# Overview of an Experimental Program for Development of Yield Surfaces Tracing Method
## 1 Introduction
Kinematic hardening mechanism that causes translation of a yield surface through stress space. The position of a YS is described with an evolving back-stress parameter, \(\mathbf{\alpha}\), representing an offset of the YS center with respect to the origin of the stress space. In this case, the yield criterion (4) takes the following more general form: \[f(\mathbf{\sigma})=\sqrt{\frac{3}{2}\left(s_{ij}-\alpha_{ij}\right):\left(s_{ij}- \alpha_{ij}\right)}-\sigma_{y}\] (7) where \(s_{ij}\) denotes the components of the deviatoric Cauchy stress tensor: \[s_{ij}=\sigma_{ij}-\frac{1}{3}\delta_{ij}\sigma_{kk}\] (8) The well-known Bauschinger effect is a typical feature of kinematic hardening. 2. Isotropic hardening is represented by a uniform expansion of the yield surface in the deviatoric stress space. Strain hardening induces an increase of yield strength not only in the primary loading mode (e.g., tension), but also in any other loading modes (e.g., compression, pure shear, etc.). The yield function of an ideal isotropically hardening material can be described as:  
\[f(\sigma)=\sigma_{\rm eq}-\sigma_{\rm y}(\kappa) \tag{9}\]  
The strain hardening parameter, \(\kappa\), can be expressed as a function of either the plastic strain energy density accumulated in the material element throughout the plastic flow history  
\[\kappa=\int\ \sigma:\hat{\epsilon}^{\rm(pl)}\ \rm dt \tag{10}\]  
or the equivalent plastic strain which is a time integral of equivalent plastic strain rate:  
\[\kappa=\int\ \sqrt{\frac{2}{3}\hat{\epsilon}^{\rm(pl)}:\hat{\epsilon}^{\rm(pl)} }\ \rm dt \tag{11}\]  
In practice, however, yield surfaces evolve in a more complicated manner. Apart from translation and expansion, distortion, rotation, or affine deformation can be discerned as separate mechanisms [5]. One of the most important features is sharpening of the forward apex (i.e., in the direction of plastic loading), and flattening of the region at the rear [6,7]. An accompanying phenomenon called cross-effect represents a change in size of the YS in directions perpendicular to the load in deviatoric stress space. A negative cross effect was reported for a low work hardening aluminum alloy in [8], while a positive cross effect was reported for a high work hardening aluminum alloy in [9]. Further evidence on evolution of YS shape under plastic flow was provided by many studies [10, 11, 12, 13, 14, 15, 16, 17, 18]. Careful experimental identification of yield surface shape is critical to the development and calibration of phenomenological models of plasticity, and their application on real engineering problems [19, 20, 21, 22]. In particular, recent work [23] uses YS shape change to help develop models aimed at improving predictions of multiaxial ratcheting behavior, since conventional models that neglect the YS distortion may not provide satisfactory predictions in this complex loading case [1]. This paper describes the advances of the Laboratory of Thermomechanics in the development of a rigorous procedure for experimental data acquisition of yield surfaces. First, a detailed description of the methods being developed is presented. In short, the YST technique under development is based on loading thin-walled tubular specimens, with a servo-hydraulic testing machine under various combinations of tension/compression and torque. Particular aspects of the experimental design are based on approaches presented by [PERSON], [PERSON] 1991 [6], and other authors (e.g., [16,24,25]). Unlike some prior work (e.g., [8,9,26]), the current technique employs proportional loading characterized by regulation of both stress components simultaneously and proportionally with each other. The onset of plastic flow is currently based on a small proof equivalent plastic strain being continuously evaluated from the output of a contact biaxial extensometer of a type similar to those employed in [2,4,16,27]. Second, suitability of the method for effective capture of YSs is demonstrated by results obtained on conventional low-alloyed ferritic steels.