Back Stress in Modeling the Response of PEEK and PC
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.
KeywordsMechanical modeling Plastic flow Equilibrium stress Thermal expansion Digital image correlation
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  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 . 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 . No thermal conditioning was done to either the PEEK or PC samples.
23.2.1 Thermal Expansion, Density Reduction and Shape Recovery
23.2.2 Equilibrium Stress Measurement
Ratchetted cyclic uniaxial compression was conducted to determine the equilibrium stress of PEEK following the procedure described in . 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 . 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
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.
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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