Yield Criteria for Incompressible Materials in the Shear Stress Space

Abstract

In the theory of plasticity different yield criteria for incompressible material behavior are used. The criteria of Tresca, von Mises and Schmidt-Ishlinsky are well known and the first two are presented in the textbooks of Strength of Materials. Both Tresca and Schmidt-Ishlinsky criteria have a hexagonal symmetry and the criterion of von Mises has a rotational symmetry in the $\uppi$-plane. These criteria do not distinguish between tension and compression (no strength differential effect), but numerous problems are treated in the engineering practice using these criteria. In this paper the yield criteria with hexagonal symmetry for incompressible material behavior are compared. For this purpose, their geometries in the $\uppi$-plane will be presented in polar coordinates. The radii at the angles of $15^\circ$ and $30^\circ$ will be related to the radius at $0^\circ$. Based on these two relations, these and other known criteria will be shown in one diagram. In this diagram the extreme shapes of the yield surfaces are restricted by two criteria: the Unified Yield Criterion (UYC) and the Multiplicative Ansatz Criterion (MAC). The models with hexagonal symmetry in the $\uppi$-plane for incompressible materials can be formulated in the shear stress space. For this formulation platonic, archimedean and catalan solids with orthogonal symmetry planes are used. The geometrical relations of such models in the $\uppi$-plane will be depicted in the above mentioned diagram. The examination of the yield surfaces leads to the generalized criterion with two parameters. This model describes all possible convex forms with hexagonal symmetry. The proposed way to look at the yield criteria simplifies the selection of a proper criterion. The extreme solutions for the analysis of structural members can be found using these criteria.

Axiatoric-Deviatoric Invariants

The model of isotropic material behavior $\Phi$ is a function of three stress invariants, e.g. of the axiatoric-deviatoric invariants [2, 48]

$$\begin{aligned} {\text{I}}_1= {\upsigma }_{{\text{I}}}+{\upsigma }_{{\text{II}}}+{\upsigma }_{{\text{III}}}, \end{aligned}$$

(26)

$$\begin{aligned} {\text{I}}_2^{\prime }=\frac{1}{2\cdot 3}\, \left[ ({\upsigma }_{{\text{I}}}-{\upsigma }_{{\text{II}}})^2+ ({\upsigma }_{{\text{II}}}-{\upsigma }_{{\text{III}}})^2+({\upsigma }_{{\text{III}}}-{\upsigma }_{{\text{I}}})^2 \right] , \end{aligned}$$

(27)

$$\begin{aligned} {\text{I}}_3^{\prime }=\left( {\upsigma }_{{\text{I}}}-\frac{1}{3}\,\text I_1 \right) \left( {\upsigma }_{{\text{II}}}-\frac{1}{3}\,\text I _1 \right) \left( {\upsigma }_{{\text{III}}}-\frac{1}{3}\,\text I _1 \right) . \end{aligned}$$

(28)

The stress angle ${\uptheta }$ [8, 28, 48], cf. [27]

$$\begin{aligned} \cos 3{\uptheta } = \frac{3\sqrt{3}}{2}\frac{\text I_3^{\prime }}{\left( \text I _2^{\prime }\right) ^{3/2}} \end{aligned}$$

(29)

is used quite often instead of $\text I _3^{\prime }$.