Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

Amstutz, Samuel; Goethem, Nicolas Van

doi:10.3934/dcdsb.2020240

Cited by 4 publications

(8 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…. We refer to Amstutz and Van Goethem [10] for the precise meaning of this latter condition and for the proof of the following result.…”

Section: Beltrami Decomposition and Related Function Spacesmentioning

confidence: 99%

“…The model obtained from the first two terms has been studied in Amstutz and Van Goethem [6, 10, 16]. It is appealing in that the mixed term in the simplified form ℓ inc E · E ^ can be derived by integration by parts from a first gradient (with respect to strain) model with natural assumptions, it leads to well-posed governing equations, and it is consistent with compatible elasticity at the limit ℓ → ± ∞ in the absence of kinematical constraint.…”

Section: Construction Of a Second-order Model Of Incompatible Elasticitymentioning

confidence: 99%

“…Indeed, it appears to us as the most natural in its expression, due to its generality, rigor in its application if attention is paid to the notion of objective descriptors (here we follow the approach advocated by Germain [7] and Maugin [8]), and appropriateness to the mathematical treatment since a variational or weak form is readily obtained. Moreover, to infer strong forms and prescribe relevant boundary conditions and loads, it is required to carefully study the associated functional spaces, as we have done in Amstutz and Van Goethem [9, 10], including orthogonal decompositions of these spaces in the spirit of Leray’s projection which is classically used in fluid mechanics. The interpretation of these boundary conditions was until now an open problem that we also address here through a mesoscopic analysis, that is at a scale where the dislocations are singularity lines, the so-called Volterra dislocations [11, 12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A second-order model of small-strain incompatible elasticity

Amstutz,

Van Goethem

2023

Mathematics and Mechanics of Solids

Self Cite

View full text Add to dashboard Cite

This work deals with the modeling of solid continua undergoing incompatible deformations due to the presence of microscopic defects like dislocations. Our approach relies on a geometrical description of the medium by the strain tensor and the representation of internal efforts using zeroth- and second-order strain gradients in an infinitesimal framework. At the same time, energetic arguments allow to monitor the corresponding moduli. We provide mathematical and numerical results to support these ideas in the framework of isotropic constitutive laws.

show abstract

“…. We refer to Amstutz and Van Goethem [10] for the precise meaning of this latter condition and for the proof of the following result.…”

Section: Beltrami Decomposition and Related Function Spacesmentioning

confidence: 99%

Section: Construction Of a Second-order Model Of Incompatible Elasticitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A second-order model of small-strain incompatible elasticity

Amstutz,

Van Goethem

2023

Mathematics and Mechanics of Solids

Self Cite

View full text Add to dashboard Cite

show abstract

“…[7][8][9]47,95,109]. Furthermore, the equation inc sym e ≡ 0 is equivalent to the Saint-Venant compatibility condition(s) 10 defining the relation between the displacement vector field u and the symmetric strain sym e, more precisely:…”

Section: Remark 25mentioning

confidence: 99%

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

2021

View full text Add to dashboard Cite

Let $$\Omega \subset \mathbb {R}^3$$ Ω ⊂ R 3 be an open and bounded set with Lipschitz boundary and outward unit normal $$\nu $$ ν . For $$1<p<\infty $$ 1 < p < ∞ we establish an improved version of the generalized $$L^p$$ L p -Korn inequality for incompatible tensor fields P in the new Banach space $$\begin{aligned}&W^{1,\,p,\,r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3}) \\&\quad = \{ P \in L^p(\Omega ; \mathbb {R}^{3 \times 3}) \mid {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \in L^r(\Omega ; \mathbb {R}^{3 \times 3}),\ {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}(P \times \nu ) = 0 \text { on }\partial \Omega \} \end{aligned}$$ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) = { P ∈ L p ( Ω ; R 3 × 3 ) ∣ dev sym Curl P ∈ L r ( Ω ; R 3 × 3 ) , dev sym ( P × ν ) = 0 on ∂ Ω } where $$\begin{aligned} r \in [1, \infty ), \qquad \frac{1}{r} \le \frac{1}{p} + \frac{1}{3}, \qquad r >1 \quad \text {if }p = \frac{3}{2}. \end{aligned}$$ r ∈ [ 1 , ∞ ) , 1 r ≤ 1 p + 1 3 , r > 1 if p = 3 2 . Specifically, there exists a constant $$c=c(p,\Omega ,r)>0$$ c = c ( p , Ω , r ) > 0 such that the inequality $$\begin{aligned} \Vert P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})}\le c\,\left( \Vert {{\,\mathrm{sym}\,}}P \Vert _{L^p(\Omega ,\mathbb {R}^{3\times 3})} + \Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^{r}(\Omega ,\mathbb {R}^{3\times 3})}\right) \end{aligned}$$ ‖ P ‖ L p ( Ω , R 3 × 3 ) ≤ c ‖ sym P ‖ L p ( Ω , R 3 × 3 ) + ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 3 ) holds for all tensor fields $$P\in W^{1,\,p, \, r}_0({{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}; \Omega ,\mathbb {R}^{3\times 3})$$ P ∈ W 0 1 , p , r ( dev sym Curl ; Ω , R 3 × 3 ) . Here, $${{\,\mathrm{dev}\,}}X :=X -\frac{1}{3} {{\,\mathrm{tr}\,}}(X)\,{\mathbb {1}}$$ dev X : = X - 1 3 tr ( X ) 1 denotes the deviatoric (trace-free) part of a $$3 \times 3$$ 3 × 3 matrix X and the boundary condition is understood in a suitable weak sense. This estimate also holds true if the boundary condition is only satisfied on a relatively open, non-empty subset $$\Gamma \subset \partial \Omega $$ Γ ⊂ ∂ Ω . If no boundary conditions are imposed then the estimate holds after taking the quotient with the finite-dimensional space $$K_{S,dSC}$$ K S , d S C which is determined by the conditions $${{\,\mathrm{sym}\,}}P =0$$ sym P = 0 and $${{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P = 0$$ dev sym Curl P = 0 . In that case one can replace $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{L^r(\Omega ,\mathbb {R}^{3\times 3})} $$ ‖ dev sym Curl P ‖ L r ( Ω , R 3 × 3 ) by $$\Vert {{\,\mathrm{dev}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}P \Vert _{W^{-1,p}(\Omega ,\mathbb {R}^{3\times 3})}$$ ‖ dev sym Curl P ‖ W - 1 , p ( Ω , R 3 × 3 ) . The new $$L^p$$ L p -estimate implies a classical Korn’s inequality with weak boundary conditions by choosing $$P=\mathrm {D}u$$ P = D u and a deviatoric-symmetric generalization of Poincaré’s inequality by choosing $$P=A\in {{\,\mathrm{\mathfrak {so}}\,}}(3)$$ P = A ∈ so ( 3 ) . The proof relies on a representation of the third derivatives $$\mathrm {D}^3 P$$ D 3 P in terms of $$\mathrm {D}^2 {{\,\mathrm{dev}\,}}{{\

show abstract

“…By showing that the curvature approximation can be analyzed via the incompatibility operator, we hope to generate new ideas for computing and analyzing approximations of the intrinsic curvature tensor of higher dimensional manifolds. The incompatibility operator also arises in modeling elastic materials with dislocations [2,3], another potential area of application.…”

Section: Introductionmentioning

confidence: 99%

Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

Gopalakrishnan¹,

Neunteufel²,

Schöberl³

et al. 2022

Preprint

View full text Add to dashboard Cite

The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

show abstract

Existence and asymptotic results for an intrinsic model of small-strain incompatible elasticity

Cited by 4 publications

References 36 publications

A second-order model of small-strain incompatible elasticity

A second-order model of small-strain incompatible elasticity

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

Analysis of curvature approximations via covariant curl and incompatibility for Regge metrics

Contact Info

Product

Resources

About