## A criterion for description of anisotropy and yield differential effects in pressure-insensitive metals.(English)Zbl 1107.74006

The authors propose two yield criteria for pressure-insensitive materials, with the goal to capture: (1) the strength differential (SD) effects, namely the strong asymmetry between compressive and tensile strengths, experimentally observed in materials that deform by twinning or by slip, (2) a pronounced anisotropy in the rolling sheets, namely higher strength in the transverse direction than in the rolling one. The isotropic criterion is written in a form that contains an odd function of $$J_{3},$$ being expressed by $$f \equiv (J_{2})^{3/2} - c J_{3}= \tau_{Y}^3.$$ The criterion has been suggested by Drucker’s criterion $$\Phi \equiv (J_{2})^3 - c J_{3}^2= \tau_{Y}^6.$$ Here $$\tau_{Y}$$ is the yield stress in pure shear, $$J_{2}$$ and $$J_{3}$$ denote the second and third invariants of stress deviator tensor, respectively. Three sequences of yield surfaces were plotted for three different values of parameter $$c$$, with $$\tau_{Y}$$ kept constant. The material parameter $$c$$ is expressed through the ratio of uniaxial tensile yield stress to the uniaxial yield stress in compression.
Let us remark that $$\tau_{Y},$$ calculated in the paper either by the formula (6) or by (5), has to be dependent on $$c$$, and it also remains dependent on one of the uniaxial yield stresses. Thus $$\tau_{Y}$$ could not be kept constant. Due to the fact that $$c$$ can be considered to be a certain function, the authors determine (in appendix) the interval of the variation for $$c$$ that assures the convexity of the yield criteria.
In appendix $$J_{3}$$ is calculated by $$J_{3}= s_{1} s_{2} s_{3},$$ in contrast with $$J_{3}= \displaystyle{\frac{1}{3}}(s_{1}^3 + s_{2}^3 + s_{3}^3)$$ that has been consistently used in previous sections (here $$s_i$$ denote the principal values of stress deviator). The proposed orthotropic yield criterion is written in the same form, this time with $$J_{2}$$ and $$J_{3}$$ replaced by $$J_{2}^0$$ and $$J_{3}^0,$$ homogeneous functions of degree two in the stress deviator (with six parameters) and of degree three, respectively, with ten parameters. $$c$$ is kept in the representation, although it could be incorporated in the material parameters involved in $$J_{3}^0.$$ Let us remark that for a fixed set of material parameters, the proposed criteria are invariant relative to the orthogonal group and to the orthotropic group, respectively.
On the other hand, the yield criteria kept the initial symmetry group (the orthogonal group and orthotropic group, respectively) if and only if the material parameters, viewed as functions of certain variables, remain invariant with respect to an appropriate symmetry group. In the paper a methodology for the determination of material parameters is proposed. Ten constants and the constant $$c$$ appear in the plane stress state representation for the yield surface; they can be reduced to 7 constants and the constant $$c$$ if a principal axis representation is adopted. Numerical values of material parameters have been determined using experimental data reported in the literature. These data correspond to measurements performed for different orientations $$\theta,$$ relative to the principal symmetry axes and refer to (a) uniaxial yield stresses in the considered direction $$\theta,$$ and to (b) the ratios between in plane transverse strain rate and the through-thickness strain rate. Three sequences of the yield surface are determined for several constant levels of the largest principal strain.
Even if in the paper there is no reference to the flow rule in order to express the ratios of the appropriate strain rates in terms of the constants that enter the yield criteria, see formula (21), a flow rule associated to the yield criteria has been employed. Excepting the case of perfect plastic material, the plastic surfaces defined by yield criteria are changing in shape and position in the appropriate stress spaces. In order to describe the deformability of the yield surface, the internal variables dependent on the history of the irreversible deformations, are traditionally inferred in the models via appropriate evolution equations. Bearing in mind that in the reported experiments the test specimens have been cut from the sheets deformed at different levels of the strain (say applied along the rolled direction), the material parameters can be viewed as functions of the appropriate plastic deformation invariants. Consequently, the influence of the history of plastic deformations could be assimilated with the prestrains applied to the sheets (elastic strains are small in metallic sheets) in the models analyzed here.
Conclusion. The theoretical criterion for orthotropic material proposed by the authors allows them, in the plane stress case, a very good agreement with experimental yield loci, calculated using the data after Kelly and Hosford (1968), that correspond to different levels of total strain. A similar good agreement is obtained for the isotropic yield criterion with the data obtained by polycrystalline simulations by Hosford and Allen (1973).

### MSC:

 74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) 74E10 Anisotropy in solid mechanics
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