Plastics Construction Materials Dercentage - of DET can therefore be ascribed to the decreased surface resulting from the corresponding loss in craze. Note that the maximum strength coincides with the minimum craze (25% DET, No. 7). Since D E T monomer alone (No. 2) gives a less heat-resistant structure than TAC alone (No. l),it might be expected that with the craze factor stabilized a t DET percentages greater than 2570, the additional D E T would act to decrease the heat resistance. A more heat-resistant monomer (TAC) is being replaced by a lesser one ( D E T ) without any compensating decrease in craze after the percentage of D E T has reached 25%. This is the situation actually observed. The monomers, acrylonitrile and triallyl aconitate, also act to decrease craze. However, they are intrinsically much poorer monomers for heat resistance than D E T (Nos. 2, 11 and 17). It would appear that this factor overshadows the improvement due to craze reduction and for this reason the synergistic effect noted with D E T is absent.
ACKNOWLEDG-VENT
Most of the work reported here was sponsored by the Materials Laboratory, Directorate of Research WADC, USAF, Contract AF 33(600)16825. LITERATURE CITED
(1) Cummings, W., and Botwick, M., Division of Paint, Plastics,
and Printing Ink Chemistry, 124th Meeting, ACS, Chicago, Ill., Sept. 9, 1953. (2) Ebers, E. S., Brucksch, W. F., Elliott, P. M., Holdsworth, R. S., and Robinson, H. W., IND.ENQ.CHEM.,42, 114-19 (1950). (3) Elliott, P. XI., Modern Plastics, 29, No. 11, 113-14, 185-7 (July 1952). (4) Knapp, R. L. (to U. S. Rubber Co.), U. S. Patent 2,671,070 (March 2, 1954). RECEIVED for review September 17, 1954.
ACCEPTSDDecember 15, 1954.
Design of Plastic Structures for Complex Static Stress Systems Design equations for plastics in static stress systems have been developed from the theory of linear viscoelasticity. The time dependent properties of plastics are characterized by two parameters that are easily measured. Development of equations as well as comparison of calculated and measured values are given for deflection of clamped beams and disks, torsion of rods, buckling of columns, and diametral expansion of pipe and tanks under hydrostatic pressure.
A. A. MACLEOD Polychemicals Department, E. I . du Pont de Nemours & Co., Inc., Wilmington, Del.
T
H E engineering design of plastics parts raises some difficult problem because many physical properties of plastics are time dependent. Since these quantities are also sensitive to temperature and frequently to humidity and since in most applications temperature and humidity are ambient, even the measurement of the properties to represent a given application is a task. The properties needed for the design of mechanical parts are the stress to cause fracture and the relation between stress and deformation. Both of these are time dependent for plastics. Conscquently, they are not determined by the usual short-time tests, such as those for elastic modulus and tensile strength. Further, because of the wide variety of plastics and conditions of operation, the data on long-time behavior for the particular design problem are seldom available. I n this article, the relations between stress and deformation are analyzed a t stresses well below the long-time fracture stress. A method of estimating the longtime behavior from quantities determinable in short-time tests is presented. The objective is to describe a way t o obtain design equations for plastics in any static stress system. Both the way to measure the quantities to characterize the time dependence of the plastic and the way to include the quantities in design formulas are described. T o be of practical value, these time dependent properties must be determinable easily in a short time. Consequently calculation of long-time behavior involves large extrapolations. The results, then, are order of magnitude values to act as a guide in design. It follows that if the plastic is to be subjected to a range of temperature and humidity, the worst possible combinaJuly 1955
tion of conditions should be chosen for the calculations. dpplication of the method requires measurement of the parameter characterizing time dependence for the worst conditions of operation followed by calculation of the long-time behavior. The assumptions made in developing this method were that the theory of linear viscoelasticity was applicable, that pure volume changes of plastics were independent of time, and that the materials were homogeneous and isotropic and remained so under stress. Applying these assumptions yielded a mathematical model of the time dependent behavior of plastics under stress a t stress levels below the long-time fracture stress. The resulting equations give calculated values in satisfactory agreement with current knowledge. As our knowledge of plastics increases, however, the model will be modified and extended. THEORETICAL BACKGROUND
Development from basic principles of the equations presented here is given by DiDonato ( 3 , 4). The equations are developed from the theory of linear viscoelasticity. The theory proposes that stress be proportional to strain a t all times and that the proportionality coefficient be a function of time. A useful model for the development of the equations is the combination of springs and dashpots. A unit is made up of a spring and a dashpot either in series or in parallel, and the model consists of a number of units in parallel or in series, respectively. For plastics an infinite number of units is desirable. The mathematical equations from this model and other related
INDUSTRIAL AND ENGINEERING CHEMISTRY
1319
analysis of the theory of linear viscoelasticity are given by Alfrey ( I ) , Gross (6, 6 ) , Leaderman (?), Sips (IO), and Tobolsky (12). However, these authors either have developed the theory only or have presented their results in the form of general equations that are difficult to apply in engineering design. Brull ( 2 ) reduced the complications greatly by the approximation that pure volume changes were independent of time. He restricted his studies to models consisting of a small number of spring and dashpot units-a satisfactory representation of time dependent elasticity in metals. For this work the simplifying assumption of Brull was combined with the model of an infinite number of springs and dashpots to obtain equations suitable for the engineering design of plastics. For static systems, the results can be generalized to a simple modification of the elastic coefficients.
Lu"l
"I
(D
C.65
I
I
CURVES CALCULATED POINTS MEASURED
EXPERIMENTAL
The quantities to characterize time dependence of plastics may be determined from data on any time dependent phenomenon. Data from either creep or stress relaxation measurements are suitable. For new materials or special operating conditions the data may have to be measured. The most common equipment available for precise measurements of time dependence of plastics is the standard tensile testing instrument that records the load as a function of time. Consequently, stress relaxation data should be easy to obtain. I n this work, stress relaxation data were made on the Instron universal testing instrument. The most reliable procedure was as follows:
A tensile specimen of the standard ASTM cross section was clamped between the fixed and moving jaws. Any tendency to load the specimen during clamping was neutralized by manual adjustment of the moving crosshead. The load was then applied in a standard time (for this work all loads were applied in 6 seconds) and the decay of load recorded for 10 hours. Throughout the tests temperature was maintained a t 23' & 1" C. and relative humidity a t 50 =t2%.
\p
0.70
1
I
I
that the time dependence of pure volume changes be negligible. The very high precision required to determine this quantity makea the measurement difficult. Values of pure volume changes for a number of plastics are reported by Weir (15). For the present, the main justification of this condition is that the resulting model gives satisfactory calculated values.
% ' dJ
0.60-
I
I
I
I
I
1
I
0 55T I M E , HOURS
0 5010-3
I
I
I
I
10-2
10-1
I00
IO I
Figure 1.
102
Stress relaxation data
-1
100 f 01
The use of the linear viscoelastic theory limits the stresses that may be studied and sets certain conditions on the plastics analyzed. As in classical elasticity, the linearity of the relation between stress and strain is a first-order approximation. Consequently stress levels that may be applied are limited to those that cause small deformations. It was found that for plastics, values calculated from the theory agreed well with measured values for stresses up to 50% of the yield stress. This limit was set arbitrarily and may be exceeded by some plastics. The designer must remember that the long-time fracture stress can be relatively low. The conditions on the plastics are that they be homogeneous and isotropic and remain so under load. These requirements are to be interpreted in the macroscopic rather than the microscopic sense. Good agreement between measured and calculated values was obtained for 30 plastics that included both amorphous and highly crystalline material, filled compositions and laminates. Therefore, aside from the limitation to small deformations, the equations developed from this theory will be widely applicable. Two other conditions are important. First, experimental considerations limit measurements to constant temperature and humidity. Measurements are not generally available that would permit calculation of the effects of variable temperature and humidity. Secondly, the equations are based on the condition
Table I. Values of m and b
1320
Materials
m
66 Nylon 610 Nylon Molded aorylio resin Tetrafluoroethylene resin Polyethylene Polyesterglass fiber laminate
0.047 0.048 0.029 0.062 0.040 0.021
b , Hours x 10-4 s x 10-4 1.9 x lo-' 3.3 x lo-' 1.0 x 10-6 2.7 X 10-6
4
STRESS = MAXIMUM FIBER
STRESS
u. n
STRESS = 5000 PSI
20
0
I IO
Figure 2.
I ZC
I 30
I I I 40 50 60 TIME, H O U R S
I
70
8
Flexural creep of glass fiber-polyester laminate clamped beam
Full lines calculated, dashed lines measured
From these measurements, values of load from 16 seconds to 10 hours were obtained. The data were fitted by the equation $)
=
(1
+;>
-m
where S ( t ) = stress remaining a t any time S ( 0 ) = initial stress t = time, hours m, b = parameters characterizing time dependence The equation was fitted to the data of all plastics tested within 1% over the range of the data. The values of the parameters were found to be independent of S(0) up t o 50% of the yield stress. The fit is shown in Figure 1. The more hiportant values of m and b are given in Table I. T h creep equation corresponding to the empirical function for relaxation data can be obtained from equations developed in the publications previously mentioned. The best ones for this purpose are those by DiDonato ( 5 ) , Gross ( 5 ) , Leaderman (7), and Sips (9). The resulting creep equation follows.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 1
Plastics Construction Materials e(t) 40)
- ;;is
$r(;)m
where e ( t ) = strain at any time e ( 0 ) = initial strain t = time, hours m,b = parameters characterizing time dependence The equation is applicable for times greater than 1 hour. Note that since the equations do not approach asymptotic values for long times they cannot represent completely cross-linked plastic materials.
_ _ _ _ _ _ _ _ - -- ----- -
40 2o 7
7
_
STRESS'GOOPSI.
1 IO
0
20
30
PROCEDURE FOR DESIGN CALCULATIONS
The general stress-strain equations for linear viscoelasticity are given by Brull ( 2 ) and DiDonato (4). For static systems, the equations for time dependent elasticity differ from those for classical elasticity in that the elastic coefficients contain a function of time. Two elastic coefficients are required to define the elastic stress response of a homogeneous, isotropic material. T o make use of the time independence of pure volume changes, the bulk modulus must be one of the coefficients. Generally the shear modulus is associated with the bulk modulus. However, for plastics, the choice of coefficients can be bulk modulus and Young's modulus without affecting the results beyond the accuracy of the experimental data. Since Young's modulus is the one usually given in handbooks, this choice of coefficients was used in this work. The design equations for plastics in any static stress system can be developed by a general procedure. First the solution of the problem for the purely elastic system must be obtained. For many problems the solutions are tabulated in handbooks. These solutions are generally expressed in terms of Young's modulus and Poisson's ratio. Expnmion of Poisson's ratio in terms of bulk modulus and Young's modulus is accomplished by the formula y
where
Y
- E) = (3B __6B
Figure 3.
I
1
40 50 60 TIME,HOURS
I
70
I
I
BO
100
90
Creep deflection .of disks of molded acrylic resin
Stresses are maximum fiber stress, full linen calculated, dashed lines measured
6o W
t
50-
-
S T R E S S , 1800 PSI
7
0 W
STRESS
~
~
1200 PSI.
-
_ _ _ _ _ _ _ _ _ _ _ _ _ -- - - - -S T R E S S , 600 PSI.
Figure 4.
Creep in torsion of 610 nylon
Stresses are maximum fiber stress, full lines calculated, dashed lines measured
= Poisson's ratio
E = Young's modulus B = bulk modulus
With the equation in terms of bulk modulus and Young's modulus, the solution for time dependent elasticity can be obtained aB follows: Wherever Young's modulus appears in the denominator of the 1 equation, replace the term - b y E
I
I
t
> 1 hour
Figure 5.
and wherever Young's modulus appears in the numerator of the equation, replace the term E by
I
2
I
3
I
4
I
I
5 6 TIME, HOURS
I
7
I
8
1
9
Buckling load as function of time for 66 nylon columns Full lines calculated, points measured
where E is the elastic modulus of the plastic determined by the ASTM method. I n the following paragraphs samples of the procedure are given with comparisons of values calculated from the equations t o experimental values. APPLICATIONS
Deflection of Beams. The equations for the elastic solution are readily obtained from handbooks. ELASTIC SOLUTION (beam fixed at both ends, load a t middle): July 1955
where D
maximum deflection load distance between supports Young's modulus Z = moment of inertia of cross section =
W
= 1 = E =
VISCOELASTIC SOLUTION:
INDUSTRIAL AND ENGINEERING CHEMISTRY
1321
70
I
I
I
I
1
I
I
I
1
I
I-
s z
0
-
+
+
5
INTERNAL PRESSURE. 50 PSI
5
F U L L LINES CALCUATEC POINTS, X = 100 PSI G 8 - 8 5 PSI G 5 0 PSI G.
-
IO+
I
IO
Figure 6.
+=
I
I
30
20
I
I
I
50 60 T I M E , HOURS
40
I
70
I
80
This system was used as a standard for comparing measured values to those calculated from the parameters determined from stress relaxation data. 811 30 plastics previously mentioned were tested. The agreement between theory and measurement is shovin in Figure 2 for a beam of polyester-glass fiber laminate. The inch by beam was inch thick, the supports were 4 inches apart, and the load was a t the center. The full line represents calculated values and the dashed line measured values. Deflection of Disks (freely supported, loaded a t the center); the elastic solution for this system was obtained from Roark's handbook (8). ELASTIC SOLUTION :
I
I
100
90
Diametral expansion of polyethylene pipe ' under hydrostatic pressure
where W = load h = thickness a = radius of support T O = radius of load application
VISCOELASTIC SOLUTIOK :
-
9.0INTERNAL PRESSURE. 100 PSI G.
_ _ _ _ L C - - - - - - -
I N T E R N A L PRESSURE,
-
85 PSI G.
33.0
t
b
I
IO
Figure 7.
I
I
20
30
40 do TIME, HOURS
$,
jo
do
&
0
Diametral expansion of tetrafluoroethylene resin pipe under hydrostatic pressure
IPS is 1 inch, full lines calculated, dashed lines measured
The comparison between calculated and measured values is shown in Figure 3. T h e disk was molded of acrylic resin. It was 1/16 inch thick, freely supported on a circumference of 4 inches in diameter and loaded on a concentric circle l/, inch in diameter. Again the full line is calculated and the dash line measured. Note that the measured and calculated curves are nearly parallel from zero time. Some of the discrepancy can be attributed t o difference in loading time of the disk and the loading time used in measuring the ASTM elastic modulus.
Torsion. ELASTIC SOLUTION:
Yz 0 f
0.4(
e=- T JP
where
0.30
iii
z
e J
=
= polar moment of inertia
Y =
3
UJ
,0.20'
z
r
1
z
2 0.10
1"
VISCOELASTIC SOLUTION:
"
1 I
I
IO'
IO2
I
TIME,
IOa
3
'1
70
IO*
HOURS
Figure 8. Diametral expansion of tank of glass reinforced polyester under hydrostatic pressure Curves calculated
1322
shear modulus
x= IO FT.
-
0.00
angle of twist/unit length
1' = torque
10'
T
The comparison between Calculated and measured values is inch in diameter shown in Figure 4. The rod of 610 nylon was and 6 inches long. The full line is calouIated and the dash line measured. Buckling of a Column. The load to cause buckling depends on the nature of the supports. The elastic solution was obtained by measuring the instantaneous buckling load. ELASTIC SOLUTION:
P = Ef (B)
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 7
Plastics Construction Materials where P = limiting load f(B) = function of end conditions
where
a h p , A S ,Ag d x
VISCOELASTIC SOLUTIOS:
P
=
E(l +
p
B
) = P o (1
+I)-%
where Po = instantaneous buckling load
~
~
+
Eh
&I
(11).
ELASTIC SOLUTION: Eh [ e - & (Aa cos pz
+ A , sin pz) + (z - d ) ]
ma
ma
(i)m[e-
Bz
( A scos pr
+ Ad sin pr) + (z - d)l
The calculated values are given in Figure 8. The base of the tank was assumed to be anchored in concrete and the walls were inch thick. CONCLUSION
The method of extending elastic equations for static stress systems to viscoelastic equations is relatively easy. Parameters to characterize viscoelasticity are readily obtained from either creep or relaxation data. Agreement between calculated and measured values is within the limits of sample variability and experimental accuracy. LITERATURE CITED
(1) Alfrey, T., “Mechanical Behavior of High Polymers,” Inter-
~
Comparison of measured and calculated values for polyethylene pipe are shown in Figure 6 and €or tetrafluoroethylene resin pipe in Figure 7 . On Figure 6 the measured points are shown for polyethylene; on Figure 7 the dashed line represents the measured values for tetrafluoroethylene resin. Water Tanks. This solution is included as another example of the procedure for obtaining design equations for viscoelastic systems. The results have not been tested against an actual installation. The elastic solution was obtained from Timoshenko
w =
= = =
p a 2 sin w =--
where subscript 1 refers to direction a t right angles to axis of pipe and subscript 2 refers to direction parallel to axis of pipe E = strain P = hydrostatic pressure T = radius of pipe t = wall thickness
r SOLUTION :~
= =
diametral expansion density of contents radius of tank wall thickness parameters and integration constants depth of tank distance above base of tank
VISCOELASTIC SOLUTION:
Comparison between measured and calculated values is shown in Figure 5. The columns were 1/2 inch diameter and 6 inches long. The full line is calculated, the crosses are measured values. Pipe under Hydrostatic Pressure. The manufacture of pipe often results in different properties parallel and perpendicular to the axis of the pipe. The following equations include the two sets of properties. ELASTIC SOLUTION:
T
w = p =
science, New York, 1948. Brull, ~M. A., Proceeding of First ~ ~ Midwestern~ Conference ~on Solid Mechanics, Engineering Experimental Station, University of Illinois, pp. 141-7, dpril 1953. (3) DiDonato, A. R . , AIacLeod, A. A , , and Webber, A. C., “Creep and Relaxation of Plastics,” unpublished. (4) DiDonato, A. R., hlacleod, A. A., and Webber, A. C., “Viscoelasticity in Complex Static Stress Systems,” unpublished. (5) Gross, B., J . A p p l . Phys., 18, 212 (1947). (6) Ibid., 22, 1035 (1935). ( 7 ) Leaderman, H., “Elastic and Creep Properties of Filamentous Materials and Other High Polymers,” Textile Foundation, Washington, D. C., 1944. (8) Roark, R. J., “Formulas for Stress and Strain,” McGraw-Hill, New York, 1943. (9) Sips, R., J. Polymer Sci., 5 , 69 (1950). (10) Ibid., 7, 191 (1952). (11) Timoshenko, S., “Theory of Plates and Shells,” MoGraw-Hill, New York, 1940. (12) Tobolsky, A., Tectile Research J., 21, 404 (1951). (13) Weir, C. E., J . Research Natl. Bur. Standards, 46, 207 (1951). (2)
RECEIVED for review September 17, 1954.
ACCEPTED May 26, 1955,
Corrugated acrylic plastic is used with movable metal partitions at Rohm & Haas’ Eiristol plant
July 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
1323
