Shocks through layered materials with SPH

Iason Zisis CASAday, 6 Nov 2013

I.Zisis dr. ir. B. vd Linden prof. dr. ir. B.Koren

Hypervelocity impacts

• Small-sized meteorites and space debris��• Impact speed higher than sound speed through target�

ü Velocity: 10 km/s�

ü Time scale: 10 μs�

ü Pressure: 100 GPa�

ü Temperature: 1,000 K �

2

Gallileo, European Space Agency

3

Phenomenology

• Shock compression�• Expansion with phase changes�• Fluid-like behavior�• Severe fragmentation�

• Stresses�

credits European Space Agency

Constitutive models Equation of State

�↵� = �P �↵� + s↵�

P = P (⇢, e)

SPH – Aluminum 7km/s impact

4

Inhomogeneous materials

•  Shields from laminated materials�

�• Delamination away from impact point�

• Homogenization�+ existing SPH context�

- only averaged delamination effects�

• Multiphase shock problem�+ reflection-transmission effects�

+ delamination effects localized�

- not thoroughly studied!�

5

O(cm�mm)

How to derive SPH schemes ?

�• Traditional SPH schemes come from function approximations on

moving points��

�

• Particles also define a particle system�

• Variational principles ?�

6

hf(x)i ⇡Z

⌦f(x

o

)W (|x� x

o

|, h)d⌦

Variational SPH framework

• Medium’s density from particles [Monaghan 2005, Price 2008]�

�

• Density evolution equation�

7

⇢i =X

j

mjWij(hi)

d⇢idt

=X

j

mjdWij

dt

hi = ⌘⇣mi

⇢i

⌘ 1d

Variational SPH framework

• Lagrange variational principle�

• Particle motion equation�

8

de = Tds+ P⇢2 d⇢

d⌦j := Vj =mj

⇢j

d

dt

⇣ @L@vi

⌘� @L

@xi= 0

depends on density estimate

@L@xi

=dejd⇢j

���s

⇣@⇢j@xi

⌘

L =

Z

⌦

⇣12⇢v2 � ⇢e(⇢, s)

⌘d⌦

L =X

j

mj

⇣12v2j � e(⇢j , sj)

⌘

More SPH schemes

• Other density estimates�

• Differential formulations are better for bounded domains��

• Derive conservative SPH schemes…�

9

⇢i =X

j

⇢jWij(hi)Vj⇢i = mi

X

j

Wij(hi)

Conservative SPH schemes

10

d⇢idt

=1

⌦i

X

j

mj(vi � vj)rWij(hi)

d⇢idt

=⇢i⌦i

X

j

(vi � vj)rWij(hi)Vj

d⇢idt

=mi

⌦i

X

j

(vi � vj)rWij(hi)

dvi

dt= �

X

j

mj

⇣ Pi

⌦i⇢2irWij(hi) +

Pj

⌦j⇢2jrWij(hj)

⌘

dvi

dt= � 1

⇢i

X

j

⇣Pi

⌦irWij(hi) +

Pj

⌦jrWij(hj)

⌘Vj

dvi

dt= � 1

mi

X

j

⇣ Pi

⇢i⌦i�irWij(hi) +

Pj

⇢j⌦j�jrWij(hj)

⌘

• Mass based�

• Volume based� •  Inverse volume based�

Lessons learned

• SPH schemes come from Lagrangian mechanics and can be fully conservative�

�• SPH momentum equation is not a free choice… depends on

density estimate equation �����

“…give me a density estimate and � I will move the SPH particles!” �

11

Still some details left…

• Which scheme to choose for shocks through layered materials ?�…test against exact solutions of Riemann problems�

• Volume based scheme performed the best�

Ø Ideal processes need numerical dissipation�

Ø New artificial mass flux term devised�

12

d⇢idt

=⇢i⌦i

X

j

(vi � vj)rWij(hi)Vj

dvi

dt= � 1

⇢i

X

j

⇣Pi

⌦irWij(hi) +

Pj

⌦jrWij(hj)

⌘Vj +⇧ij

+�ij

Inhomogeneous shock-bar

�• 1D isothermal impact into discontinuous material�

���

�

�13

Ca(x) =⇢

o

(x)v2impact

K(x)

Impact point (t = 0.1)

14

point at initial configuration

Shock on interface (t = 0.2)

15

point at initial configuration

momentum conservation�at machine precision…�

Shock tube (t=0.2)

16

Dissipative processes

• No general framework for natural dissipation in SPH�

• Towards a formulation with natural dissipation�

• GENERIC�

17

e = e(�(k), s) ! de = Tds+ �(k)d⌧ (k)

L =X

j

mj

⇣12v2 � e(⇢j , sj)

⌘

CASAsph platform

18

��

• General SPH solver �with all available schemes�

Thank you…

19