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A finite element formulation for a simplified, relaxed micromorphic continuum model
〈M ,P 〉dX , Ω ⊂ R3 ,
∇u =
u1,1 u1,2 u1,3
u2,1 u2,2 u2,3
u3,1 u3,2 u3,3
, CurlP =
curl [ P1,1 P1,2 P1,3
]T
curl [ P2,1 P2,2 P2,3
]T
curl [ P3,1 P3,2 P3,3
]T
, curlv = ∇× v , (1)
with u : [...] 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 takes [...] curl operator in 2D is given by curl ζ := ζ2,1 − ζ1,2 , ζ ∈ R2 . From Eq. (2) the variational principle yields
∀ δu :
∫
Ω
2µe〈∇u− ζ,∇δu〉 − 〈f, δu〉dX = 0 , (3)
∀ δζ :
∫
Ω
2µe〈∇u− ζ,−δζ〉+ 2µmicro〈ζ, δζ〉+ µmacro …
