/MAT/LAW103 (HENSEL-SPITTEL)

Block Format Keyword This law represents an isotropic elastic-plastic material at high temperature using Hensel-Spittel yield stress formula. The yield stress is a function of strain, strain rate and temperature. This material law can be used with an equation of state /EOS.

This material is often used in hot forging simulations. The law parameters are valid only for a given range of temperature and strain rate. This material law is compatible with solid and SPH elements only.

Format

(1)	(2)	(3)	(5)	(7)	(9)
/MAT/LAW103/`mat_ID`/`unit_ID` or /MAT/HENSEL-SPITTEL/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$ $ρ_{i}$		$ρ_{0}$ $ρ_{0}$
`E`		$ν$ $ν$
`A`₀		`m`₁	`m`₂	`m`₃	`m`₄
`m`₅		`m`₇
	`F`_smooth	`F`_cut	$ε_{0}$ $ε_{0}$	`P`_min
$ρ C_{p}$ $ρ C_{p}$		`T`₀	$η$ $η$

Definition

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Unit 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 the default equation of state. Default = $ρ_{i}$ $ρ_{i}$ (Real)	$[\frac{kg}{m^{3}}]$ $[\frac{kg}{m^{3}}]$
`E`	Initial Young's modulus. (Real)	$[Pa]$ $[Pa]$
$ν$ $ν$	Poisson's ratio. (Real)
`A`₀	Stress parameter. (Real)	$[Pa]$ $[Pa]$
`m`₁	Material parameter 1. (Real)
`m`₂	Material parameter 2. (Real)
`m`₃	Material parameter 3. (Real)
`m`₄	Material parameter 4. (Real)
`m`₅	Material parameter 5. (Real)
`m`₇	Material parameter 7. (Real)
`F`_smooth	Smooth strain rate flag. =0 No strain rate smoothing. = 1 Strain rate smoothing active. (Integer)
`F`_cut	Cutoff frequency for strain rate filtering. (Real)	$[\frac{1}{s}]$ $[\frac{1}{s}]$
$ε_{0}$ $ε_{0}$	Reference strain. (Real)
`P`_min	Pressure cutoff (< 0). Default = 10³⁰ (Real)	$[Pa]$ $[Pa]$
$ρ C_{p}$ $ρ C_{p}$	Specific heat per unit volume. (Real)	$[\frac{J}{m^{3} \cdot K}]$ $[\frac{J}{m^{3} \cdot K}]$
`T`₀	Initial temperature. (Real)	$[K]$ $[K]$
$η$ $η$	Heat conversion parameter 0 < $η$ $η$ < 1.0. (Real)

▸Example (Alloy)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW103/1/1
Magnesium alloy
#        Init. dens.          Ref. dens.
              0.0018              0.0018
#                  E                  Nu
               45000                0.28
#                 A0                  M1                  M2                  M3                  M4
               709.4             -0.0065             -0.1538                   0             -0.0261
#                 M5                  M7
                   0                   0
#            Fsmooth                Fcut                 Eps                Pmin
                   0                   0               0.010                   0
#              RhoCp                  T0                 ETA
                1.89              673.15                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

Yield stress: ¹(1)
$σ_{y} = A_{0} \exp^{m_{1} T} ε^{m_{2}} {\dot{ε}}^{m_{3}} \exp^{\frac{m_{4}}{ε}} {(1 + ε)}^{m_{5} T} \exp^{m_{7} ε}$ $σ_{y} = A_{0} \exp^{m_{1} T} ε^{m_{2}} {\dot{ε}}^{m_{3}} \exp^{\frac{m_{4}}{ε}} {(1 + ε)}^{m_{5} T} \exp^{m_{7} ε}$

Where,

$T$ $T$

Temperature in °C

$ε$ $ε$

True strain $ε = ε_{0} + {\bar{ε}}_{p}$ $ε = ε_{0} + {\bar{ε}}_{p}$

${\bar{ε}}_{p}$ ${\bar{ε}}_{p}$

Equivalent plastic strain

$\dot{ε}$ $\dot{ε}$

True strain rate in s^-1

$m_{1}$ $m_{1}$ - $m_{5}$ $m_{5}$

Material parameters
In case of purely mechanical simulation, the temperature is computed assuming adiabatic condition:(2)
$T = T_{0} + \frac{η \cdot E_{int}}{ρ C_{p} V}$ $T = T_{0} + \frac{η \cdot E_{int}}{ρ C_{p} V}$

Where,

E_int

Internal energy of the element.

$η$ $η$

Taylor-Quinney coefficient used as scale of plastic energy, which transfers into heat.

$V$ $V$

Volume of the element
There is no strain rate effect if m₃ = 0.
By default, the hydrostatic pressure is linearly proportional to volumetric strain:(3)
$P = K μ$ $P = K μ$

Where,

$K = \frac{E}{3 (1 - 2 v)}$ $K = \frac{E}{3 (1 - 2 v)}$

Bulk modulus

$μ = \frac{ρ}{ρ_{0}} - 1$ $μ = \frac{ρ}{ρ_{0}} - 1$

Volumetric strain

An additional Equation of State (/EOS) card can refer to this material to model a nonlinear dependency between hydrostatic pressure and volumetric strain.
This material can be used with the material options, /HEAT/MAT, /THERM_STRESS/MAT, /EOS, and /VISC.

A. Hensel, T. Spittel, VEB German Pushling House for Basic Industry, Leipzig, Deutschland, 1978