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Architected materials achieve unique mechanical properties through precisely engineered microstructures that minimize material usage. However, a key challenge of low-density materials is balancing high stiffness with stable deformability up to large strains. Current microstructures, which employ slender elements such as thin beams and plates arranged in periodic patterns to optimize stiffness, are largely prone to instabilities, including buckling and brittle collapse at low strains. This challenge is here addressed by introducing a new class of aperiodic architected materials inspired by quasicrystalline lattices. Beam networks derived from canonical quasicrystalline patterns, such as the Penrose tiling in 2D and icosahedral quasicrystals in 3D, are shown to create stiff, stretching-dominated topologies with non-uniform force chain distributions, effectively mitigating the global instabilities observed in periodic designs. Numerical and experimental results confirm the effectiveness of these designs in combining stiffness and stable deformability at large strains, representing a significant advancement in the development of low-density metamaterials for applications requiring high impact resistance and energy absorption. Our results demonstrate the potential of deterministic quasi-periodic topologies to bridge the gap between periodic and random structures, while branching towards uncharted territory in the property space of architected materials.

We study the rigidity properties of the $T_3$-structure for the symmetrized gradient from \cite{BFJK94} qualitatively, quantitatively and numerically. More precisely, we complement the flexibility result for approximate solutions of the associated differential inclusion which was deduced in \cite{BFJK94} by a rigidity result on the level of exact solutions and by a quantitative rigidity estimate and scaling result. The $T_3$-structure for the symmetrized gradient from \cite{BFJK94} can hence be regarded as a symmetrized gradient analogue of the Tartar square for the gradient. As such a structure cannot exist in $\mathbb{R}^{2\times 2}_{sym}$ the example from \cite{BFJK94} is in this sense minimal. We complement our theoretical findings with numerical simulations of the resulting microstructure., Comment: 41 pages, 5 figures, comments welcome

We introduce HyperCAN, a machine learning framework that utilizes hypernetworks to construct adaptable constitutive artificial neural networks for a wide range of beam-based metamaterials exhibiting diverse mechanical behavior under finite deformations. HyperCAN integrates an input convex network that models the nonlinear stress-strain map of a truss lattice, while ensuring adherence to fundamental mechanics principles, along with a hypernetwork that dynamically adjusts the parameters of the convex network as a function of the lattice topology and geometry. This unified framework demonstrates robust generalization in predicting the mechanical behavior of previously unseen metamaterial designs and loading scenarios well beyond the training domain. We show how HyperCAN can be integrated into multiscale simulations to accurately capture the highly nonlinear responses of large-scale truss metamaterials, closely matching fully resolved simulations while significantly reducing computational costs. This offers new efficient opportunities for the multiscale design and optimization of truss metamaterials.

The nucleation and propagation of disconnections play an essential role during twin growth. Atomistic methods can reveal such small structural features on twin facets and model their motion, yet are limited by the simulation length and time scales. Alternatively, mesoscale modeling approaches (such as the phase field method) address these constraints of atomistic simulations and can maintain atomic-level accuracy when integrated with atomic-level information. In this work, a phase field model is used to simulate the disconnection-mediated twinning, informed by molecular dynamics (MD) simulations. This work considers the specific case of the growth of $\{10\bar{1}2\}$ twin in magnesium. MD simulations are first conducted to obtain the orientation-dependent interface mobility and motion threshold, and to simulate twin embryo growth and collect facet velocities, which can be used for calibrating the continuum model. The phase field disconnections model, based on the principle of minimum dissipation potential, provides the theoretical framework. This model incorporates a nonconvex grain boundary energy, elasticity and shear coupling, and simulates disconnections as a natural emergence under the elastic driving force. The phase field model is further optimized by including the anisotropic interface mobility and motion threshold suggested by MD simulations. Results agree with MD simulations of twin embryo growth in the aspects of final twin thickness, twin shape, and twin size, as well as the kinetic behavior of twin boundaries and twin tips. The simulated twin microstructure is also consistent with experimental observations, demonstrating the fidelity of the model.

Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework to inform denoising diffusion models of underlying constraints on such generated samples during model training. Our approach improves the alignment of the generated samples with the imposed constraints and significantly outperforms existing methods without affecting inference speed. Additionally, our findings suggest that incorporating such constraints during training provides a natural regularization against overfitting. Our framework is easy to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives., Comment: 15 pages, 4 figures; added further theoretical motivation, new residual estimation mechanism and additional experimental study

Predicting the mechanics of large structural networks, such as beam-based architected materials, requires a multiscale computational strategy that preserves information about the discrete structure while being applicable to large assemblies of struts. Especially the fracture properties of such beam lattices necessitate a two-scale modeling strategy, since the fracture toughness depends on discrete beam failure events, while the application of remote loads requires large simulation domains. As classical homogenization techniques fail in the absence of a separation of scales at the crack tip, we present a concurrent multiscale technique: a fully-nonlocal quasicontinuum (QC) multi-lattice formulation for beam networks, based on a conforming mesh. Like the original atomistic QC formulation, we maintain discrete resolution where needed (such as around a crack tip) while efficiently coarse-graining in the remaining simulation domain. A key challenge is a suitable model in the coarse-grained domain, where classical QC uses affine interpolations. This formulation fails in bending-dominated lattices, as it overconstrains the lattice by preventing bending without stretching of beams. Therefore, we here present a beam QC formulation based on mixed-order interpolation in the coarse-grained region -- combining the efficiency of linear interpolation where possible with the accuracy advantages of quadratic interpolation where needed. This results in a powerful computational framework, which, as we demonstrate through our validation and benchmark examples, overcomes the deficiencies of previous QC formulations and enables, e.g., the prediction of the fracture toughness and the diverse nature of stress distributions of stretching- and bending-dominated beam lattices in two and three dimensions.

Grain boundary (GB) properties greatly influence the mechanical, electrical, and thermal response of polycrystalline materials. Most computational studies of GB properties at finite temperatures use molecular dynamics (MD), which is computationally expensive, limited in the range of accessible timescales, and requires cumbersome techniques like thermodynamic integration to estimate free energies. This restricts the reasonable computation (without incurring excessive computational expense) of GB properties to regimes that are often unrealistic, such as zero temperature or extremely high strain rates. Consequently, there is a need for simulation methodology that avoids the timescale limitations of MD, while providing reliable estimates of GB properties. The Gaussian Phase-Packet (GPP) method is a temporal coarse-graining technique that can predict relaxed atomic structures at finite temperature in the quasistatic limit. This work applies GPP, combined with the quasiharmonic approximation for computing the free energy, to the problem of determining the free energy and shear coupling factor of grain boundaries in metals over a range of realistic temperatures. Validation is achieved by comparison to thermodynamic integration, which confirms that the presented approach captures relaxed-energy GB structures and shear coupling factors at finite temperature with a high degree of accuracy.

Although metamaterials have been widely used for controlling elastic waves through bandgap engineering, the directed guidance of stress waves in non-periodic structures has remained a challenge. This work demonstrates that spatially graded metamaterials based on conformal mappings present a rich design space for controlling and attenuating wave motion - without the need for bandgaps. Conformal mappings transform an elementary unit cell by scaling and rotation into graded lattices with approximately geometrically similar unit cells. This self-similarity allows for control over the local wave dispersion throughout the metamaterial. As a key mechanism, it is shown that elastic waves cannot propagate through graded unit cells with significant size differences, except at low frequencies. This is exploited to create low-pass elastic wave guides, extending beyond classical bandgap engineering, since bandgaps are not required to achieve wave guiding and attenuation. Experiments confirm the low-pass elastic wave filtering capability of a planar truss metamaterial with conformal grading. Finally, a systematic design of curved metamaterial surfaces is presented, providing a flexible framework for programming low-pass attenuation and wave guiding in three dimensions.

The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials--truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of truss-based metamaterials has remained highly limited and often heuristic, due to the vast, discrete design space and the lack of a comprehensive parameterization. We here present a graph-based deep learning generative framework, which combines a variational autoencoder and a property predictor, to construct a reduced, continuous latent representation covering an enormous range of trusses. This unified latent space allows for the fast generation of new designs through simple operations (e.g., traversing the latent space or interpolating between structures). We further demonstrate an optimization framework for the inverse design of trusses with customized mechanical properties in both the linear and nonlinear regimes, including designs exhibiting exceptionally stiff, auxetic, pentamode-like, and tailored nonlinear behaviors. This generative model can predict manufacturable (and counter-intuitive) designs with extreme target properties beyond the training domain.

The accelerated inverse design of complex material properties - such as identifying a material with a given stress-strain response over a nonlinear deformation path - holds great potential for addressing challenges from soft robotics to biomedical implants and impact mitigation. While machine learning models have provided such inverse mappings, they are typically restricted to linear target properties such as stiffness. To tailor the nonlinear response, we here show that video diffusion generative models trained on full-field data of periodic stochastic cellular structures can successfully predict and tune their nonlinear deformation and stress response under compression in the large-strain regime, including buckling and contact. Unlike commonly encountered black-box models, our framework intrinsically provides an estimate of the expected deformation path, including the full-field internal stress distribution closely agreeing with finite element simulations. This work has thus the potential to simplify and accelerate the identification of materials with complex target performance., Comment: 21 pages, 3 figures

Overcoming the time scale limitations of atomistics can be achieved by switching from the state-space representation of Molecular Dynamics (MD) to a statistical-mechanics-based representation in phase space, where approximations such as maximum-entropy or Gaussian phase packets (GPP) evolve the atomistic ensemble in a time-coarsened fashion. In practice, this requires the computation of expensive high-dimensional integrals over all of phase space of an atomistic ensemble. This, in turn, is commonly accomplished efficiently by low-order numerical quadrature. We show that numerical quadrature in this context, unfortunately, comes with a set of inherent problems, which corrupt the accuracy of simulations -- especially when dealing with crystal lattices with imperfections. As a remedy, we demonstrate that Graph Neural Networks, trained on Monte-Carlo data, can serve as a replacement for commonly used numerical quadrature rules, overcoming their deficiencies and significantly improving the accuracy. This is showcased by three benchmarks: the thermal expansion of copper, the martensitic phase transition of iron, and the energy of grain boundaries. We illustrate the benefits of the proposed technique over classically used third- and fifth-order Gaussian quadrature, we highlight the impact on time-coarsened atomistic predictions, and we discuss the computational efficiency. The latter is of general importance when performing frequent evaluation of phase space or other high-dimensional integrals, which is why the proposed framework promises applications beyond the scope of atomistics.

The numerical modeling of thin shell structures is a challenge, which has been met by a variety of finite element (FE) and other formulations -- many of which give rise to new challenges, from complex implementations to artificial locking. As a potential alternative, we use machine learning and present a Physics-Informed Neural Network (PINN) to predict the small-strain response of arbitrarily curved shells. To this end, the shell midsurface is described by a chart, from which the mechanical fields are derived in a curvilinear coordinate frame by adopting Naghdi's shell theory. Unlike in typical PINN applications, the corresponding strong or weak form must therefore be solved in a non-Euclidean domain. We investigate the performance of the proposed PINN in three distinct scenarios, including the well-known Scordelis-Lo roof setting widely used to test FE shell elements against locking. Results show that the PINN can accurately identify the solution field in all three benchmarks if the equations are presented in their weak form, while it may fail to do so when using the strong form. In the thin-thickness limit, where classical methods are susceptible to locking, training time notably increases as the differences in scaling of the membrane, shear, and bending energies lead to adverse numerical stiffness in the gradient flow dynamics. Nevertheless, the PINN can accurately match the ground truth and performs well in the Scordelis-Lo roof benchmark, highlighting its potential for a drastically simplified alternative to designing locking-free shell FE formulations., Comment: 24 pages, 16 figures

We introduce a computational framework for the topology optimization of cellular structures with spatially varying architecture, which is applied to functionally graded truss lattices under quasistatic loading. We make use of a first-order homogenization approach, which replaces the discrete truss by an effective continuum description to be treated by finite elements in a macroscale boundary value problem. By defining the local truss architecture through a set of Bravais vectors, we formulate the optimization problem with regards to the spatially varying basis vectors and demonstrate its feasibility and performance through a series of benchmark problems in 2D (though the method is sufficiently general to also apply in 3D, as discussed). Both the displacement field and the topology are continuously varying unknown fields on the macroscale, and a regularization is included for well-posedness. We argue that prior solutions obtained from aligning trusses along the directions of principal stresses are included as a special case. The outlined approach results in heterogeneous truss architectures with a smoothly varying unit cell, enabling easy fabrication with a tunable length scale (the latter avoiding the ill-posedness stemming from classical nonconvex methods without an intrinsic length scale).

Surface energies and surface elasticity largely affect the mechanical response of nanostructures as well as the physical phenomenon associated with surfaces such as evaporation and adsorption. Studying surface energies at finite temperatures is therefore of immense interest for nanoscale applications. However, calculating surface energies and derived quantities from atomistic ensembles is usually limited to zero temperature or involve cumbersome thermodynamic integration techniques at finite temperature. Here, we illustrate a technique to identify the energy and elastic properties of surfaces of solids at non-zero temperature based on a Gaussian phase packets (GPP) approach (which in the isothermal limit coincides with a maximum-entropy formulation). Using this setup, we investigate the effect of temperature on the surface properties of different crystal faces for six pure metals -- copper, nickel, alumimum, iron, tungsten and vanadium -- thus covering both FCC and BCC lattice structures. While the obtained surface energies and stresses usually show a decreasing trend with increasing temperature, the elastic constants do not show such a consistent trend across the different materials and are quite sensitive to temperature changes. Validation is performed by comparing the obtained surface energy densities of selected BCC and FCC materials to those calculated via molecular dynamics.

When subjected to electro-mechanical loading, ferroelectrics see their polarization evolve through the nucleation and evolution of domains. Existing mesoscale phase-field models for ferroelectrics are typically based on a gradient-descent law for the evolution of the order parameter. While this implicitly assumes that domain walls evolve with linear kinetics, experiments instead indicate that domain wall kinetics is nonlinear. This, in turn, is an important feature for the modeling of rate-dependent effects in polarization switching. We propose a new multiple-phase-field model for ferroelectrics, which permits domain wall motion with nonlinear kinetics, with applications in other solid-solid phase transformation problems. By means of analytical traveling wave solutions, we characterize the interfacial properties (energy and width) and the interface kinetics of straight domain walls, as furnished by the general kinetics model, and compare them to those of the classical Allen--Cahn model. We show that the proposed model propagates domain walls with arbitrarily chosen nonlinear kinetic relations, which can be tuned to differ for the different types of domain walls in accordance with experimental evidence.

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function $W$. We will demonstrate these methods using the planar isotropic rank-one convex function \[ W_{\rm magic}^+(F)=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}-\log\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+\log\det F=\frac{\lambda_{\rm max}}{\lambda_{\rm min}}+2\log\lambda_{\rm min}\,, \] where $\lambda_{\rm max}\geq\lambda_{\rm min}$ are the singular values of $F$, as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies $W:\operatorname{GL}^+(2)\rightarrow\mathbb{R}$ with an additive volumetric-isochoric split of the form \[ W(F)=W_{\rm iso}(F)+W_{\rm vol}(\det F)=\widetilde W_{\rm iso}\bigg(\frac{F}{\sqrt{\det F}}\bigg)+W_{\rm vol}(\det F) \] with a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.

Machine learning has found its way into almost every area of science and engineering, and we are only at the beginning of its exploration across fields. Being a popular, versatile and powerful framework, machine learning has proven most useful where classical techniques are computationally inefficient, which applies particularly to computational solid mechanics. Here, we dare to give a non-exhaustive overview of potential avenues for machine learning in the numerical modeling of solids and structures and offer our (subjective) perspective on what is yet to come., Comment: 11 pages, 4 figures

Magnesium (Mg) has the lowest density of all structural metals and has excellent potential for wide use in structural applications. While pure Mg has inferior mechanical properties; the addition of further elements at various concentrations has produced alloys with enhanced mechanical performance and corrosion resistance. An important consequence of adding such elements is that the saturated Mg matrix can locally decompose to form solute clusters and intermetallic particles, often referred to as precipitates. Controlling the shape, number density, volume fraction, and spatial distribution of solute clusters and precipitates significantly impacts the alloy's plastic response. Conversely, plastic deformation during thermomechanical processing can dramatically impact solute clustering and precipitation. In this paper, we first discuss how solute atoms, solute clusters, and precipitates can improve the mechanical properties of Mg alloys. We do so by primarily comparing three alloy systems: Mg-Al, Mg-Zn, and Mg-Y-based alloys. In the second part, we provide strategies for optimizing such microstructures by controlling nucleation and growth of solute clusters and precipitates during thermomechanical processing. In the third part, we briefly highlight how one can enable inverse design of Mg alloys by a more robust Integrated Computational Materials Design (ICMD) approach.

The quasicontinuum method was originally introduced to bridge across length scales -- from atomistics to significantly larger continuum scales -- thus overcoming a key limitation of classical atomic-scale simulation techniques while solely relying on atomic-scale input (in the form of interatomic potentials). An associated challenge lies in bridging across time scales to overcome the time scale limitations of atomistics. To address the biggest challenge, bridging across both length and time scales, only a few techniques exist, and most of those are limited to conditions of constant temperature. Here, we present a new strategy for the space-time coarsening of an atomistic ensemble, which introduces thermomechanical coupling. We investigate the quasistatics and dynamics of a crystalline solid described as a lattice of lumped correlated Gaussian phase packets occupying atomic lattice sites. By definition, phase packets account for the dynamics of crystalline lattices at finite temperature through the statistical variances of atomic momenta and positions. We show that momentum-space correlation allows for an exchange between potential and kinetic contributions to the crystal's Hamiltonian. Consequently, local adiabatic heating due to atomic site motion is captured. Moreover, within the quasistatic approximation the governing equations reduce to the minimization of thermodynamic potentials such as Helmholtz free energy (depending on the fixed variables), and they yield the local equation of state. We further discuss opportunities for describing atomic-level thermal transport using the correlated Gaussian phase packet formulation and the importance of interatomic correlations. Such a formulation offers a promising avenue for a finite-temperature non-equilibrium quasicontinuum method that may be combined with thermal transport models., Comment: Content of the new submission has changed significantly along with the author list

We present a two-scale topology optimization framework for the design of macroscopic bodies with an optimized elastic response, which is achieved by means of a spatially-variant cellular architecture on the microscale. The chosen spinodoid topology for the cellular network on the microscale (which is inspired by natural microstructures forming during spinodal decomposition) admits a seamless spatial grading as well as tunable elastic anisotropy, and it is parametrized by a small set of design parameters associated with the underlying Gaussian random field. The macroscale boundary value problem is discretized by finite elements, which in addition to the displacement field continuously interpolate the microscale design parameters. By assuming a separation of scales, the local constitutive behavior on the macroscale is identified as the homogenized elastic response of the microstructure based on the local design parameters. As a departure from classical FE$^2$-type approaches, we replace the costly microscale homogenization by a data-driven surrogate model, using deep neural networks, which accurately and efficiently maps design parameters onto the effective elasticity tensor. The model is trained on homogenized stiffness data obtained from numerical homogenization by finite elements. As an added benefit, the machine learning setup admits automatic differentiation, so that sensitivities (required for the optimization problem) can be computed exactly and without the need for numerical derivatives - a strategy that holds promise far beyond the elastic stiffness. Therefore, this framework presents a new opportunity for multiscale topology optimization based on data-driven surrogate models., Comment: 26 pages, 16 figures