Integrating local energetics into Maxwell-Calladine constraint counting to design mechanical metamaterials

Abstract

The Maxwell-Calladine index theorem plays a central role in our current understanding of the mechanical rigidity of discrete materials. By considering the geometric constraints each material component imposes on a set of underlying degrees of freedom, the theorem relates the emergence of rigidity to constraint counting arguments. However, the Maxwell-Calladine paradigm is significantly limited-its exclusive reliance on the geometric relationships between constraints and degrees of freedom completely neglects the actual energetic costs of deforming individual components. To address this limitation, we derive a generalization of the Maxwell-Calladine index theorem based on susceptibilities that naturally incorporate local energetic properties such as stiffness and prestress. Using this extended framework, we investigate how local energetics modify the classical constraint counting picture to capture the relationship between deformations and external forces. We then combine this formalism with group representation theory to design mechanical metamaterials where differences in symmetry between local energy costs and structural geometry are exploited to control responses to external forces.

FIG. 1.. Local energetic properties, external perturbations, and elastic responses.

The linear elastic properties are shown for a simple central-force spring network. (a) The stiffnesses of the bonds $k_{\alpha }$ (top), represented by line thickness, characterize their compressibility, while the prestresses $t_{\alpha }^{\mathrm{p}\mathrm{s}}$ (bottom), characterize any geometric frustration that arises when bonds are stretched or compressed in the ground state configuration. Green arrows indicate the prestress tension exerted by each bond on the rest of the network. (b) From the perspective of the nodes, external forces ${\overrightarrow{f}}_i$ (top) result in displacements ${\overrightarrow{u}}_i$ (bottom). The notation ${\overrightarrow{u}}_i$ and ${\overrightarrow{f}}_i$ represent the $d$-dimensional vectors of displacements and forces associated with each node $i$ for a network in $d$ dimensions. (c) From the perspective of the bonds, the external forces on the nodes are equivalent to external bond tensions $t_{\alpha }$ (top), while node displacements correspond to bond extensions $\delta {\mathit{\ell }}_{\alpha }$ (bottom) indicated by the color scale. (d) The network contains a state of self-stress (top) [identical to the prestress in (a)] composed of external tensions that result in no net forces on the nodes, along with a linear zero mode (bottom) composed of displacements do result in zero extension of the bonds.

FIG. 2.. Susceptibilities and their relationships as maps between vector spaces.

The four susceptibilities, the compatibility matrix, and its transpose provide maps between forces, tensions, displacements, and extensions. While, $\mathbf{C}$ and ${\mathbf{C}}^T$ map between node space (degrees of freedom) (top row) and bond space (components/constraints) (bottom row), the susceptibilities map external perturbations (left column) to deformation responses (right column). Arrows indicate the direction of each map with dashed arrows representing operators that are compositions of other operators.

FIG. 3.. Breaking and restoring dihedral symmetry with prestress.

A diamond-shaped central-force spring network with fixed boundary conditions whose geometry exhibits four-fold dihedral symmetry $D_4$. In each row, a linear zero mode (LZM) and the linear zero-extension force (LZEF) that couples to it via the susceptibility $\partial \overrightarrow{\mathbf{u}}/\partial \overrightarrow{\mathbf{f}}$ are shown for a different choice of prestress. For each case, the rightmost column lists the symmetry groups describing the geometry, energy (prestress), elastic response, LZM, and LZEF. In all cases, the spring constants are chosen to be uniform $(\mathbf{K}=\mathbf{I})$. (a) When the prestress is zero $\left({\overrightarrow{\mathbf{t}}}^{\mathrm{p}\mathrm{s}}=0\right)$ the energy exhibits full permutation symmetry $S_{12}$ which encompasses the symmetry of the geometry, resulting in overall $D_4$ symmetry. The (a-i) LZM and its corresponding (a-ii) LZEF are identical, both exhibiting full four-fold dihedral symmetry. (b) Introducing nonzero prestress along the central line of horizontal bonds breaks the energetic symmetry to $S_4\otimes S_8$, reducing overall symmetry of the response to two-fold dihedral symmetry $D_2$. While the (b-i) LZM is the same, the (b-ii) LZEF that couples to it no longer matches. (c) Introducing additional prestress along the central line of vertical bonds restores some of the energetic symmetry to $S_4\otimes S_8$ so that the overall response exhibits four-fold dihedral symmetry $D_4$. The (c-i) LZM and the (c-ii) LZM that couples to it match once again.

FIG. 4.. Eigenmode structure and irreducible representations of diamond network.

The energy levels and eigenmodes for each configuration in Fig. 3 labeled according to IRREP. In each column, the local energetic properties are depicted in the upper-left panel. The bond stiffnesses $k_{\alpha }$ are chosen to be uniform for each network, while the prestress $t_{\alpha }^{\mathrm{p}\mathrm{s}}$ is shown with green arrows. In the bottom-right, we show the energy levels ${\omega }_{{\mathrm{\Gamma }}_s}^2$ (eigenvalues of the Hessian written as squared vibrational frequencies). Each energy is labeled with an IRREP $\mathrm{\Gamma }$ and a copy number $s$. In the right panel, we depict the eigenmodes of the Hessian $|{\mathrm{\Gamma }}_s,i\rangle$ corresponding to each energy level. Solid boxes group together modes that transform according to the same IRREP $\mathrm{\Gamma }$, while dashed boxes group together degenerate modes corresponding to a single copy $s$ of $\mathrm{\Gamma }$. In red we highlight the energy level and eigenmodes that couple to the corresponding LZMs and LZEFs in Fig. 3. (a) When the symmetry of the elastic response matches the geometric symmetry, the LZM spans the entire one-dimensional invariant subspace corresponding to the $1^{\prime }$ IRREP of $D_4$. As a result, the LZM and LZEF are guaranteed to match. (b) When the energetic symmetry is more restrictive than the geometric symmetry, the the LZM now couples to two different eigenmodes from two different copies of the $1^{\prime }$ IRREP for $D_2$, but does not span the entire two-dimensional subspace spanned by all copies of $1^{\prime }$. Therefore, the LZM and LZEF are no longer guaranteed to match. (c) The symmetry of the elastic response matches the geometric symmetry again, so the LZM again spans the entire invariant subspace corresponding to the $1^{\prime }$ IRREP of $D_4$. The LZM and LZEF now both match again.

FIG. 5.. Repeatedly breaking translational symmetry with prestress.

A central-force spring network forming a $4\times 4$ periodic square lattice. The network geometry exhibits four-fold dihedral symmetry, along with discrete translational symmetry along the $x$- and $y$-axes, forming the group $\left(Z_4\otimes Z_4\right)⋊D_4$. In each row, two linear zero modes (LZMs) and the linear zero-extension forces (LZEFs) that couple to them via the susceptibility $\partial \overrightarrow{\mathbf{u}}/\partial \overrightarrow{\mathbf{f}}$ are shown for a different choice of prestress. The central column lists the symmetry groups for each case describing the geometry, energy (prestress), elastic response, LZMs, and LZEFs. In all cases, the spring constants are chosen to be uniform ($\mathbf{K}=\mathbf{I}$). (a) For zero prestress $\left({\overrightarrow{\mathbf{t}}}^{\mathrm{p}\mathrm{s}}=0\right)$ the energy exhibits full permutation symmetry $S_{32}$, resulting in overall $\left(Z_4\otimes Z_4\right)⋊D_4$ symmetry. Each LZM matches the LZEF that couples to it, with one pair, (a-i) and (a-ii), exhibiting $x$-translational symmetry, and the other pair, (a-iii) and (a-iv), exhibiting $y$-translational symmetry. (b) Introducing nonzero prestress along two horizontal lines of bonds ($t_{\alpha }^{\text{ps}}=0.8$) breaks the $y$-translational, symmetry reducing the energetic symmetry to $S_8\otimes S_{24}$, and the overall elastic response to $Z_4⋊D_2$. The (b-iv) LZEF that previously displayed $y$-translational symmetry no longer matches its (b-iii) LZM, while the (b-ii) LZEF that displayed $x$-translational symmetry is unaffected. (c) Introducing additional prestress ($t_{\alpha }^{\mathrm{p}\mathrm{s}}=0.4$ ) along two vertical lines of bonds, but with smaller magnitude than the prestressed horizontal bonds, breaks $x$-translational symmetry, reducing the energetic symmetry to $S_8\otimes S_8\otimes S_{24}$, and the overall elastic response to two-fold dihedral symmetry $D_2$. In this case, neither LZEF matches it corresponding LZM.

FIG. 6.. Energy levels and irreducible representations of 4×4 square lattice.

The energy levels for each configuration in Fig. 5 labeled according to IRREP. The local energetic properties are depicted in the top row. The bond stiffnesses $k_{\alpha }$ are chosen to be uniform for each network, while the prestress $t_{\alpha }^{\mathrm{p}\mathrm{s}}$ is shown with green arrows. In the bottom row, we show the energy levels ${\omega }_{{\mathrm{\Gamma }}_s}^2$, each with an IRREP and copy number $s$. In red and blue we highlight the energy levels that couple to the LZMs with $x$- and $y$-translational symmetry, respectively [see Fig. 5]. (a) When the elastic response symmetry matches the geometric symmetry, the LZMs span the entire invariant subspaces corresponding to the $2^{\left(2\right)},2^{\left(6\right)}$, and $4^{\left(5\right)}$ IRREPs of $\left(Z_4\otimes Z_4\right)⋊D_4$. As a result, the LZMs and LZEFs are guaranteed to match. (b) Upon breaking $y$-translational symmetry with prestress, the LZMs with $y$-translational symmetry no longer span the entire invariant subspace for any IRREP and no longer match their LZEFs. In contrast, preservation of $x$-translational symmetry causes the LZMs with $x$-translational symmetry to fully span the subspaces corresponding to the the $1^{\left(5\right)}$ and $1^{\left(7\right)}$ IRREPs of $Z_4⋊D_2$ and therefore match their LZEFs. (c) Upon further breaking $x$-translational symmetry with prestress, the LZMs with $x$-translational symmetry also no longer span the entire invariant subspace for any IRREP, so that none of the LZMs are guaranteed to match their LZEFs.