CiteULike: norris's poisson

Extreme Lame Compliance in Anisotropic Crystals

Cliff Guo — 2008-03-11T16:46:52-00:00

Mathematics and Mechanics of Solids (11 March 2008), 1081286507080807.

For a crystalline material, Poisson�s ratio depends upon two orthogonal directions, one corresponding to the applied uniaxial stress and another for the resulting transverse strain. Interest in directions that yield a negative value leads to a consideration of extreme values and associated directions. Another indicator of a negative Poisson�s ratio is the Lame compliance, defined as the transverse strain response to a unit uniaxial stress. Where this quantity is positive, Poisson�s ratio is negative and it is natural to associate a near mnimum value of the ratio with a maximum of the Lame compliance. Moreover, the stationary directions associated with the compliance bear a clearer relationship to the crystallographic directions than those of the ratio. Indeed, many of these directions are not dependent upon the elastic constants within a given crystal symmetry class. In the case of alpha-cristobalite, the maximum value of the Lame compliance is associated with such invariant stationary points. In the present work, we describe the invariant stationary directions and touch on a few of the simplest material-dependent stationary points. 10.1177/1081286507080807

The Incremental Bulk Modulus, Young's Modulus and Poisson's Ratio in Nonlinear Isotropic Elasticity: Physically Reasonable Response

NH Scott — 2007-11-08T17:50:44-00:00

Mathematics and Mechanics of Solids, Vol. 12, No. 5. (1 October 2007), pp. 526-542.

An incremental (or tangent) bulk modulus for finite isotropic elasticity is defined which compares an increment in hydrostatic pressure with the corresponding increment in relative volume. Its positivity provides a stringent criterion for physically reasonable response involving the second derivatives of the strain energy function. Also, an average (or secant) bulk modulus is defined by comparing the current stress with the relative volume change. The positivity of this bulk modulus provides a physically reasonable response criterion less stringent than the former. The concept of incremental bulk modulus is extended to anisotropic elasticity. For states of uniaxial tension an incremental Poisson's ratio and an incremental Young's modulus are similarly defined for nonlinear isotropic elasticity and have properties similar to those of the incremental bulk modulus. The incremental Poisson's ratios for the isotropic constraints of incompressibility, Bell, Ericksen, and constant area are considered. The incremental moduli are all evaluated for a specific example of the compressible neo-Hookean solid. Bounds on the ground state Lame elastic moduli, assumed positive, are given which are sufficient to guarantee the positivity of the incremental bulk and Young's moduli for all strains. However, although the ground state Poisson's ratio is positive we find that the incremental Poisson's ratio becomes negative for large enough axial extensions. 10.1177/1081286506064719

The structure and mechanics of cork

LJ Gibson — 2007-05-18T13:30:49-00:00

Proc. R. Soc. Lond. A, Vol. 377 (1981), pp. 99-117.

Extreme Poisson's ratios and related elastic crystal properties

Cliff Guo — 2007-03-01T20:19:14-00:00

Journal of the Mechanics and Physics of Solids, Vol. 54, No. 4. (April 2006), pp. 690-707.

With the aim of understanding anisotropic crystals that possess a negative Poisson's ratio and to lay a foundation for investigating molecular mechanisms, we discuss the definition of the ratio and establish conditions on the compliance that govern its sign. We derive results on orientation averaging that are useful in the context of anisotropy and helpful in the investigation of isotropic polycrystals. We discuss [alpha]-cristobalite, a polymorph of silicon dioxide that possesses interesting negative ratio properties in single crystals and hypothetical polycrystals. In this connection, we draw attention to the transverse compliance as an alternative and simpler metric for gaging the ratio and for orientation averaging. For [alpha]-cristobalite, we arrive at new results for the directions that yield the most negative Poisson's ratio. This result should be of value in divining the underlying molecular mechanism that explains the negative values of Poisson's ratio in [alpha]-cristobalite, a crystal of tetragonal symmetry that possesses six independent elastic constants.

Poisson's ratio for anisotropic elastic materials can have no bounds

TCT Ting — 2007-02-01T17:15:48-00:00

The Quarterly Journal of Mechanics and Applied Mathematics, Vol. 58, No. 1. (2005)

Very Large Poisson?s Ratio with a Bounded Transverse Strain in Anisotropic Elastic Materials

TCT Ting — 2007-02-01T17:13:29-00:00

Journal of Elasticity, Vol. 77, No. 2. (2004), pp. 163-176.

Negative Poisson's ratios in anisotropic linear elastic media

TCT Ting — 2007-01-11T15:55:29-00:00

Vol. 72 (2005), pp. 929-931.

Extrema of Young's modulus for elastic solids with tetragonal symmetry

Antonio Cazzani — 2006-10-28T15:42:51-00:00

International Journal of Solids and Structures, Vol. 42, No. 18-19. (September 2005), pp. 5057-5096.

For a homogeneous and linearly elastic solid the general expression of Young's modulus E(n) is given, and a constrained extremum problem is formulated for the evaluation of the directions n corresponding to stationary values of the modulus. The formulation follows that presented in [International Journal of Solids and Structures 40 (2003) 1713-1744] for the cubic and transversely isotropic elastic symmetries. In this paper the tetragonal elastic symmetry class is considered, and explicit solutions for the directions n associated to critical points of E(n) are analytically evaluated. Properties of these directions and of the corresponding values of the modulus are discussed in detail. The results are presented in terms of three material parameters, which are responsible of the degree of anisotropy. For the tetragonal system, the complete description of the directional dependence of Young's modulus leads to the identification of 12 classes of behavior. For each of these classes several examples of real materials are shown and suitable graphical representations of the function E(n) are given as well.

Extrema of Young's modulus for cubic and transversely isotropic solids

Antonio Cazzani — 2006-10-28T15:02:44-00:00

International Journal of Solids and Structures, Vol. 40, No. 7. (April 2003), pp. 1713-1744.

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young's modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young's modulus on direction n are given as well.

Molecular Origin of Auxetic Behavior in Tetrahedral Framework Silicates

Andrew Alderson — 2006-08-18T13:22:14-00:00

Physical Review Letters, Vol. 89, No. 22. (11 November 2002), 225503.

Recent analytical models for the Poissonâs ratios (Î½ i j ) of tetrahedral frameworks are applied to Î±-cristobalite and Î±-quartz for the first time. Rotation and dilation of the SiO 4 tetrahedral subunits are considered. Each mechanism leads to negative Î½ 31 values; whereas negative and positive values are possible when they act concurrently. The concurrent model is in excellent agreement with experiment and explains the dichotomy between negative and positive Î½ 31 values in Î±-cristobalite and Î±-quartz; respectively. The predicted strain-dependent trends confirm those from molecular modeling.

Molecular-dynamics study of the high-temperature elasticity of quartz above the alpha-beta phase transition

Hajime Kimizuka — 2006-08-18T13:14:15-00:00

Physical Review B (Condensed Matter and Materials Physics), Vol. 67, No. 2. (2003)

We have presented the molecular-dynamics (MD) results for the temperature dependence of the adiabatic elastic constants Cij of and quartz, using a statistical fluctuation formula. It is noteworthy that the calculated Cij values are in a good agreement with the experimental values in the entire temperature range of 300–1100 K, including the - phase-transition region. We have confirmed that the net increase of bulk Cij's in the phase can be attributed to the internal relaxations, which arise from the cooperative motions of corner-linked SiO4 tetrahedra. Our MD simulations have revealed the existence of dynamical disorder in quartz at high temperatures, and its influence on the macroscopic elastic properties, in contrast to the ordered -quartz structure model.

Structural Origin of Negative Thermal Expansion in High-Temperature Silica Polymorphs

Liping Huang — 2006-08-17T16:04:05-00:00

Physical Review Letters, Vol. 95, No. 21. (2005)

The and modifications of quartz and cristobalite silica have been successfully simulated using molecular dynamics simulations based on a single parametrization of a charge transfer three-body potential. The simulated forms exhibit positive thermal expansion; it is almost zero for cristobalite up to 1500 K and slightly negative at higher temperatures, while a negative thermal expansion of quartz is observed immediately above the -to- transition. A detailed analysis of atomic trajectories reveals that the origin of negative thermal expansion in the high-temperature forms of silica is a gradual reactivation of the same displacement mode that underlies the transformation between the and modifications.

Mechanism for Negative Poisson Ratios over the alpha - beta Transition of Cristobalite, SiO_2: A Molecular-Dynamics Study

Hajime Kimizuka — 2006-08-17T13:44:39-00:00

Physical Review Letters, Vol. 84, No. 24. (12 June 2000), 5548.

The adiabatic elastic constants ( C ij ) of cristobalite have been evaluated successfully over the temperature range of 300â1800 K using the molecular-dynamics method with a fluctuation formula. Cristobalite shows a negative Poisson ratio over this temperature range. However; the mechanisms differ between the Î± and Î² phases. In the cubic Î² phase; C 44 exhibits a value extremely close to C 11 rather than C 12 ; in contrast to the Cauchy relation. This predicts a remarkable property that the longitudinal and transverse velocities coincide for the acoustic waves propagating along the [100] direction.