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A finite element formulation for a simplified, relaxed micromorphic continuum model
⊂ 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 : Ω ⊂ R3 → R3 [...] R3 and P : Ω ⊂ R3 → R3×3 being the displacement and microdistortion, respectively. External loading is denoted by f and M . Further, Ce and Cmicro are standard elasticity tensors and Cc is a positive s [...] ∀ δu :
∫
Ω
2µe〈∇u− ζ,∇δu〉 − 〈f, δu〉dX = 0 , (3)
∀ δζ :
∫
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 …
