Back Stress in Modeling the Response of PEEK and PC

  • Wenlong Li
  • George Gazonas
  • Eric N. Brown
  • Philip J. Rae
  • Mehrdad Negahban
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


With the development of new methods for the characterization of equilibrium stress through cyclic loading, it is now possible to follow the evolution of back stress during the nonlinear deformation of polymers. Experiments on PEEK and PC below the glass-transition temperature indicate a back stress that may evolve with plastic deformation, and which is substantially different from that seen during the response in the rubbery range. In particular, the back stress during the response of PC shows the characteristic post-yield softening, possibly indicating that the observed post-yield softening in the response comes from the back stress. This is not seen in PEEK, which also shows no substantial post-yield softening. The equilibrium stress plays a central role in modeling both the quasi-static and dynamic response of PEEK.


Mechanical modeling Plastic flow Equilibrium stress Thermal expansion Digital image correlation 

23.1 Introduction

Many constitutive models for the response of time dependent materials use mechanical analogs that include internal state variables, such as plastic deformation, to characterize the changing response of these materials [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. One element that is commonly seen in such models is a back stress element used in models for both polymers and metals [7, 8, 9, 10, 11, 12, 13, 14]. The back stress is frequently experimentally characterized by evaluating the equilibrium stress, which is associated with the relaxation or creep of a material toward equilibrium. The relation between the back stress and the equilibrium stress depends on the specific model, yet the equilibrium stress represents a characteristic of the response of a material that a comprehensive model should capture correctly. The element that is responsible for the non-zero equilibrium stress in polymers is normally also responsible for storing the energy that drives recovery and makes shape memory possible.

Relaxation and creep tests are normally used to determine the equilibrium stress [11, 15]. However, depending on the material and conditions, creep or relaxation may take a long time. This is particularly true for polymers below their glass transition temperature. It has been shown that a cyclic loading process can be designed that can fairly rapidly determine the equilibrium stress in tension/compression and shear [16, 17], and that this measurement correlates closely with those obtained by relaxation.

The nonzero equilibrium stress during relaxation after plastic flow, and the thermally initiated shape recovery after plastic flow [18] are studied here. First, thermal expansion after plastic deformation of poly-ether-ether-ketone (PEEK) is measured for plastically deformed samples to capture the onset of anisotropic expansion due to onset of shape recover. This is followed by evaluation of equilibrium stress after plastic flow and relaxation for a range of temperatures using the method of cyclic loading [17]. The equilibrium stress for PEEK is compared to that for polycarbonate (PC), which shows a pronounced yielding maximum followed by a post yield softening.

23.2 Materials and Experimental Methods

The experimental results are presented for as received PEEK (VICTREX 450G, 0.5 in. thickness commercial sheet) and PC (Lexan 9034). The PEEK was initially cut in the form of cylindrical samples with axis normal to the sheet. The PC sample preparation and results are described in [17]. No thermal conditioning was done to either the PEEK or PC samples.

23.2.1 Thermal Expansion, Density Reduction and Shape Recovery

Thermal expansion of samples were evaluated after room-temperature plastic compression. The compression was done using an MTS 8500 testing machine to compress the PEEK samples to 0 % (undeformed), −15, −30 and −45 % strain. After compression, from each cylindrical sample two 12.7 × 6.35 × 4 mm3 square prismatic samples, shown in Fig. 23.1a, were cut with one surface perpendicular to the axial compression. The samples were speckled with flat white and black spray (high heat, RUST-OLEUM) and placed in an oven with the front door replaced by a glass window. The pair was placed so one showed the front view and one the side view. A K-type thermocouple was inserted through a 0.5 mm hole in the side of one of the two samples and used for monitoring temperature. An ARAMIS stereo-optical digital image correlation (DIC) system (GOM, Germany) was placed 50 cm away from the samples in front of the glass window. The DIC optical system was calibrated using a 35 × 28 mm calibration panel. The ARAMIS system was used to measure the axial and both transverse strains during heating. The thermal expansion was calculated from the strain in the three directions using the well know relation for volume ratio \( J= \det \left(\mathbf{F}\right) \) in terms of the deformation gradient F [13]. For the experiment this was taken as
Fig. 23.1

Thermal expansion: (a) sample orientation after compression, and (b) testing setup and sample orientation in oven

$$ J=\left(1+{\varepsilon}_x\right)\left(1+{\varepsilon}_y\right)\left(1+{\varepsilon}_z\right), $$
where \( J=1+{\varepsilon}_V \), εV is the volumetric strain, and (εx, εy, εz) are the strains along the thee directions of the rectangular samples as measured by the ARAMIS system on the two samples. The room-temperature density ρo of the samples was measured using the weight Wa of the sample in air and the weight Ww of the sample in distilled water using a Mettler Toledo AT201 scale. The equation for this calculation is
$$ {\rho}_o={\rho}_w\times \frac{W_a}{W_a-{W}_w}, $$
where ρw is the density of the distilled water. The density ρ of the sample at an elevated temperature was calculated using the room-temperature density ρo and the measured volume ratio J due to thermal expansion using the well-known relation
$$ \rho =\frac{\rho_o}{J}. $$
Starting from 24 °C, the samples were heated to 120 °C with a step of 5 °C or 10 °C. The samples were allowed to equilibrate to the set temperature by waiting 25 min after each temperature adjustment. After this wait, the sample temperature was measured and recorded using the thermocouple, and the thermal strain was measured by the DIC system.

23.2.2 Equilibrium Stress Measurement

Ratchetted cyclic uniaxial compression was conducted to determine the equilibrium stress of PEEK following the procedure described in [17]. This process allows determining the equilibrium stress from cyclic loading involving cycles that have a large compression step followed by a small unloading step. The point of equal slope in a stress-strain plot for the unloading and subsequent loading correlates with the point at which the response becomes rate independent, indicating conditions of equilibrium.

The equilibrium stress of PEEK has been determined at room temperature [14]. For evaluation of equilibrium stress at elevated temperatures, 6.35 mm diameter and 6.35 mm length cylindrical samples were prepared and sprayed with a stochastic pattern using the same process described for thermal expansion. The cyclic compression was conducted inside a thermal oven with a glass window using an MTS 8500 testing machine. The ARAMIS stereo-optical DIC system was used to follow the sample strains. The ARAMIS system was located 40 cm in front of the oven. The system was calibrated with a 15 × 12 mm calibration panel. To reduce friction between the sample and the compression grip, a Teflon dry lubricant (PTFE spray, ANTI-SEIZE) was used to lightly coat the compression plate.

Isothermal tests were conducted from room temperature to 120 °C. In each case, the unloading cycle was around 10 % strain. The cycles were continued up to 50 % compression strain at a strain rate of 0.01 1/s. To study the effect of loading rate on the equilibrium stress, room temperature and 120 °C tests were also conducted at a strain rate of 0.0001 1/s.

23.3 Experimental Results and Discussion

Figure 23.2 shows thermal expansion after plastic flow for 0, −15, −30 and −45 % plastic strain. As can be seen, for the most part the overall volumetric strain increases linearly with temperature irrespective of extent of compression. The inserts of the figure show the individual strains in the three directions. As can be seen, the three strains are identical during the entire range for the plastically undeformed sample, while the −15, −30 and −45 % plastically deformed samples show identical thermal expansion in all three directions up to about 70 °C where the internal loads associated with shape recovery overcome the internal barriers to recovery. At this point, the expansion in the direction of compression accelerates and the other two directions start to contract. Despite the fact that the individual strains drive the sample to recover its initial shape after 70 °C, the volumetric thermal expansion of the sample continues to linearly increase. As a result, it can be seen in Fig. 23.3 that the density decreases close to linearly with increasing of sample temperature.
Fig. 23.2

Thermal expansion and shape recovery of plastically compressed samples of PEEK

Fig. 23.3

Densities of plastically compressed PEEK as a function of temperature

Figure 23.4 shows the equilibrium stress of PEEK from room temperature to 120 °C for different amounts of plastic compression. As can be seen, there is a 30 MPa drop of the equilibrium stress with 100 °C rise in temperature. In addition, the equilibrium stress is nearly constant over the entire testing range.
Fig. 23.4

Equilibrium stress in the cyclic uniaxial compression tests at different temperature (lines are added to aid visualization)

It was experimentally shown that the equilibrium stress evaluated by cyclic loading of PC is strain rate independent [17] (here shown in Fig. 23.5b). Figure 23.5a shows that the equilibrium stress evaluated by cyclic loading for PEEK under two loading rates for both room temperature and for 120 °C. As can be seen, the equilibrium stress is not affected by the loading rate of the measurement over two decades of change in strain rate.
Fig. 23.5

Rate independence of equilibrium stress: (a) for PEEK, and (b) for PC from [16] (lines are added to aid visualization)

There is a subtle difference in the shape of the equilibrium stress for PC and PEEK. As seen in Fig. 23.5, the equilibrium stress of PC, which is a glassy polymer, shows a characteristic initial drop with the increasing of plastic flow, identical to that seen during monotonic loading, followed by a steep strain hardening. In contrast, the equilibrium strain of PEEK, a semi-crystalline polymer, is fairly constant over the entire range of compressions. This is also consistent with the monotonic loading of PEEK, which shows a steady and constant flow, without the softening seen for PC.

23.4 Summary

The equilibrium stress of PEEK is evaluated by using a ratcheting cyclic loading test in compression during isothermal loading for temperatures from room temperature to 120 °C. The measurements are shown to be rate independent over the entire temperature range, at least for the rates used. It is expected that at higher rates additional relaxation processes get engaged that are not seen at the lower rates. Unlike PC, which shows a softening in the equilibrium stress followed by strain hardening similar to that seen during monotonic loading of PC, PEEK showed fairly constant equilibrium stress over the entire loading range consistent with the constant flow seen during monotonic compression of PEEK.

Even though the volumetric thermal expansion and density change of PEEK showed close to linear change with temperature, shape recovery in the plastically deformed samples dominated the directional changes of the strain. This occurred from about 50 °C above the temperature of plastic deformation. The amount of strain recovery depended on the amount of plastic strain, but for the plastic compressions tested, this recovery was substantial even at temperatures below the glass transition temperature.



The research was partially supported by the US Army Research Laboratory through Contract Number W911NF-11-D-0001-0094. The experiments were completed by utilizing the stress analysis facility at the University of Nebraska-Lincoln.


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Copyright information

© The Society for Experimental Mechanics, Inc. 2017

Authors and Affiliations

  • Wenlong Li
    • 1
  • George Gazonas
    • 2
  • Eric N. Brown
    • 3
  • Philip J. Rae
    • 3
  • Mehrdad Negahban
    • 1
  1. 1.Mechanical and Materials EngineeringUniversity of Nebraska-LincolnLincolnUSA
  2. 2.U.S. Army Research LaboratoryAberdeen Proving GroundAberdeenUSA
  3. 3.Los Alamos National LaboratoryLos AlamosUSA

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