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Research Papers

Energy-Based Strength Theory for Soft Elastic Membranes

[+] Author and Article Information
Reza Pourmodheji

Department of Mechanical Engineering,
The City College of New York,
New York, NY 10031
e-mail: rpourmodheji@ccny.cuny.edu

Shaoxing Qu

Department of Engineering Mechanics,
State Key Laboratory of Fluid Power and Mechatronic Systems,
Key Laboratory of Soft Machines and Smart Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China
e-mail: squ@zju.edu.cn

Honghui Yu

Department of Mechanical Engineering,
The City College of New York,
New York, NY 10031
e-mail: yu@ccny.cuny.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received December 4, 2018; final manuscript received March 6, 2019; published online April 19, 2019. Assoc. Editor: Pedro Reis.

J. Appl. Mech 86(7), 071008 (Apr 19, 2019) (11 pages) Paper No: JAM-18-1687; doi: 10.1115/1.4043145 History: Received December 04, 2018; Accepted March 06, 2019

In the previous studies by the authors and others, it was demonstrated that there are two possible defect growth modes and a characteristic material length for any soft material. For a pre-existing defect smaller than the material characteristic length, the energy is dissipated all around the defect as it grows and the critical load for the growth is independent of the defect size. For defects larger than the characteristic length, the growth is by cracking and the energy is dissipated along a plane. Thus, the critical load for the growth is size dependent and can be predicted by fracture mechanics. In this study, we apply the same energy-based argument to the failure of thin membranes, with the focus on the first growth mode that gives the maximum critical load. We assume that strain localization due to damage is the precursor to rupture, and hence, we model the corresponding zone as a through-thickness hole, with its size smaller than the material characteristic length. The defect grows when the elastic energy relaxed by the growth is enough to provide the energy needed for internal microstructure changes. This leads us to the size-independent failure conditions for membranes under the biaxial load. The conditions are expressed in terms of either two principal stretches or two principal stresses for two different types of materials. For verification, we test the theory using the published experimental data on natural and styrene-butadiene rubber. By using the experimental data from equal biaxial loading, we predict the critical principal stretch ratios and critical stresses for different biaxialities. The predictions agree well with the experimental results.

Copyright © 2019 by ASME
Topics: Stress , Failure , Membranes
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Figures

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Fig. 1

Two modes of defect growth: (a) cracking by dissipating energy along a layer of finite thickness and (b) omnidirectional growth by dissipating energy all around the defect

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Fig. 2

A membrane under in-plane biaxial loading: (a) microscopic strain localization by defect, damage, or material inhomogeneity and (b) circular hole as a simplified model

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Fig. 3

Remote equi-biaxial tension versus the hole stretch ratio: Seitz et al. [19], Jm = 20; Gent [17], Jlim = 92; Gent and Thomas [70], C1 = 0.5C and C2 = 1.5C; and Arruda and Boyce [71], λm = 7

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Fig. 4

Normalized driving force for hole growth versus equi-biaxial remote tension

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Fig. 5

Normalized driving force for the crack growth compared with that of a hole of same surface area

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Fig. 6

Normalized driving force versus the larger remote stress for the neo-Hookean material under biaxial loadings of different biaxialities

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Fig. 7

γ(n), as defined in Eq. (8), versus biaxiality for the neo-Hookean material

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Fig. 8

Normalized driving force versus the remote stress for Mooney's material with C1/C = 0.724 and C1/C = 0.276 in different biaxialities

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Fig. 9

The failure criteria of the membrane made of the neo-Hookean material, for Γp/C 180 ∼ 300: (a) in principle stretch plane and (b) in principle stress plane

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Fig. 10

The failure criteria of the membrane made of Gum-Stock mentioned in Ref. [69], for Γp/C 180 ∼ 300: (a) in principle stretch plane and (b) in principle stress plane. Marks are calculated values and solid lines are fitted curves.

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Fig. 11

Normalized driving force for the hole growth versus the larger remote stretch for NR. Marks of “×” show the critical points from experiments.

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Fig. 12

Normalized driving force for the hole growth versus the larger remote stretch for SBR. Marks of “×” show the critical points from experiments.

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Fig. 13

The failure criteria of NR membrane: (a) in principle stretch plane and (b) in principle stress plane. Marks of “×” are from the experiment.

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Fig. 14

The failure criteria of SBR membrane: (a) in principle stretch plane and (b) in principle stress plane. Marks of “×” are from the experiment.

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Fig. 15

(a) The current configuration of the membrane under the remote stretch of λ and cavity stretch λa. (b) The traction S is stretching the cavity to λ generating a uniform deformation

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