This study demonstrates that two- and three-dimensional spatially graded, truss-based polymeric-material metamaterials can be designed for beneficial impact mitigation and energy absorption capabilities. Through a combination of numerical and experimental techniques, we highlight the broad property space of periodic viscoelastic trusses, realized using 3D printing via selective laser sintering. Extending beyond periodic designs, we investigate the impact response of spatially variant viscoelastic lattices in both two and three dimensions. Our result reveal that introducing spatial variations in lattice topology allows for redirecting of the impact trajectory, opening new opportunities for engineering and tailoring lightweight materials with target impact functionality. This is achieved through the combined selection of base material and metamaterial design.

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 neural 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.

Natural porous materials have exceptional properties—for example, light weight, mechanical resilience, and multi-functionality. Efforts to imitate their properties in engineered structures have limited success. This, in part, is caused by the complexity of multi-phase materials composites and by the lack of quantified understanding of each component's role in overall hierarchy. This challenge is twofold: 1) computational. because non-periodicity and defects render constructing design guidelines between geometries and mechanical properties complex and expensive and 2) experimental. because the fabrication and characterization of complex, often hierarchical and non-periodic 3D architectures is non-trivial.

Although architected materials based on truss networks have been shown to possess advantageous or extreme mechanical properties, those can be highly affected by tolerances and uncertainties in the manufacturing process, which are usually neglected during the design phase. Deterministic computational tools typically design structures with the assumption of perfect, defect-free architectures, while experiments have confirmed the inevitable presence of imperfections and their possibly detrimental impact on the effective properties. Information about the nature and expected magnitude of geometric defects that emerge from the additive manufacturing processes would allow for new designs that aim to mitigate (or at least account for) the effects of defects and to reduce the uncertainty in the effective properties. To this end, we here investigate the effects of four most commonly found types of geometric imperfections in trusses, applied to eleven representative truss topologies in two and three dimensions. Through our study, we (i) quantify the impact of imperfections on the effective stiffness through computational homogenization, (ii) examine the sensitivity of the various truss topologies with respect to those imperfections, (iii) demonstrate the applicability of the model through experiments on 3D-printed trusses, and (iv) present a machine learning framework to predict the presence of defects in a given truss architecture based merely on its mechanical response.

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.

Inspired by crystallography, the periodic assembly of trusses into architected materials has enjoyed popularity for more than a decade and produced countless cellular structures with beneficial mechanical properties. Despite the successful and steady enrichment of the truss design space, the inverse design has remained a challenge: While predicting effective truss properties is now commonplace, efficiently identifying architectures that have homogeneous or spatially varying target properties has remained a roadblock to applications from lightweight structures to biomimetic implants. To overcome this gap, we propose a deep-learning framework, which combines neural networks with enforced physical constraints, to predict truss architectures with fully tailored anisotropic stiffness. Trained on millions of unit cells, it covers an enormous design space of topologically distinct truss lattices and accurately identifies architectures matching previously unseen stiffness responses. We demonstrate the application to patient-specific bone implants matching clinical stiffness data, and we discuss the extension to spatially graded cellular structures with locally optimal properties.

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 Wmagic+(F)=λmaxλmin-logλmaxλmin+logdetF=λmaxλmin+2logλmin,where λ_max≥ λ_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: GL ⁺(2) → R with an additive volumetric-isochoric split of the form W(F)=Wiso(F)+Wvol(detF)=W~iso(FdetF)+Wvol(detF)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.

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 FE²-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.

Microstructural patterns emerge ubiquitously during phase transformations, deformation twinning, or crystal plasticity. Challenges are the prediction of such microstructural patterns and the resulting effective material behavior. Mathematically, the experimentally observed patterns are energy-minimizing sequences produced by an underlying non-(quasi)convex strain energy. Therefore, identifying the microstructure and effective response is linked to finding the quasiconvex, relaxed energy. Due to its nonlocal nature, quasiconvexification has traditionally been limited to (semi-)analytical techniques or has been dealt with by numerical techniques such as the finite element method (FEM). Both have been restricted to primarily simple material models. We here contrast three numerical techniques—FEM, a Fourier-based spectral formulation, and a meshless maximum-entropy (max-ent) method. We demonstrate their performance by minimizing the energy of a representative volume element for both hyperelasticity and finite-strain phase transformations. Unlike FEM, which fails to converge in most scenarios, the Fourier-based spectral formulation (FFT) scheme captures microstructures of intriguingly high resolution, whereas max-ent is superior at approximating the relaxed energy. None of the methods are capable of accurately predicting both microstructures and relaxed energy; yet, both FFT and max-ent show significant advantages over FEM. Numerical errors are explained by the energy associated with microstructural interfaces in the numerical techniques compared here.

Severe plastic deformation (SPD), occurring ubiquitously across metal forming processes, has been utilized to significantly improve bulk material properties such as the strength of metals. The latter is achieved by ultra-fine grain refinement at the polycrystalline mesoscale via the application of large plastic strains on the macroscale. We here present a multiscale framework that aims at efficiently modeling SPD processes while effectively capturing the underlying physics across all relevant scales. At the level of the macroscale boundary value problem, an enhanced maximum-entropy (max-ent) meshless method is employed. Compared to finite elements and other meshless techniques, this method offers a stabilized finite-strain updated-Lagrangian setting for improved robustness with respect to mesh distortion arising from large plastic strains. At each material point on the macroscale, we describe the polycrystalline material response via a Taylor model at the mesoscale, which captures discontinuous dynamic recrystallization through the nucleation and growth/shrinkage of grains. Each grain, in turn, is modeled by a finite-strain crystal plasticity model at the microscale. We focus on equal-channel angular extrusion (ECAE) of polycrystalline pure copper as an application, in which severe strains are generated by extruding the specimen around a 90°-corner. Our framework describes not only the evolution of strain and stress distributions during the process but also grain refinement and texture evolution, while offering a computationally feasible treatment of the macroscale mechanical boundary value problem. Though we here focus on ECAE of copper, the numerical setup is sufficiently general for other applications including SPD and thermo-mechanical processes (e.g., rolling, high-pressure torsion, etc.) as well as other materials systems.

After a decade of periodic truss-, plate-, and shell-based architectures having dominated the design of metamaterials, we introduce the non-periodic class of spinodoid topologies. Inspired by natural self-assembly processes, spinodoid metamaterials are a close approximation of microstructures observed during spinodal phase separation. Their theoretical parametrization is so intriguingly simple that one can bypass costly phase-field simulations and obtain a rich and seamlessly tunable property space. Counter-intuitively, breaking with the periodicity of classical metamaterials is the enabling factor to the large property space and the ability to introduce seamless functional grading. We introduce an efficient and robust machine learning technique for the inverse design of (meta-)materials which, when applied to spinodoid topologies, enables us to generate uniform and functionally graded cellular mechanical metamaterials with tailored direction-dependent (anisotropic) stiffness and density. We specifically present biomimetic artificial bone architectures that not only reproduce the properties of trabecular bone accurately but also even geometrically resemble natural bone.

We introduce an improved meshfree approximation scheme which is based on the local maximum-entropy strategy as a compromise between shape function locality and entropy in an information-theoretical sense. The improved version is specifically designed for severe, finite deformation and offers significantly enhanced stability as opposed to the original formulation. This is achieved by (i) formulating the quasistatic mechanical boundary value problem in a suitable updated-Lagrangian setting, (ii) introducing anisotropy in the shape function support to accommodate directional variations in nodal spacing with increasing deformation and eliminate tensile instability, (iii) spatially bounding and evolving shape function support to restrict the domain of influence and increase efficiency, (iv) truncating shape functions at interfaces in order to stably represent multi-component systems like composites or polycrystals. The new scheme is applied to benchmark problems of severe elastic and elastoplastic deformation that demonstrate its performance both in terms of accuracy (as compared to exact solutions and, where applicable, finite element simulations) and efficiency. Importantly, the presented formulation overcomes the classical tensile instability found in most meshfree interpolation schemes, as shown for stable simulations of, e.g., the inhomogeneous extension of a hyperelastic block up to 100% or the torsion of a hyperelastic cube by 200° — both in an updated Lagrangian setting and without the need for remeshing.