CONTACT PROBLEM FOR A CLASS OF ANISOTROPIC ELASTIC CELLULAR BODIES WITH NONPOSITIVE POISSONS RATIODorota JasiskaMagorzata Janus-MichalskaInstitute of Structural MechanicsCracow University of Technology

CONTENTSIntroduction auxetic materialsdefinition, occurance, propertiesCellular materials as anisotropic continuumtwo scale modeling stiffness matricesenergy based yield criterionmaterial effort coefficientFormulation of a contact problemNumerical examplesConclusionsReferences

1. Materials with negative Poissons ratio - auxetics

Poissons ratio for isotropic materials Where: K - bulk modulus G - shear modulus d dimension of the problem For spatial problems

For planar problems

Auxetic materials from molecular to macroscopic levelMolecular auxetics single crystals (iron pyrites, cadmium)crystalline ( cristobalite)Mesoscopic auxeticspolymeric and metallic foams microporous polymerscellular materials (inverted honeycombs)Macroscopic auxetics composites

Other properties of auxeticsincreased resistance to indentation (toughness)resistance to fracture and damagedifferent deformation pattern tendency to form double curved dome shaped surfaces while bendingimproved acoustic and vibrational absorptiongreater resiliancereduced impact forces in contact

2. Cellular materials as anisotropic continuum Regular cellular plane structures and their representative unit cells

a) square cell structure with square unit cell

c) equilateral triangular structure with hexagonal unit cell

b) honeycombstructure with triangle unit cell

d) reentrant structure with trapezoid unit cell

Two scale modelinguniform strain states of equivalent continuumdisplacement affinity of nodes and midpoints

Mechanical model for skeleton - displacement method for Timoshenko beamUniform continuum strains skeleton midpoint displacements uniaxial extension x in x direction.

uniaxial extension y in y direction.

pure shear gxy in xy plane.

Resultant forces in skeletonIn arbitrary strain state:

definition of equivalent continuum - averaging of strain potential for skeleton strain potential of beam skeleton

strain potential of anisotropic continuum

stiffness matrix components for equivalent continuum

Hooke's law for anisotropic continuumstiffness tensor, compliance tensor ,where material constants:

Kelvins notation in 6-D space:

strain vector stress vectorStiffness matrix:

Energy based yield criterion for anisotropic continuum Eigenvalues of stiffness tensor

corresponding stiffness matrix eigenstrains

stiffness matrix eigenstress

the limit condition for bending and tension in skeleton

Coefficient

Limit eigenstrains and eigenstresses

Limit elastic energy stored in particular eigenstate

Energy based yield criterion for equivalent continuum (Rychlewski)

Material effort coefficientArbitrary plane stress state

Decomposition of the stress vector into stress eigenstates

Material effort coefficient

Where A,B,C functions of and

For arbitrary cell orientationwhere:

3.Unilateral static contact problem of anisotropic elastic bodyin on on boundary conditionson contact and friction conditionswhere: - initial gap, - contact pressure, displacement normal to the boundary - tangential contact force, -increment of tangential displacement

4.Numerical examples4.1 Square block made of material with different cell types under pressure Data: Skeleton data: Es=10 GPa, s=0.3, Re=10MPa, / s=0.1155 , p=25kN/m, =0.3

Structure Geometric data [mm]Ex [MPa]Ey [MPa]xyyx L0i=2.6 t=0.15 576.92 576.92 0.0 0.0 L0i=1.5 t=0.15 21.87 21.87 0.96 0.96 L0i=4.5 t=0.15 385.47 385.47 0.33 0.33 L0i=3.1t=0.15 =700 0.13 1.95 -0.26 -3.85

Deformation and contact pressure distribution [MPa] scale=1500scale=50scale=1000scale=3

Contact pressure, friction stress and contact status along contact line

Distribution of material effort coefficient .

Structure n max/pT max/pmax 1.001.E-5 3.61.08 0.22E-1 1.850.23 0.2E-3 1.090.28 0.95

Dependence of Poissons ratio on angle of tension direction

For nxy=const=0.96

For nxy=const=0.33

For nxy

4.2 Square block made of re-entrant cellular material with different location of cell axis with respect to contact line

p=4kN/m Anisotropic material constants for different angles.

EX [MPa]EY [MPa]XYYX01.9540.128 -3.85-0.26450.1040.1040.3650.365900.1281.954 -0.26-3.85

Contact pressure and friction stress along contact line

Deformation and distribution of material effort coefficient

5. conclusionsMicromechanical model of cellular material is applied to predict mechanical properties on a macroscale. Method is used for analysis of stress distribution in contact zone and material effort in the elastic range. Cellular materials due to a variety of structure topology, which results in different types of material symmetry and macroscopic properties can be tailored to special demands of the given problem.Celular materials with reentrant structure, which give negative Poissons ratio in a certain range of directions can be advantagous in contact problems (knee pads, materaces, wrestling mats)Proper choice of microstructural geometrical parameters and orientation of material symmetry axis with respect to load direction can significantly influence contact stress distribution and may play an important role in reducing contact peak pressureLinear analysis first step.

6. ReferencesL.J. Gibson, M.F. Ashby Cellular Solids, 2nd edition Cambridge University Press. (1997). R.S. Kumar, D.L.McDowell, Generalized continuum modeling of 2-D periodic cellular solids, Int. Journ. of Solids and Struct. 41 , (2004?).S. Nemat-Naser, M.Hori, Micromechanics,. 2nd edition Elsevier (1999). M. Hori, S.Nemat-Nasser, On micromechanics theories for determining micro-macro relations in heterogeneous solids, Mech. Mat., 31, 667-682, (1999).M.M.Mehrabadi, S.C.Cowin, Eigentensors of linear anisotropic elastic materials, Q. J. Mech. Appl. Math., 43, 15-41,(1990). M, Janus-Michalska , Effective Models Describing Elastic Behaviour of Cellular Materials, Archives of Mettalurgy and Materials, 3, vol.50, 596-608, (2005). J.Rychlewski, Unconventional approach to linear elasticity, Arch. Mech., 47, 149-171. (1995).J.Ostrowska-Maciejewska, J. Rychlewski, Generalized proper states for anisotropic elastic materials, Arch. Mech. 53 (4-5) 501-518 (2001).Kordzikowski, M. Janus-Michalska , R.B.Pcherski , Specification of EnergyBased Criterion of Elastic Limit States for Cellular Materials, Archives of Metallurgy and Materials, vol.50, issue 3, pp. 621-634, 2005.M, Janus-Michalska , Energy Based Approach Constructing Elastic Model of Auxetic Cellular Materials submitted I gdzieD.W. Overaker, A.M. Cuitino, N.A. Langrana, Elastoplastic Micromechanical Modeling of Two-Dimensional Irregular Convex and Nonconvex (Re-entrant) Hexagonal Foams, Transactions of ASME, Vol.65, 748-757, (1998). R.S. Lakes, Advances in Negative Poissons Ratio Materials, Advanced Materials, 5, 293-296, (1993). R.S. Lakes ,Deformation mechanisms of negative Poissons ratio materials: structural aspects, J. Mat. Science, 26, 2287-2292, (1991).Wang Y., Lakes R., Analytical parametric analysis of the contact problem of human buttocks and negative Poissons ratio foam cushions, Int.J.Sol.Struc. 39, pp. 4825-38, 2002.Szefer G., Kdzior D., Contact of Elastic Bodies with Negative Poissons Ratio, Springer V., 2002.Kikuchi N., Oden J.T., Contact Problems in Elasticity: A study of Variational Ineqalitiees and Finite Element Methods, SIAM Philadelphia, (1988).