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About this book

This book discusses bulk solids that derive their mechanical properties not from those of their base materials, but from their designed microstructures. Focusing on the negative mechanical properties, it addresses topics that reveal the counter-intuitive nature of solids, specifically the negativity of properties that are commonly positive, such as negative bulk modulus, negative compressibility, negative hygroexpansion, negative thermal expansion, negative stiffness phase, and negative Poisson’s ratio. These topics are significant not only due to the curiosity they have sparked, but also because of the possibility of designing materials and structures that can behave in ways that are not normally expected in conventional solids, and as such, of materials that can outperform solids and structures made from conventional materials.
The book includes illustrations to facilitate learning, and, where appropriate, reference tables. The presentation is didactic, starting with simple cases, followed by increasingly complex ones. It provides a solid foundation for graduate students, and a valuable resource for practicing materials engineers seeking to develop novel materials through the judicious design of microstructures and their corresponding mechanisms.

Table of Contents

Frontmatter

1. Introduction

Abstract
This chapter gives a brief introduction to metamaterials starting from its definition to the generic (including electromagnetic) metamaterials, through mechanical metamaterial, and finally to the mechanics of metamaterials with special emphasis on negative properties.
Teik-Cheng Lim

2. Auxetic Microstructures

Abstract
This chapter gives an overview of auxetic microstructures by considering the various geometries and their intermittent microstructures via gradation. This is followed by some comments on analogies between different auxetic models. Finally, an example is given for the extension of a 2D auxetic model to a 3D one, as well as an example of a rarely discussed auxetic model.
Teik-Cheng Lim

3. Analogies Across Auxetic Models

Abstract
This chapter views 2D auxetic models, including 3D deformation models with 2D auxetic behavior, from mechanism perspective instead of geometrical perspective. On this basis, auxetic models across different geometrical groups can be regrouped into clusters that exhibit analogy in deformation mechanism. Factors that are taken into consideration include the identification of corresponding rotation and non-rotation units, as well as linkages/joints between rotation and non-rotation units and non-linkages/non-joints across various auxetic models. As a result, five clusters of auxetic models have been identified, in which auxetic models within each cluster are analogous to each other. The identified clusters are those that exhibit: (1) double periodicity in the rotation direction of their rotating units, (2) synchronized rotation direction of their rotation units, (3) single periodicity in the rotation direction of their rotating units, (4) random rotation of their rotation units, and (5) non-rotation of units. Results from this analogy identification place auxetic models in a systematic representation and will enrich the future development of auxetic models, particularly, those that do not fall within these five clusters.
Teik-Cheng Lim

4. Thin Auxetic Plates

Abstract
This chapter discusses some non-circular and non-rectangular thin auxetic plates—such as equilateral triangular plates, elliptical plates, sectorial plates and rhombic plates—with special emphasis on their performances such as bending stress minimization or in terms of deflection minimization.
Teik-Cheng Lim

5. Thick Auxetic Plates

Abstract
This chapter considers shear deformation in transversely loaded thick plates—such as rectangular plates, equilateral triangular plates, isosceles right triangular plates, hexagonal plates, and regular polygonal plates—with special emphasis on the effect of auxeticity on their performances and the use of Reddy plate theory to extract the shear correction factors of Mindlin plates. Thereafter a comparison is made between the shear deformation in thick plates and buckling of thick columns to establish an analogy between them. Finally, a discussion is made on vibration of thick auxetic plates.
Teik-Cheng Lim

6. Longitudinal Elastic Waves in Auxetic Solids

Abstract
This chapter discusses the longitudinal wave speed in prismatic rods, plates, and bulk solids made from isotropic auxetic materials—as well as their intermediate structural elements—with special emphasis on the changes in cross-sectional area and density while retaining the traditional strength-of-materials flavor in order to keep the practical solutions tractable.
Teik-Cheng Lim

7. Elasticity of Auxetic Beams

Abstract
This chapter employs elasticity models to understand the effect of Poisson’s ratio negativity. Some discussions include how the use of classical elasticity models differs from the refined models as applied to auxetic beams.
Teik-Cheng Lim

8. Auxetic Composites with Mixed Auxeticity

Abstract
This chapter considers composites containing phases with opposing Poisson’s ratio signs. Special emphasis is placed on the overall sign of the composite, which differs depending on the mode of loading.
Teik-Cheng Lim

9. Auxetic Composites with Enhanced Moduli

Abstract
This chapter evaluates the extent of moduli increase, especially beyond the simple rule-of-mixture mode, for composites consisting of positive and negative Poisson’s ratio phases. Specific topics include fiber composites, laminates, and particle composites. The refined moduli models include correction terms or functions to cater for the increased stiffness.
Teik-Cheng Lim

10. Auxetic Membranes

Abstract
This chapter evaluates the deflection and stretching stresses occurring in large deflection of auxetic membranes using strain energy minimization approach. Results show that auxeticity tends to increase the deflection. Optimization study shows that the membrane stresses can be minimized by controlling the Poisson’s ratio and aspect ratio of the rectangular membranes.
Teik-Cheng Lim

11. Negative Thermal Expansion

Abstract
This chapter reviews the various 2D NTE systems, including those constructed from bimaterial strips, laminates (of various stiffness disparity), trusses (of triangular cells, Y-shaped elements, and Hoberman circle), meshes, rigid unit modes, and ring-rod assemblies (both 2D and 3D). Finally, a few examples of 3D NTE structures are briefly mentioned.
Teik-Cheng Lim

12. Negative Compressibility

Abstract
This chapter surveys work done on negative compressibility (NC) systems, which can be broadly categorized into cellular system, bimaterial strip system, and interconnected membrane system. In the category of cellular NC system, topics include deformation solely by rib stretching and those solely by joint rotation, as well as those with combined modes of deformation.
Teik-Cheng Lim

13. Negative Moisture Expansion, Negative Hygrothermal Expansion, and Negative Environmental Expansion

Abstract
This chapter introduces negative moisture expansion (NME), which is also known as negative hygroscopic expansion and negative swelling. A comparison is made between NME, negative thermal expansion (NTE), and negative compressibility (NC). Thereafter, the concepts of negative hygrothermal expansion (NHTE) and negative environmental expansion (NEE) are discussed.
Teik-Cheng Lim

14. Negative Stiffness

Abstract
This chapter reviews a few representative examples of negative stiffness structures (NS) that exhibit NS in one direction, two directions, and all three orthogonal directions based on 2D and 3D microstructures.
Teik-Cheng Lim

15. Sign-Switching of Metamaterial Properties

Abstract
Sign-switching refers to materials that can exhibit both positive and negative properties in situ as a result of changing loading direction or opposing changes in environmental conditions, without active control or the need impose changes to the microstructural architecture. Two concepts for sign-switching are briefly introduced, microstructural reshape and microstructural duality, which are elaborated in the remaining chapters of the book.
Teik-Cheng Lim

16. Sign-Switching of Poisson’s Ratio with Stress Reversal

Abstract
This chapter considers two examples of Poisson’s ratio sign-switching upon stress reversal. In the first example, the strain that is transverse to the loading direction is persistently negative for two types of microstructures. The microstructures were designed based on microstructural reshape with the direction of applied strain. In addition, these microstructures reveal NTE and ZTE properties under unconstrained and constrained boundaries when specific conditions are met. In the second example, the strain that is transverse to the loading direction is persistently positive for two types of microstructures. The microstructures were designed based on microstructural duality with the direction of applied strain. Specifically, these microstructures employ alternating lock and slide mechanism.
Teik-Cheng Lim

17. Sign-Switching of Environmental Expansion Coefficients with Environmental Change Reversals

Abstract
This chapter considers two examples of expansion coefficient sign-switching upon environmental change reversal. For the first example, the thermal and moisture strains in one direction are persistently negative. In the first instance, it is shown that these microstructures exhibit Poisson’s ratio sign-toggling with stress direction reversal in the same manner as that discussed in Sect. 16.​1. In the second example, the in-plane thermal strain is shown to be persistently positive, based on the principle of microstructural duality.
Teik-Cheng Lim

18. Sign-Switching of Poisson’s Ratio with Temperature Change Reversals

Abstract
This chapter considers two examples of Poisson’s ratio sign-switching upon temperature change reversal using bimaterial strips with alternating orientation. In the first example, alternating signs of temperature change switches the microstructure between hexagonal-like cells and re-entrant-like cells. The former and latter are known for exhibiting positive and negative Poisson’s ratios, respectively. In addition to the CTE analysis, Poisson’s ratio analysis is included. In the second example, the cell walls are also made from alternating bimaterial strips, and are arranged in rectangular array with interconnecting rigid rods that are joined at the centers of the bimaterial strips. Fluctuating temperature flips the microstructure shape between octagon-like and star-like cells. Although only the CTE analysis is furnished for the second example, it is known that the interconnected star array manifests auxetic property. In spite of only thermal analysis, the effective CTE models developed can be converted to effective compressibility and effective CME models.
Teik-Cheng Lim

19. Sign-Switching of Expansion Coefficients with Auxetic Behavior

Abstract
This chapter considers two examples of microstructures with sign-switching coefficients of expansion by microstructural reshape. However, the general shape of the environmentally deformed microstructure bears certain similarities regardless of whether the environmental (thermal, pressure, moisture concentration) changes are positive or negative. The first example is based on pin-jointed truss system that exhibits in-plane isotropy, while the second example is based on hybrid bimaterial strip and pin-jointed truss system that gives in-plane anisotropy. In both examples, the in-plane environmental strain is negative regardless of increasing or decreasing environmental parameters. Some auxetic aspects are observed for both microstructures. In the first example, the driven or secondary cells are analogous to the rotating squares while the driving or primary cells are analogous to the empty spaces between the squares. In the second example, the environmentally deformed microstructure bear resemblance with the rotating squares model, anti-tetrachiral model, and the instability-induced auxeticity of square grids.
Teik-Cheng Lim

20. Metamaterials and Islamic Geometric Patterns

Abstract
This chapter introduces the science and art of metamaterials, particularly in the design of negative materials. Proceeding from auxetic metamaterials inspired from Islamic motifs, the rest of the chapter deals with sign-switching of expansion coefficients in metamaterials whose microstructural geometries do not readily exhibit Islamic geometric patterns in their original state, but bloom into an Islamic motif as a consequence of environmental changes. Both sign-switching metamaterials deform to form 2D arrays of 8-pointed stars; one of them employs bimaterial strips while the other adopts linkage mechanism.
Teik-Cheng Lim

Backmatter

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