0
ADDITIONAL TECHNICAL PAPERS

Failure Surfaces for Finitely Strained Two-Phase Periodic Solids Under General In-Plane Loading

[+] Author and Article Information
N. Triantafyllidis1

Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140

M. D. Nestorović

Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109-2140

M. W. Schraad

Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545

Here and subsequently, Latin indexes range from 1 to 2, unless indicated differently. Einstein’s summation convention is implied over repeated indexes.

The solutions to linear equations with periodic coefficients, in more than one dimension, are commonly named “Bloch waves” in physics literature.

The absence of subscripts indicates that the compressibility ratio has the same value in each phase.

We are grateful to the authors of (15) who confirmed that the macroscopic moduli of their randomly, particle-reinforced material are strongly elliptic.

1

Corresponding author.

J. Appl. Mech 73(3), 505-515 (Sep 01, 2005) (11 pages) doi:10.1115/1.2126695 History: Received February 09, 2005; Revised September 01, 2005

For ductile solids with periodic microstructures (e.g., honeycombs, fiber-reinforced composites, cellular solids) which are loaded primarily in compression, their ultimate failure is related to the onset of a buckling mode. Consequently, for periodic solids of infinite extent, one can define as the onset of failure the first occurrence of a bifurcation in the fundamental solution, for which all cells deform identically. By following all possible loading paths in strain or stress space, one can construct onset-of-failure surfaces for finitely strained, rate-independent solids with arbitrary microstructures. The calculations required are based on a Bloch wave analysis on the deformed unit cell. The presentation of the general theory is followed by the description of a numerical algorithm which reduces the size of stability matrices by an order of magnitude, thus improving the computational efficiency for the case of continuum unit cells. The theory is subsequently applied to porous and particle-reinforced hyperelastic solids with circular inclusions of variable stiffness. The corresponding failure surfaces in strain-space, the wavelength of the instabilities, and their dependence on micro-geometry and macroscopic loading conditions are presented and discussed.

FIGURES IN THIS ARTICLE
<>
Copyright © 2006 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Schematic representation of an infinite solid with a perfectly periodic microstructure and its corresponding unit cell. The solid is deformed under finite plane strain conditions in the X1-X2 plane.

Grahic Jump Location
Figure 2

Schematic representation of an infinite, perfectly periodic solid, with (a) square and (b) diagonal microgeometry

Grahic Jump Location
Figure 3

Partition of the nodes of the discretized unit cell into internal and boundary ones, as required by the f.e.m.-based condensation technique employed in Bloch wave calculations

Grahic Jump Location
Figure 4

In (a) are plotted the macroscopic (continuous line) and microscopic (dashed line) onset-of-failure surfaces in the principal macroscopic logarithmic strain space for a perfectly periodic, nearly incompressible (κ∕μ=98) solid, with a square distribution of voids (μf∕μm=0.02) and subjected to biaxial loading along the initial axes of material orthotropy (θ=0). In (b) is plotted a blow-up of the biaxial compression region showing that the first bifurcation occurs before the macroscopic loss of ellipticity. Note that the distance between the microscopic and macroscopic onset-of-failure surfaces increases with increasing compressibility of the material.

Grahic Jump Location
Figure 5

Eigenmode of the microscopic bifurcation instability (antisymmetric in the unit cell) for balanced biaxial compression of the more compressible (κ∕μ=9.8) voided solid examined in Fig. 4 (courtesy of Dr. J. C. Michel, CNRS-LMA, Marseille, France)

Grahic Jump Location
Figure 6

In (a) are plotted the macroscopic and microscopic onset-of-failure surfaces in the principal macroscopic logarithmic strain space for a perfectly periodic, nearly incompressible (κ∕μ=98) solid, with a square distribution of inclusions (μf∕μm=50) and subjected to biaxial loading along the initial axes of material orthotropy (θ=0). Notice that the macroscopic and microscopic onset-of-failure surfaces coincide. The dotted-dashed line indicates macroscopic strains at which there is rigid inclusion contact. In (b) is plotted a blow-up of the biaxial compression region showing the influence of material compressibility.

Grahic Jump Location
Figure 7

Influence of changing the stiffness contrast on the onset-of-failure surfaces for (a) weaker than the matrix (μf∕μm=0.02,0.1,0.5) and (b) stronger than the matrix (μf∕μm=2.0,10.0,50.0) inclusions under orthotropic loading and for a square microgeometry arrangement

Grahic Jump Location
Figure 8

Influence of changing the stiffness contrast on the onset-of-failure surfaces for (a) the voided solid (μf∕μm=0.02) and (b) the solid with rigid inclusions (μf∕μm=50.0) under orthotropic loading and for a diagonal microgeometry arrangement

Grahic Jump Location
Figure 9

Influence of principal axes orientation θ on the onset-of-failure surfaces for a perfectly periodic solid, with (a) a square distribution of voids (μf∕μm=0.02) and (b) a square distribution of rigid inclusions (μf∕μm=50). Observe the increase in the critical strains from a loading at a angle θ=0 to a loading at angle θ=π∕18 with respect to the initial axes of material orthotropy.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In