# Overview of an Experimental Program for Development of Yield Surfaces Tracing Method
## 2 Materials and Methods
### Experimental Procedure
However, there is an important difference for probing that consists in the reassessment of the center-point, that is updated to \"\(C_{\mathrm{x}}\)\" showed in Figure 3. This step is necessary due to intensive drift of the initial YS through the stress space. The drift is a feature of kinematic strain hardening and was confirmed in [8; 9; 26] to be sufficiently intensive to drift the YS far away from the origin of the stress space, as schematically depicted in Figure 3. As the stress-free state is no longer enclosed by the YS, complete unload would result in an undesirable reverse plasticity. Hence, the process of subsequent YS tracing is designed into the following steps:  
* Starting from the centerpoint from the previous load, the specimen is subject to strain-controlled loading in order to achieve the desired levels of axial and shear plastic prestrains. * A relaxation period is applied by keeping strain at the desired level for a period of 10 min. The final stress state defines an approximate forward yield point (FYP in Figure 3). * An approximate reverse yield point (RYP) is evaluated by a steady unload towards the preceding centerpoint. * A new centerpoint (\(C_{\mathrm{x}}\) in Figure 3) is defined at a stress state half between RYP and FYP. * Subsequently, a probing procedure identical to that applied for the initial YS tracing is conducted and finished at the centerpoint. Careful selection of an appropriate YP definition is crucial for clear identification of yield points. The YP definition needs to be suitable for any individual probe characterized by different levels of both stress components. There were several yield criteria used in the past:  
* \"Lode backward extrapolation\" method (e.g., [32]) defines a YP as intersection of a backward extrapolation line of a stress-strain curve with the elastic line or the ordinate. This method, however, requires an excessive amount of plastic overstrain, so that a specimen is suitable only for the determination of a single YP. * \"Deviation from linearity\" method assigns the YP equal to the limit of proportionality of the stress strain curve. The method was introduced and particularly highlighted by [PERSON] [28; 30; 31] because of excellent consistency of experimental results. How  
Figure 3: Schematic drawing of an initial and a subsequent YS. Probing systems composed of series of 16 probes. ever, extremely high-level accuracy of strain measurement (on the order of units of microstrains, \(\mu\epsilon\), \(10^{-6}\)) is required. A YP definition denoted as \"small proof plastic strain\" is implemented into the current YS tracing method. A YP is identified based on the equivalent plastic strain increment developed throughout a probe and calculated using the following fundamental relationship:  
\[\epsilon_{\text{eq}}=\sqrt{\left(\epsilon_{\text{pl}}\right)^{2}+\frac{1}{3} \Big{(}\gamma_{\text{pl}}\Big{)}^{2}} \tag{18}\]  
Terms under the square root in (18), i.e., the axial and the shear plastic strain increments, can be expressed as follows:  
\[\epsilon_{\text{pl}}=\epsilon-\epsilon_{\text{el}}-\epsilon_{\text{R}};\quad \gamma_{\text{pl}}=\gamma-\gamma_{\text{el}}-\gamma_{\text{R}} \tag{19}\]  
All terms in (19) are schematically identified in Figure 4 showing an axial stress-strain record of a single probe. The total strain values, \(\epsilon\), \(\gamma\), are measured directly with the extensometer, the elastic parts are calculated from the following relationships based on the Hooke's law:  
Figure 3.0  
Figure 3.1  
A vital part of the procedure is an algorithm that continuously and automatically performs a least squares linear regression analysis to fit the elastic part of the stress-strain curve throughout a probe. There are two primary outputs: the elastic moduli (Young's modulus, \(E\), and shear modulus, \(G\)) as slopes, and the residual strains (\(\epsilon_{\text{R}}\), \(\gamma_{\text{R}}\)) as offsets.