Deviatoric section shape functions for materials exhibiting dependence on shear-to-axial yield stress ratio

Proposal of a class of smooth functions describing shape of the deviatoric section of the failure surface of isotropic materials is given. Their convexity is investigated and the resultant restrictions to the free parameters are derived. Applications to pressure sensitive or insensitive materials as yield criteria are shown. Discussion of calibration of material parameters is carried out and dependence of the failure surface on the shear-to-axial yield stress ratio is demonstrated.


Introduction
Each phenomenon of yielding, damage or failure of materials, including cast iron, porous metals, metallic foams is quite complex and is the subject of an intense research. For example, behaviour of the mentioned materials depends on the first (pressure sensitivity) and third (Lode dependence) stress invariants. To formulate mathematical models for the aforementioned phenomena an approach based on the yield, damage or failure criteria is used, predominantly relying upon the experimental evidence of material behaviour.
In the present article proposal of new functions describing shape of the deviatoric section of the failure surface is given. Yield or failure criteria are tailored to interpolate typical experimental results for certain class of isotropic materials. The regarded shape functions used for formulation of the criteria preserve smoothness and convexity of the yield surface. Moreover, they can be included in the constitutive models of materials in which apexes or intersection corners exist, for example in case of the Drucker-Prager yield surface.
Excellent review of the shape functions for deviatoric section is given in [1]. Special case of functions analysed herein was introduced by Drucker [1, 2], but generalization was proposed in [3,4]. This paper reports further development of that type of shape functions.

Shape functions of deviatoric section
The analysis is restricted to isotropic materials, therefore the Haigh-Westergaard representation of the yield surface is used. The following class of isotropic scalar-valued functions is regarded: where  , r and  are invariants of stress tensor σ defined as: To fulfil the mentioned properties invariant cos3 can be used in definition of the shape function [4].
A two-parameter smooth shape function is proposed as follows: which is a periodic function of period / 3  with axes of symmetry at / 6 m   for 0,1, 2,3, 4,5 m  .  and n are free parameters which can be determined from experimental data. Definition (3) holds true for arbitrary exponent n when inequality g  , and this case is excluded from the following analysis. Further correlation between  and n is found by imposing convexity requirement on yield function (1). Restricting attention to the following subclass of functions (1): one can specify convexity requirements on two functions   g  and   h  . Then the shape function has to fulfil the following inequality [2,4,5]: while h  has to be convex, which implies the condition [5]: " 0 h  , for every  .
Function (7) can attain local minima depending on the values of  and n parameters, for which the following equation is met: When 0   function (4)     . Moreover, the local extremum of function (7) is analysed, which is attained when: At this level of analysis function (7) Graphical representation of the obtained limits (10) on  and n is shown in Figure 1.

Application to pressure insensitive material
As the first example of application of the introduced shape functions a pressure insensitive material is considered. Then function   h  is a constant function, which can be conveniently defined for tensile meridian, 0   , or for compressive meridian, / 3    . Function (4) can be converted to the form: in which size parameter R and shape parameter  of the yield surface      , from (12) one can get the set of equations: Solution to (13) results in: parameter is introduced, which describes the shape of deviatoric section of a yield surface. Then yield function (12) can be rewritten in the form of the yield criterion: , with S f k  being the shear yield stress. Therefore for an incompressible material the ratio of shear-to-axial yield stress can be expressed as: When 3 T f k  formula (16) gives 1 t  and equation (15) reduces to the well-known yield criterion of Huber-Mises (HM) [2,4]. For example, the shear-to-axial strength ratio was considered in [6] for constitutive modelling of ductile fracture of structural metals.
Using relationship (14), which relates of the shape parameter t and  , namely , restrictions (10) can be re-casted to   t n . Hence, convexity restrictions (10) can be expressed as: 2 2 1 10 2 2 1 10 1 1 9 1 2 9 1 2 n n P n P n t n n n n where   P n is defined by (11). Limits according to (17) and accompanying region of convexity for function (15) are shown in Figure 2. Dependence of the shape of deviatoric section on the exponent is shown in Figure 3.
Extreme values of the shape parameter t are attained for ext 0.06326823 n   , which results in the values min 0.901065 t  and max 1.109798 t  (Fig. 2) Figure 4. Graphs of the yield criterion (15) with exponent 0.0625 n   and accompanying limits 1.109 t  and 0.9010 t  are shown in Fig. 5. When 0.9010 t  criterion (15) can be interpreted as a smooth approximation of the well-known Tresca-Guest criterion, while for 1.109 t  can serve as a smooth approximation of the Hill-Ivlev-Haythornthwaite surface [3,4]. By change of t from the minimum to the maximum value a smooth transition in the shape of the yield surface can be achieved, compare