/MAT/LAW163 (CRUSHABLE_FOAM)

Block Format Keyword This law models strain rate dependent crushable foam material. This law is applicable only for solid elements and is typically used to model low density, closed cell polyurethane foams.

Format


(1)	(2)	(3)	(5)	(7)	(10)
/MAT/LAW163/`mat_ID`/`unit_ID` or /MAT/CRUSHABLE_FOAM/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`		$ν$	`TSC`	`DAMP`	`NCYCLE`
	`Tab_ID`	`Epsd_ref`	`Fscale`	`SRC_LIMIT`	`NRS`

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)
$ρ_{i}$	Initial density. (Real)
`E`	Young's modulus. (Real)	$[Pa]$
$ν$	Poisson's ratio. Default = 0.0 (Real)
`TSC`	Tensile stress cutoff. Default = 10²⁰ (Real)	$[Pa]$
`DAMP`	Damping coefficient (recommended between 0.05 and 0.5). (Real)
`NCYCLE`	Number of cycles for volumetric strain rate filtering. Default = 12 (Integer)
`Tab_ID`	Yield stress table identifier versus volumetric strain and volumetric strain rate. (Integer)
`Epsd_ref`	Scale factor for strain rate in `Tab_ID` (also called reference strain rate). Default = 1.0 (Real)	$[\frac{1}{s}]$
`Fscale`	Scale factor for stress in `Tab_ID`. Default = 1.0 (Real)	$[Pa]$
`SRC_LIMIT`	Strain rate change limit. Default 10²⁰ (Real)	$[\frac{1}{s}]$
`NRS`	Strain rate dependency type flag. = 0 (Default) True volumetric strain rate. = 1 Engineering volumetric strain rate. (Integer)

Example (Crushable Foam)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit_Mg_mm_s
                  Mg                  mm                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/LAW163/1/1
Crushable foam
#              RHO_I
               1E-10                   
#                  E                  NU                 TSC                DAMP              NCYCLE
                  10                 0.1                 5.0                0.05                   1
#             TAB_ID            EPSD_REF              FSCALE           SRC_LIMIT                 NRS
                   1                                                                               0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/TABLE/1/1
Stress vs volumetric strain
         1
#                  X                   Y
                   0                   0
               0.001         0.324440816
               0.002         0.432139501
               0.003         0.511748811
               0.004         0.577416977
               0.005         0.634416215
               0.006         0.685381975
               0.007         0.731849674
               0.008         0.774804109
               0.009         0.814920344
                0.01         0.852684325
                0.02         1.153108519
                0.03         1.380738481
                0.04         1.572021977
                0.05         1.740600504
                0.06         1.893316035
                0.07         2.034165965
                0.08         2.165720524
                0.09          2.28974548
                 0.1         2.407514491
                0.11         2.519981277
                0.12         2.627881481
                0.13         2.731796386
                0.14          2.83219458
                0.15         2.929460226
                0.16         3.023912838
                0.17         3.115821497
                0.18         3.205415286
                0.19         3.292891111
                 0.2         3.378419669
                0.21         3.462150065
                0.22         3.544213433
                0.23         3.624725821
                0.24         3.703790499
                0.25         3.781499843
                0.26         3.857936868
                0.27           3.9331765
                0.28         4.007286637
                0.29          4.08032903
                 0.3         4.152360035
                0.31         4.223431248
                0.32         4.293590044
                0.33          4.36288005
                0.34         4.431341538
                0.35         4.499011781
                0.36          4.56592535
                0.37         4.632114386
                0.38         4.697608826
                0.39         4.762436612
                 0.4         4.826623873
                0.41         4.890195086
                0.42         4.953173221
                0.43         5.015579868
                0.44         5.077435358
                0.45          5.13875887
                0.46         5.199568528
                0.47         5.259881493
                0.48         5.319714057
                0.49         5.379081726
                 0.5         5.437999308
                0.51         5.496481009
                0.52         5.554540531
                0.53         5.612191185
                0.54         5.669446026
                0.55          5.72631801
                0.56         5.782820188
                0.57         5.838965953
                0.58         5.894769339
                0.59         5.950245415
                 0.6         6.005410776
                0.61         6.060284172
                0.62         6.114887315
                0.63         6.169245897
                0.64         6.223390893
                0.65         6.277360212
                0.66          6.33120078
                0.67         6.384971174
                0.68         6.438744927
                0.69         6.492614676
                 0.7         6.546697342
                0.71         6.601140582
                0.72         6.656130808
                0.73         6.711903108
                0.74         6.768753501
                0.75         6.827054026
                0.76         6.887271255
                0.77         6.949988973
                0.78         7.015935864
                0.79         7.086019231
                 0.8         7.161365954
                0.81         7.243372132
                0.82         7.333763083
                0.83         7.434665711
                0.84         7.548695574
                0.85         7.679061418
                0.86         7.829690392
                0.87         8.005377719
                0.88         8.211965236
                0.89         8.456553919
                 0.9         8.747756361
                0.91         9.095996146
                0.92         9.513862121
                0.93         10.01652688
                0.94         10.62224018
                0.95         11.35290965
                0.96         12.23478312
                0.97         13.29924886
                0.98         14.58377261
                0.99         16.13299299
                   1                  18
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

The material law is based on the volumetric strain denoted $γ$ and computed with:
$γ = 1 - \frac{ρ_{0}}{ρ}$
Where,

$ρ_{0}$

Initial density

$ρ$

Current density

Such definition of volumetric strain implies that $γ > 0$ in compression, and $γ < 0$ in tension.
The elastic behavior considers a Young's modulus denoted E and a Poisson’s ratio denoted $ν$ (considered to be close or equal to zero). Note that the Young's modulus defined in the input may be overwritten. The maximum slope of stress versus volumetric strain tabulated evolution is computed and retained for elastic stiffness to ensure stability. Using E and $ν$ , the classical Hook isotropic elasticity is considered as trial stress tensor.
$σ_{n + 1}^{r} = σ_{n} + ℂ Δ ε_{n + 1}$
For foams, the Poisson’s ratio $ν$ is often assumed to be zero. In that case, the equation above becomes:
$σ_{n + 1}^{t r} = σ_{n} + E Δ ε_{n + 1}$
The nonlinear behavior is assumed to be unsymmetric, considering a tensile and a compressive case. To do so, the scaling of trial stress tensor is based on the modification of its principal stresses. Once the three principal stresses are obtained, each one can be scaled depending on its sign as:
$σ_{i = 1, 3} = \{\begin{matrix} - |σ_{t a b}| & i f & σ_{i} \leq - |σ_{t a b}| \\ σ_{T S C} & i f & σ_{i} \geq σ_{T S C} \end{matrix}$
Thus, negative principal stresses are scaled using the tabulated stress versus volumetric strain (and volumetric strain rate if defined), and positive principal stresses are scaled to the constant cutoff value.
Figure 1.

Note: The tensile cutoff stress is not submitted to strain rate dependency. Also, the tabulated scaling stress is only used in compression.
Afterwards, the modified principal stress tensor is rotated back to the global reference system using principal vectors to obtain the current Cauchy stress tensor.
$σ = P σ_{p} P^{-1} {with σ}_{p} = [\begin{matrix} σ_{1} & 0 & 0 \\ 0 & σ_{2} & 0 \\ 0 & 0 & σ_{3} \end{matrix}]$
To improve the stability of this material law, a viscous damping can be used. Viscous over-stresses are then added to the computed Cauchy stress tensor. Their computation is governed by the equation:
$\begin{array}{l} Δ σ_{i} = a (\frac{{\dot{ε}}_{i} - {\dot{ε}}_{m}}{1 + ν} + \frac{{\dot{ε}}_{m}}{1 - 2 ν}) with {\dot{ε}}_{m} = \frac{Tr (\dot{ε})}{3} if i = 1, 2, 3 \\ Δ σ_{i} = \frac{a {\dot{ε}}_{i}}{2 (1 + ν)} if i = 4, 5, 6 \\ With a = \frac{c_{s} ρ μ_{d a m p} L_{e}}{1 + γ} \end{array}$
Where,

$c_{s}$

Material sound speed

$μ_{d a m p}$

Damping coefficient defined in the input card

$L_{e}$

Element characteristic length
To avoid noisy results due to volumetric strain rate dependency, a low-pass filter can be activated over several cycles defined by you. A default value of 12 cycles is used. The filtered volumetric strain rate is obtained with:
$\begin{array}{l} {\dot{γ}}_{n + 1}^{f i l t} = α {\dot{γ}}_{n + 1} + (1 - α) {\dot{γ}}_{n}^{f i l t} \\ With α = \frac{2 π}{2 π + N_{c y c l e}} \end{array}$
You can choose the type of the volumetric strain rate dependency defined in the input.
- If NRS = 0, true volumetric strain rate dependency is used.
  $\begin{array}{l} \dot{γ} = - \frac{\dot{V}}{V} = - Tr (\dot{ε}) \end{array}$
- If NRS = 1, engineering volumetric strain rate dependency is used.
  $\begin{array}{l} \dot{γ} = - \frac{\dot{V}}{V_{0}} \end{array}$