Abstract
Higher order gradient continuum theories have often been proposed as models for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent comparison of the solution to a boundary value problem using both the exact microscopic model and the corresponding approximate higher order gradient macroscopic model.
In the interest of simplicity the microscopic model is a discrete periodic nonlinear elastic structure. The corresponding macroscopic model derived from it is a continuum model involving higher order gradients in the displacements. Attention is focused on the simplest such model, namely the one whose energy density involves only the second order gradient of the displacement. The discrete to continuum comparisons are done for a boundary value problem involving two different types of macroscopic material behavior. In addition the issues of stability and imperfection sensitivity of the solutions are also investigated.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
R. Abeyaratne and N. Triantafyllidis, An investigation of localization in a porous elastic material using homogenization theory. J. Appl. Mech. 51 (1984) 481–486.
R. Abeyaratne and J.K. Knowles, Non-elliptic elastic materials and the modeling of elasticplastic behavior for finite deformation. J. Mech. Phys. Solids 35 (1987) 343–365.
R. Abeyaratne and J.K. Knowels, On the dissipative response due to discontinuous strains in bars of unstable elastic material. Int. J. Solids Structures 24 (1988) 1021–1044.
E.C. Aifantis and J.B. Serrin, The mechanical theory of fluid interfaces and Maxwell's rule. J. Coll. and Interf. Sci. 96 (1983) 517–529.
E.C. Aifantis and J.B. Serrin, Equilibrium solutions in the mechanical theory of fluid microstructures. J. Coll. and Interf. Sci. 96 (1983) 530–547.
E.C. Aifantis, On the microstructural origin of certain inelastic models. Transactions of ASME, J. Engng. Mat. Tech. 106 (1984) 326–330.
E.C. Aifantis, The physics of plastic deformation. Int. J. Plasticity 3 (1987) 211–247.
V. Alexiades and E.C. Aifantis, On the thermodynamic theory of fluid interfaces: infinite intervals, equilibrium solutions, and minimizers. J. Coll. Interf. Sci. 111 (1986) 119–132.
A. Askar, Lattice Dynamical Foundations of Continuum Theories. (1985) World Scientific, Singapore.
Z.P. Bažant, Softening instability: Part I-Localization into a planar band. J. Appl. Mech. 55 (1988) 517–522.
J. Carr, M. Gurtin and M. Slemrod, Structured phase transitions on a finite interval. Arch. Rat. Mech. Anal. 86 (1984) 317–351.
R.J. Clifton, High strain rate behavior of metals. Appl. Mech. Rev. 43 (1990) S9-S22.
B.D. Coleman, Necking and drawing in polymeric fibers under tension. Arch. Rat. Mech. Anal. 83 (1983) 115–137.
B.D. Coleman and M.L. Hodgdon, On shear bands in ductile materials. Arch. Rat. Mech. Anal. 90 (1985) 219–247.
A.C. Eringen and E.S. Suhubi, Nonlinear theory of simple micro-elastic solids—I. Int. J. Engng. Sci. 2 (1964) 189–203.
G. Geymonant, S. Müller and N. Triantafyllidis, Quelques remarques sur l'homogénéisation des matériaux élastiques nonlinéaires. C.R. Acad. Sci. Paris, 311 (Ser. 1) (1990) 911–916.
J. Hadamard, Lecons sur la Propagation des Ondes et les Equations de l'Hydrodynamique. Paris: Hermann (1903) Chap. 6.
R. Hill, Acceleration waves in solids. J. Mech. Phys. Solids 10 (1962) 1–16.
R.D. James, Displacive phase transformations in solids. J. Mech. Phys. Solids 34 (1986) 359–394.
J.K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatics plane strain. Arch. Rat. Mech. Anal. 63 (1977) 321–336.
I.A. Kunin, Elastic Media with Microstructure—I (One-Dimensional Models) Springer (1982).
S. Kyriakides and Y.-C. Chang, The initiation and propagation of a localized instability in an inflated elastic tube. Int. J. Solids Structures 27 (1991) 1085–1111.
D. Lasry and T. Belytschko, Localization limiters in transient problems. Int. J. Solids Structures 24 (1988) 581–597.
J. Mandel, Conditions de stabilité et postulat de drucker. In: J. Kravtchenko and P.M. Sirieys (eds), Rheology and Soil Mechanics. Berlin: Springer (1966) pp. 58–68.
Z. Marciniak and K. Kuczynski, Limit strains in the process of stretch forming sheet metal. Int. J. Mech. Sciences 9 (1967) 609–625.
R.D. Mindlin, Micro-structure in linear elasticity. Arch. Rat. Mech. Anal. 16 (1964) 51–78.
R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Structures 1 (1965) 417–438.
A. Molinari and R.J. Clifton, Analytical characterization of shear localization in thermoviscoplastic solids. J. Appl. Mech. 54 (1987) 806–812.
J.R. Rice, The localization of plastic deformation. In: W.T. Koiter (ed.), Theoretical and Applied Mechanics. Proceedings of the 14th I.U.T.A.M. Conference, Delft, August 30–September 4, 1976. Amsterdam: North-Holland (1976) pp. 207–220.
E.S. Suhubi and A.C. Eringen, Nonlinear theory of micro-elastic solids—II. Int. J. Engng. Sci. 2 (1964) 389–404.
T.Y. Thomas, Plastic Flow and Fracture in Solids. New York: Academic Press (1961).
R.A. Toupin and D.C. Gazis, Surface effects and initial stress in continuum and lattice models of elastic crystals. In: R.F. Wallis (ed.), Proceedings of the International Conference on Lattice Dynamics, Copenhagen, August 1963. Oxford: Pergamon Press (1965) pp. 597–605.
N. Triantafyllidis and B.N. Maker, On the comparison between microscopic and macroscopic instability mechanisms in a class of fiber-reinforced composites. J. Appl. Mech. 52 (1985) 794–800.
N. Triantafyllidis and E.C. Aifantis, A gradient approach to localization of deformation. I. Hyperelastic materials. J. Elasticity 16 (1986) 225–237.
V. Tvergaard, A. Needleman and K.K. Lo, Flow localization in the plain strain tensile test. J. Mech. Phys. Solids 29 (1981) 115–142.
J.D. Van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density (in Dutch). Verhandel. Konink. Akad. Weten. Amsterdam (sec. I) 1 (1893).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Triantafyllidis, N., Bardenhagen, S. On higher order gradient continuum theories in 1-D nonlinear elasticity. Derivation from and comparison to the corresponding discrete models. J Elasticity 33, 259–293 (1993). https://doi.org/10.1007/BF00043251
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00043251