# Anisotropic Yield Criterion of Rolled AZ31 Magnesium Alloy via Nanoindentation
## 1 Introduction  
Magnesium alloys have aroused great research interest and are used in many applications on account of their excellent properties, which include high strength and good wear resistance, corrosion resistance, and thermal stability. However, although magnesium alloys present many advantages over other alloys, they are not perfect metals, because of their poor formability and corrosion resistance at ambient temperature. Magnesium alloys have a hexagonal closed-packed crystal structure with a limited number of available slip systems (basal [0001], prismatic [1010], and pyramidal [1011]), which leads to their poor formability during cold working [1]. While cast magnesium alloys are used more extensively than deformed magnesium alloys, the strength, ductility, and mechanical properties of the former are poorer than those of the latter. AZ31 is one of the most widely used alloys currently available. Rolling is an important means to improve the properties of this type of alloy. Under annealing and mechanical twinning, magnesium can show anisotropy at room temperature; thus, the nature of the plastic deformation of magnesium is quite complex [2]. Studies on the deformation and damage behavior of magnesium indicate that anisotropy clearly occurs during magnesium alloy deformation. Initial textures can impact the shape and stress-strain behavior of the samples [1]. The anisotropicmechanical characteristics of AZ31 magnesium alloys have a significant influence on their plastic deformation. Therefore, obtaining a comprehensive understanding of the anisotropic mechanical behavior of magnesium alloys is important. An accurate description of the magnesium alloy yielding behavior is essential to predict its forming processes. Industrial applications require an overall understanding of the mechanical properties of rolled AZ31 alloys. An accurate mathematical description is significant for predicting material deformation and yielding behaviors; thus, researchers have exerted considerable efforts over the last several decades to establish a solid foundation through which the magnesium yielding and deformation behaviors may be described. In 1864, [PERSON] established a phenomenological model to describe the yield of materials using the concept of maximum shear stress. [PERSON] proposed an anisotropic yield criterion that satisfies the anisotropic presentation under uniaxial and equibiaxial tension [3]. [PERSON] et al. [4] concluded that taking plastic anisotropy into account obviously improves the estimation of the final width. Many reports on the measurement of mechanical properties at the macro scale have been published. However, the in situ acquisition of the micro-scale mechanical properties of AZ31 alloys has yet to be conducted. [PERSON] et al. [5] investigated size-dependent plastic responses by nanomechanical testing, high-resolution microscopy, and phase analysis and showed that the local microstructure has an obvious influence on small-scale plastic responses. Knowledge of the micro-scale mechanical properties of AZ31 alloys could provide a fundamental basis for its applications. Most traditional experimental approaches, such as compression, tension, and torsion, are based on conventional bulk-scale testing, are destructive, and require a large number and volume of materials. The nanoindentation testing technique, a non- or semi-destructive approach, is considered a reliable, convenient, and robust approach with which to study the mechanical properties of materials at the nano- and micro-scales. Many scholars have proposed a series of simple mechanical properties for AZ31 alloys determined from indentation tests. However, in-depth studies of the mechanical properties of AZ31 alloys obtained via nanoindentation must be carried out, which can solve the impact due to the local deformation of the indentation test and complex operating conditions.