# Overview of an Experimental Program for Development of Yield Surfaces Tracing Method
## 1 Introduction  
This paper establishes an experimental technique to evaluate yield surfaces of metallic materials, namely a yield surface tracing (YST) method. The concept of a yield surface (YS) is critical to phenomenological plasticity theory and it is defined as a set of yield points (YPs) within a particular stress space that represent the onset of yield in any combination of mixed-mode loading from a predefined initial point. The yield surface may change size, shape and/or location due to plastic flow. YST is an essential tool for acquisition of experimental data required for deriving and calibrating robust phenomenological models of directional distortional hardening (DDH). A fine description of yield surfaces is expected to improve the accuracy of phenomenological modelling predictions for structures experiencing complicated and demanding loading modes. Plausible applications of phenomenological DDH models include, but are not limited to, the processes of forming of metallic products, or predicting the behavior of structures under cyclic loading (e.g., during earthquakes, in service conditions, repetitive wind or wave loading) leading to the accumulation of plastic strain (a phenomenon known as ratcheting) [1]. Despite significant advances in physically-based plasticity models [2; 3; 4], the phenomenological models are still the most useful for structural applications. The phenomenological approach is based on the concept of substituting the real crystallographic structure with a continuum solid. Thus, phenomenological models tend to require significantly less computational time than the physically-based models. In phenomenological modelling, the YS defines when the material experiences plastic deformation and is often also usedas a potential to define the direction of plastic strain accumulation. The mathematical expression of a YS, i.e., a yield criterion, can be written in the following simple form:  
\[f(\mathbf{\sigma})=0 \tag{1}\]  
where \(\mathbf{\sigma}\) is the Cauchy stress tensor:  
\[\mathbf{\sigma}=\left( \begin{array}{ccc}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{12}&\sigma_{22}&\sigma_{23}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{array} \right) \tag{2}\]  
One of the first and most widely used phenomenological yield criterion is the von Mises criterion, which is based on the concept of equivalent stress (von Mises stress) and is defined as:  
\[\sigma_{\text{eq}}=\sqrt{\frac{1}{2}\left[\left(\sigma_{11}-\sigma_{22}\right) ^{2}+\left(\sigma_{22}-\sigma_{33}\right)^{2}+\left(\sigma_{33}-\sigma_{11} \right)^{2}+6\left(\sigma_{12}^{2}+\sigma_{23}^{2}+\sigma_{31}^{2}\right) \right]} \tag{3}\]  
Correspondingly, the von Mises yield criterion is written as:  
\[\sigma_{\text{eq}}-\sigma_{y}=0 \tag{4}\]  
where \(\sigma_{y}\) is equal to the yield strength under pure tension, and can be easily derived by setting \(\sigma_{11}=\sigma\) and all other stress components to zero into expression (3), resulting in:  
\[\sigma=\sigma_{y} \tag{5}\]  
By analogy, yield strength for pure shear can be derived by setting \(\sigma_{12}=\tau\), giving:  
\[\tau=\sigma_{y}/\sqrt{3} \tag{6}\]  
which indicates that the onset of plastic flow occurs at a shear stress level which is decreased by the factor of \(\sqrt{3}\) in comparison with the axial yield strength. The von [PERSON] yield criterion is usually suitable for initial yield surfaces of isotropic metallic materials with either no plastic prestrain or with prestress that has been mitigated by annealing. As the stress state is elevated beyond the yield surface, plastic flow occurs. If the material integrity is preserved throughout the plastic flow, then the YS has to change its size, shape, and /or position in the stress space in order to persistently include the actual stress state in the set of points where yielding occurs. The process is associated with permanent changes of yield strength and is referred to as strain hardening. Elementary phenomenological plasticity theory discerns the two most fundamental strain hardening mechanisms:  
1.