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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
Original

Abstract

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.

Keywords

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

List of symbols and abbreviations

Φ

Model of isotropic material behavior

I1

First invariant of the stress tensors

I2′, I3

Second and third invariants of the stress deviator

θ

Stress angle

σeq

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

\({\dot\lambda}\)

Lagrange multiplier

σijk

Stress gradient

R

Structural parameter

dkbZbDiZuD

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

\({\xi_1,\,\xi_2,\,\xi_3}\)

Cartesian co-ordinates

jlm

Powers of the terms in the compressible substitution

bd

Parameters of the UST

ψ

Inclination of the meridian with respect to the hydrostatic axis

ZBD

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

DBZ

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

KIZUD

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

AZAD

Points of hydrostatic tension and compression

π-plane

Cut of the surface Φ orthogonal to the hydrostatic axis

ΦHay

Model of Haythornthwaite

ΦSay

Model of Sayir II

BCM

Bicubic model Φ6

GMM

Geometric–mechanical model

MC

Model of Mohr–Coulomb

NSH

Normal stress hypothesis

SI

Model of Schmidt-Ishlinsky

SST

Single stress theory of Yu or MC

TST

Twin stress theory of Yu

UYC

Unified yield criterion of Yu

UST

Unified strength theory of Yu

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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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