Nonquadratic Yield

Nonquadratic yield surfaces in Abaqus/Explicit:

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.

This page discusses:

Nonquadratic Yield Surfaces
Barlat Plasticity
Progressive Damage and Failure
Elements
Output
References

Nonquadratic Yield Surfaces

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

f (σ) = \max (| σ_{1} - σ_{2} |, | σ_{1} - σ_{3} |, | σ_{2} - σ_{3} |) \leq σ^{0},

where $σ^{0}$ is the uniaxial yield stress and $σ_{1}$ , $σ_{2}$ , and $σ_{3}$ 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

f (σ) = {(\frac{1}{2} ({| σ_{1} - σ_{2} |}^{α} + {| σ_{1} - σ_{3} |}^{α} + {| σ_{2} - σ_{3} |}^{α}))}^{\frac{1}{α}} \leq σ^{0},

where $σ^{0}$ is the uniaxial yield stress; $σ_{1}$ , $σ_{2}$ , and $σ_{3}$ are the principal stresses; and $α$ is the exponent. When $α = 1$ or $α$ goes to infinity, the Hosford yield function reduces to the Tresca yield function. When $α = 2$ or $α = 4,$ 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, $σ^{0}$ . 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

\begin{matrix} f (σ) = (\frac{1}{4} ({| {\tilde{S}}_{1}^{'} - {\tilde{S}}_{1}^{"} |}^{α} + {| {\tilde{S}}_{1}^{'} - {\tilde{S}}_{2}^{"} |}^{α} + {| {\tilde{S}}_{1}^{'} - {\tilde{S}}_{3}^{"} |}^{α} + {| {\tilde{S}}_{2}^{'} - {\tilde{S}}_{1}^{"} |}^{α} + \\ {\begin{matrix} {| {\tilde{S}}_{2}^{'} - {\tilde{S}}_{2}^{"} |}^{α} + {| {\tilde{S}}_{2}^{'} - {\tilde{S}}_{3}^{"} |}^{α} + {| {\tilde{S}}_{3}^{'} - {\tilde{S}}_{1}^{"} |}^{α} + {| {\tilde{S}}_{3}^{'} - {\tilde{S}}_{2}^{"} |}^{α} + {| {\tilde{S}}_{3}^{'} - {\tilde{S}}_{3}^{"} |}^{α})) \end{matrix}}^{\frac{1}{α}} \end{matrix},

where $α$ is the exponent and ${\tilde{S}}_{i}^{'}$ , ${\tilde{S}}_{i}^{"}$ ( $i$ =1, 2, 3) are principal values of stress tensors ${\tilde{S}}^{'}$ and ${\tilde{S}}^{"}$ , respectively.

The principal values of a stress tensor $\tilde{S}$ ( ${\tilde{S}}^{'}$ or ${\tilde{S}}^{"}$ ) are the roots of the characteristic equation

- {\tilde{S}}_{k}^{3} + {\tilde{I}}_{1} {\tilde{S}}_{k}^{2} - {\tilde{I}}_{2} {\tilde{S}}_{k} + {\tilde{I}}_{3} = 0,

where the first, second, and third stress invariants are

{\tilde{I}}_{1} = {\tilde{S}}_{11} + {\tilde{S}}_{22} + {\tilde{S}}_{33},

{\tilde{I}}_{2} = {\tilde{S}}_{11} {\tilde{S}}_{22} + {\tilde{S}}_{11} {\tilde{S}}_{33} + {\tilde{S}}_{22} {\tilde{S}}_{33} - {\tilde{S}}_{12}^{2} - {\tilde{S}}_{13}^{2} - {\tilde{S}}_{23}^{2},

{\tilde{I}}_{3} = {\tilde{S}}_{11} {\tilde{S}}_{22} {\tilde{S}}_{33} + 2 {\tilde{S}}_{12} {\tilde{S}}_{13} {\tilde{S}}_{23} - {\tilde{S}}_{12}^{2} {\tilde{S}}_{33} - {\tilde{S}}_{13}^{2} {\tilde{S}}_{22} - {\tilde{S}}_{23}^{2} {\tilde{S}}_{11} .

The principal values are

{\tilde{S}}_{1} = \frac{2}{3} \tilde{q} \cos (\tilde{Θ}) - \tilde{p},

{\tilde{S}}_{2} = \frac{2}{3} \tilde{q} \cos (\tilde{Θ} - \frac{2 π}{3}) - \tilde{p},

{\tilde{S}}_{3} = \frac{2}{3} \tilde{q} \cos (\tilde{Θ} - \frac{4 π}{3}) - \tilde{p},

where the deviatoric polar angle is

\tilde{Θ} = \frac{1}{3} \arccos ({(\frac{\tilde{r}}{\tilde{q}})}^{3}),

and the stress invarients are

\tilde{p} = - \frac{1}{3} {\tilde{I}}_{1},

\tilde{q} = \sqrt{{\tilde{I}}_{1}^{2} - 3 {\tilde{I}}_{2}},

\tilde{r} = 3 {(\frac{1}{27} {\tilde{I}}_{1}^{3} - \frac{1}{6} {\tilde{I}}_{1} {\tilde{I}}_{2} + \frac{1}{2} {\tilde{I}}_{3})}^{\frac{1}{3}} .

The tensors ${\tilde{S}}^{'}$ and ${\tilde{S}}^{"}$ are defined by two linear transformations of the deviatoric stress $S$ ,

{\tilde{s}}^{'} = C^{'} S = C^{'} T σ \equiv L^{'} σ,

{\tilde{s}}^{"} = C^{"} S = C^{"} T σ \equiv L^{"} σ,

where the tensors $C^{'}$ and $C^{"}$ are transformation tensors contains anisotropy coefficients and the tensor $T$ transforms the Caushy stress, $σ$ , to the deviatoric stress. The transformations can be expressed in matrix form:

\begin{matrix} {\begin{matrix} {\tilde{S}}_{11}^{'} \\ {\tilde{S}}_{22}^{'} \\ {\tilde{S}}_{33}^{'} \\ {\tilde{S}}_{12}^{'} \\ {\tilde{S}}_{13}^{'} \\ {\tilde{S}}_{23}^{'} \end{matrix}} = [\begin{matrix} 0 & - c_{1122}^{'} & - c_{1133}^{'} & 0 & 0 & 0 \\ - c_{2211}^{'} & 0 & - c_{2233}^{'} & 0 & 0 & 0 \\ - c_{3311}^{'} & - c_{3322}^{'} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 c_{1212}^{'} & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 c_{1313}^{'} & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 c_{2323}^{'} \end{matrix}] {\begin{matrix} S_{11} \\ S_{22} \\ S_{33} \\ S_{12} \\ S_{13} \\ S_{23} \end{matrix}} = \\ \frac{1}{3} [\begin{matrix} c_{1122}^{'} + c_{1133}^{'} & - 2 c_{1122}^{'} + c_{1133}^{'} & {c_{1122}^{'} - 2 c}_{1133}^{'} & 0 & 0 & 0 \\ {- 2 c}_{2211}^{'} + c_{2233}^{'} & c_{2211}^{'} + c_{2233}^{'} & c_{2211}^{'} - 2 c_{2233}^{'} & 0 & 0 & 0 \\ - 2 c_{3311}^{'} + c_{3322}^{'} & c_{3311}^{'} - 2 c_{3322}^{'} & c_{3311}^{'} + c_{3322}^{'} & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 c_{1212}^{'} & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 c_{1313}^{'} & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 c_{2323}^{'} \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}}, \end{matrix}

and

\begin{matrix} {\begin{matrix} {\tilde{S}}_{11}^{"} \\ {\tilde{S}}_{22}^{"} \\ {\tilde{S}}_{33}^{"} \\ {\tilde{S}}_{12}^{"} \\ {\tilde{S}}_{13}^{"} \\ {\tilde{S}}_{23}^{"} \end{matrix}} = [\begin{matrix} 0 & - c_{1122}^{"} & - c_{1133}^{"} & 0 & 0 & 0 \\ - c_{2211}^{"} & 0 & - c_{2233}^{"} & 0 & 0 & 0 \\ - c_{3311}^{"} & - c_{3322}^{"} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 c_{1212}^{"} & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 c_{1313}^{"} & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 c_{2323}^{"} \end{matrix}] {\begin{matrix} S_{11} \\ S_{22} \\ S_{33} \\ S_{12} \\ S_{13} \\ S_{23} \end{matrix}} = \\ \frac{1}{3} [\begin{matrix} c_{1122}^{"} + c_{1133}^{"} & - 2 c_{1122}^{"} + c_{1133}^{"} & {c_{1122}^{"} - 2 c}_{1133}^{"} & 0 & 0 & 0 \\ {- 2 c}_{2211}^{"} + c_{2233}^{"} & c_{2211}^{"} + c_{2233}^{"} & c_{2211}^{"} - 2 c_{2233}^{"} & 0 & 0 & 0 \\ - 2 c_{3311}^{"} + c_{3322}^{"} & c_{3311}^{"} - 2 c_{3322}^{"} & c_{3311}^{"} + c_{3322}^{"} & 0 & 0 & 0 \\ 0 & 0 & 0 & 6 c_{1212}^{"} & 0 & 0 \\ 0 & 0 & 0 & 0 & 6 c_{1313}^{"} & 0 \\ 0 & 0 & 0 & 0 & 0 & 6 c_{2323}^{"} \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} \end{matrix}

The 18 anisotropy coefficients $c_{i j k l}^{'}$ , $c_{i j k l}^{"}$ can be calibrated from experiments.

If $c_{i i j j}^{'} = c_{i i j j}^{"} = 1$ and $c_{i j i j}^{'} = c_{i j i j}^{"} = 0.5$ , the yield function reduces to the Hosford isotropic yield function. In addition, when $α = 2$ or $α = 4$ , the Mises isotropic plasticity model is recovered.

The flow rule is

d ε^{p l} = d λ \frac{\partial f}{\partial σ},

where, from the definition of $f$ above,

\frac{\partial f}{\partial σ_{i j}} = \frac{\partial f}{\partial {\tilde{S}}_{p}^{'}} \frac{\partial {\tilde{S}}_{p}^{'}}{\partial {\tilde{I}}_{q}^{'}} \frac{\partial {\tilde{I}}_{q}^{'}}{\partial {\tilde{S}}_{r s}^{'}} L_{r s i j}^{'} + \frac{\partial f}{\partial {\tilde{S}}_{p}^{"}} \frac{\partial {\tilde{S}}_{p}^{"}}{\partial {\tilde{I}}_{q}^{"}} \frac{\partial {\tilde{I}}_{q}^{"}}{\partial {\tilde{S}}_{r s}^{"}} L_{r s i j}^{"} .

Barlat Plasticity (Yld91)

When the Barlat plasticity considers only one linear transformation (that is, ${\tilde{S}}^{'} = {\tilde{S}}^{"} = \tilde{S}$ ), the yield surface reduces to the Barlat Yld91 yield surface (Barlat et al., 1991) as

f (σ) = {(\frac{1}{2} ({| {\tilde{S}}_{1} - {\tilde{S}}_{2} |}^{α} + {| {\tilde{S}}_{1} - {\tilde{S}}_{3} |}^{α} + {| {\tilde{S}}_{2} - {\tilde{S}}_{3} |}^{α})}^{\frac{1}{α}},

where $α$ is the exponent; and ${\tilde{S}}_{1}$ , ${\tilde{S}}_{2}$ , and ${\tilde{S}}_{3}$ are principal values of the stress tensor $\tilde{S}$ .

The linear transformation is defined as

\tilde{S} = L σ,

or expressed in matrix form,

{\begin{matrix} S_{11} \\ S_{22} \\ S_{33} \\ S_{12} \\ S_{13} \\ {\begin{matrix} S \end{matrix}}_{23} \end{matrix}} = \frac{1}{3} [\begin{matrix} \bar{b} + \bar{c} & - \bar{c} & - \bar{b} & 0 & 0 & 0 \\ - \bar{c} & \bar{c} + \bar{a} & - \bar{a} & 0 & 0 & 0 \\ - \bar{b} & - \bar{a} & \bar{a} + \bar{b} & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 \bar{h} & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 \bar{g} & 0 \\ 0 & 0 & 0 & 0 & 0 & 3 \bar{f} \end{matrix}] {\begin{matrix} σ_{11} \\ σ_{22} \\ σ_{33} \\ σ_{12} \\ σ_{13} \\ σ_{23} \end{matrix}} .

The six anisotropy coefficients $\bar{a}$ , $\bar{b}$ , $\bar{c}$ , $\bar{f}$ , $\bar{g}$ , and $\bar{h}$ 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 $α = 2$ or $α = 4$ , 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, ${\bar{ε}}^{p l} = {{\bar{ε}}^{p l} |}_{0} + \int_{0}^{t} {\dot{\bar{ε}}}^{p l} d t = {{\bar{ε}}^{p l} |}_{0} + \int_{0}^{t} \frac{σ : {\dot{ε}}^{p l} d t}{σ^{0}},$ where ${{\bar{ε}}^{p l} |}_{0}$ is the initial equivalent plastic strain (zero or user-specified; see Initial Conditions).
YIELDS: Yield stress, $σ^{0}$ .
YIELDPOT: Yield potential, $f (σ)$ .

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.