Research Papers

Fracture of Soft Elastic Foam

[+] Author and Article Information
Zhuo Ma

Department of Aerospace Engineering,
Iowa State University,
Ames, IA 50011
e-mail: zhuoma@iastate.edu

Xiangchao Feng

Department of Aerospace Engineering,
Iowa State University,
Ames, IA 50011
e-mail: xfeng@iastate.edu

Wei Hong

Department of Aerospace Engineering,
Iowa State University,
Ames, IA 50011
e-mail: whong@iastate.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 2, 2015; final manuscript received November 14, 2015; published online December 10, 2015. Editor: Yonggang Huang.

J. Appl. Mech 83(3), 031007 (Dec 10, 2015) (7 pages) Paper No: JAM-15-1533; doi: 10.1115/1.4032050 History: Received October 02, 2015; Revised November 14, 2015

Consisting of stretchable and flexible cell walls or ligaments, soft elastic foams exhibit extremely high fracture toughness. Using the analogy between the cellular structure and the network structure of rubbery polymers, this paper proposes a scaling law for the fracture energy of soft elastic foam. To verify the scaling law, a phase-field model for the fracture processes in soft elastic structures is developed. The numerical simulations in two-dimensional foam structures of various unit-cell geometries have all achieved good agreement with the scaling law. In addition, the dependences of the macroscopic fracture energy on geometric parameters such as the network connectivity and spatial orientation have also been revealed by the numerical results. To further enhance the fracture toughness, a type of soft foam structures with nonstraight ligaments or folded cell walls has been proposed and its performance studied numerically. Simulations have shown that an effective fracture energy one order of magnitude higher than the base material can be reached by using the soft foam structure.

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Maiti, S. K. , Gibson, L. J. , and Ashby, M. F. , 1984, “ Deformation and Energy Absorption Diagrams for Cellular Solids,” Acta Metall., 32(11), pp. 1963–1975. [CrossRef]
Han, F. , Zhu, Z. , and Gao, J. , 1998, “ Compressive Deformation and Energy Absorbing Characteristic of Foamed Aluminum,” Metall. Mater. Trans. A, 29(10), pp. 2497–2502. [CrossRef]
Gibson, L. J. , and Ashby, M. F. , 1999, Cellular Solids: Structure and Properties, Cambridge University Press, Cambridge, UK.
Maiti, S. K. , Ashby, M. F. , and Gibson, L. J. , 1984, “ Fracture Toughness of Brittle Cellular Solids,” Scr. Metall., 18(3), pp. 213–217. [CrossRef]
Gibson, L. J. , Ashby, M. F. , Zhang, J. , and Triantafillou, T. C. , 1989, “ Failure Surfaces for Cellular Materials Under Multiaxial Loads—I. Modelling,” Int. J. Mech. Sci., 31(9), pp. 635–663. [CrossRef]
Brezny, R. , and Green, D. J. , 1991, “ Factors Controlling the Fracture Resistance of Brittle Cellular Materials,” J. Am. Ceram. Soc., 74(5), pp. 1061–1065. [CrossRef]
Chen, J. Y. , Huang, Y. , and Ortiz, M. , 1998, “ Fracture Analysis of Cellular Materials: A Strain Gradient Model,” J. Mech. Phys. Solids, 46(5), pp. 789–828. [CrossRef]
Bažant, Z. P. , Zhou, Y. , Zi, G. , and Daniel, I. M. , 2003, “ Size Effect and Asymptotic Matching Analysis of Fracture of Closed-Cell Polymeric Foam,” Int. J. Solids Struct., 40(25), pp. 7197–7217. [CrossRef]
McIntyre, A. , and Anderton, G. E. , 1979, “ Fracture Properties of a Rigid Polyurethane Foam Over a Range of Densities,” Polymer, 20(2), pp. 247–253. [CrossRef]
Lake, G. J. , and Thomas, A. G. , 1967, “ The Strength of Highly Elastic Materials,” Proc. R. Soc. London, Ser. A, 300(1460), pp. 108–119. [CrossRef]
Thomas, A. G. , 1955, “ Rupture of Rubber. II. The Strain Concentration at an Incision,” J. Polym. Sci., 18(88), pp. 177–188. [CrossRef]
Aranson, I. S. , Kalatsky, V. A. , and Vinokur, V. M. , 2000, “ Continuum Field Description of Crack Propagation,” Phys. Rev. Lett., 85(1), pp. 118–121. [CrossRef] [PubMed]
Karma, A. , Kessler, D. A. , and Levine, H. , 2001, “ Phase-Field Model of Mode III Dynamic Fracture,” Phys. Rev. Lett., 87(4), p. 045501. [CrossRef] [PubMed]
Hakim, V. , and Karma, A. , 2009, “ Laws of Crack Motion and Phase-Field Models of Fracture,” J. Mech. Phys. Solids, 57(2), pp. 342–368. [CrossRef]
Eastgate, L. O. , Sethna, J. P. , Rauscher, M. , Cretegny, T. , Chen, C. S. , and Myers, C. R. , 2002, “ Fracture in Mode I Using a Conserved Phase-Field Model,” Phys. Rev. E, 65(3), p. 036117. [CrossRef]
Wang, Y. U. , Jin, Y. M. , and Khachaturyan, A. G. , 2002, “ Phase Field Microelasticity Theory and Modeling of Elastically and Structurally Inhomogeneous Solid,” J. Appl. Phys., 92(3), pp. 1351–1360. [CrossRef]
Marconi, V. I. , and Jagla, E. A. , 2005, “ Diffuse Interface Approach to Brittle Fracture,” Phys. Rev. E, 71(3), p. 036110. [CrossRef]
Spatschek, R. , Hartmann, M. , Brener, E. , Müller-Krumbhaar, H. , and Kassner, K. , 2006, “ Phase Field Modeling of Fast Crack Propagation,” Phys. Rev. Lett., 96(1), p. 015502. [CrossRef] [PubMed]
Miehe, C. , Welschinger, F. , and Hofacker, M. , 2010, “ Thermodynamically Consistent Phase‐Field Models of Fracture: Variational Principles and Multi‐Field FE Implementations,” Int. J. Numer. Methods Eng., 83(10), pp. 1273–1311. [CrossRef]
Miehe, C. , and Schänzel, L. M. , 2014, “ Phase Field Modeling of Fracture in Rubbery Polymers. Part I: Finite Elasticity Coupled With Brittle Failure,” J. Mech. Phys. Solids, 65, pp. 93–113. [CrossRef]
Karma, A. , and Lobkovsky, A. E. , 2004, “ Unsteady Crack Motion and Branching in a Phase-Field Model of Brittle Fracture,” Phys. Rev. Lett., 92(24), p. 245510. [CrossRef] [PubMed]
Henry, H. , and Levine, H. , 2004, “ Dynamic Instabilities of Fracture Under Biaxial Strain Using a Phase Field Model,” Phys. Rev. Lett., 93(10), p. 105504. [CrossRef] [PubMed]
Henry, H. , 2008, “ Study of the Branching Instability Using a Phase Field Model of Inplane Crack Propagation,” Europhys. Lett., 83(1), p. 16004. [CrossRef]
Griffith, A. A. , 1921, “ The Phenomena of Rupture and Flow in Solids,” Philos. Trans. R. Soc. London, 221(582–593), pp. 163–198. [CrossRef]
Bazant, Z. P. , 2005, Scaling of Structural Strength, Butterworth-Heinemann, London.
Hoagland, R. G. , Embury, J. D. , and Green, D. J. , 1975, “ On the Density of Microcracks Formed During the Fracture of Ceramics,” Scr. Metall., 9(9), pp. 907–909. [CrossRef]
Evans, A. G. , and Fu, Y. , 1985, “ Some Effects of Microcracks on the Mechanical Properties of Brittle Solids—II. Microcrack Toughening,” Acta Metall., 33(8), pp. 1525–1531. [CrossRef]
Hutchinson, J. W. , 1987, “ Crack Tip Shielding by Micro-Cracking in Brittle Solids,” Acta Metall., 35(7), pp. 1605–1619. [CrossRef]


Grahic Jump Location
Fig. 1

The rupture and retraction process of a ligament (or cell wall) in a foam structure. Because of its slenderness, a soft filament will tend to buckle or coil and could not effectively transduce energy.

Grahic Jump Location
Fig. 2

Sketch of the loading conditions for the foam structures

Grahic Jump Location
Fig. 3

(a) Part of the 2D honeycomb foam structure being simulated. (b) The calculated deformation and damage fields of a honeycomb structure, during the propagation of a pre-existing crack. The shades represent the dimensionless strain-energy density W/Wc. The crack profile is indicated schematically by the dashed line, which goes through the transition zone from the intact to the fully damaged regions in terms of ϕ. The deformation is shown to scale, and only part of the structure near the crack tip is shown. The actual computational domain is much larger than that shown to circumvent size effect.

Grahic Jump Location
Fig. 4

(a) Undeformed coordinates of the ligaments as a function of the times of rupture. The line is the best linear fit. The slope indicates the dimensionless crack velocity. (b) Dimensionless fracture energy (energy release rate) Γ/rWc as a function of the dimensionless crack velocity v/rmWc. The line is the best linear fit, and the vertical intercept shows the quasi-static fracture energy of the soft foam.

Grahic Jump Location
Fig. 5

Calculated fracture energy of the hexagonal soft foams versus (a) the normalized ligament length l/r at constant volume fraction ψ=9%, and (b) the volume fraction of the solid phase ψ at constant ligament length l/r=4.2. Two different orientations are simulated as indicated by the insets (with horizontal cracks).

Grahic Jump Location
Fig. 6

Damage patterns of hexagonal soft foams in (a) zigzag and (b) armchair orientations. The shading represents the damage variable ϕ, plotted in the undeformed geometry. The dashed curves show the approximate paths of crack propagation.

Grahic Jump Location
Fig. 7

Geometric effect on the fracture energy of soft foams. The geometries and orientations are represented by the sketch insets, with cracks running horizontally.

Grahic Jump Location
Fig. 8

(a) Sketch of the unit cell of a soft elastic foam containing serpentine ligaments. (b) Simulated deformation and fracture process in the soft elastic foam. The shading shows the dimensionless strain-energy density W/Wc. The deformed shape is plotted by downscaling the actual displacement value to 10%.

Grahic Jump Location
Fig. 9

Calculated fracture energy of soft foam structures with serpentine ligaments, as a function of the solid volume fraction ψ. The structures have identical unit-cell size and ligament thickness. The volume fraction is controlled by changing the width of the serpentine pattern, as indicated by the insets.

Grahic Jump Location
Fig. 10

The calculated nominal stress–stretch curve of a soft foam with serpentine ligaments as shown in Fig. 8 (without a pre-existing crack). The initial stiffness of the structure is more than two orders of magnitude lower than the solid material. The stress–stretch curve exhibits a strain-stiffening behavior, even though the material is taken to be neo-Hookean.



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