Archive of Applied Mechanics

, Volume 83, Issue 7, pp 1061–1085 | Cite as

Visualization of the Unified Strength Theory

  • V. A. Kolupaev
  • M. -H. Yu
  • H. Altenbach


The Unified Strength Theory (UST) provides the fundamentals for the systematic study of various strength hypotheses and yields criteria for isotropic materials. It shows relationship between known models (Mohr-Coulomb, Pisarenko-Lebedev, Twin-Shear Theory of Yu), and apart from these known models, this model contains also classical models like the normal stress hypothesis, von Mises, Tresca and Schmidt-Ishlinsky. The UST can be adapted for different types of materials. Thus, it is a suitable tool for the analysis of experimental data.

For the UST, the inelastic Poisson’s ratio and the maximum hydrostatic tension stress will be computed as a function of model parameters which simplifies the comparison with another model. The correlations between uniaxial, biaxial and hydrostatic stress will be illustrated and compared with classical models. For all classical models and for the UST, the uniaxial and biaxial tension failure stress and also the uniaxial and biaxial compression failure stress are equal. In this sense, the UST can be classified as a classical model.

The failure behavior of new materials like some polymers and alloys differs from the classical one. The UST can be extended to such failure behavior. For this purpose, the Unified Yield Criterion (UYC) as part of the UST will be modified so that all known criteria of incompressible material behavior can be approximated.

With the help of a simple substitution, the UYC can be further developed for compressible material behavior. Different convex lines can be adjust for the form of the meridian. With this substitution, the hydrostatic tension stress will be restricted with one of the parameters. Furthermore, the model can be applied for the description of failure behavior of ceramics, hard foams and sintered materials. For this application, both the hydrostatic tension and compression stress will be restricted too. Some reference values for hydrostatic loading are established.

For the visual comparison of different parameter setting of the models, graphical methods can be used. The UST will be represented in the principal stress space. Further considerations will be carried out in the Burzyński-plane and in the π-plane. For engineering applications, the Burzyński-plane is preferred to the meridional cut. For better analysis and a direct comparison of fitted models to the experimental values, the line of the plane stress state will be shown in the Burzyński-plane and in the π-plane.


Equivalent stress Spatial representation Deviatoric plane Burzyński-plane Plane stress state Compressible generalization Inelastic Poisson’s ratio 

List of symbols and abbreviations


Model of isotropic material behavior


First invariant of the stress tensors

I2′, I3

Second and third invariants of the stress deviator


Stress angle


Equivalent stresses

σ+, σ

Limits of the uniaxial stress states (tension and compression)

τ*, σBZ, σBD, σIZ, σUD

Limits of the biaxial stress states

σAZ, σAD

Limits of the hydrostatic stress states

σI, σII, σIII

Principal stresses


Lagrange multiplier


Stress gradient


Structural parameter


Relations for the loading points of the plane stress state

\({a_{\rm +}^{\rm hyd},\,a_{\rm -}^{\rm hyd}}\)

Relations for the loading points of the hydrostatic stress state

\({\nu_+^{\rm in},\,\nu_-^{\rm in}}\)

Inelastic Poisson’s ratio at tension and compression


Cartesian co-ordinates


Powers of the terms in the compressible substitution


Parameters of the UST


Inclination of the meridian with respect to the hydrostatic axis


Points of the meridian θ = 0: tension and balanced biaxial compression


Points of the meridian θ = 60: compression and balanced biaxial tension


Points of the meridian θ = 30: torsion, thin-walled tube specimen with closed ends under inner and outer pressure


Points of hydrostatic tension and compression


Cut of the surface Φ orthogonal to the hydrostatic axis


Model of Haythornthwaite


Model of Sayir II


Bicubic model Φ6


Geometric–mechanical model


Model of Mohr–Coulomb


Normal stress hypothesis


Model of Schmidt-Ishlinsky


Single stress theory of Yu or MC


Twin stress theory of Yu


Unified yield criterion of Yu


Unified strength theory of Yu


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.German Institute for Polymers (DKI)DarmstadtGermany
  2. 2.Xi’an Jiaotong University, School of AerospaceXi’anPeople’s Republic of China
  3. 3.Otto-von-Guericke-Universität MagdeburgMagdeburgGermany

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