Hill Anisotropic Yield/Creep

Hill anisotropic yield and/or creep:

can be used for materials that exhibit different yield and/or creep behavior in different directions;
is introduced through user-defined stress ratios that are applied in the quadratic Hill's potential function;
can be used only in conjunction with the metal plasticity and, in Abaqus/Standard, the metal creep material models;
can be used in conjunction with the extended Drucker-Prager, critical state (clay) plasticity, and crushable foam plasticity models;
is available for the nonlinear isotropic/kinematic hardening model in Abaqus/Explicit (Models for Metals Subjected to Cyclic Loading); and
can be used in conjunction with the models of progressive damage and failure in Abaqus/Explicit (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:

Yield and Creep Stress Ratios
Anisotropic Yield
Anisotropic Creep
Defining Anisotropic Yield Behavior on the Basis of Strain Ratios (Lankford's r-Values)
Progressive Damage and Failure
Initial Conditions
Elements
Output

Yield and Creep Stress Ratios

Anisotropic yield or creep behavior using quadratic Hill's potential is modeled through the use of yield or creep stress ratios, $R_{i j}$ . In the case of anisotropic yield the yield ratios are defined with respect to a reference yield stress, $σ^{0}$ (given for the metal plasticity definition), such that if $σ_{i j}$ is applied as the only nonzero stress, the corresponding yield stress is $R_{i j} σ^{0}$ . The plastic flow rule is defined below.

In the case of anisotropic creep the $R_{i j}$ are creep ratios used to scale the stress value when the creep strain rate is calculated. Thus, if $σ_{11}$ is the only nonzero stress, the equivalent stress, $\tilde{q}$ , used in the user-defined creep law is $\tilde{q} = R_{11} | σ_{11} |$ .

Yield and creep stress ratios 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).

Input File Usage

Use the following option to define anisotropic yield or creep:

POTENTIAL, TYPE=HILL (default)

This option must appear immediately after the PLASTIC or the CREEP material option data to which it applies. Thus, if anisotropic metal plasticity and anisotropic creep behavior are both required, the POTENTIAL option must appear twice in the material definition, once after the metal plasticity data and again after the creep data.

Abaqus/CAE Usage

Use one of the following models:

Property module: material editor: 
MechanicalPlasticityPlastic: SuboptionsPotential
MechanicalPlasticityCreep: SuboptionsPotential

Anisotropic Yield

Hill's potential function is a simple extension of the Mises function, which can be expressed in terms of rectangular Cartesian stress components as

f (σ) = \sqrt{F {(σ_{22} - σ_{33})}^{2} + G {(σ_{33} - σ_{11})}^{2} + H {(σ_{11} - σ_{22})}^{2} + 2 L σ_{23}^{2} + 2 M σ_{31}^{2} + 2 N σ_{12}^{2}},

where $F, G, H, L, M,$ and N are constants obtained by tests of the material in different orientations. They are defined as

F = \frac{{(σ^{0})}^{2}}{2} (\frac{1}{{\bar{σ}}_{22}^{2}} + \frac{1}{{\bar{σ}}_{33}^{2}} - \frac{1}{{\bar{σ}}_{11}^{2}}) = \frac{1}{2} (\frac{1}{R_{22}^{2}} + \frac{1}{R_{33}^{2}} - \frac{1}{R_{11}^{2}}),

G = \frac{{(σ^{0})}^{2}}{2} (\frac{1}{{\bar{σ}}_{33}^{2}} + \frac{1}{{\bar{σ}}_{11}^{2}} - \frac{1}{{\bar{σ}}_{22}^{2}}) = \frac{1}{2} (\frac{1}{R_{33}^{2}} + \frac{1}{R_{11}^{2}} - \frac{1}{R_{22}^{2}}),

H = \frac{{(σ^{0})}^{2}}{2} (\frac{1}{{\bar{σ}}_{11}^{2}} + \frac{1}{{\bar{σ}}_{22}^{2}} - \frac{1}{{\bar{σ}}_{33}^{2}}) = \frac{1}{2} (\frac{1}{R_{11}^{2}} + \frac{1}{R_{22}^{2}} - \frac{1}{R_{33}^{2}}),

L = \frac{3}{2} {(\frac{τ^{0}}{{\bar{σ}}_{23}})}^{2} = \frac{3}{2 R_{23}^{2}},

M = \frac{3}{2} {(\frac{τ^{0}}{{\bar{σ}}_{13}})}^{2} = \frac{3}{2 R_{13}^{2}},

N = \frac{3}{2} {(\frac{τ^{0}}{{\bar{σ}}_{12}})}^{2} = \frac{3}{2 R_{12}^{2}},

where each ${\bar{σ}}_{i j}$ is the measured yield stress value when $σ_{i j}$ is applied as the only nonzero stress component; $σ^{0}$ is the user-defined reference yield stress specified for the metal plasticity definition; $R_{11}$ , $R_{22}$ , $R_{33}$ , $R_{12}$ , $R_{13}$ , and $R_{23}$ are anisotropic yield stress ratios; and $τ^{0} = σ^{0} / \sqrt{3}$ . Therefore, the six yield stress ratios are defined as follows (in the order in which you must provide them):

\frac{{\bar{σ}}_{11}}{σ^{0}}, \frac{{\bar{σ}}_{22}}{σ^{0}}, \frac{{\bar{σ}}_{33}}{σ^{0}}, \frac{{\bar{σ}}_{12}}{τ^{0}}, \frac{{\bar{σ}}_{13}}{τ^{0}}, \frac{{\bar{σ}}_{23}}{τ^{0}} .

Because of the form of the yield function, all of these ratios must be positive. If the constants F, G, and H are positive, the yield function is always well defined. However, if one or more of these constants is negative, the yield function might be undefined for some stress states because the quantity under the square root is negative.

The flow rule is

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

where, from the definition of f above,

b = [\begin{matrix} - G (σ_{33} - σ_{11}) + H (σ_{11} - σ_{22}) \\ F (σ_{22} - σ_{33}) - H (σ_{11} - σ_{22}) \\ - F (σ_{22} - σ_{33}) + G (σ_{33} - σ_{11}) \\ 2 N σ_{12} \\ 2 M σ_{31} \\ 2 L σ_{23} \end{matrix}] .

Input File Usage

Use both of the following options:

PLASTIC
POTENTIAL, TYPE=HILL (default)

Abaqus/CAE Usage

Property module: material editor: MechanicalPlasticityPlastic: SuboptionsPotential

Anisotropic Creep

For anisotropic creep in Abaqus/Standard Hill's function can be expressed as

\tilde{q} (σ) = \sqrt{F {(σ_{22} - σ_{33})}^{2} + G {(σ_{33} - σ_{11})}^{2} + H {(σ_{11} - σ_{22})}^{2} + 2 L σ_{23}^{2} + 2 M σ_{31}^{2} + 2 N σ_{12}^{2}},

where $\tilde{q} (σ)$ is the equivalent stress and F, G, H, L, M, and N are constants obtained by tests of the material in different orientations. The constants are defined with the same general relations as those used for anisotropic yield (above); however, the reference yield stress, $σ^{0}$ , is replaced by the uniaxial equivalent deviatoric stress, $\tilde{q}$ (found in the creep law), and $R_{11}$ , $R_{22}$ , $R_{33}$ , $R_{12}$ , $R_{13}$ , and $R_{23}$ are referred to as “anisotropic creep stress ratios.” The six creep stress ratios are, therefore, defined as follows (in the order in which they must be provided):

\frac{σ_{11}}{\tilde{q}}, \frac{σ_{22}}{\tilde{q}}, \frac{σ_{33}}{\tilde{q}}, \frac{σ_{12}}{\tilde{q} / \sqrt{3}}, \frac{σ_{13}}{\tilde{q} / \sqrt{3}}, \frac{σ_{23}}{\tilde{q} / \sqrt{3}} .

You must define the ratios $R_{i j}$ in each direction that will be used to scale the stress value when the creep strain rate is calculated. If all six $R_{i j}$ values are set to unity, isotropic creep is obtained.

Input File Usage

Use both of the following options:

CREEP
POTENTIAL

Abaqus/CAE Usage

Property module: material editor: MechanicalPlasticityCreep: SuboptionsPotential

Defining Anisotropic Yield Behavior on the Basis of Strain Ratios (Lankford's r-Values)

As discussed above, Hill's anisotropic plasticity potential is defined in Abaqus from user input consisting of ratios of yield stress in different directions with respect to a reference stress. However, in some cases, such as sheet metal forming applications, it is common to find the anisotropic material data given in terms of ratios of width strain to thickness strain. Mathematical relationships are then necessary to convert the strain ratios to stress ratios that can be input into Abaqus.

In sheet metal forming applications, we are generally concerned with plane stress conditions. Consider $x, y$ to be the “rolling” and “cross” directions in the plane of the sheet; z is the thickness direction. From a design viewpoint, the type of anisotropy usually desired is that in which the sheet is isotropic in the plane and has an increased strength in the thickness direction, which is normally referred to as transverse anisotropy. Another type of anisotropy is characterized by different strengths in different directions in the plane of the sheet, which is called planar anisotropy.

In a simple tension test performed in the x-direction in the plane of the sheet, the flow rule for this potential (given above) defines the incremental strain ratios (assuming small elastic strains) as

d ε_{11} : d ε_{22} : d ε_{33} = G + H : - H : - G .

Therefore, the ratio of width to thickness strain, often referred to as Lankford's r-value, is

r_{x} = \frac{d ε_{22}}{d ε_{33}} = \frac{H}{G} .

Similarly, for a simple tension test performed in the y-direction in the plane of the sheet, the incremental strain ratios are

d ε_{11} : d ε_{22} : d ε_{33} = - H : F + H : - F,

and

r_{y} = \frac{d ε_{11}}{d ε_{33}} = \frac{H}{F} .

Transverse Anisotropy

A transversely anisotropic material is one where $r_{x} = r_{y}$ . If we define $σ^{0}$ in the metal plasticity model to be equal to ${\bar{σ}}_{11}$ ,

R_{11} = R_{22} = 1

and, using the relationships above,

R_{33} = \sqrt{\frac{r_{x} + 1}{2}} .

If $r_{x} = 1$ (isotropic material), $R_{33} = 1$ and the Mises isotropic plasticity model is recovered.

Planar Anisotropy

In the case of planar anisotropy $r_{x}$ and $r_{y}$ are different and $R_{11}, R_{22}, R_{33}$ will all be different. If we define $σ^{0}$ in the metal plasticity model to be equal to ${\bar{σ}}_{11}$ ,

R_{11} = 1

and, using the relationships above, we obtain

R_{22} = \sqrt{\frac{r_{y} (r_{x} + 1)}{r_{x} (r_{y} + 1)}}, R_{33} = \sqrt{\frac{r_{y} (r_{x} + 1)}{(r_{x} + r_{y})}} .

Again, if $r_{x} = r_{y} = 1$ , $R_{22} = R_{33} = 1$ and the Mises isotropic plasticity model is recovered.

General Anisotropy

Thus far, we have only considered loading applied along the axes of anisotropy. To derive a more general anisotropic model in plane stress, the sheet must be loaded in one other direction in its plane. Suppose we perform a simple tension test at an angle $α$ to the x-direction; then, from equilibrium considerations we can write the nonzero stress components as

σ_{11} = σ \cos^{2} α, σ_{22} = σ \sin^{2} α, σ_{12} = σ \sin α \cos α,

where $σ$ is the applied tensile stress. Substituting these values in the flow equations and assuming small elastic strains yields

d ε_{11} = [(G + H) \cos^{2} α - H \sin^{2} α] \frac{σ}{f} d λ,

d ε_{22} = [(F + H) \sin^{2} α - H \cos^{2} α] \frac{σ}{f} d λ,

d ε_{33} = - [F \sin^{2} α + G \cos^{2} α] \frac{σ}{f} d λ, and

d γ_{12} = [2 N \sin α \cos α] \frac{σ}{f} d λ .

Assuming small geometrical changes, the width strain increment (the increment of strain at right angles to the direction of loading, $α$ ) is written as

d ε_{α + \frac{π}{2}} = d ε_{11} \sin^{2} α + d ε_{22} \cos^{2} α - d γ_{12} \sin α \cos α,

and Lankford's r-value for loading at an angle $α$ is

r_{α} = \frac{d ε_{α + \frac{π}{2}}}{d ε_{33}} = \frac{H + (2 N - F - G - 4 H) \sin^{2} α \cos^{2} α}{F \sin^{2} α + G \cos^{2} α} .

One of the more commonly performed tests is that in which the loading direction is at 45°. In this case

r_{45} = \frac{2 N - (F + G)}{2 (F + G)} or \frac{N}{G} = (r_{45} + \frac{1}{2}) (1 + \frac{r_{x}}{r_{y}}) .

If $σ^{0}$ is equal to ${\bar{σ}}_{11}$ in the metal plasticity model, $R_{11} = 1$ . $R_{22}, R_{33}$ are as defined before for transverse or planar anisotropy and, using the relationships above,

R_{12} = \sqrt{\frac{3 (r_{x} + 1) r_{y}}{(2 r_{45} + 1) (r_{x} + r_{y})}} .

Progressive Damage and Failure

In Abaqus/Explicit Hill anisotropic yield can be used in conjunction with the models of progressive damage and failure 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.

Input File Usage

Use the following options:

PLASTIC
DAMAGE INITIATION
DAMAGE EVOLUTION

Abaqus/CAE Usage

Property module: material editor: MechanicalDamage for Ductile Metalsdamage initiation type: specify the damage initiation criterion: SuboptionsDamage Evolution: specify the damage evolution parameters

Initial Conditions

When we need to study the behavior of a material that has already been subjected to some work hardening, Abaqus allows you to prescribe initial conditions for the equivalent plastic strain, ${\bar{ε}}^{p l}$ , by specifying the conditions directly (Initial Conditions).

Input File Usage

INITIAL CONDITIONS, TYPE=HARDENING

Abaqus/CAE Usage

Load module: Create Predefined Field: Step: Initial, choose Mechanical for the Category and Hardening for the Types for Selected Step

User Subroutine Specification in Abaqus/Standard

For more complicated cases, initial conditions can be defined in Abaqus/Standard through user subroutine HARDINI.

Input File Usage

INITIAL CONDITIONS, TYPE=HARDENING, USER

Abaqus/CAE Usage

Load module: Create Predefined Field: Step: Initial, choose Mechanical for the Category and Hardening for the Types for Selected Step; Definition: User-defined

Elements

You can define Hill anisotropic yield for any element that can be used with the following models:

Classical metal plasticity models in Abaqus (Classical Metal Plasticity)
Critical state (clay) plasticity model (Critical State (Clay) Plasticity Model)
Extended Drucker-Prager material models (Extended Drucker-Prager Models)
Crushable foam plasticity models (Crushable Foam Plasticity Models)

You cannot define Hill anisotropic yield for one-dimensional elements in Abaqus/Explicit (beams and trusses). In Abaqus/Standard it can also be defined for any element that can be used with the linear kinematic hardening plasticity model (Models for Metals Subjected to Cyclic Loading) but not with the nonlinear isotropic/kinematic hardening model. Likewise, anisotropic creep with Hill's function can be defined for any element that can be used with the classical metal creep model in Abaqus/Standard (Rate-Dependent Plasticity: Creep and Swelling).

Output

The standard output identifiers available in Abaqus (Abaqus/Standard Output Variable Identifiers and Abaqus/Explicit Output Variable Identifiers) and all output variables associated with the the following models are available when you define anisotropic yield and creep:

Creep model (Rate-Dependent Plasticity: Creep and Swelling)
Classical metal plasticity models (Classical Metal Plasticity)
Critical state (clay) plasticity model (Critical State (Clay) Plasticity Model)
Extended Drucker-Prager material models (Extended Drucker-Prager Models)
Crushable foam plasticity models (Crushable Foam Plasticity Models)
Linear kinematic hardening plasticity model (Models for Metals Subjected to Cyclic Loading)

The following variables have special meaning if anisotropic yield and creep are defined:

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).
CEEQ: Equivalent creep strain, ${\bar{ε}}^{c r} = \int_{0}^{t} {\dot{\bar{ε}}}^{c r} d t = \int_{0}^{t} \frac{σ : {\dot{ε}}^{c r} d t}{σ^{0}} .$
YIELDS: Yield stress, $σ^{0}$ .
YIELDPOT: Yield potential, $f (σ)$ (Abaqus/Explicit only).