RHEINISCHE FRIEDRICH-WILHELMS- UNIVERSITÄT BONN
S5E2 — Graduate Seminar on Efficient Simulation
Prof. Dr. Martin Rumpf, Dr. Martin Lenz
The seminar will be concerned with the discussion of different approaches to simula- te damage and fracture processes in materials. A unifying motif will be the treat- ment of the time evolution as a rate-independent variational problem and appropria- te discretization techniques.
Rate-independent hysteresis loop in a magnetic shape memory material.
There will be a arrangement meeting on
Friday, October 12th, 2018 at 12 c.t. in room 2.040, Endenicher Allee 60.
The seminar itself will be scheduled a a block seminar in January.
Literature
Mielke: Evolution of rate-independent systems. 2005.
Mielke, Roubíček: Numerical approaches to rate-independent processes and applications in inelasticity. 2009.
Mielke, Paoli, Petrov, Stefanelli: Error-estimates for space-time discretization of a rate-independent variational inequality. 2010.
Roubíček, Kružík, Zeman: Delamination and adhesive contact models and their mathematical analysis and numerical treatment. 2013.
Bourdin, Francfort, Marigo: Numerical experiments in revisited brittle fracture.
2000.
Allaire, Jouve, Van Goethem: Damage and fracture evolution in brittle materials by shape optimization methods. 2011.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
-1 -0.5 0 0.5 1
fraction of horizontally shorter variant
(horizontal) external magnetic field in T E = 0.25 E = 1.00 E = 4.00 0.3
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
-1 -0.5 0 0.5 1
fraction of horizontally shorter variant
(horizontal) external magnetic field in T -1 -0.5 0 0.5 1
-1 -0.5 0 0.5 1
(horizontal part of) magnetic moment relative to saturation
(horizontal) external magnetic field in T E = 0.25 E = 1.00 E = 4.00 -1
-0.5 0 0.5 1
-1 -0.5 0 0.5 1
(horizontal part of) magnetic moment relative to saturation
(horizontal) external magnetic field in T E= 0.25 A
A
B B
C C
D D
E E
F F
G G
E= 1.00
Figure 4:Simulation of the hysteresis loop (states C–G), including an initial phase (states A,B), in a MSM-polymer composite with one disc shaped particle per cell of the periodic lattice. Parameters as given at the beginning of Section 6. For the polymer elasticity modulus we compare three di↵erent values, the standard oneE= 1 M Pa, a larger valueE= 4 M Pa and a smaller valueE= 0.25 M Pa.
The plot on the left side depicts the volume fraction of one variant over the strengthH(t) of the external magnetic field in T. The plot on the right side shows the horizontal component of the average magnetization (relative to saturation) again overH(t).
Figure 5:Simulations with di↵erent particle radius. Same parameters and geometry as in Figure 4.
The left plot again shows the volume fraction of one variant, the right plot the horizontal part of the magnetization.
In the following we illustrate the e↵ect of the various parameters, which permits to better un- derstand the mechanisms behind the observed cycle, and to make the role of the di↵erent terms quantitative. The general picture in most of our simulations is similar to the one we just described, therefore we focus on the resulting diagrams and highlight the di↵erences.
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INSTITUT FÜR
NUMERISCHE SIMULATION
