## 2 Determination of Chaboche Material Parameters
### Experimental Setup  
The experiments were conducted on INSTRON 5982 universal testing machine. The machine provides the capability of testing under load or displacement control. The load readout from the Instron was monitored by another calibrated load cell, which was synced with a digital high speed data logger. The strains were measured using high elongation single element strain gauges for monotonic and uniaxial test samples and 90\({}^{\circ}\) 2-element strain rosettes for the biaxial test. The gauges were bonded to opposite sides of the test specimens to eliminate bending effects. The strains and loads were monitored and stored continuously by the data logger. The rate of loading was set at 0.75 mm/min and the stress-strain data stored at time intervals of 0.5 s. The data combination of force from the designed load cell and extension reading from the Instron machine was employed to monitor the  
Figure 3: Biaxial loading test sample: (**a**) external view (**b**) internal view.  
Figure 2: Monotonic uniaxial test sample: (**a**) external view (**b**) internal view.  
force-displacement response of test specimens and not for numerical analysis because these data lack accuracy. The setup of the test system is shown in Figures 4 and 5.  
Advanced ratcheting analysis began with [PERSON] et al. who concluded that cyclic hardening could have a significant effect on ratcheting rate in unsymmetrical stress cycling [25]. Hence, in the present study, all monotonic, uniaxial and biaxial test specimens were cyclically stabilized through uniaxial symmetric strain cycling in an attempt to reduce the coupling of the two phenomena. The strain controlled tests imparted the specimen with symmetric tension compression cycles within the strain range of \(\pm 0.75\%\). Following the strain hardening stabilization process, post-stabilized monotonic, uniaxial and biaxial tests were conducted.  
Figure 4: Sketch of biaxial testing setup.  
Figure 5: Experimental setup of specimens: (**a**) Uniaxial (**b**) Biaxial.  
The post-stabilized monotonic tension tests were continued up to 1.8% strain. In uniaxial testing, there are two concepts of conducting tests under stress control. The first concept of uniaxial testing is performing incremental loading and unloading to zero load in order to establish an elastic limit, by reading the total strain, which is recorded at every unloaded zero load. In this test, the stress increment for every subsequent loading is set at 4 MPa. The second concept of uniaxial testing is to establish the uniaxial ratcheting rate. Here, the applied stress range is set to 255 MPa under a constant mean stress of 65 MPa.  
In the biaxial tests, the pipes were pressurized using a hydraulic pump. The axial load provided by the Instron machine and internal pressure were coupled through a feedback monitoring system so that the internal pressure would only give rise to pure circumferential stress. Under any magnitude of internal pressure, the system would adjust to maintain zero axial pressure stress. Cyclic axial stress would then be provided by the testing machine to study ratcheting. The uniaxial strain of the test specimen was cycled in a symmetric mode. The axial stress and strain are known as \((\sigma_{x},~{}\varepsilon_{x})\), and the circumferential stress and strain are recognized as \((\sigma_{\theta},~{}\varepsilon_{\theta})\). The internal pressure is denoted by \(IP\) and the strain amplitude of the cycles is identified by \(\varepsilon_{xc}\). In the biaxial tests, the strain was controlled and cycled and the stress response was logged. The strain limit is \(\varepsilon_{xc}=\pm 0.4\%\). The internal pressure was set at \(IP=5\) MPa. It is worth mentioning that as with other researches, the loading rate-dependent behavior of the material and pipes was not studied [30; 34; 35].