### Tech Briefs

Upset Forging of a Cylindrical Billet Using the Anand Material Model

The Anand material model (see refs.[1] and [2]) is used to model metals at high temperatures, such as solder.

The material model incorporates temperature-dependent viscoplasticity. The key feature is the use of an internal variable, the deformation resistance, which, along with the effective stress, governs the viscoplastic strain rate, given by

The viscoplastic strain rate increases as the ratio of the effective stress, , to the
deformation resistance, *s*, increases. There is no level of non-zero effective stress for which the viscoplastic
strain rate is zero.

For details about the deformation resistance and the material constants used, see refs.[1] and [2].

The following example, taken from ref.[1], demonstrates the use of the Anand model. A cylindrical billet of Al-1100-O at 400°C is subjected to upset forging, as shown in the following figure:

The force-deflection curve from ADINA is shown in the next figure, along with experimental data digitized from ref.[1].

The evolution of the effective stress and contact tractions are shown in the movie at the top of this web page.

The following figures show the deformation resistance and accumulated effective viscoplastic strain at the end of the analysis.

It is important to note that the u/p formulation should be used with the Anand material model, because the viscoplastic strains are incompressible. To demonstrate the effect of using the u/p formulation, we show the pressures and contact tractions as computed from the above analysis:

and also the pressures and contact tractions computed from an analysis in which the u/p formulation is not used:

The pressures lock and the contact tractions are poor when the u/p formulation is not used.

**References**

- G. Weber and L. Anand, "Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids", Comp. Meth. Appl. Mech. Engng., 79 (1990) 173-202.
- S. B. Brown, K. H. Kim and L. Anand, "An internal variable constitutive model for hot working of metals", Int. J. Plasticity, 5 (1989), 95-130.