Korn inequalities for incompatible tensor fields in three space dimensions with conformally invariant dislocation energy

Abstract

Let Ω ⊂ R 3 be an open and bounded set with Lipschitz boundary and outward unit normal ν. For 1 < p < ∞ we establish an improved version of the generalized L p -Korn inequality for incompatible tensor fields P in the new Banach space W 1, p, r

Analytical solutions of the cylindrical bending problem for the relaxed micromorphic continuum and other generalized continua

“…An important advantage of the model compared to the Cosserat model is that there is no stiffness singularity in whatsoever situation and four of the eight constants (µ macro , λ macro , µ e , and λ e ) can be determined ab initio from size-independent homogeneous tests [46]. There remains to fit three curvature parameters and the Cosserat couple modulus µ c ≥ 0 (which in some situations may be chosen to be zero since the model remains well-posed) [6,36,37,38,56].…”

“…The values of A 1 , B 1 are determined from the boundary conditions (36), while, due to the divergent behaviour of the Bessel function of the second kind at r = 0, we have to set A 2 = 0 and B 2 = 0 in order to have a continuous solution. The fulfilment of the boundary conditions (36) allows us to find the expressions of the integration constants…”

Conforming Finite Elements for $H(\text{sym}\,\text{Curl})$ and $H(\text{dev}\,\text{sym}\,\text{Curl})$

“…and by Theorem 4.1 of [4] the two spaces are not equal when Ω is bounded. Curiously, however, Observation 2.3 in [4] shows that…”

“…In [4], Lewintan, Müller, and Neff introduced the spaces W 1,p (sym Curl; Ω; R 3 ) := P ∈ L p (Ω; R 3×3 ) : sym Curl P ∈ L p (Ω; R 3×3 ) and W 1,p (dev sym Curl; Ω; R 3 ) := P ∈ L p (Ω; R 3×3 ) : dev sym Curl P ∈ L p (Ω; R 3×3 ) , where L p (Ω; R 3×3 ), p ≥ 1, is the Lebesgue space of R 3×3 -valued functions that are pintegrable on a domain Ω. The operator Curl denotes the classical curl operator acting row-wise on a matrix; to distinguish the two we write the matrix form with an upper-case letter.…”

“…However, the elements do not fulfill any exact-sequence properties, and no inf-sup condition is shown either. This is because the applications envisioned in [4] are not saddle-point problems, and therefore these structural properties are of lesser importance. The gain is a vastly simplified construction compared to [3], and finite elements with fewer degrees of freedom per element.…”