# Overview of an Experimental Program for Development of Yield Surfaces Tracing Method
## 2 Materials and Methods
### Experimental Procedure  
The stress state within a thin-walled tubular TS under combined tensile and torsional loading can be described as a superposition of a pure tensile stress state and a pure shear stress state, i.e. :  
\[\sigma=\left( \begin{array}{ccc}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\end{array} \right)=\left( \begin{array}{ cccc}\sigma&\tau&0\\ \tau&0&0\\ 0&0&0\end{array} \right) \tag{12}\]  
The stress state is schematically depicted in Figure 2B. Consequently, the von Mises stress (3), and the corresponding von Mises yield criterion (4) can be simplified into the following forms:  
\[\sigma_{\text{eq}}=\sqrt{\sigma^{2}+3\tau^{2}} \tag{13}\]  
\[\sqrt{\sigma^{2}+3\tau^{2}}=\sigma_{\text{y}} \tag{14}\]  
In the subsequent procedures, both the axial and the shear true stresses are replaced by \"engineering\" quantities and expressed by the following simple relationships:  
\[\sigma=4F/\left[\pi\left(d_{\text{o}}^{2}-d_{\text{i}}^{2}\right)\right] \tag{15}\]  
\[\tau=16\ T\ d_{\text{o}}/\left[\pi\left(d_{\text{o}}^{4}-d_{\text{i}}^{4} \right)\right] \tag{16}\]  
where \(F\) and \(T\) are the axial force and the torsional moment, respectively, \(d_{\text{o}}\) is the outer diameter and \(d_{\text{i}}\) is the internal diameter of the TS's initial cross-section. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**C** & **Mn** & **P** & **S** & **N** \\ \hline
0.16 & 1.32 & 0.021 & 0.024 & 0.005 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Contents of alloying elements (wt%) in S235 JR steel investigated in the present work. As indicated in Section 2.1, the yield surface tracing procedure is performed by a series of probes, each contributing a single yield point on the current yield surface. A probe is a very slow stress-controlled loading sequence characterized by elevation of both stress components proportionally with each other. As a special case, one of the stress components could be kept at zero for achieving either pure axial or pure torsional loading. Below is a detailed procedure for tracing of the initial YS:  
* The test specimen begins unloaded (i.e., zero stress). This centerpoint of the initial yield surface is represented by \"\(C_{0}\)\" in Figure 3.
* Probe #1 is conducted in order to evaluate the first yield point. While probing, the stress is gradually increased until a yield point is clearly determined by fulfilment of a particular experimental YP definition described below. * Once a YP is detected, further loading ceases and the specimen is unloaded back to the stress level corresponding to the centerpoint. * The above described process is repeated for all the subsequent probes (1 through 16) in the order depicted in Figure 3. Design of this particular pattern was adopted from [25] and slightly modified. * The process is finished by unloading to the centerpoint. All probes are characterized by an identical magnitude of an overall stress rate given by an implicit relationship analogous to (13):  
\[\sqrt{\dot{\sigma}^{2}+3\dot{\tau}^{2}}=1.5\;\text{MPa s}^{-1} \tag{17}\]  
The factor of \(\sqrt{3}\) in the shear term is convenient for consistency with the (\(\sigma,\sqrt{3\tau}\)) stress space used for the YS, where the von Mises yield surface (14) possesses a circular shape. Figure 2: Experimental setup (**A**): a thin-walled tubular specimen clamped in hydraulic grips of the testing machine, and a contact biaxial extensometer is attached in four points on the reduced section of the TS. Stress state representation (**B**) within the TS subject to combined axial load and torque. The procedure for subsequent YS tracing is basically equivalent to that of the initial YS. It is preceded by application of a desired level of plastic prestrain to the specimen.