are available in the form of Tresca or Hosford yield surfaces with
associated plastic flow to model isotropic yield or in the form of Barlat yield
surfaces with associated plastic flow to model anisotropic yield;
are introduced through user-defined exponents and coefficients of
nonquadratic stress potential functions;
can be used with perfect plasticity or isotropic hardening behavior;
and
can be used in conjunction with progressive damage and failure models
(About Damage and Failure for Ductile Metals)
to specify different damage initiation criteria and damage evolution laws that
allow for the progressive degradation of the material stiffness and the removal
of elements from the mesh.
Nonquadratic yield surfaces with associated plastic flow for classical
metal plasticity are available. You can specify the Tresca or Hosford yield
surface for isotropic yield or the Barlat yield surface for anisotropic yield.
Tresca Yield Surface
The Tresca yield potential for isotropic yielding takes the form
where is the uniaxial yield stress and , , and are the principal stresses.
Hosford Yield Surface
The Hosford yield surface for isotropic yielding is a generalization of the
Mises yield surface (Mises Yield Surface).
The Hosford stress potential function is
where is the uniaxial yield stress; , , and are the principal stresses; and is the exponent. When or goes to infinity, the Hosford yield function reduces to the
Tresca yield function. When or the Hosford yield function reduces to the Mises yield
function.
Barlat Yield Surface
The Barlat yield surface allows you to model complex anisotropic yielding
behavior. The plastic flow rule is defined below. Anisotropic yield with
Barlat's potential is modeled through the use of anisotropy coefficients that
are defined with respect to a reference yield stress, . There are two Barlat yield potential functions: the
Yld2004-18p potential function that contains 18 coefficients and the Yld91
potential function that contains 6 coefficients. The anisotropy coefficients
can be defined as constants or as tabular functions of temperature and
predefined field variables. A local orientation must be used to define the
direction of anisotropy (see
Orientations).
Barlat Plasticity
Barlat's anisotropic yield potential is based on linear transformations of
stress. The stress potential function (Yld2004-18p) proposed by Barlat et al.
(2005) is
where is the exponent and , (=1, 2, 3) are principal values of stress tensors
and , respectively.
The principal values of a stress tensor
(
or )
are the roots of the characteristic equation
where the first, second, and third stress invariants are
The principal values are
where the deviatoric polar angle is
and the stress invarients are
The tensors and are defined by two linear transformations of the deviatoric
stress ,
where the tensors and are transformation tensors contains anisotropy coefficients
and the tensor transforms the Caushy stress, , to the deviatoric stress. The transformations can be
expressed in matrix form:
and
The 18 anisotropy coefficients , can be calibrated from experiments.
If
and ,
the yield function reduces to the Hosford isotropic yield function. In
addition, when
or ,
the Mises isotropic plasticity model is recovered.
The flow rule is
where, from the definition of above,
Barlat Plasticity (Yld91)
When the Barlat plasticity considers only one linear transformation (that
is, ), the yield surface reduces to the Barlat Yld91 yield
surface (Barlat et al., 1991) as
where is the exponent; and , , and are principal values of the stress tensor .
The linear transformation is defined as
or expressed in matrix form,
The six anisotropy coefficients , , , , , and can be calibrated from experiments. If all six coefficients
are set to unity, the yield function reduces to the Hosford isotropic yield
function. In addition, when or , the Mises isotropic plasticity model is recovered.
Progressive Damage and Failure
Nonquadratic yield can be used in conjunction with the progressive damage
and failure models discussed in
About Damage and Failure for Ductile Metals.
The capability allows for the specification of one or more damage initiation
criteria, including ductile, shear, forming limit diagram
(FLD), forming limit stress diagram
(FLSD), and Müschenborn-Sonne forming limit
diagram (MSFLD) criteria. After damage
initiation, the material stiffness is degraded progressively according to the
specified damage evolution response. The model offers two failure choices,
including the removal of elements from the mesh as a result of tearing or
ripping of the structure. The progressive damage models allow for a smooth
degradation of the material stiffness, making them suitable for both
quasi-static and dynamic situations.
Elements
Nonquadratic yield is available only with three-dimensional solid elements
and two-dimensional plane strain elements.
Output
In addition to the standard output identifiers available in
Abaqus/Explicit
(Abaqus/Explicit Output Variable Identifiers),
the following variables have special meaning for the classical metal plasticity
models:
PEEQ
Equivalent plastic strain,
where
is the initial equivalent plastic strain (zero or user-specified; see
Initial Conditions).
YIELDS
Yield stress, .
YIELDPOT
Yield potential, .
References
Barlat, F., H. Aretz, J. W. Yoon, M. E. Karabin, J. C. Brem, and R. E. Dick, “Linear
Transformation-based Anisotropic Yield Functions,” International Journal of
Plasticity, vol. 21, pp. 1009–1039, 2005.
Barlat, F., D. J. Lege, and J. C. Brem, “A
Six-Component Yield Function for Anisotropic Materials,” International Journal of
Plasticity, vol. 7, pp. 693–712, 1991.