# MECHANICS OF MATERIALS WITH NEGATIVE POISSON’S RATIO – Hire Academic Expert

MECHANICS OF MATERIALS WITH NEGATIVE POISSON’S RATIO

19891167

Project Proposal for

300597 Master Project 1

Specialisation: Mechanical Engineering

Supervisor: Prof.B.Wang

School of Engineering, Design and Built Environment

Western Sydney University

May 2021

Introduction

Poisson’s ratio is the ratio of lateral (transverse) contraction strain to longitudinal extension

strain in a simple tension experiment. The allowable range of Poisson’s ratio ν in three

dimensional isotropic solids is from –1 to 1/2 . Common materials usually have a Poisson’s ratio close to 13 . Rubbery materials, however, have values approaching 1/2.

They readily undergo shear deformations, governed by the shear modulus G, but resist volumetric (bulk) deformation governed by the bulk modulus K; for rubbery materials G K. Even though textbooks can still be found  which categorically state that Poisson’s ratios less than zero are unknown, or even impossible, there are in fact a number of examples of negative Poisson’s ratio solids. Although such behaviour is counter-intuitive, negative Poisson’s ratio structures and materials can be easily made and used in lecture demonstrations and for student projects. Such solids become fatter in cross section when stretched. A solid with ν ≈ – 1 would be the opposite of rubber: difficult to shear but easy to deform volumetrically: G >>K. Negative Poisson’s ratio materials offer a new direction for achieving unusual and improved mechanical performance.

Objective and aims of the project

We present an introduction to the use of negative Poisson’s ratio materials to illustrate

various aspects of mechanics of materials. Poisson’s ratio is defined as minus the ratio of

transverse strain to longitudinal strain in simple tension. For most materials, Poisson’s ratio

is close to 1/3. Negative Poisson’s ratios are counter-intuitive but permissible according to

the theory of elasticity.

Literature review

The elastic properties of a material are described in terms of its strain (ε) relative to an applied stress (σ), where ε is the ratio of extension (ΔL) to original length (L0), and σ is the applied force (F) per cross sectional area (A).

The Poisson’s ratio has historically been the least explored, though it is associated with some interesting and unusual properties particularly when in a range not normally encountered. Defined as the ratio of lateral to axial strain in a structure or material, the Poisson’s ratio contributes to both G and K. It has been accepted theory that ν can have negative values for over 150 years and in 1991 the term auxetic was first used to describe materials with this property. Auxetic materials may prove beneficial to current materials technology with potential improvements to mechanical properties such as hardness, indent resistance, fracture toughness, shear strength, and sound absorption,[1, 5–7] or by exploiting geometrical properties such as maintaining the shape of pores within molecular sieves, or allowing for a double curvature within a honeycomb panel.[8–10] For a material with any degree of anisotropy these four scalar constants, which can be derived from any two, are no longer sufficient to fully describe its elastic properties. Instead, a 4th order tensor is used to express stress in terms of strain (stiffness), or strain in terms of stress (compliance). It is often convenient to represent these tensors as 6×6 matrices, using the well-established Voigt notation, in order to aid visualisation and allow for easier manipulation. The maximum number of coefficients used to fully define the elastic properties of a material is 21, but this number decreases with increased crystal symmetry. As a consequence, the Poisson’s ratio of anisotropic materials is a complex function 20 (dependant on three directional parameters, two angles describing the axial vector, one describing the perpendicular lateral vector). The prevalence or even simply the existence of auxeticity depends on the complex interplay between the tensor elements. To simplify analysis of anisotropic materials, their elastic properties can be averaged to simulate how they may behave whilst part of an isotropic polycrystalline structure. The four main ways in which these properties are averaged are the Voigt, Reuss, and Hill schemes along with a direct averaging method. Both the Voigt and Reuss averaging schemes provide values for the bulk modulus K and the shear modulus G, Voigt deriving these from the stiffness matrix (with coefficients Cij) and Reuss from the compliance matrix (with coefficients Sij).

Methodology

We begin with the assumption that students have been exposed to the definitions of stress

and strain, and have further been exposed to the relationships between the two through the use of Mohr’s circle. The permissible range of Poisson’s ratio can then be explained during the introduction of Hooke’s law. We suggest proceeding as follows.

From a pedagogical perspective, it is most natural to introduce the fundamental quantities

in the isotropic constitutive equations as Young’s modulus, E, Poisson’s ratio, ν, and shear

modulus, G. The dependence of G on E and ν can be illustrated with an exercise suggested,

for instance, by McClintock and Argon . The student begins with a state of pure shear,

rotates it 45°, and writes the normal stress and strain components in terms of the corresponding shear stress and strain. Employing the Hooke’s law relations illustrates that G must be related to E and ν through G E = + 2 1( ) ν (1)

Each engineering modulus E and G must be positive if a free block of the material is to be in

stable equilibrium, therefore ν > – 1. Summing the normal components of stress and defining the bulk modulus as the ratio of pressure to volumetric strain provides a definition for bulk modulus, K E = − 31 2 ( ) ν (2) Requiring K > 0 and E > 0 for stability gives ν < 0.5. Combining these gives the range of ν from –1 to 12 . If one modulus were to be exactly zero, a condition of neutral stability would obtain, associated with Poisson’s ratio exactly attaining one of the limits.

One may alternatively use an energy approach. Observe that the elementary Hooke’s law

relations are still a bit cumbersome, requiring a set of six equations to represent all compo

nents of the stress state. A different representation may be considered desirable, first because a different set of stress measures are more physically meaningful, and second because this alternative set of measures permits the representation of Hooke’s law in a more compact and pleasing form. A brief discussion of deviatoric stress components is required, and this permits the instructor to touch on the role of such stress components in material failure as a precursor to later introduction of failure theories. With deviatoric components of stress and strain defined, the student or instructor may be asked to show that the entire set of Hooke’s law relations may be reduced to: ′ = ′ σ ε ij ij 2G (3)

The instructor is now in a position to define the range of ν based on an energy argument,

as suggested, for instance, by Malvern . We begin with the definition of a differential

element of internal strain energy density, This provides an opportunity to revisit the topic from a basic physics course, where the increment is one-dimensional, as expressed for instance through a spring constant, dW = F dx. We remind the student that forces normal to the spring do no work during the deformation. Equation (4) then represents nothing more than a spring-like statement of incremental work for a multi-dimensional spring. It is our experience that students are d d W = σ ε ij ij (4) Making and characterizing negative Poisson’s ratio materials 51International Journal of Mechanical Engineering Education Vol 30 No 1 52 R. S. Lakes and R. Witt initially intimidated by indicial notation and the implicit summation over indices. Instructors may be advised to write out the summation in full, emphasizing that only normal components of stress do work through their respective normal strain increments, and only shear components of stress do work through their respective shear strain increments. Writing both σij and dεij in terms of their deviatoric components, we then show that the energy density increment can be recast as: d dd W p =− + ij ij ′ ′ εσ ε (5) where p and dε are pressure and increments of volumetric strain respectively. Using both the definition of bulk modulus and the compact form of Hooke’s law, we integrate the expression to obtain WK G ij ij = + ′ ′ 12 2 ε εε (6) The instructor can then return to the one-dimensional spring, noting the equivalent statement W kx = 12 2 and the implications of negative k. Stability arguments may be used to make the case that the spring constant k must be positive for the spring to have any energy storage capacity (W > 0) and to be stable under a small perturbation. Similarly, the instructor can argue that the multi-dimensional statement embodied in equation (6) requires both the bulk and shear modulus to be positive, which in turn requires ν to fall in the range −< < 1 ν 12 . Specifically, one may have a purely volumetric strain state, so K > 0 for stability; one may have a purely deviatoric strain state, so G > 0 for stability. Finally, the instructor may note that many texts, while making this argument, are quick to point out that there are no known materials with negative Poisson’s ratio. Beer and Johnston  state even more emphatically ‘On the other hand, the very definition of Poisson’s ratio requires it to be a positive quantity. We thus conclude that, for any engineering material, 0 12 < < ν ’.

Time Line : Research to be done during Master project 1 and the results are to be provided during Master Project 2

REFERENCES

 Fung, Y. C., Foundation of Solid Mechanics, Prentice-Hall, Englewood, NJ, p. 353, 1968.

 Beer, F. P. and Johnston, E. R., Mechanics of Materials, McGraw Hill, New York, 1981, 2nd ed., 1992.

 McClintock, F. A. and Argon, A. S., Mechanical Behavior of Materials, Addison-Wesley, 79,

1966.

 Malvern, L. E., Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, p. 292,

1969.

 Gibson, L. J. and Ashby, M. F., Cellular Solids, Pergamon, 1988.

 Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, McGraw-Hill, 3rd edition, 1970.

 Lakes, R. S., ‘Foam structures with a negative Poisson’s ratio’, Science, 235 , 1038–1040, (1987).

 Friis, E. A., Lakes, R. S. and Park, J. B., ‘Negative Poisson’s ratio polymeric and metallic

materials’, Journal of Materials Science, 23 , 4406–4414, (1988).

 Choi, J. B. and Lakes, R. S., ‘Nonlinear properties of metallic cellular materials with a negative

Poisson’s ratio’, J. Materials Science, 27 , 5373–5381, (1992).

 Lakes, R. S., ‘Advances in negative Poisson’s ratio materials’, Advanced Materials (Weinheim,

Germany), 5, 293–296, (1993).

 Ellis, A. B., Geselbracht, M. J., Johnson, B. J., Lisensky, G. C. and Robinson, W. R., Teaching

general Chemistry, A Materials Science Companion, American Chemical Society, Washington,

DC, 1993.

 Lakes, R. S., ‘Negative Poisson’s ratio materials’, Science, 238 , 551, (1987).

 Weiner, J. H., Statistical Mechanics of Elasticity, J. Wiley, NY, 1983.

 Lakes, R. S., ‘Deformation mechanisms of negative Poisson’s ratio materials: structural aspects’,

J. Materials Science, 26 , 2287–2292 (1991).

 Caddock, B. D. and Evans, K. E., ‘Microporous materials with negative Poisson’s ratio: I.

Microstructure and mechanical properties’, J. Phys. D. Appl. Phys., 22 , 1877–1882, (1989).