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Atomistic / Nonlinear / Linear Elasticity Coupling Model

Construction of the a/n/l problem

We first extend our domain with a region which we denote \Omega^l Then we consider the linearisation of the Cauchy Born model in this region. As what follows requires indices, we briefly employ indicial summation convention. Let F_0\in \R^{m\times2} be a reference strain, then

W(F_0 + G) \approx W(F_0) + \partial_{F_{i\alpha}} W(F_0) G_{i\alpha} + \frac{1}{2}\partial_{F_{i\alpha}F_{j\beta}} W(F_0) G_{i\alpha}G_{j\beta}

thus for our problem:

 W(\nabla (u + u_0)) \approx \underbrace{W(\nabla u_0)}_{= 0} + \underbrace{\partial W (\nabla u_0)}_{= 0}\cdot\nabla u + \frac{1}{2} \nabla u\cdot\underbrace{\partial^2 W (\nabla u_0)}_{:=\mu I}\nabla u

Where \mu is known as the shear modulus of linear elasticity. If we denote the fourth order tensor A^{j\beta}_{i\alpha} := \partial_{F_{i\alpha}F_{j\beta}} W(F_0), then for a small displacement u, we obtain the linearised energy difference functional:

\mathcal{E}^l(u) = \frac{1}{2} \int_{\R^2}A^{j\beta}_{i\alpha}\nabla_\alpha u_i \nabla_\beta u_j \dd x

In particular, in our project we are concerned with 1D motion in an antiplane fashion, (i.e displacements occur perpendicular to the atomistic lattice). Under this assumption, which implies F_0 = 0 here, and if we use a potential V uses nearest neighbour interactions, the linearised energy difference functional becomes

 \mathcal{E}^l(u) := \frac{\mu}{2} \int_{\Omega^l}|\nabla u|^2

Due to the fundemental local definition of the continuum models, there are no long range interactions that are taken into account, and thus no force imbalances over the continuum interface. So coupling the energy is simply acheived by summing the energies in each domain, Hence:

 \mathcal{E}^{anl}(u) := \mathcal{E}^{a}(u) + \mathcal{E}^{\text{GRAC}}(u) + \mathcal{E}^{n}(u) + \mathcal{E}^{l}(u)

then the A/N/L coupling problem seeks to find

 u^{anl} \in \arg \min \mathcal{E}^{anl}(u)

Consistency errors and convergence rates

We now state the construction of the analytical error estimates in H^1 seminorm between this model and the ideal atomistic model. The consistency analysis is straightforward as we are able to split up the consistency difference into two parts:
\langle\partial\mathcal{E}^{\text{a}}(u)- \partial\mathcal{E}^{\text{anl}}(u),v\rangle = \langle\partial\mathcal{E}^{\text{a}}(u)- \mathcal{E}^{\text{ac}}(u),v\rangle + \langle\partial\mathcal{E}^{\text{ac}}(u)- \mathcal{E}^{\text{anl}}(u),v\rangle \hspace{15pt} \text{for } v \in \dot{\mathscr{W}}^{ac}_h

The first of these is considered in the Atomistic/Nonlinear coupling section. The second term is then dealt with an application of the taylors theorem to yield.

\langle\partial\mathcal{E}^{\text{ac}}(u)- \partial\mathcal{E}^{\text{anl}}(u),v\rangle= \int_{\Omega^l} (\nabla W(\nabla u) - \mu\nabla u) \cdot \nabla v \dd x \lesssim \tilde{C} \|(\nabla u)^2\|_{L^2} \|\nabla v\|_{L^2}

From which we can obtain a rate of convergence based on the radius of the atomistic domain R for our point defect (which satisfies  \nabla u\sim R^{-2}):

\|(\nabla u)^2\|_{L^2} \lesssim \left(\int_R^\infty r\cdot (r^{-4})^2 \dd r\right)^{\frac{1}{2}} = \left(\int_R^\infty r^{-7} \dd r\right)^{\frac{1}{2}} = \frac{C}{R^3}

So, as in the nonlinear elasticity case the boundary is the biggest source of error. In fact, our numerical tests found there is no practical difference between the nonlinear and linear elasticity minimisers in the case of this defect.