Hill Anisotropic Yield/Creep

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

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

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).

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

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

where each ${\overline{\sigma }}_{ij}$ is the measured yield stress value when ${\sigma }_{ij}$ is applied as the only nonzero stress component; ${\sigma }^0$ 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 ${\tau }^0={\sigma }^0/\sqrt{3}$ . Therefore, the six yield stress ratios are defined as follows (in the order in which you must provide them):

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

where, from the definition of f above,

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

where $\tilde{q}\left(\mathit{\sigma }\right)$ 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, ${\sigma }^0$, 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):

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

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

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

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

and

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.