Tricolor Photoviscoelastic Technique and Its Application to Moving Contact by S. Yoneyama, J. Gotoh and M. Takashi ABSTRACT--A new technique for simultaneous determination of both fringe order and principal direction of birefringence in practical photoviscoelastic analysis using white incident light with a set of the primary colors, called tricolor photoviscoelasticity, is described. This method can determine both the fringe order and principal direction of birefringence from a single-color photoviscoelastic image under plane polarization. Then, the authors evaluate time dependentstresses and strains around a contact region in a viscoelastic strip plate under nonproportional loading condition. The variations of the principal stresses and strains are easily obtained over a wide time range by use of the optical constitutive equations of photoviscoelasticity and the characteristic material property functions.
Introduction Many polymeric materials have time and temperature dependent properties in stress and strain relationships, called viscoelasticity. The wide use of polymeric materials in industrial fields has demanded better understanding of not only time dependent mechanical properties but also mechanisms of complicated behaviors, such as those in fracture and/or contact. It should be pointed out that any kind of analysis of contact between or with viscoelastic bodies suffers from difficulties arising from the fact that not only the region but conditions of contact cannot be known in advance, 1according to remarkable time dependence of constrained deformation, particularly around a contact region. Several theoretical and numerical studies on viscoelastic contact problems have been published. 2-5 In these techniques, however, certain kinds of assumptions have to be made inevitably concerning unknown boundary conditions, in particular, on the contact surface. From the viewpoint of experimental analysis of time dependent stress and strain in a viscoelastic material, photoviscoelasticity is a very useful technique. In the current photoviscoelastic technique 6'7 with monochromatic light, however, it is still impossible to determine both fringe order and principal direction of birefringence simultaneously from single-image data or measurements. Particularly in case of
s. Yoneyama (SEM Member) is a Graduate Student, Department of Mechanical Engineering, Aoyama Gakuin University, Tokyo 157, Japan. J. Gotoh (SEM Member) is Research Fellow, Department of Mechanical Engineering and Materials Science, Yokohama National University, Yokohama 240, Japan. M. Takashi (SEM Member) is Professor, Department of Mechanical Engineering, Aoyama Gakuin University, Tokyo 157, Japan. Original manuscript submitted: June 17, 199Z Final manuscript received: April 7, 1998.
the problems in which good reproducibility of experiments cannot be expected, the simultaneous measurement of fringe order and principal direction of birefringence is important and inevitable. Recently, several image-processing techniques and automated polariscope systems for analysis of photoelastic fringe patterns using monochromatic light have been developed. 8-11 White light photoelastic techniques, such as spectral analysis, have also been proposed. 12-15 However, these current photoelastic techniques cannot be applied to photoviscoelastic analysis of time-varying phenomena under nonproportional loading conditions. In this context, the authors 16,17 have already proposed a photoelastic and photoviscoelastic method for determining fringe order and principal direction of birefringence from single-color image data using three different wavelengths of light, which is called tricolor photoviscoelasticity. Then, the effectiveness of the method was demonstrated by both computer simulations and static photoelastic and photoviscoelastic analyses using a set of three images by different wavelengths. I6, I7 Also, an experimental system for the tricolor photoviscoelasticity was developed. 18,19 This method is based on the fact that the distributions of isochromatics depend on, and isoclinics are independent of, the wavelength of incident light. Combining this method with the constitutive equations of photoviscoelasticity, the differences of time dependent principal stresses and strains and their directions can be successfully calculated and analyzed. In this paper, the theory of tricolor photoviscoelastic technique is described briefly. Then, the authors discuss the firststep approach to the evaluation of time dependent stresses around a contact surface under a nonproportional loading condition in which the principal axes of birefringence, stress and strain show different variations from one another.
Theoretical Considerations of Tricolor Photoviscoelasticity Determination Method of Fringe Order and Principal Direction of Birefringence Consider a transparent specimen in a plane polariscope in dark field with three monochromatic lights of different wavelengths. The emerging light intensity of photoelastic or photoviscoelastic image data for each wavelength at a point in the specimen can be expressed as [i = X i Y ' J c Qi,
(1)
ExperimentalMechanics 9 211
where
Xi = Si sin 2 Ni
(2)
Y = sin2 2(a - 0).
Here, Si is constant and depends on the intensity of incident light and the measuring system, Ni denotes the fringe order corresponding to each wavelength, et is the principal direction of birefringence, 0 is the polarization angle and Qi are background intensities. Assuming that background intensities Qi are independent of the wavelength of light used, we have the following equation:
Ii-I) =(Xi-Xj)r
(i,j = R , G , B ) .
(3)
Eliminating Y from eq (3), we have the following equation:
IR - - IG XR - XG
IG - IB XG - XB
-- IB)XR
{Crl(t) - cr2(t)} cos 2~(t) = CUr1(t)- n(0) cos 2a(0) t
+ f c~lr(t _ .c)dn(x) COSd~2~ 0
t
+ f C~]r (t - x)dn(x)dxsin 2et(t) dx
(10)
0
{81(t) -- 82(t)} COS2y(0 = C~c1(t). n(0) cos 2c~(0) + (IB -- IR)X6
+ (IR -
IG)XB
= 0.
t
(5)
+ f C~cl(' _ ~) dn(x) COSd~2~(t)dx
Substituting eq (2) into eq (5), we have
{El (t) - ~2(t)} sin 2y(0 = C~c1(t). n(0) sin 2a(0)
+ (lit -- IG)SB sin 2 DNG= = O, (6) where C = NR/NG and D = NB/NG, which must be calibrated beforehand because of the dispersion of birefringence.2~ Equation (5) is a nonlinear equation with respect to the fringe order NG for green light. Thus, we can determine the fringe order No by solving eq (6) numerically. In rare cases, Y = 0 and/or XR = X G = X B , eq (6) does not give any solution. In these cases, however, an approximate solution can be easily obtained by adopting the B-spline function interpolation technique. 21 After determining the fringe order, the principal direction of birefringence can be calculated as c~ = O4- sin-1
1/2
I l c - IBI + l i b - l n l + IIn - I~1
u
(7)
I)
at every point where XR ~ XG ~ Xs. Here, sin -] denotes the principal value of the inverse sine function. In the case of Y = 0, the principal direction of birefringence is obtained as 1
t
+ f C~cl(t _ ,) dn ('t) dxsin2e~(t) d~,
where C~r1 (t) is the inverse relaxation stress-birefringence coefficient and C~cl(t) is the inverse creep strainbirefringenee coefficient, which must be measured and determined beforehand. Material
Properties
The material used in this study is polyurethane rubber. The composition of the material is polypropyren glycol, trimethylolpropane, tolylen diisocyanate and di-n-butyltin diaurate. This material is linearly viscoelastic, thermorheologically simple and birefringent and exhibits excellent light transmission properties without any color. Thus, the difference of the transmittance due to wavelength difference is quite small. Its glass transition temperature was measured as Tg = 223 K. Figure 1 shows the photoviscoelastic characteristics of the material, i.e., the inverse relaxation stress-birefringence coef1.e|
Adopting these identifications to the time dependent photoviscoelastic image data, the time variation of the fringe order and the principal direction of birefringence at any point over the whole field can be easily calculated. The above theory and the procedure have been discussed, and the validity and the effectiveness of the proposed method have shown in previous papers. 16-19
1.~ r
N
.E o.e I"
~
Constitutive Equations of Photoviscoelasticity The generalized two-dimensional photoviscoelastic constitutive equations6'7 are shown below. When the time variation of the fringe order per unit thickness of a specimen 212 9 VoL 38, No. 3, September 1998
(12)
0
(8)
a = 0,0 + ~rc,0-t- r~.
(11)
0
(IG - IB)SR sin 2 CNox + (IB - IR)Sa sin 2 NGrt
IxS
(9)
{Crl(t) -- cr2(t)} sin 213(t) = C~l(t) 9n(O) sin 2a(0)
(4)
Thus, we have
(IG
n (t) = N (t)/h (where h is the thickness of the specimen) and the angle c~(t) of the principal direction of birefringence are obtained experimentally, the time dependent principal stress difference cr] (t) - cr2(t) and the principal strain difference el(t) -- ~2(t), as well as the angle ~(t) of the principal stress direction and the angle ,/(t) of the principal strain direction, can be obtained:
. . . . . . . . . . . . . . . . .
/ o.e
1 o,2
k
cg,(O --
lo.
t-o.2 ~_
='.~ ..... t-,
~o.2r
~
,,"
1-o.s'd
-o., t
_........-\ ,o-,,,,, 1.1., -o., r-....................... J" ~-_1"1"' "~ .;o -~ .~ .;, -; ; ;'-l.e Redueed Time, log t ,
$
Fig. l mMaster curves of the optical characteristic material functions C~r1(t) and C~c! (t) at the reference temperature To = 273 K
Temperature controlchamber .
--or
I
I! I I '
light source G"
J
"
|'
Color CCD video camera R m
B
,,il, l
I i........ Polarizer
~
Computer
,t
Analyzer
Sp2~m/~en
Lens
%~)M3
DM4 )
B
C 9"I F3 lib 9" NDF ( 9
n u
I !
"~ G F2 NDF DM1
F: Interference filter M: Mirror DM: Dichroic mirror NDF: ND filter
CCD G
D
~o'~DM 9
CCD
.
uH Lens NO Mercury lamp
R
I
F1
n u Lens Halogen lamp
(b) Fig. 2--(a) An apparatus for the tricolor photoviscoelasticity, (b) the construction of the tricolor light source developed, (c) the three-CCD system of the camera adopted ficient C~-] (t) and the inverse creep strain-birefringence coefficient C~c] (t), which are measured and determined using light wavelength 546 nm through uniaxial tension tests under several constant strain rates and various temperatures, and constructed using the well-known WLF time-temperature shift factor.22 Then, the master curves of C~l(t) and C~c1(t) are approximated by Prony series to facilitate the calculation of the convolution integrals involved in the photoviscoelastic constitutive equations. The mechanical properties of the material, i.e., the relaxation modulus Er(t) and the creep compliance Dc(t), were also determined similarly. Same characterization tests are performed in cases of other wavelengths 436 nm and 619 nm, and then it is confirmed that the effect of light wavelength on the coefficients Cjr 1(t) and C~cl(t) appears merely in parallel shifting up or down the whole curves, i.e., the time and temperature dependence of the coefficients Cff1 (t) and C~c1(t) do not change any more. Thus, the ratios C = NR/NG and D = NIXING in eq (6) could be kept constant over a wide range of the reduced time.
Experimental Procedure
Experimental Setup Figures 2(a), 2(b) and 2(c) show an experimental setup for tricolor photoviscoelastic experiment with a newly developed tricolor light source system and the three-CCD system of a CCD video camera. The light source consists of a mercury lamp, a halogen lamp, ND filters, lenses and dichroic mirrors. This light source emits compound light of the wavelength ~-i = 6t9 nm, 546 nm and 436 nm as shown in Fig. 3. Using this light source and a color CCD camera having three independent sensors corresponding to red, green and blue lights, color image data obtained by a single shot of exposure can be separated into three different monochromatic images corresponding to each wavelength.
Material Calibration The fundamental theory of photoelasticity predicts that fringe order is inversely proportional to wavelength, i.e., ratios C and D in eq (6) are expected to be determined only
Experimental Mechanics
9 213
by wavelengths of light used. However, as described before, ratios C and D have to be measured in an actual light system, taking the effect of dispersion of birefringence2~into account. Adopting four-point bending, calibration tests are performed on a beam specimen made of the same material (polyurethane rubber). Figure 4 shows the distribution of the fringe order corresponding to wavelengths X = 436 nm, 546 nm and 619 nm along a transverse section of the calibration beam at temperature T = 245 K. In this figure, approximated curves obtained using the least squares are also plotted. The inclinations of these curves give the ratios C and D as shown in Table 1. In this table, theoretically obtained ratios C and D are also shown. The difference of values between the theory and the result is about 10 percent. The same results are obtained at temperature T = 233 K.
100 G 546 nm
B 436 nm
8O
R 619 nm
=60 ,m
O
4o!
n20
./ ~1~o
4so
soo
5so
6o0
~o
Wavelength, nm
Experiment
Figures 5(a) and 5(b) show a specimen and a loading history for an example of two-dimensional viscoelastic contact. A plate specimen of 200 mm in width, 50 nun in height and 10 mm in thickness is loaded by a rigid cylinder of 30 mm in radius under a step load P = 24.2 N as shown in Fig. 5(b). The temperature is kept constant at T = 234 K during the whole measurement, where the material shows marked viscoelastic behavior. The polarization angle is selected as rc/4 rad for easy observation around the contact point. Here, any polarization angle can be selected for observation of fringe pattern, avoiding superposition of isoclinic line and a point on which attention is paid. Then, time dependent photoviscoelastic fringe patterns are recorded using a color CCD video camera as digital images of 512 x 480 pixels and 256 gray levels of brightness for each color. A point, S, shown in the figure, is selected as an example point for evaluating the time dependent stress and strain, and their directions. Finite Element Analysis
The results of photoviscoelastic analysis are compared with those of a viscoelastic finite element (FEM) analysis. Figure 6 shows a FEM model with boundary conditions simulating the experiment. A contact model with nonpenetradon constraints is used in the MARC FEM program together with viscoelastic constitutive equations. Adopting four-node plane stress type elements, the numbers of nodes and elements are chosen as 1757 and 1639, respectively. As shown in this figure, the viscoelastic plate specimen is pressed into contact at the upper edge by a rigid cylinder. Here, friction between a cylinder and a viscoelastic plate is ignored. The mechanical properties adopted for the analysis, i.e., relaxation modulus, Er(t), is approximated with the 18-term Prony series obtained from the master curve. Results and Discussion
Figure 7 shows examples of time variation of the color photoviscoelastic image data and the separated monochro-
TABLE 1 Calibrated Value
Theoretical
C = NR/N G
0.893
0.882
D = NB/NG
1.33
1.25
214 9 Vol. 38, No. 3, September 1998
Value
Fig. 3--Spectral distribution of the light source emittance
9Fringe order, R ~ 0.0218x m Fringe order, G X
'
o.q244_x- - .
J J ,,-tiP'-" 9
~-dnge oroer, [] O
,
1
,s*/i /" , ...,,0" , , , ~
o
,.0~..~r
m.,
"
. . . .
o
~
C LI.
-1 -2
ojOJ~
-3
-4 .-
l 00
,
-50
,
0 Position, plxel
50
~
1
Fig. 4--Fringe order and approximated curves along the transverse section of the calibration beam for the determination of the ratios C and D
matic images around the contact surface measured under polarization angle of x / 4 rad at T = 234 K. Differences of isochromatic fringe patterns due to wavelength differences are obviously seen. Isochromatic fringe order is increasing, and the location of maximum fringe order moves toward the inside of the specimen with time. This phenomenon is due to creep deformation of the specimen and implies that the contact surface changes reflecting viscoelastic properties of the material. Also, we can find that isoclinics are moving toward the outside of contact surface with time. This means that the nonproportional loading state arises around the contact surface. Applying the proposed method to image data obtained, fringe order and the principal direction of birefringence are calculated. Figure 8 shows the time variation of the fringe order and the principal direction of birefringence at point S mentioned in Fig. 5(a). The differences of principal stresses and strains, and their directions at point S, are evaluated using the constitutive equations of photoviscoelasticity as shown in Figs. 9 and 10. In these figures, results obtained by FEM are also plotted for comparison. The values of the principal stress difference increase abruptly when loading, then relax. On the other hand, the principal strain differences show creep
3 s
Load
60 s
240 s
600 s
r
+
Specimen:
Viscoelastic strip
Ii i
I S (1 ~ ram, 1.25 ram) I
J
I
'
!
I. . . .
J.. . . .
~,
Region captured by a CCD camera
(b)
24.2
(c)
1 0.1
(d)
Time(s)
(h) Fig. 5--(a) Specimen and the selected points for analysis, (b) loading condition
Fig. 7--Examples of the photoviscoelastic image data at temperature T = 234 K under polarization angle rr/4 rad: (a) color image, (b) R-image (7. = 619 nm), (c) G-image (7. = 546 nm), (d) B-image (7. =436 nm)
ol
0 Nodes:
1757
Elements:
1639
S :::::::::::::::::::::::::::::::::::::::::::
Ii P
Fringe order~. Pdndpel directionof birefringenca
a: :i:i:+: : :++; : : : -: : :;;; : : : :;,: : ;;;+; : : : : : : :;;; : : ::;; : : : :;;: : ;: :: :: :: :: :: :: :: : : :: ::: ::: : - Ji , i, , ,i, i i i i i,
, i
, ,
!!!!!!=:ii;i;;;;;iiiii+;+;iii i:;;'i i~i i:~':.jIi iiiiiiiiiiiiiiiiiiiil..iiiii~.:Aiiiiii:'4
13 @
I
~~
s . lOO
Time, t, $
,,..oo
. 500
0.6~= 0.4! "O
6000
Fig. 8---Time variation of the fringe order and the principal direction of birefringence at point S
Fig. 6--Finite element meshing and boundary conditions
phenomena and a very similar variation as that of the fringe order. Also, principal direction of stresses and strains varies remarkably with time, reflecting the nonproportional loading condition and viscoelastic properties of the material. Differences of the results between experiment and FEM are due to boundary conditions on the contact surface. Because the actual boundary condition of contact surface cannot be known in advance, the friction on the contact surface is neglected in FEM analysis. However, in actual contact, it is easily ob-
served that there could be slip, adhesion and constraint of displacement. Figures ll(a) and ll(b) show examples of threedimensional diagrams of the time variation of the distribution of the principal stress difference and the principal strain difference along the line normal to the contact surface. In Fig. 1 l(a), the maximum value of the principal stress difference appears at a small distance from the contact surface on the plane of symmetry, and at this point, the principal stress difference varies rapidly. Similarly, in Fig. 1 l(b), the variation of the principal strain difference at a small distance from the contact surface is large and at the contact surface small. Small variation of the principal strain difference implies that the strains are constrained not to move at the contact surface by the cylindrical indenter. Experimental Mechanics 9
215
:E
Principal stress difference, Gl(t) - o2(t) , M P a
1.2 1.4 1.2
I
~ 0.8
Photoviscoelastic analysis~" FEM "_-:
1
F=
i
0.8 '15
~0.0
2
.l"~ 0,4 w
~ . .
~0.2 o0
0.6
~ (t) %(t)
i
0.45 " 0.4
0.35 . ~ 0.3 0.25 0.2
I
I
0A i
...................
~-
0.2
60o0
Time,t, $
Fig. 9--Time variation of the principal stress difference and its direction at point S
t, s
Distance from the co
surface, mm
Principal strain difference, Sl (t) - s2(.t)
xl#
8.0 A
7.0
~Kt)-~(t)
.............. "
I
1.4 1.2 ~-
6.0 t~
~ 5,0 .~4.0 ~ .~ 3.0 2.0
0,8 0.6
"o
"o e-
i[ 0.4 "!~
'~ 1.0
.......
0.2
O.
0o
Time,t, s
6o00
Fig. lO---Time variation of the principal strain difference and its direction at point S
7.0 6.0 5.0 4.0 3.0 2.0 1.0 0 o t
600 00
I
2 3
5 Distance from the contact
6~
surface,mm
1
8
./
300
0 100 Time,t,s
(b) Fig. 11--Three-dimensional plot of (a) the principal stress difference and (b) the principal strain difference along the normal directions to the contact surface on the plane of symmetry
Taking the above results in this experiment into account, it is pointed out that this type of problem must be treated as a time dependent mixed boundary value problem regardless of the condition of given force boundary in the experiment. Conclusions
This paper describes a new method for determining both fringe order and principal direction of birefringence simultaneously from color image data obtained by a single shot. Also, the experimental system for the method was described. Then, an example of viscoelastic contact problem in a framework of photoviscoelasticity is described. The time dependent differences of principal stresses and strains around the contact surface were evaluated under nonproportional conditions using the tricolor photoviscoelastic technique. The remarkable effects of the viscoelastic properties and behaviors of the material were observed in the experimental results. In conclusion, it is confirmed that the time dependent stress and strain analysis can be performed from single-color image data by the proposed method. Also, the method can be applied to the problems in which good reproducibility of experiments is not expected. The method can offer a new way to analyze experimental stress of viscoelastic materials.
216 9
VoL 38, No. 3, September 1998
Acknowledgments The authors appreciate the financial support of the Center for Science and Engineering Research Institute, Aoyama Gakuin University. The material used in this study was supplied by Prof. A. Misawa at the Kanagawa Institute of Technology, Japan. Our thanks are extended to Ms. Y. Tagaya; Messrs. M. Shimizu, J. Oono and T. Takenaka; and others for their help in the characterization tests of the material.
References l. Johnson, K.L, Contact Mechanics, Cambridge University Press, Cambridge (1985). 2. Lee, E.H. and Radok, J.R.M., "The Contact Problem for Viscoelastic Bodies," Trans. ASME, Ser. E, J. AppL Mech., 27, 438--444 (1960). 3. l~ng, ZC.T, "Contact Problems in the Linear Theory of Viscoelasticity," Trans. ASME, Ser. E, J. AppL Mech., 35, 248-254 (1968). 4. Chen, W.H., Chang, C.M., and Yeh, J.T., "An Incremental Relaxation Finite Element Analysis of Viscoelastic Problems with Contact and Friction," Computer Meth. Appl. Mech. Eng., 109, 315-329 (1993). 5. Raous, M., Chabrand, P., and Lebon, E, "Numerical Methods for Frictional Contact Problems and Applications," J. TheoreticaI Appl. Mech., 7, 111-128 (1988). 6. Kunio, T., Miyano, Y, and Sugimori, S., "Fundamentals of Photaviscoelastic Techniquefor Analysis of Time and Temperature Dependent Stress
and Strain," Appfied Stress Analysis, ed. T.H.Hyde and E. Ollerton, Elsevier Applied Science, London, 588-597 (1990). 7. Gotoh, J., Horie, H., Misawa, A., and Takashi, M., "Photoviscoelastic Method for Time Dependent Stress and Strain Analysis Under Nonproportional Loading," Proc. SEM Spring Conf. on Exp. Mech., Grand Rapids, MI, 234-241 (1995). 8. Voloshin, A.S. and Burger, C.P., "Half-fringe Photoelasticity: A New Approach to Whole-field Stress Analysis," EXPERIMENTAL MECHANICS, 23, 304-313 (1983). 9. Patterson, E.A. and Wang, Z.E, "Towards Full Field Automated Photoetastic Analysis of Complex Components," Strain, 27, 49-56 (1991). 10. Morimoto, Y., Morimoto, E, Jr, and Hayashi, T., "Separation of Isochromatics and Isoclinics Using Fourier Transform," Exp. Tech., 18 (5), 13-17 (1994). 11. UmezakL E., Watanabe, H., and Shimamoto, A., "Automatic Wholefield Measurement of Photoelastic Fringe Orders Using Generalized Phaseshift Method," Post Conf. Proc. VIII lnt. Cong. on Exp. Mech., 154-159 (1996). 12. Redner, A.S., "Photoelastic Measurements by Means of Computerassisted Spectral-content Analysis," EXPERIMENTAL MECHANICS, 25, 148-153 (1985). 13. Haake, S.J. and Patterson, E.A., "Photoelastic Analysis of Frozen Stressed Specimens Using Spectral-contents Analysis," EXPERIMENTAL MECHANICS, 32, 266-272 (1992).
14. Ajovalasit, A., Barone, S., and PeruccL G., "Towards RGB Photoelasticity: Full-field Automated Photoelasticity in White Light," EXPERIMENTAL MECHANICS, 35, 193-200 (1995). 15. Hay, D.P.E. and Zhao, W, "Color Digital Imaging Systemfor Whitelight lsoclinic Analysis," Post Conf. Proc. VIII Int. Cong. on Exp. Mech., 13-17 (1996). 16. Gotoh, J., Yoneyama, S., Mawatari, S., and Takashi, M., "Photoelastic Analysis from a Single Piece of lmage with Linearly Polarized RGB Lights," Post Conf. Proc. Vlll lnt. Cong. on Exp. Mech., 160-166 (1996). 17. Yoneyama, S., TakashL M., and Gotoh, J., "lime- and Temperaturedependent Stress-strain Analysis Using White Light Photoviscoelasticity," Proc. SPIE, 2921, 486-491 (1997). 18. Yoneyama, S., Gotoh, J., and TakashL M., "Photoelastic Analysis from Single Image with the Three Primary Colors," Proc. Int. Conf. on Advanced Technology in Exp. Mech., Wakayama, 429~134 (1997). 19. Yoneyama,S., Shimizu, M., Gotoh, J., and TakashL M., "Photoelastic Analysis with a Single Tricolor Image," Opt. Lasers Eng., forthcoming. 20. Pindera, J. T. and Cloud, G., "On Dispersion of Birefringence of Photoelastic Materials," EXPERIMENTAL MECHANICS, 6, 470--480 (1966). 21. lshida, K. and Yoshimoto, E, Spline Functions and Its Application, Kyoiku Shuppan, Tokyo (1979) (in Japanese). 22. Williams, M.L., Landel, R.E, and Ferry, J.D., "The Temperature Dependence of Relaxation Mechanics in Amorphous Polymers and Other Glass-forming Liquids," J. Am. Chem. Sac., 77, 3701-3707 (1955).
Experimental Mechanics 9 217