×

Finite elements for materials with strain gradient effects. (English) Zbl 0943.74072

Summary: We present finite element implementation of the Fleck-Hutchinson phenomenological strain gradient theory [N. A. Fleck and J. W. Hutchinson, J. Mech. Phys. Solids 41, 1825-1857 (1993)]. This theory fits within the Toupin-Mindlin framework [R. A. Toupin, Arch. Ration. Mech. Anal. 11, 385-414 (1962; Zbl 0112.16805); R. D. Mindlin, ibid. 16, 51-78 (1964; Zbl 0119.40302)] and deals with first-order strain gradients and the associated work-conjugate higher-order stresses. In conventional displacement-based approaches, the interpolation of displacement requires \(C^1\)-continuity in order to ensure convergence of the finite element procedure for higher-order theories. Mixed-type finite elements are developed herein for the Fleck-Hutchinson theory; these elements use standard \(C^0\)-continuous shape functions and can achieve the same convergence as \(C^1\) elements. These \(C^0\) elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems, and the comparative accuracy of various types of elements is evaluated.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74A30 Nonsimple materials
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] and Theorie des Corps Deformables, Hermann et Fils, Paris, 1909.
[2] Toupin, Arch. Rational Mech. Anal. 11 pp 385– (1962) · Zbl 0112.16805 · doi:10.1007/BF00253945
[3] Mindlin, Arch. Rational Mech. Anal. 16 pp 51– (1964) · Zbl 0119.40302 · doi:10.1007/BF00248490
[4] Aifantis, Trans. ASME J. Engng. Mater. Tech. 106 pp 326– (1984) · doi:10.1115/1.3225725
[5] Fleck, J. Mech. Phys. Solids 41 pp 1825– (1993) · Zbl 0791.73029 · doi:10.1016/0022-5096(93)90072-N
[6] Fleck, Adv. Appl. Mech. 33 pp 295– (1997) · doi:10.1016/S0065-2156(08)70388-0
[7] and The Finite Element, 4th edn., McGraw-Hill, New York, 1994.
[8] Specht, Int. J. Numer. Meth. Engng. 26 pp 705– (1988) · Zbl 0633.73082 · doi:10.1002/nme.1620260313
[9] Xia, J. Mech. Phys. Solids 44 pp 1621– (1996) · doi:10.1016/0022-5096(96)00035-X
[10] Shu, Int. J. Solids Struct. 35 pp 1363– (1998) · Zbl 0923.73056 · doi:10.1016/S0020-7683(97)00112-1
[11] ?Mixed finite elements for couple-stress analysis?, in and (eds.), Hybrid and Mixed Finite Element Methods, Wiley, New York, 1983.
[12] Shu, J. Mech. Phys. Solids (1998)
[13] Smyshlyaev, J. Mech. Phys. Solids 44 pp 465– (1996) · Zbl 1054.74553 · doi:10.1016/0022-5096(96)00009-9
[14] Zienkiewicz, Comput. Meth. Appl. Mech. Engng. 51 pp 3– (1985) · Zbl 0538.73099 · doi:10.1016/0045-7825(85)90025-8
[15] Laursen, Int. J. Numer. Meth. Engng. 12 pp 67– (1978) · Zbl 0379.65014 · doi:10.1002/nme.1620120107
[16] Mindlin, Exp. Mech. 3 pp 1– (1963) · doi:10.1007/BF02327219
[17] Hinton, Int. J. Numer. Meth. Engng. 9 pp 235– (1975) · doi:10.1002/nme.1620090117
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.