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Abstract This article introduces a computational design framework for obtaining three‐dimensional (3D) periodic elastoplastic architected materials with enhanced performance, subject to uniaxial or shear strain. A nonlinear finite element model accounting for plastic deformation is developed, where a Lagrange multiplier approach is utilized to impose periodicity constraints. The analysis assumes that the material obeys a von Mises plasticity model with linear isotropic hardening. The finite element model is combined with a corresponding path‐dependent adjoint sensitivity formulation, which is derived analytically. The optimization problem is parametrized using the solid isotropic material penalization method. Designs are optimized for either end compliance or toughness for a given prescribed displacement. Such a framework results in producing materials with enhanced performance through much better utilization of an elastoplastic material. Several 3D examples are used to demonstrate the effectiveness of the mathematical framework. 

The equine hoof wall has outstanding impact resistance, which enables high-velocity gallop over hard terrain with minimum damage. To better understand its viscoelastic behavior, complex moduli were de- termined using two complementary techniques: conventional ( ∼5 mm length scale) and nano ( ∼1 μm length scale) dynamic mechanical analysis (DMA). The evolution of their magnitudes was measured for two hydration conditions: fully hydrated and ambient. The storage modulus of the ambient hoof wall was approximately 400 MPa in macro-scale experiments, decreasing to ∼250 MPa with hydration. In contrast, the loss tangent decreased for both hydrated ( ∼0.1–0.07) and ambient ( ∼0.04–0.01) conditions, over the frequency range of 1–10 Hz. Nano-DMA indentation tests conducted up to 200 Hz showed little frequency dependence beyond 10 Hz. The loss tangent of tubular regions showed more hydration sensitivity than in intertubular regions, but no significant difference in storage modulus was observed. Loss tangent and effective stiffness were higher in indentations for both hydration levels. This behavior is attributed to the hoof wall’s hierarchical structure, which has porosity, functionally graded aspects, and material interfaces that are not captured at the scale of indentation. The hoof wall’s viscoelasticity characterized in this work has implications for the design of bioinspired impact-resistant materials and structures. 

Additive manufacturing (AM) is often used to create designs inspired by topology optimization and biological structures, yielding unique cross-sectional geometries spanning across scales. However, manufacturing defects intrinsic to AM can affect material properties, limiting the applicability of a uniform material model across diverse cross-sections. To examine this phenomenon, this paper explores the influence of specimen size and layer height on the compressive modulus of polycarbonate (PC) and thermoplastic polyurethane (TPU) specimens fabricated using fused filament fabrication (FFF). Micro-computed tomography imaging and compression testing were conducted on the printed samples. The results indicate that while variations in the modulus were statistically significant due to both layer height and size of the specimen in TPU, variations in PC were only statistically significant due to layer height. The highest elastic modulus was observed at a 0.2 mm layer height for both materials across different sizes. These findings offer valuable insights into design components for FFF, emphasizing the importance of considering mechanical property variations due to feature size, especially in TPU. Furthermore, locations with a higher probability of failure are recommended to be printed closer to the print bed, especially for TPU, because of the lower void volume fraction observed near the heated print bed. 

Biological structural designs in nature like hoof walls can be used as inspiration for generating structures with excellent mechanical properties. A common theme in these designs is the small percent porosity. Under dynamic transverse compression, we investigated the structure-property relations in low porosity structures. A diverse design space was created using polygonal tubules with different numbers of rows and columns. The volume fraction and the orientation angle of the tubules were also varied. The finite element method was used with a rate-dependent elastoplastic material model to generate the stress-strain curves in plane strain conditions. 

The equine hoof wall has a complex, hierarchical structure that can inspire designs of impact-resistant materials. In this study, we utilized micro-computed tomography (micro-CT) and serial block-face scanning electron microscopy (SBF-SEM) to image the microstructure and nanostructure of the hoof wall. We quantified the morphology of tubular medullary cavities by measuring equivalent diameter, surface area, volume, and sphericity. Highresolution micro-CT revealed that tubules are partially or fully filled with tissue near the exterior surface and become progressively empty towards the inner part of the hoof wall. Thin bridges were detected within the medullary cavity, starting in the middle section of the hoof wall and increasing in density and thickness towards the inner part. Porosity was measured using three-dimensional (3D) micro-CT, two-dimensional (2D) micro-CT, and a helium pycnometer. The highest porosity was obtained using the helium pycnometer (8.07%), followed by 3D (3.47%) and 2D (2.98%) micro-CT. SBF-SEM captured the 3D structure of the hoof wall at the nanoscale, showing that the tubule wall is not solid, but has nano-sized pores, which explains the higher porosity obtained using the helium pycnometer. The results of this investigation provide morphological information on the hoof wall for the future development of hoof-inspired materials and offer a novel perspective on how various measurement methods can influence the quantification of porosity. 

Reference point indentation (RPI) is a novel experimental technique designed to evaluate bone quality. This study utilizes two RPI instruments, BioDent and Osteoprobe, to investigate the mechanical responses of several 3D-printed polymers. We correlated the mechanical properties from a tensile test with the RPI parameters obtained from the BioDent and OsteoProbe. In addition, we tested the same polymers five years later (Age 5). The results show that for Age 0 polymers, the elastic modulus is highly correlated with average unloading slope (r = 0.87), first unloading slope (r = 0.85), bone material strength index (BMSi) (r = 0.85), average loading slope (r = 0.82), first indentation distance (r = 0.79), and total indentation distance (r = 0.76). The ultimate stress correlates significantly with first unloading slope (r = 0.85), average unloading slope (r = 0.83), BMSi (r = 0.81), first indentation distance (r = 0.73), average loading slope (r = 0.71), and total indentation distance (r = 0.70). The elongation has no significant correlation with the RPI parameters except with the average creep indentation distance (r = 0.60). For Age 5 polymers, correlations between mechanical properties and RPI parameters are low. This study illustrates the potential of RPI to assess the mechanical properties of polymers nondestructively with simple sample requirements. Furthermore, for the first time, 3D-printed polymers and aged polymers are investigated with RPI. 

Previous studies show that the properties of parts manufactured via additive manufacturing, such as selective laser melting, depend on local feature sizes like lattice wall thickness and strut diameter. Although size dependence has been studied extensively, it was not included in constitutive models for numerical simulations. In this work, flat dog-bone tensile specimens of different thicknesses were manufactured and tested under quasi-static conditions to characterize the size-dependent properties experimentally. It was observed that key mechanical properties decrease with specimen thickness. Through curve-fitting to experimental data, this work provides approximate analytical expressions for the material properties values as a function of specimen thickness, furnishing a phenomenological size-dependent constitutive model. The interpolating capability of the model is cross-validated with existing experimental data. Two numerical examples demonstrate the application of the size-dependent material model. The axial crushing of thin-walled lattices at varying wall thicknesses was simulated by the size-dependent material model and one that ignores size effects. Results show that ignoring size effects leads to overestimated peak crushing force and specific energy absorption. The two material models were also compared in the topology optimization of thin-walled structures. Results show that the size-dependent model leads to a more robust optimized design: having higher energy absorption and sustaining less material fracture. 

The equine hoof wall has a unique hierarchical structure that allows it to survive high‐impact scenarios. Previous authors have explored the compressive, viscoelastic, and fracture control properties of the hoof wall and suggested that this complex structure plays a vital role in the hoof's behavior. However, the link between the structure and the behavior of the hoof wall has been made primarily with the use of post‐fracture analysis. Here, periodic microcomputed tomography scans are used to observe the temporal behavior of the hoof's meso and microstructures during compression, fracture, and relaxation. These results shed light on the structural anisotropy of the hoof wall and how its hollow tubules behave when compressed in different directions, at different hydration levels, and in various locations within the hoof wall. The behavior of tubule bridges during compression is also reported for the first time. This study elucidates several fracture phenomena, including the way cracks are deflected at tubule interfaces and tubule bridging, tubule arresting, and fiber bridging. Finally, relaxation tests are used to show how the tubule cavities can regain their shape after compression. 

This paper investigates the structure–property relations of thin-walled lattices, characterized by their cross-sections and heights, under dynamic longitudinal compression. These relations elucidate the inter- actions of different geometric features of a design on mechanical response, including energy absorption. We proposed a combinatorial, key-based design system to generate different lattice designs and used the finite element method to simulate their response with the Johnson–Cook material model. Using an autoencoder, we encoded the cross-sectional images of the lattices into latent design feature vectors, which were supplied to the neural network model to generate predictions. The trained models can accu- rately predict lattice energy absorption curves in the key-based design system and can be extended to new designs outside of the system via transfer learning.