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CONTACT PROBLEM FOR A CLASS OF ANISOTROPIC ELASTIC CELLULAR BODIES WITH NONPOSITIVE POISSON’S RATIO

CONTACT PROBLEM FOR A CLASS OF ANISOTROPIC ELASTIC CELLULAR BODIES WITH NONPOSITIVE POISSON’S RATIO. Dorota Jasińska Małgorzata Janus-Michalska Institute of Structural Mechanics Cracow University of Technology. CONTENTS. Introduction – auxetic materials definition, occurance, properties

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CONTACT PROBLEM FOR A CLASS OF ANISOTROPIC ELASTIC CELLULAR BODIES WITH NONPOSITIVE POISSON’S RATIO

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  1. CONTACT PROBLEM FOR A CLASS OF ANISOTROPIC ELASTIC CELLULAR BODIES WITH NONPOSITIVE POISSON’S RATIO Dorota Jasińska Małgorzata Janus-Michalska Institute of Structural Mechanics Cracow University of Technology CMM-2007 – Computer Methods in MechanicsJune 19–22, 2007, Łódź–Spała, Poland

  2. CONTENTS • Introduction – auxetic materials • definition, occurance, properties • Cellular materials as anisotropic continuum • two scale modeling • stiffness matrices • energy based yield criterion • material effort coefficient • Formulation of a contact problem • Numerical examples • Conclusions • References CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  3. 1. Materials with negativePoisson’s ratio - auxetics Poisson’s ratio for isotropic materials Where: K - bulk modulus G - shear modulus d – dimension of the problem For spatial problems For planar problems CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  4. Auxetic materials from molecular to macroscopic level • Molecular auxetics • single crystals (iron pyrites, cadmium) • crystalline (α – cristobalite) • Mesoscopic auxetics • polymeric and metallic foams • microporous polymers • cellular materials (inverted honeycombs) • Macroscopic auxetics • composites CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  5. Other properties of auxetics • increased resistance to indentation (toughness) • resistance to fracture and damage • different deformation pattern • tendency to form double curved dome shaped surfaces while bending • improved acoustic and vibrational absorption • greater resiliance • reduced impact forces in contact CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  6. a) square cell structure with square unit cell c) equilateral triangular structure with hexagonal unit cell b) ‘honeycomb’structurewithtriangle unit cell d) ‘reentrant’ structure with trapezoid unit cell 2. Cellular materials as anisotropic continuumRegular cellular plane structuresand their representative unit cells CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  7. Two scale modeling • uniform strain states of equivalent continuum • displacement affinity of nodes and midpoints CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  8. Mechanical model for skeleton- displacement method for Timoshenko beam Uniform continuum strains – skeleton midpoint displacements • uniaxial extensionεx in xdirection. • uniaxial extension εy in y direction. • pure shear gxyin xy plane. CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  9. Resultant forces in skeleton In arbitrary strain state: CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  10. definition of equivalent continuum - averaging of strain potential for skeleton • strain potential of beam skeleton • strain potential of anisotropic continuum CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  11. stiffness matrix components for equivalent continuum • Hooke's law for anisotropic continuum stiffness tensor, where compliance tensor , • material constants: CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  12. Kelvin’s notation in 6-D space: strain vector stress vector Stiffness matrix: CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  13. 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 CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  14. Coefficient • Limit eigenstrains and eigenstresses • Limit elastic energy stored in particular eigenstate • Energy based yield criterion for equivalent continuum (Rychlewski) CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  15. Material effort coefficient • Arbitrary plane stress state • Decomposition of the stress vector into stress eigenstates • Material effort coefficient Where A,B,C – functions of and CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  16. For arbitrary cell orientation where: CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  17. 3.Unilateral static contact problem of anisotropic elastic body in Ω boundary conditions contact and friction conditions on on on where: - initial gap, - contact pressure, displacement normal to the boundary - tangential contact force, -increment of tangential displacement CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  18. 4.Numerical examples 4.1 Square block made of material with different cell types under pressure Data: Skeleton data: Es=10 GPa, νs=0.3, Re=10MPa, ρ/ ρs=0.1155 , p=25kN/m, μ=0.3 CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  19. Deformation and contact pressure distribution [MPa] scale=1500 scale=50 scale=1000 scale=3 CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  20. Contact pressure, friction stress and contact status along contact line CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  21. Distribution of material effort coefficient φ. CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  22. Dependence of Poisson’s ratio on angle of tension direction For nxy=const=0.96 For nxy=const=0.33 For nxy<0 for CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  23. 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 β angles. CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  24. Contact pressure and friction stress along contact line CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  25. Deformation and distribution of material effort coefficient φ CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  26. 5. conclusions • Micromechanical 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 Poisson’s 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 pressure • Linear analysis– first step. CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

  27. 6. References • L.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.Pęcherski , Specification of Energy–Based 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 gdzie • D.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 Poisson’s Ratio Materials, Advanced Materials, 5, 293-296, (1993). • R.S. Lakes ,Deformation mechanisms of negative Poisson’s 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 Poisson’s ratio foam cushions, Int.J.Sol.Struc. 39, pp. 4825-38, 2002. • Szefer G., Kędzior D., Contact of Elastic Bodies with Negative Poisson’s 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). CMM-2007 – Computer Methods in Mechanics June 19–22, 2007, Łódź–Spała, Poland

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