Modeling Viscoelastic Behavior of HDPE Pipes Subjected to a Diametral Load Using the Standard Linear Solid Model

Abstract

This paper presents the study of the viscoelastic behavior of high-density polyethylene (HDPE) ASTM 4710 pipes under diametral loads. The experimental procedure consists of applying a displacement ramp followed by a stress relaxation stage on six ring specimens extracted from pipes with varying thickness-to-diameter ratios. The proposed methodology combines the Standard Linear Solid Model (SLSM) with beam theory, introduces adjustment equations for estimating SLSM parameters, and discusses the influence of residual stresses induced during pipe manufacturing and cooling. Finally, the paper shows the validation of the modeling approach based on the results of the mechanical response of an independent test case.

1. Introduction

The extended use of High-Density Polyethylene (HDPE) 4710 in pressurized systems is due to its advantages over other alternatives, including high corrosion resistance, ease of handling and installation, high-quality butt-fusion welding, flexibility, and resistance to biological microorganisms such as fungi and bacteria.

Its important presence in the industry increases the requirement necessity for a more extensive evaluation of its mechanical behavior under both steady-state pressures and transient pressures caused by water hammer phenomena. These transients arise from wave trains propagating through pipelines as a result of sudden operational changes [1,2,3,4,5], suggesting that neglecting the viscoelastic effects of the pipe wall can lead to significant errors in estimating both the velocity and attenuation of pressure waves.

Polyethylene (PE), as a material in continuous technological development, has undergone significant changes in its physicochemical properties over time, resulting in the emergence of different types of resin. Variations in the mechanical properties intrinsic to each resin type, along with residual stresses induced during various stages of pipe fabrication, operating temperatures, and wall thickness, affect the mechanical response of HDPE pipes [6,7,8,9,10].

Polyethylene (PE) behaves mostly isotropically under low strain levels but exhibits anisotropic behavior under high strains [10]. Studies conducted under uniaxial stress states report subtle differences in tensile and compressive properties obtained under relaxation and creep conditions. However, in practical applications, the material exhibits similar behavior across these conditions [11,12,13]. In contrast, under multiaxial stress states—with circumferential stress approximately twice the axial stress—experiments involving internal pressurization of HDPE pipes and monitoring of diameter changes over time have shown that the apparent elastic modulus is about 25% higher than that obtained under uniaxial conditions. This difference is due to the combined stress components that constrain deformation [11].

To represent the time-dependent behavior of PE in water hammer studies, an increasing number of investigations rely on generalized rheological models, such as the generalized Kelvin-Voigt and Maxwell models or creep compliance functions [2,3,5,10,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].

Although these models provide high versatility in fitting numerical simulations to experimental data, the inclusion of numerous parameters can lead to overfitting [20,23,27]. In many cases, multiple parameter combinations can reproduce the experimental results with practically indistinguishable differences [5]. While several important studies [3,5,14,28] suggest that using four or five Kelvin–Voigt elements is generally sufficient to achieve good agreement with experimental data, accurately estimating the corresponding parameters remains a significant challenge [29].

There are two common approaches to determining viscoelastic parameters. The first involves calibration using unsteady flow testing. However, this approach faces a significant limitation due to the difficulty in separating the dispersive and energy dissipation effects present during the water hammer phenomenon, such as viscoelastic deformations, unsteady friction, leaks, and other fluid-structure interactions. This overlap of effects can lead to ambiguous interpretations regarding the origin of energy losses observed in experimental studies [2,3,5,10,14,15,16,17,18,30]. The second approach relies on standardized tests, such as those proposed by the American Society for Testing and Materials (ASTM) [31,32,33,34,35], which characterize material behavior through creep functions, relaxation modulus, apparent elastic modulus, long-term modulus, and flexural modulus. Although results from this testing methodology show a significant correlation with those obtained from water hammer experiments, the viscoelastic material response remains incompletely characterized because ASTM mechanical tests are designed for load or displacement rates slower than those occurring during water hammer events. Therefore, adapting and proposing alternative testing methods may improve the reliability of material characterization under actual working conditions. The work of Autrique-Ruiz et al. [36] evidences the relationship between instantaneous elastic modulus and wave celerity in HDPE 4710 pipes.

The complexity of estimating viscoelastic parameters for multimodal models encourages the use of models that better preserve the physical interpretability of mechanical behavior, rather than focusing on overly detailed fitting of hydraulic responses for a specific case study [3,28]. In this context, the Standard Linear Solid Model (SLSM) is an attractive alternative since it uses only three parameters—long-term modulus, Maxwell branch modulus, and viscous parameter—to reproduce both instantaneous and delayed deformations typical of thermoplastic materials. Its formulation leads to simple mathematical solutions with low computational costs in water hammer analysis, providing advantages when studying more complex systems composed of pipe sections with varying lengths, diameters, thicknesses, and materials, as well as heterogeneous pipeline networks [15,25,26,28,30,37].

Many studies [15,25,26,28,30,37] conclude that, within the range of short characteristic times typical of water hammer, the Standard Linear Solid Model (SLSM) can attenuate and disperse the pressure wave with sufficient accuracy for design and diagnostic applications. Moreover, as a viscoelastic model with a reduced number of parameters, it benefits the establishment of direct relationships between these parameters and the physical characteristics of the pipe system under analysis. Pezzinga et al., Carmona-Paredes et al., and Paniagua-Lovera et al. [17,18,20,27,28,30,37] suggest that a strong correlation exists between viscoelastic parameters and the characteristic time of pipeline systems, defined as the period of the pressure wave.

This paper addresses these challenges from a perspective focused on material behavior, proposing a practical methodology for viscoelastic characterization and its application to HDPE 4710. The organization of this work is as follows: (i) an experimental testing configuration consisting of a displacement ramp followed by a stress relaxation stage in rings under diametral loads; (ii) an analytical formulation of bending in viscoelastic beams using the Standard Linear Solid Model (SLSM); (iii) calibration of viscoelastic parameters by fitting the numerical solution to experimental data from six pipe ring specimens; (iv) proposal of adjustment equations to obtain the SLSM parameters; and (v) validation of the methodology by comparing the proposed method’s results with observed data from one test not included in the calibration group, demonstrating the method’s capability to correlate with known physical variables during the design stage, where the characteristic time is a dominant factor influencing the viscoelastic parameters.

2. Materials and Methods

2.1. Classical Viscoelastic Models

This study applies the three-parameter Standard Linear Solid Model (SLSM) due to its capacity to represent both instantaneous and retarded responses over time, typically observed in PE, with the fewest parameters possible. This model consists of a Maxwell element with elastic parameter $E_2$ (in Pa) and viscous parameter $\eta$ (in Pa·s), connected in parallel to another elastic element with long-term modulus $E_1$ (in Pa), as shown in Figure 1. The relationship between stress and strain ($\sigma -\epsilon$) is described by an ordinary linear first-order differential equation, as written in Equation (1), where the coefficient $\left(E_1+E_2\right)$ is the instantaneous elastic modulus $E_0$.

$${\displaystyle \frac{d\sigma }{dt}}+{\displaystyle \frac{E_2}{\eta }}\sigma =\left(E_1+E_2\right){\displaystyle \frac{d\epsilon }{dt}}+{\displaystyle \frac{E_1{E}_2}{\eta }}\epsilon$$

(1)

2.2. Mathematical Modeling

2.2.1. Bending in Viscoelastic Beams Using the SLSM

This work relies on classical bending theory to represent the mechanical behavior of pipe rings under load and to determine the viscoelastic parameters that accurately describe HDPE 4710. According to this theory, when a beam is subjected to bending moments ($M$), stress ($\sigma$), and normal strain ($\epsilon$) along the longitudinal axis ($s$) are distributed linearly over the cross-section, following the law expressed in Equation (2) [38], as illustrated in Figure 2. In this equation, $Y$ represents the distance between the neutral axis (which coincides with the centroid) and the fiber under analysis, while $I$ is the moment of inertia with respect to the neutral axis, calculated by Equation (3). This moment of inertia depends on the pipe thickness ($e$) and ring length ($L$). For curved beams, Equation (2) is valid when the curvature ratio is greater than or approximately equal to five times the beam’s depth [39,40], as presented in HDPE pipe rings.

$$\sigma ={\displaystyle \frac{M}{I}}Y$$

(2)

$$I={\displaystyle \frac{L{e}^3}{12}}$$

(3)

Equation (4) defines the relationship between normal stress and strain along the longitudinal axis ($s$) applied to fiber $A$ in the constitutive model. In this equation, $A_v$, $B_v$, $C_v$, and $D_v$ are constants that depend on the parameters of the selected first-order viscoelastic model. For the Standard Linear Solid Model (SLSM), these constants are defined as follows: $A_v=1$, $B_v=E_2/\eta$, $C_v=E_1+E_2$ y $D_v={E_1·E}_2/\eta$.

$$A_v{\dot{\sigma }}_A+B_v{\sigma }_A=C_v{\dot{\epsilon }}_A+D_v{\epsilon }_A$$

(4)

Equations (5) and (6) describe the kinematics of the deformable solid under small deformations, considering the beam geometry shown in Figure 2. These equations account for the linear deformations of the fibers (${\epsilon }_A$) and the rotations of the transverse sections with respect to the neutral axis, denoted by $\phi$. In this context, $d\overline{AB}$ represents the linear displacement of the exterior fiber along the longitudinal direction, $d\overline{AG}$ is the distance between the neutral axis and the exterior fiber, and $ds$ is the length of the beam section under study.

$${\epsilon }_A=2{\displaystyle \frac{d\overline{AB}}{ds}}$$

(5)

$$\phi ={\displaystyle \frac{d\overline{AB}}{d\overline{AG}}}$$

(6)

Substituting the bending Equation (2) and the kinematic Equations (5) and (6) into Equation (4) results in the differential Equation (7), which corresponds to the problem of the linear viscoelastic beam with three parameters subjected to pure bending.

$${\displaystyle \frac{1}{I}}\left(A_v{\displaystyle \frac{dM}{dt}}+B_{v}M\right)ds=C_v{\displaystyle \frac{d\phi }{dt}}+D_v\phi$$

(7)

2.2.2. Solution of Pipe Subjected to Diametral Load as Curved Viscoelastic Beam

In the analysis and design of pipes, it is often necessary to study the case of a circular ring subjected to a diametral load, as shown in Figure 3. In practice, the load $W$ represent a mechanical demand on the pipe or a force applied during experimental testing to obtain a mechanical characterization of the material.

Figure 4 shows the upper right quadrant of a pipe ring, with the external and internal actions on the beam in equilibrium. The only external action is the load $W$, while the internal actions are shear forces ($Q$) and bending moments ($M$), which occur at the ends of the circular sector, $P_0$ and $P_1$.

Due to the symmetry of the ring over the quadrants defined by the load $W$ and its perpendicular axis, it is possible to analyze only one quadrant of the ring to determine the internal forces and deformations. Based on the geometry and symmetry of this problem, at points $P_0$ and $P_1$, with polar coordinates ${\theta }_0=0$ and ${\theta }_1=\pi /2$, respectively, there are no rotations of cross-sections at any time; in other words, ${\phi }_0={\phi }_1=0$ and ${\dot{\phi }}_0={\dot{\phi }}_1=0$. Additionally, given the relationships established in beam theory between bending moment and shear force $dM/ds=Q$ and considering the moment–curvature relationship $d\phi /ds=M/EI$, it leads to $Q_0$ and $Q_1$ being equal to zero [41].

Evaluating the shear conditions at the ends of the beam and the vertical reactions at the supports, which are equal to $W/2$ at the horizontal extremes of the ring, Equation (8) expresses the bending moment at any position $x$; in this expression, $M_0$ is a bending moment that can be obtained from the redundancy conditions of the quadrant.

$$M\left(x\right)=-{\displaystyle \frac{Wx}{2}}+M_0$$

(8)

To analyze the rotation of the cross-section $\phi$, we first substitute Equation (8) into Equation (7). Then, the displacement and angular velocity conditions at the ends of the beam are applied, with both being set to zero. Considering the circular sector geometry of the ring with a mean radius $a$, where $ds=ad\theta$, and evaluating the moment arm with respect to point $P_0$ as $x=a\left(1-\cos \theta \right)$, we derive the expressions $M_0=\left(1/2-1/\pi \right)Wa$ and $M_1=-\left(1/\pi \right)Wa$. This leads to the viscoelastic solution of the ring subjected to diametral loads, as described by Equation (9).

$${\displaystyle \frac{1}{I}}\left[-\left({\displaystyle \frac{A_v}{2}}{\displaystyle \frac{dW}{dt}}+{\displaystyle \frac{B_v}{2}}W\right)\left(\theta -\sin \theta \right)a+\left(A_v{\displaystyle \frac{dW}{dt}}+B_{v}W\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{\pi }}\right)a\theta \right]a=C_v{\displaystyle \frac{d\phi }{dt}}+D_v\phi$$

(9)

Equation (10) represents the Navier–Bresse expression [42], which describes the relative displacement between point $P_1$ and point $P_0$ in three dimensions. It states that the total displacement of point $P_1$ is the sum of the rigid body displacements, represented by the first two terms, and the deformable body displacements, accounted for by the subsequent terms. The vectors associated with this expression include the displacement vector $\overrightarrow{u}$, rotation $\overrightarrow{\phi }$, and position $\overrightarrow{r}$.

$$\overrightarrow{u_1}=\overrightarrow{u_{P_0}}+\overrightarrow{{\phi }_{P_0}}\times \left(\overrightarrow{r_{P_1}}-\overrightarrow{r_{P_0}}\right)+{\int }_{P_0}^{P_1}d\overrightarrow{u}+{\int }_{P_0}^{P_1}d\overrightarrow{\phi }\times \overrightarrow{r}$$

(10)

Finally, by substituting Equation (9) into Equation (10) and considering that the specific problem of the viscoelastic ring subjected to loads lies in the $x-y$ plane, we obtain Equations (11) and (12). These equations correspond to the horizontal linear displacement $u$ at point $P_0$ and the vertical linear displacement $v$ at point $P_1$ as functions of time.

$$C_v{\displaystyle \frac{du}{dt}}+D_{v}u={\displaystyle \frac{1}{I}}\left({\displaystyle \frac{1}{\pi }}-{\displaystyle \frac{1}{4}}\right)\left(A_v{\displaystyle \frac{dW}{dt}}+B_{v}W\right)a^3$$

(11)

$$C_v{\displaystyle \frac{dv}{dt}}+D_{v}v={\displaystyle \frac{1}{I}}\left({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{1}{\pi }}\right)\left(A_v{\displaystyle \frac{dW}{dt}}+B_{v}W\right)a^3$$

(12)

2.2.3. Solution for Imposition Displacement Ramp and Relaxation Stage

Since the displacement imposition is not instantaneous, the parameter calibration takes place in two stages. The first stage involves the imposition ramp, where the HDPE pipe ring is subjected to a diametral vertical deflection, $v$, at point $P_1$. This deflection increases progressively until the final condition is reached at the application time, $t_c$, as illustrated in Figure 5.

The variation of the displacement v over time, along with the initial conditions, is known. Therefore, the solution to Equation (12) applying a finite differences scheme [43], results in Equation (13).

$$W_{i+1}=W_i+{\displaystyle \frac{∆t}{A_v}}\left[{\displaystyle \frac{I}{a^3\left(\frac{\pi }{8}-\frac{1}{\pi }\right)}}\left(C_v{\displaystyle \frac{v_{i+1}-v_i}{∆t}}+D_v{v}_i\right)-B_v{W}_i\right]$$

(13)

The second stage of the test involves the relaxation state, which is defined by a constant diametral deflection on the deformable solid. This stage studies the decrement in resisting load over time, as illustrated in Figure 5. In this particular case, simplifying Equation (12) leads to Equation (14), where the geometrical characteristics of the ring and the SLSM viscoelastic parameters define the constants: $\alpha =A_v\left(a^3/I\right)\left(\pi /8-1/\pi \right)$, $\beta =B_v\left(a^3/I\right)\left(\pi /8-1/\pi \right)$ and $\gamma =-D_v{v}_f$.

$$\alpha {\displaystyle \frac{dW}{dt}}+\beta W+\gamma =0$$

(14)

Equation (14) is a first-order non-homogeneous linear differential equation, whose solution is a particular case of the Bernoulli differential equation written in Equation (15), where $t^{\mathrm{*}}=t-t_c$. Wmax corresponds to the maximum load reached during the experimental testing at the beginning of the relaxation stage at $t=t_c$.

$$W=\left(W_{max}+{\displaystyle \frac{\gamma }{\beta }}\right)e^{-{\displaystyle \frac{\beta }{\alpha }}{t}^{\mathrm{*}}}-{\displaystyle \frac{\gamma }{\beta }}$$

(15)

2.3. Experimental Testing

The experimental test bench was developed and constructed at the Instituto de Ingeniería of the Universidad Nacional Autónoma de México. It consists of six HDPE 4710 ring specimens, cut from straight pipes, each approximately 30 mm in length. These specimens are subjected to diametral loading and have an outer diameter of 114.3 mm, with nominal thickness-to-outer diameter ratio $\left(RD\right)$ values ranging from 9 to 32.5. The testing temperature varied between 18.85 and 21.00 °C.

Figure 6 illustrates the experimental setup, where an actuator applies a vertical deflection, $v$, to the ring, gradually increasing until it reaches the target value within a predefined imposition time, $t_c$. After this, the constant deflection imposed on the ring induces stress relaxation in the material. The time values and imposed displacement magnitude ($\nu$) are obtained from video recordings of the experiment using a high-speed camera (240 FPS, 1080p resolution). These measurements are subsequently verified by using digital caliper and readings from a load cell, which continuously records the resisting load $W$ exerted by the tube over time. Additionally, a temperature transducer monitors the ambient temperature $T_e$ throughout the test, confirming that it remains practically constant. The sampling frequency for the resisting load $W$ and room temperature $T_e$ is 500 Hz.

Table 1. Test specimen characteristics and experimental conditions (values with expanded uncertainties, 95% confidence level).

RD	v [μm]	Wmax [N]	a [mm]	e [mm]	Dext [mm]	tc [s]	L [mm]	Te [°C]
9.80	60 ± 3.13	35.85 ± 0.41	50.984 ± 0.02	11.591 ± 0.02	113.56 ± 0.02	0.418	29.95 ± 0.02	21.00 ± 1.0
12.53	100 ± 2.61	21.02 ± 0.29	52.176 ± 0.02	9.0471 ± 0.02	113.40 ± 0.02	1.162	30.27 ± 0.02	20.00 ± 1.0
12.68	150 ± 2.61	35.49 ± 0.51	52.734 ± 0.02	9.0325 ± 0.02	114.50 ± 0.02	0.482	30.00 ± 0.02	18.85 ± 1.0
14.87	230 ± 2.34	37.95 ± 0.67	53.241 ± 0.02	7.6788 ± 0.02	114.20 ± 0.02	0.668	30.00 ± 0.02	19.00 ± 1.0
28.76	1010 ± 1.56	14.05 ± 0.36	55.269 ± 0.02	3.9813 ± 0.02	114.50 ± 0.02	0.458	29.70 ± 0.02	20.10 ± 1.0
29.16	1110 ± 1.56	13.03 ± 0.28	55.277 ± 0.02	3.9263 ± 0.02	114.80 ± 0.02	0.300	30.20 ± 0.02	19.50 ± 1.0

summarizes the experimental parameters for each pipe specimen, including imposed deflection $v$, maximum load reached $W_{max}$, mean pipe radius $a$, pipe thickness $e$, outer diameter $D_{ext}$, imposition ramp time $t_c$, pipe segment length $L$, and test temperature $T_e$, together with their respective uncertainties at a 95% confidence level. This experimental procedure offers several advantages over other testing methods [30,37], such as cost efficiency, ease of setup, minimal space requirements for specimens and instrumentation, and the availability of the stress–strain solution developed in this study [44].

3. Results

3.1. Deformation Imposition Ramp

To ensure that the experimental results remain within the limits of linear viscoelasticity [45,46] suggest that fiber strain must be below 0.01 and normal stress should not exceed 60% of the yield stress. Equation (16) estimates the strain in the outer fibers located at $Y=e/2$, considering the curvature induced by the deflection $v$ at point $P_1$ [41].

$${\epsilon }_{max}=\left({\displaystyle \frac{1}{a+v}}-{\displaystyle \frac{1}{a}}\right){\displaystyle \frac{e}{2}}$$

(16)

The study on the tensile behavior of HDPE presented in [8] provides empirical expressions, such as Equation (17), to estimate the apparent elastic modulus, yield stress, and ultimate stress. The proposed coefficients for estimating the yield stress $S_y$ (in MPa) are: $A=6.65\times {10}^{-4}$, $B=-0.189$, $C=1.22$ and $D=15.4$, where $\dot{\epsilon }$ is the strain rate in $s^{-1}$.

$$S_y=D+C\left(\log \dot{\epsilon }\right)+B\left(T_e\right)+A\left({T_e}^2\right)$$

(17)

Table 2. Stress and strain results from bending in the outer fibers of HDPE ring specimens.

RD	Te [°C]	tc [s]	με [-]	dε/dt [s−1]	Wmax [N]	M1 [N·m]	σ1 [MPa]	Sy [MPa]	σ1/Sy [%]
9.80	21.00 ± 1.0	0.418	133.62 ± 6.66	(3.20 ± 0.16) × 10−4	35.85 ± 0.41	0.5817 ± 0.0069	0.87 ± 0.01	7.46 ± 0.19	11.63 ± 0.11
12.53	20.00 ± 1.0	1.162	165.84 ± 3.86	(1.43 ± 0.03) × 10−4	21.02 ± 0.29	0.3491 ± 0.0049	0.52 ± 0.01	7.19 ± 0.17	7.23 ± 0.05
12.68	18.85 ± 1.0	0.482	242.92 ± 3.54	(5.04 ± 0.07) × 10−4	35.49 ± 0.51	0.5957 ± 0.0088	0.89 ± 0.02	8.05 ± 0.17	11.03 ± 0.03
14.87	19.00 ± 1.0	0.668	310.19 ± 2.16	(4.64 ± 0.03) × 10−4	37.95 ± 0.67	0.6431 ± 0.0116	0.96 ± 0.02	7.98 ± 0.17	12.01 ± 0.01
28.76	20.10 ± 1.0	0.458	646.36 ± 2.61	(1.41 ± 0.06) × 10−4	14.05 ± 0.36	0.2472 ± 0.0064	0.37 ± 0.01	8.39 ± 0.16	4.39 ± 0.05
29.16	19.50 ± 1.0	0.300	699.12 ± 2.97	(2.33 ± 0.01) × 10−4	13.03 ± 0.28	0.2293 ± 0.0050	0.34 ± 0.01	8.76 ± 0.16	3.90 ± 0.03

presents the results obtained from specimens tested at different temperatures and strain rates in this study. The results indicate that all pipe rings exhibit strains below 0.0007 and maximum stresses lower than 12.02% of the yield stress, confirming that the experiments fall within the applicable range of linear viscoelasticity theory.

Figure 7 illustrates the results of the application of the least squares error method on the finite difference solution of the imposition ramp state, expressed in Equation (13), to fit the numerical model to experimental data.

Table 3. Calibrated viscoelastic parameters for the displacement imposition stage of the six specimens.

RD	tc [s]	Te [°C]	E0 [GPa]	E1 [GPa]	E2 [GPa]	η [GPa·s]	RMSE [N]
9.80	0.418	21.00 ± 1.0	1.78	1.50 ± 0.11	0.28 ∓ 0.11	0.11	0.0539
12.53	1.162	20.00 ± 1.0	1.41	1.09 ± 0.11	0.32 ∓ 0.11	0.30	0.2385
12.68	0.482	18.85 ± 1.0	1.69	1.38 ± 0.05	0.32 ∓ 0.05	0.14	0.1166
14.87	0.668	19.00 ± 1.0	1.96	1.55 ± 0.07	0.40 ∓ 0.07	0.23	0.4400
28.76	0.458	20.10 ± 1.0	1.34	1.11 ± 0.11	0.23 ∓ 0.11	0.10	0.0864
29.16	0.3	19.50 ± 1.0	1.14	0.97 ± 0.05	0.17 ∓ 0.05	0.05	0.0306

presents the calibrated parameters for the six specimens in the imposition stage. The 95% confidence intervals for the fitted parameters follow the methodology described by Bates et al. [47] and Huang et al. [48]. Table 3 includes the confidence interval values for $E_1$, which range from 3.38% to 9.66% of the fitted value. The imposition stage shows practically no sensitivity to $E_2$ and $\eta$, resulting in much larger confidence intervals for these parameters. This behavior supports keeping fixed values of $E_2$ and $\eta$ without arbitrary modifications. The parameter $E_2$ adjustment is only to preserve the sum $E_1+E_2=E_0$ from the original calibration, ensuring that the instantaneous modulus remains physically consistent. The results show that the instantaneous elastic modulus $E_0$ increases as the thickness-to-outer-diameter ratio ($RD$) increases and as temperature decreases. A short imposition ramp provides a more reliable calibration of the instantaneous elastic modulus, as also noted by Bilgin [45]. Longer ramps tend to introduce greater load damping, which can lead to underestimation of instantaneous elastic modulus $E_0$. Therefore, the most accurate estimation of $E_0$ is achieved by imposing the ramp as shortly as possible and complementing the results with wave speed measurements from water hammer tests.

In contrast, Autrique et al. [36] report a single value of $E_0\approx 1.38GPa$ for HDPE 4710 pipes obtained from wave speed measurements during water hammer tests. Although the methodologies differ, the $E_0$ value in this approach fall within a similar range. Additionally, the opposition load $W$ exhibits greater damping for longer imposition ramp durations $t_c$.

3.2. Relaxation State

The first consideration in analyzing the relaxation state is that the final load $W_f$ experimentally reached at each characteristic time $T=t_c+t^{\mathrm{*}}$ is a known condition for parameter calibration. Consequently, by applying the variable transformations $y^{\prime }=\ln W$, ${\beta ’}_1=-E_2/\eta$ and ${\beta ’}_0=\ln \left|W_f\left(W_{max}-W_f\right)\right|$, Equation (15) is transformed into Equation (18), expressed as follows:

$$y^{\prime }={\beta }_1^{\prime }{t}^{\mathrm{*}}+{\beta }_0^{\prime }$$

(18)

Equation (18) corresponds to a straight line with a known intercept ${\beta ’}_0$, leading to Equation (19), which calculates the long-term elastic modulus $E_1$ for each characteristic time $T$.

$$E_1=\left({\displaystyle \frac{\pi }{8}}-{\displaystyle \frac{1}{\pi }}\right){\displaystyle \frac{W_f{a}^3}{vI}}$$

(19)

Furthermore, applying the least squares error method allows calculating the slope ${\beta }_1^{\prime }$ expressed in Equation (20), where $n$ is the number of experimental data points used for the adjustment at each characteristic time $T$.

$${\beta }_1^{\prime }={\displaystyle \frac{\sum _{i=1}^n{t}_i^{\mathrm{*}}{y}_i^{\prime }-{\beta }_0^{\prime }\sum _{i=1}^n{t}_i^{\mathrm{*}}}{\sum _{i=1}^n{\left(t_i^{\mathrm{*}}\right)}^2}}$$

(20)

Since $E_0$ and $E_1$ are known values obtained from the displacement imposition ramp and the final load analysis, and given the definition of the Standard Linear Solid Model (SLSM), the instantaneous modulus $E_0$ equals the sum of the long-term modulus $E_1$ and the Maxwell modulus $E_2$. Equations (21) and (22) are then used to calculate $E_2$ and the damping parameter $\eta$, respectively.

$$E_2=E_0-E_1$$

(21)

$$\eta =-{\displaystyle \frac{E_2}{{\beta }_1^{\prime }}}$$

(22)

Figure 8 shows the comparison of results for the numerical model with the fitted viscoelastic parameters and the recorded experimental data in the relaxation stage for the pipe with a thickness-to-outer diameter ratio $\left(RD\right)$ of 9.80.

Table 4. Calibrated viscoelastic parameters for the relaxation stage for RD 9.80 pipe ring at different characteristic times.

T [s]	t* [s]	E0 [GPa]	E1 [GPa]	E2 [GPa]	η [GPa·s]	R
0.84	0.422	1.78	1.40	0.38	0.27 ± 0.01	0.96
2.63	2.212	1.78	1.36	0.42	0.78 ± 0.02	0.94
12.63	12.212	1.78	1.31	0.47	3.26 ± 0.04	0.92
42.63	42.212	1.78	1.26	0.52	9.96 ± 0.09	0.90
92.63	92.212	1.78	1.23	0.55	20.3 ± 0.11	0.88
191.624	121.206	1.78	1.20	0.58	39.1 ± 0.10	0.87

presents the obtained parameters by applying the least squares method at different characteristic times $T$. $E_0$, $E_1$, and $E_2$ are fixed parameters for each characteristic time due to the imposition ramp analysis and the evaluation of Equations (19) and (21); therefore, the only parameter with confidence intervals at the 95% level is the viscosity parameter $\eta$.

These results, shown in Figure 9 and Table 4, indicate that the numerical model accurately reproduces the load curvature product of viscoelastic effects over time for characteristic times shorter than 5 s, gradually losing accuracy as confirmed by the correlation coefficient $R$, diminishing to 0.93 to 0.85 when $T$ reaches 200 s. The reason for this behavior is that the SLSM analytical solution for the relaxation state follows an exponential function. However, experimental data reported in [45] and obtained in this study suggest that HDPE exhibits a power-law relationship under these mechanical conditions, and the applicability of this model is considered a good approximation only for characteristic times under 5 s.

The different HDPE ring specimens subjected to diametral loads exhibit varying adjustment values for the viscoelastic parameters $E_1$, $E_2$ and $\eta$, depending on the simulated characteristic time $T$. Figure 10, Figure 11 and Figure 12 display the values of the SLSM parameters obtained from the calibration of the experimental tests. The results show that both the instantaneous and long-term elastic moduli decrease as temperature increases.

Similarly to the displacement ramp stage, in the relaxation stage, both the instantaneous and long-term moduli increase for HDPE rings with lower thickness-to-outer diameter ratios (RD). This suggests that thick-walled pipes exhibit different behavior compared to thin-walled pipes, which can be attributed to possible morphological changes in the polymeric chains caused by variations in stress magnitude and the non-homogeneous distribution of stresses and temperatures acting on the tube material during the manufacturing process, including melting of pellets, extrusion, molding, cooling, and cutting. The ratio $E_2/\eta$, which is directly related to the damping effect, shows a similar behavior across all study cases. It remains unaffected by temperature and thickness-to-outer diameter ratio, suggesting that it depends solely on the characteristic time.

4. Discussion

4.1. Parameter Adjustment Equations

The experimental and numerical results obtained from calibrations of the SLSM model for both the imposition ramp and relaxation stages show dependence on factors such as temperature, thickness-to-outer-diameter ratio, and the characteristic time. This work proposes estimating the viscoelastic parameters of the SLSM model for intermediate conditions by leveraging the analyzed cases, thereby avoiding the need for a complete analysis by using the established methodology. This methodology modifies the elastic moduli estimation proposed by [11], including material factors that account for the influence of the manufacturing processes on the mechanical behavior of HDPE 4710 pipes.

According to [11], Equation (23) estimates the instantaneous elastic modulus $E_0^{PPI}$, where the reference parameter value is the dynamic elastic modulus at 23 °C, $E_0^{\prime }=1.034 \mathrm{G}\mathrm{P}\mathrm{a}$, and $F_{td}$ is the corresponding temperature compensating multiplier expressed by Equation (24).

$$E_0^{PPI}=E_0^{\prime }{F}_{td}$$

(23)

$$F_{td}=3.1264\times {10}^{-4}{T}_e^2-5.0141\times {10}^{-2}{T}_e+1.9860$$

(24)

Table 5. Calibrated instantaneous elastic modulus $E_0$, temperature compensating multiplier $F_{td},$and instantaneous elastic modulus PPI estimation $E_0^{PPI}$ for the displacement imposition stage of the six specimens.

RD	Te [°C]	Ftd	${\mathbf{E}}_0^{\mathbf{P}\mathbf{P}\mathbf{I}}$ [GPa]	E0 [GPa]
9.80	21.00 ± 1.0	1.07 ± 0.05	1.11 ± 0.04	1.78
12.53	20.00 ± 1.0	1.11 ± 0.05	1.15 ± 0.04	1.41
12.68	18.85 ± 1.0	1.15 ± 0.05	1.19 ± 0.04	1.69
14.87	19.00 ± 1.0	1.15 ± 0.05	1.19 ± 0.04	1.96
28.76	20.10 ± 1.0	1.10 ± 0.05	1.14 ± 0.04	1.34
29.16	19.50 ± 1.0	1.13 ± 0.05	1.17 ± 0.04	1.14

compares the instantaneous elastic modulus obtained from the imposition ramp analysis $\left(E_0\right)$ with the values calculated $\left(E_0^{PPI}\right)$ using the Equations (23) and (24) proposed by [11], for each HDPE ring specimen tested.

These results show that the values obtained using the PPI equations are underestimated compared to those calibrated from the imposition ramp analysis. Additionally, since each thickness-to-outer-diameter ratio $\left(RD\right)$ exhibits a particular influence on the material behavior due to manufacturing processes, the dynamic material factor $F_{md}$ accounts for this effect in the analysis, and it is calculated using Equation (25).

Table 6. Dynamic material factor $F_{md}$ for the displacement imposition stage of the six specimens.

RD	Fmd
9.80	1.60 ∓ 0.05
12.53	1.23 ∓ 0.05
12.68	1.42 ∓ 0.05
14.87	1.68 ∓ 0.05
28.76	1.17 ∓ 0.05
29.16	0.98 ∓ 0.05

and Figure 13 show the calculated $F_{md}$ for each specimen tested.

$$F_{md}={\displaystyle \frac{E_0}{E_0^{PPI}}}$$

(25)

Equation (26) presents the $F_{md}$ adjustment equation depending on $RD$, and Equation (27) corresponds with the expression to obtain the instantaneous elastic modulus for this methodology.

$$F_{md}=-0.0227RD+1.7604$$

(26)

$$E_0=E_0^{\prime }{F}_{md}{F}_{td}$$

(27)

Following a similar procedure to obtain the instantaneous elastic modulus estimation, Ref. [11] provides Equation (28) to calculate the long-term elastic modulus $E_1^{PPI}$, where Fta is temperature compesating multiplier, which follows the adjustment Equation (29), and applies it on the reference apparent elastic modulus $E_1^{\prime }$ at 23 °C for durations of sustained loading from 0.5 h to 100 years.

$$E_1^{PPI}=E_1^{\prime }{F}_{ta}$$

(28)

$$F_{ta}=1.5658\times {10}^{-4}{T}_e^2-2.8930\times {10}^{-2}{T}_e+1.6019$$

(29)

However, the variations in behavior observed in the pipe ring specimens, shown in Figure 10, highlight the influence of residual stresses from the fabrication process on the mechanical response, depending on the $RD$ ratio. Consequently, this work proposes a modification to Equation (28), including the material factor $F_m$ representing this effect, resulting in Equation (30), where the calibrated long-term modulus referenced at 23 °C $\left(E_1^{\prime }\right)$ follows Equation (31) for characteristic times $T<11.89 s$ and Equation (32) for $T\in \left[11.89 \mathrm{s},200 \mathrm{s}\right]$. Figure 14 illustrates the calibrated $F_m$ values obtained in this study, and Figure 15 presents the long-term elastic modulus $E_1^{\prime }$, referenced at 23 °C, as a function of the characteristic time $T$ for the experimentally tested $RD$ ratios.

$$E_1=E_1^{\prime }{F}_m{F}_{ta}$$

(30)

$$E_1^{\prime }=8.2290\times {10}^8{T}^{-0.076606}$$

(31)

$$E_1^{\prime }=7.4776\times {10}^8{T}^{-0.037929}$$

(32)

Moreover, the correlation equations proposed in this study for the long-term elastic modulus of pipe rings subjected to diametral loads align closely with the values reported in [11] for the apparent elastic modulus of HDPE 4710 obtained through uniaxial testing on specimens subjected to sustained and constant loading below 2.76 MPa. The light variations in trends shown in Figure 15 are a consequence of differences in material input properties from various PE pellet manufacturers, differences in stress states across experimental setups, and the effects of residual stresses.

On the other hand, the instantaneous elastic modulus $E_0$ is defined as the sum of the long-term elastic modulus $E_1$ and the Maxwell branch elastic modulus $E_2$. Consequently, $E_2$ is the third parameter to be determined, as given by Equation (33).

$$E_2=E_0-E_1$$

(33)

Finally, the findings of this study enable the complete estimation of the SLSM parameters for HDPE 4710 pipe rings subjected to diametral loads. Figure 12 illustrates the mathematical trend of the damping ratio $E_2/\eta$, which follows a power law function with respect to the characteristic time $T$, as expressed by the adjustment Equation (34). Consequently, the viscous parameter $\eta$, expressed in Pa·s and derived through algebraic manipulation follows Equation (35).

$${\displaystyle \frac{E_2}{\eta }}=1.2241T^{-0.8387}$$

(34)

$$\eta =0.8169E_2{T}^{0.8387}$$

(35)

4.2. Validation

To validate the proposed methodology, this section presents the analysis of an independent test that is not part of the viscoelastic parameter calibration experimental data, assessing the predictive capacity of the SLSM. The HDPE pipe ring specimen characteristics are as follows: a thickness-to-outer diameter ratio of 13.02, an outer diameter of 114.9 mm, a thickness of 8.8225 ± 0.02 mm, and a length of 29.95 ± 0.02 mm. The imposed ramp displacement is 65 ± 0.02 μm, with an application time of 0.426 s and a room temperature of 21 °C.

Figure 16 compares the numerical model results and experimental data during the imposition ramp (IR) and relaxation stages at characteristic times of 1, 2, and 5 s.

Table 7. Viscoelastic parameters estimated for the validation test in the imposition and relaxation stages.

T [s]	Ftd	Fmd	Fta	Fm	E0 [GPa]	E1 [GPa]	E2 [GPa]	η [GPa·s]	RMSE [N]
0.426 (IR)	1.07	1.46	-	-	1.62	1.368 ± 0.046	0.254 ∓ 0.046	0.101	0.198
1	-	-	1.06	1.53	1.62	1.342	0.280	0.229 ± 0.011	0.184
2	-	-	1.06	1.55	1.62	1.287	0.335	0.490 ± 0.018	0.141
5	-	-	1.06	1.57	1.62	1.217	0.405	1.280 ± 0.025	0.201

presents the viscoelastic parameter values obtained by applying the proposal expressed from Equations (24), (26), (27) and (29), and from (31) to (35), using the physical properties of the ring tested and material factors $F_m$ obtained by interpolating between the 12.68 and 14.87 RD-curves, shown in Figure 14.

Additionally, Table 7 reports the RMSE between experimental data and the model predictions for characteristic times lower than 5 s. The maximum RMSE obtained is less than 0.201 N, representing a relative error of less than 2%. These results, along with the previous analysis shown in Figure 9, indicate a better fit of the SLSM model for characteristic times below 5 s. However, as the characteristic time increases, the model’s precision gradually decreases, as evidenced in Figure 8.

Although the SLSM reduces its precision over extended periods, its applicability remains for hydraulic transients with short characteristic times, as these are commonly present in many hydraulic pipelines and networks, improving water hammer simulations and facilitating the design and optimization of surge protection devices.

In the particular case of HDPE pipe with a thickness-to-outer diameter ratio of 13.02, the pressure wave celerity is approximately 380 m/s, as reported by experimental studies of the resin 4710 by Autrique et al. [36], considering that the maximum characteristic time of 5 s is equal to the pressure wave period as proposed by Carmona et al. [37], the SLSM maintains its validity to pipelines that not exceed 950 m, in order to approximate correctly the magnitude and time of damping of the pressure surges, this calculus considers the classic expression of period written in Equation (36), where $T$ is the period in second, $L_p$ is the pipeline length and $a_p$ is the wave celerity in m/s.

$$T={\displaystyle \frac{2L_p}{a_p}}$$

(36)

5. Conclusions

This paper presents a framework for estimating the viscoelastic parameters of PE pipes based on the results of a new setup for experimental testing of rings under diametral loads and its application to HDPE 4710 material represented by the SLSM. The methodology considers factors such as temperature, thickness-to-outer diameter ratio, and characteristic time. The main conclusions of this research are the following:

The experimental methodology and material response representation using the SLSM apply to other PE resins or polymers exhibiting linear viscoelastic behavior. Furthermore, the proposed preparation, installation, instrumentation, and experimental procedure offer economic and spatial advantages over other testing methods used to characterize the mechanical properties of pipes under sustained or transient pressures.

Performing the imposition ramp as briefly as possible is essential for accurately estimating the instantaneous elastic modulus. Longer ramps increase damping in the mechanical response and may lead to an underestimation of this modulus. Complementing its determination with water hammer testing and wave speed analysis further improves the reliability of the instantaneous elastic modulus estimation.

Increasing the sample size and range in key variables, such as temperature and outer diameter, reduces uncertainty in the viscoelastic parameters. Also, it provides information for further improvements in expressions, such as the compensating temperature multiplier, as this work directly accepts the proposal by PPI [11].

Based on the correlation coefficients obtained during parameter calibration in the imposition and relaxation stages, the mathematical model based on the SLSM accurately reproduces the experimental response of HDPE pipes subjected to displacement imposition through diametral loads for characteristic times of up to 5 s. The SLSM effectively models viscoelastic behavior in rapid pipeline phenomena such as water hammer, as observed for [14,30,36,37,49]. To reproduce phenomena at longer characteristic times than those captured by the SLSM, it is necessary to explore generalized Kelvin-Voigt models proposing a methodology that correlates viscoelastic parameters with known physical system characteristics, thereby enabling the correct analysis in design engineering stages and avoiding overfitting problems that can lead to ambiguous conclusions.

The variations in viscoelastic parameters observed for different thickness-to-outer-diameter ratios provide evidence of residual stresses induced during the extrusion and cooling stages of the manufacturing process. These stresses alter the morphology of HDPE polymeric chains, directly affecting mechanical behavior. Generally, residual stress effects are more pronounced in thick-walled pipes (with lower thickness-to-outer-diameter ratios), a phenomenon also reported by [8].

The long-term elastic modulus depends on characteristic time, temperature, and thickness-to-outer-diameter ratio. The apparent elastic modulus trend, along with its corresponding temperature compensation multiplier reported in [11], aligns with the long-term elastic modulus obtained through numerical calibrations of the relaxation stage in specimens subjected to a flexural mechanical state for thin-walled pipes (with higher thickness-to-outer-diameter ratios). Thick-walled pipe rings exhibit a higher long-term elastic modulus than thin-walled pipes.

Similar to the long-term elastic modulus, the dynamic elastic modulus, adjusted using temperature compensation factors reported in [11], shows consistency with the instantaneous elastic modulus calibrated during the displacement imposition ramp for thin-walled pipes. Thick-walled pipes exhibit higher instantaneous modulus values than thin-walled pipes.

This study proposes adjustment equations to estimate the viscoelastic parameters of the SLSM for HDPE 4710. These equations demonstrate a strong correlation with variables such as characteristic modeling time, temperature, and a fabrication factor representing the influence of residual stresses on the pipe’s mechanical properties.