Common Equations for the Equivalent Stress Concept

V. A. Kolupaev 1, H. Altenbach 2

1German Institute for Polymers (DKI), Darmstadt, Germany,
2Martin-Luther-Universität Halle-Wittenberg, Halle-Saale, Germany

The concept of the equivalent stress is used to deliver a “compact” form of certain relevant information about the current stress state and its limits. To make an appropriate model choice the available experimental data (e.g., at compression, torsion, hydrostatic compression, etc.) are compared with the results of tension test: , , ). In this case the tensile stress is the same as the equivalent stress.

Both the strength hypotheses and the flow criteria can be expressed as a surface in the principal stress space, and. To describe this surface, different sets of invariants are applied, for example

·  the axiatoric-deviatoric invariants

, , .

·  the cylindrical invariants (invariants due to Novozhilov)

, , .

The universal appearance of the model is or .

These phenomenological criteria result in a simple and complete description of experimental data. They can be improved or corrected by experiences. In the case of new materials the criteria admit simple engineering applications. During the last century various advanced equivalent stress expression were suggested, the number of criteria increases dramatically. The aim of this paper is to show that the number of admissible shapes of the surfaces is limited. This allows simplifying the choice of a suitable criterion.

Models of Incompressible Material Behaviour. The models of incompressible material behaviour are based on neglecting of the first invariant

, .

The most important models allow computing explicitly the equivalent stress:

·  the cubic model (CM) of SayirI

, ,

·  the bicubic model (BCM) with a hexagonal symmetry () in the -plane

, (1)

which contains the models of Tresca, Schmidt-Ishlinsky-Hill, and Drucker,

·  irregular hexagonal prism of SayirII

, , (2)

·  irregular hexagonal prism after Haythornthwaite

(3)

with , and , .

The models (2) and (3) restrict the region of convex forms in the diagram (Fig. 1).

Fig. 1. Models of incompressible material behaviour relating to the model of vonMises with M:

T – model of Tresca; S – model of Schmidt-Ishlinsky-Hill; K – model of Ko; G – maximum strain criterion of Mariotte-St.Venant with

The geomechanical model [1, 2]

. (4)

with the convexity region restricted by

, ,

provides comprehensive possibilities in the description of the material properties.

Compressible Generalisations. All models for incompressible material behaviour allow a compressible generalisation using the transformation of Sayir:

.

The models (1) and Eq. (4) can be extended using the quadratic transformation:

. (5)

In order to apply the cubic transformation

(6)

to the geomechanical model (4) the right hand side of the model should be represented in the form . Thus the model becomes suitable for a large variety of isotropic materials:

. (7)

The resulting rotationally symmetric model () provides more possibilities of fitting than the quadratic model with the transformation Eq. (5). With the compressible generalisation of the model of Drucker arises.

The parameters describe the position of the hydrostatic nodes (), whereas the forms of the meridian, which are of practical relevance, can be obtained by setting ; with or conversely with.

For closed criteria, which restrict in addition to the hydrostatic tension the hydrostatic compression too, the following parameters with; with or with, should be considered. The parameters are restricted furthermore with the Poisson’s ratio in tension results from flow rule:

(8)

with and . It follows for the transformation (6) with Eq. (8)

.

The limitation arises to restrictions on the parameters of the models for ductile materials. For brittle materials can be introduced formally: . The geometry of the models with results in for certain materials, which must be treated separately. For the closed surface can be required .

The model (7) was chosen for the analysis of the measurements. Empirically it follows the restriction for engineering materials. For the measurements of a hard thermoplastic foams (closed surface ) and non-reinforced thermoplastics (surface with ) different solutions were found, which could be compared using the relations , , and . The evaluation of the model (7) is made in certain special plane sections (Burzyński-plane, p-plane).

References

1. Kolupaev V.A., Bolchoun A., Altenbach H. New trends in application of strength hypotheses (in German) // Konstruktion. – 2009. – 5. – S. 59–66.

2. Altenbach H., Kolupaev V.A. Fundamental Forms of Strength Hypotheses // In: Proc. of XXXVI Summer School “Advanced Problems in Mechanics” (Ed. by D.A. Indeitsev & A.M. Krivtsov). Institute for Problems in Mechanical Engineering RAS, St. Petersburg, (Repino) June 30 – July 5, 2009. – P. 32–45.