Corrigendum to ‘Antipsychotic Potential and Safety Profile of TPGS-Based Mucoadhesive Aripiprazole Nanoemulsion: Development and Optimization for Nose-To-Brain Delivery’ [Journal of Pharmaceutical Sciences 110/4 (2021) 1761-1778]

Santosh Ashok Kumbhar

doi:10.1016/j.xphs.2026.104397

Hierarchical and Fractal-Inspired Mechanical Metamaterials: A Mechanics-Oriented Review of Multiscale Design, Strength, Deformation, and Energy Absorption

2026-08-13

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Saulius Diliūnas *

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Vitalis Leišis

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Tilmutė Pilkaitė

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Inga Skiedraitė

DOI：10.3390/ma19163432

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Classical continuum descriptions based on homogenized parameters, such as modulus and relative density, become inadequate when performance is governed by deliberately engineered multiscale architecture rather than by composition, as in hierarchical and fractal-inspired mechanical metamaterials. In these materials, geometry couples deformation across length scales in ways that single-scale models cannot capture. This review synthesizes theoretical, numerical, and experimental literature published mainly over the last two decades, with emphasis on the last five years. It draws preferentially on peer-reviewed journal articles and highly cited studies; strictly periodic single-scale metamaterials and general electromagnetic–metamaterial literature are cited only where needed to establish historical context and are not otherwise within the scope of this review. The specific contribution of this review is threefold: (i) a mechanics-oriented classification of hierarchical and fractal-inspired architectures according to the deformation mechanism they activate (bending-, stretching-, membrane-, or buckling-dominated), rather than by geometric appearance alone; (ii) an explicit structure-mechanism-property-function framework linking multiscale geometry to strength, failure evolution, and energy absorption, tested against automotive crashworthiness as an application case study; and (iii) a critical account of when fractal–mathematical descriptors (scale invariance, similarity ratio, iteration depth, and fractal dimension) are mechanically meaningful, since many finite hierarchical lattices are fractal-inspired rather than fractal in the strict mathematical sense. The synthesis shows that relative density alone is an insufficient performance descriptor: outcomes depend on whether deformation is bending-, stretching-, membrane-, or buckling-dominated, and hierarchy or fractal-inspired recursion improves performance only when each structural level is assigned a distinct mechanical role rather than repeating geometry without function. The crashworthiness case study confirms that no single architecture is universally optimal once force efficiency, intrusion control, and manufacturability are considered alongside energy absorption. Future progress requires integrated structure-mechanism-property-function frameworks that combine theoretical modeling, high-resolution simulation, data-driven design, and experimental validation, with complexity justified only when it is mechanically purposeful, validated, and manufacturable at scale.

Keywords:

mechanical metamaterials

hierarchical structures

fractal-inspired architectures

multiscale design

failure modes

strength-to-weight performance

energy absorption

crashworthiness

architecture-controlled mechanics

Hierarchical and Fractal-Inspired Mechanical Metamaterials: A Mechanics-Oriented Review of Multiscale Design, Strength, Deformation, and Energy Absorption

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