Appendix

Basic Relations


	$E, ν$	$E, G$	$E, B$	$G, ν$	$G, B$	$B, ν$	$λ, μ$
$E$	$E$	$E$	$E$	$2 (1 + ν) G$	$\frac{9 B G}{3 B + G}$	$3 (1 - 2 ν) B$	$\frac{(3 λ + 2 μ) μ}{λ + μ}$
$G = μ$	$\frac{E}{2 (1 + ν)}$	$G$	$\frac{3 E B}{9 B - E}$	$G$	$G$	$\frac{3 (1 - 2 ν) B}{2 (1 + ν)}$	$μ$
$B = K$	$\frac{E}{3 (1 - 2 ν)}$	$\frac{E G}{9 G - 3 E}$	$B$	$\frac{2 (1 + ν) G}{3 (1 - 2 ν)}$	$B$	$B$	$\frac{3 λ + 2 μ}{3}$
$ν$	$ν$	$\frac{E - 2 G}{2 G}$	$\frac{3 B - E}{6 B}$	$ν$	$\frac{3 B - 2 G}{6 B + 2 G}$	$ν$	$\frac{λ}{2 (λ + μ)}$
$D_{11}$	$\frac{E (1 - ν)}{(1 + ν) (1 - 2 ν)}$	$\frac{(4 G - E) G}{3 G - E}$	$\frac{3 B (3 B + E)}{9 B - E}$	$\frac{2 G (1 - ν)}{1 - 2 ν}$	$\frac{3 B + 4 G}{3}$	$\frac{3 B (1 - ν)}{1 + ν}$	$λ + 2 μ$
$D_{12} = λ$	$\frac{E ν}{(1 + ν) (1 - 2 ν)}$	$\frac{(E - 2 G) G}{3 G - E}$	$\frac{3 B (3 B - E)}{9 B - E}$	$\frac{2 G ν}{1 - 2 ν}$	$\frac{3 B - 2 G}{3}$	$\frac{3 B ν}{1 + ν}$	$λ$
$C_{11}$	$\frac{E}{1 + ν^{2}}$	$\frac{4 G G}{4 G - E}$	$\frac{36 B E}{36 - {(3 - \frac{E}{B})}^{2}}$	$\frac{2 G}{1 - ν}$	$\frac{4 G (3 B + G)}{3 B + 4 G}$	$\frac{3 B (1 - 2 ν)}{1 - ν^{2}}$
$C_{12}$	$\frac{E ν}{1 + ν^{2}}$	$\frac{(E - 2 G) 2 G}{4 G - E}$	$\frac{6 E (3 - \frac{E}{B})}{36 - {(3 - \frac{E}{B})}^{2}}$	$\frac{2 G ν}{1 - ν}$	$\frac{2 G (3 B - 2 G)}{3 B + 4 G}$	$\frac{3 B (1 - 2 ν)}{1 - ν^{2}}$

Hook Law 3D (principal stress and strain)

$σ = D ε$

$σ_{1} = D_{11} ε_{1} + D_{12} ε_{2} + D_{13} ε_{3}$

$σ_{1} = (λ + 2 μ) ε_{1} + λ (ε_{2} + ε_{3})$

$σ_{1} = λ (ε_{1} + ε_{2} + ε_{3}) + 2 μ ε_{1}$

$σ_{1} = K ε_{k k} + 2 μ e_{1}$ with $ε_{k k} = ε_{1} + ε_{2} + ε_{3}$ and $e_{1} = ε_{1} - \frac{1}{3} (ε_{1} + ε_{2} + ε_{3})$

Hook Law 2D (plane stress)

$σ = C ε$

$σ_{1} = C_{11} ε_{1} + C_{12} ε_{2}$

Unit Systems


Length	Time	Mass	Force	Pressure	Velocity	$ρ$	Energy	G
m	s	Kg	Kg m/s2	N/m2	m/s	Kg/m3	Kmg2/s2	9.81
m	s	Kg	N	Pa	m/s	m Kg/l	J	9.81
m	s	g	mN	mPa	m/s	$μ$ Kg/l	mJ	9.81
m	s	Mg (ton)	KN	KPa	m/s	Kg/l	KJ	9.81
m	ms	Kg	MN	MPa	Km/s	m Kg/l	MJ	9.81e-6
m	ms	g	KN	KPa	Km/s	$μ$ Kg/l	KJ	9.81e-6
m	ms	Mg (ton)	GN	GPa	Km/s	Kg/l	GJ	9.81e-6
mm	s	Kg	mN	KPa	mm/s	M Kg/l	mJ	9.81e+3
mm	s	g	mN	Pa	mm/s	K Kg/l	nJ	9.81e+3
mm	s	Mg (ton)	N	MPa	mm/s	G Kg/l	mJ	9.81e+3
mm	ms	Kg	KN	GPa	m/s	M Kg/l	J	9.81e-3
mm	ms	g	N	MPa	m/s	K Kg/l	mJ	9.81e-3
mm	ms	Mg (ton)	MN	TPa	m/s	G Kg/l	KJ	9.81e-3
cm	ms	g	daN	10⁵Pa bar	dam/s	Kg/l	dJ	9.81e-4
cm	ms	Kg	10⁴ N (KdaN)	10⁸Pa (Kbar)	dam/s	K Kg/l	hJ	9.81e-4
cm	ms	Mg (ton)	10⁷ N (MdaN)	10¹¹ Pa (Mbar)	dam/s	M Kg/l	10⁵ J	9.81e-10
cm	$μ_{s}$	g	10⁷ N (MdaN)	10¹¹ Pa (Mbar)	10⁴ m/s	Kg/l	10⁵ J	9.81e-10

Filtering

Often it is useful to filter results in a material or failure law to remove numerical noise. The most common filter is an exponential moving average filter. This is especially important for material models that include strain rate effects.

In most materials, the flag F_smooth = 1 must be defined to enable the filtering and the cutoff frequency entered using F_cut. For the case of filtering strain rates, use:

${\dot{ε}}_{f i l t e r e d} (t) = α \dot{ε} (t) + (1 - α) {\dot{ε}}_{f i l t e r e d} (t - d t)$

Where,

${\dot{ε}}_{f i l t e r e d} (t)$: Filtered strain rate.
$\dot{ε} (t)$: Strain rate at the current timestep before filtering.
$α$: Degree of weighting decrease, a constant smoothing factor between 0 and 1. A higher value discounts previous values faster which results in less filtering.
$d t$: Timestep of the simulation.
${\dot{ε}}_{f i l t e r e d} (t - d t)$: Filtered strain rate at the previous time step.

For materials laws where F_cut can be entered.

$α = 2 π d t F_{c u t}$

Where, F_cut is the cutoff frequency.

Thus,

$F_{c u t} = \frac{α}{2 π d t}$

The cutoff frequency is a function of the model timestep. Experience shows that the speed of the deformation is important also. For slower speeds, like a car crash, 1 – 10 kHz (1000 – 10,000 Hz) is a good value, but for high-speed events, like ballistic, less filtering should be used - so 1 – 10 GHz is appropriate. Good engineering judgment should be used to determine a reasonable value for each simulation. Refer to RD-E: 1102 Strain Rate Effect for an example of strain rate filtering.