## 6.3. Notes

1. ALPHA is a material model dependent variable, which can have the following meanings for the specified material models:

   JWL: | Burn fraction |
   Porous: | Compaction, defined as current density / solid density
   Tillotson: | Current phase of material ) |
   Puff: | Current phase of material ) | See Equation of State in the Autodyn User's Manual
   Twophase: | Current phase of material ) |

2. Damage is not used in standard material models. It will only be non-zero if defined in user subroutines (for example, EXDAM).

3. An explanation and the derivation of the equations for effective plastic strain, effective plastic strain rate and effective strain are given below:

   The plane which makes equal angles with each of the principal directions is called the octahedral plane. The shear stress on this plane is given by:

   where the second invariant of the stress deviators is given by:

   where σij is the total stress tensor in the ij direction. Directions 11, 22 and 33 are the principal stress directions.

   The Von Mises yield criterion states that yielding begins when the octahedral shearing reaches a critical value defined by:

   where k is the yield stress in pure shear

   and the yield criterion is:

   Yielding will occur in a uniaxial tension test when:

   Substituting these values in the above equations gives the uniaxial yield stress as:

   In Autodyn at each cycle the stress state is checked against the yield criterion and if the yield criterion is exceeded an increment of effective plastic strain is computed as follows:

   The effective plastic strain is the integrated value of these increments during the calculation:

   The effective plastic strain rate is given by:

   and the effective strain is given by:

   where

   where εij is the total strain tensor in the ij direction which includes elastic and plastic components

4. When using the Euler and Euler Godunov processors a given cell may contain more than one material. In such a case, there is not a single value for such variables as compression and energy. In order to obtain these “multi-material” variables one has to reference the multiple material arrays. For Autodyn-2D, a mass-weighted value for compression, internal energy, temperature, and alpha is available in the “standard” array locations (for example: XMU, EN, TEMP, and ALPHA). In Autodyn-3D, the multi-material access method is used for all processors including Lag/ALE and Shells.

   Refer to the User Subroutine tutorial for further details and examples.

5. When using the Lagrange, ALE, Shell, Beam, SPH and FCT solvers, these variables are most efficiently accessed from the multi-material array structure using the direct method: For each cell, set the material variable array pointer using

   ML => MTSUB(IJK)%V(1:NUMMLV)

   Then access/set the material data using:

   ML(`index`).

   For example, to set the cell damage to one and internal energy to zero use:

       ML(NCDM) = 1.0
       ML(NCEN) = 0.0

6. To define the pressure and energy on the Hugoniot requires the material shock velocity-particle velocity relationship as material input. In Autodyn this is specified through:

   where

   Us is the shock velocity

   up is the particle velocity

   S is the linear slope of the Us – up relationship

   C0 is the bulk acoustic sound speed

   The bulk acoustic sound speed is calculated from the bulk modulus (K) and density at initial state (ρref ) :

   The local material sound speed is calculated from the bulk modulus, shear modulus (G) and density (ρ):

   For a multi-material Euler element, the element sound speed (SOUNDSPEED) is calculated as a stiffness-weighted average of the sound speed of the different material fractions.

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