Damage mechanics incorporating two back stress
kinematic hardening constitutive models q
by damage evolution in the plastic deformation behavior. Moreover, as the nonlinear strain rate description in the back
q The views expressed in this work are those of the authors and do not reflect the official policy of the United States Air Force, the
Department of Defense, or the US Government. This material is declared a work of the US Government and is not subject to copyright
protection in the United States.
* Corresponding author. Tel.: +82 42 861 2613/821 4370; fax: +82 42 821 2392.
E-mail address: sjy4541@Yahoo.com (S. Yun).
Engineering Fracture Mechanics 74 (2007) 2844–2863
www.elsevier.com/locate/engfracmech0013-7944/$ - see front matter  2007 Elsevier Ltd. All rights reserved.stress evolution becomes dominant, the plastic strain localization becomes intensified as well as damage. It is also possible
to describe a wide range of plastic deformation and damage behavior by selecting a simple combination of two back stress
evolution rules.
 2007 Elsevier Ltd. All rights reserved.
Keywords: Finite deformation; Damage; Two-back stress; Kinematic hardeningS. Yun a,*, A. Palazotto b
a Tech-4-4, Agency for Defense Development, Yuseong-ku, P.O. Box 35, Daejon 305-600, Republic of Korea
b Department of Aeronautics and Astronautics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA
Received 11 September 2006; received in revised form 20 December 2006; accepted 26 December 2006
Available online 17 January 2007
Abstract
In the present work, damage mechanics is incorporated into the plastic constitutive relation in order to describe the
localized plastic deformation and damage experienced in a tensile test of the high strength VascoMax 300 steel specimen.
The results obtained on the VascoMax 300 steel are also compared with 1080 steel. In order to identify the effect of damage
on the shear band formation, one-dimensional simple shear analysis is carried out with non-isothermal visco-plasticity
using a finite difference modeling assuming isotropic hardening. The results depicts that the damage evolution within a
work piece accelerates the plastic deformation localization and the temperature variation plays less significant role in a
shear band formation compared with damage mechanics. A coupling between damage mechanics and isothermal rate inde-
pendent plasticity is proposed using the kinematic hardening constitutive relation which in turn is formulated by combin-
ing the nonlinear Armstrong–Frederick rule and the linear Phillips rule. The effects of the various hardening parameters on
plastic deformation localization are investigated within the J2 deformation theory. The material with a lower hardening
exponent results in a rapid shear band formation, and the results from the numerical analysis reflected closely with the
micro-structures around the fractured regime. However, the material with a higher hardening exponent is less affecteddoi:10.1016/j.engfracmech.2006.12.032

Nomenclature
A surface area
Cijkl constant of plasticity
Dij rate of deformation
E modulus of elasticity
Es effective secant modulus
f fractional factor
K modulus of plasticity
Kt modulus in tension
Lijkl constant of elasticity
sij stress deviator
T temperature
Wij continuum spin
a damage exponent
aij back stress tensor
au ultimate back stress
b 1st coefficient of Armstrong–Frederick rule
eeij elastic strain
epij plastic strain
epth threshold plastic strain
/ damage factor
l coefficient of Ziegler rule
q density
rij stress tensor
re equivalent stress
ry yield stress
Xij spin of sub-structure
A effective surface area
Dpij rate of plastic deformation
E Effective modulus of elasticity
Et effective tangential modulus
K thermal conductivity
K effective modulus of plasticity
Kc modulus in compression
sij effective stress deviator
vi,j velocity gradient
W pij plastic spin
aij effective back stress tensor
as kinematic variable at reverse loading
eeij effective elastic strain
epij effective plastic strain
epcr critical plastic strain
c 2nd coefficient of Armstrong–Frederick rule
m poison ratio
h dimensionless temperature
rij effective stress tensor
rH hydro static stress
ry effective yield stress
S. Yun, A. Palazotto / Engineering Fracture Mechanics 74 (2007) 2844–2863 2845