Archive of Applied Mechanics

, Volume 85, Issue 4, pp 455–468 | Cite as

On the application of the continuum damage mechanics to multi-axial low-cyclic damage

Original

Abstract

The paper is devoted to continuum damage mechanics with applications to low-cyclic multi-axial loadings. The development of this solid mechanics branch started more than 50 years ago with the pioneering works of Miner, Palmgren, Kachanov and Rabotnov among others. The investigations in this field are supported by results derived in material science and physics of materials and have great practical importance since cyclic loading is a typical loading case in mechanical engineering. The reason for this is the well-known fact that damage and fracture occur under cyclic loading conditions even if the values of the applied stresses are much lower in comparison with the limit stresses estimated for the static case. Three problems for metallic materials will be discussed below. The first one is related to the experimental investigation of the damage behavior. The main open question is what kind of tests is suitable and result in acceptable data sets. The second problem is connected with the modeling of the damage processes in structural materials. The last one is what kind of conclusions can be drawn structural safety. The present paper gives an overview concerning the contributions of Russian and Ukrainian scientists in this field. A review of various approaches to the damage description at static and cyclic loading is presented.

Keywords

CDM Low-cyclic loading Damage 

List of symbols

D

Damage parameter

DR

Critical value of the damage parameter

D1R

Critical value of the damage parameter for uniaxial tension

\({\tilde{A}}\)

Actual cross-sectional area

A

Initial cross-sectional area

\({\tilde{\sigma}}\)

Effective normal stress (tension)

\({\tilde{\tau}}\)

Effective shear stress (torsion)

\({\tilde{\sigma}_{ij}}\)

Components of the effective stress tensor

\({\sigma^{\rm tr}}\)

True stress

Mijkl

Fourth-rank tensor mapping the stress tensor

Dij

Damage tensor of rank two

\({D_\sigma}\)

Scalar damage parameter at tension

\({D_\tau}\)

Scalar damage parameter at torsion

\({\Omega}\)

Free energy

\({\Omega_{\rm el}}\)

Elastic part of the free energy

\({\Omega_{\rm pl}}\)

Plastic part of the free energy

\({\varepsilon^{\rm el}_{\rm eq}}\)

Function of the invariants of the elastic strain

\({\varepsilon^{\rm pl}_{\rm eq}}\)

Function of the invariants of the plastic strain

Y

Energy density release rate of the damage material

\({\nu}\)

Poisson’s coefficient

\({\sigma_0}\)

Mean stress

\({\chi, B, C, a, m}\)

Material parameters

FD

Damage potential

\({\sigma_{\rm eq}}\)

Equivalent stress

K

Parameter of the loading stiffness

\({\varepsilon^{\rm pl}_{\rm eqD}}\)

Value of the plastic von Mises strain at the moment of the damage initiation[1mm]

\({\varepsilon^{\rm pl}_{\rm eqR}}\)

Limit value of the plastic von Mises strain[1mm]

\({\varepsilon^{\rm pl}_{\rm D}}\)

Threshold of the plastic strain at damage initiation under tension[1mm]

\({\gamma^{\rm pl}_{\rm D}}\)

Threshold of the plastic strain at damage initiation under torsion[1mm]

\({\varepsilon^{\rm pl}_{\rm R}}\)

Limit plastic strain at macro-crack initiation at tension[1mm]

\({\gamma^{\rm pl}_{\rm R}}\)

Limit plastic strain at macro-crack initiation at torsion[1mm]

R

Triaxiality function

\({\delta_{ij}}\)

Kronecker delta

N

Number of cycles

\({\Omega_{\rm D}}\)

Specific energy of the damaged material

\({\Omega^\Sigma_{\rm D}}\)

Total energy of the damaged material

\({\Psi_{\rm D}}\)

Specific energy of the additional stresses

\({\sigma_x}\)

Additional tensile stress

\({\tau_x}\)

Additional torsional stress

h

Micro-defect closure parameter

\({\delta^{(+)}_\sigma}\)

Width of the plastic hysteresis loop during a half-cycle at tension

\({\delta_\tau}\)

Width of the plastic hysteresis loop during a half-cycle at torsion

k

Number of half-cycle

\({\sigma^{(+)}_{\rm pr}}\)

Proportionality limit at tension

\({\tau^{(+)}_{\rm pr}}\)

Proportionality limit at torsion

\({\sigma^{(+k)}_{\rm pr}}\)

Proportionality limit at tension for kth half-cycle

\({\sigma^{(-k)}_{\rm pr}}\)

Proportionality limit at compression for kth half-cycle

\({\rho_\sigma(\varepsilon^{\rm pl})}\)

Function of the additional stress at tension

\({\rho_\tau(\gamma^{\rm pl})}\)

Function of the additional stress at torsion

\({\Psi^\sigma_{\rm st}}\)

Specific energy of the additional stress at static loading by an axial force

\({\Psi^\tau_{\rm st}}\)

Specific energy of the additional stress at static loading by a torque

\({\Psi^\sigma_{\rm cycle}}\)

Specific energy of the additional stress during low-cyclic fatigue at tension

\({\Psi^\tau_{\rm cycle}}\)

Specific energy of the additional stress during low-cyclic fatigue at torsion

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Mykola Bobyr
    • 1
  • Holm Altenbach
    • 2
  • Oleksandr Khalimon
    • 1
  1. 1.Institute of Mechanical EngineeringNTUU “Kyiv Polytechnic Institute”KievUkraine
  2. 2.Lehrstuhl für Technische Mechanik, Institut für Machanik, Fakultät für MaschinenbauOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

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