On the Young’s modulus of a auxetic composite structure. (English) Zbl 1258.74177
Summary: In this paper, the behavior of auxetic composites is interpreted in the light of Cosserat elasticity which admits degrees of freedom not present in the classical elasticity: The rotation of points in the material, and a couple per unit area or the couple stress. The prediction of the Young’s modulus is developed for a laminated periodic material made up of alternating aluminum and auxetic material, by using the Bécus homogenization technique.
MSC:
| 74Q15 | Effective constitutive equations in solid mechanics |
| 74Q05 | Homogenization in equilibrium problems of solid mechanics |
| 74E30 | Composite and mixture properties |
Keywords:
auxetic material; Cosserat elasticity; Young’s modulus; negative Poisson ratio; Bécus homogenization techniqueReferences:
| [1] | Adomeit, G.: Determination of elastic constants of a structured material, Mechanics of generalized continua, iutam symposium, freudenstadt, Stuttgart (1967) · Zbl 0181.53801 |
| [2] | Babuška, I.: Homogenization and its application, mathematical and computational problems, Numerical solution of partial differential equations – II, synspade 1975, 89-116 (1976) · Zbl 0346.65064 |
| [3] | Bécus, G. A.: Homogenization and random evolutions: applications to the mechanics of composite materials, Quart. appl. Math. 37, No. 3, 209-217 (1979) · Zbl 0427.73062 |
| [4] | Berglund, K.: Structural models of micropolar media, Mechanics of micropolar media (1982) |
| [5] | Chiroiu, V.: Identification and inverse problems related to material properties and behaviour, Topics in applied mechanics 1, 83-126 (2003) |
| [6] | Cosserat, E., Cosserat, F., 1909. Theorie des Corps Deformables. Hermann et Fils, Paris. · JFM 40.0862.02 |
| [7] | Eringen, A. C.: Linear theory of micropolar elasticity, J. math. Mech. 15, 909-924 (1966) · Zbl 0145.21302 |
| [8] | Eringen, A. C.: Theory of micropolar elasticity, Fracture 2, 621-729 (1968) · Zbl 0266.73004 |
| [9] | Gauthier, R. D.: Experimental investigations on micropolar media, CISM courses and lectures, 395-463 (1982) |
| [10] | Hlavacek, M.: A continuum theory for fibre reinforced composites, Int. J. Solids struct. 11, 199-211 (1975) · Zbl 0315.73086 · doi:10.1016/0020-7683(75)90053-0 |
| [11] | Hlavacek, M.: On the effective moduli of elastic composite materials, Int. J. Solids struct. 12, 655-670 (1976) · Zbl 0333.73026 · doi:10.1016/0020-7683(76)90012-3 |
| [12] | Honig, G.; Hirdes, U.: A method for the numerical inversion of the Laplace transforms, J. comput. Appl. math. 10, 113-132 (1984) · Zbl 0535.65090 · doi:10.1016/0377-0427(84)90075-X |
| [13] | Kröner, E.: On the physical reality of torque stresses in continuum mechanics, Int. J. Eng. sci. 1, 261-278 (1963) |
| [14] | Kumar, R.; Choudhary, S.: Dynamical problem of micropolar viscoelasticity, Proc. indian acad. Sci. (Earth planet. Sci.) 110, No. 3, 215-223 (2001) |
| [15] | Lakes, R. S.: Experimental microelasticity of two porous solids, Int. J. Solids struct. 22, 55-63 (1986) |
| [16] | Lakes, R. S.: Foam structures with a negative Poisson’s ratio, Science 235, 1038-1040 (1987) |
| [17] | Lakes, R. S.: Experimental micro mechanics methods for conventional and negative Poisson’s ratio cellular solids as Cosserat continua, J. eng. Mater. technol. 113, 148-155 (1991) |
| [18] | Love, A. E. H.: A treatise on the mathematical theory of elasticity, (1926) |
| [19] | Maxwell, J. C.: A treatise an electricity and magnetism, A treatise an electricity and magnetism 1 (1881) · JFM 05.0556.01 |
| [20] | Mindlin, R. D.: Microstructure in linear elasticity, Arch. rat. Mech. anal. 16, 51-78 (1964) · Zbl 0119.40302 · doi:10.1007/BF00248490 |
| [21] | Mindlin, R. D.: Stress functions for a Cosserat continuum, Int. J. Solids struct. 1, 265-271 (1965) |
| [22] | Poisson, S. D.: Second mémoire sur la théorie du magnétisme, Mem. de l’acad. De France 5 (1822) |
| [23] | , The transforms and applications handbook (2000) |
| [24] | Press, W. H.; Teukolsky, S. A.; Vellerlig, W. T.; Flannery, B. P.: Numerical recipes in Fortran, (1986) |
| [25] | Rayleigh, R. S.: On the influence of obstacles arranged in rectangular order upon the properties of a medium, Philos. mag. 34, 481-502 (1872) · JFM 24.1015.02 |
| [26] | Rosakis, P.; Ruina, A.; Lakes, R. S.: Microbuckling instability in elastomeric cellular solids, J. mater. Sci. 28, 4667-4672 (1993) |
| [27] | Teodorescu, P. P.; Munteanu, L.; Chiroiu, V.: On the wave propagation in chiral media, New trends in continuum mechanics. Theta series in advanced mathematics, 303-310 (2005) · Zbl 1177.74041 |
| [28] | Teodorescu, P. P.; Badea, T.; Munteanu, L.; Onisoru, J.: On the wave propagation in composite materials with a negative stiffness phase, New trends in continuum mechanics. Theta series in advanced mathematics, 295-302 (2005) · Zbl 1177.74040 |
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