/MAT/LAW26 (SESAM)

SESAME EOS covers a wide range of phases including solids, fluids and high temperature/high density plasmas, and the well-known transitions between these various phases. It requires SESAME tables, which were developed at Los Alamos National Laboratory in USA.

Format

(1)	(3)	(5)	(7)	(9)
/MAT/LAW26/`mat_ID` or /MAT/SESAM/`mat_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$	$ρ_{0}$ $ρ_{0}$
`E`	$υ$ $υ$
`a`	`b`	`n`	$ε_{p}^{max}$ $ε_{p}^{max}$	$σ_{\max}$ $σ_{\max}$
`E`₀
SESAM301
`c`	$ε_{0}$ $ε_{0}$	`m`	`T`_melt	`T`_max

If thermal effects are taken into account (otherwise must keep the following blank):

(1)	(3)	(5)	(7)
SESAM505
`K`_LOR	$λ$ $λ$	`A`	`K`_max
`SESAM502`
$σ$ $σ$

Definition

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`mat_title`	Material title. (Character, maximum 100 characters)
$ρ_{i}$ $ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
$ρ_{0}$ $ρ_{0}$	Reference density used in E.O.S (equation of state). Default $ρ_{0}$ $ρ_{0}$ = $ρ_{i}$ $ρ_{i}$ (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
`E`	Young's modulus. (Real)	$[Pa]$ $[Pa]$
$υ$ $υ$	Poisson's ratio. (Real)
`a`	Plasticity yield stress. (Real)	$[Pa]$ $[Pa]$
`b`	Hardening hardening parameter. (Real)
`n`	Plasticity hardening exponent (must be ≤ 1). Default = 1.0 (Real)
$ε_{p}^{max}$ $ε_{p}^{max}$	Failure plastic strain. (Real)
$σ_{\max}$ $σ_{\max}$	Maximum stress. (Real)	$[Pa]$ $[Pa]$
`E`₀	Initial energy per unit volume. (Real)	$[Pa]$ $[Pa]$
SESAM301	File name of the SESAME EOS table (301). (Character, maximum 100 characters)
`c`	Strain rate coefficient. = 0 No strain rate effect. Default = 0.00 (Real)
$ε_{0}$ $ε_{0}$	Reference strain rate. Default = 10^-06 (Real)	$[\frac{1}{s}]$ $[\frac{1}{s}]$
`m`	Temperature exponent. Default = 1.00 (Real)
`T`_melt	Melting temperature. = 0 No temperature affect. Default = 10³⁰ (Real)	$[K]$ $[K]$
`T`_max	Maximum temperature. for `T` > `T`_max: `m` = 1 is used. (Real)	$[K]$ $[K]$
SESAM505	File name of the SESAME table (504). (Character, maximum 100 characters)
`K`_LOR	Lorentz conductivity. Default = 1.6833x10^-9 (Real)	$[\frac{J}{m \cdot s \cdot K^{\frac{7}{2}}}]$ $[\frac{J}{m \cdot s \cdot K^{\frac{7}{2}}}]$
$λ$ $λ$	Lamda. Default = 8.3x10⁶ (Real)	$[{(\frac{m}{K})}^{3 / 2}]$ $[{(\frac{m}{K})}^{3 / 2}]$
`A`	Atomic weight. (Real)	$[\frac{kg}{mole}]$ $[\frac{kg}{mole}]$
`K`_max	Maximum conductivity. (Real)	$[\frac{J}{m \cdot s \cdot K}]$ $[\frac{J}{m \cdot s \cdot K}]$
SESAM502	File name of the SESAME table (502). (Character, maximum 100 characters)
$σ$ $σ$	Stefan-Boltzman constant (equal to 5.6697x10^-8). (Real)	$[\frac{W}{m^{2} \cdot K^{4}}]$ $[\frac{W}{m^{2} \cdot K^{4}}]$

▸Example (Water)

#RADIOSS STARTER
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW26/1
WATER - 560 K - 130 bars - Initially Liquid - Without Plastic Behavior (Unit: kg-mm-ms)
#              RHO_I               RHO_0
              .00074                   0
#              Young             Poisson
             1.0E-20                 0.0
#                  A                   B                  N             EPSP_MAX             SIG_MAX
             1.0E+20                 0.0                0.0                  0.0                 0.0   
#                 E0
               740.0    
# SESAM301 TABLE
sesam--water_301.dat   
#                  C           EPS_DOT_0                  M               T_MELT               T_MAX
                 0.0                 0.0                0.0                  0.0                 0.0
# SESAM505 TABLE
sesam--water_505.dat
#                XKL               XLAMB               ATOM                XKMAX
                 0.0                 0.0                0.0                  0.0    
# SESAM502 TABLE
sesam--water_502.dat 
#                SIG                                                                               
                 0.0   
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The thermal conductivity flux is:(1)
$F_{e} = - K_{e} \nabla T_{e}$ $F_{e} = - K_{e} \nabla T_{e}$

Where,

K_e

Thermal conductivity

K_L

Conductivity for a Lorentz gas

(2)
$K_{e} = 0.4 δ_{T} K_{L}$ $K_{e} = 0.4 δ_{T} K_{L}$

With:(3)
$K_{L} = K_{L O R} \cdot \frac{T_{e}^{5 / 2}}{Z \ln (Λ)}$ $K_{L} = K_{L O R} \cdot \frac{T_{e}^{5 / 2}}{Z \ln (Λ)}$

and(4)
$δ_{T} = \frac{1}{1 + (3.44 \times 0.26 \frac{ln (Z)}{Z})}$ $δ_{T} = \frac{1}{1 + (3.44 \times 0.26 \frac{ln (Z)}{Z})}$

K_LOR (Lorentz conductivity) is:(5)
$K_{L O R} = 1.6833 \times 10^{- 9} [\frac{J}{m \cdot s \cdot K^{7 / 2}}]$ $K_{L O R} = 1.6833 \times 10^{- 9} [\frac{J}{m \cdot s \cdot K^{7 / 2}}]$

LAMBDA depends of the electronic temperature (T_e), the electronic density (n_e) and the coefficient $λ$ $λ$ :(6)
$Λ = λ \cdot \frac{T_{e}^{3 / 2}}{n_{e}^{1 / 2}}$ $Λ = λ \cdot \frac{T_{e}^{3 / 2}}{n_{e}^{1 / 2}}$
(7)
$n_{e} = \frac{ρ \cdot N_{A} \cdot Z}{A}$ $n_{e} = \frac{ρ \cdot N_{A} \cdot Z}{A}$

Where,

A

Atomic weight

N_A

Avogadro number N_A = 6.0225 x 10²³

Z

Atomic number obtained from the SESAME EOS table 504

$ρ$ $ρ$

Density

T_e

Electronic temperature obtained from the SESAME table
The thermal radiation flux is:(8)
$F_{r} = - K_{r} \nabla T_{r}$ $F_{r} = - K_{r} \nabla T_{r}$

With:(9)
$K_{r} = \frac{\frac{16}{3} \cdot σ \cdot L_{R} \cdot T^{3} \cdot \frac{\partial T}{\partial x}}{1 + \frac{16}{3} \cdot \frac{L_{R}}{T} \cdot | \frac{\partial T}{\partial x} |}$ $K_{r} = \frac{\frac{16}{3} \cdot σ \cdot L_{R} \cdot T^{3} \cdot \frac{\partial T}{\partial x}}{1 + \frac{16}{3} \cdot \frac{L_{R}}{T} \cdot | \frac{\partial T}{\partial x} |}$

Where,

$σ$ $σ$

Stefan-Boltzmann constant

L_R

Rosseland mean length obtained from the SESAME tables.

$T_{r}$ $T_{r}$ (equivalent to T)

Radiation temperature is obtained from the SESAME tables
Files SESAM301, SESAM505, and SESAM502 are tabulated equations from analytical model selected to describe material state in different states. Most of SESAME tables are double entry tables (two variable functions). These tabulated EOS provide pressure and energy in function of density and temperature. Tables are interpolated in both directions. With Radioss the two variables are density and specific energy.
/EOS/SESAM is able to handle SESAME tabulated EOS. This present material law also take into account Johnson-Cook yield criteria and conductive/radiative transports in potential plasma.