On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions

Abstract

We provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the order of lamination of the affine displacement data determines the complexity of the microstructure. As in \cite{RT21} we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.

Figures (6)

• Figure 1: A graphic representation of the arguments in the proof of Proposition \ref{['prop:loc2']}. The vector $b$ is the projection of $b_1$ on the space $V$.
• Figure 2: The slice orthogonal to $d$ of the sets $\Omega_i^{(j)}$.
• Figure 3: The domain subdivision of $\Omega_1^{(j)}$ into the sets $S$, $T_l$ and $T_r$. The regions $S_i$ and $\omega$ (cf. Steps 1 and 3 in the proof of Lemma \ref{['lem:2ord1']}) are also depicted. The shaded region represents the cell $\omega^{(i)}$ which is depicted in more detail in Figure \ref{['fig:oscA2']}.
• Figure 4: The domain subdivision of $\omega^{(i)}$.
• Figure 5: The domain subdivision of $T_l$ into sets $\omega_m$.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3: Upper bound construction, second order laminates
• Remark 1.1
• Theorem 4: Upper bound construction, first order laminates
• Definition 2.1: Laminates, order of lamination
• Lemma 3.1: Lemma 4.1 in KKO13
• Lemma 3.2: Lemma 4.2 in KKO13
• Proposition 3.3
• Lemma 3.4: Lemma 4.5, RT22