eldorado.tu-dortmund.de/server/api/core/bitstreams/994bd060-2871-4e3a-b525-010824d95df0/content
A finite element formulation for a simplified, relaxed micromorphic continuum model
base functions [3, 4] for ζ. For comparison, we also derive a purely H 1(Ω) × H 1(Ω) element. We set Ω = [−4, 4]× [−4, 4] ⊂ R2 and impose the lift and material constants
u(−4, y) = u(4, y) = 0 , u(−2, y) [...] solution reads
ũ =
4 + x for −4 ≤ x < −2 2 for −2 ≤ x < 2
4− x for 2 ≤ x ≤ 4 , ζ̃ =
[ 0.5 0
]T for −4 ≤ x < −2
[ 0 0
]T for −2 ≤ x < 2
[ −0.5 0
]T for 2 ≤ x ≤ 4
.
The microdistortion field [...] :
∫
Ω
2µe〈∇u− ζ,−δζ〉+ 2µmicro〈ζ, δζ〉+ µmacro L 2 c〈curl ζ, curl δζ〉 − 〈m, δζ〉dX = 0 . (4)
From Eq. (3) and Eq. (4) we obtain the weak form for further numerical investigation ∫
Ω
2µe〈∇u− ζ,∇δu− δζ〉+ 2µmicro〈ζ …
