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Equation of state

In some situations, an EOS is required in order to accurately simulate material behavior. An EOS determines the hydrostatic, or bulk, behavior of the material by calculating pressure as a function of density and perhaps, energy and/or temperature. Situations that call for an EOS are characterized by very high strain rates, material pressures far in excess of yield stress, and propagation of shock waves. Of course, these phenomena are very much interrelated.

*EOS_LINEAR_POLYNOMIAL or *EOS_GRUNEISEN are probably the most commonly used EOS forms for non-gaseous materials. Gruneisen parameters are available for many materials including metals.

Total stress is the sum of deviatoric stress and pressure. The mean stress (sig1 + sig2 + sig3)/3 is equal to the pressure. Constitutive models which do NOT employ an EOS calculate total stress directly. In these models, the pressure component of total stress is based only on volumetric strain. For instance, for an elastic material, p = K * mu where K is the bulk modulus and mu = rho/rho0 - 1.

Material models that require an accompanying EOS calculate only the deviatoric component of stress, i.e. the strength behavior, whereas the EOS calculates the pressure component of total stress, i.e., the hydrostatic behavior.

Note that an EOS can only be defined for a continuum element (*ELEMENT_SHELL with shell type 13, 14, or 15 or *ELEMENT_SOLID) which employs one of the *MAT_ earmarked for an EOS in the table starting on p. 6 of Vol. II of the 960 User's Manual.

If you're using a material model where an EOS is required, you can achieve simple bulk behavior by using *EOS_LINEAR_POLYNOMIAL and setting C1 to the bulk modulus and modulus and all the other C terms to zero. I would only recommend this approach if strain rates are low to moderate. Strain rates in an auto crash would qualify as moderate.

The book "High Velocity Impact Dynamics", edited by Zukas (1990, John Wiley and Sons) is a good reference on the subject of material behavior at high strain rates.

EOS parameters for approximately 50 materials are given in "Equation of State and Strength Properites of Selected Materials", Danial J. Steinberg, Lawrence Livermore National Laboratory, 1991 (Change 1 issued 1996), UCRL-MA-106439. (LLNL does not provide copies.)

Regarding *EOS_TABULATED_COMPACTION and *EOS_TABULATED : The manual isn't very specific. The notes I have indicate the following:

  • The eVi terms (abscissa of the curve) represent ln(relative volume) and thus are negative in compression.
  • eVi = ln(relative volume) values should be given in descending order, that is, tensile (positive) value first and largest compression (most negative) value last.
  • Pressure is positive in compression. If gamma = 0, Ci is equal to pressure on the loading curve. Thus Ci should have an algebraic sign opposite of eVi.

jpd