Micromechanical Aspects of Isotropic Granular Assemblies With Linear Contact Interactions

1988 ◽  
Vol 55 (1) ◽  
pp. 17-23 ◽  
Author(s):  
R. J. Bathurst ◽  
L. Rothenburg
Keyword(s):  
Numerical Simulations ◽  
Contact Force ◽  
Elastic Constants ◽  
Poisson’S Ratio ◽  
Three Dimensional ◽  
Poisson's Ratio ◽  
Spherical Particles ◽  
Two Dimensional ◽  
Normal Contact ◽  
Granular Assemblies

The paper presents a micromechanical analysis of plane granular assemblies of discs with a range of diameters, and interacting according to linear contact force-interparticle compliance relationships. Contacts are assumed to be fixed and indestructible. Macroscopically, the system is described in terms of a two-dimensional analogue of generalized Hooke’s law. Explicit expressions for elastic constants in terms of microstructure are derived for dense isotropic assemblies. It is shown that Poisson’s ratio for dense systems depends on the ratio of tangential to normal contact stiffnesses. The derived expression for Poisson’s ratio is verified by numerically simulating plane assemblies comprising 1000 particles. The effect of density on Poisson’s ratio is investigated using numerical simulations. The theory of dense plane systems is extended to dense three-dimensional systems comprising spheres. Finally, it is shown that Poisson’s result ν=1/4 is recovered for spherical particles with central interactions.

Effective properties of composite materials containing voids

1993 ◽  
Vol 440 (1909) ◽  
pp. 461-473 ◽  
Keyword(s):  
Young’S Modulus ◽  
Poisson’S Ratio ◽  
Effective Properties ◽  
Young's Modulus ◽  
Matrix Material ◽  
Practical Importance ◽  
Three Dimensional ◽  
Poisson's Ratio ◽  
Two Dimensional ◽  
Isotropic Composite

Recent results of theoretical and practical importance prove that the two-dimensional (in-plane) effective (average) Young’s modulus for an isotropic elastic material containing voids is independent of the Poisson’s ratio of the matrix material. This result is true regardless of the shape and morphology of the voids so long as isotropy is maintained. The present work uses this proof to obtain explicit analytical forms for the effective Young’s modulus property, forms which simplify greatly because of this characteristic. In some cases, the optimal morphology for the voids can be identified, giving the shapes of the voids, at fixed volume, that maximize the effective Young’s modulus in the two-dimensional situation. Recognizing that two-dimensional isotropy is a subset of three-dimensional transversely isotropic media, it is shown in this more general case that three of the five properties are independent of Poisson’s ratio, leaving only two that depend upon it. For three-dimensionally isotropic composite media containing voids, it is shown that a somewhat comparable situation exists whereby the three-dimensional Young’s modulus is insensitive to variations in Poisson’s ratio, v m , over the range 0 ≤ v m ≤ ½, although the same is not true for negative values of v m . This further extends the practical usefulness of the two-dimensional result to three-dimensional conditions for realistic values of v m .

Role of Elastic Constants in Certain Contact Problems

1970 ◽  
Vol 37 (4) ◽  
pp. 965-970 ◽  
Author(s):  
J. Dundurs ◽  
M. Stippes
Keyword(s):  
Elastic Constants ◽  
Poisson’S Ratio ◽  
Contact Problems ◽  
Poisson's Ratio ◽  
Two Dimensional ◽  
Frictionless Contact ◽  
Plane Contact

The dependence of stresses on the elastic constants is explored in frictionless contact problems principally for the case when the contacting bodies are made of the same material and the deformations are induced by prescribed surface tractions. The strongest results can be obtained for problems with contacts that either recede or remain stationary upon loading. In such problems, the stresses are proportional to the applied tractions and the extent of contact is independent of the level of loading. Furthermore, it is shown that the Michell result regarding the dependence of stresses on Poisson’s ratio carries over to plane contact problems with receding and stationary contacts. In three and two-dimensional problems with advancing contacts, it is possible to establish certain rules for scaling displacements and stresses.

scholarly journals Tangential Loading of General Three-Dimensional Contacts

1998 ◽  
Vol 65 (4) ◽  
pp. 998-1003 ◽  
Author(s):  
M. Ciavarella
Keyword(s):  
Pressure Distribution ◽  
Poisson’S Ratio ◽  
Normal Load ◽  
Complete Solution ◽  
Tangential Force ◽  
Three Dimensional ◽  
Poisson's Ratio ◽  
Partial Slip ◽  
Normal Contact ◽  
Slip Regime

A general three-dimensional contact, between elastically similar half-spaces, is considered. With a fixed normal load, we consider a pure relative tangential translation between the two bodies. We show that, for the case of negligible Poisson’s ratio, an exact solution is given by a single component of shearing traction, in the direction of loading. It is well known that, for full sliding conditions, the tangential force must be applied through the center of the pressure distribution. Instead, for a full stick case the tangential force must be applied through the center of the pressure distribution under a rigid flat indenter whose planform is the contact area of the problem under consideration. Finally, for finite friction a partial slip regime has to be introduced. It is shown that this problem corresponds to a difference between the actual normal contact problem, and a corrective problem corresponding to a lower load, but with same rotation of the actual normal indentation. Therefore for a pure translation to occur in the partial slip regime, the point of application of the tangential load must follow the center of the “difference” pressure. The latter also provides a complete solution of the partial slip problem. In particular, the general solution in quadrature is given for the axisymmetric case, where it is also possible to take into account of the effect of Poisson’s ratio, as shown in the Appendix.

scholarly journals The General Relationships Between the Elastic Constants of Isotropic Materials in n Dimensions

1958 ◽  
Vol 11 (2) ◽  
pp. 154 ◽  
Author(s):  
NW Tschoegl
Keyword(s):  
Elastic Constants ◽  
Poisson’S Ratio ◽  
Three Dimensional ◽  
Poisson's Ratio ◽  
Isotropic Materials ◽  
Monomolecular Films

The relationships between the elastic constants of homogeneous isotropic materials in n dimension are derived and are shown to depend on n. The maximal value of the generalized Poisson's ratio is 1/(n?1). The n-dimensional formulae reproduce the well-known three-dimensional relations for n=3, while n=2 produces the relations, appropriate for monomolecular films. The correct degeneration is shown for n= 1.

Designing a new three-dimensional periodic cellular auxetic material

2017 ◽  
Vol 31 (16-19) ◽  
pp. 1744088
Author(s):  
Yiyi Zhou ◽  
Lianmen Chen
Keyword(s):  
Poisson’S Ratio ◽  
Three Dimensional ◽  
Parametric Model ◽  
Early Research ◽  
Poisson's Ratio ◽  
Negative Poisson’S Ratio ◽  
Two Dimensional ◽  
Auxetic Materials ◽  
Negative Poisson's Ratio ◽  
Fundamental Mechanism

Auxetics are materials showing a negative Poisson’s ratio. Early research found several categories of auxetic materials in the chemical field. Later research identified the fundamental mechanism generating this behavior is rotation; a variety of two-dimensional auxetic material have been generated accordingly. Nevertheless, the successful example of three-dimensional auxetic material is still rare. This paper introduces a new design of three-dimensional periodic cellular auxetic material based on geometrical and mechanical methodology. The projections of the optimized periodic modules in two horizontal directions are geometrically same with auxetic hexahedral poem, so that the optimized periodic material can perform auxetic in both two horizontal directions under vertical compression. Parametric model is simulated to prove the design.

scholarly journals Giant negative Poisson’s ratio in two-dimensional V-shaped materials

2021 ◽  
Author(s):  
Xikui Ma ◽  
Jian Liu ◽  
Yingcai Fan ◽  
Weifeng Li ◽  
Jifan Hu ◽  
...  
Keyword(s):  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
Negative Poisson’S Ratio ◽  
Two Dimensional ◽  
Auxetic Materials ◽  
Negative Poisson's Ratio ◽  
Poisson's Ratios ◽  
Poisson’S Ratios

Two-dimensional (2D) auxetic materials with exceptional negative Poisson’s ratios (NPR) are drawing increasing interest due to the potentials in medicine, fasteners, tougher composites and many other applications. Improving the auxetic...

scholarly journals Monolayer Mg2C : Negative Poisson's ratio and unconventional two-dimensional emergent fermions

2018 ◽  
Vol 2 (10) ◽  
Author(s):  
Shan-Shan Wang ◽  
Ying Liu ◽  
Zhi-Ming Yu ◽  
Xian-Lei Sheng ◽  
Liyan Zhu ◽  
...  
Keyword(s):  
Poisson’S Ratio ◽  
Poisson's Ratio ◽  
Negative Poisson’S Ratio ◽  
Two Dimensional ◽  
Negative Poisson's Ratio

An Extension of CEB Elastic-Plastic Contact Model

2007 ◽  
Author(s):  
A. Sepehri ◽  
K. Farhang
Keyword(s):  
Contact Force ◽  
Tangential Force ◽  
Three Dimensional ◽  
Tangential Direction ◽  
Asperity Contact ◽  
Force Component ◽  
Normal Contact ◽  
Elastic Plastic ◽  
Force Components ◽  
Plastic Parts

Three dimensional elastic-plastic contact of two nominally flat rough surfaces is by developing the equations governing the shoulder-shoulder contact of asperities based on the Chang, Etsion and Bogy (CEB) model of contact in which volume conservation is assumed in the plastic flow regime. Shoulder-shoulder asperity contact yields a slanted contact force consisting of both tangential (parallel to mean plane) and normal components. Each force component comprises elastic and elastic-plastic parts. Statistical summation of normal force components leads to the derivation of the normal contact force for the elastic-plastic contact akin to the CEB model. Half-plane tangential force due to elastic-plastic contact is derived through the statistical summation of tangential force component along an arbitrary tangential direction.

scholarly journals Derivation of Equations for a Size Distribution of Spherical Particles in Non-Transparent Materials

2021 ◽  
Vol 5 (4) ◽  
pp. 53-60
Author(s):  
Daniel Gurgul ◽  
Andriy Burbelko ◽  
Tomasz Wiktor
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Size Distribution ◽  
Density Function ◽  
Cross Section ◽  
Three Dimensional ◽  
Spherical Particles ◽  
Two Dimensional ◽  
Mathematical Formulas ◽  
Transparent Materials

This paper presents a new proposition on how to derive mathematical formulas that describe an unknown Probability Density Function (PDF3) of the spherical radii (r3) of particles randomly placed in non-transparent materials. We have presented two attempts here, both of which are based on data collected from a random planar cross-section passed through space containing three-dimensional nodules. The first attempt uses a Probability Density Function (PDF2) the form of which is experimentally obtained on the basis of a set containing two-dimensional radii (r2). These radii are produced by an intersection of the space by a random plane. In turn, the second solution also uses an experimentally obtained Probability Density Function (PDF1). But the form of PDF1 has been created on the basis of a set containing chord lengths collected from a cross-section.The most important finding presented in this paper is the conclusion that if the PDF1 has proportional scopes, the PDF3 must have a constant value in these scopes. This fact allows stating that there are no nodules in the sample space that have particular radii belonging to the proportional ranges the PDF1.

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