Table 1.

Main mechanical categories of mechanical metamaterials and their typical mechanical characteristics

Formation
Material level		Material
		Metallic	Polymeric
Mechanism	Constitutive relationship	• Work hardening constitutive relationship (e.g., Johnson-Cook model and Zerilli-Armstrong model) • Dynamic recovery constitutive relationship (e.g., Arrhenius model) • Dynamic recrystallization constitutive relationship (e.g., Sellars model) • Unified constitutive relationship (e.g., Miller model and Walker model)	• Thermoviscoelasticity-based models • Phase evolution-based models
Structural level		Overall
		2D beam or plate	3D cube and others
	Origami or kirigami	• Origami metamaterials with square twist76 • Reconfigurable origami metamaterials81	• 3D transformable origami metamaterials with multiple degrees of freedom79 • Reentrant origami metamaterials71 • Programmable self-locking origami metamaterials78
Unit	Chiral	• Chiral, anti-chiral and hierarchical honeycombs59,66 • Double-negative mechanical metamaterials73,76	• 3D chiral metamaterials with a twist48 • Self-rotating 3D chiral mechanical metamaterials83 • 3D chiral metamaterials with topological design66 • 3D chiral metamaterials with modular design57
	Lattice	• Hierarchical lattice materials86 • Functionally graded cellular composites with auxetics150 • Cellular flexible metamaterials172	• Nanolattices1,85,173 • Alternating pentamode lattices87,174 • 3D plate-lattices54 • Reversibly assembled cellular composite materials88 • 3D cellular metamaterials with anti-chiral topology91

Performance
Mechanical characteristics		Current status
		Advantages	Limitations
Response	Ultra-stiffness	• Impact resistance2,99 • Energy absorption • Vibration reduction	• Difficulties in design, characterization and application175 • Difficulties in fabrication (e.g., ultra-fine complex nanostructures, multi-material systems and super-large structures)176
	Ultra-lightweight	• Sound insulation, absorption and reduction54,104 • Low consumables and cost
	Negative response	• Negative Poisson’s ratio (e.g., shear, impact and damage resistance, and energy absorption)45,69,71,177 • Negative Stiffness (e.g., large bearing capacity and small deformation, and low natural frequency)73 • Negative thermal expansion (e.g., high thermal and electrical conductivity)178,179
	Programmable response	• Controllability • Tunable stimuli