~t "y
All the previous matrix formulae remain valid. As an example, consider the data [71 for a nonvulcanized elastomer SKS-30, collected in the Table. Calculations which followed the previous scheme provided the following results: Uo = 61 "27 kJ/mole, ~n= 3 '58, log B = 13'35. The coefficient of multiple correlation R 2 =0.995 is significant: the Bartenev model explains the scatter of experimental points to within 99.5~. The residual variance is s2= 5-87 × 10-4; the 9 5 ~ confidence limits for the parameters which enter formula (2) are Uo = 61.27_ 5.12 k J/mole, rn = 3-58 + 0,27, log B = 13'35 _+ 1.62.
Translated by M. KUMN
REFERENCES 1. V. R. REGEL', A. I. S L U T S K E R and E. Ye. TOMASHEVSKII, Kineticheskaya ptiroda prochnosti tverdykh tyel (Kinetic Nature of Strength of Solids). p. 77, Moscow, 1974 2. G. i . BARTENEV, Prochnogt' i mekhanizm razrusheniya polimerov (Strength and Mechanism of Polymer Ruptule) p. 225, Moscow, 1984 3. G. m. BARTENEV and E. M. KARTASHOV, Fiz.-khim. mekhanika materialov, 5, 106, 1984 4. E. M. KARTASHOV, A. A. VALISHIN and V. V. SHEVELEV, Kauchuk i rezina, 7, 16, 1987 5. N. D R A P E R and G. S M I T H , Prikladnoi regressionnyi analiz (Applied Regression Analysis). Vol. 1, p. 314, Moscow, 1987 6. I. VUCHKOV, L. B O Y A D Z H I E V A and Ye. SOLAKOV, Prikladnoi lineinyi regressionnyl ana.liz (Applied LineaI Regression Analysis). p. 50, Moscow, 1987 7. G. M. BARTENEV and Yu. A. SINICHKINA, Mekhanika elastomerov, 2, 13, 1978
A . A . VALISHINand E. M. KARTASHOV
REFERENCES 1. R. Y. TING, J. Appl. Polym, Sci. 20: 3017, 1976 2. G. V. NESYN, V. N. MANZHAI and V. P. SHIBAYEV, Vysokomol. soyed. B28: 714, 1986 (Not translated in Polymer Sci. U.S.S.R.)
PolymerScienceU.S.S.R. Vol. 31, No. 4, pp. 970-975, 1989 Printedin Poland
0032-3950/89 $10.00+.00 © 1990PergamonPress pl¢
APPLICATION OF MATHEMATICAL STATISTICS TO A STUDY OF LONG-TERM POLYMER STRENC;TH* A. /tk. VALISHIN a n d E. M. KARTASHOV M. V. Lomonosov Institute of Light Chemical Technology, Moscow ( R e c e i v e d 28 June 1988)
A methodology based on multivariate regression analysis and statistical modelling is proposed for experimental investigation of dependence of long-term strength on stress, temperature, and other factors. A complete statistical analysis of the Zhurkov and Bartenev models has been carried out. The minimum size of experiment required for an unequivocal determination of parameters in a model has been found. THE level of statistical methods used in experimental investigation of strength is by no means satisfactory. Thus, from the whole arsenal of regression methods only the simplest type of one-dimensional least-squares analysis is used and even then without the appropriate statistical analysis. Multivariate regression ~nalysis, which forms the basis of statistical models described by functions of several variables, is almost completely ignored. In this paper we want to explain the modern approach to the establishment of dependences and to demonstrate it on the example of polymer strength considered as a function of time and temperature. Application of digital computers enables one to dispense with graphical methods and extrapolations that invariably lead to a loss of accuracy. Two formulae were proposed to describe the stress and temperature dependences of long-term strength of uniaxially stretched polymers: the Zhurkov law [1] for solid polymers r = ro exp (( Up - y t r ) / k T )
(1)
and the Bartenev formula for elastomers [2] r = B a - m exp (Uo/kT)
(2)
The formalism used in these equations is generally known; ~o, Uo, ~', B, and m are constants which must be found from experimental data. An accurate determination of these parameters is prerequisito for reaching valid physical conclusions. Long-term strength is usually measured separately as a function of stress and temperature; the latter dependence is often found by the so-called method of intersects [1]. The parameters in formulae (1) and (2) are found graphically by means of far extrapolations [1, 2]. Such methods are highly * Vysokomol. soyed. A31: No. 4, 877-882, 1989.
Study of long-term polymer strength
971
unstable: small errors in slopes of the straight lines are amplified by the extrapolation and result in low accuracy of the final characteristics of long-term strength. From this point of view the large scatter of values to be found in standard reference textbooks is understandable. A more promising approach to the evaluation of constants in the above equations consists in that the strength is considered as a function of at least two variables, which is then processed by methods of mathematical statistics. Let us rewrite formula (1) in the form
r/='go + fl2 y + flj2 x y , where x=tr, y = l / k T , r/=log r, flo=log to, f l 2 = M U o ; Formula (2) can be written as
(3)
ill2 = - M y ; with M = l o g e.
,I = 'go + 'g, x + 'g2 y
(4)
where x = l o g a, y = 1/kT, r/=log r, ,go-log B, 'gl = - m , and 'gz=MUo. In the terminology of,regression analysis the variables x and y are called predictors (or factors), and the variable r/is the response. The Bartenev formula (4) represents a model of first order since it is linear in the predictors, while the Zhurkov formula (3) is a second-degree model as it contains a quadratic cross-term. The modern regression analysis includes a check whether models such as (3) and (4) adequately describe existing experimental data. If one of them is found to be adequate, its coefficients are determined along with their accuracy. If none of the models tested is found to be adequate, a more general model is conceived and the analysis starts from the beginning. Such "tuning" then leads to the optimum model. F r o m the geometrical point of view the set of all predictor values forms what is called the factorial space, in our case two-dimensional (a surface). The response (strength) is measured at several values of stress and temperature, i.e., on an "experimental subset" of points (cells) in the factorial space. Let the number of such points be N; they can be distributed in the factorial space either regularly, i.e., lie on some curves, or chaotically. The first case corresponds to some sort of experimental design, in the second instant the measurements are performed with randomly selected values of factors. References [3, 4] present a strategy which uses two levels of temperature, i.e., the measuring points lie on two parallel lines in the factorial plane. In the extended factorial space which includes a third axis coresponding to the response, the models (3) and (4) represent a response surface. It is easy to show that the response surface is a hyperbolic paraboloid in the case of the Zhurkov model (3); the so-called pole, whereat the straight lines of the stress and tempelature dependences converge, is the saddle point of the surface which contains all the above straight lines. The response surface of the Bartenev model is a simple plane. Assume that in the j-th experimental cell the response (i.e., log r) has been measured p:times,
j=I,--N. The total number of observations then is N o = ~
pj, where obviously No>~N. Let w:j
J=~
be the l-th replica (i= 1, pj) in th.e j-th experimental cell ( j = 1, N); the quantity
PJ ~=z
designates the mean in the j-th cell. The observed responses wtj can be taken as consisting of a regular (r/j) and a random part (etj); thus,
wo=~lj+el~,
i = 1 , pj,
j=I,N
(6)
Neither of them is known, only their sum is accessible experimentally. Useful information is contained only in the regular components, whilst the random part reflects "noise" due to effects beyond the control of the experimenter. The regular part is a parameter of the model; the components e~j are assumed to be independent random variables with zero mean and variance D(e~j) given by
D(ect)= trZ/toj,
i= I, p j,
j= L-,IV,
(7)
972
A. A. VALISHINand E.'M. KARTASHOV
where o~j is the statistical weight of the j-th cell respecting the possible variation of variance between individual cells. The viariance (7) is usually unknown and only the quantities a 2 and roj can be estimated from the experimental data. Z h u r k o v model. This model can be represented by
~j=Bo+F~y~+/h~zj+~j,
i=l, ~,
(8)
where z = x y , and "~j is the mean experimental error in t h e / t h cell. The measured quantities, i.e., the individual factors and the response, usually have quite different physical meaning and different dimensions; this brings about some inconvenient computational consequences, as the necessity to operate simultaneously with very small and very large numbers leads to rounding errors. To avoid this complication the factors and the response are usually coded; for the model (8) the coded variables have the form x~ = xj. x , sx
y~= yj -- y , s~
,
~j . . . .
zj -- z , s~
wj -- w s~
w ~ = ~ ,
(9a)
where N
N
N
.,
W'~"
.,
N
J-I
S2~ -
~. eojpj(xj--~)~.
(9b)
J:l
The remaining quantities which appear in formula (ga) are defined similarly. As a result, the Zhurkov model assumes the simple form k
k
k
--
/=1, N
wj=ct2yj+ct3zj+ej,
(10)
The actual physical parameters of the Zhurkov model are related to the coded coefficients in equation (10) through Uo = M - ~ ( s w / s , ) ~2 = - M-
log~o =
~-
ICsw/s,)
(sw/s,) ~
-
~3
(11)
(s~/s,)-~3
In the following we shall assume all quantities to be coded and drop the superscripts in formulae (9) and (10). Let us now introduce the following matrix notation: w r = (~z . . . . , WN),the vector of observations whose components are the coded experimental response values (the sperscript T designates matrix transpose); the error vector T = ( ~ .-., eN); the empirical regression vector w =(w~ . . . . w~) with components given by the values of response predicted by the assumed model. Let us further introduce the design matrix F of type N x 2, consisting of components given by coded values of regressors in all experimental cells,
the vector of regression coefficients r = (e2, e3), and the vector of their estimates Q-- (e ~, e 3). The sums of elements in all columns of the coded design matrix (12) are zero, whilst their squares add to unity. The coded Zhurkov model in ~he matrix notation assumes the form w= Fa+~
(13)
The statistical estimate ~ of the vector a is constructed so as to extract maximum useful information from the measured values of the response, in other words, by minimizing the random corn-
Study of long-term polymer strength
973
ponent. To do that one defines the so-called loss function, equal to the weighted sum of squared random components for all observations, i.e., j=~rGe,
(14)
where G, a diagonal matrix of type N × N, is the weight matrix with elements ¢ojpj, j--1, N. The requirement that the loss function be minimized leads to a system of equations B a = F r Gw ,
(15)
B= : a F
(16)
where is a symmetric matrix of type 2 × 2, called the information matrix. It is then easy to find the minimum extent of experiment required for unambiguous estimation of parameters in the Zhurkov model by solving the system of equations (15): both the stress and temperature must be varied at least on two levels with replication. Assuming these conditions to be satisfied, we obtain from (15) ~t = B - 1FrGw (17) This formula defines the statistical estimates ~2 and ~3 of the sought parameters in model (10): they are usually called the least squares (LS) estimates. It can be shown [5] that, provided the structure of the model is correct, the LS estimates are unbiassed, effective, and substantial. Statistical estimates of the physical parameters in the Zhurkov model can be derived from (11) in the form Uo = M
(sw/sy) ~t2
~x
-1
y=M
A
(18)
(sw/sz)cta
logro = ( N - I l r - ~ r B - I F r G )
w,
where ~r=(~vsw/s~, -zsw/s,) and l r = (I, 1. . . . ,1). The variances of the estimates (18) are "~
D(Uo)=a
2
M
--2
2
--1
(s~/s,) b,1
D (~) = a 2 M - 2(sw/s=)2b~
(19)
D (log to) = a 2 ( N - 2 Sp G- 1+ ~r B- iO, where b~ x are elements of the matrix B-~ (the inverse of the information matrix B). The estimates Uo, Y, log ro are in general correlated. The coded values of the response predicted by the model, . wl, J = 1, N, i.e., the vector of empirical regression, can be derived from the matrix equation txk
- Fa
(20)
V ( ~ ) = a: F B ~ ~F r
(21)
The variance matrix of the vector ~ is
k . m mdiwdual . . The diagonal elements are the variances of the predicted values of the response Awj experimental cells; differences in the variances reflect the differences in the accuracy of prediction. When the off-diagonal elements are non-zero, the predicted values of the response ~,j are correlated, in contrast to the directly observed values ~,j. The variance trz which appears in the above formulae can be estimated from replicate determination of response inside the experimental cells of the factorial plane: in particular, the statistic N
$2 "-
serves as an unbiassed estimate of o 2.
J=l
pj
I=l No-N
(22)
974
A. A. VALISFIINand E, M. KARTASHOV
The LS estimates of the regression coefficients are unbiassed estimates of the actual coefficients, provided the model adequately represents the experimental data. The postulated model is said to be adequate if it yields unbiassed estimates [5]. The adequacy is checked by comparing the statistic j 2 _ equation ( 2 2 ) - with another unbiassed estimate of the variance o 5. The latter is calculated from the residual sum of squares as
s:=
(w- ~)~G (w - ~) N-2
(23)
The statistic s,2 represents art unbiassed estimate of o 2 if the model structure has been correctly selected , i.e., the model is adequate. Hence, by comparing the actual values of the statistics s 2 and s2 by means of the Fisher criterion, the adequacy of the model can. be verified. The model is adequate if the ratio of ttte two statistics is insignificant; in this case the two stat~sttcs . . s,2 and s,2 are pooled into a new unbiassed estimate of the variance as: N
pj
:=
(24) Jft
1=1
Inserting the final value of a 2 into (19) and (21), we can estimate all variances and covariances. The accuracy of the estimates (18) is expressed by the confidence interval which includes the actual value of the parameter at a significance level 6 (with confidence probability 1 - 6). The 100(1-6) ~ confidence limits for the physical parameters that enter formula (1) then are Up= U o + t ( 1 - 6 / 2 , v) tD(O'o)] t/2
r=? +_t(1-6/2, v) [b(~)l '/~
(25)
log re =logz'o + t ( 1 - - 6 / 2 , v) [D(log~'o)] U2 , where t(l -tr/2, v) is the quantil of Student distribution, corresponding to the significance level 6, v = No - 2 is the number of degrees of freedom. The confidence intervals defined by (25) are useful when the design matrix is orthogonal and, consequently, the estimates (18) are independent. The orthogonality can be assured by selecting a suitable experimental design. When the columns of the design matrix are not orthogonal, in particular in a passive experiment, the common confidence interval which includes (with a probability 1 - 6) the true values of the parameters is much more accurate. The 100(l - J ) confidence intervals for the activation energy Up and for the structure-sensitive coefficient 7 are defined by the inequalities
b~,(sr/s,)(Uo- Up) 2 - 2 b a 2 ( U o - Uo)(~-7)+b22(s=/sy)(~-7)2<~2s2M-2(sw/s,s=)F(6,
2, v)
(26)
where btj are elements of the information matrix B and F(6, 2, v) is the quantil of Fisher distribution. They define the interior of an ellipse in the plane with coordinates Up, 7, centered at (Up, ~). As an example consider the data collected in the Table, which refer to long term strength of oriented polyamide [1]. By means of formulae (16) to (18) we obtained the following estimates: Up = 189.10 kJ/mole, ~ = 2.08 x 10-as m 3, log re = - 12"03. In this case, where no replicate determinations are available, the adequacy of the model can be checked by means of the multiple correlation coefficient [6] R* =.
(~,j - ~,)2
(wj - ~')'
(27)
J=l
The resulting value of R= is 0.9998. The Fisher criterion proves it to be highly significant, so that there is no reason to reject the Zhurkov model, which explains the scatter of experimental points from the overall mean to within 99"98~o. As no replicate experiments were performed, the variance o 2 was estimated from the residual variance (23) as s2= ~ 2 = 2.94 × 10-5. The variances and the confidence limits can be estimated from (19) and (25), respectively. At the confidence level of 95~/o the actual
Study of long-term polymer strength
975
values of the parameters in formula (1) lie within the limits Uo=189.10_+2.54 kJ/mole ~,=(2.08+0.07)x 10 -2s m "s log ro = - 12.03 +__0"30
(28)
The 955/0 confidence ellipse for Uo and ?, constructed according to formula (26), is very asymmetric: its eccentricity e---0-88 shows that the estimates Uo and ~ are strongly correlated. EXPERIMENTAL DATA ON LONG-TERM STRENGTH OF ORIENTED POLYAMIDE [I] AND NON-VULCANIZED
ELASTOMERSKS-30 [7] tr, MPa 294 323.4 343 362.6 392 274"4 303.8 343 362-6 372.4
T, K
i ! : ! [
i
Polyamide 301 301 301 301 301 353 353 353 353 353
log x [see]
log a [MPa]
14.4 13.8 13.3 12.9 12"3 10.9 10'3 9"6 9'2 9"1
0"03 0"26 0'44 0"58 0.67 0'03 0,26 0"46 0.55 0.67
T, K SKS-30 333 333 333 333 333 393 393 393 393 393
I
log r [sec] 5-0 4.0 3.6 3.0 2-8 3.6 2.7 2.0 1.6 1.2
Bartenev model The data were coded exactly as in the Zhurkov model. The design matrix of the coded model has the form k
F=
k
. .
(29)
x~t "y
All the previous matrix formulae remain valid. As an example, consider the data [71 for a nonvulcanized elastomer SKS-30, collected in the Table. Calculations which followed the previous scheme provided the following results: Uo = 61 "27 kJ/mole, ~n= 3 '58, log B = 13'35. The coefficient of multiple correlation R 2 =0.995 is significant: the Bartenev model explains the scatter of experimental points to within 99.5~. The residual variance is s2= 5-87 × 10-4; the 9 5 ~ confidence limits for the parameters which enter formula (2) are Uo = 61.27_ 5.12 k J/mole, rn = 3-58 + 0,27, log B = 13'35 _+ 1.62.
Translated by M. KUMN
REFERENCES 1. V. R. REGEL', A. I. S L U T S K E R and E. Ye. TOMASHEVSKII, Kineticheskaya ptiroda prochnosti tverdykh tyel (Kinetic Nature of Strength of Solids). p. 77, Moscow, 1974 2. G. i . BARTENEV, Prochnogt' i mekhanizm razrusheniya polimerov (Strength and Mechanism of Polymer Ruptule) p. 225, Moscow, 1984 3. G. m. BARTENEV and E. M. KARTASHOV, Fiz.-khim. mekhanika materialov, 5, 106, 1984 4. E. M. KARTASHOV, A. A. VALISHIN and V. V. SHEVELEV, Kauchuk i rezina, 7, 16, 1987 5. N. D R A P E R and G. S M I T H , Prikladnoi regressionnyi analiz (Applied Regression Analysis). Vol. 1, p. 314, Moscow, 1987 6. I. VUCHKOV, L. B O Y A D Z H I E V A and Ye. SOLAKOV, Prikladnoi lineinyi regressionnyl ana.liz (Applied LineaI Regression Analysis). p. 50, Moscow, 1987 7. G. M. BARTENEV and Yu. A. SINICHKINA, Mekhanika elastomerov, 2, 13, 1978