The energy scaling behaviour of singular perturbation models of staircase type in linearized elasticity for higher order laminates

TL;DR

The paper develops a Fourier-based framework to derive sharp energy scaling laws for singular perturbation models of staircase-type microstructures in geometrically linearized elasticity with higher-order laminates and gauge invariance. It shows that two order parameters—the lamination order of the boundary data and the number of degenerate versus non-degenerate symmetrized rank-one directions—govern the scaling, with explicit lower bounds across Dirichlet and periodic settings for configurations ranging from two to $m+1$ wells. The results reveal novel scaling regimes induced by the gauge group Skew$(d)$, including exponents such as $2p/(2p+3)$ for second-order laminates and $2p/(2p+4)$ for third-order laminates, and provide upper-bound constructions in select geometries that suggest sharpness. The work advances the understanding of microstructure complexity in vector-valued, gauge-invariant elasticity models and informs how boundary data geometry and compatibility directions shape energy minimization in multi-well materials.

Abstract

We investigate the scaling behaviour of a singular perturbation model within the geometrically linearized theory of elasticity involving data of higher lamination order. We study boundary data which are of staircase type and show rather general lower scaling bounds, both in the setting of prescribed Dirichlet data and for periodic configurations with a mean value constraint. In contrast to the setting without gauge invariances, these lower scaling bounds depend on \emph{two} parameters -- the order of lamination of the boundary data as well as the number of involved (non-)degenerate symmetrized rank-one directions. By discussing upper bounds in specific geometries and for a specific constellation of wells, we give evidence of the sharpness of these lower bound estimates. Hence, it is necessary to keep track of the outlined \emph{two} parameters in deducing scaling laws within the geometrically linearized theory of elasticity.

Figures (7)

• Figure 1: A schematic two-dimensional visualization of the symmetrized lamination convex hull in the cases \ref{['eq:choiceofmatrices']} (left) and \ref{['eq:secondcase']} (right). The numbers enclosed by the circle represent the number of compatibility directions between the wells.
• Figure 2: Schematic visualization of the staircase in \ref{['eq:constructiondim']}. The nodes represent the set $K_m$ while the one-dimensional lines visualize the symmetrized lamination convex hull $K_m^{(\mathrm{lc})}$. The function $f$ represents the number of compatibility directions between matrices on these lines. Notice that in two dimensions, $K_m^{(\mathrm{lc})}$ cannot be one-dimensional, see Proposition \ref{['prop:topoprop']} below, i.e., this figure should be interpreted schematically.
• Figure 3: Visualization of the cone $C_{j, \kappa, \mu}$ in \ref{['eq:defcones']}. The horizontal axis indicates the vector space $V_j$ whereas the vertical axis represents the orthogonal vector space $V_j^\perp$. The surface in blue depicts the intersection of the unit disc with the cone.
• Figure 4: A schematic illustration of the iteration argument for the general lower bound. The circles represent cones associated with the $i$th index, $i \in \{1 ,\dots , l(\ell) \}$, where cones corresponding to two compatibility directions (green circles) are controlled via Case 2, and the ones in black via Case 1, i.e., black circles indicate cones around degenerate compatibility directions. More precisely, the Fourier coefficients outside of the cones associated to $S_1$ (green circles) are controlled via a cone corresponding either to degenerate compatibility directions (black circle with lower index) or again to non-degenerate directions (green circle with lower index), and controlled error terms (blue circles). The newly arising cones are truncated at higher frequencies, where the new effective truncation parameter is (roughly) the length of the $i$th cone divided by $\kappa_{S_1}$ or $\kappa_{S_2}$, respectively. Although all paths in the picture can arise, we note that for a given $f \in \mathcal{S}$, the iteration follows only one of the dashed arrows. In Case 1 the cones are truncated via non-degenerate ones, where the preceding (black arrow) and following index (red arrow) arises. The red arrows need a separate discussion in order to ensure that the resulting contributions remain controlled.
• Figure 5: Visualization of the constructions in Lemma \ref{['lem:firstorder2d']} for $H = L = 1$. Whereas the interfaces in Lemma \ref{['lem:firstorder2d']}(i) can be straight lines, the interfaces in (ii) necessarily need to be curved. The picture shows $j_0 =2$ layers of building blocks, where the blue part corresponds to the cut-off layer. In Lemma \ref{['lem:firstorder2d']}(i), we have $\partial_1 u_2 = -2$ on $\omega_1$ and $\omega_3$, and on $\omega_2$ and $\omega_4$, it holds that $\partial_1 u_2 = 2$.

Theorems & Definitions (53)

• Theorem 1.1: Scaling of the two-well problem CC14
• Theorem 1.2: Lower bounds for the three-well problem
• Proposition 1.3: Upper bound for a cylinder in three dimensions
• Theorem 1.4: Upper bound for a second order laminate in the case \ref{['eq:choiceofmatrices']}
• Theorem 1.5: Lower bounds for a four-well problem
• Remark 1.6: Cardinality of $\mathcal{S}_m$
• Theorem 1.7: Lower bounds for the $m+1$-well problem
• Remark 1.8
• Theorem 1.9: Lower bounds for the $m+1$-well problem in the periodic case
• Definition 2.1