### The Influence of Microstructure and Specimen Size Ratio on the Elastic Modulus of a Bar Element

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#### Abstract

By introducing higher order strain gradient terms and an internal length parameter (the gradient elastic coefficient) into the constitutive stress-strain relation for the analysis of a bar, a corresponding elastic modulus is herein derived, which includes the classical and gradient elastic terms. With this, the micro-scale influence of the material is readily captured and the Elastic modulus is shown to depend on changes in the ratio between the dimension of the microstructure and the bar element specimen the so called size effect. A numerical study is presented as illustration. *Copyright © 2014 Praise Worthy Prize - All rights reserved.*

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W. N. Sharpe, K. M. Jackson, K. J. Hemker, Z. L. Xie. Effect of Specimen size on Young’s modulus and fracture strength of polysilicon. Journal of microelectromechanical systems, 10 (3): 317-326, 2001.

T. Namazu, Y. Isono, and T. Tanaka. Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bend test using afm. Journal of microelectromechanical system, 9(4): 450-459, 2000.

A. A. Volinsky, J. Vella, I. S. Adhihelty, V. Sarihan, L. Mercado, B. H. Yeung, and W. W. Gerberich. Microstructure and Mechanical Properties of Electroplated Cu Thin Films. MRS Proceedings, 649, Q5.3 doi:10.1557/PROC-649-Q5.3, 2000.

H. Mizubayashi, Y. Yoshihara,S. and Okudu. The elastic measurements of aluminum nm-films. Physica status solids A- Applied Research, 129 (2): 475-481, 1992.

A. Latapie, D. Farkas. Effect of grain size on the elastic properties of nanocrystalline alpha-iron. Scripta Mater.. – 48, N 5. – P. 611-615, 2003.

S. J. Zhao, K. Albe, H. Hahn. Grain size dependence of the bulk modulus of nanocrystalline nickel. Scripta Mater. 55, N 5. P. 473-476, 2006.

J. E. Carsley, J. Ning, W. W. Milligan, S. A. Hackney, E. C. Aifantis. A simple mixture based model for grain size dependence of strength in nanophase metals. Nanostruct. Mater. 5, 441-448, 1995.

H. S. Kim, Y. Estrin, M. B. Bush. Constitutive Modelling of strength and plasticity of nanocrystalline metallic materials. Mater. Sci. Eng. A 316, 195-199, 2001.

B. Jiang, G. J. Weng. A theory of compressive yield strength of nanograined ceramics, Int. J. Plast. 20, 2007-2026 2004.

R. S. Lakes. Experimental microelasticity of 2 porous solids. International Journal of solids and structures, 22 (1): 55 – 63, 1986.

D. S. Potter, V. Gupta, and S. Hauert. Effects of specimen size and sample aspect ratio on the elastic stiffness of graphite/epoxy laminates. Composite science and technology 60 (12-13): 2517-2524, 2000.

F. Lagattu, J. Brillaud, M-C. Lafarie-Frenot. High strain gradient measurements by using digital image correlation technique Materials Characterization 53 17–28, 2004.

E. C. Aifantis. Strain gradient interpretation of size effect. International journal of fracture 95(1-4): 299 – 314, 1999.

E. C. Aifantis. Gradient deformation at nano, micro and macro scale. Journal of engineering material and technology – Transaction of the ASME, 121 (2): 189-202, 1999.

H. Gao, Y. Huang, W. D. Nix, and J.W. Hutchinson. Mechanism based strain gradient plasticity – I Theory, J. Mech. Phys. Sol. 47, 1239 – 1263, 1999.

N. A. Fleck, G. M. Muller, M. F. Ashby, J. W. Hutchinson, Strain gradient plasticity: theory and experiment, Acta Metall. Mater. 42, 475 – 487, 1994.

N. A. Fleck, and J. W. Hutchinson. A reformulation of strain gradient plasticity. Journal of the mechanics and physics of solids, 49 (10): 2245 – 2271, 2001.

L. B. Freund. The stability of a dislocation threading a strained layer on a substrate. Journal of applied mechanics – Transaction of ASME, 54 (3): 553 – 557, 1987.

C. V. Thompson, C. V. The yield stress of polycrystalline thin films. Journal of materials research, 8 (2): 237 – 238, 1993.

W. D. Nix. Mechanical properties of thin films. Metallugical transaction A- physical metallurgy and metals science, 20 (11): 2217 – 2245, 1998.

H. M. Zbib, and E. C. Aifantis. Size effects and length scales in gradient plasticity and dislocation dynamics. Scripta materialia 48 (2): 155-160, 2003.

A. Needleman, and E. Van der Giessen. Discrete dislocation plasticity. In engineering plasticity from macroscale to nanoscale, volume 233-2 of key Engineering materials pages 13-24, 2003.

A. G. Atkins. Scaling laws of elastoplastic fracture. International Journal of Fracture, 95 (1-4): 51-65, 1999.

Z. P. Bazant. Size effect on structural strength: A review. Archive of applied mechanics 69 (9-10): 2245-2271, 1999.

G. R. Irwin, G. R. (1964). Dimensional and geometric aspects of fracture. In fracture of engineering material, American society for metals, pages 211-230, 1964.

J. Liu, and H. Zenner. Estimation of s-n curves under consideration of geometrical and statistical size effect. Materialwissenschaft und werkstofftechnik, 26 (1): 14-21, 1995.

Seifried, A. About statistics in fatigue strength. Materialwissenschaft und werkstofftechnik, 35(2): 93-111, 2004.

A. C. Aifantis. Gradient effects at macro, micro and nano scales, J. Mech. Behavior Mats., 5, 355-375, 1994.

H. Askes, E. C. Aifantis, Gradient elasticity in statics and

dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results, International Journal of Solids and Structures, 48, 13 pp. 1962-1990, 2011.

M. R. Begley, J. W. Hutchinson. The mechanics of size dependent indentation, J. Mech. Phys. Solids 46, 2049 – 2068, 1998.

K. Mc Elhaney, J. J. Valssak, W. D. Nik. Determination of indenter trip geometry and indentation contact area for depth sensory indentation experiment, J. Mater. Res. 13, 1300 – 1306, 1998.

J. S. Stolken, A. G. Evans. A microbend test method for measuring the plasticity length-scale. Acta Mater. 46, 5109 – 5115, 1998.

B. S. Altan, and E. C. Aifantis. On some aspects in the special theory of gradient elasticity, J. Mech. Behavior Mats. 8, 231-282, 1997.

K. G. Tsepoura, S. Papargyri-Beskou, D. Polyzos, D. E. Beskos. Static and dynamic analysis of a gradient-elastic bar in tension, Achieve of Applied Mechanics, 72, 483 – 497, 2002.

O. T. Akintayo. Analytical and Numerical Study of the Behavior of Materials and Structures in Gradient Elasticity, Ph.D. dissertation, Gen. Dept. Eng. Sch., Aristotle University Thessaloniki 2011.

Akintayo, O.T., Papadopoulos, P.G., Aifantis, E.C., A note on gradient truss models, (2012) International Review of Mechanical Engineering (IREME), 6 (4), pp. 691-697.

Olufemi T. Akintayo, Towards a Gradient Truss Model. Part I: Bar Element Displacement-Force Relations, (2014) International Review of Civil Engineering (IRECE), 5 (1), pp. 32-42.

Aouissi, F., Brahma, A., Yang, C.C., Prediction of concrete mechanical properties by triphasic model, (2012) International Review of Mechanical Engineering (IREME), 6 (3), pp. 496-500.

R. D. Mindlin. Microstructure in linear elasticity, Arch. Ration. Mech. Anal., 16, 51-78, 1964.

M. Ben-Amoz. A dynamic theory for composite materials. J Appl. Math. Ph. (ZAMP) 27, 83–99, 1976.

E. C. Aifantis, On the microstructural origin of certain inelastic models, Transactions of ASME, J. Engng. Mat. Tech. 106, 326-330, 1984.

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