A Variational Approach to Single Crystals with Dislocations
Abstract: We study the graphs of maps u : Ω → R 3 whose curl is an integral 1-current with coefficients in Z 3. We characterize the graph boundary of such maps under suitable summability property. We apply these results to study a three-dimensional single crystal with dislocations forming general one-dimensional clusters in the framework of finite elasticity. By virtue of a variational approach, a free energy depending on the deformation field and its gradient is considered. The problem we address is the joint minimizat… Show more
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Cited by 10 publications
References 36 publications
In this paper we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored-elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents and the dissipation distance is chosen to be the flat distance.
In this paper we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored-elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents and the dissipation distance is chosen to be the flat distance.
The mathematical modelling in mechanics has a long-standing history as related to geometry, and significant progresses have often been achieved by the invention of new geometrical tools. Also, it happened that the elucidation of practical issues led to the invention of new scientific concepts, and possibly new paradigms, with potential impact far beyond. One such example is Riemann's intrinsic view in geometry, that offered a radically new insight in the Physics of the early 20th century. On the other hand, the rather recent intrinsic approaches in elasticity and elasto-plasticity also share this philosophical standpoint of looking from inside, i.e., from the "manifold" point of view. Of course, this approach requires smoothness, and is thus incomplete for an analyst. Nevertheless, its first aim is to highlight the concepts of metric, curvature and torsion; these notions are addressed in the first part of this survey paper. In a second part, they are given a precise functional meaning and their properties are studied systematically. Further, a novel approach to elasto-plasticity constructed upon a model of incompatible elasticity is designed, carrying this intrinsic spirit. The main mathematical object in this theory is the incompatibility operator, i.e., a linearized version of Riemann's curvature tensor. So far, this route not only has led the authors to a new model with a solid functional foundation and proof of existence results, but also to a framework with a minimal amount of ad-hoc assumptions, and complying with both the basic principles of thermodynamics and invariance principles of Physics. The questions arising from this novel approach are complex and intriguing, but we believe that the model is now sufficiently well posed to be studied simultaneously as a problem of mathematics and of mechanics. Most of the research programme remains to be done, and this survey paper is written to present our model, with a particular care to put this approach into a historical perspective.2010 Mathematics Subject Classification. 35J48,35J58,49S05,49K20,74C05,74G99,74A05,74A15, 80A17.
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}\,}}{{\
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