##
**Experiments and theory in strain gradient elasticity.**
*(English)*
Zbl 1077.74517

Summary: Conventional strain-based mechanics theory does not account for contributions from strain gradients. Failure to include strain gradient contributions can lead to underestimates of stresses and size-dependent behaviors in small-scale structures. In this paper, a new set of higher-order metrics is developed to characterize strain gradient behaviors. This set enables the application of the higher-order equilibrium conditions to strain gradient elasticity theory and reduces the number of independent elastic length scale parameters from five to three. On the basis of this new strain gradient theory, a strain gradient elastic bending theory for plane-strain beams is developed. Solutions for cantilever bending with a moment and line force applied at the free end are constructed based on the new higher-order bending theory. In classical bending theory, the normalized bending rigidity is independent of the length and thickness of the beam. In the solutions developed from the higher-order bending theory, the normalized higher-order bending rigidity has a new dependence on the thickness of the beam and on a higher-order bending parameter, \(b_{\text h}\). To determine the significance of the size dependence, we fabricated micron-sized beams and conducted bending tests using a nanoindenter. We found that the normalized beam rigidity exhibited an inverse squared dependence on the beam’s thickness as predicted by the strain gradient elastic bending theory, and that the higher-order bending parameter, \(b_{\text h}\), is on the micron-scale. Potential errors from the experiments, model and fabrication were estimated and determined to be small relative to the observed increase in beam’s bending rigidity. The present results indicate that the elastic strain gradient effect is significant in elastic deformation of small-scale structures.

### MSC:

74B99 | Elastic materials |

74A20 | Theory of constitutive functions in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74-05 | Experimental work for problems pertaining to mechanics of deformable solids |

PDF
BibTeX
XML
Cite

\textit{D. C. C. Lam} et al., J. Mech. Phys. Solids 51, No. 8, 1477--1508 (2003; Zbl 1077.74517)

Full Text:
DOI

### References:

[1] | Albrecht, T. R.; Akamine, S.; Carver, T. E.; Quate, C. F., Microfabrication of cantilever for the atomic force microscope, J. Vacuum Sci. Technol. A-Vacuum Surf. Films, 8, 3386-3396 (1990) |

[2] | Bashir, R.; Gupta, A.; Neudeck, G. W.; McElfresh, M.; Gomez, R., On the design of piezoresistive silicon cantilevers with stress concentration regions for scanning probe microscopy applications, J. Micromech. Microeng., 10, 483-491 (2000) |

[3] | Carr, D. W.; Craighead, H. G., Fabrication of nanoelectromechanical systems in single crystal silicon using silicon on insulator substrates and electron beam lithography, J. Vacuum Sci. Technol. B, 15, 2760-2763 (1997) |

[4] | Chong, A. C.M.; Lam, D. C.C., Strain gradient plasticity effect in indentation hardness of polymers, J. Mater. Res., 14, 4103-4110 (1999) |

[5] | Craighead, H. G., Nanoelectromechanical systems, Science, 290, 1532-1535 (2000) |

[7] | Fleck, N. A.; Hutchinson, J. W., A reformulation of strain gradient plasticity, J. Mech. Phys. Solids., 49, 2245-2271 (2001) · Zbl 1033.74006 |

[8] | Fleck, N. A.; Muller, G. M.; Ashby, M. F.; Hutchinson, J. W., Strain gradient plasticitytheory and experiments, Acta Metall. Mater., 42, 475-487 (1994) |

[9] | Gao, H. J.; Huang, Y.; Nix, W. D.; Hutchinson, J. W., Mechanism-based strain gradient plasticity-I. Theory, J. Mech. Phys. Solids., 47, 1239-1263 (1999) · Zbl 0982.74013 |

[10] | Koiter, W. T., Couple stresses in the theory of elasticity. I and II, Proc. K. Ned. Akad. Wet. (B), 67, 17-44 (1964) · Zbl 0124.17405 |

[11] | Lam, D. C.C.; Chong, A. C.M., Indentation model and strain gradient plasticity law for glassy polymers, J. Mater. Res., 14, 3784-3788 (1999) |

[12] | Ma, Q.; Clarke, D. R., Size dependent hardness of silver single crystals, J. Mater. Res., 10, 853-863 (1995) |

[13] | Manias, E.; Chen, J.; Fang, N.; Zhang, X., Polymeric micromechanical components with tunable stiffness, Appl. Phys. Lett., 79, 1700-1702 (2001) |

[14] | Menard, K. P., Dynamic Mechanical Analysis: a Practical Introduction (1999), CRC Press: CRC Press Boca Raton, FL |

[15] | Mindlin, R. D., Micro-structure in linear elasticity, Arch. Rational Mech. Anal., 16, 51-78 (1964) · Zbl 0119.40302 |

[16] | Mindlin, R. D., Second gradient of strain and surface tension in linear elasticity, Int. J. Solids Struct., 1, 417-438 (1965) |

[17] | Mindlin, R. D.; Tiersten, H. F., Effects of couple-stresses in linear elasticity, Arch. Rational Mech. Anal., 11, 415-448 (1962) · Zbl 0112.38906 |

[18] | Nix, W. D., Mechanical properties of thin films, Metall. Trans. A, 20, 2217-2245 (1989) |

[19] | O’Malley, R. E., Singular Perturbation Methods for Ordinary Differential Equations (1991), Springer: Springer New York · Zbl 0743.34059 |

[20] | Poole, W. J.; Ashby, M. F.; Fleck, N. A., Micro-hardness of annealed and work-hardened copper polycrystals, Scripta Metall. Mater., 34, 4, 559-564 (1996) |

[21] | Stelmashenko, N. A.; Walls, M. G.; Brown, L. M.; Milman, Y. V., Microindentations on W and Mo oriented single crystalsan STM study, Acta Metall. Mater., 41, 2, 2855-2865 (1993) |

[22] | Stölken, J. S.; Evans, A. G., A microbend test method for measuring the plasticity length scale, Acta Metall. Mater., 46, 5109-5115 (1998) |

[23] | Toupin, R. A., Elastic materials with couple stresses, Arch. Rational Mech. Anal., 11, 385-414 (1962) · Zbl 0112.16805 |

[24] | Yang, F.; Chong, A. C.M.; Lam, D. C.C.; Tong, P., Couple stress Based Strain gradient theory for elasticity, Int. J. Solids Struct., 39, 2731-2743 (2002) · Zbl 1037.74006 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.