Burst pressure prediction of polyamide pipes.
Polymer pipes are widely used in industry. They are chosen for their flexibility and their chemical and mechanical resistance. The highest pressure borne by pressurized pipes, or burst pressure, is a fundamental parameter for the applications. The designer must consider how the burst pressure changes with respect to the geometry and the materials. Guidelines have been determined by measurements of the burst pressure of many pipes of different sizes. Experiments performed under the standard DIN (1) are available for polyamide pipes (2). In this test, the specimen is plugged at both ends while the internal pressure is increased rapidly up to the ductile rupture [ILLUSTRATION FOR FIGURE 1 OMITTED]. The results shows that the relevant parameter is the hoop stress. Whatever the pipe radius and thickness, bursting occurs when the hoop stress reaches a critical value. This value is temperature dependent and varies with respect to the polymer grade.
This paper intends to show that the critical hoop stress can be calculated for a given polymer and temperature. What is required is the mechanical behavior data of the polymer itself, as will be discussed in the paper.
The burst pressure of metal pipes has been studied (3-5). The theory predicts the onset of a plastic flow instability at a critical internal pressure. Bursting is supposed to occur at this pressure. In the first part of this paper, the instability derivation is briefly reviewed. The particular case of a pipe under pressure in the conditions of the standard DIN is examined. As for metals, the deformation is large, so the behavior of the polymer is no longer elastic but plastic. The Von Mises theory of isotropic plasticity will be used. Taking into account that the thickness is much greater than the diameter, the thin pipe approximation will be considered. A close form equation gives the burst pressure and hoop stress.
In the second part, the theory is compared with experimental results. Burst pressure measurements have been done on plasticized polyamide pipes at four temperatures between 23 [degrees] C and 80 [degrees] C. Agreement between theory and experiments is discussed. Finally, the use of this approach to other polymers will be considered.
According to the DIN standard procedure (1), the pipe ends are plugged. The internal pressure acts not only toward the pipe walls but also to the ends. This creates a force or a stress along the axial direction. Thanks to the cylindrical symmetry, the stress field can be defined by the principal components [[Sigma].sub.r], [[Sigma].sub.[Theta]], [[Sigma].sub.z], respectively, the radial, hoop, and axial components; see Fig. 1. The thin pipe approximation will be considered here. Writing the balance of forces acting in the hoop and axial directions yields the expressions:
[[Sigma].sub.[Theta]] = p R/t and [[Sigma].sub.z] = p R/2t (1)
where p is the internal pressure, and R and t the radius and thickness of the pipe, respectively. Under the assumption of thin pipe, R takes a mean value, for instance, the arithmetic average of the internal and the external radius. The radial component of the stress or is of the order of the pressure p and can be neglected compared with [[Sigma].sub.[Theta]] and [[Sigma].sub.z]. This defines a plane stress state. The classical Levy-Mises plastic law for isotropic behavior (6) will be used. The rate of deformation [Mathematical Expression Omitted] is related to the stress by:
[Mathematical Expression Omitted] (2)
where G is the elastic shear modulus, [s.sub.j] the deviatoric part of the stress, [Mathematical Expression Omitted] the effective stress, and [Mathematical Expression Omitted] the effective plastic strain rate; the superposed dot denotes differentiation with respect to time. [Mathematical Expression Omitted] and [Mathematical Expression Omitted] expressions are given in the Appendix. Here, the elastoplastic deformation is supposed to be incompressible.
The deviatoric components ([s.sub.i] = [[Sigma].sub.i] - 1/3 [summation over k = 1, 3] [[Sigma].sub.[Kappa]]) of the stress are easily obtained from Eq 1. It yields [s.sub.z] = 0, and substituting in Eq 2 provides:
[Mathematical Expression Omitted] and [Mathematical Expression Omitted] (3)
The axial component of the strain rate is equal to zero. The pipe length remains constant along the deformation of the pipe. This is a consequence of the incompressibility assumption. From the definition, the strain rate components are related to the radius R and thickness t through the following expressions:
[Mathematical Expression Omitted] and [Mathematical Expression Omitted] (4)
and after integration with respect to time and introducing the effective strain [Mathematical Expression Omitted] (see Appendix), the expressions of R and t become:
[Mathematical Expression Omitted] and [Mathematical Expression Omitted] (5)
where [R.sub.0] and [t.sub.0] are the initial mean radius and thickness of the pipe, respectively.
Inserting Eq 5 in Eq 1 and using the definition of the effective stress (see Appendix), a close form expression is found that relates the internal pressure to the effective strain:
[Mathematical Expression Omitted] (6)
[Mathematical Expression Omitted] is a function of the effective strain and can be determined through any kind of mechanical testing, a tensile test for instance.
This result, Eq 6, is consistent with Durban's analysis (5). Durban deals with the general case of a thick pipe under plane strain ([[Epsilon].sub.z] = 0). An equation similar to Eq 6 is found for the limiting case of thin pipe.
From Eq 6, it can be seen that as the strain increases in the pipe, the pressure reaches a maximum. The optimum is found by the derivation of the pressure p with respect to the effective strain. The derivative is zero for a critical value of the effective strain [Mathematical Expression Omitted] that fulfills the equation:
[Mathematical Expression Omitted] (7)
This equation defines the onset of the instability. This result is a particular case of a general analysis of the plastic flow instability for several deformed structures as bars, spheres, etc., and checked for metals. A review was done by Backofen (4). The critical hoop stress can be then easily derived (see Appendix). It yields:
[Mathematical Expression Omitted] (8)
where [Mathematical Expression Omitted] is the equivalent stress at the strain [Mathematical Expression Omitted]. Equations 7 and 8 will be used in the following sections to derive the burst hoop stress. The mechanical behavior will be characterized by the tensile test.
The polymers examined herein are two grades of plasticized poly-12-amino dodecanoic acid. These are two commercial grades named PA12-P20 and PA12-P40 (2) containing, respectively, a low and high percentage of plasticizer. They are heat and light stabilized. Dogbone samples (standard ISO527) were prepared by injection molding for the tensile test analysis. The samples were 100 mm long, 10 mm wide, and 4 mm thick. The pipes were processed by extrusion.
Burst Equipment and Testing
The burst test is performed under the standard DIN (1). A 300 mm long specimen is filled with water. One end is plugged while the other one is connected to a compressor, The pipe is immersed into a temperature [+ or -] 1 [degrees] C controlled water bath. The sample is left there for one hour before any measurement. The pressure is increased at a controlled rate of one bar per second. The bursting has to occur within a period of two minutes. The burst pressure is the maximum recorded pressure, [p.sub.m]. The hoop stress is obtained using the formula in Eq 1:
[Mathematical Expression Omitted] (9)
where [D.sub.0] and [e.sub.0] are the initial mean diameter (arithmetic average between the internal and external diameter) and the initial thickness of the pipe. The dot index in Eq 9 recalls that the stress is defined with respect to the undeformed geometry.
Results at four temperatures, 23 [degrees] C, 40 [degrees] C, 60 [degrees] C, and 80 [degrees] C, for the two grades of polyamides are available (2). Data are obtained from pipes of several sizes, typically diameters ranging from 8 to 30 mm and thickness from 1 to 4 mm. Whatever the pipe radius and thickness, bursting occurs when the hoop stress, defined in Eq 9, reaches a critical value. The results are reported in Table 1. The experimental error is as large [+ or -] 10%. This is caused by variations of the pipe dimensions from one specimen to the other. The critical hoop stress decreases with content of plasticizer and increase in temperature.
Table 1. Measured Burst Hoop Stress (MPa) for the Two Grades PA12-P20 and PA12-P40 at the Four Temperatures. PA12-P20 PA12-P40 23 [degrees] C 32 24 40 [degrees] C 21.5 17 60 [degrees] C 16 13.5 80 [degrees] C 13.5 12
Tensile Test Analysis
The tensile test apparatus is a MTS 810 from MTS Systems Co. including an oven for temperature controlled tests. The elongation is measured by an extensometer fixed on the sample. The tensile speed is maintained for all the temperatures at a value of 50 mm/mm, which corresponds to a strain rate of about 0.005 [s.sup.-1]. This is consistent with the strain rate imposed on the pipe in the burst test. This estimation will be justified later.
The engineering data (force and deplacement) have been corrected to get the true stress [Sigma](t) and strain [Epsilon](t). As long as the deformation is uniform over the sample without localized necking, the true strain is derived from the elongation by the expression:
[Epsilon](t) = Log [l(t)/[l.sub.0]] (10)
where [l.sub.0] and l(t) are the sample length at time zero and time t. Log is the natural logarithm. Thanks to the incompressibility, the true stress is derived from the force by:
[Sigma](t) = F/[S.sub.0] l(t)/[l.sub.0] (11)
where F and [S.sub.0] are the tensile force and initial sample section, respectively. For these polyamide grades, uniform deformation is verified up to [Epsilon] = 0.4.
Figures 2 and 3 give the results for the two grades, respectively, at four temperatures: 23 [degrees] C, 40 [degrees] C, 60 [degrees] C, and 80 [degrees] C. Mathematical expressions of the equivalent stress versus equivalent strain are required in order to solve Eq 7. From the definition (see the Appendix), the equivalent strain and stress in Eq 7 are readily the true strain and stress, Eqs 10 and 11. The following function [Mathematical Expression Omitted] provides avery good fitting. The Mathematica package (7) allows the calculation of the four coefficients a, b, c, d as well as the solution of Eq 7.
RESULTS AND DISCUSSION
Table 2 provides the results of the theory at each temperature and for the two grades of polymer. Three parameters are reported. The critical equivalent strain [Mathematical Expression Omitted] is the solution of Eq 7. The hoop stress [[Sigma].sub.[Theta]] has been defined earlier, in Eq 8. What can be compared with the experiments is the hoop stress, defined with respect to the initial dimension of the pipe, [Mathematical Expression Omitted], Eq 9. Using Eqs 6 and 10, this factor is connected to [[Sigma].sub.[Theta]] by:
[Mathematical Expression Omitted] (12)
The experimental data Table 1 have been added in Table 2 in the last column for comparison with the calculated [Mathematical Expression Omitted]. The agreement is good. The discrepancy is of the same magnitude as the experimental errors. The strain at burst is almost constant with respect to the temperature. Values of about 0.24 are obtained for PA12-P20 and 0.28 for PA12-P40. Returning to the estimation of the tensile speed previously defined (see Experimental section), the strain rate in the pipe can be evaluated as the ratio of [Mathematical Expression Omitted] over the time to burst. This latter is of the order of one minute as specified in the Standard procedure. This yields a value of 0.005 [s.sup.-1], as previously stated. This justifies a posteriori the chosen tensile speed in the tensile test.
A significant difference between the hoop stresses [Mathematical Expression Omitted] and [[Sigma].sub.[Theta]] can be noticed. These relate the same state of stress but with respect to the initial and deformed geometry, respectively.
The theory agrees well with the experiments because some basic assumptions on the material behavior are checked in the polymers studied here. First, the mechanical properties are isotropic. In effect, it has been verified on large injection molded plates that the properties along the parallel and perpendicular [TABULAR DATA FOR TABLE 2 OMITTED] directions to the injection flow are quite similar. A weak sensitivity to the process conditions is noticed. The constitutive equation has been identified from injection molded samples although it has been used successfully for the deformation analysis of extruded pipe. Therefore, the plastic behavior is safely represented by an isotropic flow rule. This cannot be the general case. Polyvinyl chloride (PVC), for instance, is known to be very sensitive to the process history and conditions (8). Such a study on PVC pipes, albeit possible, would require special attention to the preparation and reliability of the samples.
The incompressibility condition is also fulfilled by these plasticized polyamides. This assumption may not be corroborated in all polymers. When the deformation proceeds by crazing, incompressibility is not verified, as in the case of amorphous polymers (9). This theoretical model is not relevant and should be modified.
We have shown that burst pressure of plasticized PA12 pipes can be predicted at room and elevated temperature. A few data - from tensile testing here - on the mechanical behavior of the materials are required. The theoretical model reviewed has been developed for metals and but has applied well for these polymers. This approach is a way to study new polymer compounds. Simple tensile tests only are required. Improvement of the burst resistance can be quickly assessed with a small quantity of product and without costly pipe processing.
The Von Mises theory of isotropic plasticity introduces two scalars denoted the equivalent strain rate and equivalent stress. These expressions are the following in the present case:
[Mathematical Expression Omitted]
According to Eq 3, it can be simplified to:
[Mathematical Expression Omitted]
The equivalent strain is the cumulative effective strain rate, i.e:
[Mathematical Expression Omitted]
The equivalent stress is defined using the principal components as:
[Mathematical Expression Omitted]
and can be reduced using Eq 1 in:
[Mathematical Expression Omitted]
The author thanks Mr. P. Kerelo and Dr. P. Dang for discussions during the course of this study.
1. Standard DIN 53 758, Epreuve de pression interne de courte duree sur corps creux ("Short term test of internal pressure resistance of vessels") (1975).
2. Technical Data Sheet, Rilsan 12 pipes, Elf Atochem Co. (1996).
3. H. W. Swift, J. Mech. Phys. Solids, 1, 1 (1952).
4. W. Backofen, Deformation Processing, Chap. 10. Addison-Wesley Publishing Co. (1972).
5. D. Durban, Trans. ASME, 46, 228 (1979)
6. W. H. Johnson and P. B. Mellor, Engineering Plasticity, pp 86-87, Van Nostrand, London (1973).
7. Wofram Research, Mathematica Package (1992).
8. B. J. Lanham and W. V. Titow, PVC Technology, Chap. 14, Elsevier Applied Science Publishers, London and New York (1984).
9. J. G. Williams, Fracture Mechanics of Polymers, Chap. 6, Ellis Horwood Limited (1984).
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|Publication:||Polymer Engineering and Science|
|Date:||Apr 1, 1998|
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