# A practical approach to modeling time-dependent nonlinear creep behavior of polyethylene for structural applications.

INTRODUCTION

Polyethylenes (PEs) are increasingly used in many aspects of our lives, including infrastructure and building construction. The rationale for the research presented herein is the need for rational structural analysis and design tools for polymeric structures and structural components. Such analytical tools should be capable of capturing essential features of the material behavior, being at the same time relatively simple, versatile, and computationally efficient.

One of the common construction applications of PE is as pipes for gas, water, and sewer systems. Specifically, immediate need for research presented herein comes from related work on modeling PE pipes installed and rehabilitated using trenchless methods [1]. PE pipes have been used for their combination of strength, flexibility, corrosion resistance, and lightweight. In modern pipeline installation practices, pipes are often pulled underground through horizontal boreholes, which in order to minimize pulling loads, require low stiffness of a pipe and PE is very well suited for this application. Loading of the pipe consists of short-term longitudinal stresses due to pulling forces and, long-term, variable hoop stresses due to internal and external pressures, which combined with scratching of the outer surface can result in slow crack growth and premature failure. These pipes are subject to a complex 3D load history, making hereditary effects very important in effective constitutive modeling. Adequate long-term strength and performance of the PE pipes are probably the major concerns in design and construction; however, no adequate predictive models exist that can link slow crack growth in real pipes with environmental stress crack resistance (ESCR) determined using standard laboratory tests.

Creep of polymers is related to their long-term strength and to ESCR. It also best represents the time-dependent nature of the behavior of polymeric structures (e.g. pipes) subject to complex load history. Based on creep test results, hereditary-type constitutive models can be developed, which form a part of structural analysis formulations. Many constitutive models for PE have been proposed and will be reviewed briefly in this paper. Modeling often addresses either viscoelastic (defined rather arbitrarily as behavior at low strain, before yield) or post yield before necking, or at very large deformations and strains during necking. The last two problems are not really applicable for structural design; structures are never allowed to carry loads which cause such excessive deformations. In fact, from a structural analysis point of view, yield or slow stress cracking should be considered a failure. PE is a semicrystalline polymer with large hydrocarbon chain molecules which are capable of adopting two distinct arrangements, crystalline and amorphous. The material behavior of semicrystalline polymers at low strains is often categorized as viscoelastic or viscoplastic, which suggests a combination of viscous flow typical for fluids, with either elastic or plastic characteristics typical for solids. The proportion of the viscous, elastic, and plastic characteristics depends on the rate of loading, time, loading history, stress level, and temperature. It also depends strongly on the molecular structure including crystallinity, molecular weight, molecular weight distribution, and short- and long-chain branching. This is why different PEs respond differently to loads and should be modeled using different material parameters specific for a given material. Most of the approaches published in the literature are material specific (developed for one analyzed material) and because they utilize rather complex mathematical expressions with several material parameters they are difficult to apply in practical structural analysis cases. Since every PE behaves differently under the same loads, a material model developed and calibrated for one PE does not necessarily work well for another one.

Constitutive modeling of the PE falls under two main categories: micromechanical or macromechanical. Micromechanical approaches start with the analysis of microstructure on the molecular level; the crystalline and amorphous phases are modeled separately considering their specific characteristics and then combined to form a homogenized model of a bulk material. These models (e.g. [2, 3]) are very important for studies on the behavior of PEs under simple tension, compression, or shear loads and can also serve as an excellent research tool for linking chemical composition of the polymer with its mechanical response to applied loads [4].

For practical analysis of real structures, macromechanical, phenomenological models are needed, which consist of mathematical equations relating strains to stresses at the macrostructural level. Such formulations can form part of structural analysis procedures. This paper presents such a modeling approach. The macromechanical modeling generally uses the experimental behavior under simple loads to define material factors to be used in mathematical equations describing the relations between stresses and strains. Within the macromodeling approaches, a large number of papers have been written on time-dependent, nonlinear behavior before yield, where researchers derived their relationships in the form of viscoelastic or viscoplastic equations; dependent on time, loading history, loading rate; callibrated based on uniaxial tensile or compressive tests. Well-known nonlinear time-dependent formulations were proposed by Schapery [5], Krishnaswamy et al. [6], and Zhang and Moore [7, 8], who developed phenomenological integral time-dependent models applicable for finite element structural analysis. They addressed the non-linearity of PE behavior by formulating material coefficients as functions of stress. Lu et al. [9] proposed phenomenological creep modeling for ABS pipes, using different mathematical formulae depending on stress and temperature levels. An adaptive link theory was proposed by Drozdov [10] and Drozdov and Kalamkarov [11]; integral form viscoelastic formulations were obtained from these models. The model proposed by Popelar et al. [12] is based on relaxation tests. It predicts well the stress-strain response in uniaxial constant strain rate test at small strain rate (<[10.sup.-3] [s.sup.-1]). Beijer and Spoormaker [13] investigated the performance of integral model formulations under small strain (less than 5%) and under strains closer to yield and including yield. They pointed out the difference in the character of time-dependent behavior of HDPE with increasing stress approaching yield. Lai and Bakker [14] investigated creep of PE, and their formulation includes aging through the time shift function. A theoretical method for predicting hoop stresses in PE pipes, using a visco-hyperelastic constitutive model, was presented by Guan and Boot [15]. A model proposed by Duan et al. [16] follows the phenomenological unified approach to predict the stress-strain relationship for semicrystalline and glassy polymers for a large spectrum of deformations; before yield, during viscoplastic phase after yield and strain hardening. It was fit on test data on poly-methyl-methacrylate (PMMA).

Although substantial work on nonlinear time-dependent modeling has been done, design engineers have considerable difficulty finding an easy and yet rational way for performing structural analysis which captures both time and nonlinear effects of polymeric structures (see also comments by Drozdov [10]). Linear viscoelastic, or even simple linear elastic analysis, is often used instead. Such analysis procedures are readily available in many commercial finite element packages. The objective of the present work is to develop a practical method for constitutive modeling for PE that would include time and nonlinear effects with the accuracy acceptable for analysis of structures. The method should be easily calibrated for a given material based on tests and then developed into a format that can be applied for structural analysis (e.g. finite element analysis). The paper presents the proposed formulation and the relevant material testing conducted on samples from four different PEs. Applicability of the formulation for modeling a variety of loading scenarios is shown. The theoretical background used in constitutive modeling is discussed first. Classical formulations based on integral equations are adopted for linear viscoelastic modeling, based on either multiple Kelvin elements or power law functions. Creep testing is used for experimental determination of material parameters which are calculated using a numerical procedure. The method for including nonlinearity and the procedure for modeling load rate effects are also discussed. The models are verified by comparison with creep responses at different stress levels, with step loading test results and load rate testing. The potential of this work for practical structural analysis is discussed.

MATERIAL TESTING

Four different PE materials (Table 1) were tested in creep. Three of the materials were specified as HDPE, one as MDPE. Two materials, HDPE-Pipe and MDPE-Pipe, were cut from pipe samples donated for this work by KWH Pipe Canada. HDPE-1 and HDPE-2 were donated by Imperial Oil in pellet form and were subsequently hot pressed into plates and cut into dog-bone shaped specimens. The first set of tests was conducted for calibrating constitutive formulations (estimating material parameters) for five different stress levels, each for 24 h. The 24-h time frame was chosen as reasonable for practical industrial applications. The tests were conducted at room temperature (22[degrees]C). The engineering strain versus time was measured using a 1/2" clip gauge attached to the side of a specimen. Load was applied by dead weights through a lever arm. The test setup is a four-pole frame with the vertical tensile rod-clipper device in the center. The example of measured creep curves, for HDPE-Pipe, is shown in Fig. 1. The examples of experimental creep compliance curves are shown in Figs. 2 and 3. These figures are based on measurements of engineering stress and strains. Two general observations can be made from Figs. 2 and 3. First, each PE material behaves differently under similar load levels. There are noticeable differences in creep responses for the two HDPE materials (compliance curves for HDPE Resin 1 were also different). Second, the materials exhibit responses that are both time dependent and nonlinear since a single creep compliance curve cannot describe the behavior of a given material.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The second set of tests involved creep under intermediate stresses (stresses different from the ones used for estimating material parameters) and were used for corroboration of the modeling approach for nonlinear creep (dependence of creep compliance on stress). The third type of tests involved creep under step loading. These tests were used to corroborate a numerical routine used to evaluate hereditary effects in the behavior of PE. Finally, in the fourth type of tests, which were done in the MTS 810 tester, different load rates, strain rates, and relaxation responses were investigated. All tests were conducted at room temperature (22[degrees]C).

[FIGURE 3 OMITTED]

LINEAR TIME-DEPENDENT MODELING

In linear time-dependent modeling the material response is dependent on time only. Boltzmann superposition principle of linear systems can be applied to study the material behavior. Linear viscoelastic models are used in either integral or differential equation form [17, 18]. From a practical point of view, integral form models can be discretized and the resulting formulation implemented into numerical routines. A standard modeling approach for viscoelastic modeling for the stress-strain relationship for time-dependent materials, ignoring aging effects, can take on the integral form

[sigma](t) = [[integral].sub.0.sup.t] [phi](t - [tau]) [dot.[epsilon]]([tau]) d[tau]. (1)

Alternatively, the strain of the material at time t can also be expressed as

[epsilon](t) = [[integral].sub.0.sup.t] [psi](t - [tau]) [dot.[sigma]]([tau]) d[tau], (2)

where functions [phi] and [psi] are known as the stress-relaxation function and the creep function, respectively. Material properties should be independent of testing procedures, which follows that either 1 or 2 uniquely defines the material response.

Phenomenological modeling of the behavior of viscoelastic solids requires determining creep or relaxation functions to be used in Eq. 1 or 2. Creep functions are generally easier to obtain from experiments and are used in this work. In a creep test, the stress in the tested sample is kept constant and the strain is recorded with time. For a constant stress Eq. 2 becomes

[epsilon](t) = [[sigma].sub.c][psi](t), (3)

where [psi](t) is the creep compliance. The material modeling task is to find a function [psi](t) that best fits test results.

Multi-Kelvin Approach

For a linear viscoelastic material subject to a constant stress applied at time [t.sub.0], an elastic strain occurs instantaneously, followed by the viscous strain growth. The strain growth rate slows down with time and it can become constant at a certain time. When the loading is removed, there is an instantaneous elastic strain recovery, followed by gradual recovery of the viscous strain.

When the material behaves as aforementioned, its behavior can be modeled by the exponential creep function, [psi](t):

[psi](t) = [[psi].sub.e] + [[psi].sub.v](t) = [1/[E.sub.0]] + [n.summation over (i=1)] [1/[E.sub.i]]{1 - exp(-[t/[[tau].sub.i]])}, (4)

where [[psi].sub.e] represents the instantaneous elastic component, and [[psi].sub.v](t) represents the time-dependent viscoelastic effects. Material constants are the elastic modulus [E.sub.0], which defines the instantaneous response, and the moduli [E.sub.i], with corresponding relaxation times [[tau].sub.i]. Each pair of [E.sub.i] and [[tau].sub.i] defines the material response mainly at times smaller than [[tau].sub.i]. Since the creep function is independent of stress or strain, the model defined in 4 is linear. The constitutive equation can be obtained by substituting Eq. 4 into Eq. 2,

[FIGURE 4 OMITTED]

[epsilon](t) = [[integral].sub.0.sup.t]{[1/[E.sub.0]] + [n.summation over (i=1)] [1/[E.sub.i]]{1 - exp(-[[t - [tau]]/[[tau].sub.i]])}} [dot.[sigma]]([tau]) d[tau]. (5)

The creep compliance in Eq. 4 can also be written as

[psi](t) = [[psi].sub.e] + [[psi].sub.v](t) = [1/[E.sub.0]] + [n.summation over (i=1)] [1/[E.sub.i]] [W.sub.i](t). (6)

The values of the relaxation times [[tau].sub.i] determine the values of multiplication factors [W.sub.i] for the creep coefficients 1/[E.sub.i]. Figure 4 shows the variation of these multiplication factors with time, for different relaxation times [[tau].sub.i]. Each [[tau].sub.i] controls a separate time window. When W becomes equal to unity, the viscoelastic behavior becomes elastic. Thus, using the formulation in Eq. 6, the viscoelastic behavior of the material will become elastic with time. In the present work, to simplify calculations, the magnitudes of the relaxation times [[tau].sub.i] are taken as constants with the shortest time being 500 s. The next relaxation time is taken as 20 times the value of the previous one, and so on. By assigning constant values to the relaxation time, linear least squares equations are obtained for determining the creep coefficients 1/[E.sub.i].

It should also be pointed out that the multi-Kelvin viscoelastic modeling can also be obtained from rheological models that consist of one spring and a series of Kelvin elements (see Fig. 5). A Kelvin element consists of one spring and one dashpot parallel to each other; the deformations of the spring and dashpot under loading are assumed to be equal at any time. Material constants are the elastic moduli of the springs [E.sub.0], [E.sub.i], and the viscous moduli of the dashpots, [[eta].sub.i], with [[tau].sub.i] = [[eta].sub.i]/[E.sub.i]. Therefore, the exponential form of the creep compliance function is referred to as multi-Kelvin approach.

[FIGURE 5 OMITTED]

Power Law

Power law functions can also be used for modeling creep (e.g. [6]). This type of modeling is often referred as viscoplastic because the modeled deformation (strain) keeps growing, at decreasing rate, and does not approach an asymptotic value; even after a long time the material remains time-dependent.

To model the behavior described above, the creep compliance, [psi](t), can be expressed by a power law function as

[psi](t) = [[psi].sub.e] + [[psi].sub.v](t) = [1/[E.sub.0]] + [C.sub.0] [t.sup.[C.sub.1]], (7)

where [[psi].sub.e] is the instantaneous elastic component; [[psi].sub.v](t) represents the time-dependent component. [E.sub.0], [C.sub.0], and [C.sub.1] are material constants obtained from the least squares estimation. The constitutive equation for the viscoplastic model can be obtained by substituting Eq. 7 into Eq. 2,

[epsilon](t) = [[integral].sub.0.sup.t] {[1/[E.sub.0]] + [C.sub.0] (t - [tau])[.sup.[C.sub.1]]}[dot.[sigma]]([tau])d[tau]. (8)

Curve Fitting of the Creep Response

Both equations, Eqs. 4 and 7, can be used for parameter estimation based on experimentally obtained creep curves. In fact, for the time frame corresponding to the test, the result of curve fitting will be satisfactory regardless of which equation is chosen. However, the predictions by the two equations for longer times will differ. The creep behavior of semicrystalline polymers depends, among other factors, on the level of loading and it has been experimentally observed that under lower levels of loading the behavior is characterized by initial creep which decays with time (see Fig. 6), and hence the viscoelastic behavior becomes elastic with time. This type of behavior can be well modeled using the multi-Kelvin approach with properly chosen relaxation times.

In creep tests under higher loads (for PE material these loads would be larger than ~10 MPa), the behavior changes and strains continue to grow under these higher loads. The creep compliance does not approach an asymptotic value but grows continuously with time. Therefore, the power law of Eq. 7 will better reflect actual physical behavior of the material at longer times [9].

NONLINEAR TIME-DEPENDENT MODELING

Most polymeric materials exhibit behavior that is dependent, at a given temperature, on both time and stress. As shown in Figs. 2 and 3, creep behavior cannot be described by a single compliance-time curve. For each load level, a different curve is obtained and thus different set of material parameters should be defined.

Formulations that incorporate stress dependence in the creep compliance include, e.g., the Schapery model [5], Krishnaswamy et al. [6], Zhang and Moore [8], and Beijer and Spoormaker [13]. A simple method to incorporate stress dependence in the material behavior is to formulate a creep compliance [psi] with the material parameters that depend on stress. This creates a nonseparable form of the nonlinear constitutive relationship (e.g. [6, 13]). For a nonlinear multi-Kelvin model, the creep compliance, [psi]([sigma],t), can then be written as

[psi]([sigma], t) = [[psi].sub.e]([sigma]) + [[psi].sub.v]([sigma], t)

= [1/[[E.sub.0]([sigma])]] + [n.summation over (i=1)] [1/[[E.sub.i]([sigma])]]{1 - exp(-[t/[[[tau].sub.i]([sigma])]])}. (9)

For a nonlinear power law model, the creep compliance, [psi]([sigma],t), becomes

[psi]([sigma], t) = [[psi].sub.e]([sigma]) + [[psi].sub.p]([sigma], t) = [1/[[E.sub.0]([sigma])]] + [C.sub.0]([sigma]) [t.sup.[C.sub.1]([sigma])], (10)

where [E.sub.0]([sigma]), [E.sub.i]([sigma]), [[tau].sub.i]([sigma]), [C.sub.0]([sigma]), and [C.sub.1]([sigma]) are functions of stress. The modeling requires finding appropriate functions that describe the dependence of the material coefficients on stress.

[FIGURE 6 OMITTED]

The nonlinear constitutive equation is then obtained by substituting either Eq. 9 or 10 into Eq. 2:

[epsilon](t) = [[integral].sub.0.sup.t] {[psi]([sigma], t - [tau])}[dot.[sigma]]([tau])d[tau]. (11)

Proposed Nonlinear Modeling Procedure

To incorporate stress dependence on material properties, a simple approach is taken herein where a linear interpolation is used to include stress influence on the material parameter values. Thus, instead of finding the best function to fit the relationships, e.g. [E.sub.i]([sigma]), a piecewise linear function is adopted. It is assumed that such approach provides adequate accuracy for most structural analysis problems. It also allows using a numerical computer procedure to find these material constants and their stress dependence making the modeling very easy to apply. The numerical procedure generates the appropriate model for a given material. The procedure consists of the following steps.

Step 1. Creep tests are done first for a few selected stress levels, [[sigma].sub.1],..., [[sigma].sub.n].

Step 2. For each stress level a separate set of material parameters is obtained. The sets of constants for the creep tests generate an array of material constants. The model is presented as a matrix of material parameters.

Step 3. The material parameters for stresses other than the tested stresses are obtained by linear interpolation. The modeling is implemented into a numerical procedure. It can be repeated for any material.

For the multi-Kelvin model, the relaxation times are assumed independent of stress. This simplifies the least squares estimation, making it linear, which is used to determine the creep coefficients [x.sub.i] = 1/[E.sub.i]. Least squares estimation for the multi-Kelvin model is done by assuming first that one relaxation time (one Kelvin element and a spring), N = 1, is satisfactory to model the whole creep response. The fitting error is evaluated and if a certain stopping criterion is not satisfied, the number of Kelvin elements is increased to 2, N = 2. Again, the fitting error is evaluated and N is continuously increased until satisfactory curve fitting is obtained. In this work, the fitting error of 1-1.3% was used. It resulted, in all presented cases, in the selection of three Kelvin elements. The procedure is implemented in a computer routine.

The number of stress levels for creep tests used for model development, and thus creep tests which need to be done, and the magnitudes of stresses should be determined depending on the type of practical problem to be analyzed. Then, the creep coefficients for any intermediate stress are found from

[x.sub.i]([sigma]) = [1/[[E.sub.i]([sigma])]] = [x.sub.i]([[sigma].sub.m]) + [[[sigma] - [[sigma].sub.m]]/[[[sigma].sub.n] - [[sigma].sub.m]]][[x.sub.i]([[sigma].sub.n]) - [x.sub.i]([[sigma].sub.m])], (12)

where [x.sub.i] = 1/[E.sub.i]; [[sigma].sub.m] and [[sigma].sub.n] are the stresses used for model development; for [sigma] is: [[sigma].sub.m] < [sigma] < [[sigma].sub.n].

Similarly, linear interpolation for a viscoplastic model takes the form

[C.sub.i]([sigma]) = [C.sub.i]([[sigma].sub.m]) + [[[sigma] - [[sigma].sub.m]]/[[[sigma].sub.n] - [[sigma].sub.m]]][[C.sub.i]([[sigma].sub.n]) - [C.sub.i]([[sigma].sub.m])]. (13)

The elastic stiffness terms, [E.sub.0]([sigma]), for different stresses, for both models, are found directly from

[E.sub.0]([sigma]) = [E.sub.0]([[sigma].sub.m]) + [[[sigma] - [[sigma].sub.m]]/[[[sigma].sub.n] - [[sigma].sub.m]]][[E.sub.0]([[sigma].sub.n]) - [E.sub.0]([[sigma].sub.m])]. (14)

MODELING LOAD HISTORY EFFECTS

Viscoelastic constitutive equations contain time dependency in their formulation, where strains and stresses depend on load history. Boltzman superposition principle can then be applied for calculating strain (or stress) at time t when the material is subject to the given stress (or strain) history. In case of nonlinear material behavior, the creep compliance depends on both time and stress, and thus, strictly speaking, superposition is not valid. In the present work, the modified superposition principle is adopted where the requirement for linearity in stress is relaxed and nonlinearity effects are treated in the following approximate sense [19], as described briefly below.

Response to Multiple Loading Steps

When the load history includes stress changing with time, it can be approximated as a series of small time steps in which constant values of stress are applied. The stress [[sigma].sub.1] is applied abruptly at time [t.sub.0] and held constant till time [t.sub.1]. At time [t.sub.1], the strain is equal to:

[[epsilon].sub.1] = [[[psi].sup.e]([[sigma].sub.1]) + [[psi].sup.v]([[sigma].sub.1], [t.sub.1] - [t.sub.0])][[sigma].sub.1] (15)

At time [t.sub.1] a new stress [[sigma].sub.2] is applied in a jump and held constant till time [t.sub.2]. At time [t.sub.1] the creep behavior is calculated as if at this instant, stress [[sigma].sub.1] is removed and at the same time stress [[sigma].sub.2] is applied. Both stresses are considered as independent and calculation of strain at time [t.sub.2] includes strain recovery from time [t.sub.1] to [t.sub.2]:

[[epsilon].sub.2] = [[psi].sup.e]([[sigma].sub.2])[[sigma].sub.2] + [[psi].sup.v]([[sigma].sub.1], [t.sub.2] - [t.sub.0])[[sigma].sub.1] + [[[psi].sup.v]([[sigma].sub.2], [t.sub.2] - [t.sub.1])[[sigma].sub.2] - [[psi].sup.v]([[sigma].sub.1], [t.sub.2] - [t.sub.1])[[sigma].sub.1]]. (16)

[FIGURE 7 OMITTED]

For the N increments of stress applied in N time steps the following general formula can be written:

[[epsilon].sub.N] = [[psi].sup.e]([[sigma].sub.N])[[sigma].sub.N] + [N.summation over (n=1)] [[[psi].sup.v]([[sigma].sub.n], [t.sub.N] - [t.sub.n-1])[[sigma].sub.n] - [[psi].sup.v]([[sigma].sub.n-1], [t.sub.N] - [t.sub.n-1])[[sigma].sub.n-1]] (17)

in which the total time t is divided into N discrete intervals.

NUMERICAL RESULTS

The proposed procedure was employed for the analysis of PE samples under a variety of loading conditions. Results and comparisons between experimental and numerical results are presented herein. Both multi-Kelvin and power-law approaches are shown.

Creep Tests for Model Generation

Creep tests were conducted on all four PEs. For each PE, five creep compliances representing five different stress levels were obtained. These experimental results were used for generating material models according to Eqs. 5 and 8. The modeling parameters are given in Tables 2-5.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

The simulation results for all four PE materials follow almost exactly the test results for both multi-Kelvin and power-law models and therefore they are not presented for the sake of brevity.

Creep Tests at Intermediate Stress Levels

To check the performance of the proposed approach a comparison is done between simulated and test results for creep loading at stress levels different from the ones which were used to calibrate the model. The comparison provides information on how the linear interpolation of material parameters between different stress levels works for nonlinear behavior of PE. The results for HDPE-Pipe is shown in Fig. 7 and for HDPE Resin 2 are shown in Fig. 8. There is some difference between the predicted and test results; however, this difference is acceptable for engineering purposes.

Step Loading and Load Rates

Figure 9 shows the comparison of simulated and test results for multistep creep loading for multi-Kelvin approach. Predictions using power law approach are almost exactly the same and therefore are not shown. Both modeling equations simulate the experimental results with an accuracy acceptable for engineering purposes. Load rate tests were also conducted on HDPE-Pipe samples using the MTS 810 testing machine. The results of experiments and numerical simulations are presented in Figs. 10 and 11. Testing involved control of load rate which was chosen to correspond to the appropriate engineering stress rate. Two stress rates are shown, namely, 0.01 MPa/s in Fig. 10 and 0.1 MPa/s in Fig. 11.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The effects of step loading and load rate are captured by both models implemented based on the modified superposition principle. Generally it can be observed that in these analyses the main error is at the beginning of each new load step (see Fig. 4). Since the test creep data were collected over a 24-h period (which was considered reasonable for structural applications), the curve fitting procedure minimizes fitting error over the whole 24 h, thus reducing the accuracy for very short times. This is also the reason why the models perform better for slower loading (see Fig. 10) but predict too soft response for faster loading (see Fig. 11).

Complex Loading Results

The HDPE-Pipe samples were subjected to complex loading which included combinations of loading and unloading rates and creep at different stress levels. An example is shown in Fig. 12 (loading history) and Fig. 13 (experimental and theoretical responses). The theoretical procedure provided results with acceptable accuracy for both loading and unloading sequences in the load history.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

CONCLUSIONS

A practical modeling approach is introduced for PE materials. The proposed approach is based on existing theoretical principles; namely, integral formulation for modeling hereditary behavior of PE before yield. The goal was to create a time-dependent and stress-dependent approach that can easily be applied to any viscoelastic material, provided creep testing has been done for few selected, representative stress levels.

The approach can easily be repeated for other materials. Standard Fortran subroutines were written for generating a creep compliance function by least squares estimation based on exponential (multi-Kelvin) and power law models. Stress dependence of material parameters was incorporated using a piece-wise linear representation. Models were generated for four PE materials and successfully compared to experimental data (creep and load history effects).

ACKNOWLEDGMENTS

Material samples used in the experiments were donated by Imperial Oil and KWH Pipe Canada.

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Hongtao Liu, (1) Maria Anna Polak, (1) Alexander Penlidis (2)

(1) Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(2) Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Correspondence to: Maria Anna Polak; e-mail: polak@uwaterloo.ca

Contract grant sponsor: Imperial Oil and the Natural Sciences and Engineering Research Council (NSERC).

Polyethylenes (PEs) are increasingly used in many aspects of our lives, including infrastructure and building construction. The rationale for the research presented herein is the need for rational structural analysis and design tools for polymeric structures and structural components. Such analytical tools should be capable of capturing essential features of the material behavior, being at the same time relatively simple, versatile, and computationally efficient.

One of the common construction applications of PE is as pipes for gas, water, and sewer systems. Specifically, immediate need for research presented herein comes from related work on modeling PE pipes installed and rehabilitated using trenchless methods [1]. PE pipes have been used for their combination of strength, flexibility, corrosion resistance, and lightweight. In modern pipeline installation practices, pipes are often pulled underground through horizontal boreholes, which in order to minimize pulling loads, require low stiffness of a pipe and PE is very well suited for this application. Loading of the pipe consists of short-term longitudinal stresses due to pulling forces and, long-term, variable hoop stresses due to internal and external pressures, which combined with scratching of the outer surface can result in slow crack growth and premature failure. These pipes are subject to a complex 3D load history, making hereditary effects very important in effective constitutive modeling. Adequate long-term strength and performance of the PE pipes are probably the major concerns in design and construction; however, no adequate predictive models exist that can link slow crack growth in real pipes with environmental stress crack resistance (ESCR) determined using standard laboratory tests.

Creep of polymers is related to their long-term strength and to ESCR. It also best represents the time-dependent nature of the behavior of polymeric structures (e.g. pipes) subject to complex load history. Based on creep test results, hereditary-type constitutive models can be developed, which form a part of structural analysis formulations. Many constitutive models for PE have been proposed and will be reviewed briefly in this paper. Modeling often addresses either viscoelastic (defined rather arbitrarily as behavior at low strain, before yield) or post yield before necking, or at very large deformations and strains during necking. The last two problems are not really applicable for structural design; structures are never allowed to carry loads which cause such excessive deformations. In fact, from a structural analysis point of view, yield or slow stress cracking should be considered a failure. PE is a semicrystalline polymer with large hydrocarbon chain molecules which are capable of adopting two distinct arrangements, crystalline and amorphous. The material behavior of semicrystalline polymers at low strains is often categorized as viscoelastic or viscoplastic, which suggests a combination of viscous flow typical for fluids, with either elastic or plastic characteristics typical for solids. The proportion of the viscous, elastic, and plastic characteristics depends on the rate of loading, time, loading history, stress level, and temperature. It also depends strongly on the molecular structure including crystallinity, molecular weight, molecular weight distribution, and short- and long-chain branching. This is why different PEs respond differently to loads and should be modeled using different material parameters specific for a given material. Most of the approaches published in the literature are material specific (developed for one analyzed material) and because they utilize rather complex mathematical expressions with several material parameters they are difficult to apply in practical structural analysis cases. Since every PE behaves differently under the same loads, a material model developed and calibrated for one PE does not necessarily work well for another one.

Constitutive modeling of the PE falls under two main categories: micromechanical or macromechanical. Micromechanical approaches start with the analysis of microstructure on the molecular level; the crystalline and amorphous phases are modeled separately considering their specific characteristics and then combined to form a homogenized model of a bulk material. These models (e.g. [2, 3]) are very important for studies on the behavior of PEs under simple tension, compression, or shear loads and can also serve as an excellent research tool for linking chemical composition of the polymer with its mechanical response to applied loads [4].

For practical analysis of real structures, macromechanical, phenomenological models are needed, which consist of mathematical equations relating strains to stresses at the macrostructural level. Such formulations can form part of structural analysis procedures. This paper presents such a modeling approach. The macromechanical modeling generally uses the experimental behavior under simple loads to define material factors to be used in mathematical equations describing the relations between stresses and strains. Within the macromodeling approaches, a large number of papers have been written on time-dependent, nonlinear behavior before yield, where researchers derived their relationships in the form of viscoelastic or viscoplastic equations; dependent on time, loading history, loading rate; callibrated based on uniaxial tensile or compressive tests. Well-known nonlinear time-dependent formulations were proposed by Schapery [5], Krishnaswamy et al. [6], and Zhang and Moore [7, 8], who developed phenomenological integral time-dependent models applicable for finite element structural analysis. They addressed the non-linearity of PE behavior by formulating material coefficients as functions of stress. Lu et al. [9] proposed phenomenological creep modeling for ABS pipes, using different mathematical formulae depending on stress and temperature levels. An adaptive link theory was proposed by Drozdov [10] and Drozdov and Kalamkarov [11]; integral form viscoelastic formulations were obtained from these models. The model proposed by Popelar et al. [12] is based on relaxation tests. It predicts well the stress-strain response in uniaxial constant strain rate test at small strain rate (<[10.sup.-3] [s.sup.-1]). Beijer and Spoormaker [13] investigated the performance of integral model formulations under small strain (less than 5%) and under strains closer to yield and including yield. They pointed out the difference in the character of time-dependent behavior of HDPE with increasing stress approaching yield. Lai and Bakker [14] investigated creep of PE, and their formulation includes aging through the time shift function. A theoretical method for predicting hoop stresses in PE pipes, using a visco-hyperelastic constitutive model, was presented by Guan and Boot [15]. A model proposed by Duan et al. [16] follows the phenomenological unified approach to predict the stress-strain relationship for semicrystalline and glassy polymers for a large spectrum of deformations; before yield, during viscoplastic phase after yield and strain hardening. It was fit on test data on poly-methyl-methacrylate (PMMA).

Although substantial work on nonlinear time-dependent modeling has been done, design engineers have considerable difficulty finding an easy and yet rational way for performing structural analysis which captures both time and nonlinear effects of polymeric structures (see also comments by Drozdov [10]). Linear viscoelastic, or even simple linear elastic analysis, is often used instead. Such analysis procedures are readily available in many commercial finite element packages. The objective of the present work is to develop a practical method for constitutive modeling for PE that would include time and nonlinear effects with the accuracy acceptable for analysis of structures. The method should be easily calibrated for a given material based on tests and then developed into a format that can be applied for structural analysis (e.g. finite element analysis). The paper presents the proposed formulation and the relevant material testing conducted on samples from four different PEs. Applicability of the formulation for modeling a variety of loading scenarios is shown. The theoretical background used in constitutive modeling is discussed first. Classical formulations based on integral equations are adopted for linear viscoelastic modeling, based on either multiple Kelvin elements or power law functions. Creep testing is used for experimental determination of material parameters which are calculated using a numerical procedure. The method for including nonlinearity and the procedure for modeling load rate effects are also discussed. The models are verified by comparison with creep responses at different stress levels, with step loading test results and load rate testing. The potential of this work for practical structural analysis is discussed.

MATERIAL TESTING

Four different PE materials (Table 1) were tested in creep. Three of the materials were specified as HDPE, one as MDPE. Two materials, HDPE-Pipe and MDPE-Pipe, were cut from pipe samples donated for this work by KWH Pipe Canada. HDPE-1 and HDPE-2 were donated by Imperial Oil in pellet form and were subsequently hot pressed into plates and cut into dog-bone shaped specimens. The first set of tests was conducted for calibrating constitutive formulations (estimating material parameters) for five different stress levels, each for 24 h. The 24-h time frame was chosen as reasonable for practical industrial applications. The tests were conducted at room temperature (22[degrees]C). The engineering strain versus time was measured using a 1/2" clip gauge attached to the side of a specimen. Load was applied by dead weights through a lever arm. The test setup is a four-pole frame with the vertical tensile rod-clipper device in the center. The example of measured creep curves, for HDPE-Pipe, is shown in Fig. 1. The examples of experimental creep compliance curves are shown in Figs. 2 and 3. These figures are based on measurements of engineering stress and strains. Two general observations can be made from Figs. 2 and 3. First, each PE material behaves differently under similar load levels. There are noticeable differences in creep responses for the two HDPE materials (compliance curves for HDPE Resin 1 were also different). Second, the materials exhibit responses that are both time dependent and nonlinear since a single creep compliance curve cannot describe the behavior of a given material.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The second set of tests involved creep under intermediate stresses (stresses different from the ones used for estimating material parameters) and were used for corroboration of the modeling approach for nonlinear creep (dependence of creep compliance on stress). The third type of tests involved creep under step loading. These tests were used to corroborate a numerical routine used to evaluate hereditary effects in the behavior of PE. Finally, in the fourth type of tests, which were done in the MTS 810 tester, different load rates, strain rates, and relaxation responses were investigated. All tests were conducted at room temperature (22[degrees]C).

[FIGURE 3 OMITTED]

LINEAR TIME-DEPENDENT MODELING

In linear time-dependent modeling the material response is dependent on time only. Boltzmann superposition principle of linear systems can be applied to study the material behavior. Linear viscoelastic models are used in either integral or differential equation form [17, 18]. From a practical point of view, integral form models can be discretized and the resulting formulation implemented into numerical routines. A standard modeling approach for viscoelastic modeling for the stress-strain relationship for time-dependent materials, ignoring aging effects, can take on the integral form

[sigma](t) = [[integral].sub.0.sup.t] [phi](t - [tau]) [dot.[epsilon]]([tau]) d[tau]. (1)

Alternatively, the strain of the material at time t can also be expressed as

[epsilon](t) = [[integral].sub.0.sup.t] [psi](t - [tau]) [dot.[sigma]]([tau]) d[tau], (2)

where functions [phi] and [psi] are known as the stress-relaxation function and the creep function, respectively. Material properties should be independent of testing procedures, which follows that either 1 or 2 uniquely defines the material response.

Phenomenological modeling of the behavior of viscoelastic solids requires determining creep or relaxation functions to be used in Eq. 1 or 2. Creep functions are generally easier to obtain from experiments and are used in this work. In a creep test, the stress in the tested sample is kept constant and the strain is recorded with time. For a constant stress Eq. 2 becomes

[epsilon](t) = [[sigma].sub.c][psi](t), (3)

where [psi](t) is the creep compliance. The material modeling task is to find a function [psi](t) that best fits test results.

Multi-Kelvin Approach

For a linear viscoelastic material subject to a constant stress applied at time [t.sub.0], an elastic strain occurs instantaneously, followed by the viscous strain growth. The strain growth rate slows down with time and it can become constant at a certain time. When the loading is removed, there is an instantaneous elastic strain recovery, followed by gradual recovery of the viscous strain.

When the material behaves as aforementioned, its behavior can be modeled by the exponential creep function, [psi](t):

[psi](t) = [[psi].sub.e] + [[psi].sub.v](t) = [1/[E.sub.0]] + [n.summation over (i=1)] [1/[E.sub.i]]{1 - exp(-[t/[[tau].sub.i]])}, (4)

where [[psi].sub.e] represents the instantaneous elastic component, and [[psi].sub.v](t) represents the time-dependent viscoelastic effects. Material constants are the elastic modulus [E.sub.0], which defines the instantaneous response, and the moduli [E.sub.i], with corresponding relaxation times [[tau].sub.i]. Each pair of [E.sub.i] and [[tau].sub.i] defines the material response mainly at times smaller than [[tau].sub.i]. Since the creep function is independent of stress or strain, the model defined in 4 is linear. The constitutive equation can be obtained by substituting Eq. 4 into Eq. 2,

[FIGURE 4 OMITTED]

[epsilon](t) = [[integral].sub.0.sup.t]{[1/[E.sub.0]] + [n.summation over (i=1)] [1/[E.sub.i]]{1 - exp(-[[t - [tau]]/[[tau].sub.i]])}} [dot.[sigma]]([tau]) d[tau]. (5)

The creep compliance in Eq. 4 can also be written as

[psi](t) = [[psi].sub.e] + [[psi].sub.v](t) = [1/[E.sub.0]] + [n.summation over (i=1)] [1/[E.sub.i]] [W.sub.i](t). (6)

The values of the relaxation times [[tau].sub.i] determine the values of multiplication factors [W.sub.i] for the creep coefficients 1/[E.sub.i]. Figure 4 shows the variation of these multiplication factors with time, for different relaxation times [[tau].sub.i]. Each [[tau].sub.i] controls a separate time window. When W becomes equal to unity, the viscoelastic behavior becomes elastic. Thus, using the formulation in Eq. 6, the viscoelastic behavior of the material will become elastic with time. In the present work, to simplify calculations, the magnitudes of the relaxation times [[tau].sub.i] are taken as constants with the shortest time being 500 s. The next relaxation time is taken as 20 times the value of the previous one, and so on. By assigning constant values to the relaxation time, linear least squares equations are obtained for determining the creep coefficients 1/[E.sub.i].

It should also be pointed out that the multi-Kelvin viscoelastic modeling can also be obtained from rheological models that consist of one spring and a series of Kelvin elements (see Fig. 5). A Kelvin element consists of one spring and one dashpot parallel to each other; the deformations of the spring and dashpot under loading are assumed to be equal at any time. Material constants are the elastic moduli of the springs [E.sub.0], [E.sub.i], and the viscous moduli of the dashpots, [[eta].sub.i], with [[tau].sub.i] = [[eta].sub.i]/[E.sub.i]. Therefore, the exponential form of the creep compliance function is referred to as multi-Kelvin approach.

[FIGURE 5 OMITTED]

Power Law

Power law functions can also be used for modeling creep (e.g. [6]). This type of modeling is often referred as viscoplastic because the modeled deformation (strain) keeps growing, at decreasing rate, and does not approach an asymptotic value; even after a long time the material remains time-dependent.

To model the behavior described above, the creep compliance, [psi](t), can be expressed by a power law function as

[psi](t) = [[psi].sub.e] + [[psi].sub.v](t) = [1/[E.sub.0]] + [C.sub.0] [t.sup.[C.sub.1]], (7)

where [[psi].sub.e] is the instantaneous elastic component; [[psi].sub.v](t) represents the time-dependent component. [E.sub.0], [C.sub.0], and [C.sub.1] are material constants obtained from the least squares estimation. The constitutive equation for the viscoplastic model can be obtained by substituting Eq. 7 into Eq. 2,

[epsilon](t) = [[integral].sub.0.sup.t] {[1/[E.sub.0]] + [C.sub.0] (t - [tau])[.sup.[C.sub.1]]}[dot.[sigma]]([tau])d[tau]. (8)

Curve Fitting of the Creep Response

Both equations, Eqs. 4 and 7, can be used for parameter estimation based on experimentally obtained creep curves. In fact, for the time frame corresponding to the test, the result of curve fitting will be satisfactory regardless of which equation is chosen. However, the predictions by the two equations for longer times will differ. The creep behavior of semicrystalline polymers depends, among other factors, on the level of loading and it has been experimentally observed that under lower levels of loading the behavior is characterized by initial creep which decays with time (see Fig. 6), and hence the viscoelastic behavior becomes elastic with time. This type of behavior can be well modeled using the multi-Kelvin approach with properly chosen relaxation times.

In creep tests under higher loads (for PE material these loads would be larger than ~10 MPa), the behavior changes and strains continue to grow under these higher loads. The creep compliance does not approach an asymptotic value but grows continuously with time. Therefore, the power law of Eq. 7 will better reflect actual physical behavior of the material at longer times [9].

NONLINEAR TIME-DEPENDENT MODELING

Most polymeric materials exhibit behavior that is dependent, at a given temperature, on both time and stress. As shown in Figs. 2 and 3, creep behavior cannot be described by a single compliance-time curve. For each load level, a different curve is obtained and thus different set of material parameters should be defined.

Formulations that incorporate stress dependence in the creep compliance include, e.g., the Schapery model [5], Krishnaswamy et al. [6], Zhang and Moore [8], and Beijer and Spoormaker [13]. A simple method to incorporate stress dependence in the material behavior is to formulate a creep compliance [psi] with the material parameters that depend on stress. This creates a nonseparable form of the nonlinear constitutive relationship (e.g. [6, 13]). For a nonlinear multi-Kelvin model, the creep compliance, [psi]([sigma],t), can then be written as

[psi]([sigma], t) = [[psi].sub.e]([sigma]) + [[psi].sub.v]([sigma], t)

= [1/[[E.sub.0]([sigma])]] + [n.summation over (i=1)] [1/[[E.sub.i]([sigma])]]{1 - exp(-[t/[[[tau].sub.i]([sigma])]])}. (9)

For a nonlinear power law model, the creep compliance, [psi]([sigma],t), becomes

[psi]([sigma], t) = [[psi].sub.e]([sigma]) + [[psi].sub.p]([sigma], t) = [1/[[E.sub.0]([sigma])]] + [C.sub.0]([sigma]) [t.sup.[C.sub.1]([sigma])], (10)

where [E.sub.0]([sigma]), [E.sub.i]([sigma]), [[tau].sub.i]([sigma]), [C.sub.0]([sigma]), and [C.sub.1]([sigma]) are functions of stress. The modeling requires finding appropriate functions that describe the dependence of the material coefficients on stress.

[FIGURE 6 OMITTED]

The nonlinear constitutive equation is then obtained by substituting either Eq. 9 or 10 into Eq. 2:

[epsilon](t) = [[integral].sub.0.sup.t] {[psi]([sigma], t - [tau])}[dot.[sigma]]([tau])d[tau]. (11)

Proposed Nonlinear Modeling Procedure

To incorporate stress dependence on material properties, a simple approach is taken herein where a linear interpolation is used to include stress influence on the material parameter values. Thus, instead of finding the best function to fit the relationships, e.g. [E.sub.i]([sigma]), a piecewise linear function is adopted. It is assumed that such approach provides adequate accuracy for most structural analysis problems. It also allows using a numerical computer procedure to find these material constants and their stress dependence making the modeling very easy to apply. The numerical procedure generates the appropriate model for a given material. The procedure consists of the following steps.

Step 1. Creep tests are done first for a few selected stress levels, [[sigma].sub.1],..., [[sigma].sub.n].

Step 2. For each stress level a separate set of material parameters is obtained. The sets of constants for the creep tests generate an array of material constants. The model is presented as a matrix of material parameters.

Step 3. The material parameters for stresses other than the tested stresses are obtained by linear interpolation. The modeling is implemented into a numerical procedure. It can be repeated for any material.

For the multi-Kelvin model, the relaxation times are assumed independent of stress. This simplifies the least squares estimation, making it linear, which is used to determine the creep coefficients [x.sub.i] = 1/[E.sub.i]. Least squares estimation for the multi-Kelvin model is done by assuming first that one relaxation time (one Kelvin element and a spring), N = 1, is satisfactory to model the whole creep response. The fitting error is evaluated and if a certain stopping criterion is not satisfied, the number of Kelvin elements is increased to 2, N = 2. Again, the fitting error is evaluated and N is continuously increased until satisfactory curve fitting is obtained. In this work, the fitting error of 1-1.3% was used. It resulted, in all presented cases, in the selection of three Kelvin elements. The procedure is implemented in a computer routine.

The number of stress levels for creep tests used for model development, and thus creep tests which need to be done, and the magnitudes of stresses should be determined depending on the type of practical problem to be analyzed. Then, the creep coefficients for any intermediate stress are found from

[x.sub.i]([sigma]) = [1/[[E.sub.i]([sigma])]] = [x.sub.i]([[sigma].sub.m]) + [[[sigma] - [[sigma].sub.m]]/[[[sigma].sub.n] - [[sigma].sub.m]]][[x.sub.i]([[sigma].sub.n]) - [x.sub.i]([[sigma].sub.m])], (12)

where [x.sub.i] = 1/[E.sub.i]; [[sigma].sub.m] and [[sigma].sub.n] are the stresses used for model development; for [sigma] is: [[sigma].sub.m] < [sigma] < [[sigma].sub.n].

Similarly, linear interpolation for a viscoplastic model takes the form

[C.sub.i]([sigma]) = [C.sub.i]([[sigma].sub.m]) + [[[sigma] - [[sigma].sub.m]]/[[[sigma].sub.n] - [[sigma].sub.m]]][[C.sub.i]([[sigma].sub.n]) - [C.sub.i]([[sigma].sub.m])]. (13)

The elastic stiffness terms, [E.sub.0]([sigma]), for different stresses, for both models, are found directly from

[E.sub.0]([sigma]) = [E.sub.0]([[sigma].sub.m]) + [[[sigma] - [[sigma].sub.m]]/[[[sigma].sub.n] - [[sigma].sub.m]]][[E.sub.0]([[sigma].sub.n]) - [E.sub.0]([[sigma].sub.m])]. (14)

MODELING LOAD HISTORY EFFECTS

Viscoelastic constitutive equations contain time dependency in their formulation, where strains and stresses depend on load history. Boltzman superposition principle can then be applied for calculating strain (or stress) at time t when the material is subject to the given stress (or strain) history. In case of nonlinear material behavior, the creep compliance depends on both time and stress, and thus, strictly speaking, superposition is not valid. In the present work, the modified superposition principle is adopted where the requirement for linearity in stress is relaxed and nonlinearity effects are treated in the following approximate sense [19], as described briefly below.

Response to Multiple Loading Steps

When the load history includes stress changing with time, it can be approximated as a series of small time steps in which constant values of stress are applied. The stress [[sigma].sub.1] is applied abruptly at time [t.sub.0] and held constant till time [t.sub.1]. At time [t.sub.1], the strain is equal to:

[[epsilon].sub.1] = [[[psi].sup.e]([[sigma].sub.1]) + [[psi].sup.v]([[sigma].sub.1], [t.sub.1] - [t.sub.0])][[sigma].sub.1] (15)

At time [t.sub.1] a new stress [[sigma].sub.2] is applied in a jump and held constant till time [t.sub.2]. At time [t.sub.1] the creep behavior is calculated as if at this instant, stress [[sigma].sub.1] is removed and at the same time stress [[sigma].sub.2] is applied. Both stresses are considered as independent and calculation of strain at time [t.sub.2] includes strain recovery from time [t.sub.1] to [t.sub.2]:

[[epsilon].sub.2] = [[psi].sup.e]([[sigma].sub.2])[[sigma].sub.2] + [[psi].sup.v]([[sigma].sub.1], [t.sub.2] - [t.sub.0])[[sigma].sub.1] + [[[psi].sup.v]([[sigma].sub.2], [t.sub.2] - [t.sub.1])[[sigma].sub.2] - [[psi].sup.v]([[sigma].sub.1], [t.sub.2] - [t.sub.1])[[sigma].sub.1]]. (16)

[FIGURE 7 OMITTED]

For the N increments of stress applied in N time steps the following general formula can be written:

[[epsilon].sub.N] = [[psi].sup.e]([[sigma].sub.N])[[sigma].sub.N] + [N.summation over (n=1)] [[[psi].sup.v]([[sigma].sub.n], [t.sub.N] - [t.sub.n-1])[[sigma].sub.n] - [[psi].sup.v]([[sigma].sub.n-1], [t.sub.N] - [t.sub.n-1])[[sigma].sub.n-1]] (17)

in which the total time t is divided into N discrete intervals.

NUMERICAL RESULTS

The proposed procedure was employed for the analysis of PE samples under a variety of loading conditions. Results and comparisons between experimental and numerical results are presented herein. Both multi-Kelvin and power-law approaches are shown.

Creep Tests for Model Generation

Creep tests were conducted on all four PEs. For each PE, five creep compliances representing five different stress levels were obtained. These experimental results were used for generating material models according to Eqs. 5 and 8. The modeling parameters are given in Tables 2-5.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

The simulation results for all four PE materials follow almost exactly the test results for both multi-Kelvin and power-law models and therefore they are not presented for the sake of brevity.

Creep Tests at Intermediate Stress Levels

To check the performance of the proposed approach a comparison is done between simulated and test results for creep loading at stress levels different from the ones which were used to calibrate the model. The comparison provides information on how the linear interpolation of material parameters between different stress levels works for nonlinear behavior of PE. The results for HDPE-Pipe is shown in Fig. 7 and for HDPE Resin 2 are shown in Fig. 8. There is some difference between the predicted and test results; however, this difference is acceptable for engineering purposes.

Step Loading and Load Rates

Figure 9 shows the comparison of simulated and test results for multistep creep loading for multi-Kelvin approach. Predictions using power law approach are almost exactly the same and therefore are not shown. Both modeling equations simulate the experimental results with an accuracy acceptable for engineering purposes. Load rate tests were also conducted on HDPE-Pipe samples using the MTS 810 testing machine. The results of experiments and numerical simulations are presented in Figs. 10 and 11. Testing involved control of load rate which was chosen to correspond to the appropriate engineering stress rate. Two stress rates are shown, namely, 0.01 MPa/s in Fig. 10 and 0.1 MPa/s in Fig. 11.

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The effects of step loading and load rate are captured by both models implemented based on the modified superposition principle. Generally it can be observed that in these analyses the main error is at the beginning of each new load step (see Fig. 4). Since the test creep data were collected over a 24-h period (which was considered reasonable for structural applications), the curve fitting procedure minimizes fitting error over the whole 24 h, thus reducing the accuracy for very short times. This is also the reason why the models perform better for slower loading (see Fig. 10) but predict too soft response for faster loading (see Fig. 11).

Complex Loading Results

The HDPE-Pipe samples were subjected to complex loading which included combinations of loading and unloading rates and creep at different stress levels. An example is shown in Fig. 12 (loading history) and Fig. 13 (experimental and theoretical responses). The theoretical procedure provided results with acceptable accuracy for both loading and unloading sequences in the load history.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

CONCLUSIONS

A practical modeling approach is introduced for PE materials. The proposed approach is based on existing theoretical principles; namely, integral formulation for modeling hereditary behavior of PE before yield. The goal was to create a time-dependent and stress-dependent approach that can easily be applied to any viscoelastic material, provided creep testing has been done for few selected, representative stress levels.

The approach can easily be repeated for other materials. Standard Fortran subroutines were written for generating a creep compliance function by least squares estimation based on exponential (multi-Kelvin) and power law models. Stress dependence of material parameters was incorporated using a piece-wise linear representation. Models were generated for four PE materials and successfully compared to experimental data (creep and load history effects).

ACKNOWLEDGMENTS

Material samples used in the experiments were donated by Imperial Oil and KWH Pipe Canada.

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Hongtao Liu, (1) Maria Anna Polak, (1) Alexander Penlidis (2)

(1) Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(2) Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Correspondence to: Maria Anna Polak; e-mail: polak@uwaterloo.ca

Contract grant sponsor: Imperial Oil and the Natural Sciences and Engineering Research Council (NSERC).

TABLE 1. Description of the tested materials. Material Description 1 HDPE-Pipe Sample cut from a 24" HDPE-Pipe through the pipe wall, along the pipe radius. Tested along the pipe longitudinal axis. 2 HDPE-Resin 1 Hot pressed into plates from a blow molding resin. Samples cut from the plates. 3 HDPE-Resin 2 Hot pressed into plates from a blow molding resin. Samples cut from the plates. 4 MDPE-Pipe Sample cut from a 4" MDPE-Pipe through the pipe wall, normal to the pipe radius. Tested along the pipe longitudinal axis. TABLE 2. Multi-Kelvin model coefficients ([E.sub.1], [E.sub.2], [E.sub.3]) and Power-law model coefficients ([C.sub.0], [C.sub.1]) for HDPE-Pipe for five different stresses. Stress [E.sub.0] [E.sub.1] (a) [E.sub.2] (b) [E.sub.3] (c) (MPa) (MPa) (MPa) (MPa) (MPa) 2.97 650 797.3889 2320.3566 925.0882 5.97 580 913.5936 1212.2605 695.0461 7.71 520 1224.7911 1104.9922 385.8572 10.31 500 1034.2045 694.1084 226.4555 12.19 470 1128.4448 806.0972 140.6875 Stress (MPa) [C.sub.0] (M[Pa.sup.-1]) [C.sub.1] 2.97 0.4960 E-3 0.1254 5.97 0.2956 E-3 0.1872 7.71 0.1232 E-3 0.2706 10.31 0.1130 E-3 0.3145 12.19 0.5517 E-4 0.3893 Number of Kelvin elements is three. (a) [[tau].sub.1] = 500 s. (b) [[tau].sub.2] = 10,000 s. (c) [[tau].sub.3] = 200,000 s. TABLE 3. Multi-Kelvin model coefficients ([E.sub.1], [E.sub.2], [E.sub.3]) and Power-law model coefficients ([C.sub.0], [C.sub.1]) for HDPE-Resin 1 for five different stresses. Stress [E.sub.0] [E.sub.1] (a) [E.sub.2] (b) [E.sub.3] (c) (MPa) (MPa) (MPa) (MPa) (MPa) 2.67 990 2473.5339 1434.3650 1.0 E8 5.15 830 2153.6304 1319.8418 949.4745 7.14 790 2614.5305 993.8024 747.7686 7.58 770 1771.7237 959.6445 537.9008 10.58 730 1153.4563 706.9109 352.5731 Stress (MPa) [C.sub.0] (M[Pa.sup.-1]) [C.sub.1] 2.67 0.1242 E-3 0.2012 5.15 0.8856 E-4 0.2587 7.14 0.6390 E-4 0.3019 7.58 0.7807 E-4 0.3007 10.58 0.1298 E-3 0.2876 Number of Kelvin elements is three. (a) [[tau].sub.1] = 500 s. (b) [[tau].sub.2] = 10,000 s. (c) [[tau].sub.3] = 200,000 s. TABLE 4. Multi-Kelvin model coefficients ([E.sub.1], [E.sub.2], [E.sub.3]) and Power-law model coefficients ([C.sub.0], [C.sub.1]) for HDPE-Resin 2 for five different stresses. Stress [E.sub.0] [E.sub.1] (a) [E.sub.2] (b) [E.sub.3] (c) (MPa) (MPa) (MPa) (MPa) (MPa) 2.68 2500 2848.6134 3650.6457 1053.8829 5.58 2300 2125.6411 1811.4240 696.3469 7.28 1700 1515.4295 1537.4866 603.9634 10.60 1200 1180.3846 1111.9421 405.5838 13.72 1100 999.9933 810.1940 145.0453 Stress (MPa) [C.sub.0] (M[Pa.sup.-1]) [C.sub.1] 2.68 0.5515 E-4 0.2517 5.58 0.6842 E-4 0.2753 7.28 0.1143 E-3 0.2485 10.60 0.1346 E-3 0.2623 13.72 0.7354 E-4 0.3642 Number of Kelvin elements is three. (a) [[tau].sub.1] = 500 s. (b) [[tau].sub.2] = 10,000 s. (c) [[tau].sub.3] = 200,000 s. TABLE 5. Multi-Kelvin model coefficients ([E.sub.1], [E.sub.2], [E.sub.3]) and Power-law model coefficients ([C.sub.0], [C.sub.1]) for MDPE-Pipe for five different stresses. Stress [E.sub.0] [E.sub.1] (a) [E.sub.2] (b) [E.sub.3] (c) (MPa) (MPa) (MPa) (MPa) (MPa) 3.12 640 1137.4169 1067.2127 1168.6089 5.10 470 804.3798 718.0750 588.7810 6.23 420 813.3631 668.5170 422.0754 8.40 410 690.8382 572.2448 224.6822 9.32 390 419.6169 363.3388 106.4053 Stress (MPa) [C.sub.0] (M[Pa.sup.-1]) [C.sub.1] 3.12 0.2341 E-3 0.1983 5.10 0.2927 E-3 0.2153 6.23 0.2385 E-3 0.2414 8.40 0.1946 E-3 0.2839 9.32 0.3080 E-3 0.2918 Number of Kelvin elements is three. (a) [[tau].sub.1] = 500 s. (b) [[tau].sub.2] = 10,000 s. (c) [[tau].sub.3] = 200,000 s.

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Author: | Liu, Hongtao; Polak, Maria Anna; Penlidis, Alexander |
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Publication: | Polymer Engineering and Science |

Article Type: | Technical report |

Geographic Code: | 1USA |

Date: | Jan 1, 2008 |

Words: | 5895 |

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