Homogenization of Edge-Dislocations as a Weak Limit of de-Rham Currents

Abstract

We consider the geometric homogenization of edge-dislocations as their number tends to infinity. The material structure is represented by 1-forms and their singular counterparts, de-Rham currents. Isolated dislocations are represented by closed 1-forms with singularities concentrated on submanifolds of co-dimension one (the defect locus), whereas a continuous distribution of dislocations is represented by smooth, non-closed 1-forms. We prove that every smooth distribution of dislocations is a limit, in the sense of weak convergence of currents, of increasingly dense and properly scaled isolated edge-dislocations. We also define a notion of singular torsion current (associated with isolated dislocations), and prove that the torsion currents converge, in the homogenization limit, to the smooth torsion field which is the continuum measure of the dislocation density.

Appendix: Gluing Constructions

The homogenization procedure presented in Sects. 4 and 6 relies on gluing diffeomorphic copies of single isolated dislocations and their structure forms. To this end, we review some basic definitions and facts, following [10, Chap. 9] and prove a gluing lemma for 1-forms.

Let ${\mathcal M}$ be a smooth manifold with boundary. A neighborhood of $\partial {\mathcal M}$ is called a collar neighborhood if it is the image of a smooth embedding $\iota :[0,1)\times \partial {\mathcal M}\hookrightarrow {\mathcal M}$ sending (identically) $\{0\}\times \partial {\mathcal M}$ to $\partial {\mathcal M}$. It follows from the theory of flows that every smooth manifold with boundary admits a collar neighborhood; see [10, Theorem 9.25].

Let ${\mathcal M}_1$ and ${\mathcal M}_2$ be smooth manifolds with boundary of the same dimension, and let $A\subset \partial {\mathcal M}_1$, and $B\subset \partial {\mathcal M}_2$ be nonempty connected (possibly closed) submanifolds. Suppose that h : B → A is a diffeomorphism. h defines an equivalence relation on the disjoint union whereby p ∼_h q if and only if p = h(q). Let

where [p]_h is the ∼_h-equivalence class of p. Then, is a topological manifold (possibly with boundary and corners); it admits a smooth structure such that the natural embeddings

are smooth, and $[{\mathcal M}_1]_h\cap [{\mathcal M}_2]_h=[A]_h = [B]_h$. We will denote by

the projection map sending every point to its equivalence class .

The construction of the smooth structure relies on gluing collar neighborhoods of A and B along h. In particular the smooth structure depends on the chosen collar neighborhoods; see [10, Theorem 9.29] for details.

Let $\iota _A:[0,1)\times A\to {\mathcal M}_1$ and $\iota _B:[0,1)\times B\to {\mathcal M}_2$ be collar neighborhoods for A and B; define also the inclusions $\eta _A:A\hookrightarrow {\mathcal M}_1$ and $\eta _B:B\hookrightarrow {\mathcal M}_1$ by η _A(p) = ι _A(0, p) and η _B(p) = ι _B(0, p); see diagram below.

For later use, we note that

$$\displaystyle \begin{aligned} \pi\circ \eta_B = \pi\circ\eta_A\circ h,\end{aligned} $$

hence, differentiating, for p ∈ B,

$$\displaystyle \begin{aligned} d\pi_{\eta_B(p)}\circ (d\eta_B)_p = d\pi_{\eta_A(h(p))}\circ (d\eta_A)_{h(p)}\circ dh_p.\end{aligned} $$

(18)

The collar neighborhoods define a decomposition of $T{\mathcal M}_1$ and $T{\mathcal M}_2$ at A and B: for example,

$$\displaystyle \begin{aligned} T{\mathcal M}_1|{}_{\eta(A)} = T{\mathcal M}_1^\parallel \oplus T{\mathcal M}_1^\perp, \end{aligned}$$

where

$$\displaystyle \begin{aligned} T{\mathcal M}_1^\parallel = (\eta_A)_\star TA, \end{aligned}$$

and

$$\displaystyle \begin{aligned} T{\mathcal M}_1^\perp = \operatorname{span}(n_A), \end{aligned}$$

where

$$\displaystyle \begin{aligned} n_A=(\iota_A)_\star(\partial_t)|{}_{A\times\{0\}} \end{aligned} $$

(19)

is a vector field normal to $T{\mathcal M}_1^\parallel$ with respect to the collar neighborhood ι _A. Similar definitions apply for the tangent bundle of ${\mathcal M}_2$ at B.

We turn to characterize tangent vectors on the quotient space . Suppose first that . Then, π is a local diffeomorphism in a neighborhood of p, hence dπ _p is a linear isomorphism. In other words, tangent vectors at [p]_h can be identified with tangent vectors at p.

In contrast, let p ∈ B, i.e.,

$$\displaystyle \begin{aligned} \pi^{-1}(\pi(p)) = \{h(p),p\}, \end{aligned}$$

and let . Then, dπ ⁻¹(v) = {v ₁, v ₂}, where $v_1\in T_{h(p)}{\mathcal M}_1$ and $v_2\in T_p{\mathcal M}_2$. Each of the two vectors can be written in the form

$$\displaystyle \begin{aligned} v_1 = (\eta_A)_\star(v_1^\parallel) + v_1^\perp\, n_A \qquad \text{ and }\qquad v_2 = (\eta_B)_\star(v_2^\parallel) + v_2^\perp\, n_B, \end{aligned}$$

where $v_1^\parallel \in TA$, $v_2^\parallel \in TB$ and .

We state without a proof:

Lemma A.1

The following relations hold:

$$\displaystyle \begin{aligned} v_1^\parallel = h_\star(v_2^\parallel), {} \end{aligned} $$

(20)

and

$$\displaystyle \begin{aligned} v_1^\perp = - v_2^\perp. {} \end{aligned} $$

(21)

Moreover,

$$\displaystyle \begin{aligned} \pi_\star(n_A) = -\pi_\star(n_B). {} \end{aligned} $$

(22)

Our next goal is to glue together 1-forms along $A\subset \partial {\mathcal M}_1$ and $B\subset \partial {\mathcal M}_2$:

Lemma A.2 (Gluing of Forms)

Let $\omega _1\in \varOmega ^1({\mathcal M}_1)$ and $\omega _2\in \varOmega ^1({\mathcal M}_2)$ satisfy the following conditions:

• (i) Equality of tangential component:

  $$\displaystyle \begin{aligned} h^\star (\eta_A^\star\omega_1)=\eta_B^\star\omega_2 {} \end{aligned} $$

  (23)

  (this is an equality of 1-forms on B).

• (ii) Matching of normal component:

  $$\displaystyle \begin{aligned} \omega_1(n_A)\circ h=-\omega_2(n_B) {} \end{aligned} $$

  (24)

  (this is an equality of functions on B).

• (iii) Matching of normal derivative:

  $$\displaystyle \begin{aligned} ({\mathcal L}_{n_A}\omega_1(n_A)) \circ h = -{\mathcal L}_{n_B}\omega_2(n_B), \end{aligned}$$

  and

  $$\displaystyle \begin{aligned} h^\star\left(\eta_A^\star({\mathcal L}_{n_A}\omega_1)\right)=-\eta_B^\star({\mathcal L}_{n_B}\omega_2), \end{aligned}$$

  where ${\mathcal L}$ is the Lie derivative and n _A and n _B are extended to neighborhoods of $A\subset {\mathcal M}_1$ and $B\subset {\mathcal M}_2$ via (19).

Then, there exists a 1-form ω on which is C ¹ with respect to the smooth structure induced by ι _A and ι _B , such that the restrictions of ω to ${\mathcal M}_1$ and ${\mathcal M}_2$ coincide with ω ₁ and ω ₂.

Proof

Let be the induced form on the disjoint union. We first show that Conditions (i) and (ii) imply that projects to a well-defined 1-form ω on .

Consider first , and let . Since π ⁻¹(π(p)) = {p} and dπ _p is an isomorphism, we may define

Next, let p ∈ B and let . Now π ⁻¹(π(p)) = {h(p), p} and dπ ⁻¹(v) = {v ₁, v ₂}, where $v_1\in T_{h(p)}{\mathcal M}_1$ and $v_2\in T_p{\mathcal M}_2$. In order to define ω _π(p)(v) unambiguously, it suffices to show that ω ₁(v ₁) = ω ₂(v ₂).

Write as above

$$\displaystyle \begin{aligned} \begin{aligned} v_1 &= d\eta_A (v_1^\parallel) + v_1^\perp\, (n_A)_{h(p)} \\ v_2 &= d\eta_B(v_2^\parallel) + v_2^\perp\, (n_B)_p, \end{aligned} \end{aligned}$$

where $v_1^\parallel \in T_{h(p)}A$ and $v_2^\parallel \in T_pB$. Then,

$$\displaystyle \begin{aligned} \begin{aligned} \omega_1(d\eta_A (v_1^\parallel)) &= \eta_A^\star\omega_1 (v_1^\parallel) \stackrel{(A.3)}{=} \eta_A^\star\omega_1 (dh(v_2^\parallel)) = h^\star \eta_A^\star\omega_1 (v_2^\parallel) \\ &\stackrel{(A.6)}{=} \eta_B^\star\omega_2 (v_2^\parallel) = \omega_2 (d\eta_B(v_2^\parallel)), \end{aligned} \end{aligned}$$

and

$$\displaystyle \begin{aligned} \begin{aligned} \omega_1(v_1^\perp\, (n_A)_{h(p)}) &= v_1^\perp\, \omega_1(n_A)_{h(p)} \stackrel{(A.4)}{=} -v_2^\perp\, \omega_1(n_A)_{h(p)} \\ &\stackrel{(A.7)}{=} v_2^\perp\, \omega_2(n_B)_p = \omega_2(v_2^\perp\, n_B)_p. \end{aligned} \end{aligned}$$

We have thus proved that ω is well-defined. It remains to show that ω (or equivalently Φ _⋆ ω) is continuously differentiable. For (t, p) ∈ (−1, 1) × A and α∂ _t ⊕ v ∈ T((−1, 1) × A) ≃ T _t(−1, 1) ⊕ T _p A,

$$\displaystyle \begin{aligned} \left(\varPhi_\star\omega\right)|{}_{(t,p)}(\alpha\partial_t\oplus v)=\begin{cases} \omega_1|{}_{\iota_A(-t,p)}(-\alpha n_A+v) & t< 0\\ \omega_2|{}_{\iota_B(t,p)}(\alpha n_B+dh(v))& t\geq 0. \end{cases} \end{aligned}$$

Condition (i) then implies that the tangential (to A) derivatives of Φ _⋆ ω are continuous and Condition (iii) shows (by a similar calculation) that it is continuously differentiable in the “t” direction (one-sided limits coincide). This completes the proof. □